| import OAI.Combinatorics.FiniteCircle.OffsetRecovery |
|
|
| namespace OAI |
|
|
| namespace FiniteCircle |
|
|
| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
|
|
| theorem exists_disjoint_pair_subfamily {ι J : Type*} [DecidableEq ι] [DecidableEq J] |
| (f g : ι → J) (hf : Function.Injective f) (hg : Function.Injective g) |
| (I : Finset ι) (hne : ∀ i ∈ I, f i ≠ g i) : |
| ∃ S ⊆ I, I.card ≤ 3*S.card ∧ ∀ i ∈ S, ∀ j ∈ S, f i ≠ g j := by |
| induction I using Finset.strongInductionOn |
| rename_i I ih |
| by_cases he : I.Nonempty |
| · obtain ⟨k, hk⟩ := he |
| let N := I.filter (fun i => i = k ∨ f i = g k ∨ g i = f k) |
| let T := I \ N |
| have hkN : k ∈ N := Finset.mem_filter.mpr ⟨hk, Or.inl rfl⟩ |
| have hN : N ⊆ I := Finset.filter_subset _ _ |
| have hT : T ⊂ I := Finset.ssubset_iff_subset_ne.mpr ⟨Finset.sdiff_subset, by |
| intro heq |
| have hmem : k ∈ T := heq.symm ▸ hk |
| exact (Finset.mem_sdiff.mp hmem).2 hkN⟩ |
| have hca : (I.filter (fun i => f i = g k)).card ≤ 1 := by |
| apply Finset.card_le_one.mpr |
| intro i hi j hj |
| exact hf ((Finset.mem_filter.mp hi).2.trans (Finset.mem_filter.mp hj).2.symm) |
| have hcb : (I.filter (fun i => g i = f k)).card ≤ 1 := by |
| apply Finset.card_le_one.mpr |
| intro i hi j hj |
| exact hg ((Finset.mem_filter.mp hi).2.trans (Finset.mem_filter.mp hj).2.symm) |
| have hsub : N ⊆ {k} ∪ I.filter (fun i => f i = g k) ∪ I.filter (fun i => g i = f k) := by |
| intro i hi |
| obtain ⟨hi, hn⟩ := Finset.mem_filter.mp hi |
| rcases hn with rfl | ha | hb |
| · simp |
| · exact Finset.mem_union_left _ (Finset.mem_union_right _ (Finset.mem_filter.mpr ⟨hi, ha⟩)) |
| · exact Finset.mem_union_right _ (Finset.mem_filter.mpr ⟨hi, hb⟩) |
| have hNc : N.card ≤ 3 := by |
| have h1 := Finset.card_le_card hsub |
| have h2 := Finset.card_union_le ({k} ∪ I.filter (fun i => f i = g k)) (I.filter (fun i => g i = f k)) |
| have h3 := Finset.card_union_le ({k} : Finset ι) (I.filter (fun i => f i = g k)) |
| simp only [Finset.card_singleton] at h3 |
| omega |
| obtain ⟨S, hST, hc, hcross⟩ := ih T hT (fun i hi => hne i (Finset.sdiff_subset hi)) |
| have hkS : k ∉ S := fun hks => (Finset.mem_sdiff.mp (hST hks)).2 hkN |
| refine ⟨insert k S, Finset.insert_subset_iff.mpr ⟨hk, hST.trans Finset.sdiff_subset⟩, ?_, ?_⟩ |
| · rw [Finset.card_insert_of_notMem hkS] |
| have hIc : T.card = I.card - N.card := Finset.card_sdiff_of_subset hN |
| have hNi := Finset.card_le_card hN |
| omega |
| · intro i hi j hj |
| rcases Finset.mem_insert.mp hi with hi | hi |
| · subst i |
| rcases Finset.mem_insert.mp hj with hj | hj |
| · subst j |
| exact hne k hk |
| · intro hh |
| have hjI := (Finset.mem_sdiff.mp (hST hj)).1 |
| have hjN := (Finset.mem_sdiff.mp (hST hj)).2 |
| exact hjN (Finset.mem_filter.mpr ⟨hjI, Or.inr (Or.inr hh.symm)⟩) |
| · rcases Finset.mem_insert.mp hj with hj | hj |
| · subst j |
| intro hh |
| have hiI := (Finset.mem_sdiff.mp (hST hi)).1 |
| have hiN := (Finset.mem_sdiff.mp (hST hi)).2 |
| exact hiN (Finset.mem_filter.mpr ⟨hiI, Or.inr (Or.inl hh)⟩) |
| · exact hcross i hi j hj |
| · refine ⟨∅, Finset.empty_subset _, ?_, by simp⟩ |
| simp [Finset.not_nonempty_iff_eq_empty.mp he] |
|
|
| theorem card_bit_equalities_le {ι J : Type*} [Fintype J] [DecidableEq J] [DecidableEq ι] |
| (I : Finset ι) (f g : ι → J) (hg : Function.Injective g) |
| (hcross : ∀ i ∈ I, ∀ j ∈ I, f i ≠ g j) : |
| (Finset.univ.filter (fun ω : J → Bool => ∀ i ∈ I, ω (f i) = ω (g i))).card ≤ |
| 2 ^ (Fintype.card J - I.card) := by |
| classical |
| let T := I.image g |
| let π : (J → Bool) → (↥(Tᶜ : Finset J) → Bool) := fun ω j => ω j.val |
| have hinj : Set.InjOn π (↑(Finset.univ.filter (fun ω : J → Bool => ∀ i ∈ I, ω (f i) = ω (g i))) : Set (J → Bool)) := by |
| intro ω hω η hη he |
| have hω := (Finset.mem_filter.mp hω).2 |
