| import Mathlib |
| import OAI.Probability.SKGap.Matrix.PolynomialCoeListSum |
|
|
| namespace OAI |
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|
| section |
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|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGap.Noncrossing |
| open Diagram InverseDiagram WordSeries |
| variable {ι : Type*} [Fintype ι] |
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|
| 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] |
|
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| 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 |
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| end OAI |
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