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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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tick
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryIteration
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable Classical.decEq
section Presentation
variable {F : Type*} [Field F] {SX SY SZ X Y Z : Type}
[Fintype SX] [Fintype SY] [Fintype SZ]
{source : Tensor F SX SY SZ} {target next : Tensor F X Y Z}
private def retarget (E : Execution source target) (h : target = next) :
Execution source next where
Copies := E.Copies
copies_positive := E.copies_positive
map := {
x := E.map.x
y := E.map.y
z := E.map.z
coefficient := E.map.coefficient.trans h
}
@[simp] private theorem retarget_copies (E : Execution source target) (h : target = next) :
Fintype.card (retarget E h).Copies = Fintype.card E.Copies := rfl
end Presentation
variable {K : ℕ} (allocation : Allocation) (m : ℕ) (ε slack : ℝ)
abbrev ScheduleChoices := ∀ t : Fin (K + 2),
AllFieldHistoryTick.Choices (K := K) (tick := t.val) allocation m ε slack
def Labels (InitialTag : Type)
(chosen : ScheduleChoices (K := K) allocation m ε slack) :
(n : ℕ) → n ≤ K + 2 → Type
| 0, _ => InitialTag
| n + 1, hn =>
Labels InitialTag chosen n (Nat.le_of_succ_le hn) ×
AllFieldHistoryTick.Tags allocation m ε slack (chosen ⟨n, hn⟩)
instance labelsFintype (InitialTag : Type) [Fintype InitialTag]
(chosen : ScheduleChoices (K := K) allocation m ε slack) :
(n : ℕ) → (hn : n ≤ K + 2) →
Fintype (Labels allocation m ε slack InitialTag chosen n hn)
| 0, _ => inferInstanceAs (Fintype InitialTag)
| n + 1, hn => by
let := labelsFintype InitialTag chosen n (Nat.le_of_succ_le hn)
exact inferInstanceAs (Fintype
(Labels allocation m ε slack InitialTag chosen n (Nat.le_of_succ_le hn) ×
AllFieldHistoryTick.Tags allocation m ε slack (chosen ⟨n, hn⟩)))
theorem card_labels (InitialTag : Type) [Fintype InitialTag]
(chosen : ScheduleChoices (K := K) allocation m ε slack)
(n : ℕ) (hn : n ≤ K + 2) :
Fintype.card (Labels allocation m ε slack InitialTag chosen n hn) =
Fintype.card InitialTag * ∏ t : Fin n, ∏ _sigma : Placement,
⌊Real.exp ((m : ℝ) *
(AllFieldGroupSelection.nativeCapacity (K := K) (tick := t.val) allocation - slack))⌋₊ := by
induction n with
| zero =>
change Fintype.card InitialTag = Fintype.card InitialTag * 1
exact (Nat.mul_one _).symm
| succ n ih =>
change Fintype.card
(Labels allocation m ε slack InitialTag chosen n (Nat.le_of_succ_le hn) ×
AllFieldHistoryTick.Tags allocation m ε slack (chosen ⟨n, hn⟩)) = _
rw [Fintype.card_prod, AllFieldHistoryTick.card_tags, ih, Fin.prod_univ_castSucc]
exact Nat.mul_assoc _ _ _
abbrev Tags (InitialTag : Type)
(chosen : ScheduleChoices (K := K) allocation m ε slack) :=
Labels allocation m ε slack InitialTag chosen (K + 2) le_rfl
theorem card_tags (InitialTag : Type) [Fintype InitialTag]
(chosen : ScheduleChoices (K := K) allocation m ε slack) :
Fintype.card (Tags allocation m ε slack InitialTag chosen) =
Fintype.card InitialTag * ∏ t : Fin (K + 2), ∏ _sigma : Placement,
⌊Real.exp ((m : ℝ) *
(AllFieldGroupSelection.nativeCapacity (K := K) (tick := t.val) allocation - slack))⌋₊ :=
card_labels allocation m ε slack InitialTag chosen (K + 2) le_rfl
variable (F : Type*) [Field F] (hε : 0 < ε)
(hm : ∀ t : Fin (K + 2),
AllFieldHistoryRecovery.minimumDilation (K := K) (tick := t.val) allocation ε ≤ m)
(chosen : ScheduleChoices (K := K) allocation m ε slack)
{InitialTag : Type} [Fintype InitialTag]
(start : Execution (cwSource F K (populationLength (K := K) allocation m))
(Tensor.directSum (fun _ : InitialTag => stateTensor F (K := K) allocation m ε 0)))
def run : (n : ℕ) → (hn : n ≤ K + 2) →
Execution (cwSource F K (populationLength (K := K) allocation m))
(Tensor.directSum (fun _ : Labels allocation m ε slack InitialTag chosen n hn =>
stateTensor F (K := K) allocation m ε n))
| 0, _ => start
| n + 1, hn =>
let step := AllFieldHistoryTick.execution allocation m ε slack F hε
(hm ⟨n, hn⟩) (chosen ⟨n, hn⟩)
let operation := Execution.branchOperation
(stateTensor F (K := K) allocation m ε n)
(stateTensor F (K := K) allocation m ε (n + 1))
(run n (Nat.le_of_succ_le hn)) step.copies_positive step.map
retarget operation (by
funext x y z
simp only [Tensor.directSum]
split_ifs <;> first | rfl | contradiction)
theorem run_copies (n : ℕ) (hn : n ≤ K + 2) :
Fintype.card (run allocation m ε slack F hε hm chosen start n hn).Copies =
Fintype.card start.Copies * ∏ t : Fin n,
2 ^ (3 * AllFieldHistoryRecovery.shiftCount (K := K) (tick := t.val) allocation ε m) := by
induction n with
| zero =>
change Fintype.card start.Copies = Fintype.card start.Copies * 1
exact (Nat.mul_one _).symm
| succ n ih =>
rw [run]
erw [retarget_copies]
rw [Execution.branchOperation_copies, AllFieldHistoryTick.execution_copies,
ih, Fin.prod_univ_castSucc]
ring
def execution :
Execution (cwSource F K (populationLength (K := K) allocation m))
(Tensor.directSum (fun _ : Tags allocation m ε slack InitialTag chosen =>
stateTensor F (K := K) allocation m ε (K + 2))) :=
run allocation m ε slack F hε hm chosen start (K + 2) le_rfl
@[simp] theorem execution_copies :
Fintype.card (execution allocation m ε slack F hε hm chosen start).Copies =
Fintype.card start.Copies * ∏ t : Fin (K + 2),
2 ^ (3 * AllFieldHistoryRecovery.shiftCount (K := K) (tick := t.val) allocation ε m) :=
run_copies allocation m ε slack F hε hm chosen start (K + 2) le_rfl
end MatrixMultiplication.AllFieldHistoryIteration
end
end OAI
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