openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Separation /ComplexSeparation.lean
| import Mathlib.Algebra.BigOperators.Group.Finset.Basic | |
| import Mathlib.Algebra.BigOperators.Ring.Finset | |
| import Mathlib.Algebra.Order.BigOperators.Group.Finset | |
| import Mathlib.Basic.Real.Basic | |
| import Mathlib.Data.Int.ModEq | |
| import Mathlib.Tactic.Linarith | |
| import Mathlib.Tactic.Ring | |
| namespace OAI | |
| /-! Finite type counts, hierarchy separation and tensor execution bounds. -/ | |
| namespace MatrixMultiplication.Foundation.Separation | |
| theorem grid_modular_collision (Q c t u v : ℤ) | |
| (hc : 0 ≤ c ∧ c < Q) (ht : 0 ≤ t ∧ t < Q) | |
| (hu : 0 ≤ u ∧ u < Q) (hv : 0 ≤ v ∧ v < Q) | |
| (hmod : Int.ModEq (4 * Q) c (t + u - v)) : c = t + u - v := by | |
| have hdvd : 4 * Q ∣ t + u - v - c := hmod.dvd | |
| have hz : t + u - v - c = 0 := by | |
| rcases le_total 0 (t + u - v - c) with h | h | |
| · exact Int.eq_zero_of_dvd_of_nonneg_of_lt h (by omega) hdvd | |
| · have hn : -(t + u - v - c) = 0 := | |
| Int.eq_zero_of_dvd_of_nonneg_of_lt (by omega) (by omega) (Int.dvd_neg.mpr hdvd) | |
| omega | |
| omega | |
| variable {ι : Type*} [Fintype ι] | |
| def squaredNorm (t : ι → ℝ) : ℝ := ∑ i, (t i) ^ 2 | |
| def dot (t u : ι → ℝ) : ℝ := ∑ i, t i * u i | |
| theorem squaredNorm_displacement (t u v : ι → ℝ) : | |
| squaredNorm (fun i => t i + u i - v i) = | |
| squaredNorm t + 2 * (dot t u - dot t v) + | |
| squaredNorm (fun i => u i - v i) := by | |
| calc | |
| squaredNorm (fun i => t i + u i - v i) = | |
| ∑ i, ((t i) ^ 2 + 2 * (t i * u i - t i * v i) + | |
| (u i - v i) ^ 2) := by | |
| apply Finset.sum_congr rfl | |
| intro i hi | |
| ring | |
| _ = _ := by | |
| simp only [squaredNorm, dot, Finset.sum_add_distrib, | |
| mul_sub, Finset.mul_sum, Finset.sum_sub_distrib] | |
| theorem sphere_slice_collision (t c u v : ι → ℝ) | |
| (hsphere : squaredNorm c = squaredNorm t) | |
| (hslice : dot t u = dot t v) | |
| (hcollision : ∀ i, c i = t i + u i - v i) : u = v ∧ c = t := by | |
| have hc : c = fun i => t i + u i - v i := funext hcollision | |
| have hzero : squaredNorm (fun i => u i - v i) = 0 := by | |
| have hid := squaredNorm_displacement t u v | |
| rw [← hc, hsphere, hslice] at hid | |
| linarith | |
| have huv : u = v := by | |
| funext i | |
| have hle : (u i - v i) ^ 2 ≤ squaredNorm (fun j => u j - v j) := | |
| Finset.single_le_sum (fun j _ => sq_nonneg (u j - v j)) (Finset.mem_univ i) | |
| rw [hzero] at hle | |
| nlinarith [sq_nonneg (u i - v i)] | |
| refine ⟨huv, ?_⟩ | |
| funext i | |
| rw [hcollision, huv] | |
| ring | |
| end MatrixMultiplication.Foundation.Separation | |
| end OAI | |