openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Transfinite /GroundExternalBirthLayers.lean
| import OAI.SetTheory.PartitionPrinciple.Definability.ExternalFamilySlotCodes | |
| import OAI.SetTheory.PartitionPrinciple.Geometry.GroundPreparationFamilyImages | |
| namespace OAI | |
| namespace PartitionPilot.Forcing | |
| open SetFormula | |
| universe u | |
| theorem pushedBirthGraph_oldMeet {A B T rA rB μA μB e old : ZFSet.{u}} | |
| (source : MeetOrderCode A rA μA) (target : MeetOrderCode B rB μB) | |
| (embedding : MeetEmbeddingCode A B rA rB μA μB e) | |
| (born : ZFSet.IsFunc A T old) (oldMeet : OldMeetCode rA μA old) : | |
| OldMeetCode rB μB (pushedHeightGraph A B T e old) := by | |
| intro x y z a b c bornY bornX earlier notLower meetXY bornZ | |
| have function := pushedHeightGraph_function embedding.injection born | |
| have hx := (ZFSet.pair_mem_prod.1 (function.1 bornX)).1 | |
| have hy := (ZFSet.pair_mem_prod.1 (function.1 bornY)).1 | |
| obtain ⟨_, x0, hx0, copyX⟩ := (mem_relationImage _ _ _ _).1 hx | |
| obtain ⟨_, y0, hy0, copyY⟩ := (mem_relationImage _ _ _ _).1 hy | |
| obtain ⟨z0, sourceMeet, _⟩ := source.operation.2 (ZFSet.pair x0 y0) (ZFSet.pair_mem_prod.2 ⟨hx0, hy0⟩) | |
| have hz0 := (ZFSet.pair_mem_prod.1 (source.operation.1 sourceMeet)).2 | |
| obtain ⟨z1, copyZ, _⟩ := embedding.injection.function.2 z0 hz0 | |
| have targetMeet := embedding.map_meet x0 y0 z0 x y z1 sourceMeet copyX copyY copyZ | |
| obtain ⟨_, _, unique⟩ := target.operation.2 (ZFSet.pair x y) | |
| (ZFSet.pair_mem_prod.1 (target.operation.1 meetXY)).1 | |
| have equal := (unique z1 targetMeet).trans (unique z meetXY).symm | |
| subst z1 | |
| exact oldMeet x0 y0 z0 a b c | |
| ((pushedHeightGraph_entry embedding.injection born copyY).1 bornY) | |
| ((pushedHeightGraph_entry embedding.injection born copyX).1 bornX) earlier | |
| (fun lower => notLower (embedding.monotone x0 y0 x y lower copyX copyY)) sourceMeet | |
| ((pushedHeightGraph_entry embedding.injection born copyZ).1 bornZ) | |
| end PartitionPilot.Forcing | |
| section | |
| namespace PartitionPilot.Forcing.GroundCompressionCompletion | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m one T τ : ZFSet.{u}} | |
| variable (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ) | |
| noncomputable def externalSeedBirth (P r μ A times birth : ZFSet.{u}) : ZFSet.{u} := | |
| pushedHeightGraph (completion.algebra.union.fields 0) (completion.externalPointCarrier P r μ A) | |
| times (completion.externalIdentityCopyGraph P r μ A) birth | |
| end PartitionPilot.Forcing.GroundCompressionCompletion | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetFormula | |
| universe u | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m inv one T τ P0 r0 μ0 A0 times birth : 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 "old" => completion.externalSeedBirth P0 r0 μ0 A0 times birth | |
| local notation "P" => completion.externalSeedCarrier P0 r0 μ0 A0 | |
| include data group oldAction pairing unions powers separation infinite hG | |
| theorem externalSeedBirth_spec (hP0 : P0 ∈ M.sets) (hTimes : times ∈ M.sets) (hBirth : birth ∈ M.sets) | |
| (born : ZFSet.IsFunc (completion.algebra.union.fields 0) times birth) | |
| (oldMeet : OldMeetCode (completion.algebra.union.fields 1) (completion.algebra.union.fields 2) birth) : | |
