openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Decoding /CleanEndpointProjection.lean
| import OAI.Computability.PerfectCompleteness.Decoding.CutProjectionAssembly | |
| import OAI.Computability.PerfectCompleteness.Foundations.SourceOddListsLemmas | |
| import OAI.Computability.PerfectCompleteness.Repetition.CleanEndpointSlots | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CleanEndpointProjection | |
| open scoped Classical | |
| open RecursiveSpaces MixedSupport SourceQuestionReconstruction | |
| noncomputable section | |
| private theorem reconstruct_designated {I E Q : Type*} {Leaf : I → Type*} | |
| (designated : (i : I) → Leaf i) (clean : I → Prop) [DecidablePred clean] | |
| [∀ i, DecidableEq (Leaf i)] (endpoint : I → E → Q) | |
| (visible : Visible (E := E) designated clean) (own : {i : I // clean i} → Q) | |
| (i : {i : I // clean i}) : | |
| reconstruct designated clean endpoint visible own i.val (designated i.val) = own i := by | |
| exact dite_eq_left ⟨i.property, rfl⟩ | |
| private theorem dite_heq_of_pos {α β : Type} {P : Prop} [Decidable P] | |
| (f : P → α) (g : ¬ P → α) (b : β) (hP : P) | |
| (h : HEq (f hP) b) : HEq (dite P f g) b := by | |
| rw [dite_eq_left hP] | |
| exact h | |
| variable {v m t : Nat} (clauses : Fin m → SourceClause.NormalizedClause v) | |
| def sourceProjection (projected : Bool) (e : SourceOddLists.Occurrences m t) : | |
| ∀ k, Projection (SourceAnswerEquiv.leftSlots clauses (SourceOddLists.left e) k) | |
| (SourceKeys.slot clauses (rightTuple clauses projected e k)) := | |
| match projected with | |
| | false => fun k => Projection.keep | |
| (SourceAnswerEquiv.leftSlots clauses (SourceOddLists.left e) k) | |
| | true => SourceAnswerEquiv.slotProjection clauses e | |
| theorem sourceProjection_of_true (projected : Bool) | |
| (e : SourceOddLists.Occurrences m t) (k : Fin t) (hp : projected = true) : | |
| HEq (sourceProjection clauses projected e k) | |
| (SourceAnswerEquiv.slotProjection clauses e k) := by | |
| subst projected | |
| rfl | |
| variable {branch : Nat → Nat} {h : Nat} | |
| (designated : Fin (branch h) → Slots branch h) | |
| (clean : Fin (branch h) → Prop) | |
| (visible : Visible (E := SourceTuple m t) designated clean) | |
| theorem leftInside_designated | |
| (own : {i : Fin (branch h) // clean i} → Fin t → Fin m) | |
| (i : {i : Fin (branch h) // clean i}) : | |
| CleanEndpointSlots.leftInside clauses designated clean visible own | |
| (i.val, designated i.val) = | |
| SourceAnswerEquiv.leftSlots clauses (own i) := by | |
| funext k | |
| exact congrArg | |
| (fun question : Fin t → SourceKeys.Endpoint v m => SourceKeys.slot clauses (question k)) | |
| (reconstruct_designated designated clean (fun _ => leftTuple (v := v)) visible | |
| (fun j k => SourceKeys.Endpoint.clause (v := v) (own j k)) i) | |
| variable (projected : Fin (branch h) → Bool) | |
| theorem rightInside_designated | |
| (own : {i : Fin (branch h) // clean i} → Fin t → Fin v) | |
| (i : {i : Fin (branch h) // clean i}) : | |
| CleanEndpointSlots.rightInside clauses designated clean visible projected own | |
| (i.val, designated i.val) = | |
| SourceAnswerEquiv.rightSlots clauses (own i) := by | |
| funext k | |
| exact congrArg | |
| (fun question : Fin t → SourceKeys.Endpoint v m => SourceKeys.slot clauses (question k)) | |
| (reconstruct_designated designated clean (fun j => rightTuple clauses (projected j)) visible | |
| (fun j k => SourceKeys.Endpoint.variable (m := m) (own j k)) i) | |
| def insideProjection (cleanProjected : ∀ i, clean i → projected i = true) | |
| (e : {i : Fin (branch h) // clean i} → SourceOddLists.Occurrences m t) : | |
| ∀ s k, Projection | |
| (CleanEndpointSlots.leftInside clauses designated clean visible | |
| (fun i => SourceOddLists.left (e i)) s k) | |
| (CleanEndpointSlots.rightInside clauses designated clean visible projected | |
| (fun i => SourceOddLists.right clauses (e i)) s k) := | |
| fun s k => if hs : clean s.1 ∧ s.2 = designated s.1 then | |
| CutProjectionAssembly.castProjection | |
| (by simp only [SourceAnswerEquiv.leftSlots, SourceOddLists.left, | |
| CleanEndpointSlots.leftInside, reconstructLeft, reconstruct, dite_eq_left hs]) | |
| (by simp only [rightTuple, cleanProjected s.1 hs.1, ite_true, | |
| CleanEndpointSlots.rightInside, reconstructRight, reconstruct, dite_eq_left hs, | |
| SourceOddLists.right]) | |
| (sourceProjection clauses (projected s.1) (e ⟨s.1, hs.1⟩) k) | |
| else | |
| CutProjectionAssembly.castProjection | |
| (by simp only [SourceAnswerEquiv.leftSlots, SourceOddLists.left, | |
| CleanEndpointSlots.leftInside, reconstructLeft, reconstruct, dite_eq_right hs, leftTuple]) | |
| (by simp only [CleanEndpointSlots.rightInside, reconstructRight, reconstruct, dite_eq_right hs]) | |
| (sourceProjection clauses (projected s.1) (visible s.1 ⟨s.2, hs⟩) k) | |
| theorem insideProjection_designated | |
| (cleanProjected : ∀ i, clean i → projected i = true) | |
| (e : {i : Fin (branch h) // clean i} → SourceOddLists.Occurrences m t) | |
| (i : {i : Fin (branch h) // clean i}) (k : Fin t) : | |
| HEq (insideProjection clauses designated clean visible projected cleanProjected e | |
| (i.val, designated i.val) k) | |
| (SourceAnswerEquiv.slotProjection clauses (e i) k) := by | |
| refine @dite_heq_of_pos _ _ | |
| (clean i.val ∧ designated i.val = designated i.val) _ _ _ | |
| (SourceAnswerEquiv.slotProjection clauses (e i) k) ⟨i.property, rfl⟩ ?_ | |
| exact (CutProjectionAssembly.castProjection_heq _ _ _).trans | |
| (sourceProjection_of_true clauses (projected i.val) (e i) k | |
| (cleanProjected i.val i.property)) | |
| end | |
| end PerfectCompleteness.CleanEndpointProjection | |
| end | |
| end OAI | |