| import OAI.Combinatorics.PerfectMatching.DyadicEnvelope |
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| namespace OAI |
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| section |
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| namespace BinaryMatching.Algorithm |
| noncomputable section |
| open Finset |
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| 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 |
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| 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) |
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| 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 |
|
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| 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⟩ |
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| def phi (t : ℝ) : ℝ := Real.negMulLog t / Real.log 2 |
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| 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 |
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| lemma log_dyadic (j : ℕ) : Real.log (dyadic j)=-(j:ℝ)*Real.log 2 := by |
| simp [dyadic,Real.log_pow,Real.log_inv] |
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| 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'] |
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| lemma phi_zero : phi 0=0 := by simp [phi] |
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| 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 |
|
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| 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 |
|
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| lemma phi_mul (u v : ℝ) : phi (u*v)=v*phi u+u*phi v := by |
| simp only [phi,Real.negMulLog_mul]; ring |
|
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| 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) |
|
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| 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 |
|
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| 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] |
|
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| 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] |
|
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| 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 |
|
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| end OAI |
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