A Lean Companion to Conceptual Mathematics

Session 11: Ascending to categories of richer structures

1. A category of richer structures: Endomaps of sets

Exercise 1 (p. 153)

How many maps can you find? (There are fewer than seven.)

Solution: Exercise 1

Label the elements in the three-element set A as a_1, a_2, a_3 (clockwise starting with any element); label the elements in the set X as x_1 for the element forming the one-element loop and x_2, x_3, x_4 (anticlockwise starting with any element) for the elements forming the three-element cycle, ignoring all other elements.

inductive A | a₁ | a₂ | a₃ inductive X | x₁ | x₂ | x₃ | x₄ def A' : SetWithEndomap := { t := A carrier := Set.univ toEnd := fun | A.a₁ => A.a₂ | A.a₂ => A.a₃ | A.a₃ => A.a₁ toEnd_mem := fun _ ↦ Set.mem_univ _ } def X' : SetWithEndomap := { t := X carrier := Set.univ toEnd := fun -- a restriction of α to the subset {x₁, x₂, x₃, x₄} | X.x₁ => X.x₁ | X.x₂ => X.x₃ | X.x₃ => X.x₄ | X.x₄ => X.x₂ toEnd_mem := fun _ ↦ Set.mem_univ _ }

Then there are four structure-preserving maps from A to X, which we call f_1, f_2, f_3, f_4 below.

def f₁ : A ⟶ X | A.a₁ => X.x₁ | A.a₂ => X.x₁ | A.a₃ => X.x₁ -- f₁ is structure-preserving example : A' ⟶ X' := ⟨ f₁, ⊢ (∀ x ∈ A'.carrier, f₁ x ∈ X'.carrier) ∧ f₁ ⊚ A'.toEnd = X'.toEnd ⊚ f₁ ⊢ ∀ x ∈ A'.carrier, f₁ x ∈ X'.carrier⊢ f₁ ⊚ A'.toEnd = X'.toEnd ⊚ f₁ ⊢ ∀ x ∈ A'.carrier, f₁ x ∈ X'.carrier All goals completed! 🐙 ⊢ f₁ ⊚ A'.toEnd = X'.toEnd ⊚ f₁ a:A'.t⊢ (f₁ ⊚ A'.toEnd) a = (X'.toEnd ⊚ f₁) a ⊢ (f₁ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₁) A.a₁⊢ (f₁ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₁) A.a₂⊢ (f₁ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₁) A.a₃ ⊢ (f₁ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₁) A.a₁⊢ (f₁ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₁) A.a₂⊢ (f₁ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₁) A.a₃ All goals completed! 🐙 ⟩ def f₂ : A ⟶ X | A.a₁ => X.x₂ | A.a₂ => X.x₃ | A.a₃ => X.x₄ -- f₂ is structure-preserving example : A' ⟶ X' := ⟨ f₂, ⊢ (∀ x ∈ A'.carrier, f₂ x ∈ X'.carrier) ∧ f₂ ⊚ A'.toEnd = X'.toEnd ⊚ f₂ ⊢ ∀ x ∈ A'.carrier, f₂ x ∈ X'.carrier⊢ f₂ ⊚ A'.toEnd = X'.toEnd ⊚ f₂ ⊢ ∀ x ∈ A'.carrier, f₂ x ∈ X'.carrier All goals completed! 🐙 ⊢ f₂ ⊚ A'.toEnd = X'.toEnd ⊚ f₂ a:A'.t⊢ (f₂ ⊚ A'.toEnd) a = (X'.toEnd ⊚ f₂) a ⊢ (f₂ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₂) A.a₁⊢ (f₂ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₂) A.a₂⊢ (f₂ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₂) A.a₃ ⊢ (f₂ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₂) A.a₁⊢ (f₂ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₂) A.a₂⊢ (f₂ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₂) A.a₃ All goals completed! 🐙 ⟩ def f₃ : A ⟶ X | A.a₁ => X.x₃ | A.a₂ => X.x₄ | A.a₃ => X.x₂ -- f₃ is structure-preserving example : A' ⟶ X' := ⟨ f₃, ⊢ (∀ x ∈ A'.carrier, f₃ x ∈ X'.carrier) ∧ f₃ ⊚ A'.toEnd = X'.toEnd ⊚ f₃ ⊢ ∀ x ∈ A'.carrier, f₃ x ∈ X'.carrier⊢ f₃ ⊚ A'.toEnd = X'.toEnd ⊚ f₃ ⊢ ∀ x ∈ A'.carrier, f₃ x ∈ X'.carrier All goals completed! 🐙 ⊢ f₃ ⊚ A'.toEnd = X'.toEnd ⊚ f₃ a:A'.t⊢ (f₃ ⊚ A'.toEnd) a = (X'.toEnd ⊚ f₃) a ⊢ (f₃ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₃) A.a₁⊢ (f₃ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₃) A.a₂⊢ (f₃ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₃) A.a₃ ⊢ (f₃ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₃) A.a₁⊢ (f₃ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₃) A.a₂⊢ (f₃ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₃) A.a₃ All goals completed! 🐙 ⟩ def f₄ : A ⟶ X | A.a₁ => X.x₄ | A.a₂ => X.x₂ | A.a₃ => X.x₃ -- f₄ is structure-preserving example : A' ⟶ X' := ⟨ f₄, ⊢ (∀ x ∈ A'.carrier, f₄ x ∈ X'.carrier) ∧ f₄ ⊚ A'.toEnd = X'.toEnd ⊚ f₄ ⊢ ∀ x ∈ A'.carrier, f₄ x ∈ X'.carrier⊢ f₄ ⊚ A'.toEnd = X'.toEnd ⊚ f₄ ⊢ ∀ x ∈ A'.carrier, f₄ x ∈ X'.carrier All goals completed! 🐙 ⊢ f₄ ⊚ A'.toEnd = X'.toEnd ⊚ f₄ a:A'.t⊢ (f₄ ⊚ A'.toEnd) a = (X'.toEnd ⊚ f₄) a ⊢ (f₄ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₄) A.a₁⊢ (f₄ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₄) A.a₂⊢ (f₄ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₄) A.a₃ ⊢ (f₄ ⊚ A'.toEnd) A.a₁ = (X'.toEnd ⊚ f₄) A.a₁⊢ (f₄ ⊚ A'.toEnd) A.a₂ = (X'.toEnd ⊚ f₄) A.a₂⊢ (f₄ ⊚ A'.toEnd) A.a₃ = (X'.toEnd ⊚ f₄) A.a₃ All goals completed! 🐙 ⟩

3. The category of graphs

Exercise 2 (p. 158)inductive X | a | b | c def α : X ⟶ X | X.a => X.c | X.b => X.a | X.c => X.b def Xα : SetWithEndomap := { t := X carrier := Set.univ toEnd := α toEnd_mem := fun _ ↦ Set.mem_univ _ } inductive Y | p | q | r def β : Y ⟶ Y | Y.p => Y.q | Y.q => Y.r | Y.r => Y.p def Yβ : SetWithEndomap := { t := Y carrier := Set.univ toEnd := β toEnd_mem := fun _ ↦ Set.mem_univ _ }

Find an isomorphism from X^{↻\alpha} to Y^{↻\beta}. How many such isomorphisms are there?

Hint: You need to find {X \xrightarrow{f} Y} such that {f \alpha = \beta f}, and check that f has an inverse {Y \xrightarrow{f^{-1}} X} (meaning {f^{-1}f = 1_X} and {ff^{-1} = 1_Y}). Then you'll still need to check that f^{-1} is a map in 𝑺↻ (meaning {f^{-1} \beta = \alpha f^{-1}}), but see Exercise 4, below.

Solution: Exercise 2

There are three such isomorphisms, which we call f_1, f_2, f_3 below.

