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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Matrix.PolynomialCoeListSum
namespace OAI
section
noncomputable section
open scoped BigOperators
namespace SKGap.Noncrossing
open Diagram InverseDiagram WordSeries
variable {ι : Type*} [Fintype ι]
lemma mean_finset_sum {α : Type*} (s : Finset α) (f : α→ι→ℝ) :
Diagram.mean (fun i =>∑ a∈s,f a i)=∑ a∈s,Diagram.mean (f a) := by
simp only [Diagram.mean]
rw [Finset.sum_comm,Finset.sum_div]
lemma ordinary_mean_noise_rotate (j : ℝ) (F : List (Letter (ι→ℝ))) :
Diagram.mean (ordinary j (.noise::F))=Diagram.mean (ordinary j (F++[.noise])) := by
have h1 := funext (expect_at j (id : (ι→ℝ)→ι→ℝ) [] F)
have h2 := funext (expect_at j (id : (ι→ℝ)→ι→ℝ) F [])
simp only [ordinary]
simp only [List.nil_append,List.append_nil,Fintype.sum_empty,zero_add,add_zero] at h1 h2
rw [h1,h2,mean_finset_sum,mean_finset_sum]
apply Finset.sum_congr rfl
intro s _
rw [mean_const_mul,mean_const_mul]
ring
lemma ordinary_mean_cons_rotate (j : ℝ) (a : Letter (ι→ℝ)) (F : List (Letter (ι→ℝ))) :
Diagram.mean (ordinary j (a::F))=Diagram.mean (ordinary j (F++[a])) := by
cases a with
| noise => exact ordinary_mean_noise_rotate j F
| diag d => congr 1; funext i; simp [mul_comm]
theorem ordinary_mean_cyclic (j : ℝ) (P Q : List (Letter (ι→ℝ))) :
Diagram.mean (ordinary j (P++Q))=Diagram.mean (ordinary j (Q++P)) := by
induction P generalizing Q with
| nil => simp
| cons a P ih =>
rw [List.cons_append,ordinary_mean_cons_rotate,List.append_assoc,ih]
simp [List.append_assoc]
lemma meanSeries_cyclic (j : ℝ) (a : ι→ℝ) (P Q : List (WordLetter ι)) :
meanSeries (literalSeries j a (P++Q))=meanSeries (literalSeries j a (Q++P)) := by
have hi : inverseCount (P++Q)=inverseCount (Q++P) := by simp [inverseCount_append,add_comm]
ext m
simp only [coeff_meanSeries]
by_cases h : inverseCount (P++Q)=0
· have h' : inverseCount (Q++P)=0 := hi ▸ h
simp only [literalSeries,h,h',ite_true,PowerSeries.coeff_C]
by_cases hm : m=0
· simp only [hm,ite_true]
simpa only [expandWord,List.flatMap_append] using
ordinary_mean_cyclic j (expandWord j a [] P) (expandWord j a [] Q)
· simp [hm,Diagram.mean]
· have h' : inverseCount (Q++P)≠0 := hi ▸ h
simp only [literalSeries,h,h',ite_false,PowerSeries.coeff_mk]
rw [mean_finset_sum,mean_finset_sum]
apply Finset.sum_congr rfl
intro bs _
simpa only [expandWord,List.flatMap_append] using
ordinary_mean_cyclic j (expandWord j a bs.val P) (expandWord j a bs.val Q)
theorem wordPrediction_mean_cyclic (j : ℝ) (a : ι→ℝ) (z : ℝ)
(P Q : List (WordLetter ι)) (hI : inverseCount (P++Q)≤1) :
(∑ i,wordPrediction j a z (P++Q) i)=(∑ i,wordPrediction j a z (Q++P) i) := by
classical
by_cases hn : Fintype.card ι=0
· have : IsEmpty ι := Fintype.card_eq_zero_iff.mp hn
simp
have hc := meanSeries_cyclic j a P Q
have hI' : inverseCount (Q++P)≤1 := by simpa only [inverseCount_append,add_comm] using hI
unfold meanSeries at hc
simp_rw [literalSeries_eq_wordPredictionPolynomial j a (P++Q) hI,
literalSeries_eq_wordPredictionPolynomial j a (Q++P) hI'] at hc
have hmult : (PowerSeries.C ((Fintype.card ι:ℝ)⁻¹))≠0 := by
exact (map_ne_zero PowerSeries.C).mpr (inv_ne_zero (Nat.cast_ne_zero.mpr hn))
have hs := mul_left_cancel₀ hmult hc
have hp : (∑ i,wordPredictionPolynomial j a (P++Q) i)=
∑ i,wordPredictionPolynomial j a (Q++P) i := by
apply Polynomial.coe_injective ℝ
simpa only [polynomial_coe_finset_sum] using hs
have he := congrArg (Polynomial.eval z) hp
simpa only [Polynomial.eval_finsetSum,fun F i => (wordPredictionPolynomial_spec j a F i).2 z] using he
end SKGap.Noncrossing
end
end
end OAI
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