| 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 |
|
|
