| import OAI.Combinatorics.FiniteCircle.Distortion |
|
|
| namespace OAI |
|
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| namespace FiniteCircle |
|
|
| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
|
|
| theorem StandingPrimes.small_prime {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) (l : ℕ) (hl : 1≤l) (hlh : l≤h) (k : Fin h) : |
| p l k ≤ h^4 := by |
| apply (hp.atLevel l hl hlh k).2.2.trans |
| calc |
| 2*h^3 ≤ h*h^3 := Nat.mul_le_mul_right _ hh |
| _ = h^4 := by ring |
|
|
| theorem StandingPrimes.period_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) (l : ℕ) (hl : 1≤l) (hlh : l≤h) : |
| period h p l ≤ h^(4*h) := by |
| calc |
| _ ≤ ∏ _k : Fin h, h^4 := Finset.prod_le_prod (fun k _ => hp.small_prime hh l hl hlh k) |
| _ = _ := by simp [← pow_mul] |
|
|
| theorem StandingPrimes.circle_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) : circleSize h p ≤ h^(5*h^2) := by |
| calc |
| _ ≤ h*(∏ _l : Fin h, h^(4*h)) := Nat.mul_le_mul hh |
| (Finset.prod_le_prod (fun l _ => hp.period_bound hh (l.val+1) (by omega) (by omega))) |
| _ = h^(4*h^2+1) := by simp only [Finset.prod_const, Finset.card_univ, Fintype.card_fin, ← pow_mul]; ring |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith) |
|
|
| theorem StandingPrimes.log_circle_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) : Nat.log 2 (circleSize h p)+1 ≤ 6*h^3 := by |
| have ht : h ≤ 2^h := (Nat.lt_two_pow_self (n:=h)).le |
| have hm : circleSize h p ≤ 2^(5*h^3) := by |
| calc |
| _ ≤ h^(5*h^2) := hp.circle_bound hh |
| _ ≤ (2^h)^(5*h^2) := Nat.pow_le_pow_left ht _ |
| _ = _ := by rw [← pow_mul]; congr 1; ring |
| have hn := Nat.log_mono_right hm (b:=2) |
| rw [Nat.log_pow (by norm_num : 1<2)] at hn |
| have hp3 : 0<h^3 := pow_pos (by omega) _ |
| nlinarith |
|
|
| theorem StandingPrimes.tags_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) (l : ℕ) (hl : 1≤l) (hlh : l≤h) : |
| (maskTags h p l).length ≤ h^5 := by |
| rw [length_maskTags] |
| calc |
| _ ≤ ∑ _k : Fin h, h^4 := Finset.sum_le_sum (fun k _ => hp.small_prime hh l hl hlh k) |
| _ = _ := by simp; ring |
|
|
| theorem StandingPrimes.tag_card_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) (l : ℕ) (hl : 1≤l) (hlh : l≤h) : |
| Fintype.card (MaskTag h p l) ≤ h^5 := by |
| simpa [Fintype.card_sigma, MaskTag, length_maskTags] using hp.tags_bound hh l hl hlh |
|
|
| theorem recursive_length_bound (h M : ℕ) (P T L : ℕ → ℕ) (hh : 3≤h) |
| (h0 : L 0=M) (hs : ∀ l, L (l+1)=(h^(2*h)*P (l+1))*(T (l+1)*L l)) |
| (hP : ∀ l, 1≤l → l≤h → P l≤h^(4*h)) |