def f₁ : X ⟶ Y | X.a => Y.r | X.b => Y.q | X.c => Y.p --f₁ is an isomorphism example : ∃ f : Xα ⟶ Yβ, IsIso f := ⊢ ∃ f, IsIso f let f : Xα ⟶ Yβ := ⟨ f₁, ⊢ (∀ x ∈ Xα.carrier, f₁ x ∈ Yβ.carrier) ∧ f₁ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₁ ⊢ ∀ x ∈ Xα.carrier, f₁ x ∈ Yβ.carrier⊢ f₁ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₁ ⊢ ∀ x ∈ Xα.carrier, f₁ x ∈ Yβ.carrier All goals completed! 🐙 ⊢ f₁ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₁ -- fα = βf x:Xα.t⊢ (f₁ ⊚ Xα.toEnd) x = (Yβ.toEnd ⊚ f₁) x ⊢ (f₁ ⊚ Xα.toEnd) X.a = (Yβ.toEnd ⊚ f₁) X.a⊢ (f₁ ⊚ Xα.toEnd) X.b = (Yβ.toEnd ⊚ f₁) X.b⊢ (f₁ ⊚ Xα.toEnd) X.c = (Yβ.toEnd ⊚ f₁) X.c ⊢ (f₁ ⊚ Xα.toEnd) X.a = (Yβ.toEnd ⊚ f₁) X.a⊢ (f₁ ⊚ Xα.toEnd) X.b = (Yβ.toEnd ⊚ f₁) X.b⊢ (f₁ ⊚ Xα.toEnd) X.c = (Yβ.toEnd ⊚ f₁) X.c All goals completed! 🐙 ⟩ let finv : Yβ ⟶ Xα := ⟨ fun | Y.p => X.c | Y.q => X.b | Y.r => X.a, f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ (∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) x ∈ Xα.carrier) ∧ (fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) x ∈ Xα.carrierf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ (fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) x ∈ Xα.carrier All goals completed! 🐙 f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ (fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a -- f⁻¹β = αf⁻¹ f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩y:Yβ.t⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) y = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) y f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) Y.p = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) Y.pf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) Y.q = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) Y.qf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) Y.r = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) Y.r f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) Y.p = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) Y.pf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) Y.q = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) Y.qf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) ⊚ Yβ.toEnd) Y.r = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a) Y.r All goals completed! 🐙 ⟩ f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ IsIso f f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ finv ⊚ f = 𝟙 Xα ∧ f ⊚ finv = 𝟙 Yβ f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ finv ⊚ f = 𝟙 Xαf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ f ⊚ finv = 𝟙 Yβ f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ finv ⊚ f = 𝟙 Xα -- f⁻¹f = 𝟙 X have h : finv.val ⊚ f.val = 𝟙 X := ⊢ ∃ f, IsIso f f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩x:Xα.t⊢ (↑finv ⊚ ↑f) x = 𝟙 X x f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑finv ⊚ ↑f) X.a = 𝟙 X X.af:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑finv ⊚ ↑f) X.b = 𝟙 X X.bf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑finv ⊚ ↑f) X.c = 𝟙 X X.c f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑finv ⊚ ↑f) X.a = 𝟙 X X.af:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑finv ⊚ ↑f) X.b = 𝟙 X X.bf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑finv ⊚ ↑f) X.c = 𝟙 X X.c All goals completed! 🐙 All goals completed! 🐙 f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ f ⊚ finv = 𝟙 Yβ -- ff⁻¹ = 𝟙 Y have h : f.val ⊚ finv.val = 𝟙 Y := ⊢ ∃ f, IsIso f f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩y:Yβ.t⊢ (↑f ⊚ ↑finv) y = 𝟙 Y y f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.p = 𝟙 Y Y.pf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.q = 𝟙 Y Y.qf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.r = 𝟙 Y Y.r f:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.p = 𝟙 Y Y.pf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.q = 𝟙 Y Y.qf:Xα ⟶ Yβ := ⟨f₁, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.c | Y.q => X.b | Y.r => X.a, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.r = 𝟙 Y Y.r All goals completed! 🐙 All goals completed! 🐙 def f₂ : X ⟶ Y | X.a => Y.q | X.b => Y.p | X.c => Y.r --f₂ is an isomorphism example : ∃ f : Xα ⟶ Yβ, IsIso f := ⊢ ∃ f, IsIso f let f : Xα ⟶ Yβ := ⟨ f₂, ⊢ (∀ x ∈ Xα.carrier, f₂ x ∈ Yβ.carrier) ∧ f₂ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₂ ⊢ ∀ x ∈ Xα.carrier, f₂ x ∈ Yβ.carrier⊢ f₂ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₂ ⊢ ∀ x ∈ Xα.carrier, f₂ x ∈ Yβ.carrier All goals completed! 🐙 ⊢ f₂ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₂ -- fα = βf x:Xα.t⊢ (f₂ ⊚ Xα.toEnd) x = (Yβ.toEnd ⊚ f₂) x ⊢ (f₂ ⊚ Xα.toEnd) X.a = (Yβ.toEnd ⊚ f₂) X.a⊢ (f₂ ⊚ Xα.toEnd) X.b = (Yβ.toEnd ⊚ f₂) X.b⊢ (f₂ ⊚ Xα.toEnd) X.c = (Yβ.toEnd ⊚ f₂) X.c ⊢ (f₂ ⊚ Xα.toEnd) X.a = (Yβ.toEnd ⊚ f₂) X.a⊢ (f₂ ⊚ Xα.toEnd) X.b = (Yβ.toEnd ⊚ f₂) X.b⊢ (f₂ ⊚ Xα.toEnd) X.c = (Yβ.toEnd ⊚ f₂) X.c All goals completed! 🐙 ⟩ let finv : Yβ ⟶ Xα := ⟨ fun | Y.p => X.b | Y.q => X.a | Y.r => X.c, f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ (∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) x ∈ Xα.carrier) ∧ (fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) x ∈ Xα.carrierf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ (fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) x ∈ Xα.carrier All goals completed! 🐙 f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ (fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c -- f⁻¹β = αf⁻¹ f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩y:Yβ.t⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) y = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) y f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) Y.p = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) Y.pf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) Y.q = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) Y.qf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) Y.r = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) Y.r f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) Y.p = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) Y.pf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) Y.q = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) Y.qf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) ⊚ Yβ.toEnd) Y.r = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c) Y.r All goals completed! 🐙 ⟩ f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ IsIso f f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ finv ⊚ f = 𝟙 Xα ∧ f ⊚ finv = 𝟙 Yβ f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ finv ⊚ f = 𝟙 Xαf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ f ⊚ finv = 𝟙 Yβ f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ finv ⊚ f = 𝟙 Xα -- f⁻¹f = 𝟙 X have h : finv.val ⊚ f.val = 𝟙 X := ⊢ ∃ f, IsIso f f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩x:Xα.t⊢ (↑finv ⊚ ↑f) x = 𝟙 X x f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑finv ⊚ ↑f) X.a = 𝟙 X X.af:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑finv ⊚ ↑f) X.b = 𝟙 X X.bf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑finv ⊚ ↑f) X.c = 𝟙 X X.c f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑finv ⊚ ↑f) X.a = 𝟙 X X.af:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑finv ⊚ ↑f) X.b = 𝟙 X X.bf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑finv ⊚ ↑f) X.c = 𝟙 X X.c All goals completed! 🐙 All goals completed! 🐙 f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ f ⊚ finv = 𝟙 Yβ -- ff⁻¹ = 𝟙 Y have h : f.val ⊚ finv.val = 𝟙 Y := ⊢ ∃ f, IsIso f f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩y:Yβ.t⊢ (↑f ⊚ ↑finv) y = 𝟙 Y y f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.p = 𝟙 Y Y.pf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.q = 𝟙 Y Y.qf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.r = 𝟙 Y Y.r f:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.p = 𝟙 Y Y.pf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.q = 𝟙 Y Y.qf:Xα ⟶ Yβ := ⟨f₂, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.b | Y.q => X.a | Y.r => X.c, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.r = 𝟙 Y Y.r All goals completed! 🐙 All goals completed! 🐙 def f₃ : X ⟶ Y | X.a => Y.p | X.b => Y.r | X.c => Y.q --f₃ is an isomorphism example : ∃ f : Xα ⟶ Yβ, IsIso f := ⊢ ∃ f, IsIso f let f : Xα ⟶ Yβ := ⟨ f₃, ⊢ (∀ x ∈ Xα.carrier, f₃ x ∈ Yβ.carrier) ∧ f₃ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₃ ⊢ ∀ x ∈ Xα.carrier, f₃ x ∈ Yβ.carrier⊢ f₃ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₃ ⊢ ∀ x ∈ Xα.carrier, f₃ x ∈ Yβ.carrier All goals completed! 🐙 ⊢ f₃ ⊚ Xα.toEnd = Yβ.toEnd ⊚ f₃ -- fα = βf x:Xα.t⊢ (f₃ ⊚ Xα.toEnd) x = (Yβ.toEnd ⊚ f₃) x ⊢ (f₃ ⊚ Xα.toEnd) X.a = (Yβ.toEnd ⊚ f₃) X.a⊢ (f₃ ⊚ Xα.toEnd) X.b = (Yβ.toEnd ⊚ f₃) X.b⊢ (f₃ ⊚ Xα.toEnd) X.c = (Yβ.toEnd ⊚ f₃) X.c ⊢ (f₃ ⊚ Xα.toEnd) X.a = (Yβ.toEnd ⊚ f₃) X.a⊢ (f₃ ⊚ Xα.toEnd) X.b = (Yβ.toEnd ⊚ f₃) X.b⊢ (f₃ ⊚ Xα.toEnd) X.c = (Yβ.toEnd ⊚ f₃) X.c All goals completed! 🐙 ⟩ let finv : Yβ ⟶ Xα := ⟨ fun | Y.p => X.a | Y.q => X.c | Y.r => X.b, f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ (∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) x ∈ Xα.carrier) ∧ (fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) x ∈ Xα.carrierf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ (fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ∀ x ∈ Yβ.carrier, (fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) x ∈ Xα.carrier All goals completed! 🐙 f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ (fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd = Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b -- f⁻¹β = αf⁻¹ f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩y:Yβ.t⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) y = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) y f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) Y.p = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) Y.pf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) Y.q = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) Y.qf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) Y.r = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) Y.r f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) Y.p = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) Y.pf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) Y.q = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) Y.qf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩⊢ ((fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) ⊚ Yβ.toEnd) Y.r = (Xα.toEnd ⊚ fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b) Y.r All goals completed! 🐙 ⟩ f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ IsIso f f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ finv ⊚ f = 𝟙 Xα ∧ f ⊚ finv = 𝟙 Yβ f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ finv ⊚ f = 𝟙 Xαf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ f ⊚ finv = 𝟙 Yβ f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ finv ⊚ f = 𝟙 Xα -- f⁻¹f = 𝟙 X have h : finv.val ⊚ f.val = 𝟙 X := ⊢ ∃ f, IsIso f f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩x:Xα.t⊢ (↑finv ⊚ ↑f) x = 𝟙 X x f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑finv ⊚ ↑f) X.a = 𝟙 X X.af:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑finv ⊚ ↑f) X.b = 𝟙 X X.bf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑finv ⊚ ↑f) X.c = 𝟙 X X.c f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑finv ⊚ ↑f) X.a = 𝟙 X X.af:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑finv ⊚ ↑f) X.b = 𝟙 X X.bf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑finv ⊚ ↑f) X.c = 𝟙 X X.c All goals completed! 🐙 All goals completed! 🐙 f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ f ⊚ finv = 𝟙 Yβ -- ff⁻¹ = 𝟙 Y have h : f.val ⊚ finv.val = 𝟙 Y := ⊢ ∃ f, IsIso f f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩y:Yβ.t⊢ (↑f ⊚ ↑finv) y = 𝟙 Y y f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.p = 𝟙 Y Y.pf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.q = 𝟙 Y Y.qf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.r = 𝟙 Y Y.r f:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.p = 𝟙 Y Y.pf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.q = 𝟙 Y Y.qf:Xα ⟶ Yβ := ⟨f₃, ⋯⟩finv:Yβ ⟶ Xα := ⟨fun x ↦ match x with | Y.p => X.a | Y.q => X.c | Y.r => X.b, ⋯⟩⊢ (↑f ⊚ ↑finv) Y.r = 𝟙 Y Y.r All goals completed! 🐙 All goals completed! 🐙
Exercise 3 (p. 159)

Prove that there is no isomorphism (in 𝑺↻)

from

inductive X | x₁ | x₂ | x₃ | x₄ def α : X ⟶ X | X.x₁ => X.x₂ | X.x₂ => X.x₃ | X.x₃ => X.x₄ | X.x₄ => X.x₂ def Xα : SetWithEndomap := { t := X carrier := Set.univ toEnd := α toEnd_mem := fun _ ↦ Set.mem_univ _ }

to

inductive Y | y₁ | y₂ | y₃ | y₄ def β : Y ⟶ Y | Y.y₁ => Y.y₂ | Y.y₂ => Y.y₃ | Y.y₃ => Y.y₄ | Y.y₄ => Y.y₁ def Yβ : SetWithEndomap := { t := Y carrier := Set.univ toEnd := β toEnd_mem := fun _ ↦ Set.mem_univ _ }

Hint: In fact, more is true: there is no map (in 𝑺↻) from X^{↻\alpha} to Y^{↻\beta}.

