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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 | import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.Product
namespace OAI
/-!
# The scalar simultaneous induction in Section 6
The two diagonal inequalities imply, simultaneously, lower bounds for the
limiting row slopes and for the diagonal profile. All profile properties
needed for this argument are expressed as hypotheses below.
-/
namespace MatrixMultiplication.AuxiliarySeparation
/-- One step of the Section 6 simultaneous induction, with a positive real
index. The proof uses the diagonal increment inequality to obtain the second
conclusion as well as the first. -/
theorem growthRecurrence_step_real
{n Dprev Dnext gprev gnext Gprev : ℝ} (hn : 0 < n)
(hupper : Dnext ≤ ((3 * (n + 1) - 1) / 2) * gnext)
(hincrement : Dprev + gprev + gnext ≤ Dnext)
(hgprev : Gprev ≤ gprev)
(hDprev : ((3 * n - 1) / 2) * Gprev ≤ Dprev) :
(1 + 1 / (3 * n)) * Gprev ≤ gnext ∧
((3 * (n + 1) - 1) / 2) * ((1 + 1 / (3 * n)) * Gprev) ≤ Dnext := by
have hden : 0 < 3 * n := by positivity
have hbound : (3 * n + 1) * Gprev ≤ (3 * n) * gnext := by
nlinarith
have hidentity : (1 + 1 / (3 * n)) * Gprev =
((3 * n + 1) * Gprev) / (3 * n) := by
field_simp
have hgnew : (1 + 1 / (3 * n)) * Gprev ≤ gnext := by
rw [hidentity]
exact (div_le_iff₀ hden).mpr (by simpa only [mul_comm] using hbound)
refine ⟨hgnew, ?_⟩
have hdiagonal : ((3 * (n + 1) - 1) / 2) *
((1 + 1 / (3 * n)) * Gprev) =
((3 * n - 1) / 2) * Gprev + Gprev +
(1 + 1 / (3 * n)) * Gprev := by
field_simp
ring
rw [hdiagonal]
linarith
/-- The natural-number form of the scalar induction step, indexed by `n + 1`
so that no truncated subtraction occurs in the recurrence. -/
theorem growthRecurrence_step
{n : ℕ} (hn : 1 ≤ n) {Dprev Dnext gprev gnext Gprev : ℝ}
(hupper : Dnext ≤ ((3 * ((n : ℝ) + 1) - 1) / 2) * gnext)
(hincrement : Dprev + gprev + gnext ≤ Dnext)
(hgprev : Gprev ≤ gprev)
(hDprev : ((3 * (n : ℝ) - 1) / 2) * Gprev ≤ Dprev) :
(1 + 1 / (3 * (n : ℝ))) * Gprev ≤ gnext ∧
((3 * ((n : ℝ) + 1) - 1) / 2) *
((1 + 1 / (3 * (n : ℝ))) * Gprev) ≤ Dnext := by
apply growthRecurrence_step_real
· exact_mod_cast (show 0 < n by omega)
· exact hupper
· exact hincrement
· exact hgprev
· exact hDprev
/-- The simultaneous slope and diagonal estimates, conditional only on the
initial values, the recurrence for `G`, and the two diagonal inequalities. -/
theorem growthRecurrence_induction
(D g G : ℕ → ℝ)
(hGbase : G 1 = 1)
(hgbase : 1 ≤ g 1)
(hDbase : 1 ≤ D 1)
(hGstep : ∀ n : ℕ, 1 ≤ n →
G (n + 1) = (1 + 1 / (3 * (n : ℝ))) * G n)
(hupper : ∀ n : ℕ, 1 ≤ n →
D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n)
(hincrement : ∀ n : ℕ, 1 ≤ n →
D n + g n + g (n + 1) ≤ D (n + 1)) :
∀ n : ℕ, 1 ≤ n → G n ≤ g n ∧ ((3 * (n : ℝ) - 1) / 2) * G n ≤ D n := by
intro n hn
induction n, hn using Nat.le_induction with
| base =>
norm_num [hGbase]
exact ⟨hgbase, hDbase⟩
| succ n hn ih =>
have hu : D (n + 1) ≤
((3 * ((n : ℝ) + 1) - 1) / 2) * g (n + 1) := by
simpa only [Nat.cast_add, Nat.cast_one] using hupper (n + 1) (by omega)
have hs := growthRecurrence_step hn hu (hincrement n hn) ih.1 ih.2
simpa only [hGstep n hn, Nat.cast_add, Nat.cast_one] using hs
/-- A zero-indexed version of `growthRecurrence_induction`, convenient when the
initial term of the comparison sequence is stored at index zero. -/
