| import Mathlib |
| import OAI.Probability.BinarySweep.SparseBounds.SparseNumerics |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
|
|
| namespace BinaryCoordinateSweeps.Sparse |
|
|
| lemma sparse_amplitudes_small (b h k : ℕ) {L : ℝ} (hb : b≤16000) |
| (hh : (h:ℝ)≤100000*k) (hL : (10:ℝ)^26≤L) : |
| ((2:ℝ)^(k+k*b)*Real.exp ((b*h+k+k*b:ℕ):ℝ))^2≤Real.exp ((k:ℝ)*L/2000000000) ∧ |
| ((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2≤Real.exp ((k:ℝ)*L/2000000000) ∧ |
| (2:ℝ)^k≤Real.exp ((k:ℝ)*L/2000000000) := by |
| have hbR : (b:ℝ)≤16000 := by exact_mod_cast hb |
| have h2 : Real.log 2≤2 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<2)).trans (by norm_num) |
| have h4 : Real.log 4≤4 := (Real.log_le_sub_one_of_pos (by norm_num : (0:ℝ)<4)).trans (by norm_num) |
| have hbK := mul_le_mul_of_nonneg_right hbR (show (0:ℝ)≤k by positivity) |
| have hhB := mul_le_mul_of_nonneg_left hh (show (0:ℝ)≤b by positivity) |
| have hLK := mul_le_mul_of_nonneg_right hL (show (0:ℝ)≤k by positivity) |
| have hp (n : ℕ) : (2:ℝ)^n=Real.exp ((n:ℝ)*Real.log 2) := by |
| rw [Real.exp_nat_mul,Real.exp_log (by norm_num)] |
| constructor |
| · rw [hp,←Real.exp_add,←Real.exp_nat_mul] |
| apply Real.exp_le_exp.mpr |
| have h2K := mul_le_mul_of_nonneg_left h2 (show (0:ℝ)≤(k+k*b:ℕ) by positivity) |
| push_cast at h2K ⊢ |
| nlinarith only [h2K,hhB,hbK,hLK] |
| constructor |
| · rw [hp,←Real.exp_add,←Real.exp_nat_mul] |
| apply Real.exp_le_exp.mpr |
| have h2K := mul_le_mul_of_nonneg_left h2 (show (0:ℝ)≤k by positivity) |
| have h4K := mul_le_mul_of_nonneg_right h4 (show (0:ℝ)≤k*b by positivity) |
| norm_num |
| nlinarith only [h2K,h4K,hhB,hbK,hLK] |
| · rw [hp] |
| apply Real.exp_le_exp.mpr |
| have h2K := mul_le_mul_of_nonneg_left h2 (show (0:ℝ)≤k by positivity) |
| nlinarith only [h2K,hLK] |
|
|
| lemma few_hs_scalar (b h k wv wl : ℕ) {L g v x y : ℝ} (C : Fin b → ℝ) |
| (hb : b≤16000) (hk : 1≤k) (hh : (h:ℝ)≤100000*k) (hL : (10:ℝ)^26≤L) |
| (htv : (k:ℝ)≤8*(((wv+1)/2:ℕ):ℝ)) (htl : (k:ℝ)≤128000*wl) |
| (hg0 : 0≤g) (hg : g≤Real.exp (-L/640000)) |
| (hv0 : 0≤v) (hv : v≤Real.exp (-L/3200)) |
| (hx0 : 0≤x) (hx : x≤Real.exp (-L/3200)) |
| (hy0 : 0≤y) (hy : y≤Real.exp (-L/3200)) |
| (hC0 : ∀j,0≤C j) (hC : ∀j,C j≤Real.exp ((k:ℝ)*L/2000000000)) : |
| ((2:ℝ)^(k+k*b)*Real.exp ((b*h+k+k*b:ℕ):ℝ)*g^((k+1)/2))^2+ |
| ((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 * |
| ((2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl))≤ |
| Real.exp (-(k:ℝ)*L/1000000000000) := by |
| have hL0 : 0≤L := (by positivity : (0:ℝ)≤10^26).trans hL |
| have hkR : (1:ℝ)≤k := by exact_mod_cast hk |
| have hbR : (b:ℝ)≤16000 := by exact_mod_cast hb |
| have hKL := mul_nonneg (Nat.cast_nonneg k) hL0 |
| have hkt : (k:ℝ)≤2*(((k+1)/2:ℕ):ℝ) := by exact_mod_cast (show k≤2*((k+1)/2) from by omega) |
| obtain ⟨hA,hB,h2⟩ := sparse_amplitudes_small b h k hb hh hL |
| have hgpow : (g^((k+1)/2))^2≤Real.exp (-(k:ℝ)*L/1000000000) := by |
| have he := nat_pow_le_exp_of_le hg0 hg ((k+1)/2) |
| have he2 := pow_le_pow_left₀ (pow_nonneg hg0 _) he 2 |
| rw [←Real.exp_nat_mul] at he2 |
| refine he2.trans (Real.exp_le_exp.mpr ?_) |
| have ht := mul_le_mul_of_nonneg_right hkt hL0 |
