| import OAI.Combinatorics.FiniteCircle.Primes | |
| namespace OAI | |
| namespace FiniteCircle | |
| open scoped BigOperators List | |
| theorem halfturn_ne_zero {M : ℕ} [NeZero M] (he : M=2*(M/2)) : | |
| ((M/2:ℕ) : ZMod M) ≠ 0 := by | |
| intro hz | |
| have hd := (CharP.cast_eq_zero_iff (ZMod M) M (M/2)).mp hz | |
| have hm := NeZero.pos M | |
| have hh : 0 < M/2 := by omega | |
| have hw := Nat.le_of_dvd hh hd | |
| omega | |
| theorem halfturn_supported {M : ℕ} (he : M=2*(M/2)) : | |
| PhaseSupported {2} ((M/2:ℕ) : ZMod M) := by | |
| simp only [PhaseSupported, Finset.prod_singleton, nsmul_eq_mul] | |
| rw [← Nat.cast_mul, ← he] | |
| simp | |
| theorem StandingPrimes.mask_tags_ge {h : ℕ} {p : ℕ → ℕ → ℕ} | |
| (hp : StandingPrimes h p) (hh : 4 ≤ h) (l : ℕ) : 4 ≤ (maskTags h p l).length := by | |
| rw [length_maskTags] | |
| calc | |
| 4 ≤ h := hh | |
| _ = ∑ _k : Fin h, 1 := by simp | |
| _ ≤ _ := Finset.sum_le_sum (fun k _ => hp.positive l k) | |
| theorem StandingPrimes.half_fresh {h : ℕ} {p : ℕ → ℕ → ℕ} | |
| (hp : StandingPrimes h p) (hh : 2 ≤ h) (l : ℕ) (hl : l ≤ h) : LowerFresh h p l {2} := by | |
| intro j hj hjl k hk | |
| have ht := (hp.atLevel j hj (hjl.trans hl) k).2.1 | |
| have hh3 := Nat.pow_le_pow_left hh 3 | |
| simp only [Finset.mem_singleton] at hk | |
| omega | |
| theorem mask_long_deficit {n : ℕ} (hn : 1000000 ≤ n) (he : n^4=2*(n^4/2)) | |
| {p : ℕ → ℕ → ℕ} (hp : StandingPrimes (n^4) p) {R : ℕ} (hR : 0<R) | |
| (v : MaskState (n^4) p (circleSize (n^4) p) (n^4)) : | |
| (maskLength (n^4) p R (circleSize (n^4) p) (n^4):ℝ)/(n:ℝ)^(3*n^4) ≤ | |
| (deficit (maskWord (n^4) p R (circleSize (n^4) p) (n^4) v) | |
| (maskWord (n^4) p R (circleSize (n^4) p) (n^4) | |
| (shiftMaskState (n^4) p (circleSize (n^4) p) (n^4) v ((circleSize (n^4) p/2 : ℕ) : ZMod (circleSize (n^4) p)))) : ℝ) := by | |
| let h := n^4 | |
| let M := circleSize h p | |
| let : NeZero M := ⟨hp.circle_pos.ne'⟩ | |
| have hh : 1000000 ≤ h := hn.trans (le_self_pow₀ (by omega) (by norm_num)) | |
| have ht : 2 ≤ n^2 := by nlinarith | |
| have hht : h ≤ (n^2)^2 := by dsimp [h]; ring_nf; rfl | |
| have hth : 2*n^2 ≤ h := by dsimp [h]; nlinarith | |
| have hK : 2 ≤ h/100 := by omega | |
| have hbound := mask_denominator_descent ht hht hth hK hp.divides_circle hp.positive | |
| (fun j hj hjh k => (hp.atLevel j hj hjh k).1) | |
| (fun j hj hjh k => (hp.atLevel j hj hjh k).2.1.le) | |
| (fun j hj hjh k => (hp.atLevel j hj hjh k).2.2) hp.level_distinct | |
| h (le_refl _) {2} ((M/2:ℕ) : ZMod M) (by simp; omega) | |
| (by intro q hq; simp only [Finset.mem_singleton] at hq; subst q | |
| exact ⟨Nat.prime_two, (show 2≤h by omega).trans (le_self_pow₀ (by omega) (by norm_num))⟩) | |
| (hp.half_fresh (by omega) h (le_refl _)) | |
| (halfturn_ne_zero (StandingPrimes.circle_even (h:=h) (p:=p))) | |
| (halfturn_supported (StandingPrimes.circle_even (h:=h) (p:=p))) | |
| simp only [Finset.card_singleton] at hbound | |
| have hsep := mask_separation (StandingPrimes.circle_even (h:=h) (p:=p)) (by omega : 0<h) | |
| hR hp.divides_circle hp.positive (hp.mask_tags_ge (by omega)) h v ((M/2:ℕ) : ZMod M) | |
| have hnum := mask_descent_numeric hn he | |
| have hL : (0:ℝ) ≤ maskLength h p R M h := by positivity | |
