| import OAI.Combinatorics.FiniteCircle.Upper |
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| namespace OAI |
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| namespace FiniteCircle |
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| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
| open Classical |
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|
| theorem upperRates_sup_nonneg {α X : Type*} (word : X → List α) (f : X → RealL1) : |
| 0 ≤ sSup (upperRates word f) := by |
| apply Real.sSup_nonneg |
| rintro r ⟨x,y,_,rfl⟩ |
| positivity |
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| theorem lowerRates_sup_nonneg {α X : Type*} (word : X → List α) (f : X → RealL1) : |
| 0 ≤ sSup (lowerRates word f) := by |
| apply Real.sSup_nonneg |
| rintro r ⟨x,y,_,rfl⟩ |
| positivity |
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|
| theorem distortion_isometry_restrict {α β X Y : Type*} [Finite Y] |
| (wordX : X → List α) (wordY : Y → List β) (g : X → Y) (hg : Function.Injective g) |
| (he : ∀ x y, editDistance (wordY (g x)) (wordY (g y)) = editDistance (wordX x) (wordX y)) |
| (f : Y → RealL1) : distortion wordX (f ∘ g) ≤ distortion wordY f := by |
| have hu : sSup (upperRates wordX (f ∘ g)) ≤ sSup (upperRates wordY f) := by |
| apply Real.sSup_le _ (upperRates_sup_nonneg wordY f) |
| rintro r ⟨x,y,hxy,rfl⟩ |
| apply le_csSup (upperRates_finite wordY f).bddAbove |
| exact ⟨g x,g y,fun h => hxy (hg h),by simp only [Function.comp_apply,he]⟩ |
| have hl : sSup (lowerRates wordX (f ∘ g)) ≤ sSup (lowerRates wordY f) := by |
| apply Real.sSup_le _ (lowerRates_sup_nonneg wordY f) |
| rintro r ⟨x,y,hxy,rfl⟩ |
| apply le_csSup (lowerRates_finite wordY f).bddAbove |
| exact ⟨g x,g y,fun h => hxy (hg h),by simp only [Function.comp_apply,he]⟩ |
| exact mul_le_mul hu hl (lowerRates_sup_nonneg wordX (f ∘ g)) (upperRates_sup_nonneg wordY f) |
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| theorem optimalDistortion_isometry_mono {α β X Y : Type*} [Fintype Y] |
| (wordX : X → List α) (wordY : Y → List β) (g : X → Y) (hg : Function.Injective g) |
| (he : ∀ x y, editDistance (wordY (g x)) (wordY (g y)) = editDistance (wordX x) (wordX y)) : |
| optimalDistortion wordX ≤ optimalDistortion wordY := by |
| apply optimalDistortion_lower |
| intro f hf |
| exact (optimalDistortion_le wordX (f ∘ g) (hf.comp hg)).trans |
| (distortion_isometry_restrict wordX wordY g hg he f) |
|
|
| instance wordsFinite (α : Type*) [Finite α] (d : ℕ) : Finite (Words α d) := |
| (List.finite_length_le α d).to_subtype |
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| theorem words_nontrivial (α : Type*) [Nonempty α] {d : ℕ} (hd : 1≤d) : Nontrivial (Words α d) := by |
| classical |
| let a : α := Classical.ofNonempty |
| refine ⟨⟨⟨[],by simp⟩,⟨[a],by simpa using hd⟩,?_⟩⟩ |
| intro he |
| have ht := congrArg (fun x : Words α d => x.val.length) he |
| simp at ht |
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|
| theorem binary_alphabet_embedding (α : Type*) [Fintype α] (hα : 2≤Fintype.card α) : |
| ∃ (f : Bool → α) (g : α → Bool), Function.LeftInverse g f := by |
| classical |
| let : Nontrivial α := Fintype.one_lt_card_iff_nontrivial.mp hα |
| obtain ⟨a,b,hab⟩ := exists_pair_ne α |
| refine ⟨fun t => cond t b a,fun x => decide (x=b),?_⟩ |
| intro t |
| cases t <;> simp [hab] |
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| theorem setDistortion_le_E (α : Type*) [Fintype α] (hα : 2≤Fintype.card α) |
| {S : Set (List Bool)} {d : ℕ} (hS : ∀ x∈S, x.length≤d) : setDistortion S≤E α d := by |
| classical |
