| import Mathlib |
| import OAI.LinearAlgebra.MatrixFields.Construction.InitialState |
| import OAI.LinearAlgebra.MatrixFields.Entropy.InitialLeafRates |
| import OAI.LinearAlgebra.MatrixFields.Entropy.TerminalStatisticLaws |
|
|
| namespace OAI |
|
|
| namespace MatrixAllFields |
|
|
| open scoped BigOperators Topology Polynomial |
|
|
| noncomputable section |
|
|
| namespace MatrixMultiplication.AllFieldFinalState |
|
|
| open MatrixMultiplication.Foundation AllFieldHistory AllFieldFiniteFamily |
| open AllFieldTerminalStatisticLaws |
| open scoped BigOperators Classical |
| attribute [local instance] Classical.propDecidable Classical.decEq |
|
|
| abbrev InitialIndex (K : ℕ) := AllFieldInitialLeafRates.PopulationHistory K |
| abbrev CIndex (K : ℕ) := AfterC K × Placement |
| abbrev Index (K : ℕ) := InitialIndex K ⊕ (TerminalZero K ⊕ CIndex K) |
|
|
| def history {K : ℕ} : Index K → History K |
| | .inl h => (.initial h.1.val, h.2) |
| | .inr (.inl h) => AllFieldTerminalStatisticLaws.history h |
| | .inr (.inr h) => (.afterC h.1, h.2) |
|
|
| theorem history_terminal {K : ℕ} (h : Index K) : terminal (history h).1 := by |
| rcases h with h | h | h |
| · exact h.1.property |
| · rcases h with h | h |
| · exact h.1.property |
| · exact h.1.property |
| · trivial |
|
|
| theorem terminal_resident_final {K : ℕ} (h : CanonicalHistory K) (ht : terminal h) : |
| resident (K + 2) h := by |
| have hlot := (initialAncestor h).1.isLt |
| cases h with |
| | initial h => exact fun hp => False.elim (ht hp) |
| | afterA h => |
| refine ⟨?_, fun hp => False.elim (ht hp)⟩ |
| change h.1.val.1.val < K at hlot |
| omega |
| | afterB h => |
| refine ⟨?_, ht⟩ |
| change h.1.val.1.val.1.val < K at hlot |
| omega |
| | partC h => exact False.elim ht |
| | afterC h => |
| change h.1.1.val.1.val.1.val.1.val < K at hlot |
| change h.1.1.val.1.val.1.val.1.val + 3 ≤ K + 2 |
| omega |
|
|
| def toState {K : ℕ} (h : Index K) : State K (K + 2) := |
| ⟨history h, terminal_resident_final _ (history_terminal h)⟩ |
|
|
| def fromState {K : ℕ} (h : State K (K + 2)) : Index K := by |
| have ht := final_state_terminal h |
| rcases h with ⟨⟨h, phi⟩, hr⟩ |
| cases h with |
| | initial h => exact .inl (⟨h, ht⟩, phi) |
| | afterA h => exact .inr (.inl (.inl (⟨h, ht⟩, phi))) |
| | afterB h => exact .inr (.inl (.inr (⟨h, ht⟩, phi))) |
| | partC h => exact False.elim ht |
| | afterC h => exact .inr (.inr (h, phi)) |
|
|
| def indexEquiv (K : ℕ) : Index K ≃ State K (K + 2) where |
| toFun := toState |
| invFun := fromState |
| left_inv h := by |
| rcases h with h | h | h |
| · rfl |
| · cases h <;> rfl |
| · rfl |
| right_inv h := by |
| have ht := final_state_terminal h |
| rcases h with ⟨⟨h, phi⟩, hr⟩ |
| cases h with |
| | initial h => rfl |
| | afterA h => rfl |
| | afterB h => rfl |
| | partC h => exact False.elim ht |
| | afterC h => rfl |
|
|
| variable {K : ℕ} (allocation : Allocation) (dilation : ℕ) |
|
|
| abbrev Words := ∀ h : Index K, HistoryWord allocation dilation (history h) |
| abbrev InitialWords := ∀ h : InitialIndex K, |
| HistoryWord allocation dilation (.initial h.1.val, h.2) |
| abbrev ZeroWords := ∀ h : TerminalZero K, |
| HistoryWord allocation dilation (AllFieldTerminalStatisticLaws.history h) |
| abbrev CWords := ∀ h : CIndex K, HistoryWord allocation dilation (.afterC h.1, h.2) |
| abbrev Parts := InitialWords (K := K) allocation dilation × |
| (ZeroWords (K := K) allocation dilation × CWords (K := K) allocation dilation) |
|
|
| instance wordsFintype : Fintype (Words (K := K) allocation dilation) := |
| Fintype.ofFinite _ |
|
|
| def grouped (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (Words (K := K) allocation dilation) (Words (K := K) allocation dilation) |
| (Words (K := K) allocation dilation) := |
| CommonDimensions.familyProduct (fun h : Index K => |
| historyTensor F allocation dilation ε (history h)) |
|
|
| def stateWord (w : Words (K := K) allocation dilation) : |
| ∀ h : State K (K + 2), HistoryWord allocation dilation h.val := |
| Equiv.piCongrLeft (fun h : State K (K + 2) => HistoryWord allocation dilation h.val) |
| (indexEquiv K) w |
|
|
| theorem grouped_coefficient (F : Type*) [Field F] (ε : ℝ) |
