| import OAI.Geometry.HeilbronnTriangle.ConditionalDigits |
| import OAI.Geometry.HeilbronnTriangle.RankOneMinors |
| import OAI.Geometry.HeilbronnTriangle.DigitUnits |
|
|
| namespace OAI |
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|
|
| namespace Problem355.ConditionalMatrix |
|
|
| open scoped BigOperators |
|
|
| abbrev Header (X : Type*) (k : ℕ) := |
| (Fin 3 → Fin k → X) × (Fin 2 → Fin k → X) |
|
|
| abbrev Draw (X : Type*) (k : ℕ) := |
| Header X k × ((Fin 2 × Fin 2) → Fin k → X) |
|
|
| def picked {X : Type*} {k : ℕ} (z : Draw X k) (a c : Fin 3) (v : Fin k) : X := |
| Fin.cases (z.1.1 c v) |
| (fun i => Fin.cases (z.1.2 i v) (fun l => z.2 (i, l) v) c) a |
|
|
| def drawEquiv {X : Type*} {k : ℕ} : Draw X k ≃ (Fin 3 → Fin 3 → Fin k → X) where |
| toFun z := picked z |
| invFun f := ((fun c v => f 0 c v, fun i v => f i.succ 0 v), |
| fun i v => f i.1.succ i.2.succ v) |
| left_inv z := by |
| apply Prod.ext |
| · apply Prod.ext <;> rfl |
| · funext i v |
| rfl |
| right_inv f := by |
| funext a c v |
| refine Fin.cases ?_ (fun i => ?_) a |
| · rfl |
| · refine Fin.cases ?_ (fun l => ?_) c <;> rfl |
|
|
| def entryValue {X : Type*} {B k : ℕ} |
| (digit : Fin 3 → Fin 3 → Fin k → X → Fin B) |
| (z : Draw X k) (a c : Fin 3) : ℕ := |
| ∑ v : Fin k, (digit a c v (picked z a c v)).val * B ^ v.val |
|
|
| def residueMatrix {X : Type*} {B k : ℕ} |
| (digit : Fin 3 → Fin 3 → Fin k → X → Fin B) |
| (z : Draw X k) (j : ℕ) : Matrix (Fin 3) (Fin 3) (ZMod (B ^ j)) := |
| fun a c => (entryValue digit z a c : ZMod (B ^ j)) |
|
|
| def arrayMatrix {X : Type*} {B k : ℕ} |
| (digit : Fin 3 → Fin 3 → Fin k → X → Fin B) |
| (f : Fin 3 → Fin 3 → Fin k → X) (j : ℕ) : |
| Matrix (Fin 3) (Fin 3) (ZMod (B ^ j)) := |
| fun a c => ((∑ v : Fin k, (digit a c v (f a c v)).val * B ^ v.val : ℕ) : ZMod (B ^ j)) |
|
|
| lemma residueMatrix_eq_arrayMatrix {X : Type*} {B k : ℕ} |
| (digit : Fin 3 → Fin 3 → Fin k → X → Fin B) (z : Draw X k) (j : ℕ) : |
| residueMatrix digit z j = arrayMatrix digit (drawEquiv z) j := rfl |
|
|
| theorem encoded_isUnit {B k j : ℕ} (hB : B.Prime) (hk : 0 < k) (hj : 0 < j) |
| (d : Fin k → Fin B) (hpos : ∀ v, 0 < (d v).val) : |
| IsUnit (((∑ v : Fin k, (d v).val * B ^ v.val : ℕ)) : ZMod (B ^ j)) := by |
| apply (ZMod.isUnit_natCast_iff_not_dvd_pow hB hj).mpr |
| intro hdiv |
| have hm := Nat.mod_eq_zero_of_dvd hdiv |
| cases k with |
| | zero => omega |
| | succ k => |
| rw [← Digits.ofDigits_ofFn_eq_sum, Nat.ofDigits_mod_eq_head!, List.ofFn_succ] at hm |
| have heq : (d 0).val % B = 0 := by simpa using hm |
