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cdec422 | 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 | import Mathlib.Data.Int.Basic
import Mathlib.Tactic
namespace OAI
/-!+# The square weights in auxiliary separation
Sections 4.1–4.3 use the phase `u - v + 2 * (g - h)` to select monomials.
On a selected monomial, the three integer variable weights add to `(g - h)^2`.
Thus the surviving monomials have nonnegative total degree, although individual
variable weights may be negative. Their degree is zero precisely when the guess
`g` matches the sector `h` and the auxiliary labels agree.
-/
namespace MatrixMultiplication.AuxiliarySeparation
/-- The exponent occurring in the finite Fourier projection. -/
def phase (g h u v : ℤ) : ℤ := u - v + 2 * (g - h)
/-- Weight on the first tensor leg, which carries the guessed sector label. -/
def firstWeight (g : ℤ) : ℤ := g ^ 2
/-- Weight on the second tensor leg. -/
def secondWeight (h u : ℤ) : ℤ := h * u - h ^ 2
/-- Weight on the third tensor leg. -/
def thirdWeight (h v : ℤ) : ℤ := -(h * v)
/-- Total weight of a target monomial after the three independent substitutions. -/
def totalWeight (g h u v : ℤ) : ℤ :=
firstWeight g + secondWeight h u + thirdWeight h v
theorem totalWeight_eq_square_add_phase (g h u v : ℤ) :
totalWeight g h u v = (g - h) ^ 2 + h * phase g h u v := by
unfold totalWeight firstWeight secondWeight thirdWeight phase
ring
/-- The projection equation converts the total weight into a square. -/
theorem totalWeight_eq_sq_of_phase_eq_zero {g h u v : ℤ}
(hphase : phase g h u v = 0) :
totalWeight g h u v = (g - h) ^ 2 := by
rw [totalWeight_eq_square_add_phase, hphase, mul_zero, add_zero]
theorem totalWeight_nonneg_of_phase_eq_zero {g h u v : ℤ}
(hphase : phase g h u v = 0) : 0 ≤ totalWeight g h u v := by
rw [totalWeight_eq_sq_of_phase_eq_zero hphase]
exact sq_nonneg (g - h)
/-- On the Fourier support, precisely the correctly matched terms have weight zero. -/
theorem totalWeight_eq_zero_iff_of_phase_eq_zero {g h u v : ℤ}
(hphase : phase g h u v = 0) :
totalWeight g h u v = 0 ↔ g = h ∧ u = v := by
rw [totalWeight_eq_sq_of_phase_eq_zero hphase, sq_eq_zero_iff]
constructor
· intro hzero
have hgh : g = h := sub_eq_zero.mp hzero
refine ⟨hgh, ?_⟩
unfold phase at hphase
omega
· rintro ⟨hgh, _⟩
exact sub_eq_zero.mpr hgh
theorem phase_and_totalWeight_eq_zero_iff (g h u v : ℤ) :
(phase g h u v = 0 ∧ totalWeight g h u v = 0) ↔ g = h ∧ u = v := by
constructor
· rintro ⟨hphase, hweight⟩
exact (totalWeight_eq_zero_iff_of_phase_eq_zero hphase).mp hweight
· rintro ⟨rfl, rfl⟩
simp [phase, totalWeight, firstWeight, secondWeight, thirdWeight]
theorem totalWeight_pos_iff_of_phase_eq_zero {g h u v : ℤ}
(hphase : phase g h u v = 0) :
0 < totalWeight g h u v ↔ g ≠ h := by
rw [totalWeight_eq_sq_of_phase_eq_zero hphase, sq_pos_iff, sub_ne_zero]
/-- The degree bound in Section 4.3, for sector labels in `1, …, M`. -/
theorem totalWeight_le_of_labels {M g h u v : ℤ}
(hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M)
(hphase : phase g h u v = 0) :
totalWeight g h u v ≤ (M - 1) ^ 2 := by
rw [totalWeight_eq_sq_of_phase_eq_zero hphase]
have hprod : 0 ≤ ((M - 1) - (g - h)) * ((M - 1) + (g - h)) :=
mul_nonneg (by omega) (by omega)
nlinarith
/-- The polynomial degree attached to a surviving monomial. -/
def totalDegree (g h u v : ℤ) : ℕ := (totalWeight g h u v).toNat
theorem totalDegree_cast_of_phase_eq_zero {g h u v : ℤ}
(hphase : phase g h u v = 0) :
(totalDegree g h u v : ℤ) = totalWeight g h u v := by
exact Int.toNat_of_nonneg (totalWeight_nonneg_of_phase_eq_zero hphase)
theorem totalDegree_eq_zero_iff_of_phase_eq_zero {g h u v : ℤ}
(hphase : phase g h u v = 0) :
totalDegree g h u v = 0 ↔ g = h ∧ u = v := by
have hcast := totalDegree_cast_of_phase_eq_zero hphase
have hzero : totalDegree g h u v = 0 ↔ totalWeight g h u v = 0 := by omega
exact hzero.trans (totalWeight_eq_zero_iff_of_phase_eq_zero hphase)
theorem totalDegree_le_of_labels {M : ℕ} {g h u v : ℤ}
(hg : 1 ≤ g ∧ g ≤ M) (hh : 1 ≤ h ∧ h ≤ M)
(hphase : phase g h u v = 0) :
totalDegree g h u v ≤ (M - 1) ^ 2 := by
have hM : 1 ≤ M := by omega
have hbound := totalWeight_le_of_labels hg hh hphase
have hcast := totalDegree_cast_of_phase_eq_zero hphase
have hsub : ((M - 1 : ℕ) : ℤ) = (M : ℤ) - 1 := by omega
exact_mod_cast (show (totalDegree g h u v : ℤ) ≤ ((M - 1 : ℕ) : ℤ) ^ 2 by
rw [hcast, hsub]
exact hbound)
end MatrixMultiplication.AuxiliarySeparation
end OAI
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