openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Reduction /SourceCompleteness.lean
| import OAI.Computability.PerfectCompleteness.Foundations.SourceStrategy | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.SourceCompleteness | |
| open SourceClause ClauseSupport MixedSupport CanonicalKeys SourceKeys | |
| variable {v m n : Nat} {Y Z : Type*} | |
| structure Presentation (clauses : Fin m → NormalizedClause v) (n : Nat) (Y : Type*) where | |
| endpoints : Fin n → Endpoint v m | |
| joint : Assignment (slot clauses ∘ endpoints) → Y | |
| variable {clauses : Fin m → NormalizedClause v} {side : Side} | |
| noncomputable def presentationKey (side : Side) (P : Presentation clauses n Y) : Key n := | |
| key side (slot clauses ∘ P.endpoints) P.joint | |
| noncomputable def generatedKeys (side : Side) (clauses : Fin m → NormalizedClause v) | |
| (n : Nat) (Y : Type*) : Set (Key n) := | |
| Set.range (presentationKey (clauses := clauses) (n := n) (Y := Y) side) | |
| abbrev Vertex (side : Side) (clauses : Fin m → NormalizedClause v) (n : Nat) (Y : Type*) := | |
| ↥(generatedKeys side clauses n Y) | |
| noncomputable def vertexOf (side : Side) (P : Presentation clauses n Y) : | |
| Vertex side clauses n Y := | |
| ⟨presentationKey side P, ⟨P, rfl⟩⟩ | |
| abbrev VertexLabel (vertex : Vertex side clauses n Y) := | |
| {part : Set (Fin n → ReducedValue) // part ∈ vertex.val.partition} | |
| noncomputable def representation (vertex : Vertex side clauses n Y) : | |
| Presentation clauses n Y := | |
| vertex.property.choose | |
| theorem representation_key (vertex : Vertex side clauses n Y) : | |
| presentationKey side (representation vertex) = vertex.val := | |
| vertex.property.choose_spec | |
| noncomputable def evaluatedLabel (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) | |
| (P : Presentation clauses n Y) : Label (slot clauses ∘ P.endpoints) P.joint := | |
| evaluateLabel (slot clauses ∘ P.endpoints) P.joint | |
| (realizingAssignment clauses assignment hsat P.endpoints) | |
| noncomputable def strategy (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) | |
| (vertex : Vertex side clauses n Y) : VertexLabel vertex := by | |
| refine ⟨(evaluatedLabel assignment hsat (representation vertex)).val, ?_⟩ | |
| rw [← representation_key vertex] | |
| exact (evaluatedLabel assignment hsat (representation vertex)).property | |
| theorem strategy_value_eq (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) | |
| (vertex : Vertex side clauses n Y) (P : Presentation clauses n Z) | |
| (hkey : presentationKey side P = vertex.val) : | |
| (strategy assignment hsat vertex).val = (evaluatedLabel assignment hsat P).val := by | |
| change (evaluatedLabel assignment hsat (representation vertex)).val = | |
| (evaluatedLabel assignment hsat P).val | |
| exact evaluated_label_eq_of_key_eq clauses assignment hsat side side | |
| (representation vertex).endpoints P.endpoints | |
| (representation vertex).joint P.joint | |
| ((representation_key vertex).trans hkey.symm) | |
| def labelInPresentation (vertex : Vertex side clauses n Y) | |
| (P : Presentation clauses n Z) (hkey : presentationKey side P = vertex.val) | |
| (label : VertexLabel vertex) : Label (slot clauses ∘ P.endpoints) P.joint := | |
| ⟨label.val, by | |
| change label.val ∈ (presentationKey side P).partition | |
| rw [hkey] | |
| exact label.property⟩ | |
| theorem strategy_in_presentation (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) | |
| (vertex : Vertex side clauses n Y) (P : Presentation clauses n Z) | |
| (hkey : presentationKey side P = vertex.val) : | |
| labelInPresentation vertex P hkey (strategy assignment hsat vertex) = | |
| evaluatedLabel assignment hsat P := by | |
| apply Subtype.ext | |
| exact strategy_value_eq assignment hsat vertex P hkey | |
| theorem restore_strategy_in_presentation (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) | |
| (vertex : Vertex side clauses n Y) (P : Presentation clauses n Z) | |
| (hkey : presentationKey side P = vertex.val) : | |
| restore (slot clauses ∘ P.endpoints) P.joint | |
| (labelInPresentation vertex P hkey (strategy assignment hsat vertex)) = | |
| P.joint (realizingAssignment clauses assignment hsat P.endpoints) := by | |
| rw [strategy_in_presentation] | |
| exact restore_evaluateLabel _ _ _ | |
| def Presentation.postcompose (P : Presentation clauses n Y) (p : Y → Z) : | |
| Presentation clauses n Z where | |
| endpoints := P.endpoints | |
| joint := p ∘ P.joint | |
| noncomputable def coarseningEdge (sourceSide targetSide : Side) | |
| (P : Presentation clauses n Y) (p : Y → Z) | |
| (label : VertexLabel (vertexOf sourceSide P)) : | |
| VertexLabel (vertexOf targetSide (P.postcompose p)) := by | |
| let original := labelInPresentation (vertexOf sourceSide P) P rfl label | |
| let coarse := CanonicalEdges.coarsen (slot clauses ∘ P.endpoints) P.joint p original | |
| exact ⟨coarse.val, coarse.property⟩ | |
| theorem coarseningEdge_strategy (assignment : Fin v → Bool) | |
| (hsat : ∀ c, (clauses c).clause.eval assignment = true) | |
| (sourceSide targetSide : Side) (P : Presentation clauses n Y) (p : Y → Z) : | |
| coarseningEdge sourceSide targetSide P p | |
| (strategy assignment hsat (vertexOf sourceSide P)) = | |
| strategy assignment hsat (vertexOf targetSide (P.postcompose p)) := by | |
| have hsource : presentationKey sourceSide P = (vertexOf sourceSide P).val := rfl | |
| have htarget : presentationKey targetSide (P.postcompose p) = | |
| (vertexOf targetSide (P.postcompose p)).val := rfl | |
| have hlabel : labelInPresentation (vertexOf sourceSide P) P hsource | |
| (strategy assignment hsat (vertexOf sourceSide P)) = evaluatedLabel assignment hsat P := | |
| strategy_in_presentation assignment hsat (vertexOf sourceSide P) P hsource | |
| apply Subtype.ext | |
| calc | |
| _ = (CanonicalEdges.coarsen (slot clauses ∘ P.endpoints) P.joint p | |
| (evaluatedLabel assignment hsat P)).val := | |
| congrArg (fun L : Label (slot clauses ∘ P.endpoints) P.joint => | |
| (CanonicalEdges.coarsen (slot clauses ∘ P.endpoints) P.joint p L).val) hlabel | |
| _ = (evaluatedLabel assignment hsat (P.postcompose p)).val := | |
| congrArg (fun L : Label (slot clauses ∘ P.endpoints) (p ∘ P.joint) => L.val) | |
| (CanonicalEdges.coarsen_evaluateLabel (slot clauses ∘ P.endpoints) P.joint p | |
| (realizingAssignment clauses assignment hsat P.endpoints)) | |
| _ = _ := (strategy_value_eq assignment hsat (vertexOf targetSide (P.postcompose p)) | |
| (P.postcompose p) htarget).symm | |
| end PerfectCompleteness.SourceCompleteness | |
| end | |
| end OAI | |