| 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 |
|
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| end |
|
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| end OAI |
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|
