File size: 3,587 Bytes
f4b3ad1 | 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 | import Mathlib.Algebra.Field.ZMod
import OAI.NumberTheory.Ostmann.Construction.TransferHistoryUniqueness
import OAI.NumberTheory.Ostmann.Characters.TreeCoordinates
namespace OAI
/-! # The actual integer pivot gives the spectator's rational reconstruction -/
namespace Ostmann
theorem ValidTransferNode.spectator_composite_reconstruction {State : Type*}
{sys : TransferHistorySystem State} {σ : State} {s v w : ℤ} {P : ℕ}
(h : ValidTransferNode sys σ s v w P) {q : ℕ} [Fact q.Prime]
(u : (ZMod q)ˣ) (hs : (s : ZMod q) ≠ 0) :
reconstructedEntry (s : ZMod q) (v : ZMod q) (w : ZMod q) u
(sys.leftProduct σ : ZMod q) (sys.rightProduct σ : ZMod q) = (P : ZMod q) / u := by
have hr : (v : ZMod q) * sys.rightProduct σ - (w : ZMod q) * sys.leftProduct σ =
(s : ZMod q) * P := by
have hz := congrArg (fun z : ℤ => (z : ZMod q)) h.relation
simpa only [Int.cast_sub, Int.cast_mul, Int.cast_natCast] using hz
unfold reconstructedEntry
rw [hr]
field_simp
theorem ValidTransferNode.spectator_composite_zero_iff {State : Type*}
{sys : TransferHistorySystem State} {σ : State} {s v w : ℤ} {P : ℕ}
(h : ValidTransferNode sys σ s v w P) {q : ℕ} [Fact q.Prime]
(u : (ZMod q)ˣ) (hs : (s : ZMod q) ≠ 0) :
reconstructedEntry (s : ZMod q) (v : ZMod q) (w : ZMod q) u
(sys.leftProduct σ : ZMod q) (sys.rightProduct σ : ZMod q) = 0 ↔ q ∣ P := by
rw [h.spectator_composite_reconstruction u hs, div_eq_zero_iff,
or_iff_left (Units.ne_zero u), ZMod.natCast_eq_zero_iff]
theorem spectator_frequency_ne_zero {q : ℕ} [Fact q.Prime] (s : ℤ)
(hs : s ≠ 0) (hsmall : s.natAbs < q) : (s : ZMod q) ≠ 0 := by
intro hz
have hd : q ∣ s.natAbs := Int.natCast_dvd.mp ((ZMod.intCast_zmod_eq_zero_iff_dvd s q).mp hz)
exact (not_le_of_gt hsmall) (Nat.le_of_dvd (Int.natAbs_pos.mpr hs) hd)
theorem ValidTransferNode.spectator_reconstruction {State : Type*}
{sys : TransferHistorySystem State} {σ : State} {s v w : ℤ} {P : ℕ}
(h : ValidTransferNode sys σ s v w P) {q : ℕ} [Fact q.Prime]
(u p : ℕ) (hP : P = u * p) (hs : (s : ZMod q) ≠ 0) (hu : (u : ZMod q) ≠ 0) :
reconstructedEntry (s : ZMod q) (v : ZMod q) (w : ZMod q) (u : ZMod q)
(sys.leftProduct σ : ZMod q) (sys.rightProduct σ : ZMod q) = (p : ZMod q) := by
have hr : (v : ZMod q) * sys.rightProduct σ - (w : ZMod q) * sys.leftProduct σ =
(s : ZMod q) * (u * p) := by
have hi := h.relation
rw [hP] at hi
have hz := congrArg (fun z : ℤ => (z : ZMod q)) hi
simpa only [Int.cast_sub, Int.cast_mul, Int.cast_natCast, Nat.cast_mul] using hz
unfold reconstructedEntry
apply (div_eq_iff (mul_ne_zero hs hu)).mpr
rw [hr]
ring
theorem ValidTransferNode.spectator_zero_iff {State : Type*}
{sys : TransferHistorySystem State} {σ : State} {s v w : ℤ} {P : ℕ}
(h : ValidTransferNode sys σ s v w P) {q : ℕ} [Fact q.Prime]
(u p : ℕ) (hP : P = u * p) (hs : (s : ZMod q) ≠ 0) (hu : (u : ZMod q) ≠ 0) :
reconstructedEntry (s : ZMod q) (v : ZMod q) (w : ZMod q) (u : ZMod q)
(sys.leftProduct σ : ZMod q) (sys.rightProduct σ : ZMod q) = 0 ↔ q ∣ p := by
rw [h.spectator_reconstruction u p hP hs hu, ZMod.natCast_eq_zero_iff]
theorem spectator_child_argument {q : ℕ} [Fact q.Prime] (s : ℤ) (D H : ZMod q)
(u p P : ℕ) (hP : P = u * p) :
childArgument (s : ZMod q) D (u : ZMod q) H (p : ZMod q) =
(s : ZMod q) / (D * (P : ZMod q) * H) := by
rw [hP, Nat.cast_mul]
unfold childArgument
congr 1
ring
end Ostmann
end OAI
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