Solution: Exercise 3

We give a proof by contradiction below.

example : ¬(∃ f : Xα ⟶ Yβ, IsIso f) := ⊢ ¬∃ f, IsIso f f:Xα ⟶ Yβh✝:IsIso f⊢ False -- X.x₂ is a fixed point of α ⊚ α ⊚ α, have h₁ : (Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := ⊢ ¬∃ f, IsIso f All goals completed! 🐙 -- but β ⊚ β ⊚ β has no fixed points in Y have h₂ : ∀ y : Y, (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y := ⊢ ¬∃ f, IsIso f f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)y:Y⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₁ ≠ Y.y₁f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₂ ≠ Y.y₂f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₃ ≠ Y.y₃f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₄ ≠ Y.y₄ f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₁ ≠ Y.y₁f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₂ ≠ Y.y₂f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₃ ≠ Y.y₃f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)⊢ (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₄ ≠ Y.y₄ exact fun h ↦ f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)h:(Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) Y.y₄ = Y.y₄⊢ False All goals completed! 🐙 -- Since f ⊚ (α ⊚ α ⊚ α) = (β ⊚ β ⊚ β) ⊚ f, we can derive a -- contradiction have h_contra : f.val ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) (f.val X.x₂) := ⊢ ¬∃ f, IsIso f f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)h₂:∀ (y : Y), (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y := fun y ↦ Y.casesOn (motive := fun t ↦ y = t → (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y) y (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (Eq.refl y)⊢ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ Yβ.toEnd ⊚ ↑f ⊚ Xα.toEnd) X.x₂ f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)h₂:∀ (y : Y), (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y := fun y ↦ Y.casesOn (motive := fun t ↦ y = t → (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y) y (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (Eq.refl y)⊢ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ (↑f ⊚ Xα.toEnd) ⊚ Xα.toEnd) X.x₂ f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)h₂:∀ (y : Y), (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y := fun y ↦ Y.casesOn (motive := fun t ↦ y = t → (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y) y (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (Eq.refl y)⊢ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (((↑f ⊚ Xα.toEnd) ⊚ Xα.toEnd) ⊚ Xα.toEnd) X.x₂ All goals completed! 🐙 have : (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) (f.val X.x₂) = f.val X.x₂ := ⊢ ¬∃ f, IsIso f All goals completed! 🐙 f:Xα ⟶ Yβh✝:IsIso fh₁:(Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂ = X.x₂ := Eq.refl ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂)h₂:∀ (y : Y), (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y := fun y ↦ Y.casesOn (motive := fun t ↦ y = t → (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) y ≠ y) y (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (fun h ↦ Eq.symm h ▸ fun h ↦ Y.noConfusion h) (Eq.refl y)h_contra:↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) (↑f X.x₂) := Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = _a) (Eq.symm (types_comp_apply (↑f) (Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) X.x₂)))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = _a X.x₂) (Eq.symm (Category.assoc (↑f) (Yβ.toEnd ⊚ Yβ.toEnd) Yβ.toEnd)))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ _a) X.x₂) (Eq.symm (Category.assoc (↑f) Yβ.toEnd Yβ.toEnd)))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ Yβ.toEnd ⊚ _a) X.x₂) (Eq.symm f.property.right))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ _a) X.x₂) (Category.assoc Xα.toEnd (↑f) Yβ.toEnd))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (Yβ.toEnd ⊚ _a ⊚ Xα.toEnd) X.x₂) (Eq.symm f.property.right))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = _a X.x₂) (Category.assoc Xα.toEnd (↑f ⊚ Xα.toEnd) Yβ.toEnd))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = (_a ⊚ Xα.toEnd) X.x₂) (Category.assoc Xα.toEnd (↑f) Yβ.toEnd))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = ((_a ⊚ Xα.toEnd) ⊚ Xα.toEnd) X.x₂) (Eq.symm f.property.right))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = _a X.x₂) (Eq.symm (Category.assoc Xα.toEnd Xα.toEnd (↑f ⊚ Xα.toEnd))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = _a X.x₂) (Eq.symm (Category.assoc (Xα.toEnd ⊚ Xα.toEnd) Xα.toEnd ↑f)))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂) = _a) (types_comp_apply (Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) (↑f) X.x₂))) (Eq.refl (↑f ((Xα.toEnd ⊚ Xα.toEnd ⊚ Xα.toEnd) X.x₂))))))))))))))this✝:(Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) (↑f X.x₂) = ↑f X.x₂ := Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f X.x₂) (Eq.symm h_contra))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a = ↑f X.x₂) h₁)) (Eq.refl (↑f X.x₂)))this:(Yβ.toEnd ⊚ Yβ.toEnd ⊚ Yβ.toEnd) (↑f X.x₂) ≠ ↑f X.x₂ := h₂ (↑f X.x₂)⊢ False All goals completed! 🐙
Exercise 4 (p. 159)

Suppose {A^{↻\alpha} \xrightarrow{f} B^{↻\beta}} is a map in 𝑺↻, and that as a map of sets, {A \xrightarrow{f} B} has an inverse {B \xrightarrow{f^{-1}} A}. Show that f^{-1} is automatically a map in 𝑺↻.

Solution: Exercise 4

Let {g = f^{-1}}.

example {Aα Bβ : SetWithEndomap} (f : Aα ⟶ Bβ) (g : Bβ.t ⟶ Aα.t) (hg_mtc : ∀ b ∈ Bβ.carrier, g b ∈ Aα.carrier) (h : g ⊚ f.val = 𝟙 Aα.t ∧ f.val ⊚ g = 𝟙 Bβ.t) : ∃ finv : Bβ ⟶ Aα, finv.val = g := Aα:SetWithEndomapBβ:SetWithEndomapf:Aα ⟶ Bβg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierh:g ⊚ ↑f = 𝟙 Aα.t ∧ ↑f ⊚ g = 𝟙 Bβ.t⊢ ∃ finv, ↑finv = g Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.t⊢ ∃ finv, ↑finv = g use ⟨ g, Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.t⊢ (∀ x ∈ Bβ.carrier, g x ∈ Aα.carrier) ∧ g ⊚ Bβ.toEnd = Aα.toEnd ⊚ g Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.t⊢ ∀ x ∈ Bβ.carrier, g x ∈ Aα.carrierAα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.t⊢ g ⊚ Bβ.toEnd = Aα.toEnd ⊚ g Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.t⊢ ∀ x ∈ Bβ.carrier, g x ∈ Aα.carrier All goals completed! 🐙 Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.t⊢ g ⊚ Bβ.toEnd = Aα.toEnd ⊚ g Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.t⊢ (g ⊚ Bβ.toEnd) b = (Aα.toEnd ⊚ g) b Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.t⊢ f ((g ⊚ Bβ.toEnd) b) = f ((Aα.toEnd ⊚ g) b)Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.t⊢ Function.Injective f Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.t⊢ f ((g ⊚ Bβ.toEnd) b) = f ((Aα.toEnd ⊚ g) b) Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.t⊢ Bβ.toEnd b = f ((Aα.toEnd ⊚ g) b) All goals completed! 🐙 Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.t⊢ Function.Injective f intro a₁ Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.ta₁:Aα.ta₂:Aα.t⊢ f a₁ = f a₂ → a₁ = a₂ Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.ta₁:Aα.ta₂:Aα.thf:f a₁ = f a₂⊢ a₁ = a₂ Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.ta₁:Aα.ta₂:Aα.thf:f a₁ = f a₂hgf:g (f a₁) = g (f a₂) := congrArg g hf⊢ a₁ = a₂ repeat Aα:SetWithEndomapBβ:SetWithEndomapg:Bβ.t ⟶ Aα.thg_mtc:∀ b ∈ Bβ.carrier, g b ∈ Aα.carrierf:Aα.t ⟶ Bβ.tleft✝:∀ x ∈ Aα.carrier, f x ∈ Bβ.carrierhf_comm:f ⊚ Aα.toEnd = Bβ.toEnd ⊚ fh:g ⊚ ↑⟨f, ⋯⟩ = 𝟙 Aα.t ∧ ↑⟨f, ⋯⟩ ⊚ g = 𝟙 Bβ.tb:Bβ.ta₁:Aα.ta₂:Aα.thf:f a₁ = f a₂hgf:𝟙 Aα.t a₁ = 𝟙 Aα.t a₂⊢ a₁ = a₂ All goals completed! 🐙 ⟩
Exercise 5 (p. 159)

{\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, 3, \ldots\}} is the set of integers, and \mathbb{Z}^{↻\alpha} and \mathbb{Z}^{↻\beta} are the maps which add 2 and 3: {\alpha(n) = n + 2}, {\beta(n) = n + 3}. Is \mathbb{Z}^{↻\alpha} isomorphic to \mathbb{Z}^{↻\beta}? (If so, find an isomorphism {\mathbb{Z}^{↻\alpha} \xrightarrow{f} \mathbb{Z}^{↻\beta}}; if not, explain how you know they are not isomorphic.)