| (x y z : Words (K := K) allocation dilation) : |
| stateTensor F allocation dilation ε (K + 2) (stateWord allocation dilation x) |
| (stateWord allocation dilation y) (stateWord allocation dilation z) = |
| grouped allocation dilation F ε x y z := by |
| change (∏ h : State K (K + 2), historyTensor F allocation dilation ε h.val |
| (stateWord allocation dilation x h) (stateWord allocation dilation y h) |
| (stateWord allocation dilation z h)) = _ |
| rw [← (indexEquiv K).prod_comp] |
| unfold grouped CommonDimensions.familyProduct |
| apply Finset.prod_congr rfl |
| intro h _ |
| have hstate (w : Words (K := K) allocation dilation) : |
| stateWord allocation dilation w ((indexEquiv K) h) = w h := |
| Equiv.piCongrLeft_apply_apply |
| (fun state : State K (K + 2) => HistoryWord allocation dilation state.val) |
| (indexEquiv K) w h |
| rw [hstate, hstate, hstate] |
| rfl |
|
|
| def groupedMap (F : Type*) [Field F] (ε : ℝ) : |
| LocalMap (stateTensor F (K := K) allocation dilation ε (K + 2)) |
| (grouped (K := K) allocation dilation F ε) := by |
| apply LocalMap.ofExists |
| refine ⟨_, _, _, (Tensor.pullback_eq_restrict |
| (stateWord (K := K) allocation dilation) (stateWord (K := K) allocation dilation) |
| (stateWord (K := K) allocation dilation) _).symm.trans ?_⟩ |
| funext x y z |
| exact grouped_coefficient allocation dilation F ε x y z |
|
|
| def initialTensor (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (InitialWords (K := K) allocation dilation) (InitialWords (K := K) allocation dilation) |
| (InitialWords (K := K) allocation dilation) := |
| CommonDimensions.familyProduct (fun h : InitialIndex K => |
| historyTensor F allocation dilation ε (.initial h.1.val, h.2)) |
|
|
| def zeroTensor (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (ZeroWords (K := K) allocation dilation) (ZeroWords (K := K) allocation dilation) |
| (ZeroWords (K := K) allocation dilation) := |
| CommonDimensions.familyProduct (fun h : TerminalZero K => |
| historyTensor F allocation dilation ε (AllFieldTerminalStatisticLaws.history h)) |
|
|
| def cTensor (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (CWords (K := K) allocation dilation) (CWords (K := K) allocation dilation) |
| (CWords (K := K) allocation dilation) := |
| CommonDimensions.familyProduct (fun h : CIndex K => |
| historyTensor F allocation dilation ε (.afterC h.1, h.2)) |
|
|
| def partsTensor (F : Type*) [Field F] (ε : ℝ) : |
| Tensor F (Parts (K := K) allocation dilation) (Parts (K := K) allocation dilation) |
| (Parts (K := K) allocation dilation) := |
| Tensor.product (initialTensor (K := K) allocation dilation F ε) |
| (Tensor.product (zeroTensor (K := K) allocation dilation F ε) |
| (cTensor (K := K) allocation dilation F ε)) |
|
|
| def joinParts (w : Parts (K := K) allocation dilation) : Words (K := K) allocation dilation |
| | .inl h => w.1 h |
| | .inr (.inl h) => w.2.1 h |
| | .inr (.inr h) => w.2.2 h |
|
|
| theorem parts_coefficient (F : Type*) [Field F] (ε : ℝ) |
| (x y z : Parts (K := K) allocation dilation) : |
| grouped allocation dilation F ε (joinParts allocation dilation x) |
| (joinParts allocation dilation y) (joinParts allocation dilation z) = |
| partsTensor allocation dilation F ε x y z := by |
| unfold grouped CommonDimensions.familyProduct |
| rw [Fintype.prod_sum_type, Fintype.prod_sum_type] |
| rfl |
|
|
| def partsMap (F : Type*) [Field F] (ε : ℝ) : |
| LocalMap (grouped (K := K) allocation dilation F ε) |
| (partsTensor (K := K) allocation dilation F ε) := by |
| apply LocalMap.ofExists |
| refine ⟨_, _, _, (Tensor.pullback_eq_restrict |
| (joinParts (K := K) allocation dilation) (joinParts (K := K) allocation dilation) |
| (joinParts (K := K) allocation dilation) _).symm.trans ?_⟩ |
| funext x y z |
| exact parts_coefficient allocation dilation F ε x y z |
|
|
| def map (F : Type*) [Field F] (ε : ℝ) : |
| LocalMap (stateTensor F (K := K) allocation dilation ε (K + 2)) |
| (partsTensor (K := K) allocation dilation F ε) := |
| (groupedMap allocation dilation F ε).comp (partsMap allocation dilation F ε) |
|
|
| end MatrixMultiplication.AllFieldFinalState |
|
|
| end |
|
|
| end MatrixAllFields |
|
|
| end OAI |
|
|
