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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 | import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Growth.ConcaveSlopes
namespace OAI
/-!
# Shifted tripling and diagonal increments
These are the two estimates on the limiting row slopes used in Section 6.
Iteration retains the additive shift in the tripling inequality. Symmetry
then expresses a diagonal step as one increment in each of two adjacent rows.
-/
namespace MatrixMultiplication.AuxiliarySeparation
open Filter
open scoped Topology
/-- The indices obtained by repeatedly applying the affine tripling map. -/
def shiftedTriplingIndex (c h : ℕ) : ℕ → ℕ
| 0 => h
| j + 1 => 3 * shiftedTriplingIndex c h j + c
/-- Starting at a positive index, affine tripling escapes to infinity. -/
theorem shiftedTriplingIndex_lower (c h : ℕ) (hh : 1 ≤ h) (j : ℕ) :
h + j ≤ shiftedTriplingIndex c h j := by
induction j with
| zero => simp [shiftedTriplingIndex]
| succ j ih =>
simp only [shiftedTriplingIndex]
omega
theorem shiftedTriplingIndex_tendsto (c h : ℕ) (hh : 1 ≤ h) :
Tendsto (shiftedTriplingIndex c h) atTop atTop := by
apply tendsto_atTop_mono _ tendsto_id
intro j
change j ≤ shiftedTriplingIndex c h j
have := shiftedTriplingIndex_lower c h hh j
omega
/-- The exact affine solution, written without truncated natural subtraction. -/
theorem shiftedTriplingIndex_cast (c h j : ℕ) :
(shiftedTriplingIndex c h j : ℝ) + (c : ℝ) / 2 =
(3 : ℝ) ^ j * ((h : ℝ) + (c : ℝ) / 2) := by
induction j with
| zero => simp [shiftedTriplingIndex]
| succ j ih =>
simp only [shiftedTriplingIndex, Nat.cast_add, Nat.cast_mul, Nat.cast_ofNat,
pow_succ]
nlinarith
/-- Iterating the tripling inequality multiplies the profile by powers of three. -/
theorem shiftedTripling_profile_lower {P : ℕ → ℝ} {c h : ℕ} (hh : 1 ≤ h)
(htrip : ∀ n : ℕ, 1 ≤ n → 3 * P n ≤ P (3 * n + c)) (j : ℕ) :
(3 : ℝ) ^ j * P h ≤ P (shiftedTriplingIndex c h j) := by
induction j with
| zero => simp [shiftedTriplingIndex]
| succ j ih =>
have hj : 1 ≤ shiftedTriplingIndex c h j := by
have := shiftedTriplingIndex_lower c h hh j
omega
have hstep := htrip (shiftedTriplingIndex c h j) hj
simp only [shiftedTriplingIndex, pow_succ]
nlinarith
/-- A row with a limiting linear slope and shifted tripling lies below the
corresponding affine multiple of that slope. -/
theorem profile_le_affine_slope {P : ℕ → ℝ} {g : ℝ} {c h : ℕ}
(hh : 1 ≤ h)
(htrip : ∀ n : ℕ, 1 ≤ n → 3 * P n ≤ P (3 * n + c))
(hlim : Tendsto (fun n : ℕ => P n / (n : ℝ)) atTop (𝓝 g)) :
P h ≤ ((h : ℝ) + (c : ℝ) / 2) * g := by
have hpow : Tendsto (fun j : ℕ => (3 : ℝ) ^ j) atTop atTop :=
tendsto_pow_atTop_atTop_of_one_lt (by norm_num)
have hzero : Tendsto (fun j : ℕ => ((c : ℝ) / 2) / (3 : ℝ) ^ j)
atTop (𝓝 0) := hpow.const_div_atTop _
have hindex : Tendsto
(fun j : ℕ => (shiftedTriplingIndex c h j : ℝ) / (3 : ℝ) ^ j)
atTop (𝓝 ((h : ℝ) + (c : ℝ) / 2)) := by
have heq (j : ℕ) :