| rw [Nat.mod_eq_of_lt (d 0).isLt] at heq |
| exact (Nat.ne_of_gt (hpos 0)) heq |
|
|
| theorem moment_le_two {X : Type*} [Fintype X] [Nonempty X] |
| {B k : ℕ} (hB : B.Prime) (hk : 0 < k) |
| (digit : Fin 3 → Fin 3 → Fin k → X → Fin B) |
| (hinj : ∀ a c v, Function.Injective (digit a c v)) |
| (hpos : ∀ a c v x, 0 < (digit a c v x).val) |
| (b : Draw X k → ℕ) (hb : ∀ z, b z ≤ k) |
| (hminor : ∀ j, 1 ≤ j → j ≤ k → ∀ z, j ≤ b z → |
| UnitPivot.MinorsVanish (residueMatrix digit z j)) |
| (hsmall : (B : ℝ) * (1 / (Fintype.card X : ℝ)) ^ 4 ≤ 1 / 2) : |
| (∑ z : Draw X k, (1 / (Fintype.card (Draw X k) : ℝ)) * (B : ℝ) ^ b z) ≤ 2 := by |
| classical |
| apply ConditionalDigits.conditional_divisor_moment_le_two hB.one_lt |
| (fun i v x => digit i.1.succ i.2.succ v x) |
| (fun i v => hinj i.1.succ i.2.succ v) b hb |
| · intro j hjpos hj |
| let row : Header X k → Fin 3 → ℕ := |
| fun a c => ∑ v : Fin k, (digit 0 c v (a.1 c v)).val * B ^ v.val |
| let col : Header X k → Fin 2 → ℕ := |
| fun a i => ∑ v : Fin k, (digit i.succ 0 v (a.2 i v)).val * B ^ v.val |
| have hunit (a : Header X k) : IsUnit ((row a 0 : ℕ) : ZMod (B ^ j)) := |
| encoded_isUnit hB hk hjpos (fun v => digit 0 0 v (a.1 0 v)) |
| (fun v => hpos 0 0 v (a.1 0 v)) |
| let pivot : Header X k → (ZMod (B ^ j))ˣ := fun a => (hunit a).unit |
| refine ⟨pivot, fun a i => |
| (col a i.1 : ZMod (B ^ j)) * (row a i.2.succ : ZMod (B ^ j)), ?_⟩ |
| intro z hz i |
| have hpivot : (pivot z.1 : ZMod (B ^ j)) = (row z.1 0 : ZMod (B ^ j)) := |
| (hunit z.1).unit_spec |
| rw [hpivot] |
| have h := hminor j hjpos hj z hz 0 i.1.succ 0 i.2.succ |
| change (row z.1 0 : ZMod (B ^ j)) * |
| ((∑ v : Fin k, (digit i.1.succ i.2.succ v (z.2 i v)).val * B ^ v.val : ℕ) : ZMod (B ^ j)) = |
| (row z.1 i.2.succ : ZMod (B ^ j)) * (col z.1 i.1 : ZMod (B ^ j)) at h |
| exact h.trans (mul_comm _ _) |
| · simpa using hsmall |
|
|
| theorem array_moment_le_two {X : Type*} [Fintype X] [Nonempty X] |
| {B k : ℕ} (hB : B.Prime) (hk : 0 < k) |
| (digit : Fin 3 → Fin 3 → Fin k → X → Fin B) |
| (hinj : ∀ a c v, Function.Injective (digit a c v)) |
| (hpos : ∀ a c v x, 0 < (digit a c v x).val) |
| (b : (Fin 3 → Fin 3 → Fin k → X) → ℕ) (hb : ∀ f, b f ≤ k) |
| (hminor : ∀ j, 1 ≤ j → j ≤ k → ∀ f, j ≤ b f → |
| UnitPivot.MinorsVanish (arrayMatrix digit f j)) |
| (hsmall : (B : ℝ) * (1 / (Fintype.card X : ℝ)) ^ 4 ≤ 1 / 2) : |
| (∑ f : Fin 3 → Fin 3 → Fin k → X, |
| (1 / (Fintype.card (Fin 3 → Fin 3 → Fin k → X) : ℝ)) * (B : ℝ) ^ b f) ≤ 2 := by |
| have h := moment_le_two hB hk digit hinj hpos (fun z => b (drawEquiv z)) |