| old ∈ M.sets ∧ ZFSet.IsFunc P times old ∧ | |
| OldMeetCode (completion.externalPointOrder P0 r0 μ0 A0) (completion.externalPointMeet P0 r0 μ0 A0) old := by | |
| have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP0 | |
| have embedding := data.externalIdentityCopy_ideal group oldAction pairing unions powers separation infinite hG | |
| exact ⟨M.pushedHeightGraph_mem pairing unions powers separation (completion.algebra.union.ground 0) | |
| geometry.carrier_ground hTimes (geometry.copies_ground _ data.group.group_code.identity_mem) hBirth, | |
| pushedHeightGraph_function embedding.injection born, | |
| pushedBirthGraph_oldMeet completion.algebra.union.meet geometry.meet embedding.toMeetEmbeddingCode born oldMeet⟩ | |
| theorem externalSeedBirth_entry | |
| (born : ZFSet.IsFunc (completion.algebra.union.fields 0) times birth) | |
| {x y t : ZFSet.{u}} (copied : ZFSet.pair x y ∈ completion.externalIdentityCopyGraph P0 r0 μ0 A0) : | |
| ZFSet.pair y t ∈ old ↔ ZFSet.pair x t ∈ birth := | |
| pushedHeightGraph_entry | |
| (data.externalIdentityCopy_ideal group oldAction pairing unions powers separation infinite hG).injection born copied | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| end | |
| namespace PartitionPilot.Forcing | |
| universe u | |
| namespace GroundCompressionCompletion | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m one T τ : ZFSet.{u}} | |
| variable (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ) | |
| noncomputable def externalBirthGraph (P r μ A oldTimes oldBirth limit stamps : ZFSet.{u}) : ZFSet.{u} := | |
| ownerBirthGraph (completion.externalSeedCarrier P r μ A) (completion.externalPointCarrier P r μ A) | |
| limit (completion.externalSeedBirth P r μ A oldTimes oldBirth) | |
| (completion.externalOwnerLevels P r μ A) stamps | |
| end GroundCompressionCompletion | |
| structure GroundExternalBirthData (M : TransitiveGround.{u}) (initial : RecursiveStateData.{u}) | |
| (I Sources Reps G m one T τ P r μ A oldTimes oldBirth : ZFSet.{u}) | |
| (completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ) where | |
| stamps : ZFSet.{u} | |
| limit : ZFSet.{u} | |
| stamps_ground : stamps ∈ M.sets | |
| limit_ground : limit ∈ M.sets | |
| times : CompressionStampCode oldTimes stamps limit | |
| seed_function : ZFSet.IsFunc (completion.externalSeedCarrier P r μ A) oldTimes | |
| (completion.externalSeedBirth P r μ A oldTimes oldBirth) | |
| birth_ground : completion.externalBirthGraph P r μ A oldTimes oldBirth limit stamps ∈ M.sets | |
| function : ZFSet.IsFunc (completion.externalPointCarrier P r μ A) limit | |
| (completion.externalBirthGraph P r μ A oldTimes oldBirth limit stamps) | |
| old_meet : OldMeetCode (completion.externalPointOrder P r μ A) (completion.externalPointMeet P r μ A) | |
| (completion.externalBirthGraph P r μ A oldTimes oldBirth limit stamps) | |
| preserves_seed : ∀ x y t, ZFSet.pair x y ∈ completion.externalIdentityCopyGraph P r μ A → | |
| (ZFSet.pair y t ∈ completion.externalBirthGraph P r μ A oldTimes oldBirth limit stamps ↔ | |
| ZFSet.pair x t ∈ oldBirth) | |
| namespace GroundExternalBirthData | |
| variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}} | |
| variable {I Sources Reps G m one T τ P r μ A oldTimes oldBirth : ZFSet.{u}} | |
| variable {completion : GroundCompressionCompletion M initial I Sources Reps G m one T τ} | |
