File size: 3,132 Bytes
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 73 74 75 76 | import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Card
namespace OAI
/-! Finite type counts, hierarchy separation and tensor execution bounds. -/
namespace MatrixMultiplication.Foundation.Separation
abbrev Grid (d Q : ℕ) := Fin d → Fin Q
def gridSquaredNorm {d Q : ℕ} (x : Grid d Q) : ℕ := ∑ i, (x i : ℕ) ^ 2
def gridDot {d Q : ℕ} (t u : Grid d Q) : ℕ := ∑ i, (t i : ℕ) * (u i : ℕ)
theorem gridSquaredNorm_le {d Q : ℕ} (x : Grid d Q) : gridSquaredNorm x ≤ d * Q ^ 2 := by
calc
gridSquaredNorm x ≤ ∑ _i : Fin d, Q ^ 2 := by
apply Finset.sum_le_sum
intro i hi
exact Nat.pow_le_pow_left (Nat.le_of_lt (x i).isLt) 2
_ = d * Q ^ 2 := by simp
theorem gridDot_le {d Q : ℕ} (t u : Grid d Q) : gridDot t u ≤ d * Q ^ 2 := by
calc
gridDot t u ≤ ∑ _i : Fin d, Q ^ 2 := by
apply Finset.sum_le_sum
intro i hi
simpa only [pow_two] using
Nat.mul_le_mul (Nat.le_of_lt (t i).isLt) (Nat.le_of_lt (u i).isLt)
_ = d * Q ^ 2 := by simp
def normColor {d Q : ℕ} (x : Grid d Q) : Fin (d * Q ^ 2 + 1) :=
⟨gridSquaredNorm x, Nat.lt_succ_of_le (gridSquaredNorm_le x)⟩
def dotColor {d Q : ℕ} (t u : Grid d Q) : Fin (d * Q ^ 2 + 1) :=
⟨gridDot t u, Nat.lt_succ_of_le (gridDot_le t u)⟩
theorem two_stage_pigeonhole {A : Type*} [Fintype A] [DecidableEq A]
{M K m : ℕ} (hM : 0 < M) (f : A → Fin M) (g : A → A → Fin M)
(hK : M * K ≤ Fintype.card A) (hm : M * m ≤ Fintype.card A) :
∃ tags : Finset A, tags.card = K ∧
(∃ radius : Fin M, ∀ t ∈ tags, f t = radius) ∧
∀ t ∈ tags, ∃ slice : Finset A, slice.card = m ∧
∃ level : Fin M, ∀ u ∈ slice, g t u = level := by
let : Nonempty (Fin M) := ⟨⟨0, hM⟩⟩
obtain ⟨radius, hr⟩ := Fintype.exists_le_card_fiber_of_mul_le_card f
(by simpa only [Fintype.card_fin] using hK)
obtain ⟨tags, htags, hcard⟩ := Finset.exists_subset_card_eq hr
refine ⟨tags, hcard, ⟨radius, ?_⟩, ?_⟩
· intro t ht
exact (Finset.mem_filter.mp (htags ht)).2
· intro t ht
obtain ⟨level, hl⟩ := Fintype.exists_le_card_fiber_of_mul_le_card (g t)
(by simpa only [Fintype.card_fin] using hm)
obtain ⟨slice, hslice, hsize⟩ := Finset.exists_subset_card_eq hl
refine ⟨slice, hsize, level, ?_⟩
intro u hu
exact (Finset.mem_filter.mp (hslice hu)).2
theorem grid_tags_slices {d Q K m : ℕ}
(hK : (d * Q ^ 2 + 1) * K ≤ Q ^ d)
(hm : (d * Q ^ 2 + 1) * m ≤ Q ^ d) :
∃ tags : Finset (Grid d Q), tags.card = K ∧
(∃ radius : Fin (d * Q ^ 2 + 1), ∀ t ∈ tags, normColor t = radius) ∧
∀ t ∈ tags, ∃ slice : Finset (Grid d Q), slice.card = m ∧
∃ level : Fin (d * Q ^ 2 + 1), ∀ u ∈ slice, dotColor t u = level := by
exact two_stage_pigeonhole (Nat.succ_pos _) normColor dotColor
(by simpa only [Grid, Fintype.card_fun, Fintype.card_fin] using hK)
(by simpa only [Grid, Fintype.card_fun, Fintype.card_fin] using hm)
end MatrixMultiplication.Foundation.Separation
end OAI
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