| (hT : ∀ l, 1≤l → l≤h → T l≤h^5) : |
| ∀ l, l≤h → L l ≤ M*h^(8*h*l) := by |
| intro l |
| induction l with |
| | zero => intro _; simp [h0] |
| | succ l ih => |
| intro hl |
| rw [hs] |
| calc |
| _ ≤ (h^(2*h)*h^(4*h))*(h^5*(M*h^(8*h*l))) := |
| Nat.mul_le_mul (Nat.mul_le_mul_left _ (hP _ (by omega) hl)) |
| (Nat.mul_le_mul (hT _ (by omega) hl) (ih (by omega))) |
| _ = M*h^(2*h+4*h+5+8*h*l) := by ring |
| _ ≤ _ := Nat.mul_le_mul_left _ (Nat.pow_le_pow_right (by omega) (by nlinarith)) |
|
|
| theorem StandingPrimes.mask_length_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 3≤h) : |
| maskLength h p (h^(2*h)) (circleSize h p) h ≤ h^(13*h^2) := by |
| have ht := recursive_length_bound h (circleSize h p) (period h p) |
| (fun l => (maskTags h p l).length) (maskLength h p (h^(2*h)) (circleSize h p)) hh rfl |
| (fun _ => rfl) (hp.period_bound (by omega)) (hp.tags_bound (by omega)) h (le_refl _) |
| calc |
| _ ≤ circleSize h p*h^(8*h*h) := ht |
| _ ≤ h^(5*h^2)*h^(8*h*h) := Nat.mul_le_mul_right _ (hp.circle_bound (by omega)) |
| _ = _ := by ring |
|
|
| theorem StandingPrimes.excluded_length_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 3≤h) : |
| excludedLength h p (h^(2*h)) (circleSize h p) h ≤ h^(13*h^2) := by |
| have ht := recursive_length_bound h (circleSize h p) (period h p) |
| (fun _ => h) (excludedLength h p (h^(2*h)) (circleSize h p)) hh rfl |
| (fun l => by simp only [excludedLength,length_excludedTags]) (hp.period_bound (by omega)) |
| (fun _ _ _ => le_self_pow₀ (by omega) (by norm_num)) h (le_refl _) |
| calc |
| _ ≤ circleSize h p*h^(8*h*h) := ht |
| _ ≤ h^(5*h^2)*h^(8*h*h) := Nat.mul_le_mul_right _ (hp.circle_bound (by omega)) |
| _ = _ := by ring |
|
|
| theorem StandingPrimes.mask_alphabet_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2≤h) : |
| Fintype.card (MaskAlphabet h p h) ≤ h^(6*h^2) := by |
| have hc : ∀ l, l≤h → Fintype.card (MaskAlphabet h p l) ≤ 2*h^(5*l) := by |
| intro l |
| induction l with |
| | zero => intro _; change Fintype.card Bool ≤ 2*h^(5*0); norm_num |
| | succ l ih => |
| intro hl |
| calc |
| _ = Fintype.card (MaskTag h p (l+1))*Fintype.card (MaskAlphabet h p l) := Fintype.card_prod _ _ |
| _ ≤ h^5*(2*h^(5*l)) := Nat.mul_le_mul (hp.tag_card_bound hh _ (by omega) hl) (ih (by omega)) |
| _ = _ := by ring |
| calc |
| _ ≤ 2*h^(5*h) := hc h (le_refl _) |
| _ ≤ h*h^(5*h) := Nat.mul_le_mul_right _ hh |
| _ = h^(5*h+1) := by ring |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith) |
|
|
| theorem StandingPrimes.excluded_alphabet_bound {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 2≤h) : |
