openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /DecoderSourcePullback.lean
| import OAI.Computability.PerfectCompleteness.Decoding.ChildBlockProjection | |
| import OAI.Computability.PerfectCompleteness.Decoding.HierarchicalLeftDecoder | |
| import OAI.Computability.PerfectCompleteness.Decoding.RightDecoderLemmas | |
| namespace OAI | |
| namespace PerfectCompleteness.DecoderSourcePullback | |
| open RecursiveSpaces TreeSourceSpaces HierarchicalArrays PointwiseSpaces OwnInputReference | |
| open ChildBlockProjection | |
| noncomputable section | |
| abbrev F2 := ZMod 2 | |
| variable {branch : Nat → Nat} {n t : Nat} | |
| {slots projected : Slots branch n → Fin t → MixedSupport.Slot} | |
| def relativeCut : | |
| {n : Nat} → (upper : Nodes branch n) → | |
| (lower : Nodes branch (Nodes.height upper)) → | |
| Nodes.height lower < Nodes.height upper → | |
| Cut upper (RelativeDescendantProducts.relativeNode upper lower) | |
| | 0, upper, _, _ => nomatch upper | |
| | _ + 1, .inl _, lower, h => | |
| { proper := h | |
| path := Nodes.path lower | |
| slots_agree := fun _ => rfl } | |
| | _ + 1, .inr (i, upper), lower, h => | |
| let c := relativeCut upper lower h | |
| { proper := c.proper | |
| path := c.path | |
| slots_agree := fun s => congrArg (fun z => (i, z)) (c.slots_agree s) } | |
| def selectedCut (upper : Nodes branch n) (level : Nat) | |
| (d : HierarchicalFrozenTables.LowerNodes upper level) : | |
| Cut upper (HierarchicalLeftDecoder.LowerNode upper level d) := | |
| relativeCut upper d.val.val d.val.property | |
| theorem upperBranch (upper : Nodes branch n) (hbranch : ∀ k < n, 0 < branch k) : | |
| ∀ k < Nodes.height upper, 0 < branch k := | |
| fun k hk => hbranch k (Nat.lt_of_lt_of_le hk (Nodes.height_le upper)) | |
| variable (p : ∀ s k, MixedSupport.Projection (slots s k) (projected s k)) | |
| (upper lower : Nodes branch n) (cut : Cut upper lower) | |
| theorem restriction_projection (x : Domain (nodeSlots slots upper)) : | |
| RightDecoderDescendant.restriction projected upper lower cut | |
| (sourceProjection (nodeProjection p upper) x) = | |
| sourceProjection (nodeProjection p lower) | |
| (RightDecoderDescendant.restriction slots upper lower cut x) := by | |
| obtain ⟨y, rfl⟩ := restrictNode_surjective slots upper x | |
| rw [restrictNode_projection, RightDecoderDescendant.restriction_restrictNode, | |
| RightDecoderDescendant.restriction_restrictNode] | |
| exact (restrictNode_projection p lower y).symm | |
| theorem squareEmbedding_pullback | |
| (hbranch : ∀ k < Nodes.height upper, 0 < branch k) : | |
| (HPullback (nodeProjection p upper)).comp | |
| (RightDecoderDescendant.squareEmbedding projected upper lower cut hbranch) = | |
| (RightDecoderDescendant.squareEmbedding slots upper lower cut hbranch).comp | |
| (squarePullback (nodeProjection p lower)) := by | |
| apply LinearMap.ext | |
| intro f | |
| apply Subtype.ext | |
| funext x | |
| change f.val (RightDecoderDescendant.restriction projected upper lower cut | |
| (sourceProjection (nodeProjection p upper) x)) = | |
| f.val (sourceProjection (nodeProjection p lower) | |
| (RightDecoderDescendant.restriction slots upper lower cut x)) | |
| rw [restriction_projection p upper lower cut] | |
| theorem squarePullback_product (f g : H projected) : | |
| squarePullback p (OddListExtraction.productElement (H projected) f g) = | |
| OddListExtraction.productElement (H slots) (HPullback p f) (HPullback p g) := by | |
| apply Subtype.ext | |
| rfl | |
| def upperTarget (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) : | |
| Module.Dual F2 (UpperSpace projected upper) := | |
| (q.comp (NodeEmbedding.equiv slots upper).toLinearMap).comp | |
| (HPullback (nodeProjection p upper)) | |
| @[simp] theorem upperTarget_apply | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) | |
| (f : UpperSpace projected upper) : | |
| upperTarget p upper q f = | |
| q (NodeEmbedding.embed slots upper (HPullback (nodeProjection p upper) f)) := rfl | |
| variable (level : Nat) (hbranch : ∀ k < n, 0 < branch k) | |
| (d : HierarchicalFrozenTables.LowerNodes upper level) | |
| theorem embed_squareEmbedding | |
| (cut : Cut upper (HierarchicalLeftDecoder.LowerNode upper level d)) : | |
| (NodeEmbedding.embed slots upper).comp | |
| (RightDecoderDescendant.squareEmbedding slots upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch)) = | |
| HierarchicalLeftDecoder.squareEmbedding slots upper level hbranch d := by | |
| apply LinearMap.ext | |
| intro f | |
| apply Subtype.ext | |
| funext x | |
| change (RightDecoderDescendant.squareEmbedding slots upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) f).val | |
