| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Histories.ZeroHistorySums |
| import OAI.LinearAlgebra.MatrixFields.Parameters.TerminalPopulations |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| section |
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldTerminalRates |
|
|
| open AllFieldHistory AllFieldParameters AllFieldTerminalStatisticLaws |
| open AllFieldZeroLeafRates AllFieldZeroHistorySums Filter |
| open scoped BigOperators Topology |
| attribute [local instance] Classical.propDecidable |
|
|
| variable {K : ℕ} |
|
|
| def statisticCounts (allocation : Allocation) (h : TerminalZero K) : Slot h → ℕ := |
| AllFieldTerminalPopulations.count allocation history law 1 h |
|
|
| def terminalDilation (K : ℕ) (m : ℕ) : ℕ := |
| AllFieldTerminalPopulations.dilation (law (K := K)) m |
|
|
| def terminalLength (allocation : Allocation) (K : ℕ) (m : ℕ) : ℕ := |
| populationLength (K := K) allocation (terminalDilation K m) |
|
|
| theorem terminalDilation_pos {m : ℕ} (hm : 0 < m) : |
| 0 < terminalDilation K m := |
| AllFieldTerminalPopulations.dilation_pos law hm |
|
|
| theorem terminalLength_pos (allocation : Allocation) {m : ℕ} (hm : 0 < m) : |
| 0 < terminalLength allocation K m := |
| AllFieldPopulationCounts.blockLength_pos _ (terminalDilation_pos hm) |
|
|
| theorem terminalLength_mul_one (allocation : Allocation) (m : ℕ) : |
| terminalLength allocation K m = m * terminalLength allocation K 1 := by |
| simp only [terminalLength, terminalDilation, AllFieldTerminalPopulations.dilation, |
| populationLength, AllFieldPopulationCounts.blockLength, one_mul, mul_assoc] |
|
|
| theorem statisticCounts_pos (allocation : Allocation) (h : TerminalZero K) : |
| 0 < ∑ s, statisticCounts allocation h s := |
| AllFieldTerminalPopulations.sum_count_pos allocation history law law_nonneg law_total |
| (by decide) h (history_amount_pos allocation h) |
|
|
| theorem statisticCounts_ratio (allocation : Allocation) (h : TerminalZero K) (s : Slot h) : |
| (statisticCounts allocation h s : ℝ) / |
| (∑ s, statisticCounts allocation h s : ℕ) = (law h s : ℝ) := |
| AllFieldTerminalPopulations.count_ratio_real allocation history law law_nonneg law_total |
| (by decide) h (history_amount_pos allocation h) s |
|
|
| theorem statisticCounts_sum_inl (allocation : Allocation) (h : AZero K) : |
| (∑ s : Slot (.inl h), statisticCounts allocation (.inl h) s) = |
| ∑ s : PairSlot, statisticCounts allocation (.inl h) s := by |
| exact Fintype.sum_equiv (Equiv.refl _) _ _ (fun _ => rfl) |
|
|
| theorem statisticCounts_sum_inr (allocation : Allocation) (h : BZero K) : |
| (∑ s : Slot (.inr h), statisticCounts allocation (.inr h) s) = |
| ∑ s : Fin 6, statisticCounts allocation (.inr h) s := by |
| exact Fintype.sum_equiv (Equiv.refl _) _ _ (fun _ => rfl) |
|
|
| theorem statisticCounts_sum (allocation : Allocation) (h : TerminalZero K) : |
| (∑ s, statisticCounts allocation h s) = |
| population allocation (terminalDilation K 1) (history h) := |
| AllFieldTerminalPopulations.sum_count allocation history law law_nonneg law_total 1 h |
|
|
| theorem statisticCounts_repeat (allocation : Allocation) (m : ℕ) |
| (h : TerminalZero K) (s : Slot h) : |
| m * statisticCounts allocation h s = |
| AllFieldTerminalPopulations.count allocation history law m h s := |
| (AllFieldTerminalPopulations.count_mul_one allocation history law m h s).symm |
|
|
| theorem statisticCounts_repeat_sum (allocation : Allocation) (m : ℕ) |
| (h : TerminalZero K) : |
| (∑ s, m * statisticCounts allocation h s) = |
| population allocation (terminalDilation K m) (history h) := by |
