| import Mathlib |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Real |
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| def logMeshLength (T : ℝ) : ℝ := -Real.log (1-T) |
| def logMeshCount (T h : ℝ) : ℕ := ⌈logMeshLength T/h⌉₊ |
| def logMeshStep (T h : ℝ) : ℝ := logMeshLength T/(logMeshCount T h:ℝ) |
| def logMeshNode (T h : ℝ) (j : ℕ) : ℝ := 1-Real.exp (-(j:ℝ)*logMeshStep T h) |
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| lemma logMeshLength_pos {T : ℝ} (hT0 : 0<T) (hT1 : T<1) : 0<logMeshLength T := by |
| dsimp [logMeshLength] |
| exact neg_pos.mpr (Real.log_neg (by linarith) (by linarith)) |
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| lemma logMeshCount_pos {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) : |
| 0<logMeshCount T h := by |
| have hp := div_pos (logMeshLength_pos hT0 hT1) hh |
| have he := Nat.le_ceil (logMeshLength T/h) |
| have hc : (0:ℝ)<logMeshCount T h := hp.trans_le he |
| exact_mod_cast hc |
|
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| lemma logMeshCount_bound {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) : |
| (logMeshCount T h:ℝ)<logMeshLength T/h+1 := |
| Nat.ceil_lt_add_one (le_of_lt (div_pos (logMeshLength_pos hT0 hT1) hh)) |
|
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| lemma logMeshStep_pos {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) : |
| 0<logMeshStep T h := |
| div_pos (logMeshLength_pos hT0 hT1) (by exact_mod_cast logMeshCount_pos hT0 hT1 hh) |
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| lemma logMeshStep_le {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) : |
| logMeshStep T h≤h := by |
| have hN : (0:ℝ)<logMeshCount T h := by exact_mod_cast logMeshCount_pos hT0 hT1 hh |
| apply (div_le_iff₀ hN).mpr |
| have he := (div_le_iff₀ hh).mp (Nat.le_ceil (logMeshLength T/h)) |
| change logMeshLength T≤(logMeshCount T h:ℝ)*h at he |
| nlinarith |
|
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| lemma logMeshStep_lower {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (hL : h≤logMeshLength T) : h/2≤logMeshStep T h := by |
| have hN : (0:ℝ)<logMeshCount T h := by exact_mod_cast logMeshCount_pos hT0 hT1 hh |
| have he : (logMeshCount T h:ℝ)<(logMeshLength T+h)/h := by |
| simpa only [add_div,div_self (ne_of_gt hh)] using logMeshCount_bound hT0 hT1 hh |
| have hN' := (lt_div_iff₀ hh).mp he |
| apply (le_div_iff₀ hN).mpr |
| nlinarith |
|
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| @[simp] lemma logMeshNode_zero (T h : ℝ) : logMeshNode T h 0=0 := by |
| simp [logMeshNode] |
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| lemma logMeshNode_last {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) : |
| logMeshNode T h (logMeshCount T h)=T := by |
| have hN : (logMeshCount T h:ℝ)≠0 := by |
| exact_mod_cast ne_of_gt (logMeshCount_pos hT0 hT1 hh) |
| dsimp [logMeshNode,logMeshStep,logMeshLength] |
| rw [neg_mul, mul_div_cancel₀ _ hN,neg_neg,Real.exp_log (by linarith : 0<1-T)] |
| ring |
|
|
| lemma logMeshNode_sub {T h : ℝ} (j : ℕ) : |
| logMeshNode T h (j+1)-logMeshNode T h j= |
| (1-logMeshNode T h j)*(1-Real.exp (-logMeshStep T h)) := by |
| have he : -((j+1:ℕ):ℝ)*logMeshStep T h= |