| Fintype.card (ExcludedAlphabet h (circleSize h p) h) ≤ h^(6*h^2) := by |
| have hc : ∀ l, Fintype.card (ExcludedAlphabet h (circleSize h p) l) = h^l*(circleSize h p*2) := by |
| intro l |
| induction l with |
| | zero => change Fintype.card (Fin (circleSize h p) × Bool) = 1*(circleSize h p*2); simp |
| | succ l ih => change Fintype.card (Fin h × ExcludedAlphabet h (circleSize h p) l) = _ |
| rw [Fintype.card_prod,Fintype.card_fin,ih]; ring |
| rw [hc] |
| calc |
| _ ≤ h^h*(h^(5*h^2)*h) := Nat.mul_le_mul_left _ (Nat.mul_le_mul (hp.circle_bound hh) hh) |
| _ = h^(5*h^2+h+1) := by ring |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith) |
|
|
| theorem StandingPrimes.mask_alphabet_ge {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 1≤h) (l : ℕ) : 2≤Fintype.card (MaskAlphabet h p l) := by |
| have hT (l : ℕ) : 1≤Fintype.card (MaskTag h p l) := by |
| have ht : h≤∑ k : Fin h, p l k := by |
| calc |
| _ = ∑ _k : Fin h, 1 := by simp |
| _ ≤ _ := Finset.sum_le_sum (fun k _ => hp.positive l k) |
| simpa only [MaskTag,Fintype.card_sigma,Fintype.card_fin] using hh.trans ht |
| induction l with |
| | zero => change 2≤Fintype.card Bool; norm_num |
| | succ l ih => |
| change 2≤Fintype.card (MaskTag h p (l+1) × MaskAlphabet h p l) |
| rw [Fintype.card_prod] |
| nlinarith [hT (l+1)] |
|
|
| theorem StandingPrimes.excluded_alphabet_ge {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 1≤h) (l : ℕ) : |
| 2≤Fintype.card (ExcludedAlphabet h (circleSize h p) l) := by |
| induction l with |
| | zero => |
| change 2≤Fintype.card (Fin (circleSize h p) × Bool) |
| rw [Fintype.card_prod,Fintype.card_fin,Fintype.card_bool] |
| have ht := hp.circle_pos |
| omega |
| | succ l ih => |
| change 2≤Fintype.card (Fin h × ExcludedAlphabet h (circleSize h p) l) |
| rw [Fintype.card_prod,Fintype.card_fin] |
| nlinarith |
|
|
| theorem StandingPrimes.mask_length_pos {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 1≤h) {R : ℕ} (hR : 0<R) (l : ℕ) : |
| 0 < maskLength h p R (circleSize h p) l := by |
| induction l with |
| | zero => exact hp.circle_pos |
| | succ l ih => |
| have hT : h≤(maskTags h p (l+1)).length := by |
| rw [length_maskTags] |
| calc |
| _ = ∑ _k : Fin h, 1 := by simp |
| _ ≤ _ := Finset.sum_le_sum (fun k _ => hp.positive _ k) |
| exact Nat.mul_pos (Nat.mul_pos hR (hp.period_pos _)) (Nat.mul_pos (by omega) ih) |
|
|
| theorem StandingPrimes.excluded_length_pos {h : ℕ} {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes h p) (hh : 1≤h) {R : ℕ} (hR : 0<R) (l : ℕ) : |