| norm_num |
| nlinarith only [ht,hKL] |
| have hpow (a : ℝ) (ha0 : 0≤a) (ha : a≤Real.exp (-L/3200)) |
| (t : ℕ) (ht : (k:ℝ)≤128000*t) : |
| a^t≤Real.exp (-2*(k:ℝ)*L/1000000000) := by |
| apply (nat_pow_le_exp_of_le ha0 ha t).trans (Real.exp_le_exp.mpr ?_) |
| have he := mul_le_mul_of_nonneg_right ht hL0 |
| nlinarith only [he,hKL] |
| have hlx := hpow x hx0 hx wl htl |
| have hly := hpow y hy0 hy wl htl |
| have hlv := hpow v hv0 hv ((wv+1)/2) (htv.trans (by nlinarith only [show (0:ℝ)≤(((wv+1)/2:ℕ):ℝ) by positivity])) |
| have hgood : ((2:ℝ)^(k+k*b)*Real.exp ((b*h+k+k*b:ℕ):ℝ)*g^((k+1)/2))^2 ≤ |
| Real.exp (-(k:ℝ)*L/2000000000) := by |
| rw [mul_pow] |
| refine (mul_le_mul hA hgpow (by positivity) (Real.exp_pos _).le).trans_eq ?_ |
| rw [←Real.exp_add] |
| congr 1 |
| ring |
| have hvertex : (2:ℝ)^k*v^((wv+1)/2)≤Real.exp (-(k:ℝ)*L/1000000000) := by |
| refine (mul_le_mul h2 hlv (by positivity) (Real.exp_pos _).le).trans ?_ |
| rw [←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| nlinarith only [hKL] |
| have hholes : (2:ℝ)^k*x^wl≤Real.exp (-(k:ℝ)*L/1000000000) := by |
| refine (mul_le_mul h2 hlx (by positivity) (Real.exp_pos _).le).trans ?_ |
| rw [←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| nlinarith only [hKL] |
| have hparticles (j : Fin b) : C j*(2:ℝ)^k*y^wl≤Real.exp (-(k:ℝ)*L/1000000000) := by |
| have hc0 := hC0 j |
| refine (mul_le_mul (mul_le_mul (hC j) h2 (by positivity) (Real.exp_pos _).le) |
| hly (by positivity) (by positivity)).trans_eq ?_ |
| rw [←Real.exp_add,←Real.exp_add] |
| congr 1 |
| ring |
| have hevents : (2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl) ≤ |
| (1+2*b)*Real.exp (-(k:ℝ)*L/1000000000) := by |
| refine (add_le_add hvertex (Finset.sum_le_sum (fun j _ => add_le_add hholes (hparticles j)))).trans_eq ?_ |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
| ring |
| have hbad : ((2:ℝ)^k*Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 * |
| ((2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl)) ≤ |
| (1+2*b)*Real.exp (-(k:ℝ)*L/2000000000) := by |
| have hev0 : 0≤ (2:ℝ)^k*v^((wv+1)/2)+∑j : Fin b, ((2:ℝ)^k*x^wl+C j*(2:ℝ)^k*y^wl) := by |
| apply add_nonneg (by positivity) |
| apply Finset.sum_nonneg |
| intro j _ |
| have hc0 := hC0 j |
| positivity |
| refine (mul_le_mul hB hevents hev0 (Real.exp_pos _).le).trans_eq ?_ |
| rw [mul_left_comm,←Real.exp_add] |
| congr 1 |
| ring_nf |
| have hcount : 2+2*(b:ℝ)≤Real.exp ((k:ℝ)*L/4000000000) := by |
| apply (show 2+2*(b:ℝ)≤(k:ℝ)*L/4000000000+1 from ?_).trans (Real.add_one_le_exp _) |
| have hh := mul_le_mul_of_nonneg_right hkR hL0 |
| nlinarith only [hh,hL,hbR] |
| calc |
| _ ≤ (2+2*b)*Real.exp (-(k:ℝ)*L/2000000000) := by nlinarith only [hgood,hbad] |
| _ ≤ Real.exp ((k:ℝ)*L/4000000000)*Real.exp (-(k:ℝ)*L/2000000000) := |
| mul_le_mul_of_nonneg_right hcount (Real.exp_pos _).le |
| _ ≤ _ := by |
| rw [←Real.exp_add] |
| apply Real.exp_le_exp.mpr |
| nlinarith only [hKL] |
|
|
| end BinaryCoordinateSweeps.Sparse |
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| end |
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| end OAI |
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