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e5a97c6 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 | import OAI.Combinatorics.PerfectMatching.DyadicEnvelope
namespace OAI
section
namespace BinaryMatching.Algorithm
noncomputable section
open Finset
lemma concave_affine_chord {S : Set ℝ} {f : ℝ → ℝ} (hf : ConcaveOn ℝ S f)
{l u t p c : ℝ} (hl : l∈S) (hu : u∈S) (hlu : l<u) (ht : t∈Set.Icc l u)
(hfl : f l=p*l+c) (hfu : f u=p*u+c) : p*t+c≤f t := by
let b := (t-l)/(u-l)
have hd : 0<u-l := sub_pos.mpr hlu
have hb : 0≤b := div_nonneg (sub_nonneg.mpr ht.1) hd.le
have hb1 : b≤1 := (div_le_one hd).2 (by linarith [ht.2])
have he : (1-b)*l+b*u=t := by dsimp [b]; field_simp; ring
have hc := hf.2 hl hu (sub_nonneg.mpr hb1) hb (sub_add_cancel 1 b)
simp only [smul_eq_mul] at hc
rw [he,hfl,hfu] at hc
calc
p*t+c=(1-b)*(p*l+c)+b*(p*u+c) := by rw [←he]; ring
_ ≤ f t := hc
lemma psiReal_on_first (k : ℕ) {t : ℝ} (ht : t∈Set.Icc 0 (dyadic k)) :
psiReal k t=(k:ℝ)*t := by
apply le_antisymm
· simpa only [first_eval] using psiReal_le_piece k t _ (first_mem_pieces k)
· simpa only [add_zero] using concave_affine_chord (psiReal_concave k)
(Set.mem_univ 0) (Set.mem_univ (dyadic k)) (dyadic_pos k) ht
(p:=(k:ℝ)) (c:=0) (by simp [psiReal_zero]) (by rw [psiReal_knot k k le_rfl]; ring)
lemma psiReal_on_chord (k j : ℕ) (hj : j<k) {t : ℝ}
(ht : t∈Set.Icc (dyadic (j+1)) (dyadic j)) :
psiReal k t=((j:ℝ)-1)*t+dyadic j := by
apply le_antisymm
· simpa only [chord_eval] using psiReal_le_piece k t _ (chord_mem_pieces k j hj)
· apply concave_affine_chord (psiReal_concave k) (Set.mem_univ _) (Set.mem_univ _)
(show dyadic (j+1)<dyadic j by rw [dyadic_succ]; linarith [dyadic_pos j]) ht
· rw [psiReal_knot k (j+1) hj,dyadic_succ,Nat.cast_add,Nat.cast_one]; ring
· rw [psiReal_knot k j hj.le]; ring
lemma dyadic_cover (k : ℕ) {t : ℝ} (ht : t∈Set.Icc 0 1) :
t≤dyadic k ∨ ∃ j<k,t∈Set.Icc (dyadic (j+1)) (dyadic j) := by
induction k with
| zero => exact Or.inl (by simpa only [dyadic_zero] using ht.2)
| succ k ih =>
by_cases h : t≤dyadic (k+1)
· exact Or.inl h
· right
rcases ih with h'|⟨j,hj,hjt⟩
· exact ⟨k,Nat.lt_succ_self k,le_of_not_ge h,h'⟩
· exact ⟨j,Nat.lt_succ_of_lt hj,hjt⟩
def phi (t : ℝ) : ℝ := Real.negMulLog t / Real.log 2
lemma log_two_pos : 0<Real.log 2 := Real.log_pos (by norm_num)
lemma half_le_log_two : (1/2:ℝ)≤Real.log 2 := by
have h := Real.log_le_sub_one_of_pos (show (0:ℝ)<2⁻¹ by norm_num)
rw [Real.log_inv] at h
norm_num at h
linarith
lemma log_dyadic (j : ℕ) : Real.log (dyadic j)=-(j:ℝ)*Real.log 2 := by
simp [dyadic,Real.log_pow,Real.log_inv]
lemma phi_dyadic (j : ℕ) : phi (dyadic j)=(j:ℝ)*dyadic j := by
simp only [phi,Real.negMulLog_def,log_dyadic]
field_simp [log_two_pos.ne']
lemma phi_zero : phi 0=0 := by simp [phi]
lemma phi_concave : ConcaveOn ℝ (Set.Ici 0) phi := by
refine ⟨convex_Ici 0,?_⟩
intro x hx y hy a b ha hb hab
have hh := Real.concaveOn_negMulLog.2 hx hy ha hb hab
