openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Forcing /GroundExternalLayerWitnessRanks.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.ExternalStarCarrierRanks | |
| import OAI.SetTheory.PartitionPrinciple.Transfinite.ExternalBirthLayerSeparation | |
| namespace OAI | |
| namespace PartitionPilot.Forcing.GroundExternalBirthData | |
| open SetFormula SetRealization Compression 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 oldTimes oldBirth : ZFSet.{u}} | |
| variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ} | |
| variable (births : GroundExternalBirthData M initial I Sources Reps G m one T τ P0 r0 μ0 A0 oldTimes oldBirth completion) | |
| 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) (hP0 : P0 ∈ M.sets) | |
| local notation "family" => completion.externalFamily group oldAction data.original | |
| local notation "s" => completion.externalFamilyGroupPresentation group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "U" => completion.algebra.union.fields 0 | |
| local notation "B" => completion.externalPointCarrier P0 r0 μ0 A0 | |
| local notation "P" => completion.externalSeedCarrier P0 r0 μ0 A0 | |
| local notation "owner" => completion.externalStarOwner P0 r0 μ0 A0 | |
| local notation "levels" => completion.externalOwnerLevels P0 r0 μ0 A0 | |
| include births data group oldAction pairing unions powers separation infinite hG hP0 | |
| variable {HQs Hrs Hμs HTime : ZFSet.{u}} | |
| variable (historySpaces : FunctionGraphOn HTime HQs) | |
| variable (historyOrders : FunctionGraphOn HTime Hrs) | |
| variable (historyMeets : FunctionGraphOn HTime Hμs) | |
| variable (source : ZFSet.pair oldTimes (completion.algebra.union.fields 0) ∈ HQs ∧ | |
| ZFSet.pair oldTimes (completion.algebra.union.fields 1) ∈ Hrs ∧ | |
| ZFSet.pair oldTimes (completion.algebra.union.fields 2) ∈ Hμs) | |
| include historySpaces historyOrders historyMeets source | |
| theorem layerSeparatedCode (k : Nat) : | |
| SeparatedLayerCode HQs Hrs Hμs B (completion.externalPointOrder P0 r0 μ0 A0) | |
| (completion.externalPointMeet P0 r0 μ0 A0) births.birth (compressionStamp oldTimes k) | |
| (births.layerOwners k) (births.layerIndices k) (births.layerMaps k) | |
| (M.groundRelations U B) (births.layerComponents k) (births.layerFiber k) := by | |
| refine ⟨births.layerIndices_function k, | |
| (births.layerMaps_spec data group oldAction pairing unions powers separation infinite hG hP0 k).1, | |
| rfl, births.layerComponents_function data group oldAction pairing unions powers separation infinite hG hP0 k, | |
| ?_, ?_, ?_⟩ | |
| · intro i _ t _ e Q R V indexEntry mapEntry spaceEntry orderEntry meetEntry | |
| obtain rfl := ((pair_mem_constantFamilyGraph _ _ _ _).1 indexEntry).2 | |
| have qeq := historySpaces.unique spaceEntry source.1 | |
| have req := historyOrders.unique orderEntry source.2.1 | |
| have veq := historyMeets.unique meetEntry source.2.2 | |
| subst Q R V | |
| exact births.layerMaps_embedding data group oldAction pairing unions powers separation infinite hG hP0 k mapEntry | |
| · intro i _ t _ e Q x _ indexEntry mapEntry spaceEntry componentEntry | |
| obtain rfl := ((pair_mem_constantFamilyGraph _ _ _ _).1 indexEntry).2 | |
| have qeq := historySpaces.unique spaceEntry source.1 | |
| subst Q | |
| have owned := (pair_mem_graphRestriction.1 componentEntry).1 | |
| have hx := (pair_mem_graphRestriction.1 componentEntry).2 | |
| obtain ⟨_, _, _, _, covered⟩ := births.layerMaps_copy data group oldAction pairing unions powers separation infinite hG hP0 k hx owned mapEntry | |
| exact covered | |
| · intro x hx y hy i _ j _ first second different | |
| exact Or.inl (births.layerComponents_earlier_meet data group oldAction pairing unions powers separation infinite hG hP0 k hx hy first second different) | |
| theorem cutoffLayerSeparatedCode (k n : Nat) (before : k ≤ n) : | |