Solution: Exercise 5

Define the maps \mathbb{Z}^{↻\alpha} and \mathbb{Z}^{↻\beta} as SetWithEndomap structures.

def α := (· + (2 : ℤ)) def β := (· + (3 : ℤ)) abbrev ℤα : SetWithEndomap := { t := ℤ carrier := Set.univ toEnd := α toEnd_mem := fun _ ↦ Set.mem_univ _ } abbrev ℤβ : SetWithEndomap := { t := ℤ carrier := Set.univ toEnd := β toEnd_mem := fun _ ↦ Set.mem_univ _ }

We show that \mathbb{Z}^{↻\alpha} is not isomorphic to \mathbb{Z}^{↻\beta}.

example (f : ℤα ⟶ ℤβ) : ¬(IsIso f) := f:ℤα ⟶ ℤβ⊢ ¬IsIso f -- Assume f is an isomorphism, and derive a contradiction f:ℤα ⟶ ℤβhf_iso:IsIso f⊢ False -- We begin by extracting the structure-preserving property of f have hf_comm : ∀ x : ℤ, (f.val ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ f.val) x := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fx:ℤ⊢ (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x All goals completed! 🐙 -- and unfolding the definitions of α and β f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_comm⊢ False -- The key observation: f.val(x + 2) and f.val(x) have the same -- remainder when divided by 3 have hf_mod_3_eq : ∀ x : ℤ, f.val (x + 2) ≡ f.val x [ZMOD 3] := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commx:ℤ⊢ ↑f (x + 2) ≡ ↑f x [ZMOD 3] f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commx:ℤ⊢ ↑f (x + 2) % 3 = ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commx:ℤ⊢ (↑f x + 3) % 3 = ↑f x % 3 All goals completed! 🐙 -- Hence all even numbers map to values with the same remainder mod 3 -- as f.val(0), have hf_even_congr : ∀ x : ℤ, f.val (2 * x) ≡ f.val 0 [ZMOD 3] := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))x:ℤ⊢ ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))x:ℤ⊢ ↑f (2 * x) % 3 = ↑f 0 % 3 induction x with f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))⊢ ↑f (2 * 0) % 3 = ↑f 0 % 3 All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))x':ℕih:↑f (2 * ↑x') % 3 = ↑f 0 % 3⊢ ↑f (2 * (↑x' + 1)) % 3 = ↑f 0 % 3 All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))x':ℕih:↑f (2 * -↑x') % 3 = ↑f 0 % 3⊢ ↑f (2 * (-↑x' - 1)) % 3 = ↑f 0 % 3 All goals completed! 🐙 -- and all odd numbers map to values with the same remainder mod 3 as -- f.val(1) have hf_odd_congr : ∀ x : ℤ, f.val (2 * x + 1) ≡ f.val 1 [ZMOD 3] := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x:ℤ⊢ ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x:ℤ⊢ ↑f (2 * x + 1) % 3 = ↑f 1 % 3 induction x with f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))⊢ ↑f (2 * 0 + 1) % 3 = ↑f 1 % 3 All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x':ℕih:↑f (2 * ↑x' + 1) % 3 = ↑f 1 % 3⊢ ↑f (2 * (↑x' + 1) + 1) % 3 = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x':ℕih:↑f (2 * ↑x' + 1) % 3 = ↑f 1 % 3⊢ ↑f (2 * ↑x' + 2 + 1) % 3 = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x':ℕih:↑f (2 * ↑x' + 1) % 3 = ↑f 1 % 3⊢ ↑f (2 * ↑x' + 1 + 2) % 3 = ↑f 1 % 3 All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x':ℕih:↑f (2 * -↑x' + 1) % 3 = ↑f 1 % 3⊢ ↑f (2 * (-↑x' - 1) + 1) % 3 = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x':ℕih:↑f (2 * -↑x' + 1) % 3 = ↑f 1 % 3⊢ ↑f (2 * -↑x' - 2 * 1 + 1 + 2) % 3 = ↑f 1 % 3 have : 2 * (-x' : ℤ) - 2 * 1 + 1 + 2 = 2 * (-x') +1 := f:ℤα ⟶ ℤβ⊢ ¬IsIso f All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))x':ℕih:↑f (2 * -↑x' + 1) % 3 = ↑f 1 % 3this:2 * -↑x' - 2 * 1 + 1 + 2 = 2 * -↑x' + 1 := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))⊢ ↑f (2 * -↑x' + 1) % 3 = ↑f 1 % 3 All goals completed! 🐙 -- So the image of f.val can have at most two distinct remainders -- mod 3 have hf_img_set₁ : Set.range (fun x ↦ f.val x % 3) = {f.val 0 % 3, f.val 1 % 3} := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.t⊢ (b ∈ Set.range fun x ↦ ↑f x % 3) ↔ b ∈ {↑f 0 % 3, ↑f 1 % 3} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.t⊢ (b ∈ Set.range fun x ↦ ↑f x % 3) → b ∈ {↑f 0 % 3, ↑f 1 % 3}f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.t⊢ b ∈ {↑f 0 % 3, ↑f 1 % 3} → b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.t⊢ (b ∈ Set.range fun x ↦ ↑f x % 3) → b ∈ {↑f 0 % 3, ↑f 1 % 3} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:(fun x ↦ ↑f x % 3) a = b⊢ b ∈ {↑f 0 % 3, ↑f 1 % 3} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = b⊢ b ∈ {↑f 0 % 3, ↑f 1 % 3} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = b⊢ b = ↑f 0 % 3 ∨ b = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = bha_even:Even a⊢ b = ↑f 0 % 3 ∨ b = ↑f 1 % 3f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = bha_odd:Odd a⊢ b = ↑f 0 % 3 ∨ b = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = bha_even:Even a⊢ b = ↑f 0 % 3 ∨ b = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = bha_even:Even a⊢ b = ↑f 0 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = ba':ℤha_even:a = a' + a'⊢ b = ↑f 0 % 3 All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = bha_odd:Odd a⊢ b = ↑f 0 % 3 ∨ b = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = bha_odd:Odd a⊢ b = ↑f 1 % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.ta:ℤα.thfa:↑f a % 3 = ba':ℤha_odd:a = 2 * a' + 1⊢ b = ↑f 1 % 3 All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.t⊢ b ∈ {↑f 0 % 3, ↑f 1 % 3} → b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.thb:b ∈ {↑f 0 % 3, ↑f 1 % 3}⊢ b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.thb_mem:b = ↑f 0 % 3⊢ b ∈ Set.range fun x ↦ ↑f x % 3f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.thb_mem:b ∈ {↑f 1 % 3}⊢ b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.thb_mem:b = ↑f 0 % 3⊢ b ∈ Set.range fun x ↦ ↑f x % 3f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.thb_mem:b ∈ {↑f 1 % 3}⊢ b ∈ Set.range fun x ↦ ↑f x % 3 (f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))b:ℤβ.thb_mem:b ∈ {↑f 1 % 3}⊢ ↑f 1 % 3 ∈ Set.range fun x ↦ ↑f x % 3; All goals completed! 🐙) -- Now, since f is an isomorphism, f.val is bijective f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso f⊢ False -- and hence surjective, f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.right⊢ False -- but a surjective function on ℤ must hit all three remainders mod 3 have hf_img_set₂ : Set.range (fun x ↦ f.val x % 3) = {0, 1, 2} := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.t⊢ (b ∈ Set.range fun x ↦ ↑f x % 3) ↔ b ∈ {0, 1, 2} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.t⊢ (b ∈ Set.range fun x ↦ ↑f x % 3) → b ∈ {0, 1, 2}f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.t⊢ b ∈ {0, 1, 2} → b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.t⊢ (b ∈ Set.range fun x ↦ ↑f x % 3) → b ∈ {0, 1, 2} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.thfa:(fun x ↦ ↑f x % 3) a = b⊢ b ∈ {0, 1, 2} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.thfa:↑f a % 3 = b⊢ b ∈ {0, 1, 2} have hf_lbound : 0 ≤ f.val a % 3 := Int.emod_nonneg (f.val a) (f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.thfa:↑f a % 3 = b⊢ 3 ≠ 0 All goals completed! 🐙 : (3 : ℤ) ≠ 0) have hf_ubound : f.val a % 3 < 3 := Int.emod_lt_of_pos (f.val a) (f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.thfa:↑f a % 3 = bhf_lbound:0 ≤ ↑f a % 3 := Int.emod_nonneg (↑f a) (of_decide_eq_true (id (Eq.refl true)))⊢ 0 < 3 All goals completed! 🐙 : (0 < (3 : ℤ))) f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 0hfa:0 = bhf_lbound:0 ≤ 0hf_ubound:0 < 3⊢ b ∈ {0, 1, 2}f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 1hfa:1 = bhf_lbound:0 ≤ 1hf_ubound:1 < 3⊢ b ∈ {0, 1, 2}f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 2hfa:2 = bhf_lbound:0 ≤ 2hf_ubound:2 < 3⊢ b ∈ {0, 1, 2} f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 0hfa:0 = bhf_lbound:0 ≤ 0hf_ubound:0 < 3⊢ b ∈ {0, 1, 2}f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 1hfa:1 = bhf_lbound:0 ≤ 1hf_ubound:1 < 3⊢ b ∈ {0, 1, 2}f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 2hfa:2 = bhf_lbound:0 ≤ 2hf_ubound:2 < 3⊢ b ∈ {0, 1, 2} (f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.ta:ℤα.tk:↑f a % 3 = 2hfa:2 = bhf_lbound:0 ≤ 2hf_ubound:2 < 3⊢ 2 ∈ {0, 1, 2}; All goals completed! 🐙) f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.t⊢ b ∈ {0, 1, 2} → b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b = 0⊢ b ∈ Set.range fun x ↦ ↑f x % 3f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b = 1⊢ b ∈ Set.range fun x ↦ ↑f x % 3f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b ∈ {2}⊢ b ∈ Set.range fun x ↦ ↑f x % 3 all_goals f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b ∈ {2}a:ℤα.thfa:↑f a = b⊢ b ∈ Set.range fun x ↦ ↑f x % 3 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b ∈ {2}a:ℤα.thfa:↑f a = b⊢ (fun x ↦ ↑f x % 3) a = b f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b ∈ {2}a:ℤα.thfa:↑f a = b⊢ ↑f a % 3 = b f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.rightb:ℤβ.thb:b ∈ {2}a:ℤα.thfa:↑f a = b⊢ 2 % 3 = 2 All goals completed! 