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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Support
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedOrbitProfiles
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryOrbitCount
open AllFieldHistory AllFieldHistorySupport PermutationMatching
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
abbrev Classes (K tick : ℕ) := ActiveClass K tick
abbrev Positions {K tick : ℕ} (allocation : Allocation) (m : ℕ)
(c : Classes K tick) :=
JointCanonicalization.ClassPositions (activeCounts allocation m) c
abbrev Letters {K tick : ℕ} (c : Classes K tick) :=
Fin (activeHalfLength c.1) → Fin 7
abbrev ActualOrbit {K tick : ℕ} (allocation : Allocation) (m : ℕ) :=
PairOrbit (Positions (K := K) (tick := tick) allocation m)
(Letters (K := K) (tick := tick)) (Letters (K := K) (tick := tick))
noncomputable instance actualOrbitFintype {K tick : ℕ}
(allocation : Allocation) (m : ℕ) :
Fintype (ActualOrbit (K := K) (tick := tick) allocation m) := by
haveI : Finite (ActualOrbit (K := K) (tick := tick) allocation m) :=
Finite.of_injective pairOrbitProfile pairOrbitProfile_injective
exact Fintype.ofFinite _
def basePopulationBound {K tick : ℕ} (allocation : Allocation) : ℕ :=
∑ c : Classes K tick, activeCounts allocation 1 c.1 c.2
def orbitDegree (K tick : ℕ) : ℕ :=
2 * ∑ c : Classes K tick, 7 ^ activeHalfLength c.1
theorem card_letters {K tick : ℕ} (c : Classes K tick) :
Fintype.card (Letters c) = 7 ^ activeHalfLength c.1 := by
simp only [Letters, Fintype.card_fun, Fintype.card_fin]
theorem class_card_le {K tick : ℕ} (allocation : Allocation) (m : ℕ)
(c : Classes K tick) :
Fintype.card (Positions allocation m c) ≤
basePopulationBound (K := K) (tick := tick) allocation * m := by
have hc : activeCounts allocation 1 c.1 c.2 ≤
basePopulationBound (K := K) (tick := tick) allocation := by
unfold basePopulationBound
exact Finset.single_le_sum
(fun (d : Classes K tick) _ => Nat.zero_le (activeCounts allocation 1 d.1 d.2))
(Finset.mem_univ c)
simp only [Positions, JointCanonicalization.ClassPositions, Fintype.card_fin,
activeCounts_dilation allocation m]
simpa only [Nat.mul_comm] using Nat.mul_le_mul_left m hc
theorem card_actualOrbit_le {K tick : ℕ} (allocation : Allocation) (m : ℕ) :
Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) ≤
(basePopulationBound (K := K) (tick := tick) allocation * m + 1) ^
orbitDegree K tick := by
simpa only [ActualOrbit, orbitDegree, card_letters, two_mul] using
(card_pairOrbits_le_common_bound
(P := Positions (K := K) (tick := tick) allocation m)
(X := Letters (K := K) (tick := tick))
(Y := Letters (K := K) (tick := tick))
(basePopulationBound (K := K) (tick := tick) allocation * m)
(class_card_le allocation m))
theorem card_actualOrbit_le_polynomial {K tick : ℕ}
(allocation : Allocation) (m : ℕ) :
Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) ≤
(basePopulationBound (K := K) (tick := tick) allocation + 1) ^ orbitDegree K tick *
(m + 1) ^ orbitDegree K tick := by
refine (card_actualOrbit_le allocation m).trans ?_
rw [← mul_pow]
apply Nat.pow_le_pow_left
nlinarith [Nat.zero_le (basePopulationBound (K := K) (tick := tick) allocation)]
def orbitPrefactor {K tick : ℕ} (allocation : Allocation) : ℝ :=
((basePopulationBound (K := K) (tick := tick) allocation + 1 : ℕ) : ℝ) ^
orbitDegree K tick
theorem orbitPrefactor_pos {K tick : ℕ} (allocation : Allocation) :
0 < orbitPrefactor (K := K) (tick := tick) allocation := by
unfold orbitPrefactor
positivity
theorem card_actualOrbit_le_polynomial_real {K tick : ℕ}
(allocation : Allocation) (m : ℕ) :
(Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) : ℝ) ≤
orbitPrefactor (K := K) (tick := tick) allocation *
((m : ℝ) + 1) ^ orbitDegree K tick := by
unfold orbitPrefactor
exact_mod_cast card_actualOrbit_le_polynomial (K := K) (tick := tick) allocation m
theorem usedOrbits_card_le {K tick : ℕ} (allocation : Allocation) (m : ℕ)
(s : Finset (ActualOrbit (K := K) (tick := tick) allocation m)) :
s.card ≤ (basePopulationBound (K := K) (tick := tick) allocation + 1) ^
orbitDegree K tick * (m + 1) ^ orbitDegree K tick := by
have hs : s.card ≤ Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) := by
rw [Nat.card_eq_fintype_card]
exact s.card_le_univ
exact hs.trans (card_actualOrbit_le_polynomial allocation m)
theorem usedOrbits_card_le_real {K tick : ℕ} (allocation : Allocation) (m : ℕ)
(s : Finset (ActualOrbit (K := K) (tick := tick) allocation m)) :
(s.card : ℝ) ≤ orbitPrefactor (K := K) (tick := tick) allocation *
((m : ℝ) + 1) ^ orbitDegree K tick := by
unfold orbitPrefactor
exact_mod_cast usedOrbits_card_le allocation m s
theorem taggedUsedOrbits_card_le_real {K tick : ℕ} (allocation : Allocation) (m : ℕ)
(s : Finset (Fin 3 × ActualOrbit (K := K) (tick := tick) allocation m)) :
(s.card : ℝ) ≤ (3 * orbitPrefactor (K := K) (tick := tick) allocation) *
((m : ℝ) + 1) ^ orbitDegree K tick := by
have hs : s.card ≤ 3 * Nat.card
(ActualOrbit (K := K) (tick := tick) allocation m) := by
simpa only [Fintype.card_prod, Fintype.card_fin, Nat.card_eq_fintype_card] using
s.card_le_univ
have hsreal : (s.card : ℝ) ≤ 3 * (Nat.card
(ActualOrbit (K := K) (tick := tick) allocation m) : ℝ) := by
exact_mod_cast hs
calc
_ ≤ 3 * (Nat.card (ActualOrbit (K := K) (tick := tick) allocation m) : ℝ) := hsreal
_ ≤ 3 * (orbitPrefactor (K := K) (tick := tick) allocation *
((m : ℝ) + 1) ^ orbitDegree K tick) :=
mul_le_mul_of_nonneg_left (card_actualOrbit_le_polynomial_real allocation m)
(by norm_num)
_ = _ := (mul_assoc _ _ _).symm
end MatrixMultiplication.AllFieldHistoryOrbitCount
end
end OAI
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