| simp only [statisticCounts_repeat] |
| exact AllFieldTerminalPopulations.sum_count allocation history law law_nonneg law_total m h |
|
|
| theorem statisticCounts_sum_cast (allocation : Allocation) (h : TerminalZero K) : |
| ((∑ s, statisticCounts allocation h s : ℕ) : ℝ) = |
| (terminalLength allocation K 1 : ℝ) * (amount allocation (history h) : ℝ) := by |
| rw [statisticCounts_sum] |
| exact_mod_cast population_cast allocation (terminalDilation K 1) (history h) |
|
|
| def ZeroWords (allocation : Allocation) (m : ℕ) : TerminalZero K → Type |
| | .inl h => fourWords (aShape h.1.val) (statisticCounts allocation (.inl h)) m |
| | .inr h => twoWords (bWeight h) (statisticCounts allocation (.inr h)) m |
|
|
| local instance terminalTwoWordsFintype (allocation : Allocation) (m : ℕ) (h : BZero K) : |
| Fintype (twoWords (bWeight h) (statisticCounts allocation (.inr h)) m) := |
| twoWordsFintype _ _ _ |
|
|
| instance zeroWordsFintype (allocation : Allocation) (m : ℕ) (h : TerminalZero K) : |
| Fintype (ZeroWords allocation m h) := by |
| cases h <;> dsimp [ZeroWords] <;> infer_instance |
|
|
| theorem zeroWords_card_inl (allocation : Allocation) (m : ℕ) (h : AZero K) : |
| Fintype.card (ZeroWords allocation m (.inl h)) = |
| Fintype.card (fourWords (aShape h.1.val) (statisticCounts allocation (.inl h)) m) := by |
| exact Fintype.card_congr (Equiv.refl _) |
|
|
| theorem zeroWords_card_inr (allocation : Allocation) (m : ℕ) (h : BZero K) : |
| Fintype.card (ZeroWords allocation m (.inr h)) = |
| Fintype.card (twoWords (bWeight h) (statisticCounts allocation (.inr h)) m) := by |
| exact Fintype.card_congr (Equiv.refl _) |
|
|
| theorem zeroWords_card_pos (allocation : Allocation) (m : ℕ) (h : TerminalZero K) : |
| 0 < Fintype.card (ZeroWords allocation m h) := by |
| cases h with |
| | inl h => |
| rw [zeroWords_card_inl] |
| exact four_card_pos (aShape h.1.val) (statisticCounts allocation (.inl h)) |
| (by simpa only [statisticCounts_sum_inl] using statisticCounts_pos allocation (.inl h)) |
| (by |
| intro slot |
| simpa only [statisticCounts_sum_inl, law_inl] using |
| statisticCounts_ratio allocation (.inl h) slot) m |
| | inr h => |
| rw [zeroWords_card_inr] |
| exact two_card_pos (binaryParameter (bParent h) (bShape h.1.val)) |
| (bWeight h) (statisticCounts allocation (.inr h)) |
| (by simpa only [statisticCounts_sum_inr] using statisticCounts_pos allocation (.inr h)) |
| (by |
| intro slot |
| simpa only [statisticCounts_sum_inr, law_inr] using |
| statisticCounts_ratio allocation (.inr h) slot) m |
|
|
| theorem tendsto_zeroWords_local (allocation : Allocation) (h : TerminalZero K) : |
| Tendsto (fun m : ℕ => Real.log (Fintype.card (ZeroWords allocation m h) : ℝ) / |
| ((m : ℝ) * (∑ s, statisticCounts allocation h s : ℕ))) atTop |
| (𝓝 (zeroRate h)) := by |
| cases h with |
| | inl h => |
| simp only [zeroWords_card_inl, zeroRate, statisticCounts_sum_inl] |
| simpa only [Fintype.card_eq_nat_card] using |
| four_tendsto_log_card (aShape h.1.val) (statisticCounts allocation (.inl h)) |
| (by simpa only [statisticCounts_sum_inl] using statisticCounts_pos allocation (.inl h)) |
| (by |
| intro slot |
| simpa only [statisticCounts_sum_inl, law_inl] using |
| statisticCounts_ratio allocation (.inl h) slot) |
| | inr h => |
| simp only [zeroWords_card_inr, zeroRate, statisticCounts_sum_inr] |
| simpa only [Fintype.card_eq_nat_card] using |
| two_tendsto_log_card (binaryParameter (bParent h) (bShape h.1.val)) |
| (bWeight h) (statisticCounts allocation (.inr h)) |