🐙 -- Since we have found that the image of f.val can have at most two -- distinct elements and must also have exactly three distinct -- elements, we have a contradiction have h_card₁ : Set.ncard ({f.val 0 % 3, f.val 1 % 3} : Set ℤ) ≤ 2 := f:ℤα ⟶ ℤβ⊢ ¬IsIso f f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.righthf_img_set₂:(Set.range fun x ↦ ↑f x % 3) = {0, 1, 2} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ have hf_lbound := Int.emod_nonneg (↑f a) (of_decide_eq_true (id (Eq.refl true))); have hf_ubound := Int.emod_lt_of_pos (↑f a) (of_decide_eq_true (id (Eq.refl true))); if x : 1 ≤ ↑f a % 3 then if x : 2 ≤ ↑f a % 3 then Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Eq.refl false)))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Eq.refl false)))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)))))) (or_true False)) (eq_true True.intro)))) (or_true False)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le (Mathlib.Tactic.IntervalCases.of_lt_right hf_ubound (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))))) x)) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound else Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Eq.refl false)))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))))) (or_false True)) (eq_true True.intro)))) (or_true False)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le x) x)) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound else Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Eq.refl false)))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))))) (or_self False)) (eq_false not_false)))) (or_false True)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le x) (Mathlib.Tactic.IntervalCases.of_le_left hf_lbound (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound, mpr := fun a ↦ Or.casesOn a (fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 0)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))))) fun h ↦ Or.casesOn h (fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 1)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))))))) fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 2)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)))))) }h:{↑f 0 % 3, ↑f 1 % 3}.ncard ≤ {↑f 1 % 3}.ncard + 1 := Set.ncard_insert_le (↑f 0 % 3) {↑f 1 % 3}⊢ {↑f 0 % 3, ↑f 1 % 3}.ncard ≤ 2 rwa [Set.ncard_singletonf:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.righthf_img_set₂:(Set.range fun x ↦ ↑f x % 3) = {0, 1, 2} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ have hf_lbound := Int.emod_nonneg (↑f a) (of_decide_eq_true (id (Eq.refl true))); have hf_ubound := Int.emod_lt_of_pos (↑f a) (of_decide_eq_true (id (Eq.refl true))); if x : 1 ≤ ↑f a % 3 then if x : 2 ≤ ↑f a % 3 then Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Eq.refl false)))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Eq.refl false)))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)))))) (or_true False)) (eq_true True.intro)))) (or_true False)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le (Mathlib.Tactic.IntervalCases.of_lt_right hf_ubound (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))))) x)) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound else Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Eq.refl false)))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))))) (or_false True)) (eq_true True.intro)))) (or_true False)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le x) x)) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound else Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Eq.refl false)))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))))) (or_self False)) (eq_false not_false)))) (or_false True)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le x) (Mathlib.Tactic.IntervalCases.of_le_left hf_lbound (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound, mpr := fun a ↦ Or.casesOn a (fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 0)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))))) fun h ↦ Or.casesOn h (fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 1)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))))))) fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 2)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)))))) }h:{↑f 0 % 3, ↑f 1 % 3}.ncard ≤ 1 + 1⊢ {↑f 0 % 3, ↑f 1 % 3}.ncard ≤ 2 at h have h_card₂ : Set.ncard ({0, 1, 2} : Set ℤ) = 3 := f:ℤα ⟶ ℤβ⊢ ¬IsIso f All goals completed! 🐙 f:ℤα ⟶ ℤβhf_iso:IsIso fhf_comm:∀ (x : ℤ), (↑f ∘ ℤα.toEnd) x = (ℤβ.toEnd ∘ ↑f) x := fun x ↦ congrFun f.property.right xhf_comm':∀ (x : ℤ), ↑f (x + 2) = ↑f x + 3 := hf_commhf_mod_3_eq:∀ (x : ℤ), ↑f (x + 2) ≡ ↑f x [ZMOD 3] := fun x ↦ id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = ↑f x % 3) (hf_comm' x))) (Decidable.byContradiction fun a ↦ _example._proof_1 f x a))hf_even_congr:∀ (x : ℤ), ↑f (2 * x) ≡ ↑f 0 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 0 % 3) (mul_zero 2)) (eq_self (↑f 0 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_mod_3_eq (2 * ↑x')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1))))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * -↑x' - _a + 2) % 3 = ↑f 0 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (sub_add_cancel (2 * -↑x') 2))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) ih)) (Eq.refl (↑f 0 % 3)))))))hf_odd_congr:∀ (x : ℤ), ↑f (2 * x + 1) ≡ ↑f 1 [ZMOD 3] := fun x ↦ id (Int.induction_on x (of_eq_true (Eq.trans (congrArg (fun x ↦ ↑f x % 3 = ↑f 1 % 3) (Eq.trans (congrArg (fun x ↦ x + 1) (mul_zero 2)) (zero_add 1))) (eq_self (↑f 1 % 3)))) (fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_add 2 (↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a + 1) % 3 = ↑f 1 % 3) (mul_one 2))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (add_assoc (2 * ↑x') 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f (2 * ↑x' + _a) % 3 = ↑f 1 % 3) (add_comm 2 1))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) (Eq.symm (add_assoc (2 * ↑x') 1 2)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_mod_3_eq (2 * ↑x' + 1)))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) ih)) (Eq.refl (↑f 1 % 3))))))))) fun x' ih ↦ Eq.mpr (id (congrArg (fun _a ↦ ↑f (_a + 1) % 3 = ↑f 1 % 3) (mul_sub 2 (-↑x') 1))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm (hf_mod_3_eq (2 * -↑x' - 2 * 1 + 1))))) (have this := Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.negOfNat 2))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_gt (Int.negOfNat 2).rawCast (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 2)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Tactic.Ring.neg_congr (Mathlib.Tactic.Ring.atom_pf ↑x') (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_mul (↑x') (Nat.rawCast 1) (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 1)) (Eq.refl (Int.negOfNat 1)))))) Mathlib.Tactic.Ring.neg_zero)) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑x') (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℤ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℤ (Int.negOfNat 1)) (Eq.refl (Int.negOfNat 2))))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0))) (Mathlib.Tactic.Ring.zero_mul (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 1).rawCast + 0)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑x' ^ Nat.rawCast 1 * (Int.negOfNat 2).rawCast + 0)))); Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) this)) ih)))hf_img_set₁:(Set.range fun x ↦ ↑f x % 3) = {↑f 0 % 3, ↑f 1 % 3} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ Eq.mpr (id (congrArg (fun _a ↦ _a) (propext Set.mem_insert_iff))) (Eq.mpr (id (congrArg (fun _a ↦ b = ↑f 0 % 3 ∨ _a) (propext Set.mem_singleton_iff))) (Or.casesOn (Int.even_or_odd a) (fun ha_even ↦ Or.inl (Exists.casesOn ha_even fun a' ha_even ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) ha_even)) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 0 % 3) (Eq.symm (two_mul a')))) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 0 % 3) (hf_even_congr a'))) (Eq.refl (↑f 0 % 3))))))) fun ha_odd ↦ Or.inr (Exists.casesOn ha_odd fun a' ha_odd ↦ Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (Eq.symm hfa))) (Eq.mpr (id (congrArg (fun _a ↦ ↑f _a % 3 = ↑f 1 % 3) ha_odd)) (Eq.mpr (id (congrArg (fun _a ↦ _a = ↑f 1 % 3) (hf_odd_congr a'))) (Eq.refl (↑f 1 % 3))))))), mpr := fun hb ↦ Or.casesOn hb (fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 0)))) fun hb_mem ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ Set.range fun x ↦ ↑f x % 3) hb_mem)) (of_eq_true (Eq.trans Set.mem_range._simp_1 (exists_apply_eq_apply._simp_1 (fun a ↦ ↑f a % 3) 1))) }hf_bij:Function.Bijective ↑f := ConcreteCategory.bijective_of_isIso fhf_surj:Function.Surjective ↑f := hf_bij.righthf_img_set₂:(Set.range fun x ↦ ↑f x % 3) = {0, 1, 2} := Set.ext fun b ↦ { mp := fun a ↦ Exists.casesOn a fun a hfa ↦ have hf_lbound := Int.emod_nonneg (↑f a) (of_decide_eq_true (id (Eq.refl true))); have hf_ubound := Int.emod_lt_of_pos (↑f a) (of_decide_eq_true (id (Eq.refl true))); if x : 1 ≤ ↑f a % 3 then if x : 2 ≤ ↑f a % 3 then Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Eq.refl false)))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Eq.refl false)))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)))))) (or_true False)) (eq_true True.intro)))) (or_true False)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le (Mathlib.Tactic.IntervalCases.of_lt_right hf_ubound (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3))))) x)) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound else Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Eq.refl false)))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))))) (or_false True)) (eq_true True.intro)))) (or_true False)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le x) x)) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound else Eq.ndrec (motive := fun x ↦ ↑f a % 3 = x → x = b → 0 ≤ x → x < 3 → b ∈ {0, 1, 2}) (fun k hfa hf_lbound hf_ubound ↦ Eq.mpr (id (congrArg (fun _a ↦ _a ∈ {0, 1, 2}) (Eq.symm hfa))) (of_eq_true (Eq.trans (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_true (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))) (Eq.trans Set.mem_insert_iff._simp_1 (Eq.trans (Eq.trans (congr (congrArg Or (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1)) (Eq.refl false)))) (Eq.trans Set.mem_singleton_iff._simp_1 (eq_false (Mathlib.Meta.NormNum.isNat_eq_false (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)) (Eq.refl false))))) (or_self False)) (eq_false not_false)))) (or_false True)) (eq_true True.intro))) (eq_true True.intro)))) (Eq.symm (le_antisymm (Int.le_sub_one_of_not_le x) (Mathlib.Tactic.IntervalCases.of_le_left hf_lbound (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0)))))) (Eq.refl (↑f a % 3)) hfa hf_lbound hf_ubound, mpr := fun a ↦ Or.casesOn a (fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 0)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 0))))))) fun h ↦ Or.casesOn h (fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 1)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 1))))))) fun hb ↦ Exists.casesOn (hf_surj b) fun a hfa ↦ Exists.intro a (id (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = b) hfa)) (Eq.mpr (id (congrArg (fun _a ↦ _a % 3 = _a) hb)) (Mathlib.Meta.NormNum.isNat_eq_true (Mathlib.Meta.NormNum.isInt_emod (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2))) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 3)) (Eq.refl (Int.ofNat 0)) (Eq.refl (Int.ofNat 2)) (Eq.refl true)) (Mathlib.Meta.NormNum.isNat_ofNat ℤ (Eq.refl 2)))))) }h_card₁:3 ≤ 2h_card₂:{0, 1, 2}.ncard = 3⊢ False All goals completed! 🐙
Exercise 6 (p. 159)

Each of the following graphs is isomorphic to exactly one of the others. Which?

