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d03762b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | /- Copyright (c) Heather Macbeth, 2023. All rights reserved. -/
import Mathlib.Tactic.Choose
open Function
def Inverse (f : X → Y) (g : Y → X) : Prop := g ∘ f = id ∧ f ∘ g = id
theorem bijective_iff_exists_inverse (f : X → Y) :
Bijective f ↔ ∃ g : Y → X, Inverse f g := by
constructor
· rintro ⟨h_inj, h_surj⟩
choose g hg using h_surj
refine ⟨g, ?_, funext hg⟩
funext x
exact h_inj (hg _)
· rintro ⟨g, hgf, hfg⟩
constructor
· intro x1 x2 hx
have H : (g ∘ f) x1 = (g ∘ f) x2 := by simp [hx]
simpa only [hgf] using H
· intro y
refine ⟨g y, ?_⟩
simpa using congr_fun hfg y
theorem surjective_of_intertwining {f : X → ℕ} {x0 : X} (h0 : f x0 = 0) {i : X → X}
(hi : ∀ x, f (i x) = f x + 1) :
Function.Surjective f
| 0 => ⟨x0, h0⟩
| k + 1 => by
obtain ⟨x, hx⟩ := surjective_of_intertwining h0 hi k
refine ⟨i x, ?_⟩
simp [hi, hx]
|