| let : Fintype (Words α d) := Fintype.ofFinite _ |
| obtain ⟨f,g,hgf⟩ := binary_alphabet_embedding α hα |
| let e : S → Words α d := fun x => ⟨x.val.map f,by simpa using hS x.val x.property⟩ |
| have he : Function.Injective e := by |
| intro x y hxy |
| exact Subtype.ext (hgf.injective.list_map (congrArg Subtype.val hxy)) |
| apply optimalDistortion_isometry_mono (fun x : S => x.val) (fun x : Words α d => x.val) e he |
| intro x y |
| exact editDistance_map_eq f g hgf x.val y.val |
|
|
| theorem maskWitness_lower_E {a C δ c : ℝ} {d : ℕ} (hw : MaskWitness a C δ c d) |
| (α : Type*) [Fintype α] (hα : 2≤Fintype.card α) : Real.exp (c*exponentScale d)≤E α d := by |
| rcases hw with ⟨h,_,p,_,w,code,_,_,n,_,hnd,hlen,hD⟩ |
| apply hD.trans (setDistortion_le_E α hα _) |
| rintro x ⟨v,rfl⟩ |
| exact (hlen v).le.trans hnd |
|
|
| theorem uniformUpper_E {C : ℝ} {d₀ : ℕ} (hu : UniformUpper C d₀) |
| (α : Type) [Fintype α] (hα : 2≤Fintype.card α) {d : ℕ} (hd : d₀≤d) (hd1 : 1≤d) : |
| E α d≤Real.exp (C*exponentScale d) := by |
| classical |
| let : Nontrivial α := Fintype.one_lt_card_iff_nontrivial.mp hα |
| let : Nontrivial (Words α d) := words_nontrivial α hd1 |
| obtain ⟨M,F,hF⟩ := hu α hα d hd |
| let f : Words α d → RealL1 := fun x => finiteL1 M (F x) |
| have hb := distortion_upper (fun x : Words α d => x.val) Subtype.val_injective |
| f (Real.exp (C*exponentScale d)) 1 (Real.exp_pos _).le |
| (fun x y => by simpa only [f,finiteL1_norm_sub] using (hF x y).2) |
| (fun x y => by simpa only [f,one_mul,finiteL1_norm_sub] using (hF x y).1) |
| exact (optimalDistortion_le (fun x : Words α d => x.val) f hb.1).trans (by simpa only [mul_one] using hb.2) |
|
|
| theorem alphabetSupremum_bounds {d : ℕ} {lo hi : ℝ} |
| (hb : ∀ A : ℕ, 2≤A → lo≤E (Fin A) d ∧ E (Fin A) d≤hi) : |
| lo≤alphabetSupremum d ∧ alphabetSupremum d≤hi := by |
| have hne : {r : ℝ | ∃ A : ℕ, 2≤A ∧ r=E (Fin A) d}.Nonempty := ⟨_,2,by norm_num,rfl⟩ |
| have hub : BddAbove {r : ℝ | ∃ A : ℕ, 2≤A ∧ r=E (Fin A) d} := by |
| refine ⟨hi,?_⟩ |
| rintro r ⟨A,hA,rfl⟩ |
| exact (hb A hA).2 |
| refine ⟨(hb 2 (by norm_num)).1.trans (le_csSup hub ⟨2,by norm_num,rfl⟩),?_⟩ |
| apply csSup_le hne |
| rintro r ⟨A,hA,rfl⟩ |
| exact (hb A hA).2 |
|
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| theorem full_main : FullMain := by |
| obtain ⟨u,hu⟩ := uniform_upper |
| let d₀ := max u (lowerCap 1000000) |
| have hlow {d : ℕ} (hd : d₀≤d) : lowerCap 1000000≤d := (le_max_right _ _).trans hd |
| have hu' : UniformUpper 32 d₀ := by |
| intro α _ hα d hd |
| exact hu α hα d ((le_max_left _ _).trans hd) |
| have hb : ∀ (α : Type) [Fintype α], 2≤Fintype.card α → ∀ d : ℕ, d₀≤d → |
| Real.exp ((1/8192)*exponentScale d)≤E α d ∧ E α d≤Real.exp (32*exponentScale d) := by |
| intro α _ hα d hd |
| have hd1 : 1≤d := by have := lowerCap_ge 1000000; have := hlow hd; omega |
| exact ⟨maskWitness_lower_E (both_witnesses (hlow hd)).1 α hα, |
| uniformUpper_E hu' α hα hd hd1⟩ |
| refine ⟨1/8192,32,1/268435456,67108928,1/1048576,1/1048576,d₀, |
| by norm_num,by norm_num,by norm_num,by norm_num,by norm_num,by norm_num, |
| two_codes,hu',hb,?_⟩ |
| intro d hd |
| have hw := both_witnesses (hlow hd) |
| refine ⟨hw.1,hw.2,?_⟩ |
| exact alphabetSupremum_bounds (fun A hA => hb (Fin A) (by simpa using hA) d hd) |
|
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| end FiniteCircle |
|
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| theorem finiteCircle_fullMain : FiniteCircle.FullMain := FiniteCircle.full_main |
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| end OAI |
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