| (by simpa only [statisticCounts_sum_inr] using statisticCounts_pos allocation (.inr h)) |
| (by |
| intro slot |
| simpa only [statisticCounts_sum_inr, law_inr] using |
| statisticCounts_ratio allocation (.inr h) slot) |
|
|
| theorem tendsto_zeroWords (hK : 0 < K) (allocation : Allocation) (h : TerminalZero K) : |
| Tendsto (fun m : ℕ => Real.log (Fintype.card (ZeroWords allocation m h) : ℝ) / |
| ((K : ℝ) * terminalLength allocation K m)) atTop |
| (𝓝 ((amount allocation (history h) : ℝ) * zeroRate h / K)) := by |
| have ht := (tendsto_zeroWords_local allocation h).mul_const |
| ((amount allocation (history h) : ℝ) / K) |
| have hr : zeroRate h * ((amount allocation (history h) : ℝ) / K) = |
| (amount allocation (history h) : ℝ) * zeroRate h / K := by ring |
| rw [hr] at ht |
| apply ht.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm |
| have hm' : (m : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hm |
| have hk' : (K : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hK |
| have hn' : (terminalLength allocation K 1 : ℝ) ≠ 0 := by |
| exact_mod_cast Nat.ne_of_gt (terminalLength_pos allocation (K := K) (m := 1) (by decide)) |
| have ha' : (amount allocation (history h) : ℝ) ≠ 0 := by |
| exact_mod_cast ne_of_gt (history_amount_pos allocation h) |
| rw [terminalLength_mul_one, Nat.cast_mul, statisticCounts_sum_cast] |
| field_simp [hm', hk', hn', ha'] |
|
|
| def zeroVolume (allocation : Allocation) (m : ℕ) : ℕ := |
| Fintype.card (∀ h : TerminalZero K, ZeroWords allocation m h) |
|
|
| theorem zeroVolume_pos (allocation : Allocation) (m : ℕ) : |
| 0 < zeroVolume (K := K) allocation m := by |
| simp only [zeroVolume, Fintype.card_pi] |
| exact Finset.prod_pos fun h _ => zeroWords_card_pos allocation m h |
|
|
| theorem log_zeroVolume (allocation : Allocation) (m : ℕ) : |
| Real.log (zeroVolume (K := K) allocation m : ℝ) = |
| ∑ h : TerminalZero K, Real.log (Fintype.card (ZeroWords allocation m h) : ℝ) := by |
| rw [zeroVolume, Fintype.card_pi, Nat.cast_prod] |
| exact Real.log_prod (fun h _ => by |
| exact_mod_cast Nat.ne_of_gt (zeroWords_card_pos allocation m h)) |
|
|
| theorem tendsto_zeroVolume (hK : 0 < K) (allocation : Allocation) : |
| Tendsto (fun m : ℕ => Real.log (zeroVolume (K := K) allocation m : ℝ) / |
| ((K : ℝ) * terminalLength allocation K m)) atTop |
| (𝓝 (S1 + S2 - AllFieldStageCTerminalRates.interiorS2)) := by |
| have ht := tendsto_finsetSum Finset.univ |
| (fun h _ => tendsto_zeroWords hK allocation h) |
| have hr : (∑ h : TerminalZero K, (amount allocation (history h) : ℝ) * zeroRate h / K) = |
| S1 + S2 - AllFieldStageCTerminalRates.interiorS2 := by |
| rw [← Finset.sum_div, zero_history_sum K allocation] |
| exact mul_div_cancel_left₀ _ (by exact_mod_cast Nat.ne_of_gt hK) |
| rw [hr] at ht |
| convert ht using 1 |
| ext m |
| rw [log_zeroVolume, Finset.sum_div] |
|
|
| def terminalVolume (allocation : Allocation) (m : ℕ) : ℕ := |
| AllFieldInitialLeafRates.volume (K := K) allocation (terminalDilation K m) * |
| zeroVolume (K := K) allocation m * |
| CWStageCProducts.populationVolume (K := K) allocation (terminalDilation K m) |
|
|
| theorem stageCVolume_pos (allocation : Allocation) (dilation : ℕ) : |
| 0 < CWStageCProducts.populationVolume (K := K) allocation dilation := by |
| unfold CWStageCProducts.populationVolume |
| rw [CWStageCProducts.historyVolume_eq_prod] |
| exact Finset.prod_pos fun h _ => CWStageCProducts.volume_pos _ _ |
|
|
| theorem terminalVolume_pos (allocation : Allocation) (m : ℕ) : |
| 0 < terminalVolume (K := K) allocation m := by |
| exact Nat.mul_pos |
