openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Arithmetic /CharacterRounding.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.ExponentComparison | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Character.Basic | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.RankExponent | |
| import Mathlib.Algebra.Order.Floor.Ring | |
| namespace OAI | |
| /-! | |
| # Integer rounding in the detecting-character argument | |
| The character used at a matrix size may depend on that size. A uniform bound | |
| on all those characters is enough: the loss of one in rounding their target | |
| values cannot change a positive power exponent. The character-existence and | |
| character-growth assumptions remain explicit in the final implications. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open MatrixMultiplication.Foundation | |
| /-- Rounding strictly down works at integers too: use `ceil x - 1`. -/ | |
| theorem exists_nat_sub_one_le_lt {x : ℝ} (hx : 0 < x) : | |
| ∃ k : ℕ, x - 1 ≤ (k : ℝ) ∧ (k : ℝ) < x := by | |
| have hc : 1 ≤ ⌈x⌉₊ := Nat.one_le_ceil_iff.mpr hx | |
| refine ⟨⌈x⌉₊ - 1, ?_, ?_⟩ | |
| · rw [Nat.cast_sub hc] | |
| norm_num only [Nat.cast_one] | |
| linarith [Nat.le_ceil x] | |
| · exact ((Nat.ceil_eq_iff (by omega : ⌈x⌉₊ ≠ 0)).1 rfl).1 | |
| /-- An additive loss of one does not affect a comparison to a nonnegative | |
| power exponent. No sign assumption on the exponent on the left is needed. -/ | |
| theorem rpow_exponent_le_of_nat_sub_one_bound {ν τ : ℝ} (hτ : 0 ≤ τ) | |
| (hbound : ∀ d : ℕ, 2 ≤ d → (d : ℝ) ^ ν - 1 ≤ (d : ℝ) ^ τ) : | |
| ν ≤ τ := by | |
| apply rpow_exponent_le_of_nat_bound (C := 2) | |
| intro d hd | |
| by_cases hd2 : 2 ≤ d | |
| · have h := hbound d hd2 | |
| have hdreal : (1 : ℝ) ≤ d := by exact_mod_cast hd | |
| have hpow : 1 ≤ (d : ℝ) ^ τ := Real.one_le_rpow hdreal hτ | |
| linarith | |
| · have hd1 : d = 1 := by omega | |
| simp [hd1] | |
| /-- The integer witnesses in the final rounding argument can vary with `d`. -/ | |
| theorem exponent_le_of_integer_rounding {ν τ : ℝ} (hτ : 0 ≤ τ) | |
| (hround : ∀ d : ℕ, 2 ≤ d → | |
| ∃ k : ℕ, (d : ℝ) ^ ν - 1 ≤ k ∧ (k : ℝ) ≤ (d : ℝ) ^ τ) : | |
| ν ≤ τ := by | |
| apply rpow_exponent_le_of_nat_sub_one_bound hτ | |
| intro d hd | |
| obtain ⟨k, hk, hk'⟩ := hround d hd | |
| exact hk.trans hk' | |
| /-- A positive exponent on the left makes the target exponent nonnegative | |
| automatically, so it is an alternative to that hypothesis above. -/ | |
| theorem exponent_le_of_integer_rounding_of_pos {ν τ : ℝ} (hν : 0 < ν) | |
| (hround : ∀ d : ℕ, 2 ≤ d → | |
| ∃ k : ℕ, (d : ℝ) ^ ν - 1 ≤ k ∧ (k : ℝ) ≤ (d : ℝ) ^ τ) : | |
| ν ≤ τ := by | |
| have hτ : 0 ≤ τ := by | |
| by_contra hneg | |
| obtain ⟨k, hk, hk'⟩ := hround 2 le_rfl | |