| have hη := (Finset.mem_filter.mp hη).2 |
| funext j |
| by_cases hj : j ∈ T |
| · obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hj |
| have hf : f i ∈ Tᶜ := by |
| apply Finset.mem_compl.mpr |
| intro hh |
| obtain ⟨k, hk, heq⟩ := Finset.mem_image.mp hh |
| exact hcross i hi k hk heq.symm |
| have heq : ω (f i) = η (f i) := congrFun he ⟨f i, hf⟩ |
| exact (hω i hi).symm.trans (heq.trans (hη i hi)) |
| · exact congrFun he ⟨j, Finset.mem_compl.mpr hj⟩ |
| have hc := Finset.card_le_card_of_injOn π (fun _ _ => Finset.mem_univ _) hinj |
| simpa only [Finset.card_univ, Fintype.card_fun, Fintype.card_bool, Fintype.card_coe, |
| Finset.card_compl, Finset.card_image_of_injective I hg, T] using hc |
|
|
| theorem card_matching_equalities_le {ι J : Type*} [Fintype J] [DecidableEq J] [DecidableEq ι] |
| (I : Finset ι) (f g : ι → J) (hf : Function.Injective f) (hg : Function.Injective g) |
| (hne : ∀ i ∈ I, f i ≠ g i) (w : ℕ) (hw : w ≤ 16*I.card) : |
| (Finset.univ.filter (fun ω : J → Bool => ∀ i ∈ I, ω (f i) = ω (g i))).card ≤ |
| 2 ^ (Fintype.card J - w/64) := by |
| classical |
| obtain ⟨S, hSI, hcard, hcross⟩ := exists_disjoint_pair_subfamily f g hf hg I hne |
| have hsub : (Finset.univ.filter (fun ω : J → Bool => ∀ i ∈ I, ω (f i) = ω (g i))) ⊆ |
| Finset.univ.filter (fun ω : J → Bool => ∀ i ∈ S, ω (f i) = ω (g i)) := by |
| intro ω hω |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, fun i hi => (Finset.mem_filter.mp hω).2 i (hSI hi)⟩ |
| have hh : w/64 ≤ S.card := by omega |
| exact (Finset.card_le_card hsub).trans ((card_bit_equalities_le S f g hg hcross).trans |
| (Nat.pow_le_pow_right (by omega) (Nat.sub_le_sub_left hh _))) |
|
|
| theorem card_small_subsets_le (w : ℕ) : |
| ((Finset.univ.filter (fun S : Finset (Fin w) => (S.card : ℝ) ≤ (w : ℝ)/1048576)).card : ℝ) ≤ |
| Real.exp ((w : ℝ)/512) := by |
| classical |
| let K := Finset.univ.filter (fun S : Finset (Fin w) => (S.card : ℝ) ≤ (w : ℝ)/1048576) |
| let z : ℝ := 1/1024 |
| have hz : 0 < z := by norm_num [z] |
| have hlog : -1024 ≤ Real.log z := by |
| have h := Real.log_le_sub_one_of_pos (x := 1024) (by norm_num) |
| simp only [z, one_div, Real.log_inv] |
| linarith |
| have hlow (S : Finset (Fin w)) (hS : S ∈ K) : Real.exp (-(w : ℝ)/1024) ≤ z^S.card := by |
| have hs := (Finset.mem_filter.mp hS).2 |
| have hmul := mul_le_mul_of_nonneg_left hlog (Nat.cast_nonneg S.card) |
| have he : -(w : ℝ)/1024 ≤ (S.card : ℝ)*Real.log z := by nlinarith |
| have h := Real.exp_le_exp.mpr he |
| simpa only [Real.exp_nat_mul, Real.exp_log hz] using h |
| have htotal : ∑ S : Finset (Fin w), z^S.card = (z+1)^w := by |
| simpa using Finset.sum_pow_mul_eq_add_pow z (1 : ℝ) (Finset.univ : Finset (Fin w)) |
| have hsum : (K.card : ℝ) * Real.exp (-(w : ℝ)/1024) ≤ (z+1)^w := calc |
| _ = ∑ _S ∈ K, Real.exp (-(w : ℝ)/1024) := by simp |
| _ ≤ ∑ S ∈ K, z^S.card := Finset.sum_le_sum hlow |
| _ ≤ ∑ S : Finset (Fin w), z^S.card := Finset.sum_le_sum_of_subset_of_nonneg |
| (Finset.subset_univ _) (fun _ _ _ => pow_nonneg hz.le _) |
| _ = _ := htotal |
| have hupper : (z+1)^w ≤ Real.exp ((w : ℝ)/1024) := by |
| have hl := Real.log_le_sub_one_of_pos (x := z+1) (by positivity) |
| have hm := mul_le_mul_of_nonneg_left hl (Nat.cast_nonneg w) |
| have hh : (w : ℝ)*Real.log (z+1) ≤ (w : ℝ)/1024 := by dsimp [z] at *; nlinarith |
| have hh := Real.exp_le_exp.mpr hh |
| simpa only [Real.exp_nat_mul, Real.exp_log (show 0 < z+1 by positivity)] using hh |
| calc |
| (K.card : ℝ) ≤ (z+1)^w / Real.exp (-(w : ℝ)/1024) := (le_div_iff₀ (Real.exp_pos _)).mpr hsum |
| _ ≤ Real.exp ((w : ℝ)/1024) / Real.exp (-(w : ℝ)/1024) := div_le_div_of_nonneg_right hupper (Real.exp_pos _).le |
| _ = Real.exp ((w : ℝ)/512) := by rw [← Real.exp_sub]; congr 1; ring |
| |
| end FiniteCircle |
| |
| end OAI |
| |