| (fun z => hb (drawEquiv z)) |
| (fun j hj hk z hz => hminor j hj hk (drawEquiv z) hz) hsmall |
| have hcard := Fintype.card_congr (drawEquiv (X := X) (k := k)) |
| rw [hcard] at h |
| have hsum := (drawEquiv (X := X) (k := k)).sum_comp |
| (fun f => (1 / (Fintype.card (Fin 3 → Fin 3 → Fin k → X) : ℝ)) * (B : ℝ) ^ b f) |
| rw [← hsum] |
| exact h |
|
|
| noncomputable def prescribedDigit {r m B k : ℕ} [NeZero r] (hbound : r * m < B) |
| (residue : Fin 3 → Fin 3 → Fin k → ZMod r) |
| (a c : Fin 3) (v : Fin k) (x : Fin m) : Fin B := |
| ⟨(DigitUnits.digitFiberEquiv (residue a c v) m x).val, |
| lt_of_le_of_lt (DigitUnits.digitFiberEquiv_bounds (residue a c v) m x).2.1 hbound⟩ |
|
|
| lemma prescribedDigit_injective {r m B k : ℕ} [NeZero r] (hbound : r * m < B) |
| (residue : Fin 3 → Fin 3 → Fin k → ZMod r) (a c : Fin 3) (v : Fin k) : |
| Function.Injective (prescribedDigit hbound residue a c v) := by |
| intro x y hxy |
| apply (DigitUnits.digitFiberEquiv (residue a c v) m).injective |
| apply Subtype.ext |
| exact congrArg Fin.val hxy |
|
|
| lemma prescribedDigit_positive {r m B k : ℕ} [NeZero r] (hbound : r * m < B) |
| (residue : Fin 3 → Fin 3 → Fin k → ZMod r) (a c : Fin 3) (v : Fin k) (x : Fin m) : |
| 0 < (prescribedDigit hbound residue a c v x).val := |
| (DigitUnits.digitFiberEquiv_bounds (residue a c v) m x).1 |
|
|
| lemma prescribedDigit_residue {r m B k : ℕ} [NeZero r] (hbound : r * m < B) |
| (residue : Fin 3 → Fin 3 → Fin k → ZMod r) (a c : Fin 3) (v : Fin k) (x : Fin m) : |
| ((prescribedDigit hbound residue a c v x).val : ZMod r) = residue a c v := |
| (DigitUnits.digitFiberEquiv_bounds (residue a c v) m x).2.2 |
|
|
| theorem prescribed_moment_le_two {r m B k : ℕ} [NeZero r] [NeZero m] |
| (hB : B.Prime) (hk : 0 < k) (hbound : r * m < B) |
| (residue : Fin 3 → Fin 3 → Fin k → ZMod r) |
| (b : Draw (Fin m) k → ℕ) (hb : ∀ z, b z ≤ k) |
| (hminor : ∀ j, 1 ≤ j → j ≤ k → ∀ z, j ≤ b z → |
| UnitPivot.MinorsVanish (residueMatrix (prescribedDigit hbound residue) z j)) |
| (hsmall : (B : ℝ) * (1 / (m : ℝ)) ^ 4 ≤ 1 / 2) : |
| (∑ z : Draw (Fin m) k, (1 / (Fintype.card (Draw (Fin m) k) : ℝ)) * |
| (B : ℝ) ^ b z) ≤ 2 := by |
| exact moment_le_two hB hk (prescribedDigit hbound residue) |
| (prescribedDigit_injective hbound residue) (prescribedDigit_positive hbound residue) |
| b hb hminor (by simpa using hsmall) |
|
|
| end Problem355.ConditionalMatrix |
|
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| end OAI |
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|