| (Nat.mul_pos (AllFieldInitialLeafRates.volume_pos allocation _) |
| (zeroVolume_pos allocation m)) (stageCVolume_pos allocation _) |
|
|
| theorem log_terminalVolume (allocation : Allocation) (m : ℕ) : |
| Real.log (terminalVolume (K := K) allocation m : ℝ) = |
| Real.log (AllFieldInitialLeafRates.volume (K := K) allocation |
| (terminalDilation K m) : ℝ) + |
| Real.log (zeroVolume (K := K) allocation m : ℝ) + |
| Real.log (CWStageCProducts.populationVolume (K := K) allocation |
| (terminalDilation K m) : ℝ) := by |
| have hI : (AllFieldInitialLeafRates.volume (K := K) allocation |
| (terminalDilation K m) : ℝ) ≠ 0 := by |
| exact_mod_cast Nat.ne_of_gt (AllFieldInitialLeafRates.volume_pos allocation _) |
| have hZ : (zeroVolume (K := K) allocation m : ℝ) ≠ 0 := by |
| exact_mod_cast Nat.ne_of_gt (zeroVolume_pos allocation m) |
| have hC : (CWStageCProducts.populationVolume (K := K) allocation |
| (terminalDilation K m) : ℝ) ≠ 0 := by |
| exact_mod_cast Nat.ne_of_gt (stageCVolume_pos allocation _) |
| simp only [terminalVolume, Nat.cast_mul, Real.log_mul (mul_ne_zero hI hZ) hC, |
| Real.log_mul hI hZ] |
|
|
| theorem normalized_log_initialVolume (hK : 0 < K) (allocation : Allocation) |
| {m : ℕ} (hm : 0 < m) : |
| Real.log (AllFieldInitialLeafRates.volume (K := K) allocation |
| (terminalDilation K m) : ℝ) / ((K : ℝ) * terminalLength allocation K m) = S0 := by |
| rw [AllFieldInitialLeafRates.log_volume_eq_S0] |
| change (terminalLength allocation K m : ℝ) * ((K : ℝ) * S0) / |
| ((K : ℝ) * terminalLength allocation K m) = S0 |
| have hk : (K : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hK |
| have hn : (terminalLength allocation K m : ℝ) ≠ 0 := by |
| exact_mod_cast Nat.ne_of_gt (terminalLength_pos allocation hm) |
| field_simp [hk, hn] |
|
|
| theorem normalized_log_stageCVolume (hK : 0 < K) (allocation : Allocation) |
| {m : ℕ} (hm : 0 < m) : |
| Real.log (CWStageCProducts.populationVolume (K := K) allocation |
| (terminalDilation K m) : ℝ) / ((K : ℝ) * terminalLength allocation K m) = |
| AllFieldStageCTerminalRates.interiorS2 := |
| AllFieldStageCTerminalRates.normalized_log_populationVolume_eq_interior |
| hK allocation _ (terminalDilation_pos hm) |
|
|
| theorem tendsto_terminalVolume (hK : 0 < K) (allocation : Allocation) : |
| Tendsto (fun m : ℕ => Real.log (terminalVolume (K := K) allocation m : ℝ) / |
| ((K : ℝ) * terminalLength allocation K m)) atTop (𝓝 S) := by |
| have hI : Tendsto (fun m : ℕ => |
| Real.log (AllFieldInitialLeafRates.volume (K := K) allocation |
| (terminalDilation K m) : ℝ) / ((K : ℝ) * terminalLength allocation K m)) |
| atTop (𝓝 S0) := by |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm |
| exact (normalized_log_initialVolume hK allocation hm).symm |
| have hC : Tendsto (fun m : ℕ => |
| Real.log (CWStageCProducts.populationVolume (K := K) allocation |
| (terminalDilation K m) : ℝ) / ((K : ℝ) * terminalLength allocation K m)) |
| atTop (𝓝 AllFieldStageCTerminalRates.interiorS2) := by |
| apply tendsto_const_nhds.congr' |
| filter_upwards [eventually_gt_atTop (0 : ℕ)] with m hm |
| exact (normalized_log_stageCVolume hK allocation hm).symm |
| have ht := (hI.add (tendsto_zeroVolume hK allocation)).add hC |
| have hr : S0 + (S1 + S2 - AllFieldStageCTerminalRates.interiorS2) + |
| AllFieldStageCTerminalRates.interiorS2 = S := by unfold S; ring |
| rw [hr] at ht |
| convert ht using 1 |
| ext m |
| rw [log_terminalVolume, add_div, add_div] |
|
|
| end MatrixMultiplication.AllFieldTerminalRates |
|
|
| end |
| end |
|
|
| end MatrixAllFields |
|
|
| end OAI |
|
|