Solution: Exercise 6

We label the arrows in each graph from top to bottom.

inductive A | a₁ | a₂ | a₃

We label the dots in each graph from left to right.

inductive D | d₁ | d₂ | d₃

Then the six graphs (a) to (f) are as follows:

def graph_a : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₂ | A.a₂ => D.d₁ | A.a₃ => D.d₃ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₃ | A.a₂ => D.d₂ | A.a₃ => D.d₂ toTgt_mem := fun _ ↦ Set.mem_univ _ } def graph_b : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₁ | A.a₂ => D.d₂ | A.a₃ => D.d₂, toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₂ | A.a₂ => D.d₃ | A.a₃ => D.d₁ toTgt_mem := fun _ ↦ Set.mem_univ _ } def graph_c : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₁ | A.a₂ => D.d₁ | A.a₃ => D.d₁ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₃ | A.a₂ => D.d₂ | A.a₃ => D.d₃ toTgt_mem := fun _ ↦ Set.mem_univ _ } def graph_d : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₃ | A.a₂ => D.d₁ | A.a₃ => D.d₂ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₂ | A.a₂ => D.d₂ | A.a₃ => D.d₃ toTgt_mem := fun _ ↦ Set.mem_univ _ } def graph_e : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₁ | A.a₂ => D.d₁ | A.a₃ => D.d₃ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₃ | A.a₂ => D.d₂ | A.a₃ => D.d₁ toTgt_mem := fun _ ↦ Set.mem_univ _ } def graph_f : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₁ | A.a₂ => D.d₁ | A.a₃ => D.d₁ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₂ | A.a₂ => D.d₃ | A.a₃ => D.d₃ toTgt_mem := fun _ ↦ Set.mem_univ _ }

Graph (a) is isomorphic to graph (d).

def f₁ : graph_a ⟶ graph_d := { val := ( fun -- maps arrows | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun -- maps dots | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃ ) property := ⊢ (∀ x ∈ graph_a.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_d.carrierA) ∧ (∀ x ∈ graph_a.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_d.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toSrc = graph_d.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt = graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ⊢ ∀ x ∈ graph_a.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_d.carrierA⊢ ∀ x ∈ graph_a.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_d.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toSrc = graph_d.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt = graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt = graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 | x:graph_a.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) x = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) A.a₁ = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) A.a₂ = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) A.a₃ = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) A.a₁ = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) A.a₂ = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_a.toTgt) A.a₃ = (graph_d.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ All goals completed! 🐙 } def finv₁ : graph_d ⟶ graph_a := { val := ( fun -- maps arrows | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun -- maps dots | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃ ) property := ⊢ (∀ x ∈ graph_d.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_a.carrierA) ∧ (∀ x ∈ graph_d.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_a.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toSrc = graph_a.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt = graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ⊢ ∀ x ∈ graph_d.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_a.carrierA⊢ ∀ x ∈ graph_d.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_a.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toSrc = graph_a.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt = graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt = graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 | x:graph_d.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) x = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) A.a₁ = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) A.a₂ = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) A.a₃ = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) A.a₁ = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) A.a₂ = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_d.toTgt) A.a₃ = (graph_a.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ All goals completed! 🐙 } example : graph_a ≅ graph_d := { hom := f₁, inv := finv₁, hom_inv_id := ⊢ finv₁ ⊚ f₁ = 𝟙 graph_a have h₁ : (finv₁.val.1 ⊚ f₁.val.1 = 𝟙 graph_a.tA) ∧ (finv₁.val.2 ⊚ f₁.val.2 = 𝟙 graph_a.tD) := ⊢ finv₁ ⊚ f₁ = 𝟙 graph_a ⊢ (↑finv₁).1 ⊚ (↑f₁).1 = 𝟙 graph_a.tA⊢ (↑finv₁).2 ⊚ (↑f₁).2 = 𝟙 graph_a.tD ⊢ (↑finv₁).1 ⊚ (↑f₁).1 = 𝟙 graph_a.tA⊢ (↑finv₁).2 ⊚ (↑f₁).2 = 𝟙 graph_a.tD (x:graph_a.tD⊢ ((↑finv₁).2 ⊚ (↑f₁).2) x = 𝟙 graph_a.tD x; ⊢ ((↑finv₁).2 ⊚ (↑f₁).2) D.d₁ = 𝟙 graph_a.tD D.d₁⊢ ((↑finv₁).2 ⊚ (↑f₁).2) D.d₂ = 𝟙 graph_a.tD D.d₂⊢ ((↑finv₁).2 ⊚ (↑f₁).2) D.d₃ = 𝟙 graph_a.tD D.d₃ ⊢ ((↑finv₁).2 ⊚ (↑f₁).2) D.d₁ = 𝟙 graph_a.tD D.d₁⊢ ((↑finv₁).2 ⊚ (↑f₁).2) D.d₂ = 𝟙 graph_a.tD D.d₂⊢ ((↑finv₁).2 ⊚ (↑f₁).2) D.d₃ = 𝟙 graph_a.tD D.d₃ All goals completed! 🐙) have h₂ : (finv₁.val.1 ⊚ f₁.val.1, finv₁.val.2 ⊚ f₁.val.2) = (𝟙 graph_a.tA, 𝟙 graph_a.tD) := ⊢ finv₁ ⊚ f₁ = 𝟙 graph_a All goals completed! 🐙 All goals completed! 🐙, inv_hom_id := ⊢ f₁ ⊚ finv₁ = 𝟙 graph_d have h₁ : (f₁.val.1 ⊚ finv₁.val.1 = 𝟙 graph_d.tA) ∧ (f₁.val.2 ⊚ finv₁.val.2 = 𝟙 graph_d.tD) := ⊢ f₁ ⊚ finv₁ = 𝟙 graph_d ⊢ (↑f₁).1 ⊚ (↑finv₁).1 = 𝟙 graph_d.tA⊢ (↑f₁).2 ⊚ (↑finv₁).2 = 𝟙 graph_d.tD ⊢ (↑f₁).1 ⊚ (↑finv₁).1 = 𝟙 graph_d.tA⊢ (↑f₁).2 ⊚ (↑finv₁).2 = 𝟙 graph_d.tD (x:graph_d.tD⊢ ((↑f₁).2 ⊚ (↑finv₁).2) x = 𝟙 graph_d.tD x; ⊢ ((↑f₁).2 ⊚ (↑finv₁).2) D.d₁ = 𝟙 graph_d.tD D.d₁⊢ ((↑f₁).2 ⊚ (↑finv₁).2) D.d₂ = 𝟙 graph_d.tD D.d₂⊢ ((↑f₁).2 ⊚ (↑finv₁).2) D.d₃ = 𝟙 graph_d.tD D.d₃ ⊢ ((↑f₁).2 ⊚ (↑finv₁).2) D.d₁ = 𝟙 graph_d.tD D.d₁⊢ ((↑f₁).2 ⊚ (↑finv₁).2) D.d₂ = 𝟙 graph_d.tD D.d₂⊢ ((↑f₁).2 ⊚ (↑finv₁).2) D.d₃ = 𝟙 graph_d.tD D.d₃ All goals completed! 🐙) have h₂ : (f₁.val.1 ⊚ finv₁.val.1, f₁.val.2 ⊚ finv₁.val.2) = (𝟙 graph_d.tA, 𝟙 graph_d.tD) := ⊢ f₁ ⊚ finv₁ = 𝟙 graph_d All goals completed! 🐙 All goals completed! 🐙 }

Graph (b) is isomorphic to graph (e).