| variable (data : GroundExternalBirthData M initial I Sources Reps G m one T τ P r μ A oldTimes oldBirth completion) | |
| noncomputable def birth : ZFSet.{u} := | |
| completion.externalBirthGraph P r μ A oldTimes oldBirth data.limit data.stamps | |
| include data | |
| theorem old_before {x a : ZFSet.{u}} | |
| (entry : ZFSet.pair x a ∈ completion.externalSeedBirth P r μ A oldTimes oldBirth) (n : Nat) : | |
| a ∈ compressionStamp oldTimes n := | |
| compressionStamp_old_mem (Or.inl (ZFSet.pair_mem_prod.1 (data.seed_function.1 entry)).2) n | |
| theorem stamp_fiber (n : Nat) (x : ZFSet.{u}) : | |
| ZFSet.pair x (compressionStamp oldTimes n) ∈ data.birth ↔ | |
| x ∈ ownerFresh (completion.externalSeedCarrier P r μ A) (completion.externalPointCarrier P r μ A) ∧ | |
| ZFSet.pair x (natSetCode n) ∈ completion.externalOwnerLevels P r μ A := by | |
| apply ownerBirthGraph_stamp_fiber data.seed_function data.times.ownerStampCode | |
| · intro y a k t entry stamp | |
| rw [(data.times.entries k t).1 stamp] | |
| exact data.old_before entry k | |
| · exact (data.times.entries n _).2 rfl | |
| theorem no_limit_birth (x : ZFSet.{u}) : ZFSet.pair x data.limit ∉ data.birth := | |
| fun entry => ZFSet.mem_irrefl data.limit (ZFSet.pair_mem_prod.1 (data.function.1 entry)).2 | |
| end GroundExternalBirthData | |
| end PartitionPilot.Forcing | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetRealization SetFormula 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 : 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 "action" => completion.algebra.union.action.toOrderAction group completion.algebra.union.meet | |
| local notation "s" => completion.externalFamilyGroupPresentation group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| local notation "q" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "Index" => ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0) | |
| local notation "U" => completion.algebra.union.fields 0 | |
| local notation "B" => completion.externalPointCarrier P0 r0 μ0 A0 | |
| local notation "vcode" => completion.externalFamilyVertexCode group oldAction data.original | |
| noncomputable def externalBirthCopyVertex (k : Nat) (r : Index) : Option ((family).Child r) := by | |
| classical | |
| exact if h : ∃ i, ((family).request r).vertexSlot i = k % 3 then Classical.choose h else none | |
| theorem externalBirthCopyVertex_eq (k : Nat) (r : Index) (i : Option ((family).Child r)) | |
| (slot : ((family).request r).vertexSlot i = k % 3) : | |
| data.externalBirthCopyVertex group oldAction k r = i := by | |
| classical | |
| have existsSlot : ∃ j, ((family).request r).vertexSlot j = k % 3 := ⟨i, slot⟩ | |
| simp only [externalBirthCopyVertex, dite_eq_left existsSlot] | |
| exact ((family).request r).vertexSlot_injective ((Classical.choose_spec existsSlot).trans slot.symm) | |
| include data group oldAction pairing unions powers separation infinite hG | |
| theorem externalBirthCopy_witness (hP0 : P0 ∈ M.sets) (k : Nat) {x : ZFSet.{u}} | |
| (level : ZFSet.pair x (natSetCode k) ∈ completion.externalOwnerLevels P0 r0 μ0 A0) : | |
| letI := group.toGroup | |
| ∃ w r, ZFSet.pair x ((vpres).encode (.inr (w, r))) ∈ completion.externalStarOwner P0 r0 μ0 A0 ∧ | |
| x ∈ relationImage U B (completion.externalCopyGraph P0 r0 μ0 A0 | |
| ((s).encode ((family).flatWord w r (data.externalBirthCopyVertex group oldAction k r)))) := by | |
| let := group.toGroup | |
| have owner := (data.externalStarOwner_ground_function group oldAction pairing unions powers separation infinite hG hP0).2 | |
| obtain ⟨_, v, n, j, hj, owned, _, slot, rank⟩ := | |
| (data.externalOwnerLevels_entry_iff group oldAction pairing unions powers separation infinite hG hP0 x k).1 level | |
| have selected := (pair_mem_leastSlotOwnerGraph_of_owner owner owned).1 slot | |
| obtain ⟨w, r, i, a, eqV, eqJ, eqX⟩ := | |
| data.externalTileSlotPoints_decode group oldAction pairing unions powers separation infinite hG selected.2.2.1 | |
| have ji : j = ((family).request r).vertexSlot i := by | |
| apply natSetCode_injective | |
| exact eqJ.trans (completion.externalFamilyVertexCode_eq_slot group oldAction data.original r i) | |
| have chosen : data.externalBirthCopyVertex group oldAction k r = i := by | |
| apply data.externalBirthCopyVertex_eq group oldAction | |
| rw [← ji, rank, depthSlotIndex, Nat.add_mod] | |
| simp only [Nat.mul_mod, Nat.mod_self, Nat.zero_mul, Nat.zero_mod, Nat.zero_add, Nat.mod_eq_of_lt hj] | |
| refine ⟨w, r, eqV ▸ owned, ?_⟩ | |
| rw [chosen, eqX, (family).flatVertex_copy] | |
| have qc := data.externalPointPresentation_carrier group oldAction pairing unions powers separation infinite hG | |
| rw [data.externalCopyGraph_eq_graph group oldAction pairing unions powers separation infinite hG] | |
| apply (mem_relationImage _ _ _ _).2 | |
| exact ⟨qc ▸ (q).encode_mem _, SetMembers.raw a, SetMembers.raw_mem a, | |
| ((SetMembers.presentation U).pair_mem_graph (q) _ a _).2 rfl⟩ | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetFormula | |
| 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 (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 (collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0) | |
| variable (hG : G ∈ M.sets) (hP0 : P0 ∈ M.sets) | |
| local notation "P" => completion.externalSeedCarrier P0 r0 μ0 A0 | |
| local notation "B" => completion.externalPointCarrier P0 r0 μ0 A0 | |
| local notation "old" => completion.externalSeedBirth P0 r0 μ0 A0 oldTimes oldBirth | |
| include data group oldAction pairing unions powers separation collection infinite hG hP0 | |
| theorem exists_externalBirths_of_seed (hOldTimes : oldTimes ∈ M.sets) | |
| (ordinal : ordinalSetProperty oldTimes) (hOldBirth : oldBirth ∈ M.sets) | |
| (born : ZFSet.IsFunc (completion.algebra.union.fields 0) oldTimes oldBirth) | |
| (oldMeet : OldMeetCode (completion.algebra.union.fields 1) (completion.algebra.union.fields 2) oldBirth) : | |
| Nonempty (GroundExternalBirthData M initial I Sources Reps G m one T τ | |
| P0 r0 μ0 A0 oldTimes oldBirth completion) := by | |
| obtain ⟨stamps, hStamps, limit, hLimit, _, stampCode⟩ := | |
| M.exists_groundCompressionStamps pairing unions separation collection infinite hOldTimes ordinal | |
| have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP0 | |
| have seed := data.externalSeedBirth_spec group oldAction pairing unions powers separation infinite hG hP0 | |
| hOldTimes hOldBirth born oldMeet | |
| have levels := data.externalOwnerLevels_ground_function group oldAction pairing unions powers separation infinite hG hP0 | |
| have hP := data.externalSeedCarrier_ground group oldAction pairing unions powers separation infinite hG hP0 | |
