openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /LowerCutNodeCoordinates.lean
| import OAI.Computability.PerfectCompleteness.Decoding.LowerDirectionFiber | |
| import OAI.Computability.PerfectCompleteness.Foundations.WholeArrayInteriorExteriorLemmas | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.LowerCutNodeCoordinates | |
| open RecursiveSpaces DescendantSpaces TreeSourceSpaces HierarchicalArrays | |
| open WholeArraySubtreeSplit WholeArrayInteriorExterior | |
| open scoped Classical | |
| noncomputable section | |
| variable {branch : Nat → Nat} {n m t : Nat} | |
| def scalarEquiv : {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) → | |
| H (subtreeSlots p slots) ≃ₗ[F2] H (nodeSlots slots (upperNode p)) | |
| | _, _, .refl _, _ => LinearEquiv.refl F2 _ | |
| | _, _, .step i p, slots => scalarEquiv p (childSlots slots i) | |
| def rowIndexEquiv (rows : Nat → Nat) : {n m : Nat} → | |
| (p : Path branch n (m + 1)) → | |
| Fin (rows (m + 1)) ≃ Fin (rows (Nodes.height (upperNode p))) | |
| | _, _, .refl _ => Equiv.refl _ | |
| | _, _, .step _ p => rowIndexEquiv rows p | |
| def directionEquiv (rows : Nat → Nat) : {n m : Nat} → | |
| (p : Path branch n (m + 1)) → | |
| BucketSampler.Direction (rows (m + 1)) ≃ | |
| BucketSampler.Direction (rows (Nodes.height (upperNode p))) | |
| | _, _, .refl _ => Equiv.refl _ | |
| | _, _, .step _ p => directionEquiv rows p | |
| def rowEquiv (rows : Nat → Nat) : {n m : Nat} → | |
| (p : Path branch n (m + 1)) → | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (Fin (rows (m + 1)) → H (subtreeSlots p slots)) ≃ₗ[F2] | |
| (Fin (rows (Nodes.height (upperNode p))) → H (nodeSlots slots (upperNode p))) | |
| | _, _, .refl _, _ => LinearEquiv.refl F2 _ | |
| | _, _, .step i p, slots => rowEquiv rows p (childSlots slots i) | |
| theorem directionEquiv_apply (rows : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (a : BucketSampler.Direction (rows (m + 1))) → (i : Fin (rows (m + 1))) → | |
| (directionEquiv rows p a).val (rowIndexEquiv rows p i) = a.val i | |
| | _, _, .refl _, _, _ => rfl | |
| | _, _, .step _ p, a, i => directionEquiv_apply rows p a i | |
| theorem rowEquiv_apply (rows : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (S : Fin (rows (m + 1)) → H (subtreeSlots p slots)) → | |
| (i : Fin (rows (m + 1))) → | |
| rowEquiv rows p slots S (rowIndexEquiv rows p i) = scalarEquiv p slots (S i) | |
| | _, _, .refl _, _, _, _ => rfl | |
| | _, _, .step i p, slots, S, j => rowEquiv_apply rows p (childSlots slots i) S j | |
| theorem pivot_transport (rows : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (a : BucketSampler.Direction (rows (m + 1))) → | |
| rowIndexEquiv rows p (LowerDirectionFiber.pivot a) = | |
| LowerDirectionFiber.pivot (directionEquiv rows p a) | |
| | _, _, .refl _, _ => rfl | |
| | _, _, .step _ p, a => pivot_transport rows p a | |
| theorem rowEquiv_selectedRows (rows : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (arrays : Arrays slots rows) → | |
| rowEquiv rows p slots (SelectedArrayReplacement.selectedRows rows p slots arrays) = | |
| arrays (upperNode p) | |
| | _, _, .refl _, _, _ => rfl | |
| | _, _, .step i p, slots, arrays => | |
| rowEquiv_selectedRows rows p (childSlots slots i) (fun node => arrays (.inr (i, node))) | |
| theorem replace_eq_update (rows : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (arrays : Arrays slots rows) | |
| (fresh : Fin (rows (m + 1)) → H (subtreeSlots p slots)) : | |
| SelectedArrayReplacement.replace rows p slots arrays fresh = | |
| Function.update arrays (upperNode p) (rowEquiv rows p slots fresh) := by | |
| funext node | |
| by_cases hnode : node = upperNode p | |
| · subst node | |
| rw [Function.update_self] | |
| have h := rowEquiv_selectedRows rows p slots | |
| (SelectedArrayReplacement.replace rows p slots arrays fresh) | |
| rw [SelectedArrayReplacement.selectedRows_replace] at h | |
| exact h.symm | |
| · rw [Function.update_of_ne hnode] | |
| exact SelectedArrayReplacement.replace_outside rows p slots arrays fresh node hnode | |
| theorem rowEquiv_replace (rows : Nat → Nat) : | |
| {n m : Nat} → (p : Path branch n (m + 1)) → | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) → | |
| (a : BucketSampler.Direction (rows (m + 1))) → | |
| (S : Fin (rows (m + 1)) → H (subtreeSlots p slots)) → | |
| (fresh : H (subtreeSlots p slots)) → | |
| rowEquiv rows p slots (LowerDirectionFiber.replace a S fresh) = | |
| LowerDirectionFiber.replace (directionEquiv rows p a) | |
| (rowEquiv rows p slots S) (scalarEquiv p slots fresh) | |
| | _, _, .refl _, _, _, _, _ => rfl | |
| | _, _, .step i p, slots, a, S, fresh => | |
| rowEquiv_replace rows p (childSlots slots i) a S fresh | |
| theorem replace_fiber_eq_update (rows : Nat → Nat) (p : Path branch n (m + 1)) | |
| (slots : Slots branch n → Fin t → MixedSupport.Slot) (arrays : Arrays slots rows) | |
| (a : BucketSampler.Direction (rows (m + 1))) (fresh : H (subtreeSlots p slots)) : | |
| SelectedArrayReplacement.replace rows p slots arrays | |
| (LowerDirectionFiber.replace a | |
| (SelectedArrayReplacement.selectedRows rows p slots arrays) fresh) = | |
| Function.update arrays (upperNode p) | |
| (LowerDirectionFiber.replace (directionEquiv rows p a) | |
| (arrays (upperNode p)) (scalarEquiv p slots fresh)) := by | |
| rw [replace_eq_update, rowEquiv_replace, rowEquiv_selectedRows] | |
| end | |
| end PerfectCompleteness.LowerCutNodeCoordinates | |
| end | |
| end OAI | |