openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /GroundExternalOldAction.lean
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundExternalLocalHeightSelection | |
| namespace OAI | |
| section | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetRealization SetFormula Compression.RecursiveMaps.CosetGeometry | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 : ZFSet.{u}} | |
| variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ} | |
| variable (data : GroundExternalGeometryInputs M initial I Sources Reps G m inv one T τ P0 r0 μ0 A0 completion) | |
| variable (group : GroupCode G m inv one) (oldAction : PosetActionCode G P0 m one r0 A0) | |
| variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| variable (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| variable (infinite : Realize M.sets infinity Fin.elim0) (hG : G ∈ M.sets) | |
| local notation "family" => completion.externalFamily group oldAction data.original | |
| local notation "U" => completion.algebra.union.fields 0 | |
| local notation "sGroup" => completion.externalFamilyGroupPresentation group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| local notation "qPoint" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "letters" => externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0) | |
| theorem externalCopyGraph_eq_graph : | |
| letI := group.toGroup | |
| let a := completion.algebra.union.action.toOrderAction group completion.algebra.union.meet | |
| ∀ w, completion.externalCopyGraph P0 r0 μ0 A0 ((sGroup).encode w) = | |
| (SetMembers.presentation U).graph (qPoint) ((family).copy a w).toFun := by | |
| let := group.toGroup | |
| dsimp only | |
| intro w | |
| have classed := data.externalPointClassifier_entry_iff group oldAction pairing unions powers separation infinite hG | |
| have graph := flatCopyEmbeddingGraph_eq_graph (family) | |
| (completion.algebra.union.action.toOrderAction group completion.algebra.union.meet) | |
| (sGroup) (SetMembers.presentation U) (qPoint) classed w | |
| rw [data.externalPointPresentation_carrier] at graph | |
| exact graph | |
| include data group oldAction pairing unions powers separation infinite hG in | |
| theorem externalCopy_ideal {w : ZFSet.{u}} (hw : w ∈ M.externalGroupCarrier G letters m one) : | |
| IdealEmbeddingCode U (completion.externalPointCarrier P0 r0 μ0 A0) | |
| (completion.algebra.union.fields 1) (completion.externalPointOrder P0 r0 μ0 A0) | |
| (completion.algebra.union.fields 2) (completion.externalPointMeet P0 r0 μ0 A0) | |
| (completion.externalCopyGraph P0 r0 μ0 A0 w) := by | |
| let := group.toGroup | |
| let a := completion.algebra.union.action.toOrderAction group completion.algebra.union.meet | |
| let s := sGroup | |
| let q := qPoint | |
| have sCarrier := completion.externalFamilyGroupPresentation_carrier group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| obtain ⟨w, rfl⟩ := (s.mem_carrier w).1 (sCarrier.symm ▸ hw) | |
| have entries := data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG | |
| have code := (SetMembers.presentation U).idealEmbeddingCode q ((family).copy a w) | |
| (fun _ _ => Iff.rfl) entries.1 | |
| (fun x y z => (completion.algebra.union.meet.toSmallMeetOrder_meet_eq_iff x y z).symm) entries.2.1 | |
| rw [← data.externalCopyGraph_eq_graph group oldAction pairing unions powers separation infinite hG w] at code | |
| dsimp only [q] at code | |
| rw [data.externalPointPresentation_carrier] at code | |
| exact code | |
| include data group oldAction pairing unions powers separation infinite hG in | |
| theorem externalIdentityCopy_ideal : | |
| IdealEmbeddingCode U (completion.externalPointCarrier P0 r0 μ0 A0) | |
| (completion.algebra.union.fields 1) (completion.externalPointOrder P0 r0 μ0 A0) | |
| (completion.algebra.union.fields 2) (completion.externalPointMeet P0 r0 μ0 A0) | |
| (completion.externalCopyGraph P0 r0 μ0 A0 (M.externalGroupIdentity G letters m one)) := | |
| data.externalCopy_ideal group oldAction pairing unions powers separation infinite hG | |
| data.group.group_code.identity_mem | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetRealization SetFormula Compression.RecursiveMaps.CosetGeometry | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 : ZFSet.{u}} | |
| variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ} | |
| variable (data : GroundExternalGeometryInputs M initial I Sources Reps G m inv one T τ P0 r0 μ0 A0 completion) | |
| variable (group : GroupCode G m inv one) (oldAction : PosetActionCode G P0 m one r0 A0) | |
| variable (pairing : Realize M.sets SetFormula.pairing Fin.elim0) | |
| variable (unions : Realize M.sets SetFormula.union Fin.elim0) | |
| variable (powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets) | |
| variable (infinite : Realize M.sets infinity Fin.elim0) (hG : G ∈ M.sets) | |
| local notation "family" => completion.externalFamily group oldAction data.original | |
| local notation "U" => completion.algebra.union.fields 0 | |