| have subset : P ⊆ B := fun _ hx => ((mem_relationImage _ _ _ _).1 hx).1 | |
| have times : oldTimes ⊆ limit := fun _ ht => stampCode.union.subset 0 (compressionStamp_old_mem (Or.inl ht) 0) | |
| have oldBefore : ∀ x a n t, ZFSet.pair x a ∈ old → ZFSet.pair (natSetCode n) t ∈ stamps → a ∈ t := by | |
| intro x a n t entry stamp | |
| rw [(stampCode.entries n t).1 stamp] | |
| exact compressionStamp_old_mem (Or.inl (ZFSet.pair_mem_prod.1 (seed.2.1.1 entry)).2) n | |
| refine ⟨{ | |
| stamps := stamps | |
| limit := limit | |
| stamps_ground := hStamps | |
| limit_ground := hLimit | |
| times := stampCode | |
| seed_function := seed.2.1 | |
| birth_ground := M.ownerBirthGraph_mem pairing unions powers separation hP geometry.carrier_ground hLimit | |
| seed.1 levels.1 hStamps (M.omega_mem separation infinite) | |
| function := ownerBirthGraph_function seed.2.1 subset times levels.2 stampCode.function | |
| old_meet := ?_ | |
| preserves_seed := ?_ }⟩ | |
| · apply ownerBirthGraph_oldMeet geometry.meet seed.2.1 subset times levels.2 stampCode.ownerStampCode seed.2.2 | |
| · intro x hx y hy lower | |
| exact data.externalSeedCarrier_lower group oldAction pairing unions powers separation infinite hG hx hy lower | |
| · exact oldBefore | |
| · intro x _ y _ n k first second lower | |
| exact data.externalOwnerLevels_monotone group oldAction pairing unions powers separation infinite hG hP0 lower first second | |
| · intro x y t copied | |
| have embedding := data.externalIdentityCopy_ideal group oldAction pairing unions powers separation infinite hG | |
| have bounds := ZFSet.pair_mem_prod.1 (embedding.injection.function.1 copied) | |
| have hy : y ∈ P := (mem_relationImage _ _ _ _).2 ⟨bounds.2, x, bounds.1, copied⟩ | |
| exact (ownerBirthGraph_old_iff hy).trans | |
| (data.externalSeedBirth_entry group oldAction pairing unions powers separation infinite hG born copied) | |
| theorem exists_externalBirths {base originalBirth : ZFSet.{u}} | |
| (births : GroundCompressionBirthData M initial I Sources Reps G m one T | |
| completion.algebra.union base originalBirth) : | |
| Nonempty (GroundExternalBirthData M initial I Sources Reps G m one T τ | |
| P0 r0 μ0 A0 births.limit births.birth completion) := | |
| data.exists_externalBirths_of_seed group oldAction pairing unions powers separation collection infinite hG hP0 | |
| births.limit_ground births.times.ordinal births.birth_ground births.function births.old_meet | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetFormula SetRealization 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 "P" => completion.externalSeedCarrier P0 r0 μ0 A0 | |
| local notation "B" => completion.externalPointCarrier P0 r0 μ0 A0 | |
| local notation "U" => completion.algebra.union.fields 0 | |
| local notation "letters" => externalChildLetters I P0 (externalRectangles P0 G r0 μ0 A0) | |
| local notation "requests" => externalRequestCarrier I P0 (externalRectangles P0 G r0 μ0 A0) | |
| local notation "owners" => completion.externalStarOwner P0 r0 μ0 A0 | |
| local notation "depths" => M.externalStarDepth G letters requests m one | |
| local notation "levels" => completion.externalOwnerLevels P0 r0 μ0 A0 | |
| local notation "family" => completion.externalFamily group oldAction data.original | |
| local notation "action" => completion.algebra.union.action.toOrderAction group completion.algebra.union.meet | |
