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import OAI.LinearAlgebra.MatrixFields.Construction.InitialState
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
noncomputable section
namespace MatrixMultiplication.AllFieldRateLimit
open Filter
open scoped Topology
variable {F : Type*} [Field F]
structure HasRates (W : ℕ → AllFieldWitness F) (scale : ℕ → ℝ)
(Y S R : ℝ) : Prop where
scale_positive : ∀ᶠ n in atTop, 0 < scale n
yield_limit : Tendsto (fun n => Real.log ((W n).multiplicity : ℝ) / scale n) atTop (𝓝 Y)
volume_limit : Tendsto (fun n => Real.log ((W n).volume : ℝ) / scale n) atTop (𝓝 S)
rank_limit : Tendsto (fun n => Real.log ((W n).rankBound : ℝ) / scale n) atTop (𝓝 R)
structure HasRateBounds (W : ℕ → AllFieldWitness F) (scale : ℕ → ℝ)
(Y S R : ℝ) : Prop where
scale_positive : ∀ᶠ n in atTop, 0 < scale n
yield_lower : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop,
Y - ε ≤ Real.log ((W n).multiplicity : ℝ) / scale n
volume_lower : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop,
S - ε ≤ Real.log ((W n).volume : ℝ) / scale n
rank_upper : ∀ ε : ℝ, 0 < ε → ∀ᶠ n in atTop,
Real.log ((W n).rankBound : ℝ) / scale n ≤ R + ε
theorem HasRates.toBounds {W : ℕ → AllFieldWitness F} {scale : ℕ → ℝ}
{Y S R : ℝ} (h : HasRates W scale Y S R) : HasRateBounds W scale Y S R where
scale_positive := h.scale_positive
yield_lower ε hε := (h.yield_limit.eventually_const_lt (by linarith : Y - ε < Y)).mono
(fun _ hn => hn.le)
volume_lower ε hε := (h.volume_limit.eventually_const_lt (by linarith : S - ε < S)).mono
(fun _ hn => hn.le)
rank_upper ε hε := (h.rank_limit.eventually_lt_const (by linarith : R < R + ε)).mono
(fun _ hn => hn.le)
theorem inequality_of_rates {W : ℕ → AllFieldWitness F} {scale : ℕ → ℝ}
{Y S R : ℝ} (h : HasRates W scale Y S R) :
Y + (Arithmetic.omega F / 3) * S ≤ R := by
apply le_of_tendsto_of_tendsto
(h.yield_limit.add (h.volume_limit.const_mul (Arithmetic.omega F / 3))) h.rank_limit
filter_upwards [h.scale_positive] with n hn
have hb := div_le_div_of_nonneg_right (W n).log_inequality hn.le
simpa only [add_div, mul_div_assoc] using hb
theorem inequality_of_rateBounds {W : ℕ → AllFieldWitness F} {scale : ℕ → ℝ}
{Y S R : ℝ} (h : HasRateBounds W scale Y S R) :
Y + (Arithmetic.omega F / 3) * S ≤ R := by
apply le_of_forall_pos_le_add
intro ε hε
have hδ : 0 < ε / 3 := by positivity
obtain ⟨n, hnscale, hnyield, hnvolume, hnrank⟩ :=
(h.scale_positive.and ((h.yield_lower (ε / 3) hδ).and
((h.volume_lower (ε / 3) hδ).and (h.rank_upper (ε / 3) hδ)))).exists
have hfinite := div_le_div_of_nonneg_right (W n).log_inequality hnscale.le
simp only [add_div, mul_div_assoc] at hfinite
have hω : 0 ≤ Arithmetic.omega F / 3 := div_nonneg Arithmetic.omega_nonneg (by norm_num)
have hvolume := mul_le_mul_of_nonneg_left hnvolume hω
have hωupper := mul_le_mul_of_nonneg_right
(Arithmetic.omega_le_three (F := F)) hδ.le
nlinarith
theorem omega_le_of_rates {W : ℕ → AllFieldWitness F} {scale : ℕ → ℝ}
{Y S R : ℝ} (h : HasRates W scale Y S R) (positive : 0 < S) :
Arithmetic.omega F ≤ 3 * (R - Y) / S := by
apply (le_div_iff₀ positive).mpr
have hb := inequality_of_rates h
nlinarith
theorem omega_le_of_rateBounds {W : ℕ → AllFieldWitness F} {scale : ℕ → ℝ}
{Y S R : ℝ} (h : HasRateBounds W scale Y S R) (positive : 0 < S) :
Arithmetic.omega F ≤ 3 * (R - Y) / S := by
apply (le_div_iff₀ positive).mpr
have hb := inequality_of_rateBounds h
nlinarith
structure FixedLotFamily (F : Type*) [Field F] where
witness : ℕ → ℕ → AllFieldWitness F
scale : ℕ → ℕ → ℝ
yieldRate : ℕ → ℝ
volumeRate : ℕ → ℝ
rankRate : ℕ → ℝ
bounds : ∀ j, HasRateBounds (witness j) (scale j)
(yieldRate j) (volumeRate j) (rankRate j)
theorem inequality_of_fixedLotFamily (family : FixedLotFamily F) {Y S R : ℝ}
(yield_limit : Tendsto family.yieldRate atTop (𝓝 Y))