| -(j:ℝ)*logMeshStep T h+(-logMeshStep T h) := by push_cast; ring |
| simp only [logMeshNode,he,Real.exp_add] |
| ring |
|
|
| lemma logMeshNode_strictMono {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) : |
| StrictMono (logMeshNode T h) := by |
| apply strictMono_nat_of_lt_succ |
| intro j |
| have hs := logMeshStep_pos hT0 hT1 hh |
| have he : Real.exp (-logMeshStep T h)<1 := by |
| rw [Real.exp_lt_one_iff]; linarith |
| have hp : 0<1-logMeshNode T h j := by |
| simpa only [logMeshNode,sub_sub_cancel] using Real.exp_pos (-(j:ℝ)*logMeshStep T h) |
| have hd : 0<logMeshNode T h (j+1)-logMeshNode T h j := by |
| rw [logMeshNode_sub] |
| exact mul_pos hp (by linarith) |
| linarith |
|
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| lemma logMeshNode_mem {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| {j : ℕ} (hj : j≤logMeshCount T h) : logMeshNode T h j∈Set.Icc 0 T := by |
| have hm := (logMeshNode_strictMono hT0 hT1 hh).monotone |
| constructor |
| · simpa only [logMeshNode_zero] using hm (Nat.zero_le j) |
| · simpa only [logMeshNode_last hT0 hT1 hh] using hm hj |
|
|
| lemma logMeshCell_upper {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) (j : ℕ) : |
| logMeshNode T h (j+1)-logMeshNode T h j≤h*(1-logMeshNode T h j) := by |
| have hb : 1-Real.exp (-logMeshStep T h)≤h := by |
| have hx := Real.add_one_le_exp (-logMeshStep T h) |
| have hs := logMeshStep_le hT0 hT1 hh |
| linarith |
| have hp : 0≤1-logMeshNode T h j := by |
| simpa only [logMeshNode,sub_sub_cancel] using (Real.exp_pos (-(j:ℝ)*logMeshStep T h)).le |
| rw [logMeshNode_sub,mul_comm h] |
| exact mul_le_mul_of_nonneg_left hb hp |
|
|
| lemma logMeshCell_lower {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (hL : h≤logMeshLength T) (hsmall : h≤Real.log 2) (j : ℕ) : |
| h/4*(1-logMeshNode T h j)≤logMeshNode T h (j+1)-logMeshNode T h j := by |
| have hs0 := logMeshStep_pos hT0 hT1 hh |
| have hs := logMeshStep_le hT0 hT1 hh |
| have hs' := logMeshStep_lower hT0 hT1 hh hL |
| have he : Real.exp (logMeshStep T h)≤2 := by |
| simpa only [Real.exp_log (by norm_num : (0:ℝ)<2)] using Real.exp_le_exp.mpr (hs.trans hsmall) |
| have he' : (1:ℝ)/2≤Real.exp (-logMeshStep T h) := by |
| rw [Real.exp_neg] |
| simpa only [one_div] using (one_div_le_one_div_of_le (Real.exp_pos _) he) |
| have hw : logMeshStep T h*Real.exp (-logMeshStep T h)≤1-Real.exp (-logMeshStep T h) := by |
| have hx := mul_le_mul_of_nonneg_right (Real.add_one_le_exp (logMeshStep T h)) |
| (Real.exp_pos (-logMeshStep T h)).le |
| rw [←Real.exp_add,add_neg_cancel,Real.exp_zero] at hx |
| nlinarith |
| have hb : h/4≤1-Real.exp (-logMeshStep T h) := by |
| have hx := mul_le_mul_of_nonneg_left he' hs0.le |
| nlinarith |
| have hp : 0≤1-logMeshNode T h j := by |
| simpa only [logMeshNode,sub_sub_cancel] using (Real.exp_pos (-(j:ℝ)*logMeshStep T h)).le |
| rw [logMeshNode_sub,mul_comm (h/4)] |
| exact mul_le_mul_of_nonneg_left hb hp |
| end LogConcaveSampling.Quadrature |
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| end |
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| end |
|
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| section |
|
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| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