| local notation "vpres" => data.externalStarVertexPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "q" => data.externalPointPresentation group oldAction pairing unions powers separation infinite hG | |
| local notation "s" => completion.externalFamilyGroupPresentation group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| local notation "Index" => ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0) | |
| include data group oldAction pairing unions powers separation infinite hG | |
| theorem externalBirthCopy_level_slot (hP0 : P0 ∈ M.sets) : | |
| letI := group.toGroup | |
| ∀ (w : External (SetMembers G) (family).Letters) (r : Index) (k : Nat) {x : ZFSet.{u}}, | |
| ZFSet.pair x ((vpres).encode (.inr (w, r))) ∈ owners → | |
| ZFSet.pair x (natSetCode k) ∈ levels → | |
| ∃ n, ZFSet.pair ((vpres).encode (.inr (w, r))) (natSetCode n) ∈ depths ∧ | |
| k = depthSlotIndex n (((family).request r).vertexSlot (data.externalBirthCopyVertex group oldAction k r)) := by | |
| let := group.toGroup | |
| intro w r k x active level | |
| obtain ⟨_, v, n, j, hj, owned, depth, slot, rank⟩ := | |
| (data.externalOwnerLevels_entry_iff group oldAction pairing unions powers separation infinite hG hP0 x k).1 level | |
| have owner := (data.externalStarOwner_ground_function group oldAction pairing unions powers separation infinite hG hP0).2 | |
| have sameOwner := (show FunctionGraphOn B owners from ⟨_, owner⟩).unique owned active | |
| subst v | |
| have selected := (pair_mem_leastSlotOwnerGraph_of_owner owner active).1 slot | |
| obtain ⟨w', r', i, _, eqV, eqJ, _⟩ := | |
| data.externalTileSlotPoints_decode group oldAction pairing unions powers separation infinite hG selected.2.2.1 | |
| have same := Sum.inr.inj ((vpres).injective eqV) | |
| obtain ⟨rfl, rfl⟩ := Prod.mk.inj same | |
| have ji : j = ((family).request r).vertexSlot i := | |
| natSetCode_injective (eqJ.trans (completion.externalFamilyVertexCode_eq_slot group oldAction data.original r i)) | |
| have chosen : data.externalBirthCopyVertex group oldAction k r = i := by | |
| apply data.externalBirthCopyVertex_eq group oldAction | |
| rw [← ji, rank, depthSlotIndex, Nat.add_mod] | |
| simp only [Nat.mul_mod, Nat.mod_self, Nat.zero_mul, Nat.zero_mod, Nat.zero_add, Nat.mod_eq_of_lt hj] | |
| exact ⟨n, depth, by rw [chosen, ← ji]; exact rank⟩ | |
| theorem externalBirthCopy_level_bound (hP0 : P0 ∈ M.sets) : | |
| letI := group.toGroup | |
| ∀ (w : External (SetMembers G) (family).Letters) (r : Index) (k : Nat) {x u y : ZFSet.{u}}, | |
| ZFSet.pair x ((vpres).encode (.inr (w, r))) ∈ owners → | |
| ZFSet.pair x (natSetCode k) ∈ levels → | |
| ZFSet.pair u y ∈ completion.externalCopyGraph P0 r0 μ0 A0 | |
| ((s).encode ((family).flatWord w r (data.externalBirthCopyVertex group oldAction k r))) → | |
| y ∈ P ∨ ∃ l ≤ k, ZFSet.pair y (natSetCode l) ∈ levels := by | |
| let := group.toGroup | |
| intro w r k x u y active level copied | |
| obtain ⟨n, depth, rank⟩ := data.externalBirthCopy_level_slot group oldAction pairing unions powers separation infinite hG hP0 | |
| w r k active level | |
| let i := data.externalBirthCopyVertex group oldAction k r | |
| rw [data.externalCopyGraph_eq_graph group oldAction pairing unions powers separation infinite hG] at copied | |
| obtain ⟨a, equal⟩ := ((SetMembers.presentation U).mem_graph (q) _ _).1 copied | |
