openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /CountableFormulaFamilies.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.CountableSetFormulas | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.Name | |
| open SetRealization | |
| universe u | |
| variable {P : Type u} [Preorder P] [Countable P] {s : Presentation P} | |
| theorem atomicFamily_countable (D : NameDomain s) (small : D.names.Countable) : | |
| (atomicFamily D).Countable := by | |
| let : Countable D.names := small | |
| let memSet (z : P × D.names × D.names) := Localized z.1 (MemWitness z.2.1.val z.2.2.val) | |
| let eqSet (z : D.names × D.names) := EqDecisionWitness z.1.val z.2.val | |
| apply ((Set.countable_range memSet).union (Set.countable_range eqSet)).mono | |
| intro E hE | |
| rcases hE with ⟨p, σ, τ, hσ, hτ, rfl⟩ | ⟨σ, τ, hσ, hτ, rfl⟩ | |
| · exact Or.inl ⟨(p, ⟨σ, hσ⟩, ⟨τ, hτ⟩), rfl⟩ | |
| · exact Or.inr ⟨(⟨σ, hσ⟩, ⟨τ, hτ⟩), rfl⟩ | |
| end PartitionPilot.Forcing.Name | |
| namespace PartitionPilot.Forcing.SetFormula | |
| open SetRealization | |
| universe u | |
| variable {P : Type u} [Preorder P] [Countable P] {s : Presentation P} | |
| theorem formulaFamily_countable (D : Name.NameDomain s) (small : D.names.Countable) : | |
| (formulaFamily D).Countable := by | |
| let : Countable D.names := small | |
| let decisions (z : Σ n, SetFormula n × Assignment D n) := DecisionWitness D z.2.1 z.2.2 | |
| let universals (z : Σ n, SetFormula (n + 1) × Assignment D n) := AllDecisionWitness D z.2.1 z.2.2 | |
| apply ((Name.atomicFamily_countable D small).union | |
| ((Set.countable_range decisions).union (Set.countable_range universals))).mono | |
| intro E hE | |
| rcases hE with atomic | (⟨n, φ, env, rfl⟩ | ⟨n, φ, env, rfl⟩) | |
| · exact Or.inl atomic | |
| · exact Or.inr (Or.inl ⟨⟨n, φ, env⟩, rfl⟩) | |
| · exact Or.inr (Or.inr ⟨⟨n, φ, env⟩, rfl⟩) | |
| theorem exists_generic_formulaFamily (D : Name.NameDomain s) (small : D.names.Countable) (p : P) : | |
| ∃ H : ConditionFilter P, H.mem p ∧ H.GenericFor (formulaFamily D) := | |
| exists_generic_for_countable_family p _ (formulaFamily_countable D small) | |
| theorem exists_interpretation_truth (D : Name.NameDomain s) (small : D.names.Countable) (p : P) : | |
| ∃ H : ConditionFilter P, H.mem p ∧ | |
| ∀ {n} (φ : SetFormula n) (env : Assignment D n), | |
| Realize (D.values {q | H.mem q}) φ (evalAssignment {q | H.mem q} env) ↔ | |
| ∃ q, H.mem q ∧ Forces D φ env q := by | |
| obtain ⟨H, hp, generic⟩ := exists_generic_formulaFamily D small p | |
| exact ⟨H, hp, fun φ env => forcing_truth D H generic φ env⟩ | |
| end PartitionPilot.Forcing.SetFormula | |
| end | |
| end OAI | |