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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.Tensors
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryRegrouping
open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily
open scoped BigOperators Classical
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ} (allocation : Allocation) (m : ℕ)
abbrev StateWords (tick : ℕ) := ∀ h : State K tick, HistoryWord allocation m h.val
abbrev DueWords := ∀ h : DueState K tick, HistoryWord allocation m h.val.val
abbrev CarriedWords := ∀ h : Carried K tick, HistoryWord allocation m h.val.val
abbrev ActiveWords := ∀ h : Active K tick,
HistoryWord allocation m (activeSourceEquiv h).val.val
abbrev ProducedWords := ∀ h : Produced K tick, HistoryWord allocation m h.val
instance dueWordsFintype : Fintype (DueWords (K := K) (tick := tick) allocation m) :=
inferInstanceAs (Fintype (∀ h : DueState K tick, HistoryWord allocation m h.val.val))
instance carriedWordsFintype : Fintype (CarriedWords (K := K) (tick := tick) allocation m) :=
inferInstanceAs (Fintype (∀ h : Carried K tick, HistoryWord allocation m h.val.val))
instance activeWordsFintype : Fintype (ActiveWords (K := K) (tick := tick) allocation m) :=
inferInstanceAs (Fintype (∀ h : Active K tick,
HistoryWord allocation m (activeSourceEquiv h).val.val))
instance producedWordsFintype : Fintype (ProducedWords (K := K) (tick := tick) allocation m) :=
inferInstanceAs (Fintype (∀ h : Produced K tick, HistoryWord allocation m h.val))
def carriedTensor (F : Type*) [Field F] (ε : ℝ) :
Tensor F (CarriedWords (K := K) (tick := tick) allocation m)
(CarriedWords (K := K) (tick := tick) allocation m) (CarriedWords (K := K) (tick := tick) allocation m) :=
CommonDimensions.familyProduct (fun h : Carried K tick =>
historyTensor F allocation m ε h.val.val)
def activeTensor (F : Type*) [Field F] (ε : ℝ) :
Tensor F (ActiveWords (K := K) (tick := tick) allocation m)
(ActiveWords (K := K) (tick := tick) allocation m) (ActiveWords (K := K) (tick := tick) allocation m) :=
CommonDimensions.familyProduct (fun h : Active K tick =>
historyTensor F allocation m ε (activeSourceEquiv h).val.val)
def producedTensor (F : Type*) [Field F] (ε : ℝ) :
Tensor F (ProducedWords (K := K) (tick := tick) allocation m)
(ProducedWords (K := K) (tick := tick) allocation m) (ProducedWords (K := K) (tick := tick) allocation m) :=
CommonDimensions.familyProduct (fun h : Produced K tick =>
historyTensor F allocation m ε h.val)
def toState (x : DueWords (K := K) (tick := tick) allocation m ×
CarriedWords (K := K) (tick := tick) allocation m) : StateWords (K := K) allocation m tick :=
fun h => if hd : due tick h.val.1 then x.1 ⟨h, hd⟩ else x.2 ⟨h, hd⟩
@[simp] theorem toState_due
(x : DueWords (K := K) (tick := tick) allocation m × CarriedWords (K := K) (tick := tick) allocation m)
(h : DueState K tick) : toState (K := K) (tick := tick) allocation m x h.val = x.1 h := by
simp only [toState, dite_eq_left h.property]
@[simp] theorem toState_carried
(x : DueWords (K := K) (tick := tick) allocation m × CarriedWords (K := K) (tick := tick) allocation m)
(h : Carried K tick) : toState (K := K) (tick := tick) allocation m x h.val = x.2 h := by
simp only [toState, dite_eq_right h.property]
def dueTensorAt (F : Type*) [Field F] (ε : ℝ) :
Tensor F (DueWords (K := K) (tick := tick) allocation m)
(DueWords (K := K) (tick := tick) allocation m) (DueWords (K := K) (tick := tick) allocation m) :=
CommonDimensions.familyProduct (fun h : DueState K tick =>
historyTensor F allocation m ε h.val.val)
theorem split_coefficient (F : Type*) [Field F] (ε : ℝ)
(x y z : DueWords (K := K) (tick := tick) allocation m × CarriedWords (K := K) (tick := tick) allocation m) :
stateTensor F (K := K) allocation m ε tick (toState (K := K) (tick := tick) allocation m x)
(toState (K := K) (tick := tick) allocation m y) (toState (K := K) (tick := tick) allocation m z) =
Tensor.product (dueTensorAt (K := K) (tick := tick) allocation m F ε) (carriedTensor (K := K) (tick := tick) allocation m F ε) x y z := by
change (∏ h : State K tick, historyTensor F allocation m ε h.val
(toState (K := K) (tick := tick) allocation m x h) (toState (K := K) (tick := tick) allocation m y h) (toState (K := K) (tick := tick) allocation m z h)) = _
