openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /AdditiveKernelTransport.lean
| import Mathlib | |
| namespace OAI | |
| namespace PiExponent.AdditiveKernelTransport | |
| variable {A A' B B' : Type*} | |
| [AddCommGroup A] [AddCommGroup A'] [AddCommGroup B] [AddCommGroup B'] | |
| theorem equiv_zero (e : A ≃+ B) : e 0 = 0 := map_zero e | |
| def kernelEquiv (e : A ≃+ B) (e' : A' ≃+ B') | |
| (d : A →+ A') (D : B →+ B') (h : ∀ x, e' (d x) = D (e x)) : | |
| d.ker ≃+ D.ker where | |
| toFun c := ⟨e c.val, by | |
| change D (e c.val) = 0 | |
| rw [← h, c.property, map_zero]⟩ | |
| invFun c := ⟨e.symm c.val, by | |
| change d (e.symm c.val) = 0 | |
| apply e'.injective | |
| rw [h, e.apply_symm_apply, map_zero] | |
| exact c.property⟩ | |
| left_inv c := Subtype.ext (e.symm_apply_apply c.val) | |
| right_inv c := Subtype.ext (e.apply_symm_apply c.val) | |
| map_add' c b := Subtype.ext (map_add e c.val b.val) | |
| theorem kernelEquiv_val (e : A ≃+ B) (e' : A' ≃+ B') | |
| (d : A →+ A') (D : B →+ B') (h : ∀ x, e' (d x) = D (e x)) | |
| (c : d.ker) : (kernelEquiv e e' d D h c).val = e c.val := rfl | |
| theorem transport_primitives {A₂ B₂ : Type*} [AddCommGroup A₂] [AddCommGroup B₂] | |
| (e₀ : A ≃+ B) (e₁ : A' ≃+ B') (e₂ : A₂ ≃+ B₂) | |
| (d₀ : A →+ A') (d₁ : A' →+ A₂) (D₀ : B →+ B') (D₁ : B' →+ B₂) | |
| (h₀ : ∀ x, e₁ (d₀ x) = D₀ (e₀ x)) | |
| (h₁ : ∀ x, e₂ (d₁ x) = D₁ (e₁ x)) | |
| (hp : ∀ c, D₁ c = 0 → ∃ b, D₀ b = c) : | |
| ∀ c, d₁ c = 0 → ∃ b, d₀ b = c := by | |
| intro c hc | |
| have hclosed : D₁ (e₁ c) = 0 := by rw [← h₁, hc, map_zero] | |
| obtain ⟨b, hb⟩ := hp (e₁ c) hclosed | |
| refine ⟨e₀.symm b, e₁.injective ?_⟩ | |
| rw [h₀, e₀.apply_symm_apply, hb] | |
| end PiExponent.AdditiveKernelTransport | |
| end OAI | |