openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /BoundedDirectedOrders.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundContainmentOrders | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.BoundedFormula | |
| open SetFormula | |
| universe u | |
| def upwardDirectedFormula {n} (l r : Fin n) : BoundedFormula n := | |
| .allMem l (.allMem l.succ (exMem l.succ.succ | |
| (.conj (pairMembershipFormula 2 0 r.succ.succ.succ) | |
| (pairMembershipFormula 1 0 r.succ.succ.succ)))) | |
| theorem truth_upwardDirectedFormula {n} (l r : Fin n) (env : Fin n → ZFSet.{u}) : | |
| Truth (upwardDirectedFormula l r) env ↔ | |
| ∀ x ∈ env l, ∀ y ∈ env l, ∃ z ∈ env l, | |
| ZFSet.pair x z ∈ env r ∧ ZFSet.pair y z ∈ env r := by | |
| simp only [upwardDirectedFormula, Truth, truth_exMem, truth_pairMembershipFormula] | |
| rfl | |
| end PartitionPilot.Forcing.BoundedFormula | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| theorem realize_upwardDirectedCode {L R : ZFSet.{u}} (hL : L ∈ M.sets) (hR : R ∈ M.sets) : | |
| Realize M.sets (BoundedFormula.upwardDirectedFormula 0 1).toFormula | |
| (extend L (extend R Fin.elim0)) ↔ | |
| ∀ x ∈ L, ∀ y ∈ L, ∃ z ∈ L, ZFSet.pair x z ∈ R ∧ ZFSet.pair y z ∈ R := | |
| (BoundedFormula.realize_toFormula M _ _ | |
| (Fin.cases hL (Fin.cases hR (fun j => Fin.elim0 j)))).trans | |
| (BoundedFormula.truth_upwardDirectedFormula 0 1 _) | |
| theorem decoded_upward_directed {L R : ZFSet.{u}} (order : PartialOrderCode L R) | |
| (hL : L ∈ M.sets) (hR : R ∈ M.sets) | |
| (laws : Realize M.sets (BoundedFormula.upwardDirectedFormula 0 1).toFormula | |
| (extend L (extend R Fin.elim0))) : | |
| letI := order.toPartialOrder | |
| ∀ C D : SetMembers L, ∃ E, C ≤ E ∧ D ≤ E := by | |
| let := order.toPartialOrder | |
| intro C D | |
| obtain ⟨z, hz, cz, dz⟩ := (M.realize_upwardDirectedCode hL hR).1 laws | |
| _ (SetMembers.raw_mem C) _ (SetMembers.raw_mem D) | |
| refine ⟨SetMembers.ofMem z hz, ?_, ?_⟩ | |
| · change ZFSet.pair (SetMembers.raw C) (SetMembers.raw (SetMembers.ofMem z hz)) ∈ R | |
| rw [SetMembers.raw_ofMem] | |
| exact cz | |
| · change ZFSet.pair (SetMembers.raw D) (SetMembers.raw (SetMembers.ofMem z hz)) ∈ R | |
| rw [SetMembers.raw_ofMem] | |
| exact dz | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |