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cdec422 | 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 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 | 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
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