| have hc : (0:ℝ) ≤ 1/((2:ℝ)^h*4) := by positivity | |
| calc | |
| _ = (maskLength h p R M h:ℝ)*(1/(n:ℝ)^(3*h)) := by dsimp [h,M]; ring | |
| _ ≤ (maskLength h p R M h:ℝ)*((1/((2:ℝ)^h*4))*descentBound h (h/100) (n^2) 1 h) := | |
| mul_le_mul_of_nonneg_left hnum hL | |
| _ ≤ (1/((2:ℝ)^h*4))*(maskLength h p R M h)*maskCost h p M hp.divides_circle hp.positive h ((M/2:ℕ) : ZMod M) := by | |
| nlinarith [mul_le_mul_of_nonneg_left hbound (mul_nonneg hL hc)] | |
| _ ≤ _ := hsep | |
| theorem excluded_long_deficit {n : ℕ} (hn : 1000000 ≤ n) (he : n^4=2*(n^4/2)) | |
| {p : ℕ → ℕ → ℕ} (hp : StandingPrimes (n^4) p) {R : ℕ} (hR : 0<R) | |
| (v : ExcludedState (n^4) (circleSize (n^4) p) (n^4)) : | |
| (excludedLength (n^4) p R (circleSize (n^4) p) (n^4):ℝ)/(n:ℝ)^(3*n^4) ≤ | |
| (deficit (excludedWord (n^4) p R (circleSize (n^4) p) (n^4) v) | |
| (excludedWord (n^4) p R (circleSize (n^4) p) (n^4) | |
| (shiftExcludedState (n^4) (circleSize (n^4) p) (n^4) v ((circleSize (n^4) p/2 : ℕ) : ZMod (circleSize (n^4) p)))) : ℝ) := by | |
| let h := n^4 | |
| let M := circleSize h p | |
| let : NeZero M := ⟨hp.circle_pos.ne'⟩ | |
| have hh : 1000000 ≤ h := hn.trans (le_self_pow₀ (by omega) (by norm_num)) | |
| have ht : 2 ≤ n^3 := by nlinarith [Nat.pow_le_pow_left (show 2≤n by omega) 3] | |
| have htt : h ≤ n^3*n := by dsimp [h]; ring_nf; rfl | |
| have hsmall : 2*n^3 ≤ h := by | |
| have ht := Nat.mul_le_mul_right (n^3) (show 2≤n by omega) | |
| dsimp [h]; nlinarith [show n*n^3=n^4 by ring] | |
| have hbound := excluded_denominator_descent ht (by omega : 1≤n) (le_refl (n^4)) htt hsmall | |
| (by omega : 3≤h/100) hp.divides_circle hp.positive | |
| (fun j hj hjh k => (hp.atLevel j hj hjh k).1) | |
| (fun j hj hjh k => by | |
| have ht := ((hp.excludedPrimes (by omega)).2.2 ⟨j-1,by omega⟩ k).2 | |
| simpa only [show j-1+1=j by omega] using ht) hp.level_injective hp.level_distinct | |
| h (le_refl _) {2} ((M/2:ℕ) : ZMod M) (by simp; omega) | |
| (by intro q hq; simp only [Finset.mem_singleton] at hq; subst q | |
| exact ⟨Nat.prime_two, (show 2≤h by omega).trans (le_self_pow₀ (by omega) (by norm_num))⟩) | |
| (hp.half_fresh (by omega) h (le_refl _)) | |
| (halfturn_ne_zero (StandingPrimes.circle_even (h:=h) (p:=p))) | |
| (halfturn_supported (StandingPrimes.circle_even (h:=h) (p:=p))) | |
| simp only [Finset.card_singleton] at hbound | |
| have hsep := excluded_separation (StandingPrimes.circle_even (h:=h) (p:=p)) (by omega : 0<h) | |
| hR hp.divides_circle hp.positive (fun _ => by simp; omega) h v ((M/2:ℕ) : ZMod M) | |
| have hnum := excluded_descent_numeric hn he | |
| have hL : (0:ℝ) ≤ excludedLength h p R M h := by positivity | |
| have hc : (0:ℝ) ≤ 1/((2:ℝ)^h*4) := by positivity | |
| calc | |
| _ = (excludedLength h p R M h:ℝ)*(1/(n:ℝ)^(3*h)) := by dsimp [h,M]; ring | |
| _ ≤ (excludedLength h p R M h:ℝ)*((1/((2:ℝ)^h*4))*pairDescentBound h (h/100) (n^3) n 1 h) := | |
| mul_le_mul_of_nonneg_left hnum hL | |
| _ ≤ (1/((2:ℝ)^h*4))*(excludedLength h p R M h)*excludedCost h p M hp.divides_circle hp.positive h ((M/2:ℕ) : ZMod M) := by | |
| nlinarith [mul_le_mul_of_nonneg_left hbound (mul_nonneg hL hc)] | |
| _ ≤ _ := hsep | |
| end FiniteCircle | |
| end OAI | |