| import OAI.Combinatorics.FiniteCircle.Parameters |
|
|
| namespace OAI |
|
|
| namespace FiniteCircle |
|
|
| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
|
|
| theorem coded_range_lower {I α : Type*} [Fintype I] [DecidableEq I] [Fintype α] |
| {n k M : ℕ} [NeZero M] (hn : 1000000≤n) (hk : n^4=4*k) |
| (hJ : Nat.log 2 M+1≤n^13) |
| (H : AddSubgroup (I → ZMod M)) [Fintype H] (θ : I → ZMod M) |
| (hb : ∀ χ : AddChar (I → ZMod M) ℂ, χ.compAddMonoidHom H.subtype=0 → χ θ≠1 → |
| (((n^4:ℕ):ℝ)/2)^(n^4) ≤ ∑ i, ((coordinateFrequency χ i).valMinAbs.natAbs:ℝ)) |
| (W : (I → ZMod M) → List α) (L copies : ℕ) (hL : 0<L) |
| (hcL : L=M*Fintype.card I*copies) (hLen : ∀ v, (W v).length=L) |
| (hcheap : ∀ v (c : H), (n^4)^(2*n^4)*editDistance (W v) (W (v+c)) ≤ 2*n^4*L) |
| (hsingle : ∀ v i (j : Fin (Nat.log 2 M+1)), |
| editDistance (W v) (W (v+Pi.single i ((2^j.val:ℕ):ZMod M))) ≤ 2*copies*2^j.val) |
| (hlong : ∀ v, (L:ℝ)/(n:ℝ)^(3*n^4) ≤ (deficit (W v) (W (v+θ)):ℝ)) |
| {w : ℕ} (c : α → Fin w → Bool) (hc : SubstitutionBounds (1/268435456) 67108928 c) : |
| (n:ℝ)^k ≤ optimalDistortion (fun x : Set.range (fun v => encode c (W v)) => x.val) := by |
| classical |
| have hn0 : 0<n := by omega |
| have hR : (0:ℝ)<((n^4)^(2*n^4):ℕ) := by positivity |
| apply range_distortion_lower H θ ((((n^4:ℕ):ℝ)/2)^(n^4)) (by positivity) hb |
| (fun v => encode c (W v)) (n^4) ((n^4)^(2*n^4):ℕ) L copies w |
| ((1/268435456)*w/(n:ℝ)^(3*n^4)) ((n:ℝ)^k) (by positivity) (by have := hc.1; positivity) |
| (by exact_mod_cast hcL) |
| · intro v z |
| apply (le_div_iff₀ hR).mpr |
| have he := hc.2.2.2 (W v) (W (v+z)) |
| have ht := Nat.mul_le_mul_left ((n^4)^(2*n^4)) he |
| have hh := Nat.mul_le_mul_left w (hcheap v z) |
| have hn : editDistance (encode c (W v)) (encode c (W (v+z)))*((n^4)^(2*n^4)) ≤ w*2*n^4*L := by |
| nlinarith |
| exact_mod_cast hn |
| · intro v i j |
| have he := (hc.2.2.2 (W v) (W (v+Pi.single i ((2^j.val:ℕ):ZMod M)))).trans |
| (Nat.mul_le_mul_left w (hsingle v i j)) |
| have hn : editDistance (encode c (W v)) (encode c (W (v+Pi.single i ((2^j.val:ℕ):ZMod M)))) ≤ |
| w*2*copies*2^j.val := by nlinarith |
| exact_mod_cast hn |
| · intro v |
| have hd := (hc.2.2.1 L (W v) (W (v+θ)) (hLen v) (hLen (v+θ))).1 |
| have he := deficit_le_editDistance (x:=encode c (W v)) (y:=encode c (W (v+θ))) (by simp only [length_encode,hLen]) |
| calc |
| _ = (1/268435456)*w*((L:ℝ)/(n:ℝ)^(3*n^4)) := by ring |
| _ ≤ (1/268435456)*w*(deficit (W v) (W (v+θ)):ℝ) := |
| mul_le_mul_of_nonneg_left (hlong v) (by positivity) |
| _ ≤ (deficit (encode c (W v)) (encode c (W (v+θ))):ℝ) := hd |
| _ ≤ _ := by exact_mod_cast he |
| · intro t _ ht |
| apply lower_numeric hn hk hJ hc.1 t |
| simpa only [Nat.cast_pow] using ht |
| |
| theorem flatMaskState_surjective (h : ℕ) (p : ℕ → ℕ → ℕ) (M l : ℕ) : |
| Function.Surjective (flatMaskState h p M l) := by |
| classical |
| induction l with |
| | zero => intro v; exact ⟨fun _ => v,rfl⟩ |
| | succ l ih => |
| intro v |
| choose f hf using fun j => ih (v j) |
| refine ⟨fun i => f i.1 i.2, ?_⟩ |
| funext j |
| exact hf j |
|
|
| theorem flatExcludedState_surjective (h M l : ℕ) : |
