| 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 | |