| import OAI.Computability.DepthThree.FiniteProbability |
| import Mathlib.Algebra.Group.Hom.Defs |
| import Mathlib.Data.Fintype.Card |
|
|
| namespace OAI |
|
|
| universe uDepth1 uDepth2 |
|
|
| noncomputable section |
|
|
| open scoped BigOperators |
|
|
| namespace DepthThreeLowerBound.AlgebraFiniteFibers |
|
|
| variable {A : Type uDepth1} {B : Type uDepth2} [AddCommGroup A] [AddCommGroup B] |
|
|
| def fiberEquivKernel (L : A →+ B) (hL : Function.Surjective L) (b : B) : |
| {a : A // L a = b} ≃ {a : A // L a = 0} := by |
| let x : A := Classical.choose (hL b) |
| have hx : L x = b := Classical.choose_spec (hL b) |
| exact { |
| toFun := fun a => ⟨a.val - x, by rw [map_sub, a.property, hx, sub_self]⟩ |
| invFun := fun a => ⟨a.val + x, by rw [map_add, a.property, hx, zero_add]⟩ |
| left_inv := by |
| intro a |
| apply Subtype.ext |
| exact sub_add_cancel a.val x |
| right_inv := by |
| intro a |
| apply Subtype.ext |
| exact add_sub_cancel_right a.val x |
| } |
|
|
| variable [Fintype A] [instFintypeB : Fintype B] [instDecidableEqA : DecidableEq A] [DecidableEq B] |
|
|
| theorem card_fiber_eq |
| {A : Type uDepth1} |
| {B : Type uDepth2} |
| [AddCommGroup A] |
| [AddCommGroup B] |
| [Fintype A] |
| [Fintype B] |
| [DecidableEq A] |
| [DecidableEq B] |
| (L : A →+ B) (hL : Function.Surjective L) (b : B) : |
| Fintype.card {a : A // L a = b} = Fintype.card {a : A // L a = 0} := |
| Fintype.card_congr (fiberEquivKernel L hL b) |
|
|
| theorem card_filter_fiber_eq (L : A →+ B) (hL : Function.Surjective L) (b : B) : |
| (Finset.univ.filter (fun a : A => L a = b)).card = |
| (Finset.univ.filter (fun a : A => L a = 0)).card := by |
| simpa only [Fintype.card_subtype] using card_fiber_eq L hL b |
|
|
| theorem card_eq_mul_card_kernel (L : A →+ B) (hL : Function.Surjective L) : |
| Fintype.card A = Fintype.card B * Fintype.card {a : A // L a = 0} := by |
| calc |
| Fintype.card A = ∑ b : B, Fintype.card {a : A // L a = b} := by |
| simpa using |
| (Fintype.sum_fiberwise' (fun a : A => L a) (fun _ : B => (1 : ℕ))).symm |
| _ = ∑ _b : B, Fintype.card {a : A // L a = 0} := by |
| exact Finset.sum_congr rfl (fun b _ => card_fiber_eq L hL b) |
| _ = Fintype.card B * Fintype.card {a : A // L a = 0} := by simp |
|
|
| theorem sum_comp (L : A →+ B) (hL : Function.Surjective L) (f : B → ℝ) : |
| (∑ a : A, f (L a)) = |
| (Fintype.card {a : A // L a = 0} : ℝ) * ∑ b : B, f b := by |
| calc |
| (∑ a : A, f (L a)) = ∑ b : B, ∑ _a : {a : A // L a = b}, f b := |
| (Fintype.sum_fiberwise' (fun a : A => L a) f).symm |
| _ = ∑ b : B, (Fintype.card {a : A // L a = 0} : ℝ) * f b := by |
| apply Finset.sum_congr rfl |
| intro b _ |
| simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul, |
| card_fiber_eq L hL b] |
| _ = (Fintype.card {a : A // L a = 0} : ℝ) * ∑ b : B, f b := |
| (Finset.mul_sum Finset.univ f _).symm |
|
|
| theorem finiteAvg_comp (L : A →+ B) (hL : Function.Surjective L) (f : B → ℝ) : |
| finiteAvg (fun a : A => f (L a)) = finiteAvg f := by |
| let : Nonempty {a : A // L a = 0} := ⟨⟨0, map_zero L⟩⟩ |
| have hk : (Fintype.card {a : A // L a = 0} : ℝ) ≠ 0 := by |
| exact_mod_cast (Fintype.card_ne_zero : Fintype.card {a : A // L a = 0} ≠ 0) |
| unfold finiteAvg |
| rw [card_eq_mul_card_kernel L hL, Nat.cast_mul, sum_comp L hL f, mul_inv] |
| calc |
| (Fintype.card B : ℝ)⁻¹ * (Fintype.card {a : A // L a = 0} : ℝ)⁻¹ * |
| ((Fintype.card {a : A // L a = 0} : ℝ) * ∑ b : B, f b) = |
| (Fintype.card B : ℝ)⁻¹ * |
| ((Fintype.card {a : A // L a = 0} : ℝ)⁻¹ * |
| (Fintype.card {a : A // L a = 0} : ℝ)) * ∑ b : B, f b := by ring |
| _ = (Fintype.card B : ℝ)⁻¹ * ∑ b : B, f b := by |
| rw [inv_mul_cancel₀ hk, mul_one] |
|
|
| end DepthThreeLowerBound.AlgebraFiniteFibers |
|
|
| end |
|
|
| end OAI |
|
|