| SeparatedLayerCode HQs Hrs Hμs (births.cutoff n) (births.cutoffOrder n) | |
| (births.cutoffMeet n) (births.cutoffBirth n) (compressionStamp oldTimes k) | |
| (births.layerOwners k) (births.layerIndices k) (births.layerMaps k) | |
| (M.groundRelations U B) (births.layerComponents k) (births.layerFiber k) := by | |
| have full := births.layerSeparatedCode data group oldAction pairing unions powers separation infinite hG hP0 | |
| historySpaces historyOrders historyMeets source k | |
| refine ⟨full.indices, full.maps, births.layerFiber_cutoff k n before, full.components, ?_, ?_, ?_⟩ | |
| · intro i _ t _ e Q R V indexEntry mapEntry spaceEntry orderEntry meetEntry | |
| obtain rfl := ((pair_mem_constantFamilyGraph _ _ _ _).1 indexEntry).2 | |
| have qeq := historySpaces.unique spaceEntry source.1 | |
| have req := historyOrders.unique orderEntry source.2.1 | |
| have veq := historyMeets.unique meetEntry source.2.2 | |
| subst Q R V | |
| exact births.layerMaps_cutoff_embedding data group oldAction pairing unions powers separation infinite hG hP0 k n before mapEntry | |
| · intro i hi t ht e Q x hx indexEntry mapEntry spaceEntry componentEntry | |
| obtain ⟨_, u, hu, copied⟩ := (mem_relationImage _ _ _ _).1 | |
| (full.cover i hi t ht e Q x (births.cutoff_subset n hx) indexEntry mapEntry spaceEntry componentEntry) | |
| exact (mem_relationImage _ _ _ _).2 ⟨hx, u, hu, copied⟩ | |
| · intro x hx y hy i _ j _ first second different | |
| have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP0 | |
| obtain ⟨z, a, meetEntry, born, earlier⟩ := births.layerComponents_earlier_meet data group oldAction pairing unions powers separation infinite hG hP0 k hx hy first second different | |
| have hxCut := ((mem_layerBirthFiber _ _ _ _).1 (births.layerFiber_cutoff k n before ▸ hx)).1 | |
| have hyCut := ((mem_layerBirthFiber _ _ _ _).1 (births.layerFiber_cutoff k n before ▸ hy)).1 | |
| have hz := (ZFSet.pair_mem_prod.1 (geometry.meet.operation.1 meetEntry)).2 | |
| have hzCut := births.cutoff_lower data group oldAction pairing unions powers separation infinite hG hP0 n z hz x hxCut | |
| (geometry.meet.lower_left x y z meetEntry) | |
| exact Or.inl ⟨z, a, pair_mem_graphRestriction.2 ⟨meetEntry, ZFSet.pair_mem_prod.2 ⟨hxCut, hyCut⟩⟩, | |
| pair_mem_graphRestriction.2 ⟨born, hzCut⟩, earlier⟩ | |
| theorem layerWitness_spec (k : Nat) {N : ZFSet.{u}} : | |
| births.layerWitness k ∈ M.sets ∧ | |
| GrowthLayerWitnessCode HQs Hrs Hμs B (completion.externalPointOrder P0 r0 μ0 A0) | |
| (completion.externalPointMeet P0 r0 μ0 A0) births.birth (compressionStamp oldTimes k) N | |
| (births.layerWitness k) := | |
| ⟨births.layerWitness_ground data group oldAction pairing unions powers separation infinite hG hP0 k, | |
| Or.inr (Or.inr ⟨_, _, _, _, _, _, rfl, | |
| births.layerSeparatedCode data group oldAction pairing unions powers separation infinite hG hP0 | |
| historySpaces historyOrders historyMeets source k⟩)⟩ | |
| theorem cutoffLayerWitness_spec (k n : Nat) (before : k ≤ n) {N : ZFSet.{u}} : | |
| births.layerWitness k ∈ M.sets ∧ | |
| GrowthLayerWitnessCode HQs Hrs Hμs (births.cutoff n) (births.cutoffOrder n) | |
| (births.cutoffMeet n) (births.cutoffBirth n) (compressionStamp oldTimes k) N | |
| (births.layerWitness k) := | |
| ⟨births.layerWitness_ground data group oldAction pairing unions powers separation infinite hG hP0 k, | |
| Or.inr (Or.inr ⟨_, _, _, _, _, _, rfl, | |
| births.cutoffLayerSeparatedCode data group oldAction pairing unions powers separation infinite hG hP0 | |
| historySpaces historyOrders historyMeets source k n before⟩)⟩ | |
| end PartitionPilot.Forcing.GroundExternalBirthData | |
| namespace PartitionPilot.Forcing.GroundExternalBirthData | |
| open SetFormula SetRealization | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 oldTimes oldBirth : ZFSet.{u}} | |
| variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ} | |
