openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Models /GroundFiniteWordInjections.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundFiniteWordLaws | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| theorem finiteWordMapGraph_injection | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| (separation : SeparationSchema M.sets) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {A B e : ZFSet.{u}} (hA : A ∈ M.sets) (hB : B ∈ M.sets) | |
| (code : InjectionCode A B e) : | |
| InjectionCode (M.finiteWords A) (M.finiteWords B) (M.finiteWordMapGraph A B e) := by | |
| let s := SetMembers.presentation A | |
| let t := SetMembers.presentation B | |
| let map := s.decodeFunction t e code.function | |
| have graph : s.graph t map = e := s.graph_decodeFunction t e code.function | |
| rw [← graph] | |
| change InjectionCode (M.finiteWords s.carrier) (M.finiteWords t.carrier) | |
| (M.finiteWordMapGraph s.carrier t.carrier (s.graph t map)) | |
| rw [M.finiteWordMapGraph_eq_graph pairing unions separation infinite s t map hA hB] | |
| exact (finiteWordPresentation s M pairing unions separation infinite hA).graph_injection | |
| (finiteWordPresentation t M pairing unions separation infinite hB) (List.map map) | |
| (List.map_injective_iff.2 code.decoded_injective) | |
| theorem finiteWordMapGraph_ground_injection | |
| (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) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {A B e : ZFSet.{u}} (hA : A ∈ M.sets) (hB : B ∈ M.sets) (he : e ∈ M.sets) | |
| (code : InjectionCode A B e) : | |
| M.finiteWordMapGraph A B e ∈ M.sets ∧ | |
| InjectionCode (M.finiteWords A) (M.finiteWords B) (M.finiteWordMapGraph A B e) := | |
| ⟨M.finiteWordMapGraph_mem pairing unions powers separation infinite hA hB he, | |
| M.finiteWordMapGraph_injection pairing unions separation infinite hA hB code⟩ | |
| theorem groundEmbeddable_finiteWords | |
| (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) | |
| (infinite : Realize M.sets infinity Fin.elim0) | |
| {A B : ZFSet.{u}} (hA : A ∈ M.sets) (hB : B ∈ M.sets) | |
| (injection : M.groundEmbeddable A B) : | |
| M.groundEmbeddable (M.finiteWords A) (M.finiteWords B) := by | |
| obtain ⟨e, he, code⟩ := injection | |
| exact ⟨M.finiteWordMapGraph A B e, | |
| M.finiteWordMapGraph_ground_injection pairing unions powers separation infinite hA hB he code⟩ | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |