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