openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /BoundedMeetOrderCodes.lean
| import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundMainStageFromCodes | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.BoundedFormula | |
| open SetFormula | |
| universe u | |
| def meetOrderFormula {n} (P r μ : Fin n) : BoundedFormula n := | |
| .conj (partialOrderFormula P r) (.conj (binaryFunctionFormula P μ) | |
| (.allMem P (.allMem P.succ (.allMem P.succ.succ | |
| (imp (tripleMembershipFormula 2 1 0 μ.succ.succ.succ) | |
| (.conj (pairMembershipFormula 0 2 r.succ.succ.succ) | |
| (.conj (pairMembershipFormula 0 1 r.succ.succ.succ) | |
| (.allMem P.succ.succ.succ (imp | |
| (.conj (pairMembershipFormula 0 3 r.succ.succ.succ.succ) | |
| (pairMembershipFormula 0 2 r.succ.succ.succ.succ)) | |
| (pairMembershipFormula 0 1 r.succ.succ.succ.succ)))))))))) | |
| theorem truth_meetOrderFormula {n} (P r μ : Fin n) (env : Fin n → ZFSet.{u}) : | |
| Truth (meetOrderFormula P r μ) env ↔ MeetOrderCode (env P) (env r) (env μ) := by | |
| simp only [meetOrderFormula, Truth, truth_partialOrderFormula, | |
| truth_binaryFunctionFormula, truth_imp, truth_pairMembershipFormula, | |
| truth_tripleMembershipFormula] | |
| constructor | |
| · rintro ⟨order, operation, laws⟩ | |
| have bounds (x y z : ZFSet.{u}) (entry : ZFSet.pair (ZFSet.pair x y) z ∈ env μ) : | |
| x ∈ env P ∧ y ∈ env P ∧ z ∈ env P := by | |
| have shape := ZFSet.pair_mem_prod.1 (operation.1 entry) | |
| exact ⟨(ZFSet.pair_mem_prod.1 shape.1).1, | |
| (ZFSet.pair_mem_prod.1 shape.1).2, shape.2⟩ | |
| refine ⟨order, operation, ?_, ?_, ?_⟩ | |
| · intro x y z entry | |
| have h := bounds x y z entry | |
| exact (laws x h.1 y h.2.1 z h.2.2 entry).1 | |
| · intro x y z entry | |
| have h := bounds x y z entry | |
| exact (laws x h.1 y h.2.1 z h.2.2 entry).2.1 | |
| · intro x y z w hw entry wx wy | |
| have h := bounds x y z entry | |
| exact (laws x h.1 y h.2.1 z h.2.2 entry).2.2 w hw ⟨wx, wy⟩ | |
| · intro meet | |
| exact ⟨meet.order, meet.operation, fun x _ y _ z _ entry => | |
| ⟨meet.lower_left x y z entry, meet.lower_right x y z entry, | |
| fun w hw ⟨wx, wy⟩ => meet.greatest x y z w hw entry wx wy⟩⟩ | |
| theorem realize_meetOrderFormula (M : TransitiveGround.{u}) {n} | |
| (P r μ : Fin n) (env : Fin n → ZFSet.{u}) (members : ∀ i, env i ∈ M.sets) : | |
| Realize M.sets (meetOrderFormula P r μ).toFormula env ↔ | |
| MeetOrderCode (env P) (env r) (env μ) := | |
| (realize_toFormula M _ env members).trans (truth_meetOrderFormula P r μ env) | |
| end PartitionPilot.Forcing.BoundedFormula | |
| end | |
| end OAI | |