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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 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | import Mathlib
namespace OAI
namespace Ostmann.FiniteField
noncomputable section
open scoped BigOperators
variable {I C : Type*} [Fintype I] [DecidableEq I] [CommGroup C] [Fintype C] [DecidableEq C]
theorem product_split_coordinate {M : Type*} [CommMonoid M] (j : I) (f : I → M) :
(∏ i,f i)=f j*∏ i : {i : I // i≠j},f i := by
have he : (∏ i : {i : I // i≠j},f i)=∏ i ∈ Finset.univ.erase j,f i := by
exact (Finset.prod_subtype (Finset.univ.erase j) (fun i => by simp) f).symm
rw [he]
exact (Finset.mul_prod_erase Finset.univ f (Finset.mem_univ j)).symm
theorem constrained_character_sum_bound (j : I) (e : I → C ≃ C)
(w : I → C → ℝ) (hw : ∀ i χ,0≤w i χ) (δ : ℝ) (hδ : ∀ χ,w j χ≤δ) :
(∑ ρ : I → C, if (∏ i,e i (ρ i))=1 then ∏ i,w i (ρ i) else 0) ≤
δ*∏ i : {i : I // i≠j},∑ χ : C,w i χ := by
classical
let E := Equiv.funSplitAt j C
let rest (t : ({i : I // i≠j} → C)) : C := ∏ i : {i : I // i≠j},e i (t i)
let sol (t : ({i : I // i≠j} → C)) : C := (e j).symm ((rest t)⁻¹)
let W (t : ({i : I // i≠j} → C)) : ℝ := ∏ i : {i : I // i≠j},w i (t i)
have hE0 (z : C) (t : ({i : I // i≠j} → C)) : E.symm (z,t) j=z := by simp [E]
have hEi (z : C) (t : ({i : I // i≠j} → C)) (i : {i : I // i≠j}) : E.symm (z,t) i=t i := by simp [E,i.property]
have hp (z : C) (t : ({i : I // i≠j} → C)) :
(∏ i,e i (E.symm (z,t) i))=(e j z)*rest t := by
rw [product_split_coordinate j]
simp only [hE0,hEi,rest]
have hwprod (z : C) (t : ({i : I // i≠j} → C)) :
(∏ i,w i (E.symm (z,t) i))=w j z*W t := by
rw [product_split_coordinate j]
simp only [hE0,hEi,W]
have hconstraint (z : C) (t : ({i : I // i≠j} → C)) : (e j z)*rest t=1 ↔ z=sol t := by
rw [mul_eq_one_iff_eq_inv]
exact (e j).eq_symm_apply.symm
have hs :
(∑ ρ : I → C, if (∏ i,e i (ρ i))=1 then ∏ i,w i (ρ i) else 0) =
∑ t : ({i : I // i≠j} → C),w j (sol t)*W t := by
rw [← E.symm.sum_comp (fun ρ : I → C => if (∏ i,e i (ρ i))=1 then ∏ i,w i (ρ i) else 0),Fintype.sum_prod_type]
simp_rw [hp,hwprod]
have hh (z : C) (t : ({i : I // i≠j} → C)) :
(if (e j z)*rest t=1 then w j z*W t else 0)=
if z=sol t then w j z*W t else 0 := by simp only [hconstraint]
simp_rw [hh]
rw [Finset.sum_comm]
simp
rw [hs]
calc
_ ≤ ∑ t : ({i : I // i≠j} → C),δ*W t := by
apply Finset.sum_le_sum
intro t _
exact mul_le_mul_of_nonneg_right (hδ _) (Finset.prod_nonneg (fun i _ => hw i (t i)))
_ = δ*(∑ t : ({i : I // i≠j} → C),∏ i : {i : I // i≠j},w i (t i)) := by rw [Finset.mul_sum]
_ = _ := by rw [← Fintype.prod_sum]
end
end Ostmann.FiniteField
end OAI
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