def f₂ : graph_b ⟶ graph_e := { val := ( fun -- maps arrows | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun -- maps dots | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂ ) property := ⊢ (∀ x ∈ graph_b.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1 x ∈ graph_e.carrierA) ∧ (∀ x ∈ graph_b.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 x ∈ graph_e.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toSrc = graph_e.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt = graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1 ⊢ ∀ x ∈ graph_b.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1 x ∈ graph_e.carrierA⊢ ∀ x ∈ graph_b.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 x ∈ graph_e.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toSrc = graph_e.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt = graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt = graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1 | x:graph_b.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) x = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) A.a₁ = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) A.a₂ = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) A.a₃ = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) A.a₃ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) A.a₁ = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) A.a₂ = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).2 ⊚ graph_b.toTgt) A.a₃ = (graph_e.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₁ | D.d₃ => D.d₂).1) A.a₃ All goals completed! 🐙 } def finv₂ : graph_e ⟶ graph_b := { val := ( fun -- maps arrows | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun -- maps dots | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁ ) property := ⊢ (∀ x ∈ graph_e.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1 x ∈ graph_b.carrierA) ∧ (∀ x ∈ graph_e.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 x ∈ graph_b.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toSrc = graph_b.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt = graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1 ⊢ ∀ x ∈ graph_e.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1 x ∈ graph_b.carrierA⊢ ∀ x ∈ graph_e.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 x ∈ graph_b.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toSrc = graph_b.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt = graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt = graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1 | x:graph_e.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) x = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) A.a₁ = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) A.a₂ = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) A.a₃ = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) A.a₃ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) A.a₁ = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) A.a₂ = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).2 ⊚ graph_e.toTgt) A.a₃ = (graph_b.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₂ | A.a₃ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₂ | D.d₂ => D.d₃ | D.d₃ => D.d₁).1) A.a₃ All goals completed! 🐙 } example : graph_b ≅ graph_e := { hom := f₂, inv := finv₂, hom_inv_id := ⊢ finv₂ ⊚ f₂ = 𝟙 graph_b have h₁ : (finv₂.val.1 ⊚ f₂.val.1 = 𝟙 graph_b.tA) ∧ (finv₂.val.2 ⊚ f₂.val.2 = 𝟙 graph_b.tD) := ⊢ finv₂ ⊚ f₂ = 𝟙 graph_b ⊢ (↑finv₂).1 ⊚ (↑f₂).1 = 𝟙 graph_b.tA⊢ (↑finv₂).2 ⊚ (↑f₂).2 = 𝟙 graph_b.tD ⊢ (↑finv₂).1 ⊚ (↑f₂).1 = 𝟙 graph_b.tA⊢ (↑finv₂).2 ⊚ (↑f₂).2 = 𝟙 graph_b.tD (x:graph_b.tD⊢ ((↑finv₂).2 ⊚ (↑f₂).2) x = 𝟙 graph_b.tD x; ⊢ ((↑finv₂).2 ⊚ (↑f₂).2) D.d₁ = 𝟙 graph_b.tD D.d₁⊢ ((↑finv₂).2 ⊚ (↑f₂).2) D.d₂ = 𝟙 graph_b.tD D.d₂⊢ ((↑finv₂).2 ⊚ (↑f₂).2) D.d₃ = 𝟙 graph_b.tD D.d₃ ⊢ ((↑finv₂).2 ⊚ (↑f₂).2) D.d₁ = 𝟙 graph_b.tD D.d₁⊢ ((↑finv₂).2 ⊚ (↑f₂).2) D.d₂ = 𝟙 graph_b.tD D.d₂⊢ ((↑finv₂).2 ⊚ (↑f₂).2) D.d₃ = 𝟙 graph_b.tD D.d₃ All goals completed! 🐙) have h₂ : (finv₂.val.1 ⊚ f₂.val.1, finv₂.val.2 ⊚ f₂.val.2) = (𝟙 graph_b.tA, 𝟙 graph_b.tD) := ⊢ finv₂ ⊚ f₂ = 𝟙 graph_b All goals completed! 🐙 All goals completed! 🐙, inv_hom_id := ⊢ f₂ ⊚ finv₂ = 𝟙 graph_e have h₁ : (f₂.val.1 ⊚ finv₂.val.1 = 𝟙 graph_e.tA) ∧ (f₂.val.2 ⊚ finv₂.val.2 = 𝟙 graph_e.tD) := ⊢ f₂ ⊚ finv₂ = 𝟙 graph_e ⊢ (↑f₂).1 ⊚ (↑finv₂).1 = 𝟙 graph_e.tA⊢ (↑f₂).2 ⊚ (↑finv₂).2 = 𝟙 graph_e.tD ⊢ (↑f₂).1 ⊚ (↑finv₂).1 = 𝟙 graph_e.tA⊢ (↑f₂).2 ⊚ (↑finv₂).2 = 𝟙 graph_e.tD (x:graph_e.tD⊢ ((↑f₂).2 ⊚ (↑finv₂).2) x = 𝟙 graph_e.tD x; ⊢ ((↑f₂).2 ⊚ (↑finv₂).2) D.d₁ = 𝟙 graph_e.tD D.d₁⊢ ((↑f₂).2 ⊚ (↑finv₂).2) D.d₂ = 𝟙 graph_e.tD D.d₂⊢ ((↑f₂).2 ⊚ (↑finv₂).2) D.d₃ = 𝟙 graph_e.tD D.d₃ ⊢ ((↑f₂).2 ⊚ (↑finv₂).2) D.d₁ = 𝟙 graph_e.tD D.d₁⊢ ((↑f₂).2 ⊚ (↑finv₂).2) D.d₂ = 𝟙 graph_e.tD D.d₂⊢ ((↑f₂).2 ⊚ (↑finv₂).2) D.d₃ = 𝟙 graph_e.tD D.d₃ All goals completed! 🐙) have h₂ : (f₂.val.1 ⊚ finv₂.val.1, f₂.val.2 ⊚ finv₂.val.2) = (𝟙 graph_e.tA, 𝟙 graph_e.tD) := ⊢ f₂ ⊚ finv₂ = 𝟙 graph_e All goals completed! 🐙 All goals completed! 🐙 }

Graph (c) is isomorphic to graph (f).

def f₃ : graph_c ⟶ graph_f := { val := ( fun -- maps arrows | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun -- maps dots | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃ ) property := ⊢ (∀ x ∈ graph_c.carrierA, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_f.carrierA) ∧ (∀ x ∈ graph_c.carrierD, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_f.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toSrc = graph_f.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt = graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ⊢ ∀ x ∈ graph_c.carrierA, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_f.carrierA⊢ ∀ x ∈ graph_c.carrierD, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_f.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toSrc = graph_f.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1⊢ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt = graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt = graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 | x:graph_c.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) x = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) A.a₁ = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) A.a₂ = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) A.a₃ = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) A.a₁ = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) A.a₂ = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_c.toTgt) A.a₃ = (graph_f.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ All goals completed! 🐙 } def finv₃ : graph_f ⟶ graph_c := { val := ( fun -- maps arrows | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun -- maps dots | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃ ) property := ⊢ (∀ x ∈ graph_f.carrierA, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_c.carrierA) ∧ (∀ x ∈ graph_f.carrierD, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_c.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toSrc = graph_c.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt = graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 ⊢ ∀ x ∈ graph_f.carrierA, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 x ∈ graph_c.carrierA⊢ ∀ x ∈ graph_f.carrierD, (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 x ∈ graph_c.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toSrc = graph_c.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1⊢ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt = graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt = graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1 | x:graph_f.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) x = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) A.a₁ = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) A.a₂ = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) A.a₃ = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) A.a₁ = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) A.a₂ = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).2 ⊚ graph_f.toTgt) A.a₃ = (graph_c.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₂ | A.a₂ => A.a₁ | A.a₃ => A.a₃, fun x ↦ match x with | D.d₁ => D.d₁ | D.d₂ => D.d₂ | D.d₃ => D.d₃).1) A.a₃ All goals completed! 🐙 } example : graph_c ≅ graph_f := { hom := f₃, inv := finv₃, hom_inv_id := ⊢ finv₃ ⊚ f₃ = 𝟙 graph_c have h₁ : (finv₃.val.1 ⊚ f₃.val.1 = 𝟙 graph_c.tA) ∧ (finv₃.val.2 ⊚ f₃.val.2 = 𝟙 graph_c.tD) := ⊢ finv₃ ⊚ f₃ = 𝟙 graph_c ⊢ (↑finv₃).1 ⊚ (↑f₃).1 = 𝟙 graph_c.tA⊢ (↑finv₃).2 ⊚ (↑f₃).2 = 𝟙 graph_c.tD ⊢ (↑finv₃).1 ⊚ (↑f₃).1 = 𝟙 graph_c.tA⊢ (↑finv₃).2 ⊚ (↑f₃).2 = 𝟙 graph_c.tD (x:graph_c.tD⊢ ((↑finv₃).2 ⊚ (↑f₃).2) x = 𝟙 graph_c.tD x; ⊢ ((↑finv₃).2 ⊚ (↑f₃).2) D.d₁ = 𝟙 graph_c.tD D.d₁⊢ ((↑finv₃).2 ⊚ (↑f₃).2) D.d₂ = 𝟙 graph_c.tD D.d₂⊢ ((↑finv₃).2 ⊚ (↑f₃).2) D.d₃ = 𝟙 graph_c.tD D.d₃ ⊢ ((↑finv₃).2 ⊚ (↑f₃).2) D.d₁ = 𝟙 graph_c.tD D.d₁⊢ ((↑finv₃).2 ⊚ (↑f₃).2) D.d₂ = 𝟙 graph_c.tD D.d₂⊢ ((↑finv₃).2 ⊚ (↑f₃).2) D.d₃ = 𝟙 graph_c.tD D.d₃ All goals completed! 🐙) have h₂ : (finv₃.val.1 ⊚ f₃.val.1, finv₃.val.2 ⊚ f₃.val.2) = (𝟙 graph_c.tA, 𝟙 graph_c.tD) := ⊢ finv₃ ⊚ f₃ = 𝟙 graph_c All goals completed! 🐙 All goals completed! 🐙, inv_hom_id := ⊢ f₃ ⊚ finv₃ = 𝟙 graph_f have h₁ : (f₃.val.1 ⊚ finv₃.val.1 = 𝟙 graph_f.tA) ∧ (f₃.val.2 ⊚ finv₃.val.2 = 𝟙 graph_f.tD) := ⊢ f₃ ⊚ finv₃ = 𝟙 graph_f ⊢ (↑f₃).1 ⊚ (↑finv₃).1 = 𝟙 graph_f.tA⊢ (↑f₃).2 ⊚ (↑finv₃).2 = 𝟙 graph_f.tD ⊢ (↑f₃).1 ⊚ (↑finv₃).1 = 𝟙 graph_f.tA⊢ (↑f₃).2 ⊚ (↑finv₃).2 = 𝟙 graph_f.tD (x:graph_f.tD⊢ ((↑f₃).2 ⊚ (↑finv₃).2) x = 𝟙 graph_f.tD x; ⊢ ((↑f₃).2 ⊚ (↑finv₃).2) D.d₁ = 𝟙 graph_f.tD D.d₁⊢ ((↑f₃).2 ⊚ (↑finv₃).2) D.d₂ = 𝟙 graph_f.tD D.d₂⊢ ((↑f₃).2 ⊚ (↑finv₃).2) D.d₃ = 𝟙 graph_f.tD D.d₃ ⊢ ((↑f₃).2 ⊚ (↑finv₃).2) D.d₁ = 𝟙 graph_f.tD D.d₁⊢ ((↑f₃).2 ⊚ (↑finv₃).2) D.d₂ = 𝟙 graph_f.tD D.d₂⊢ ((↑f₃).2 ⊚ (↑finv₃).2) D.d₃ = 𝟙 graph_f.tD D.d₃ All goals completed! 🐙) have h₂ : (f₃.val.1 ⊚ finv₃.val.1, f₃.val.2 ⊚ finv₃.val.2) = (𝟙 graph_f.tA, 𝟙 graph_f.tD) := ⊢ f₃ ⊚ finv₃ = 𝟙 graph_f All goals completed! 🐙 All goals completed! 🐙 }
Exercise 7 (p. 160)

If these two graphs are isomorphic, find an isomorphism between them; if they are not isomorphic, explain how you know they are not.

