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9dbc64e | 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 | import OAI.Computability.DepthThree.Hash
import Mathlib.Data.Finset.Max
import Mathlib.Data.Fintype.EquivFin
namespace OAI
namespace DepthThreeLowerBound.BinaryHash
open scoped BigOperators
theorem exists_last_nonzero {d : ℕ} (w : Data d) (hw : w ≠ 0) :
∃ s : Fin d, w s ≠ 0 ∧ ∀ a : Fin d, s.val < a.val → w a = 0 := by
classical
have hex : ∃ a : Fin d, w a ≠ 0 := by
by_contra h
apply hw
funext a
by_contra ha
exact h ⟨a, ha⟩
obtain ⟨a, ha⟩ := hex
obtain ⟨s, hs, hmax⟩ := Finset.exists_max_image
(Finset.univ.filter fun a : Fin d => w a ≠ 0) Fin.val
⟨a, by simp [ha]⟩
refine ⟨s, (Finset.mem_filter.mp hs).2, ?_⟩
intro b hsb
by_contra hb
have hle := hmax b (by simp [hb])
omega
noncomputable def lastIndex {d : ℕ} (w : Data d) (hw : w ≠ 0) : Fin d :=
(exists_last_nonzero w hw).choose
theorem lastIndex_nonzero {d : ℕ} (w : Data d) (hw : w ≠ 0) :
w (lastIndex w hw) ≠ 0 :=
(exists_last_nonzero w hw).choose_spec.1
theorem zero_after_lastIndex {d : ℕ} (w : Data d) (hw : w ≠ 0)
(a : Fin d) (ha : (lastIndex w hw).val < a.val) : w a = 0 :=
(exists_last_nonzero w hw).choose_spec.2 a ha
theorem data_dim_pos {d : ℕ} (w : Data d) (hw : w ≠ 0) : 0 < d :=
Nat.zero_le (lastIndex w hw).val |>.trans_lt (lastIndex w hw).isLt
def Nonpivot {d r : ℕ} (w : Data d) (hw : w ≠ 0) :=
{j : Fin (d + r - 1) // ¬ ∃ i : Fin r, sumIndex i (lastIndex w hw) = j}
noncomputable instance {d r : ℕ} (w : Data d) (hw : w ≠ 0) :
Fintype (Nonpivot (r := r) w hw) := by
classical
unfold Nonpivot
infer_instance
theorem card_nonpivot {d r : ℕ} (w : Data d) (hw : w ≠ 0) :
Fintype.card (Nonpivot (r := r) w hw) = d - 1 := by
classical
have hp : Fintype.card
{j : Fin (d + r - 1) // ∃ i : Fin r, sumIndex i (lastIndex w hw) = j} = r := by
let e : Fin r ≃
{j : Fin (d + r - 1) // ∃ i : Fin r, sumIndex i (lastIndex w hw) = j} :=
Equiv.ofInjective (fun i : Fin r => sumIndex i (lastIndex w hw))
(sumIndex_left_injective (lastIndex w hw))
simpa only [Fintype.card_fin] using (Fintype.card_congr e).symm
calc
Fintype.card (Nonpivot (r := r) w hw) = Fintype.card
{j : Fin (d + r - 1) // ¬ ∃ i : Fin r, sumIndex i (lastIndex w hw) = j} :=
Fintype.card_congr (Equiv.refl _)
_ = (d + r - 1) - r := by
rw [Fintype.card_subtype_compl, hp, Fintype.card_fin]
_ = d - 1 := by
have hd := data_dim_pos w hw
omega
def triangularMap {d r : ℕ} (w : Data d) (hw : w ≠ 0) (u : Seed d r) :
Output r × (Nonpivot (r := r) w hw → F2) :=
(hashF2 u w, fun j => u j.val)
theorem triangularMap_injective {d r : ℕ} (w : Data d) (hw : w ≠ 0) :
Function.Injective (triangularMap (r := r) w hw) := by
classical
intro u v huv
have hhash : hashF2 u w = hashF2 v w := congrArg Prod.fst huv
have hfree : (fun j : Nonpivot (r := r) w hw => u j.val) =
(fun j : Nonpivot (r := r) w hw => v j.val) := congrArg Prod.snd huv
have heq : ∀ n : ℕ, ∀ j : Fin (d + r - 1), j.val = n → u j = v j := by
intro n
induction n using Nat.strong_induction_on with
| h n ih =>
intro j hj
by_cases hp : ∃ i : Fin r, sumIndex i (lastIndex w hw) = j
· obtain ⟨i, rfl⟩ := hp
have hrow : (∑ a : Fin d,
(u (sumIndex i a) - v (sumIndex i a)) * w a) = 0 := by
calc
(∑ a : Fin d, (u (sumIndex i a) - v (sumIndex i a)) * w a) =
hashF2 u w i - hashF2 v w i := by
simp only [hashF2, sub_mul, Finset.sum_sub_distrib]
_ = 0 := sub_eq_zero.mpr (congrFun hhash i)
have hsingle : (∑ a : Fin d,
(u (sumIndex i a) - v (sumIndex i a)) * w a) =
(u (sumIndex i (lastIndex w hw)) -
v (sumIndex i (lastIndex w hw))) * w (lastIndex w hw) := by
apply Finset.sum_eq_single_of_mem (lastIndex w hw) (Finset.mem_univ _)
intro a _ ha
by_cases hlt : a.val < (lastIndex w hw).val
· have hidx : (sumIndex i a).val < n := by
simp only [sumIndex_val] at hj ⊢
omega
rw [ih (sumIndex i a).val hidx (sumIndex i a) rfl, sub_self, zero_mul]
· have hne : a.val ≠ (lastIndex w hw).val := by
intro he
exact ha (Fin.ext he)
have hgt : (lastIndex w hw).val < a.val := by omega
rw [zero_after_lastIndex w hw a hgt, mul_zero]
rw [hsingle] at hrow
exact sub_eq_zero.mp ((mul_eq_zero.mp hrow).resolve_right (lastIndex_nonzero w hw))
· exact congrFun hfree ⟨j, hp⟩
exact funext fun j => heq j.val j rfl
theorem triangularMap_bijective {d r : ℕ} (w : Data d) (hw : w ≠ 0) :
Function.Bijective (triangularMap (r := r) w hw) := by
classical
apply (Fintype.bijective_iff_injective_and_card _).2
refine ⟨triangularMap_injective w hw, ?_⟩
have hd := data_dim_pos w hw
simp only [Seed, Output, Fintype.card_prod, Fintype.card_fun,
F2, BinaryAlgebra.F2, ZMod.card, Fintype.card_fin, card_nonpivot]
rw [← pow_add]
congr 1
omega
noncomputable def triangularEquiv {d r : ℕ} (w : Data d) (hw : w ≠ 0) :
Seed d r ≃ Output r × (Nonpivot (r := r) w hw → F2) :=
Equiv.ofBijective (triangularMap w hw) (triangularMap_bijective w hw)
@[simp] theorem triangularEquiv_fst {d r : ℕ} (w : Data d) (hw : w ≠ 0)
(u : Seed d r) : (triangularEquiv w hw u).1 = hashF2 u w := rfl
@[simp] theorem triangularEquiv_snd {d r : ℕ} (w : Data d) (hw : w ≠ 0)
(u : Seed d r) (j : Nonpivot (r := r) w hw) :
(triangularEquiv w hw u).2 j = u j.val := rfl
end DepthThreeLowerBound.BinaryHash
end OAI
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