| local notation "sGroup" => completion.externalFamilyGroupPresentation group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| local notation "qPoint" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "letters" => externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0) | |
| local notation "copyGraph" => completion.externalCopyGraph P0 r0 μ0 A0 (M.externalGroupIdentity G letters m one) | |
| include data group oldAction pairing unions powers separation infinite hG in | |
| theorem externalIdentityCopy_equivariant {g g' x y x' y' : ZFSet.{u}} | |
| (actor : ZFSet.pair g g' ∈ M.externalOldInclusion G letters m one) | |
| (first : ZFSet.pair x x' ∈ copyGraph) (second : ZFSet.pair y y' ∈ copyGraph) | |
| (acted : ZFSet.pair (ZFSet.pair g x) y ∈ completion.algebra.union.fields 12) : | |
| ZFSet.pair (ZFSet.pair g' x') y' ∈ completion.externalPointAction P0 r0 μ0 A0 := by | |
| let := group.toGroup | |
| let a := completion.algebra.union.action.toOrderAction group completion.algebra.union.meet | |
| rw [completion.externalFamilyGroupPresentation_old group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground] at actor | |
| obtain ⟨g, equal⟩ := ((SetMembers.presentation G).mem_graph (sGroup) _ _).1 actor | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal | |
| have identity := completion.externalFamilyGroupPresentation_identity group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| rw [identity, data.externalCopyGraph_eq_graph group oldAction pairing unions powers separation infinite hG 1] at first second | |
| obtain ⟨x, equal⟩ := ((SetMembers.presentation U).mem_graph (qPoint) _ _).1 first | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal | |
| obtain ⟨y, equal⟩ := ((SetMembers.presentation U).mem_graph (qPoint) _ _).1 second | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal | |
| have actionEq := (completion.algebra.union.action.toOrderAction_eq_iff group completion.algebra.union.meet g x y).2 acted | |
| have entries := data.externalGeometry_entries group oldAction pairing unions powers separation infinite hG | |
| apply (entries.2.2.1 _ _ _).2 | |
| change (((family).externalAction a).iso (old (SetMembers G) (family).Letters g)).toFun | |
| (((family).copy a 1).toFun x) = ((family).copy a 1).toFun y | |
| rw [(family).extends_old_action, actionEq] | |
| include data group oldAction pairing unions powers separation infinite hG in | |
| theorem externalIdentityCopy_agreement (hP : P0 ∈ M.sets) | |
| {g h g' h' v v' : ZFSet.{u}} | |
| (firstActor : ZFSet.pair g g' ∈ M.externalOldInclusion G letters m one) | |
| (secondActor : ZFSet.pair h h' ∈ M.externalOldInclusion G letters m one) | |
| (image : ZFSet.pair v v' ∈ copyGraph) | |
| (agree : ActionAgreementBelow U (completion.algebra.union.fields 1) | |
| (completion.algebra.union.fields 12) g h v) : | |
| ActionAgreementBelow (completion.externalPointCarrier P0 r0 μ0 A0) | |
| (completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointAction P0 r0 μ0 A0) g' h' v' := by | |
| have ideal := data.externalIdentityCopy_ideal group oldAction pairing unions powers separation infinite hG | |
| have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP | |
| have firstShape := ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 firstActor) | |
| have secondShape := ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 secondActor) | |
| exact ideal.actionAgreementBelow completion.algebra.union.action geometry.action | |
| firstShape.1 firstShape.2 secondShape.2 image agree | |
| (fun _ _ _ _ _ _ first second acted => data.externalIdentityCopy_equivariant group oldAction | |
| pairing unions powers separation infinite hG firstActor first second acted) | |
| (fun _ _ _ _ _ _ first second acted => data.externalIdentityCopy_equivariant group oldAction | |
| pairing unions powers separation infinite hG secondActor first second acted) | |
| include data group oldAction pairing unions powers separation infinite hG in | |
| theorem externalOriginal_agreement (hP : P0 ∈ M.sets) | |
| {g h g' h' v v' : ZFSet.{u}} | |
| (firstActor : ZFSet.pair g g' ∈ M.externalOldInclusion G letters m one) | |
| (secondActor : ZFSet.pair h h' ∈ M.externalOldInclusion G letters m one) | |
| (hv : v ∈ P0) (image : ZFSet.pair v v' ∈ copyGraph) | |
| (agree : ActionAgreementBelow P0 r0 A0 g h v) : | |
| ActionAgreementBelow (completion.externalPointCarrier P0 r0 μ0 A0) | |
| (completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointAction P0 r0 μ0 A0) g' h' v' := by | |
| have hg := (ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 firstActor)).1 | |
| have hh := (ZFSet.pair_mem_prod.1 (data.old_inclusion_injection.function.1 secondActor)).1 | |
| exact data.externalIdentityCopy_agreement group oldAction pairing unions powers separation infinite hG hP | |
| firstActor secondActor image (data.original.agreement g hg h hh v hv agree) | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| end | |
| end OAI | |