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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 | import Mathlib.Algebra.Order.Group.Unbundled.Int
import Lean.Elab.Tactic.Omega
namespace OAI
/-!
# The Fourier projection has no modular wrap
Section 4.1 of *Matrix Multiplication via Auxiliary Separation and Polynomial
Multiplication* uses exactly `5 * M` Fourier phases. The exponent
`u - v + 2 * (g - h)` has absolute value at most `3 * (M - 1)` when all four
labels lie in `{1, ..., M}`. Consequently, divisibility by `5 * M` is equivalent
to the exact integer support equation.
-/
namespace MatrixMultiplication.AuxiliarySeparation
/-- The explicit bound for the exponent in the finite Fourier projection. -/
theorem fourierPhase_abs_le {M h g u v : ℤ}
(hh : 1 ≤ h ∧ h ≤ M) (hg : 1 ≤ g ∧ g ≤ M)
(hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) :
|u - v + 2 * (g - h)| ≤ 3 * (M - 1) := by
rw [abs_le']
constructor <;> omega
/-- The Fourier modulus exceeds the full possible range of the exponent. -/
theorem fourierRadius_lt_modulus {M : ℤ} (hM : 1 ≤ M) :
3 * (M - 1) < 5 * M := by
omega
/-- Every possible exponent lies strictly between minus and plus the modulus. -/
theorem fourierPhase_abs_lt {M h g u v : ℤ}
(hM : 1 ≤ M) (hh : 1 ≤ h ∧ h ≤ M) (hg : 1 ≤ g ∧ g ≤ M)
(hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) :
|u - v + 2 * (g - h)| < 5 * M :=
lt_of_le_of_lt (fourierPhase_abs_le hh hg hu hv) (fourierRadius_lt_modulus hM)
/-- Modular Fourier support is exact integer support for the allowed labels. -/
theorem fourierPhase_eq_zero_of_dvd {M h g u v : ℤ}
(hM : 1 ≤ M) (hh : 1 ≤ h ∧ h ≤ M) (hg : 1 ≤ g ∧ g ≤ M)
(hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M)
(hdiv : 5 * M ∣ u - v + 2 * (g - h)) :
u - v + 2 * (g - h) = 0 :=
Int.eq_zero_of_abs_lt_dvd hdiv (fourierPhase_abs_lt hM hh hg hu hv)
/-- Characterization of the surviving Fourier support without modular arithmetic. -/
theorem fourierPhase_dvd_iff_eq_zero {M h g u v : ℤ}
(hM : 1 ≤ M) (hh : 1 ≤ h ∧ h ≤ M) (hg : 1 ≤ g ∧ g ≤ M)
(hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) :
(5 * M ∣ u - v + 2 * (g - h)) ↔ u - v + 2 * (g - h) = 0 := by
constructor
· exact fourierPhase_eq_zero_of_dvd hM hh hg hu hv
· intro hzero
rw [hzero]
exact ⟨0, by simp⟩
/-- The same exponent bound for natural-number labels, interpreted in `ℤ`. -/
theorem fourierPhase_abs_le_nat {M h g u v : ℕ}
(hh : 1 ≤ h ∧ h ≤ M) (hg : 1 ≤ g ∧ g ≤ M)
(hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) :
|(u : ℤ) - v + 2 * ((g : ℤ) - h)| ≤ 3 * ((M : ℤ) - 1) := by
apply fourierPhase_abs_le <;> omega
/-- Natural-number labels admit no additional solutions modulo `5 * M`. -/
theorem fourierPhase_dvd_iff_eq_zero_nat {M h g u v : ℕ}
(hM : 1 ≤ M) (hh : 1 ≤ h ∧ h ≤ M) (hg : 1 ≤ g ∧ g ≤ M)
(hu : 1 ≤ u ∧ u ≤ M) (hv : 1 ≤ v ∧ v ≤ M) :
((5 : ℤ) * M ∣ (u : ℤ) - v + 2 * ((g : ℤ) - h)) ↔
(u : ℤ) - v + 2 * ((g : ℤ) - h) = 0 := by
apply fourierPhase_dvd_iff_eq_zero <;> omega
/-- Zero-based finite labels have the same exponent as the paper's labels plus one. -/
theorem fourierPhase_abs_le_fin {M : ℕ} (h g u v : Fin M) :
|(u.val : ℤ) - v.val + 2 * ((g.val : ℤ) - h.val)| ≤
3 * ((M : ℤ) - 1) := by
have hh := h.isLt
have hg := g.isLt
have hu := u.isLt
have hv := v.isLt
rw [abs_le']
constructor <;> omega
/-- The support test for a Fourier projection indexed by `Fin M`. -/
theorem fourierPhase_dvd_iff_eq_zero_fin {M : ℕ} (h g u v : Fin M) :
((5 : ℤ) * M ∣ (u.val : ℤ) - v.val + 2 * ((g.val : ℤ) - h.val)) ↔
(u.val : ℤ) - v.val + 2 * ((g.val : ℤ) - h.val) = 0 := by
constructor
· intro hdiv
apply Int.eq_zero_of_abs_lt_dvd hdiv
apply lt_of_le_of_lt (fourierPhase_abs_le_fin h g u v)
apply fourierRadius_lt_modulus
have hh := h.isLt
omega
· intro hzero
rw [hzero]
exact ⟨0, by simp⟩
end MatrixMultiplication.AuxiliarySeparation
end OAI
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