(volume_limit : Tendsto family.volumeRate atTop (𝓝 S))
(rank_limit : Tendsto family.rankRate atTop (𝓝 R)) :
Y + (Arithmetic.omega F / 3) * S ≤ R := by
apply le_of_tendsto_of_tendsto'
(yield_limit.add (volume_limit.const_mul (Arithmetic.omega F / 3))) rank_limit
intro j
exact inequality_of_rateBounds (family.bounds j)
theorem inequality_of_outer_families
(family : ℕ → FixedLotFamily F) {Y S R : ℕ → ℝ} {Yfinal Sfinal Rfinal : ℝ}
(yield_limit : ∀ K, Tendsto (family K).yieldRate atTop (𝓝 (Y K)))
(volume_limit : ∀ K, Tendsto (family K).volumeRate atTop (𝓝 (S K)))
(rank_limit : ∀ K, Tendsto (family K).rankRate atTop (𝓝 (R K)))
(outer_yield : Tendsto Y atTop (𝓝 Yfinal))
(outer_volume : Tendsto S atTop (𝓝 Sfinal))
(outer_rank : Tendsto R atTop (𝓝 Rfinal)) :
Yfinal + (Arithmetic.omega F / 3) * Sfinal ≤ Rfinal := by
apply le_of_tendsto_of_tendsto'
(outer_yield.add (outer_volume.const_mul (Arithmetic.omega F / 3))) outer_rank
intro K
exact inequality_of_fixedLotFamily (family K)
(yield_limit K) (volume_limit K) (rank_limit K)
theorem omega_le_of_outer_families
(family : ℕ → FixedLotFamily F) {Y S R : ℕ → ℝ} {Yfinal Sfinal Rfinal : ℝ}
(yield_limit : ∀ K, Tendsto (family K).yieldRate atTop (𝓝 (Y K)))
(volume_limit : ∀ K, Tendsto (family K).volumeRate atTop (𝓝 (S K)))
(rank_limit : ∀ K, Tendsto (family K).rankRate atTop (𝓝 (R K)))
(outer_yield : Tendsto Y atTop (𝓝 Yfinal))
(outer_volume : Tendsto S atTop (𝓝 Sfinal))
(outer_rank : Tendsto R atTop (𝓝 Rfinal)) (positive : 0 < Sfinal) :
Arithmetic.omega F ≤ 3 * (Rfinal - Yfinal) / Sfinal := by
apply (le_div_iff₀ positive).mpr
have hb := inequality_of_outer_families family yield_limit volume_limit rank_limit
outer_yield outer_volume outer_rank
nlinarith
theorem staggered_rank_constraint
(family : ∀ K : ℕ, 2 ≤ K → FixedLotFamily F) {H0 Cstar Sstar beta : ℝ}
(yield_limit : ∀ K hK, Tendsto (family K hK).yieldRate atTop
(𝓝 (H0 + Cstar - beta / (K : ℝ))))
(volume_limit : ∀ K hK, Tendsto (family K hK).volumeRate atTop (𝓝 Sstar))
(rank_limit : ∀ K hK, Tendsto (family K hK).rankRate atTop (𝓝 (8 * Real.log 7))) :
H0 + Cstar + (Arithmetic.omega F / 3) * Sstar ≤ 8 * Real.log 7 := by
have hfixed (K : ℕ) (hK : 2 ≤ K) :
H0 + Cstar - beta / (K : ℝ) + Arithmetic.omega F * Sstar / 3 ≤
8 * Real.log 7 := by
have hb := inequality_of_fixedLotFamily (family K hK)
(yield_limit K hK) (volume_limit K hK) (rank_limit K hK)
convert hb using 1; ring
have hb := FiniteSchedule.fixed_lot_boundary_limit hfixed
convert hb using 1; ring
theorem omega_le_of_staggered_families
(family : ∀ K : ℕ, 2 ≤ K → FixedLotFamily F) {H0 Cstar Sstar beta : ℝ}
(yield_limit : ∀ K hK, Tendsto (family K hK).yieldRate atTop
(𝓝 (H0 + Cstar - beta / (K : ℝ))))
(volume_limit : ∀ K hK, Tendsto (family K hK).volumeRate atTop (𝓝 Sstar))
(rank_limit : ∀ K hK, Tendsto (family K hK).rankRate atTop (𝓝 (8 * Real.log 7)))
(positive : 0 < Sstar) :
Arithmetic.omega F ≤ 3 * (8 * Real.log 7 - (H0 + Cstar)) / Sstar := by
apply (le_div_iff₀ positive).mpr
have hb := staggered_rank_constraint family yield_limit volume_limit rank_limit
nlinarith
theorem omega_lt_of_staggered_families
(family : ∀ K : ℕ, 2 ≤ K → FixedLotFamily F) {H0 Cstar Sstar beta target : ℝ}
(yield_limit : ∀ K hK, Tendsto (family K hK).yieldRate atTop
(𝓝 (H0 + Cstar - beta / (K : ℝ))))
(volume_limit : ∀ K hK, Tendsto (family K hK).volumeRate atTop (𝓝 Sstar))
(rank_limit : ∀ K hK, Tendsto (family K hK).rankRate atTop (𝓝 (8 * Real.log 7)))
(positive : 0 < Sstar)
(gap : 3 * (8 * Real.log 7 - (H0 + Cstar)) / Sstar < target) :
Arithmetic.omega F < target :=
lt_of_le_of_lt (omega_le_of_staggered_families family yield_limit volume_limit
rank_limit positive) gap
end MatrixMultiplication.AllFieldRateLimit
end
end MatrixAllFields
end OAI
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