| import Mathlib |
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| namespace OAI |
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| namespace Ostmann.FiniteField |
| noncomputable section |
| open scoped BigOperators |
| variable {I C : Type*} [Fintype I] [DecidableEq I] [CommGroup C] [Fintype C] [DecidableEq C] |
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| 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 |
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| 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] |
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| end |
| end Ostmann.FiniteField |
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| end OAI |
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