openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /CodedEqualityRelations.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundEqualityStep | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.SetFormula | |
| def tripleMembershipFormula {n : Nat} (p x y r : Fin n) : SetFormula n := | |
| ex (.conj (orderedPairFormula 0 x.succ y.succ) | |
| (pairMembershipFormula p.succ 0 r.succ)) | |
| end PartitionPilot.Forcing.SetFormula | |
| namespace PartitionPilot.Forcing.Name | |
| open SetRealization | |
| universe u | |
| variable {P : Type u} (s : Presentation P) | |
| def codedEqualityRelation (S : ZFSet.{u}) (p : P) (σ τ : Name s) : Prop := | |
| ZFSet.pair (s.encode p) (ZFSet.pair σ.val τ.val) ∈ S | |
| theorem codedEqualityRelation_mono {S T : ZFSet.{u}} (h : S ⊆ T) (p : P) (σ τ : Name s) : | |
| codedEqualityRelation s S p σ τ → codedEqualityRelation s T p σ τ := | |
| fun member => h member | |
| end PartitionPilot.Forcing.Name | |
| namespace PartitionPilot.Forcing.TransitiveGround | |
| open SetRealization SetFormula | |
| universe u | |
| variable (M : TransitiveGround.{u}) | |
| theorem realize_tripleMembershipFormula | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| {n : Nat} (p x y r : Fin n) (env : Fin n → ZFSet.{u}) | |
| (hp : env p ∈ M.sets) (hx : env x ∈ M.sets) | |
| (hy : env y ∈ M.sets) (hr : env r ∈ M.sets) : | |
| Realize M.sets (tripleMembershipFormula p x y r) env ↔ | |
| ZFSet.pair (env p) (ZFSet.pair (env x) (env y)) ∈ env r := by | |
| rw [tripleMembershipFormula, realize_ex] | |
| constructor | |
| · rintro ⟨z, hz, shape, member⟩ | |
| have eq := (M.realize_orderedPairFormula pairing 0 x.succ y.succ | |
| (extend z env) hz hx hy).1 shape | |
| have raw := (M.realize_pairMembershipFormula pairing p.succ 0 r.succ | |
| (extend z env) hp hz hr).1 member | |
| change z = ZFSet.pair (env x) (env y) at eq | |
| change ZFSet.pair (env p) z ∈ env r at raw | |
| exact eq ▸ raw | |
| · intro member | |
| have hz := M.orderedPair_mem pairing hx hy | |
| refine ⟨ZFSet.pair (env x) (env y), hz, ?_, ?_⟩ | |
| · exact (M.realize_orderedPairFormula pairing 0 x.succ y.succ | |
| (extend (ZFSet.pair (env x) (env y)) env) hz hx hy).2 rfl | |
| · exact (M.realize_pairMembershipFormula pairing p.succ 0 r.succ | |
| (extend (ZFSet.pair (env x) (env y)) env) hp hz hr).2 member | |
| variable {P : Type u} (s : Presentation P) | |
| theorem relationMeaning_coded (D : Name.NameDomain s) | |
| (ground : ∀ τ ∈ D.names, τ.val ∈ M.sets) | |
| (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| (carrier : s.carrier ∈ M.sets) | |
| {k : Nat} (r : Fin k) (constants : Fin k → ZFSet.{u}) | |
| (hr : constants r ∈ M.sets) : | |
| M.RelationMeaning s D (Name.codedEqualityRelation s (constants r)) | |
| (tripleMembershipFormula 0 1 2 r.succ.succ.succ) constants := by | |
| intro p σ τ | |
| exact M.realize_tripleMembershipFormula pairing 0 1 2 r.succ.succ.succ _ | |
| (M.transitive carrier (s.encode_mem p)) (ground σ.val σ.property) | |
| (ground τ.val τ.property) hr | |
| theorem coded_postfixed_sound [Preorder P] (D : Name.NameDomain s) (S : ZFSet.{u}) | |
| (post : ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, | |
| Name.codedEqualityRelation s S p σ τ → | |
| Name.EqualityStep (Name.codedEqualityRelation s S) p σ τ) | |
| {σ τ : Name s} (hσ : σ ∈ D.names) (hτ : τ ∈ D.names) (p : P) : | |
| Name.codedEqualityRelation s S p σ τ → Name.ForcesEq p σ τ := | |
| Name.postfixed_equality_sound D (Name.codedEqualityRelation s S) post σ hσ τ hτ p | |
| theorem coded_fixed_iff [Preorder P] (D : Name.NameDomain s) (S : ZFSet.{u}) | |
| (fixed : ∀ σ ∈ D.names, ∀ τ ∈ D.names, ∀ p, | |
| Name.codedEqualityRelation s S p σ τ ↔ | |
| Name.EqualityStep (Name.codedEqualityRelation s S) p σ τ) | |
| {σ τ : Name s} (hσ : σ ∈ D.names) (hτ : τ ∈ D.names) (p : P) : | |
| Name.codedEqualityRelation s S p σ τ ↔ Name.ForcesEq p σ τ := | |
| Name.fixed_equality_iff D (Name.codedEqualityRelation s S) fixed hσ hτ p | |
| end PartitionPilot.Forcing.TransitiveGround | |
| end | |
| end OAI | |