rw [← Fintype.prod_subtype_mul_prod_subtype (fun h : State K tick => due tick h.val.1)]
simp only [toState_due, toState_carried]
rfl
def splitMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (stateTensor F (K := K) allocation m ε tick)
(Tensor.product (dueTensorAt (K := K) (tick := tick) allocation m F ε)
(carriedTensor (K := K) (tick := tick) allocation m F ε)) where
x := fun x s => if s = toState (K := K) (tick := tick) allocation m x then 1 else 0
y := fun y s => if s = toState (K := K) (tick := tick) allocation m y then 1 else 0
z := fun z s => if s = toState (K := K) (tick := tick) allocation m z then 1 else 0
coefficient := by
rw [← Tensor.pullback_eq_restrict]
funext x y z
exact split_coefficient allocation m F ε x y z
def toDue (x : ActiveWords (K := K) (tick := tick) allocation m) :
DueWords (K := K) (tick := tick) allocation m :=
Equiv.piCongrLeft (fun h : DueState K tick => HistoryWord allocation m h.val.val)
activeSourceEquiv x
theorem active_coefficient (F : Type*) [Field F] (ε : ℝ)
(x y z : ActiveWords (K := K) (tick := tick) allocation m) :
dueTensorAt (K := K) (tick := tick) allocation m F ε (toDue (K := K) (tick := tick) allocation m x) (toDue (K := K) (tick := tick) allocation m y)
(toDue (K := K) (tick := tick) allocation m z) = activeTensor (K := K) (tick := tick) allocation m F ε x y z := by
change (∏ h : DueState K tick, historyTensor F allocation m ε h.val.val
(toDue (K := K) (tick := tick) allocation m x h) (toDue (K := K) (tick := tick) allocation m y h) (toDue (K := K) (tick := tick) allocation m z h)) = _
rw [← activeSourceEquiv.prod_comp]
unfold toDue
simp only [Equiv.piCongrLeft_apply_apply]
rfl
def activeMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (dueTensorAt (K := K) (tick := tick) allocation m F ε)
(activeTensor (K := K) (tick := tick) allocation m F ε) where
x := fun x s => if s = toDue (K := K) (tick := tick) allocation m x then 1 else 0
y := fun y s => if s = toDue (K := K) (tick := tick) allocation m y then 1 else 0
z := fun z s => if s = toDue (K := K) (tick := tick) allocation m z then 1 else 0
coefficient := by
rw [← Tensor.pullback_eq_restrict]
funext x y z
exact active_coefficient allocation m F ε x y z
def activeCarriedMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (stateTensor F (K := K) allocation m ε tick)
(Tensor.product (activeTensor (K := K) (tick := tick) allocation m F ε)
(carriedTensor (K := K) (tick := tick) allocation m F ε)) :=
(splitMap (K := K) (tick := tick) allocation m F ε).comp
((activeMap (K := K) (tick := tick) allocation m F ε).product
(LocalMap.identity (carriedTensor (K := K) (tick := tick) allocation m F ε)))
def fromNext (x : StateWords (K := K) allocation m (tick + 1)) :
ProducedWords (K := K) (tick := tick) allocation m × CarriedWords (K := K) (tick := tick) allocation m :=
(fun h => x (stateTransition.symm (.inr h)),
fun h => x (stateTransition.symm (.inl h)))
theorem next_coefficient (F : Type*) [Field F] (ε : ℝ)
(x y z : StateWords (K := K) allocation m (tick + 1)) :
Tensor.product (producedTensor (K := K) (tick := tick) allocation m F ε)
(carriedTensor (K := K) (tick := tick) allocation m F ε) (fromNext (K := K) (tick := tick) allocation m x)
(fromNext (K := K) (tick := tick) allocation m y) (fromNext (K := K) (tick := tick) allocation m z) =
stateTensor F (K := K) allocation m ε (tick + 1) x y z := by
change _ = ∏ h : State K (tick + 1), historyTensor F allocation m ε h.val
(x h) (y h) (z h)
rw [← stateTransition.symm.prod_comp]
rw [Fintype.prod_sum_type]
exact mul_comm _ _
def nextMap (F : Type*) [Field F] (ε : ℝ) :
LocalMap (Tensor.product (producedTensor (K := K) (tick := tick) allocation m F ε)
(carriedTensor (K := K) (tick := tick) allocation m F ε))
(stateTensor F (K := K) allocation m ε (tick + 1)) := by
apply LocalMap.ofExists
refine ⟨_, _, _, (Tensor.pullback_eq_restrict
(fromNext (K := K) (tick := tick) allocation m)
(fromNext (K := K) (tick := tick) allocation m)
(fromNext (K := K) (tick := tick) allocation m) _).symm.trans ?_⟩
funext x y z
exact next_coefficient allocation m F ε x y z
end MatrixMultiplication.AllFieldHistoryRegrouping
end
end OAI
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