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e320e82 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 | 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
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