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f18e7de | 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 | 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
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