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