| norm_num only [Nat.cast_ofNat] at hk hk' | |
| have hpτ : (2 : ℝ) ^ τ < 1 := | |
| Real.rpow_lt_one_of_one_lt_of_neg (by norm_num) (lt_of_not_ge hneg) | |
| have hklt : (k : ℝ) < 1 := hk'.trans_lt hpτ | |
| have hk0 : k = 0 := by | |
| have : k < 1 := by exact_mod_cast hklt | |
| omega | |
| rw [hk0, Nat.cast_zero] at hk | |
| have hpν : 1 < (2 : ℝ) ^ ν := Real.one_lt_rpow (by norm_num) hν | |
| linarith | |
| exact exponent_le_of_integer_rounding hτ hround | |
| /-- Explicitly conditional passage from detecting characters to the exponent | |
| bound. The detecting character is allowed to depend on both `d` and `k`. -/ | |
| theorem exponent_le_of_detecting_characters {ν τ : ℝ} (hτ : 0 ≤ τ) | |
| (detect : ∀ d : ℕ, 2 ≤ d → ∀ k : ℕ, (k : ℝ) < (d : ℝ) ^ ν → | |
| ∃ χ : Character, (k : ℝ) ≤ χ.value (Tensor.matrixMultiplication d d d)) | |
| (bound : ∀ χ : Character, ∀ d : ℕ, 2 ≤ d → | |
| χ.value (Tensor.matrixMultiplication d d d) ≤ (d : ℝ) ^ τ) : | |
| ν ≤ τ := by | |
| apply exponent_le_of_integer_rounding hτ | |
| intro d hd | |
| have hdpos : (0 : ℝ) < d := by exact_mod_cast (show 0 < d by omega) | |
| obtain ⟨k, hk, hk'⟩ := | |
| exists_nat_sub_one_le_lt (Real.rpow_pos_of_pos hdpos ν) | |
| obtain ⟨χ, hχ⟩ := detect d hd k hk' | |
| exact ⟨k, hk, hχ.trans (bound χ d hd)⟩ | |
| /-- The character growth bound can be supplied as a uniform bound on its | |
| exponent sum, together with the matrix-value identity. -/ | |
| theorem exponent_le_of_detecting_character_exponents {ν τ : ℝ} (hτ : 0 ≤ τ) | |
| (exponentSum : Character → ℝ) | |
| (value : ∀ χ : Character, ∀ d : ℕ, 2 ≤ d → | |
| χ.value (Tensor.matrixMultiplication d d d) = (d : ℝ) ^ exponentSum χ) | |
| (upper : ∀ χ : Character, exponentSum χ ≤ τ) | |
| (detect : ∀ d : ℕ, 2 ≤ d → ∀ k : ℕ, (k : ℝ) < (d : ℝ) ^ ν → | |
| ∃ χ : Character, (k : ℝ) ≤ χ.value (Tensor.matrixMultiplication d d d)) : | |
| ν ≤ τ := by | |
| apply exponent_le_of_detecting_characters hτ detect | |
| intro χ d hd | |
| rw [value χ d hd] | |
| exact Real.rpow_le_rpow_of_exponent_le (by exact_mod_cast (show 1 ≤ d by omega)) | |
| (upper χ) | |
| /-- The paper's final `ν ≤ 9/4` implication, with its two substantive character | |
| inputs visible rather than incorporated into the definition of `ν`. -/ | |
| theorem exactRankExponent_le_nine_quarters_of_characters | |
| (detect : ∀ d : ℕ, 2 ≤ d → ∀ k : ℕ, | |
| (k : ℝ) < (d : ℝ) ^ exactRankExponent → | |
| ∃ χ : Character, (k : ℝ) ≤ χ.value (Tensor.matrixMultiplication d d d)) | |
| (bound : ∀ χ : Character, ∀ d : ℕ, 2 ≤ d → | |
| χ.value (Tensor.matrixMultiplication d d d) ≤ (d : ℝ) ^ (9 / 4 : ℝ)) : | |
| exactRankExponent ≤ 9 / 4 := | |
| exponent_le_of_detecting_characters (by norm_num) detect bound | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |