openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /ForcingWitnessFormulas.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundDefinableSelections | |
| namespace OAI | |
| namespace PartitionPilot.Forcing.SetFormula | |
| def forcingWitnessFormula {k n m : Nat} (A : ForcingPrimitives k) (φ : SetFormula (n + 2)) | |
| (params : Fin k → Fin m) (names : Fin n → Fin m) (z y : Fin m) : SetFormula m := | |
| ex (exConditions (params A.carrier).succ | |
| (.conj (orderedPairFormula z.succ.succ 1 0) | |
| (.conj (domainApplication A.domain (fun j => (params j).succ.succ) 1) | |
| (.conj (domainApplication A.domain (fun j => (params j).succ.succ) y.succ.succ) | |
| (forcingTranslation A φ (fun j => (params j).succ.succ) 0 | |
| (extend 1 (extend y.succ.succ (fun j => (names j).succ.succ)))))))) | |
| end PartitionPilot.Forcing.SetFormula | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) {P : Type u} [Preorder P] (s : Presentation P) | |
| theorem realize_forcingWitnessFormula (D : Name.NameDomain s) | |
| (ground : ∀ τ ∈ D.names, τ.val ∈ M.sets) | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (carrier : s.carrier ∈ M.sets) (order : s.conditionOrder ∈ M.sets) | |
| {k n m : Nat} (A : ForcingPrimitives k) (φ : SetFormula (n + 2)) | |
| (params : Fin k → Fin m) (names : Fin n → Fin m) (zi yi : Fin m) (env : Fin m → ZFSet.{u}) | |
| (hc : env (params A.carrier) = s.carrier) (ho : env (params A.order) = s.conditionOrder) | |
| (atomic : M.EqualityMeaning s D A.equality (fun j => env (params j))) | |
| (domain : M.DomainMeaning s D A.domain (fun j => env (params j))) | |
| (nameEnv : Assignment D n) (hnames : ∀ i, env (names i) = (nameEnv i).val.val) | |
| (hz : env zi ∈ M.sets) (hy : env yi ∈ M.sets) : | |
| Realize M.sets (forcingWitnessFormula A φ params names zi yi) env ↔ | |
| ∃ σ : D.names, ∃ p, env zi = ZFSet.pair σ.val.val (s.encode p) ∧ | |
| ∃ ρ : D.names, ρ.val.val = env yi ∧ Forces D φ (extend σ (extend ρ nameEnv)) p := by | |
| rw [forcingWitnessFormula, realize_ex] | |
| constructor | |
| · rintro ⟨x, hx, body⟩ | |
| obtain ⟨p, shape, source, target, forced⟩ := (M.realize_exConditions s (params A.carrier).succ _ | |
| (extend x env) carrier hc).1 body | |
| rw [realize_domainApplication] at source target | |
| obtain ⟨σ, eq⟩ := (domain x hx).1 source | |
| obtain ⟨ρ, eqρ⟩ := (domain (env yi) hy).1 target | |
| let e := extend (s.encode p) (extend x env) | |
| have raw := (M.realize_orderedPairFormula pairing zi.succ.succ 1 0 e hz hx | |
| (M.transitive carrier (s.encode_mem p))).1 shape | |
| refine ⟨σ, p, raw.trans (congrArg (fun z => ZFSet.pair z (s.encode p)) eq.symm), ρ, eqρ, ?_⟩ | |
| exact (M.realize_forcingTranslation s D ground pairing carrier order A φ | |
| (fun j => (params j).succ.succ) 0 (extend 1 (extend yi.succ.succ (fun j => (names j).succ.succ))) | |
| e hc ho atomic domain p (extend σ (extend ρ nameEnv)) rfl | |
| (Fin.cases eq.symm (Fin.cases eqρ.symm hnames))).1 forced | |
| · rintro ⟨σ, p, shape, ρ, eqρ, forced⟩ | |
| refine ⟨σ.val.val, ground σ.val σ.property, ?_⟩ | |
| apply (M.realize_exConditions s (params A.carrier).succ _ (extend σ.val.val env) carrier hc).2 | |
| let e := extend (s.encode p) (extend σ.val.val env) | |
| refine ⟨p, ?_, ?_, ?_, ?_⟩ | |
| · exact (M.realize_orderedPairFormula pairing zi.succ.succ 1 0 e hz (ground σ.val σ.property) | |
| (M.transitive carrier (s.encode_mem p))).2 shape | |
| · rw [realize_domainApplication] | |
| exact (domain σ.val.val (ground σ.val σ.property)).2 ⟨σ, rfl⟩ | |
| · rw [realize_domainApplication] | |
| exact (domain (env yi) hy).2 ⟨ρ, eqρ⟩ | |
| · exact (M.realize_forcingTranslation s D ground pairing carrier order A φ | |
| (fun j => (params j).succ.succ) 0 (extend 1 (extend yi.succ.succ (fun j => (names j).succ.succ))) | |
| e hc ho atomic domain p (extend σ (extend ρ nameEnv)) rfl | |
| (Fin.cases rfl (Fin.cases eqρ.symm hnames))).2 forced | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end OAI | |