| 0<excludedLength h p R (circleSize h p) l := by |
| induction l with |
| | zero => exact hp.circle_pos |
| | succ l ih => |
| simp only [excludedLength,length_excludedTags] |
| exact Nat.mul_pos (Nat.mul_pos hR (hp.period_pos _)) (Nat.mul_pos (by omega) ih) |
|
|
| theorem logarithmic_width_bound {h A w : ℕ} (hh : 100000000≤h) (hA : 0<A) |
| (hAb : A≤h^(6*h^2)) (hw : (w:ℝ)≤67108928*Real.log (2*(A:ℝ))) : |
| w≤h^(7*h^2) := by |
| have hn : Real.log (2*(A:ℝ))≤2*A := by |
| have ht := Real.log_le_sub_one_of_pos (show 0<2*(A:ℝ) by positivity) |
| linarith |
| have hwr : (w:ℝ)≤67108928*(2*A) := hw.trans (mul_le_mul_of_nonneg_left hn (by norm_num)) |
| have hwn : w≤67108928*(2*A) := by exact_mod_cast hwr |
| calc |
| _ ≤ 67108928*(2*A) := hwn |
| _ ≤ h*(h*h^(6*h^2)) := Nat.mul_le_mul (by omega) (Nat.mul_le_mul (by omega) hAb) |
| _ = h^(6*h^2+2) := by ring |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) (by nlinarith) |
|
|
| theorem coded_length_cap {h A w L : ℕ} (hh : 100000000≤h) (hA : 0<A) |
| (hAb : A≤h^(6*h^2)) (hw : (w:ℝ)≤67108928*Real.log (2*(A:ℝ))) |
| (hL : L≤h^(13*h^2)) : w*L≤h^(32*h^2) := by |
| calc |
| _ ≤ h^(7*h^2)*h^(13*h^2) := Nat.mul_le_mul (logarithmic_width_bound hh hA hAb hw) hL |
| _ = h^(20*h^2) := by ring |
| _ ≤ _ := Nat.pow_le_pow_right (by omega) (by omega) |
|
|
| theorem StandingPrimes.log_circle_fourth {n : ℕ} (hn : 1000000≤n) {p : ℕ → ℕ → ℕ} |
| (hp : StandingPrimes (n^4) p) : Nat.log 2 (circleSize (n^4) p)+1≤n^13 := by |
| have nh : 2≤n^4 := (show 2≤n by omega).trans (le_self_pow₀ (by omega) (by norm_num)) |
| calc |
| _ ≤ 6*(n^4)^3 := hp.log_circle_bound nh |
| _ ≤ n*(n^4)^3 := Nat.mul_le_mul_right _ (by omega) |
| _ = _ := by ring |
|
|
| theorem lower_coefficient_budget {n k J : ℕ} (hn : 1000000≤n) (hk : n^4=4*k) |
| (hJ : J≤n^13) : |
| (n:ℝ)^k * (4*(n^4:ℕ)/((n^4:ℕ):ℝ)^(2*n^4)+ |
| 2*J/(((n^4:ℕ):ℝ)/2)^(n^4)) ≤ (1/268435456)/(n:ℝ)^(3*n^4) := by |
| have hn1 : (1:ℝ)≤n := by exact_mod_cast (show 1≤n by omega) |
| have hnp : (0:ℝ)<n := lt_of_lt_of_le zero_lt_one hn1 |
| have hk16 : 16≤k := by |
| have hh : 64≤n^4 := by |
| have ht := Nat.pow_le_pow_left (show 3≤n by omega) 4 |
| norm_num at ht ⊢ |
| omega |
| omega |
| have hp : 0<(n:ℝ)^(14*k) := pow_pos hnp _ |
| have hn4 : (n:ℝ)^4≤(n:ℝ)^13 := pow_le_pow_right₀ hn1 (by norm_num) |
| have h14 : (n:ℝ)^(14*k) ≤ ((n^4:ℕ):ℝ)^(2*n^4) := by |
| push_cast |
| rw [← pow_mul] |
| apply pow_le_pow_right₀ hn1 |
| omega |
| have hbbase : (n:ℝ)^7 ≤ (((n^4:ℕ):ℝ)/2)^2 := by |
| have hn4r : (4:ℝ)≤n := by exact_mod_cast (show 4≤n by omega) |