|
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| lemma logMeshCell_upper_right {T h : ℝ} (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (hsmall : h≤Real.log 2) (j : ℕ) : |
| logMeshNode T h (j+1)-logMeshNode T h j≤2*h*(1-logMeshNode T h (j+1)) := by |
| have hs := logMeshStep_le hT0 hT1 hh |
| have he : Real.exp (logMeshStep T h)≤2 := by |
| simpa only [Real.exp_log (by norm_num : (0:ℝ)<2)] using Real.exp_le_exp.mpr (hs.trans hsmall) |
| have hid : (1-logMeshNode T h (j+1))*Real.exp (logMeshStep T h)=1-logMeshNode T h j := by |
| simp only [logMeshNode,sub_sub_cancel,←Real.exp_add] |
| congr 1 |
| push_cast |
| ring |
| have hn : 0≤1-logMeshNode T h (j+1) := by |
| simpa only [logMeshNode,sub_sub_cancel] using (Real.exp_pos _).le |
| have hb := mul_le_mul_of_nonneg_left he hn |
| rw [hid] at hb |
| calc |
| _ ≤ h*(1-logMeshNode T h j) := logMeshCell_upper hT0 hT1 hh j |
| _ ≤ h*((1-logMeshNode T h (j+1))*2) := mul_le_mul_of_nonneg_left hb hh.le |
| _ = _ := by ring |
|
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| lemma cell_singularity_cancel {a b t h : ℝ} (ha : 0≤a) (hat : a≤t) (htb : t≤b) |
| (hb : b<1) (hh : 0≤h) (hcell : b-a≤h*(1-b)) (n : ℕ) : |
| (t-a)^(n+1)*((Real.sqrt (1-b^2))⁻¹)^(2*n+2)≤h^(n+1) := by |
| have hb0 : 0≤b := ha.trans (hat.trans htb) |
| have hp : 0<1-b^2 := by nlinarith |
| have hlen : t-a≤h*(1-b^2) := by |
| have he : 1-b≤1-b^2 := by nlinarith |
| have hm := mul_le_mul_of_nonneg_left he hh |
| linarith |
| have hr : (t-a)*(1-b^2)⁻¹≤h := by |
| rw [←div_eq_mul_inv,div_le_iff₀ hp] |
| exact hlen |
| rw [show 2*n+2=2*(n+1) by omega,pow_mul,inv_pow,Real.sq_sqrt hp.le,←mul_pow] |
| exact pow_le_pow_left₀ (mul_nonneg (sub_nonneg.mpr hat) (inv_nonneg.mpr hp.le)) hr _ |
|
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| lemma cell_taylor_budget {a b t h A : ℝ} (ha : 0≤a) (hat : a≤t) (htb : t≤b) |
| (hb : b<1) (hh : 0≤h) (hcell : b-a≤h*(1-b)) (hA : 0≤A) (n : ℕ) : |
| ((t-a)^(n+1)/(n.factorial:ℝ))^2*(A*((Real.sqrt (1-b^2))⁻¹)^(2*n+2))^2≤ |
| (A*h^(n+1))^2 := by |
| have hfac : (1:ℝ)≤n.factorial := by exact_mod_cast Nat.factorial_pos n |
| have hfp : (0:ℝ)<n.factorial := by positivity |
| have hd : (t-a)^(n+1)/(n.factorial:ℝ)≤(t-a)^(n+1) := by |
| rw [div_le_iff₀ hfp] |
| exact le_mul_of_one_le_right (pow_nonneg (sub_nonneg.mpr hat) _) hfac |
| have hB := cell_singularity_cancel ha hat htb hb hh hcell n |
| have hR : 0≤((Real.sqrt (1-b^2))⁻¹)^(2*n+2) := by positivity |
| have he : ((t-a)^(n+1)/(n.factorial:ℝ))*(A*((Real.sqrt (1-b^2))⁻¹)^(2*n+2))≤ |
| A*h^(n+1) := by |
| calc |
| _ ≤ (t-a)^(n+1)*(A*((Real.sqrt (1-b^2))⁻¹)^(2*n+2)) := |
| mul_le_mul_of_nonneg_right hd (mul_nonneg hA hR) |
| _ = A*((t-a)^(n+1)*((Real.sqrt (1-b^2))⁻¹)^(2*n+2)) := by ring |
| _ ≤ _ := mul_le_mul_of_nonneg_left hB hA |
| rw [←mul_pow] |
| apply pow_le_pow_left₀ _ he 2 |
| positivity |
| end LogConcaveSampling.Quadrature |
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| end |
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| end |
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| end |
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| end OAI |
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