Solution: Exercise 7

We label the arrows in each graph starting with the arrow at the bottom left and moving clockwise through the four outer arrows (a_1 to a_4) and then vertically upwards through the two inner arrows (a_5 and a_6).

inductive A | a₁ | a₂ | a₃ | a₄ | a₅ | a₆

We label the dots in each graph starting with the bottom dot and moving clockwise through the four outer dots (d_1 to d_4) and then finishing with the centre dot (d_5).

inductive D | d₁ | d₂ | d₃ | d₄ | d₅

Then the two graphs are as follows:

def graph_L : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₁ | A.a₂ => D.d₂ | A.a₃ => D.d₃ | A.a₄ => D.d₄ | A.a₅ => D.d₁ | A.a₆ => D.d₅ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₂ | A.a₂ => D.d₃ | A.a₃ => D.d₄ | A.a₄ => D.d₁ | A.a₅ => D.d₅ | A.a₆ => D.d₃ toTgt_mem := fun _ ↦ Set.mem_univ _ } def graph_R : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => D.d₂ | A.a₂ => D.d₃ | A.a₃ => D.d₃ | A.a₄ => D.d₄ | A.a₅ => D.d₁ | A.a₆ => D.d₅ toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => D.d₁ | A.a₂ => D.d₂ | A.a₃ => D.d₄ | A.a₄ => D.d₁ | A.a₅ => D.d₅ | A.a₆ => D.d₃ toTgt_mem := fun _ ↦ Set.mem_univ _ }

The two graphs are isomorphic, with an isomorphism between them given by f below.

def f : graph_L ⟶ graph_R := { val := ( fun -- maps arrows | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁ , fun -- maps dots | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂ ) property := ⊢ (∀ x ∈ graph_L.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1 x ∈ graph_R.carrierA) ∧ (∀ x ∈ graph_L.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 x ∈ graph_R.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toSrc = graph_R.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt = graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1 ⊢ ∀ x ∈ graph_L.carrierA, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1 x ∈ graph_R.carrierA⊢ ∀ x ∈ graph_L.carrierD, (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 x ∈ graph_R.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toSrc = graph_R.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt = graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt = graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1 | x:graph_L.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) x = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₁ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₂ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₃ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₃⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₄ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₄⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₅ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₅⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₆ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₆ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₁ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₂ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₃ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₃⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₄ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₄⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₅ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₅⊢ ((fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).2 ⊚ graph_L.toTgt) A.a₆ = (graph_R.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₃ | A.a₂ => A.a₄ | A.a₃ => A.a₅ | A.a₄ => A.a₆ | A.a₅ => A.a₂ | A.a₆ => A.a₁, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₄ | D.d₃ => D.d₁ | D.d₄ => D.d₅ | D.d₅ => D.d₂).1) A.a₆ All goals completed! 🐙 } def finv : graph_R ⟶ graph_L := { val := ( fun -- maps arrows | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun -- maps dots | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄ ) property := ⊢ (∀ x ∈ graph_R.carrierA, (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1 x ∈ graph_L.carrierA) ∧ (∀ x ∈ graph_R.carrierD, (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 x ∈ graph_L.carrierD) ∧ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toSrc = graph_L.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1 ∧ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt = graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1 ⊢ ∀ x ∈ graph_R.carrierA, (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1 x ∈ graph_L.carrierA⊢ ∀ x ∈ graph_R.carrierD, (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 x ∈ graph_L.carrierD⊢ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toSrc = graph_L.toSrc ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1⊢ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt = graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1 all_goals first | ⊢ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt = graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1 | x:graph_R.tA⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) x = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) x; ⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₁ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₂ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₃ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₃⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₄ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₄⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₅ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₅⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₆ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₆ ⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₁ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₁⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₂ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₂⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₃ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₃⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₄ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₄⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₅ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₅⊢ ((fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).2 ⊚ graph_R.toTgt) A.a₆ = (graph_L.toTgt ⊚ (fun x ↦ match x with | A.a₁ => A.a₆ | A.a₂ => A.a₅ | A.a₃ => A.a₁ | A.a₄ => A.a₂ | A.a₅ => A.a₃ | A.a₆ => A.a₄, fun x ↦ match x with | D.d₁ => D.d₃ | D.d₂ => D.d₅ | D.d₃ => D.d₁ | D.d₄ => D.d₂ | D.d₅ => D.d₄).1) A.a₆ All goals completed! 🐙 } example : graph_L ≅ graph_R := { hom := f, inv := finv, hom_inv_id := ⊢ finv ⊚ f = 𝟙 graph_L have h₁ : (finv.val.1 ⊚ f.val.1 = 𝟙 graph_L.tA) ∧ (finv.val.2 ⊚ f.val.2 = 𝟙 graph_L.tD) := ⊢ finv ⊚ f = 𝟙 graph_L ⊢ (↑finv).1 ⊚ (↑f).1 = 𝟙 graph_L.tA⊢ (↑finv).2 ⊚ (↑f).2 = 𝟙 graph_L.tD ⊢ (↑finv).1 ⊚ (↑f).1 = 𝟙 graph_L.tA⊢ (↑finv).2 ⊚ (↑f).2 = 𝟙 graph_L.tD (x:graph_L.tD⊢ ((↑finv).2 ⊚ (↑f).2) x = 𝟙 graph_L.tD x; ⊢ ((↑finv).2 ⊚ (↑f).2) D.d₁ = 𝟙 graph_L.tD D.d₁⊢ ((↑finv).2 ⊚ (↑f).2) D.d₂ = 𝟙 graph_L.tD D.d₂⊢ ((↑finv).2 ⊚ (↑f).2) D.d₃ = 𝟙 graph_L.tD D.d₃⊢ ((↑finv).2 ⊚ (↑f).2) D.d₄ = 𝟙 graph_L.tD D.d₄⊢ ((↑finv).2 ⊚ (↑f).2) D.d₅ = 𝟙 graph_L.tD D.d₅ ⊢ ((↑finv).2 ⊚ (↑f).2) D.d₁ = 𝟙 graph_L.tD D.d₁⊢ ((↑finv).2 ⊚ (↑f).2) D.d₂ = 𝟙 graph_L.tD D.d₂⊢ ((↑finv).2 ⊚ (↑f).2) D.d₃ = 𝟙 graph_L.tD D.d₃⊢ ((↑finv).2 ⊚ (↑f).2) D.d₄ = 𝟙 graph_L.tD D.d₄⊢ ((↑finv).2 ⊚ (↑f).2) D.d₅ = 𝟙 graph_L.tD D.d₅ All goals completed! 🐙) have h₂ : (finv.val.1 ⊚ f.val.1, finv.val.2 ⊚ f.val.2) = (𝟙 graph_L.tA, 𝟙 graph_L.tD) := ⊢ finv ⊚ f = 𝟙 graph_L All goals completed! 🐙 All goals completed! 🐙, inv_hom_id := ⊢ f ⊚ finv = 𝟙 graph_R have h₁ : (f.val.1 ⊚ finv.val.1 = 𝟙 graph_R.tA) ∧ (f.val.2 ⊚ finv.val.2 = 𝟙 graph_R.tD) := ⊢ f ⊚ finv = 𝟙 graph_R ⊢ (↑f).1 ⊚ (↑finv).1 = 𝟙 graph_R.tA⊢ (↑f).2 ⊚ (↑finv).2 = 𝟙 graph_R.tD ⊢ (↑f).1 ⊚ (↑finv).1 = 𝟙 graph_R.tA⊢ (↑f).2 ⊚ (↑finv).2 = 𝟙 graph_R.tD (x:graph_R.tD⊢ ((↑f).2 ⊚ (↑finv).2) x = 𝟙 graph_R.tD x; ⊢ ((↑f).2 ⊚ (↑finv).2) D.d₁ = 𝟙 graph_R.tD D.d₁⊢ ((↑f).2 ⊚ (↑finv).2) D.d₂ = 𝟙 graph_R.tD D.d₂⊢ ((↑f).2 ⊚ (↑finv).2) D.d₃ = 𝟙 graph_R.tD D.d₃⊢ ((↑f).2 ⊚ (↑finv).2) D.d₄ = 𝟙 graph_R.tD D.d₄⊢ ((↑f).2 ⊚ (↑finv).2) D.d₅ = 𝟙 graph_R.tD D.d₅ ⊢ ((↑f).2 ⊚ (↑finv).2) D.d₁ = 𝟙 graph_R.tD D.d₁⊢ ((↑f).2 ⊚ (↑finv).2) D.d₂ = 𝟙 graph_R.tD D.d₂⊢ ((↑f).2 ⊚ (↑finv).2) D.d₃ = 𝟙 graph_R.tD D.d₃⊢ ((↑f).2 ⊚ (↑finv).2) D.d₄ = 𝟙 graph_R.tD D.d₄⊢ ((↑f).2 ⊚ (↑finv).2) D.d₅ = 𝟙 graph_R.tD D.d₅ All goals completed! 🐙) have h₂ : (f.val.1 ⊚ finv.val.1, f.val.2 ⊚ finv.val.2) = (𝟙 graph_R.tA, 𝟙 graph_R.tD) := ⊢ f ⊚ finv = 𝟙 graph_R All goals completed! 🐙 All goals completed! 🐙 }
Exercise 8 (p. 160)

(Impossible journeys) J is the graph

inductive A | a₁ | a₂ | a₃ abbrev D := Fin 2 def J : IrreflexiveGraph := { tA := A carrierA := Set.univ tD := D carrierD := Set.univ toSrc := fun | A.a₁ => 0 | A.a₂ => 1 | A.a₃ => 1 toSrc_mem := fun _ ↦ Set.mem_univ _ toTgt := fun | A.a₁ => 0 | A.a₂ => 0 | A.a₃ => 1 toTgt_mem := fun _ ↦ Set.mem_univ _ }

G is any graph, and b and e are dots of G.

variable (G : IrreflexiveGraph) (b e : G.tD) (hb : b ∈ G.carrierD) (he : e ∈ G.carrierD)

(a) Suppose that {G \xrightarrow{f} J} is a map of graphs with {fb = 0} and {fe = 1}. Show that there is no path in G that begins at b and ends at e.

(b) Conversely, suppose that there is no path in G that begins at b and ends at e. Show that there is a map {G \xrightarrow{f} J} with {fb = 0} and {fe = 1}.

Solution: Exercise 8

TODO Exercise 11.8