Datasets:
| /- 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] | |