(shiftedTriplingIndex c h j : ℝ) / (3 : ℝ) ^ j =
(h : ℝ) + (c : ℝ) / 2 - ((c : ℝ) / 2) / (3 : ℝ) ^ j := by
have hj := shiftedTriplingIndex_cast c h j
have hp : (3 : ℝ) ^ j ≠ 0 := pow_ne_zero _ (by norm_num)
field_simp
nlinarith
simpa only [heq, sub_zero] using tendsto_const_nhds.sub hzero
have hproduct := (hlim.comp (shiftedTriplingIndex_tendsto c h hh)).mul hindex
have hprofile : Tendsto
(fun j : ℕ => P (shiftedTriplingIndex c h j) / (3 : ℝ) ^ j)
atTop (𝓝 (g * ((h : ℝ) + (c : ℝ) / 2))) := by
apply hproduct.congr'
apply Eventually.of_forall
intro j
have hj : (shiftedTriplingIndex c h j : ℝ) ≠ 0 := by
have := shiftedTriplingIndex_lower c h hh j
exact_mod_cast (show shiftedTriplingIndex c h j ≠ 0 by omega)
dsimp only [Function.comp_def]
field_simp
have hbound : ∀ j : ℕ, P h ≤ P (shiftedTriplingIndex c h j) / (3 : ℝ) ^ j := by
intro j
apply (le_div_iff₀ (pow_pos (by norm_num : (0 : ℝ) < 3) j)).mpr
simpa only [mul_comm] using shiftedTripling_profile_lower hh htrip j
simpa only [mul_comm] using ge_of_tendsto hprofile (Eventually.of_forall hbound)
/-- The row estimate in the positive-index notation of the paper. -/
theorem profile_row_le_slope {P : ℕ → ℕ → ℝ} {g : ℕ → ℝ}
(htrip : ∀ a : ℕ, 1 ≤ a → ∀ h : ℕ, 1 ≤ h →
3 * P a h ≤ P a (3 * h + a - 1))
(hlim : ∀ a : ℕ, 1 ≤ a →
Tendsto (fun h : ℕ => P a h / (h : ℝ)) atTop (𝓝 (g a)))
(a h : ℕ) (ha : 1 ≤ a) (hh : 1 ≤ h) :
P a h ≤ ((h : ℝ) + ((a : ℝ) - 1) / 2) * g a := by
have ht : ∀ n : ℕ, 1 ≤ n → 3 * P a n ≤ P a (3 * n + (a - 1)) := by
intro n hn
have hi : 3 * n + (a - 1) = 3 * n + a - 1 := by omega
rw [hi]
exact htrip a ha n hn
have hcast : ((a - 1 : ℕ) : ℝ) = (a : ℝ) - 1 := by
rw [Nat.cast_sub ha, Nat.cast_one]
simpa only [hcast] using profile_le_affine_slope hh ht (hlim a ha)
/-- The diagonal is bounded above by the limiting row slope with the precise
coefficient `(3a-1)/2`. -/
theorem profile_diagonal_le_slope {P : ℕ → ℕ → ℝ} {g : ℕ → ℝ}
(htrip : ∀ a : ℕ, 1 ≤ a → ∀ h : ℕ, 1 ≤ h →
3 * P a h ≤ P a (3 * h + a - 1))
(hlim : ∀ a : ℕ, 1 ≤ a →
Tendsto (fun h : ℕ => P a h / (h : ℝ)) atTop (𝓝 (g a)))
(a : ℕ) (ha : 1 ≤ a) :
P a a ≤ ((3 * (a : ℝ) - 1) / 2) * g a := by
have hc : (a : ℝ) + ((a : ℝ) - 1) / 2 = (3 * (a : ℝ) - 1) / 2 := by ring
simpa only [hc] using profile_row_le_slope htrip hlim a a ha ha
/-- Symmetry identifies a diagonal step with one increment in each adjacent
row. Only positive-index symmetry and slope bounds are used. -/
theorem diagonal_increment_of_slope_lower {P : ℕ → ℕ → ℝ} {g : ℕ → ℝ}
(hsymm : ∀ a : ℕ, 1 ≤ a → ∀ b : ℕ, 1 ≤ b → P a b = P b a)
(hinc : ∀ a : ℕ, 1 ≤ a → ∀ h : ℕ, 1 ≤ h →
g a ≤ P a (h + 1) - P a h)
(n : ℕ) (hn : 1 ≤ n) :
P n n + g n + g (n + 1) ≤ P (n + 1) (n + 1) := by
have hfirst := hinc n hn n hn
have hsecond := hinc (n + 1) (by omega) n hn
have hsym := hsymm n hn (n + 1) (by omega)
linarith
end MatrixMultiplication.AuxiliarySeparation
end OAI
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