openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Models /GroundNameSupportEnumerationChoices.lean
| import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundOrdinalEnumerationFamilies | |
| namespace OAI | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) {G P L : Type u} | |
| variable [Group G] [PartialOrder P] [PartialOrder L] [OrderBot L] | |
| variable (a : PosetAction G P) (s : Presentation P) (t : Presentation G) | |
| variable (b : Presentation L) (d : PosetAction G L) | |
| variable (directed : ∀ x y : L, ∃ z, x ≤ z ∧ y ≤ z) | |
| theorem exists_groundSupportEnumerationChoices | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| (collection : CollectionSchema M.sets) (choice : Realize M.sets SetFormula.choice Fin.elim0) | |
| (carrier : s.carrier ∈ M.sets) (group : t.carrier ∈ M.sets) | |
| (nodes : b.carrier ∈ M.sets) (nodeOrder : b.conditionOrder ∈ M.sets) | |
| (supports : supportBasisCode b t d.principalFix ∈ M.sets) | |
| (maps : a.conditionMapFamily s t ∈ M.sets) | |
| (W : ZFSet.{u}) (hW : W ∈ M.sets) (valid : ∀ z ∈ W, IsName s z) | |
| (symmetric : ∀ tau : Name s, tau.val ∈ W → | |
| HereditarilySymmetric a s (d.principalSupportFamily ⊥ directed) tau) : | |
| ∃ supportChoice ∈ M.sets, ∃ B ∈ M.sets, ∃ enumerationChoice ∈ M.sets, | |
| ZFSet.IsFunc W b.carrier supportChoice ∧ ZFSet.IsFunc b.carrier B enumerationChoice ∧ | |
| let selected := relationCompose W b.carrier B supportChoice enumerationChoice | |
| selected ∈ M.sets ∧ ZFSet.IsFunc W B selected ∧ | |
| ∀ (tau : Name s) z, ZFSet.pair tau.val z ∈ selected → | |
| ∃ D : L, ZFSet.pair tau.val (b.encode D) ∈ supportChoice ∧ | |
| ZFSet.pair (b.encode D) z ∈ enumerationChoice ∧ | |
| d.principalFix D ≤ a.stabilizer s tau ∧ | |
| ∃ delta ∈ M.sets, ∃ f ∈ M.sets, z = ZFSet.pair delta f ∧ | |
| ordinalSetProperty delta ∧ | |
| InjectionCode delta (relationFiber b.nonbottom.carrier | |
| (relationConverse b.carrier b.carrier b.conditionOrder) (b.encode D)) f ∧ | |
| ∀ v : L, v ≠ ⊥ → v ≤ D → ∃ i ∈ delta, ZFSet.pair i (b.encode v) ∈ f := by | |
| have hNonbottom := M.nonbottom_carrier_mem b separation nodes | |
| let reverse := relationConverse b.carrier b.carrier b.conditionOrder | |
| have hReverse := M.relationConverse_mem pairing unions powers separation nodes nodes nodeOrder | |
| let fibers := M.relationFiberFamily b.carrier b.nonbottom.carrier reverse | |
| have hFibers : fibers ∈ M.sets := M.relationFiberFamily_mem pairing unions powers separation | |
| nodes hNonbottom hReverse | |
| have fibersFunction := M.relationFiberFamily_function separation nodes hNonbottom hReverse | |
| have hPower := M.groundPower_mem powers hNonbottom | |
| obtain ⟨B, hB, enumerations, hEnumerations, enumFunction, enumValues⟩ := | |
| M.exists_groundOrdinalEnumerationFamily pairing unions powers separation collection choice | |
| (M.groundPower b.nonbottom.carrier) hPower | |
| let enumerationChoice := relationCompose b.carrier (M.groundPower b.nonbottom.carrier) B fibers enumerations | |
| have enumerationGround : enumerationChoice ∈ M.sets := M.relationCompose_mem pairing unions powers | |
| separation nodes hPower hB hFibers hEnumerations | |
| have enumerationFunction : ZFSet.IsFunc b.carrier B enumerationChoice := | |
| relationCompose_function_of_subset fibersFunction enumFunction (fun _ h => h) | |
| obtain ⟨supportChoice, hSupportChoice, supportFunction, supported⟩ := | |
| M.exists_groundNameSupportChoice a s t b d ⊥ directed pairing unions powers separation collection | |
| choice carrier group nodes supports maps W hW valid symmetric | |
| refine ⟨supportChoice, hSupportChoice, B, hB, enumerationChoice, enumerationGround, | |
| supportFunction, enumerationFunction, ?_⟩ | |
| dsimp only | |
| refine ⟨M.relationCompose_mem pairing unions powers separation hW nodes hB hSupportChoice enumerationGround, | |
| relationCompose_function_of_subset supportFunction enumerationFunction (fun _ h => h), ?_⟩ | |
| intro tau z entry | |
| obtain ⟨_, _, v, hv, supportEdge, enumEdge⟩ := (pair_mem_relationCompose _ _ _ _ _ _ _).1 entry | |
| obtain ⟨D, rfl⟩ := (b.mem_carrier v).1 hv | |
| have fixes := (supported tau D supportEdge).2 | |
| obtain ⟨_, _, ideal, _, idealEdge, chosen⟩ := (pair_mem_relationCompose _ _ _ _ _ _ _).1 enumEdge | |
| have idealEq := ((M.pair_mem_relationFiberFamily b.carrier b.nonbottom.carrier reverse | |
| (b.encode D) ideal).1 idealEdge).2.2 | |
| obtain ⟨delta, hDelta, f, hf, shape, ordinal, injection, onto⟩ := enumValues ideal z chosen | |
| rw [idealEq] at injection onto | |
| refine ⟨D, supportEdge, enumEdge, fixes, delta, hDelta, f, hf, shape, ordinal, injection, ?_⟩ | |
| intro v nonzero below | |
| have member : b.encode v ∈ relationFiber b.nonbottom.carrier reverse (b.encode D) := by | |
| apply (mem_relationFiber _ _ _ _).2 | |
| refine ⟨?_, (pair_mem_relationConverse _ _ _ _ _).2 | |
| ⟨b.encode_mem v, b.encode_mem D, (b.pair_mem_conditionOrder v D).2 below⟩⟩ | |
| exact (b.nonbottom.mem_carrier _).2 ⟨⟨v, nonzero⟩, rfl⟩ | |
| exact onto (b.encode v) member | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end OAI | |