openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Separation /NoWrap.lean
| 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 | |