simp only [smul_eq_mul] at hh ⊢
have hd := div_le_div_of_nonneg_right hh log_two_pos.le
simpa only [phi,add_div,mul_div_assoc] using hd
lemma phi_le_one {t : ℝ} (ht : t∈Set.Icc 0 1) : phi t≤1 := by
have hp : (fun b : Bool => if b then t else 1-t) ∈ MatchingEntropy.probabilitySimplex Bool := by
constructor
· intro b; cases b <;> simp [ht.1,sub_nonneg.mpr ht.2]
· simp
have hh := MatchingEntropy.entropy_le_log_card hp
simp only [MatchingEntropy.entropy,Fintype.sum_bool,Fintype.card_bool,Bool.false_eq_true,
↓reduceIte,Nat.cast_ofNat] at hh
have hn := Real.negMulLog_nonneg (sub_nonneg.mpr ht.2) (by linarith [ht.1] : 1-t≤1)
unfold phi
apply (div_le_one log_two_pos).2
linarith
lemma phi_mul (u v : ℝ) : phi (u*v)=v*phi u+u*phi v := by
simp only [phi,Real.negMulLog_mul]; ring
lemma phi_lower_first (k : ℕ) {t : ℝ} (ht : t∈Set.Icc 0 (dyadic k)) :
(k:ℝ)*t ≤ phi t := by
simpa only [add_zero] using concave_affine_chord phi_concave
(Set.mem_Ici.mpr le_rfl) (Set.mem_Ici.mpr (dyadic_nonneg k)) (dyadic_pos k) ht
(p:=(k:ℝ)) (c:=0) (by simp [phi_zero]) (by rw [phi_dyadic]; ring)
lemma phi_lower_chord (j : ℕ) {t : ℝ} (ht : t∈Set.Icc (dyadic (j+1)) (dyadic j)) :
((j:ℝ)-1)*t+dyadic j≤phi t := by
apply concave_affine_chord phi_concave (Set.mem_Ici.mpr (dyadic_nonneg _))
(Set.mem_Ici.mpr (dyadic_nonneg _))
(show dyadic (j+1)<dyadic j by rw [dyadic_succ]; linarith [dyadic_pos j]) ht
· rw [phi_dyadic,dyadic_succ,Nat.cast_add,Nat.cast_one]; ring
· rw [phi_dyadic]; ring
lemma phi_upper_chord (j : ℕ) {t : ℝ} (ht : t∈Set.Icc (dyadic (j+1)) (dyadic j)) :
phi t≤((j:ℝ)+1)*t := by
have ht0 : 0<t := (dyadic_pos _).trans_le ht.1
have hh := Real.log_le_log (dyadic_pos (j+1)) ht.1
rw [log_dyadic,Nat.cast_add,Nat.cast_one] at hh
unfold phi
apply (div_le_iff₀ log_two_pos).2
rw [Real.negMulLog_def]
nlinarith [mul_le_mul_of_nonneg_left hh ht0.le]
lemma phi_error_first (k : ℕ) {t : ℝ} (ht : t∈Set.Icc 0 (dyadic k)) :
phi t-(k:ℝ)*t≤dyadic k := by
let r := t/dyadic k
have hr : r∈Set.Icc (0:ℝ) 1 :=
⟨div_nonneg ht.1 (dyadic_nonneg k),(div_le_one (dyadic_pos k)).2 ht.2⟩
have he : dyadic k*r=t := by dsimp [r]; field_simp [ne_of_gt (dyadic_pos k)]
have hh := phi_mul (dyadic k) r
rw [he,phi_dyadic] at hh
have hu := mul_le_mul_of_nonneg_left (phi_le_one hr) (dyadic_nonneg k)
nlinarith [hh,he]
theorem dyadic_entropy_error (k : ℕ) {t : ℝ} (ht : t∈Set.Icc 0 1) :
0≤phi t-psiReal k t ∧ phi t-psiReal k t≤t+dyadic k := by
rcases dyadic_cover k ht with h|⟨j,hj,hjt⟩
· rw [psiReal_on_first k ⟨ht.1,h⟩]
exact ⟨sub_nonneg.mpr (phi_lower_first k ⟨ht.1,h⟩),
(phi_error_first k ⟨ht.1,h⟩).trans (by linarith [ht.1])⟩
· rw [psiReal_on_chord k j hj hjt]
have hlo := phi_lower_chord j hjt
have hhi := phi_upper_chord j hjt
have hd := dyadic_nonneg k
constructor <;> linarith [hjt.2]
end
end BinaryMatching.Algorithm
end
end OAI
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