| Function.Surjective (flatExcludedState h M l) := by |
| classical |
| induction l with |
| | zero => intro v; exact ⟨fun _ => v,rfl⟩ |
| | succ l ih => |
| intro v |
| choose f hf using fun j => ih (v j) |
| refine ⟨fun i => f i.1 i.2, ?_⟩ |
| funext j |
| exact hf j |
|
|
| theorem range_comp_surjective {X Y Z : Type*} (f : X → Y) (g : Y → Z) (hf : Function.Surjective f) : |
| Set.range (fun x => g (f x)) = Set.range g := by |
| ext z |
| constructor |
| · rintro ⟨x,rfl⟩; exact ⟨f x,rfl⟩ |
| · rintro ⟨y,rfl⟩; obtain ⟨x,rfl⟩ := hf y; exact ⟨x,rfl⟩ |
|
|
| theorem mask_length_factor (h : ℕ) (p : ℕ → ℕ → ℕ) (R M l : ℕ) : |
| maskLength h p R M l = M*Fintype.card (TreeLeaf (MaskChildren h p) l)*treeCopies R (period h p) l := by |
| apply tree_length_eq (MaskChildren h p) M R (period h p) (maskLength h p R M) rfl |
| intro l |
| simp only [maskLength, length_maskTags] |
| congr 2 |
| simp only [MaskChildren, Fintype.card_sigma, Fintype.card_fin] |
|
|
| theorem excluded_length_factor (h : ℕ) (p : ℕ → ℕ → ℕ) (R M l : ℕ) : |
| excludedLength h p R M l = M*Fintype.card (TreeLeaf (fun _ => Fin h) l)*treeCopies R (period h p) l := by |
| apply tree_length_eq (fun _ => Fin h) M R (period h p) (excludedLength h p R M) rfl |
| intro l |
| simp only [excludedLength,length_excludedTags,Fintype.card_fin] |
|
|
| theorem mask_binary_distortion {n k : ℕ} (hn : 1000000≤n) (hk : n^4=4*k) |
| {p : ℕ → ℕ → ℕ} (hp : StandingPrimes (n^4) p) {w : ℕ} |
| (c : MaskAlphabet (n^4) p (n^4) → Fin w → Bool) |
| (hc : SubstitutionBounds (1/268435456) 67108928 c) : |
| (n:ℝ)^k ≤ setDistortion (Set.range fun v : MaskState (n^4) p (circleSize (n^4) p) (n^4) => |
| encode c (maskWord (n^4) p ((n^4)^(2*n^4)) (circleSize (n^4) p) (n^4) v)) := by |
| classical |
| let h := n^4 |
| let M := circleSize h p |
| let R := h^(2*h) |
| let : NeZero M := ⟨hp.circle_pos.ne'⟩ |
| have hh : 2≤h := (show 2≤n by omega).trans (le_self_pow₀ (by omega) (by norm_num)) |
| have hR : 0<R := pow_pos (by omega) _ |
| let H := maskCheap h p M hp.divides_circle h |
| let : Fintype H := Fintype.ofFinite H |
| have hhb (χ : AddChar (TreeLeaf (MaskChildren h p) h → ZMod M) ℂ) |
| (ha : χ.compAddMonoidHom H.subtype=0) (hw : χ (fun _ => ((M/2:ℕ):ZMod M))≠1) : |
| ((h:ℝ)/2)^h ≤ ∑ i, ((coordinateFrequency χ i).valMinAbs.natAbs:ℝ) := by |
| apply mask_character_branching hh hp.divides_circle hp.positive |
| (fun l k hl hlh => (hp.atLevel l hl hlh k).1) |
| hp.level_injective (fun l k hl hlh => (hp.atLevel l hl hlh k).2.1.le) χ ha _ hw |
| have hlong (v : TreeLeaf (MaskChildren h p) h → ZMod M) : |
| (maskLength h p R M h:ℝ)/(n:ℝ)^(3*n^4) ≤ |
| (deficit (flatMaskWord h p R M h v) (flatMaskWord h p R M h (v+(fun _ => ((M/2:ℕ):ZMod M)))):ℝ) := by |
| have he : n^4=2*(n^4/2) := by omega |
| have ht := mask_long_deficit hn he hp hR (flatMaskState h p M h v) |
| have hf := flatMaskState_shift h p M h v ((M/2:ℕ):ZMod M) |
| change flatMaskState h p M h (v+(fun _ => ((M/2:ℕ):ZMod M))) = _ at hf |
| simpa only [flatMaskWord,hf] using ht |
| have ht := coded_range_lower hn hk (hp.log_circle_fourth hn) H (fun _ => ((M/2:ℕ):ZMod M)) hhb |
| (flatMaskWord h p R M h) (maskLength h p R M h) (treeCopies R (period h p) h) |
