openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /FlatSuccessorConditions.lean
| import OAI.SetTheory.PartitionPrinciple.Transfinite.StageOneDataBoundLaws | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| open SetRealization | |
| universe u | |
| variable (M : TransitiveGround.{u}) {R P : Type u} | |
| variable (c : Presentation R) (s : Presentation P) | |
| variable (I T levels alpha : ZFSet.{u}) (tests : ZFSet.{u} → P → Prop) | |
| structure FlatSuccessorCondition where | |
| row : R | |
| assignments : ZFSet.{u} | |
| assignments_mem : assignments ∈ | |
| M.groundSmallSubsets (stageOneAssignments I T) ZFSet.omega | |
| lower : P | |
| lower_code : s.encode lower = ZFSet.pair (c.encode row) | |
| (assignmentCut levels alpha assignments) | |
| tests : tests assignments lower | |
| namespace FlatSuccessorCondition | |
| local notation "Condition" => FlatSuccessorCondition M c s I T levels alpha tests | |
| @[ext] theorem ext {p q : Condition} | |
| (rows : p.row = q.row) (data : p.assignments = q.assignments) : p = q := by | |
| cases p with | |
| | mk rp Ap smallp lp codep testp => | |
| cases q with | |
| | mk rq Aq smallq lq codeq testq => | |
| dsimp only at rows data | |
| subst rq | |
| subst Aq | |
| have equal : lp = lq := s.injective (codep.trans codeq.symm) | |
| subst lq | |
| rfl | |
| noncomputable def code (p : Condition) : ZFSet.{u} := ZFSet.pair (c.encode p.row) p.assignments | |
| theorem code_eq_iff (p q : Condition) : | |
| code M c s I T levels alpha tests p = | |
| code M c s I T levels alpha tests q ↔ p.row = q.row ∧ p.assignments = q.assignments := by | |
| rw [code, code, ZFSet.pair_inj] | |
| exact and_congr c.injective.eq_iff Iff.rfl | |
| theorem code_injective : Function.Injective (code M c s I T levels alpha tests) := by | |
| intro p q equal | |
| obtain ⟨rows, data⟩ := (code_eq_iff M c s I T levels alpha tests p q).1 equal | |
| exact ext M c s I T levels alpha tests rows data | |
| theorem code_mem_candidateBound (p : Condition) : | |
| code M c s I T levels alpha tests p ∈ M.stageOneCandidateBound c.carrier I T := | |
| ZFSet.pair_mem_prod.2 ⟨c.encode_mem p.row, p.assignments_mem⟩ | |
| instance small : Small.{u} Condition := by | |
| let bound := M.stageOneCandidateBound c.carrier I T | |
| let f : Condition → bound := fun p => | |
| ⟨code M c s I T levels alpha tests p, | |
| code_mem_candidateBound M c s I T levels alpha tests p⟩ | |
| apply small_of_injective (f := f) | |
| intro p q equal | |
| exact code_injective M c s I T levels alpha tests (congrArg Subtype.val equal) | |
| end FlatSuccessorCondition | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |