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a72760f | 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 34 35 36 37 38 39 40 41 42 43 44 45 46 47 | 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
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