openAI-math-dataset / lean /OAI /Computability /DegreeRigidity /CohenForcing /CohenPartitionRealization.lean
| import OAI.Computability.DegreeRigidity.CohenForcing.CohenPartitionJoint | |
| import OAI.Computability.DegreeRigidity.CohenForcing.CohenColumnRealValue | |
| namespace OAI | |
| namespace TuringRigidity.CohenPartitionRealization | |
| open TransitiveNameModel BoundedSetTheory CountableForcing CohenGroundPoset | |
| open InternalCohenFactor InternalCohenProjectedGeneric TaggedProductConditions CohenProductSplitting | |
| attribute [local instance] InternalCollapse.order InternalCollapse.collapsePreorder | |
| theorem reassemble_projected (M A B : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M) | |
| (hA : A ∈ M) (hB : B ∈ M) (hBA : B ⊆ A) | |
| (J : GenericFilter (Conditions (conditions B))) (hJ : AtomicForcing.GroundGeneric M J) | |
| (R : GenericFilter (Conditions (conditions (A \ B)))) | |
| (hR : AtomicForcing.GroundGeneric (genericExtensionSet M (conditions B) J.carrier) R) : | |
| ∃ U : GenericFilter (Conditions (conditions A)), AtomicForcing.GroundGeneric M U ∧ | |
| genericExtensionSet M (conditions A) U.carrier = | |
| genericExtensionSet (genericExtensionSet M (conditions B) J.carrier) | |
| (conditions (A \ B)) R.carrier ∧ projected A B hBA U = J := by | |
| have hfac := factor_internal M A B hM hT hA hB | |
| have hcA := conditions_mem M A hM hT hA | |
| obtain ⟨c,hc,hcs,f,hf,_,hfs,_⟩ := internal_product M _ _ hM hT hfac.1 hfac.2.1 | |
| let T := TaggedProductGeneric.joint _ _ c hcs J R | |
| have hTg := TaggedProductGeneric.joint_ground_generic M _ _ c hM hT hfac.1 hfac.2.1 hcs J hJ R hR | |
| have hTe := TaggedProductExtension.extension_eq M _ _ c hM hT hfac.1 hfac.2.1 hc hcs J hJ R hR | |
| let e := (productIso (conditions B) (conditions (A \ B)) c hcs).symm.trans (factorIso A B hBA).symm | |
| obtain ⟨g,hg,hge⟩ := CohenPartitionBridge.internal_union_iso M A B c f hM hT hA hB hBA hc hf hcs hfs | |
| let U := AutomorphismName.mapFilter e T | |
| have hU := InternalOrderIsoTransport.map_ground_generic M _ _ g hM hT hc hcA hg e hge T hTg | |
| have hUe := (InternalOrderIsoTransport.extension_eq M _ _ g hM hT hc hcA hg e hge T hTg).trans hTe | |
| have hmem (p : Conditions (conditions A)) : p ∈ U.carrier ↔ | |
| project A B hBA p ∈ J.carrier ∧ | |
| project A (A \ B) (fun _ h => (ZFSet.mem_sdiff.mp h).1) p ∈ R.carrier := by | |
| change (productIso _ _ c hcs).symm ((productIso _ _ c hcs) ((factorIso A B hBA) p)) ∈ | |
| (SourceEquationTail.productFilter J R).carrier ↔ _ | |
| rw [OrderIso.symm_apply_apply] | |
| rfl | |
| refine ⟨U,hU,hUe,InternalCohen.filter_ext _ _ ?_⟩ | |
| apply Set.ext | |
| intro p | |
| rw [mem_projected] | |
| constructor | |
| · rintro ⟨q,hq,rfl⟩ | |
| exact ((hmem q).mp hq).1 | |
| · intro hp | |
| obtain ⟨r,hr⟩ := R.nonempty | |
| let q := (factorIso A B hBA).symm (p,r) | |
| have hq : factorIso A B hBA q = (p,r) := (factorIso A B hBA).apply_symm_apply _ | |
| have hq1 : project A B hBA q = p := congrArg Prod.fst hq | |
| have hq2 : project A (A \ B) (fun _ h => (ZFSet.mem_sdiff.mp h).1) q = r := congrArg Prod.snd hq | |
| exact ⟨q,(hmem q).mpr ⟨hq1.symm ▸ hp,hq2.symm ▸ hr⟩,hq1⟩ | |
| theorem reassemble_column (M K a : ZFSet.{0}) (hM : Transitive M) (hT : SourceT M) | |
| (hK : K ∈ M) (ha : a ∈ K) (X : Oracle) | |
| (J : GenericFilter (Conditions (CohenColumnRealName.poset {a}))) | |
| (hJ : AtomicForcing.GroundGeneric M J) | |
| (hJX : (CohenColumnRealName.realName {a} a).val J.carrier = realCode X) | |
| (R : GenericFilter (Conditions (conditions (ZFSet.prod K ZFSet.omega \ ZFSet.prod {a} ZFSet.omega)))) | |
| (hR : AtomicForcing.GroundGeneric (genericExtensionSet M (CohenColumnRealName.poset {a}) J.carrier) R) : | |
| ∃ U : GenericFilter (Conditions (CohenColumnRealName.poset K)), | |
| AtomicForcing.GroundGeneric M U ∧ | |
| genericExtensionSet M (CohenColumnRealName.poset K) U.carrier = | |
| genericExtensionSet (genericExtensionSet M (CohenColumnRealName.poset {a}) J.carrier) | |
| (conditions (ZFSet.prod K ZFSet.omega \ ZFSet.prod {a} ZFSet.omega)) R.carrier ∧ | |
| (CohenColumnRealName.realName K a).val U.carrier = realCode X := by | |
| have hω := sourceT_omega_mem M hM hT | |
| have hsingle := singleton_mem M hM hT.pairing (hM K hK a ha) | |
| have hA := product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded hK hω | |
| have hB := product_mem M hM hT.pairing hT.union hT.powerSet hT.separation.finitePrefix.bounded hsingle hω | |
| obtain ⟨U,hU,hUe,hUJ⟩ := reassemble_projected M _ _ hM hT hA hB | |
| (CohenColumnRealName.singleton_coordinates_subset K a ha) J hJ R hR | |
| obtain ⟨Y,hYJ,hYU,_,_⟩ := CohenColumnRealName.projected_real_value M K a hM hT hK ha U hU | |
| rw [hUJ,hJX] at hYJ | |
| exact ⟨U,hU,hUe,hYU.trans hYJ.symm⟩ | |
| end TuringRigidity.CohenPartitionRealization | |
| end OAI | |