| import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore |
|
|
| namespace OAI |
|
|
| /-! Finite extraction histories, inherited masks and recovery bounds. -/ |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldHistoryRawMasks |
|
|
| open AllFieldHistory AllFieldHistoryChildLaws AllFieldHistorySupport AllFieldHistoryMasks |
| open JointCanonicalization PermutationMatching |
| open scoped BigOperators |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| variable {K tick : ℕ} |
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|
| def rawPairCount (allocation : Allocation) (m : ℕ) (i : MaskIndex K tick) |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : ℝ := |
| ∑ j : JointPopulation.Positions (activeCounts allocation m) i.1, |
| if statistic i.1 (w i.1 j).1 = i.2.1 i.1 ∧ |
| statistic i.1 (w i.1 j).2 = i.2.2 i.1 then 1 else 0 |
|
|
| def rawPass (allocation : Allocation) (m : ℕ) (ε : ℝ) (side : Fin 3) |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : Prop := |
| ∀ i : MaskIndex K tick, |
| |rawPairCount allocation m i w / |
| (population allocation m (i.1.val.1.source, i.1.val.2) : ℝ) - |
| center allocation side i| ≤ pairWidth ε i.1 |
|
|
| theorem sum_positiveClasses (allocation : Allocation) (m : ℕ) (h : Active K tick) |
| (f : ∀ c : Classes K tick, Positions allocation m c → ℝ) : |
| (∑ c : positiveClasses allocation h, ∑ j, f c.val j) = |
| ∑ u : JointPopulation.Shape, ∑ j, f (h, u) j := by |
| have hzero (c : Classes K tick) (hc : ¬0 < activeCounts allocation 1 c.1 c.2) : |
| (∑ j, f c j) = 0 := by |
| have hcount : activeCounts allocation m c.1 c.2 = 0 := by |
| rw [activeCounts_dilation, Nat.eq_zero_of_not_pos hc, mul_zero] |
| apply Finset.sum_eq_zero |
| intro j _ |
| have hj : j.val < 0 := by simpa only [hcount] using j.isLt |
| exact (Nat.not_lt_zero _ hj).elim |
| rw [Finset.sum_coe_sort (positiveClasses allocation h) |
| (fun c : Classes K tick => ∑ j, f c j)] |
| unfold positiveClasses |
| rw [Finset.sum_filter] |
| calc |
| (∑ c : Classes K tick, |
| if c.1 = h ∧ 0 < activeCounts allocation 1 c.1 c.2 then ∑ j, f c j else 0) = |
| ∑ c : Classes K tick, if c.1 = h then ∑ j, f c j else 0 := by |
| apply Finset.sum_congr rfl |
| intro c _ |
| by_cases hc : 0 < activeCounts allocation 1 c.1 c.2 |
| · simp only [hc, and_true] |
| · rw [hzero c hc] |
| simp |
| _ = ∑ u : JointPopulation.Shape, ∑ j, f (h, u) j := by |
| rw [Fintype.sum_prod_type, Fintype.sum_eq_single h] |
| · simp only [ite_true] |
| · intro h' hh' |
| apply Finset.sum_eq_zero |
| intro u _ |
| exact ite_eq_right hh' |
|
|
| theorem selectedWordPairCount_pairEquiv (allocation : Allocation) (m : ℕ) |
| (e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m)) |
| (i : MaskIndex K tick) (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : |
| selectedWordPairCount (testedClasses allocation i) |
| (pairEquiv (activeCounts allocation m) |
| (JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w) |
| classStatistic classStatistic (leftSymbol i) (rightSymbol i) = |
| rawPairCount allocation m i w := by |
| let v : Words (K := K) (tick := tick) allocation m := |
| pairEquiv (activeCounts allocation m) |
| (JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w |
| have hsum := sum_positiveClasses allocation m i.1 (fun c j => |
| if classStatistic c (v.left c j) = leftSymbol i c ∧ |
| classStatistic c (v.right c j) = rightSymbol i c then (1 : ℝ) else 0) |
| unfold selectedWordPairCount |
| rw [testedClasses, hsum] |
| unfold rawPairCount |
| rw [← Fintype.sum_sigma'] |
| apply Fintype.sum_equiv (positionEquiv (activeCounts allocation m) e i.1).symm |
| rintro ⟨u, j⟩ |
| rfl |
|
|
| theorem testedClasses_populationSize (allocation : Allocation) (m : ℕ) |
| (i : MaskIndex K tick) : |
| populationSize (fun c : testedClasses allocation i => Positions allocation m c.val) = |
| (population allocation m (i.1.val.1.source, i.1.val.2) : ℝ) := by |
| change populationSize (selectedPositions allocation m (positiveClasses allocation i.1)) = _ |
| rw [positiveClasses_populationSize, population_dilation allocation m, Nat.cast_mul] |
| exact mul_comm _ _ |
|
|
| theorem raw_population_pos (allocation : Allocation) {m : ℕ} (hm : 0 < m) |
| (h : Active K tick) : |
| 0 < (population allocation m (h.val.1.source, h.val.2) : ℝ) := by |
| exact_mod_cast work_source_population_pos allocation hm h.val |
|
|
| theorem passes_pairEquiv_iff_rawPass (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (side : Fin 3) |
| (e : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m)) |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : |
| passes allocation m ε side |
| (pairEquiv (activeCounts allocation m) |
| (JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w) ↔ |
| rawPass allocation m ε side w := by |
| simp only [passes, selectedWordPairMaskBad, selectedWordPairCount_pairEquiv, |
| testedClasses_populationSize, rawPass, not_lt] |
|
|
| theorem passes_pairEquiv_target_independent (allocation : Allocation) (m : ℕ) (ε : ℝ) |
| (side : Fin 3) |
| (e f : JointPopulation.Target (activeCounts (K := K) (tick := tick) allocation m)) |
| (w : ActiveRawPairs (K := K) (tick := tick) allocation m) : |
| passes allocation m ε side |
| (pairEquiv (activeCounts allocation m) |
| (JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) e w) ↔ |
| passes allocation m ε side |
| (pairEquiv (activeCounts allocation m) |
| (JointCanonicalCW.Left activeHalfLength) (JointCanonicalCW.Right activeHalfLength) f w) := |
| (passes_pairEquiv_iff_rawPass allocation m ε side e w).trans |
| (passes_pairEquiv_iff_rawPass allocation m ε side f w).symm |
|
|
| end MatrixMultiplication.AllFieldHistoryRawMasks |
|
|
| end |
|
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| end OAI |
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|
