File size: 3,031 Bytes
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 | import OAI.Computability.DepthThree.AlgebraBits
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
import Lean.Elab.Tactic.Omega
namespace OAI
namespace DepthThreeLowerBound.BinaryHash
open scoped BigOperators
abbrev F2 := BinaryAlgebra.F2
abbrev Data (d : ℕ) := Fin d → F2
abbrev Seed (d r : ℕ) := Fin (d + r - 1) → F2
abbrev Output (r : ℕ) := Fin r → F2
def sumIndex {d r : ℕ} (i : Fin r) (s : Fin d) : Fin (d + r - 1) :=
⟨i.val + s.val, by have := i.isLt; have := s.isLt; omega⟩
@[simp] theorem sumIndex_val {d r : ℕ} (i : Fin r) (s : Fin d) :
(sumIndex i s).val = i.val + s.val := rfl
theorem sumIndex_left_injective {d r : ℕ} (s : Fin d) :
Function.Injective (fun i : Fin r => sumIndex i s) := by
intro i j h
apply Fin.ext
have hval := congrArg Fin.val h
simpa only [sumIndex_val, Nat.add_right_cancel_iff] using hval
def hashF2 {d r : ℕ} (u : Seed d r) (x : Data d) : Output r :=
fun i => ∑ s : Fin d, u (sumIndex i s) * x s
@[simp] theorem hashF2_apply {d r : ℕ} (u : Seed d r) (x : Data d) (i : Fin r) :
hashF2 u x i = ∑ s : Fin d, u (sumIndex i s) * x s := rfl
@[simp] theorem hashF2_zero_data {d r : ℕ} (u : Seed d r) :
hashF2 u 0 = 0 := by
funext i
simp [hashF2]
@[simp] theorem hashF2_zero_seed {d r : ℕ} (x : Data d) :
hashF2 (0 : Seed d r) x = 0 := by
funext i
simp [hashF2]
theorem hashF2_add_data {d r : ℕ} (u : Seed d r) (x y : Data d) :
hashF2 u (x + y) = hashF2 u x + hashF2 u y := by
funext i
simp [hashF2, mul_add, Finset.sum_add_distrib]
theorem hashF2_sub_data {d r : ℕ} (u : Seed d r) (x y : Data d) :
hashF2 u (x - y) = hashF2 u x - hashF2 u y := by
funext i
simp [hashF2, mul_sub, Finset.sum_sub_distrib]
theorem hashF2_add_seed {d r : ℕ} (u v : Seed d r) (x : Data d) :
hashF2 (u + v) x = hashF2 u x + hashF2 v x := by
funext i
simp [hashF2, add_mul, Finset.sum_add_distrib]
theorem hashF2_sub_seed {d r : ℕ} (u v : Seed d r) (x : Data d) :
hashF2 (u - v) x = hashF2 u x - hashF2 v x := by
funext i
simp [hashF2, sub_mul, Finset.sum_sub_distrib]
def hashBool {d r : ℕ} (u : Fin (d + r - 1) → Bool) (x : Fin d → Bool) :
Fin r → Bool :=
fun i => BinaryAlgebra.fieldBit
(hashF2 (fun j => BinaryAlgebra.bitValue (u j))
(fun s => BinaryAlgebra.bitValue (x s)) i)
@[simp] theorem bitValue_hashBool {d r : ℕ}
(u : Fin (d + r - 1) → Bool) (x : Fin d → Bool) (i : Fin r) :
BinaryAlgebra.bitValue (hashBool u x i) =
hashF2 (fun j => BinaryAlgebra.bitValue (u j))
(fun s => BinaryAlgebra.bitValue (x s)) i := by
simp [hashBool]
theorem bitsEquivF2_hashBool {d r : ℕ}
(u : Fin (d + r - 1) → Bool) (x : Fin d → Bool) :
BinaryAlgebra.bitsEquivF2 (Fin r) (hashBool u x) =
hashF2 (BinaryAlgebra.bitsEquivF2 (Fin (d + r - 1)) u)
(BinaryAlgebra.bitsEquivF2 (Fin d) x) := by
funext i
exact bitValue_hashBool u x i
end DepthThreeLowerBound.BinaryHash
end OAI
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