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9dbc64e | 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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | 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
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