| variable (births : GroundExternalBirthData M initial I Sources Reps G m one T τ P0 r0 μ0 A0 oldTimes oldBirth completion) | |
| 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) (action : 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) (hP0 : P0 ∈ M.sets) | |
| variable {o : Ordinal.{u}} (limit : Order.IsSuccLimit o) | |
| variable (omega : ZFSet.rank ZFSet.omega < o) (indices : ZFSet.rank I < o) | |
| variable (points : ZFSet.rank P0 < o) (actors : ZFSet.rank G < o) | |
| variable (completed : ZFSet.rank (completion.algebra.union.fields 0) < o) | |
| variable (time : ZFSet.rank oldTimes < o) | |
| include data group action pairing unions powers separation infinite hG hP0 | |
| limit omega indices points actors completed time | |
| theorem layerWitness_rank_lt (k : Nat) : ZFSet.rank (births.layerWitness k) < o := by | |
| have carrier := completion.externalPointCarrier_rank_lt_of_inputs limit r0 μ0 A0 | |
| omega indices points actors completed | |
| have vertices := M.externalStarVertices_rank_lt_of_inputs limit r0 μ0 A0 m one | |
| omega indices points actors | |
| have fiber := (ZFSet.rank_mono (fun x hx => (births.mem_layerFiber k x).1 hx |>.1)).trans_lt carrier | |
| have owners := (ZFSet.rank_mono (fun v hv => (births.layerOwners_cases (k := k) hv).1)).trans_lt vertices | |
| have bound := M.groundRelations_rank_lt limit completed carrier | |
| exact separatedLayerWitness_rank_lt limit owners | |
| (rank_constantFamilyGraph_lt limit owners time) | |
| (rank_functionGraph_lt limit | |
| (births.layerMaps_spec data group action pairing unions powers separation infinite hG hP0 k).1 | |
| owners bound) bound | |
| (rank_functionGraph_lt limit | |
| (births.layerComponents_function data group action pairing unions powers separation infinite hG hP0 k) | |
| fiber owners) fiber | |
| variable {HQs Hrs Hμs HTime : ZFSet.{u}} | |
| variable (historySpaces : FunctionGraphOn HTime HQs) | |
| variable (historyOrders : FunctionGraphOn HTime Hrs) | |
| variable (historyMeets : FunctionGraphOn HTime Hμs) | |
| variable (source : ZFSet.pair oldTimes (completion.algebra.union.fields 0) ∈ HQs ∧ | |
| ZFSet.pair oldTimes (completion.algebra.union.fields 1) ∈ Hrs ∧ | |
| ZFSet.pair oldTimes (completion.algebra.union.fields 2) ∈ Hμs) | |
| include historySpaces historyOrders historyMeets source | |
| theorem ranked_layerWitness_spec (k : Nat) {N : ZFSet.{u}} : | |
| births.layerWitness k ∈ M.sets ∧ ZFSet.rank (births.layerWitness k) < o ∧ | |
| GrowthLayerWitnessCode HQs Hrs Hμs (completion.externalPointCarrier P0 r0 μ0 A0) | |
| (completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointMeet P0 r0 μ0 A0) | |
| births.birth (compressionStamp oldTimes k) N (births.layerWitness k) := by | |
| have code := births.layerWitness_spec data group action pairing unions powers separation infinite hG hP0 | |
| historySpaces historyOrders historyMeets source k (N := N) | |
| exact ⟨code.1, births.layerWitness_rank_lt data group action pairing unions powers separation infinite | |
| hG hP0 limit omega indices points actors completed time k, code.2⟩ | |
| theorem ranked_cutoffLayerWitness_spec (k n : Nat) (before : k ≤ n) {N : ZFSet.{u}} : | |
| births.layerWitness k ∈ M.sets ∧ ZFSet.rank (births.layerWitness k) < o ∧ | |
| GrowthLayerWitnessCode HQs Hrs Hμs (births.cutoff n) (births.cutoffOrder n) | |
| (births.cutoffMeet n) (births.cutoffBirth n) (compressionStamp oldTimes k) N | |
| (births.layerWitness k) := by | |
| have code := births.cutoffLayerWitness_spec data group action pairing unions powers separation infinite hG hP0 | |
| historySpaces historyOrders historyMeets source k n before (N := N) | |
| exact ⟨code.1, births.layerWitness_rank_lt data group action pairing unions powers separation infinite | |
| hG hP0 limit omega indices points actors completed time k, code.2⟩ | |
| end PartitionPilot.Forcing.GroundExternalBirthData | |
| end OAI | |