| have ht := mul_le_mul_of_nonneg_right hn4r (pow_nonneg hnp.le 7) |
| push_cast |
| nlinarith [show (n:ℝ)*(n:ℝ)^7=(n:ℝ)^8 by ring] |
| have hb : (n:ℝ)^(14*k) ≤ (((n^4:ℕ):ℝ)/2)^(n^4) := by |
| calc |
| _ = ((n:ℝ)^7)^(2*k) := by rw [← pow_mul]; congr 1; ring |
| _ ≤ ((((n^4:ℕ):ℝ)/2)^2)^(2*k) := pow_le_pow_left₀ (by positivity) hbbase _ |
| _ = _ := by rw [← pow_mul]; congr 1; omega |
| have hc : (4*((n^4:ℕ):ℝ)/((n^4:ℕ):ℝ)^(2*n^4)+ |
| 2*(J:ℝ)/(((n^4:ℕ):ℝ)/2)^(n^4)) ≤ 6*(n:ℝ)^13/(n:ℝ)^(14*k) := by |
| have hJr : (J:ℝ)≤(n:ℝ)^13 := by exact_mod_cast hJ |
| calc |
| _ ≤ (4*(n:ℝ)^13)/(n:ℝ)^(14*k)+(2*(n:ℝ)^13)/(n:ℝ)^(14*k) := by |
| apply add_le_add |
| · exact div_le_div₀ (by positivity) (by push_cast; linarith) hp h14 |
| · exact div_le_div₀ (by positivity) (by linarith) hp hb |
| _ = _ := by ring |
| have hconst : (6*268435456:ℝ)≤(n:ℝ)^3 := by |
| have hr : (1000000:ℝ)≤n := by exact_mod_cast hn |
| nlinarith [sq_nonneg ((n:ℝ)-1000000)] |
| have hpow : (6*268435456:ℝ)*(n:ℝ)^(13*k+13) ≤ (n:ℝ)^(14*k) := by |
| calc |
| _ ≤ (n:ℝ)^3*(n:ℝ)^(13*k+13) := mul_le_mul_of_nonneg_right hconst (by positivity) |
| _ = (n:ℝ)^(13*k+16) := by rw [← pow_add]; congr 1; ring |
| _ ≤ _ := pow_le_pow_right₀ hn1 (by omega) |
| have hlast : (n:ℝ)^k*(6*(n:ℝ)^13/(n:ℝ)^(14*k)) ≤ (1/268435456)/(n:ℝ)^(3*n^4) := by |
| apply (le_div_iff₀ (pow_pos hnp _)).mpr |
| rw [div_eq_mul_inv] |
| have he : (n:ℝ)^k*(6*(n:ℝ)^13*((n:ℝ)^(14*k))⁻¹)*(n:ℝ)^(3*n^4) = |
| 6*(n:ℝ)^(13*k+13)/(n:ℝ)^(14*k) := by |
| rw [hk] |
| ring |
| rw [he] |
| apply (div_le_iff₀ hp).mpr |
| nlinarith |
| exact (mul_le_mul_of_nonneg_left hc (by positivity)).trans hlast |
|
|
| theorem lower_numeric {n k M : ℕ} (hn : 1000000≤n) (hk : n^4=4*k) |
| (hJ : Nat.log 2 M+1≤n^13) {w : ℕ} (hw : 0<w) (t : ℝ) |
| (ht : (1/268435456)*(w:ℝ)/(n:ℝ)^(3*n^4) ≤ t*w* |
| (4*(n^4:ℕ)/((n^4:ℕ):ℝ)^(2*n^4)+2*(Nat.log 2 M+1:ℕ)/(((n^4:ℕ):ℝ)/2)^(n^4))) : |
| (n:ℝ)^k ≤ t := by |
| let q : ℝ := 4*(n^4:ℕ)/((n^4:ℕ):ℝ)^(2*n^4)+2*(Nat.log 2 M+1:ℕ)/(((n^4:ℕ):ℝ)/2)^(n^4) |
| have hn0 : (0:ℝ)<n := by exact_mod_cast (show 0<n by omega) |
| have hq : 0<q := by dsimp [q]; positivity |
| have hwq : 0<(w:ℝ)*q := mul_pos (by exact_mod_cast hw) hq |
| apply (mul_le_mul_iff_left₀ hwq).mp |
| calc |
| _ = w*((n:ℝ)^k*q) := by ring |
| _ ≤ w*((1/268435456)/(n:ℝ)^(3*n^4)) := |
| mul_le_mul_of_nonneg_left (lower_coefficient_budget hn hk hJ) (by positivity) |
| _ = (1/268435456)*w/(n:ℝ)^(3*n^4) := by ring |
| _ ≤ t*w*q := ht |
| _ = _ := by ring |
|
|
| end FiniteCircle |
|
|
| end OAI |
|
|