| (restrictNode slots upper ((TreeCanonical.assignmentEquiv slots).symm x)) = | |
| f.val (restrictNode slots (HierarchicalLeftDecoder.LowerNode upper level d) | |
| ((TreeCanonical.assignmentEquiv slots).symm x)) | |
| exact RightDecoderDescendant.squareEmbedding_restrictNode slots upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) f | |
| ((TreeCanonical.assignmentEquiv slots).symm x) | |
| theorem functional_upperTarget | |
| (cut : Cut upper (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) : | |
| RightDecoderDescendant.functional projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) | |
| (upperTarget p upper q) = | |
| (HierarchicalLeftDecoder.output slots upper level hbranch d q).comp | |
| (squarePullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d))) := by | |
| apply LinearMap.ext | |
| intro f | |
| change q (NodeEmbedding.embed slots upper | |
| (HPullback (nodeProjection p upper) | |
| (RightDecoderDescendant.squareEmbedding projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) f))) = | |
| q (HierarchicalLeftDecoder.squareEmbedding slots upper level hbranch d | |
| (squarePullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) f)) | |
| calc | |
| _ = q (NodeEmbedding.embed slots upper | |
| (RightDecoderDescendant.squareEmbedding slots upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) | |
| (squarePullback (nodeProjection p | |
| (HierarchicalLeftDecoder.LowerNode upper level d)) f))) := | |
| congrArg (fun g => q (NodeEmbedding.embed slots upper g)) | |
| (LinearMap.congr_fun (squareEmbedding_pullback p upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch)) f) | |
| _ = _ := congrArg q (LinearMap.congr_fun | |
| (embed_squareEmbedding (slots := slots) upper level hbranch d cut) | |
| (squarePullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) f)) | |
| theorem form_upperTarget | |
| (cut : Cut upper (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) | |
| (f g : H (nodeSlots projected (HierarchicalLeftDecoder.LowerNode upper level d))) : | |
| RightDecoderDescendant.form projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) | |
| (upperTarget p upper q) f g = | |
| OddListExtraction.multiplicationForm | |
| (H (nodeSlots slots (HierarchicalLeftDecoder.LowerNode upper level d))) | |
| (HierarchicalLeftDecoder.output slots upper level hbranch d q) | |
| (HPullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) f) | |
| (HPullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) g) := by | |
| change RightDecoderDescendant.functional projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) | |
| (upperTarget p upper q) | |
| (OddListExtraction.productElement | |
| (H (nodeSlots projected (HierarchicalLeftDecoder.LowerNode upper level d))) f g) = _ | |
| rw [functional_upperTarget p upper level hbranch d cut q] | |
| change (HierarchicalLeftDecoder.output slots upper level hbranch d q) | |
| (squarePullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (OddListExtraction.productElement | |
| (H (nodeSlots projected (HierarchicalLeftDecoder.LowerNode upper level d))) f g)) = _ | |
| rw [squarePullback_product] | |
| rfl | |
| theorem form_eq_of_target_error | |
| (cut : Cut upper (HierarchicalLeftDecoder.LowerNode upper level d)) | |
| (q : Module.Dual F2 (NodeEmbedding.RowSpace slots upper)) | |
| (Q : Submodule F2 (Module.Dual F2 (UpperSpace projected upper))) | |
| (decoded : Module.Dual F2 (UpperSpace projected upper)) | |
| (herror : decoded - upperTarget p upper q ∈ Q) | |
| (hvanish : ∀ z ∈ Q, z.comp | |
| (RightDecoderDescendant.squareEmbedding projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch)) = 0) | |
| (f g : H (nodeSlots projected (HierarchicalLeftDecoder.LowerNode upper level d))) : | |
| RightDecoderDescendant.form projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) | |
| decoded f g = | |
| OddListExtraction.multiplicationForm | |
| (H (nodeSlots slots (HierarchicalLeftDecoder.LowerNode upper level d))) | |
| (HierarchicalLeftDecoder.output slots upper level hbranch d q) | |
| (HPullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) f) | |
| (HPullback (nodeProjection p (HierarchicalLeftDecoder.LowerNode upper level d)) g) := by | |
| rw [RightDecoderDescendant.form_eq_of_sub_mem projected upper | |
| (HierarchicalLeftDecoder.LowerNode upper level d) cut (upperBranch upper hbranch) | |
| Q decoded (upperTarget p upper q) herror hvanish] | |
| exact form_upperTarget p upper level hbranch d cut q f g | |
| end | |
| end PerfectCompleteness.DecoderSourcePullback | |
| end OAI | |