| (hp.mask_length_pos (by omega) hR h) (mask_length_factor h p R M h) |
| (fun v => length_maskWord h p R M h (flatMaskState h p M h v)) |
| (fun v z => mask_cheap_ed hR hp.divides_circle hp.positive hp.coprime h v z z.property) |
| (fun v i j => mask_single_ed (StandingPrimes.circle_even (h:=h) (p:=p)) h v i (2^j.val)) hlong c hc |
| have he := range_comp_surjective (flatMaskState h p M h) |
| (fun v => encode c (maskWord h p R M h v)) (flatMaskState_surjective h p M h) |
| unfold flatMaskWord at ht |
| rw [he] at ht |
| exact ht |
| theorem excluded_binary_distortion {n k : ℕ} (hn : 1000000≤n) (hk : n^4=4*k) |
| {p : ℕ → ℕ → ℕ} (hp : StandingPrimes (n^4) p) {w : ℕ} |
| (c : ExcludedAlphabet (n^4) (circleSize (n^4) p) (n^4) → Fin w → Bool) |
| (hc : SubstitutionBounds (1/268435456) 67108928 c) : |
| (n:ℝ)^k ≤ setDistortion (Set.range fun v : ExcludedState (n^4) (circleSize (n^4) p) (n^4) => |
| encode c (excludedWord (n^4) p ((n^4)^(2*n^4)) (circleSize (n^4) p) (n^4) v)) := by |
| classical |
| let h := n^4 |
| let M := circleSize h p |
| let R := h^(2*h) |
| let : NeZero M := ⟨hp.circle_pos.ne'⟩ |
| have hh : 2≤h := (show 2≤n by omega).trans (le_self_pow₀ (by omega) (by norm_num)) |
| have hR : 0<R := pow_pos (by omega) _ |
| let H := excludedCheap h p M hp.divides_circle h |
| let : Fintype H := Fintype.ofFinite H |
| have hhb (χ : AddChar (TreeLeaf (fun _ => Fin h) h → ZMod M) ℂ) |
| (ha : χ.compAddMonoidHom H.subtype=0) (hw : χ (fun _ => ((M/2:ℕ):ZMod M))≠1) : |
| ((h:ℝ)/2)^h ≤ ∑ i, ((coordinateFrequency χ i).valMinAbs.natAbs:ℝ) := by |
| apply excluded_character_branching hh hp.divides_circle hp.positive |
| (fun l k hl hlh => (hp.atLevel l hl hlh k).1) |
| hp.level_injective (fun l k hl hlh => (hp.atLevel l hl hlh k).2.1.le) χ ha _ hw |
| have hlong (v : TreeLeaf (fun _ => Fin h) h → ZMod M) : |
| (excludedLength h p R M h:ℝ)/(n:ℝ)^(3*n^4) ≤ |
| (deficit (flatExcludedWord h p R M h v) (flatExcludedWord h p R M h (v+(fun _ => ((M/2:ℕ):ZMod M)))):ℝ) := by |
| have he : n^4=2*(n^4/2) := by omega |
| have ht := excluded_long_deficit hn he hp hR (flatExcludedState h M h v) |
| have hf := flatExcludedState_shift h M h v ((M/2:ℕ):ZMod M) |
| change flatExcludedState h M h (v+(fun _ => ((M/2:ℕ):ZMod M))) = _ at hf |
| simpa only [flatExcludedWord,hf] using ht |
| have ht := coded_range_lower hn hk (hp.log_circle_fourth hn) H (fun _ => ((M/2:ℕ):ZMod M)) hhb |
| (flatExcludedWord h p R M h) (excludedLength h p R M h) (treeCopies R (period h p) h) |
| (hp.excluded_length_pos (by omega) hR h) (excluded_length_factor h p R M h) |
| (fun v => length_excludedWord h p R M h (flatExcludedState h M h v)) |
| (fun v z => excluded_cheap_ed hR hp.divides_circle hp.positive hp.coprime h v z z.property) |
| (fun v i j => excluded_single_ed (StandingPrimes.circle_even (h:=h) (p:=p)) h v i (2^j.val)) hlong c hc |
| have he := range_comp_surjective (flatExcludedState h M h) |
| (fun v => encode c (excludedWord h p R M h v)) (flatExcludedState_surjective h M h) |
| unfold flatExcludedWord at ht |
| rw [he] at ht |
| exact ht |
| |
| end FiniteCircle |
| |
| end OAI |
| |