theorem growthRecurrence_induction_shifted
(D g H : ℕ → ℝ)
(hHbase : H 0 = 1)
(hgbase : 1 ≤ g 1)
(hDbase : 1 ≤ D 1)
(hHstep : ∀ n : ℕ,
H (n + 1) = (1 + 1 / (3 * ((n : ℝ) + 1))) * H n)
(hupper : ∀ n : ℕ, 1 ≤ n →
D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n)
(hincrement : ∀ n : ℕ, 1 ≤ n →
D n + g n + g (n + 1) ≤ D (n + 1)) :
∀ n : ℕ, H n ≤ g (n + 1) ∧
((3 * ((n : ℝ) + 1) - 1) / 2) * H n ≤ D (n + 1) := by
intro n
induction n with
| zero =>
norm_num [hHbase]
exact ⟨hgbase, hDbase⟩
| succ n ih =>
have hn : 1 ≤ n + 1 := by omega
have hu : D (n + 1 + 1) ≤
((3 * (((n + 1 : ℕ) : ℝ) + 1) - 1) / 2) * g (n + 1 + 1) := by
simpa only [Nat.cast_add, Nat.cast_one] using
hupper (n + 1 + 1) (by omega)
have hp : ((3 * ((n + 1 : ℕ) : ℝ) - 1) / 2) * H n ≤ D (n + 1) := by
simpa only [Nat.cast_add, Nat.cast_one] using ih.2
have hs := growthRecurrence_step hn hu (hincrement (n + 1) hn) ih.1 hp
simpa only [hHstep, Nat.cast_add, Nat.cast_one] using hs
/-- The simultaneous bounds for the explicit comparison product from Section 6. -/
theorem growthProduct_le_slope_and_diagonal
(D g : ℕ → ℝ)
(hgbase : 1 ≤ g 1)
(hDbase : 1 ≤ D 1)
(hupper : ∀ n : ℕ, 1 ≤ n →
D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n)
(hincrement : ∀ n : ℕ, 1 ≤ n →
D n + g n + g (n + 1) ≤ D (n + 1)) :
∀ n : ℕ, growthProduct n ≤ g (n + 1) ∧
((3 * ((n : ℝ) + 1) - 1) / 2) * growthProduct n ≤ D (n + 1) := by
apply growthRecurrence_induction_shifted D g growthProduct
growthProduct_zero hgbase hDbase
· intro n
rw [growthProduct_succ, mul_comm]
· exact hupper
· exact hincrement
/-- Combining the scalar induction with the elementary product estimate gives
an exact fourth-power lower bound on the cube of each diagonal value. -/
theorem diagonal_fourth_power_lower
(D g : ℕ → ℝ)
(hgbase : 1 ≤ g 1)
(hDbase : 1 ≤ D 1)
(hupper : ∀ n : ℕ, 1 ≤ n →
D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n)
(hincrement : ∀ n : ℕ, 1 ≤ n →
D n + g n + g (n + 1) ≤ D (n + 1))
(n : ℕ) :
((n : ℝ) + 1) ^ 4 ≤ D (n + 1) ^ 3 := by
have hdiag := (growthProduct_le_slope_and_diagonal D g hgbase hDbase
hupper hincrement n).2
have hpositive := growthProduct_pos n
have hlinear : ((n : ℝ) + 1) * growthProduct n ≤ D (n + 1) := by
have hcoeff : (n : ℝ) + 1 ≤ (3 * ((n : ℝ) + 1) - 1) / 2 := by
have hn : (0 : ℝ) ≤ n := Nat.cast_nonneg n
linarith
exact (mul_le_mul_of_nonneg_right hcoeff (le_of_lt hpositive)).trans hdiag
calc
((n : ℝ) + 1) ^ 4 = ((n : ℝ) + 1) ^ 3 * ((n : ℝ) + 1) := by ring
_ ≤ ((n : ℝ) + 1) ^ 3 * growthProduct n ^ 3 :=
mul_le_mul_of_nonneg_left (growthProduct_cube_lower n) (by positivity)
_ = (((n : ℝ) + 1) * growthProduct n) ^ 3 := by ring
_ ≤ D (n + 1) ^ 3 := pow_le_pow_left₀ (by positivity) hlinear 3
/-- The diagonal fourth-power lower bound using the paper's positive indices. -/
theorem diagonal_fourth_power_lower_of_one_le
(D g : ℕ → ℝ)
(hgbase : 1 ≤ g 1)
(hDbase : 1 ≤ D 1)
(hupper : ∀ n : ℕ, 1 ≤ n →
D n ≤ ((3 * (n : ℝ) - 1) / 2) * g n)
(hincrement : ∀ n : ℕ, 1 ≤ n →
D n + g n + g (n + 1) ≤ D (n + 1))
(n : ℕ) (hn : 1 ≤ n) : (n : ℝ) ^ 4 ≤ D n ^ 3 := by
cases n with
| zero => omega
| succ n =>
simpa only [Nat.cast_add, Nat.cast_one] using
diagonal_fourth_power_lower D g hgbase hDbase hupper hincrement n
end MatrixMultiplication.AuxiliarySeparation
end OAI
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