openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /CodedLayerRestrictions.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GrowthLayerSourceCongruence | |
| import OAI.SetTheory.PartitionPrinciple.Geometry.TemplateHeightRangeCodes | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| theorem MeetEmbeddingCode.restrictTarget {A L Q rA rL μA μL e : ZFSet.{u}} | |
| (code : MeetEmbeddingCode A L rA rL μA μL e) | |
| (values : ∀ x y, ZFSet.pair x y ∈ e → y ∈ Q) : | |
| MeetEmbeddingCode A Q rA (heightOrderRestriction Q rL) μA | |
| (graphRestriction (ZFSet.prod Q Q) μL) e := by | |
| refine ⟨⟨⟨?_, code.injection.function.2⟩, code.injection.injective⟩, ?_, ?_⟩ | |
| · intro z hz | |
| obtain ⟨x, hx, y, _, rfl⟩ := ZFSet.mem_prod.1 (code.injection.function.1 hz) | |
| exact ZFSet.pair_mem_prod.2 ⟨hx, values x y hz⟩ | |
| · intro x y x' y' lower first second | |
| exact (pair_mem_heightOrderRestriction _ _ _ _).2 | |
| ⟨code.monotone x y x' y' lower first second, values x x' first, values y y' second⟩ | |
| · intro x y z x' y' z' meet first second third | |
| exact pair_mem_graphRestriction.2 | |
| ⟨code.map_meet x y z x' y' z' meet first second third, | |
| ZFSet.pair_mem_prod.2 ⟨values x x' first, values y y' second⟩⟩ | |
| def CodedGrowthLayer.restrictBirth | |
| {M : TransitiveGround.{u}} {Space sourceOrder sourceMeet : ZFSet.{u} → ZFSet.{u}} | |
| {L r μ birth t : ZFSet.{u}} | |
| (layer : CodedGrowthLayer M Space sourceOrder sourceMeet L r μ birth t) | |
| (meet : MeetOrderCode L r μ) : | |
| CodedGrowthLayer M Space sourceOrder sourceMeet L r μ (graphRestriction L birth) t := by | |
| cases layer with | |
| | countablePoints D ground small cover => | |
| exact .countablePoints D ground small | |
| (fun x hx entry => cover x hx (pair_mem_graphRestriction.1 entry).1) | |
| | copies I parts groundI small groundParts index e earlier groundCopy embedding cover inCopy => | |
| exact .copies I parts groundI small groundParts index e earlier groundCopy embedding | |
| (fun x hx entry => cover x hx (pair_mem_graphRestriction.1 entry).1) inCopy | |
| | separated I index e earlier groundCopy embedding component cover separated => | |
| let original (x : {x : ZFSet.{u} // x ∈ L ∧ ZFSet.pair x t ∈ graphRestriction L birth}) : | |
| {x : ZFSet.{u} // x ∈ L ∧ ZFSet.pair x t ∈ birth} := | |
| ⟨x.val, x.property.1, (pair_mem_graphRestriction.1 x.property.2).1⟩ | |
| refine .separated I index e earlier groundCopy embedding (fun x => component (original x)) | |
| (fun x => cover (original x)) ?_ | |
| intro x y different | |
| rcases separated (original x) (original y) different with | |
| ⟨z, a, entry, born, before⟩ | ⟨z, hz, a, born, before, between⟩ | |
| · exact Or.inl ⟨z, a, entry, pair_mem_graphRestriction.2 | |
| ⟨born, (ZFSet.pair_mem_prod.1 (meet.operation.1 entry)).2⟩, before⟩ | |
| · exact Or.inr ⟨z, hz, a, pair_mem_graphRestriction.2 ⟨born, hz⟩, before, between⟩ | |
| end PartitionPilot.Forcing | |
| end | |
| end OAI | |