| obtain ⟨rfl, rfl⟩ := ZFSet.pair_inj.1 equal | |
| have supported := (data.externalTileSlotPoints_entry group oldAction pairing unions powers separation infinite hG | |
| w r i ((family).copy (action) ((family).flatWord w r i) |>.toFun a)).2 | |
| ⟨a, (family).flatVertex_copy (action) w r i a⟩ | |
| rw [completion.externalFamilyVertexCode_eq_slot group oldAction data.original r i] at supported | |
| have bound := data.externalSlotCopy_old_or_level_bound group oldAction pairing unions powers separation infinite hG hP0 | |
| supported depth | |
| rw [← rank] at bound | |
| exact bound | |
| theorem externalBirthCopy_image_level_bound (hP0 : P0 ∈ M.sets) : | |
| letI := group.toGroup | |
| ∀ (w : External (SetMembers G) (family).Letters) (r : Index) (k : Nat) {x y : ZFSet.{u}}, | |
| ZFSet.pair x ((vpres).encode (.inr (w, r))) ∈ owners → | |
| ZFSet.pair x (natSetCode k) ∈ levels → | |
| y ∈ relationImage U B (completion.externalCopyGraph P0 r0 μ0 A0 | |
| ((s).encode ((family).flatWord w r (data.externalBirthCopyVertex group oldAction k r)))) → | |
| y ∈ P ∨ ∃ l ≤ k, ZFSet.pair y (natSetCode l) ∈ levels := by | |
| let := group.toGroup | |
| intro w r k x y active level image | |
| obtain ⟨_, u, _, copied⟩ := (mem_relationImage _ _ _ _).1 image | |
| exact data.externalBirthCopy_level_bound group oldAction pairing unions powers separation infinite hG hP0 | |
| w r k active level copied | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| namespace PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| open SetRealization SetFormula 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 (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) | |
| variable (births : GroundExternalBirthData M initial I Sources Reps G m one T τ P0 r0 μ0 A0 oldTimes oldBirth completion) | |
| 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 "Index" => ExternalRequestIndex I P0 (externalRectangles P0 G r0 μ0 A0) | |
| 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 "levels" => completion.externalOwnerLevels P0 r0 μ0 A0 | |
| local notation "owner" => completion.externalStarOwner P0 r0 μ0 A0 | |
| include data group oldAction pairing unions powers separation infinite hG hP0 births | |
| noncomputable def externalBirth_fresh_layer | |
| (Space sourceOrder sourceMeet : ZFSet.{u} → ZFSet.{u}) | |
| (atSpace : Space oldTimes = U) | |
| (atOrder : sourceOrder oldTimes = completion.algebra.union.fields 1) | |
| (atMeet : sourceMeet oldTimes = completion.algebra.union.fields 2) (k : Nat) : | |
| CodedGrowthLayer M Space sourceOrder sourceMeet B (completion.externalPointOrder P0 r0 μ0 A0) | |
| (completion.externalPointMeet P0 r0 μ0 A0) births.birth (compressionStamp oldTimes k) := by | |
| classical | |
| letI := group.toGroup | |
| let Copies := External (SetMembers G) (family).Letters × Index | |
| let e (v : Copies) := completion.externalCopyGraph P0 r0 μ0 A0 | |
| ((s).encode ((family).flatWord v.1 v.2 (data.externalBirthCopyVertex group oldAction k v.2))) | |
| let Slice := {x : ZFSet.{u} // x ∈ ownerFresh P B ∧ ZFSet.pair x (natSetCode k) ∈ levels} | |
| have witnesses (x : Slice) : ∃ v : Copies, | |
| ZFSet.pair x.val ((vpres).encode (.inr v)) ∈ owner ∧ x.val ∈ relationImage U B (e v) := by | |
| obtain ⟨w, r, owned, covered⟩ := | |
| data.externalBirthCopy_witness group oldAction pairing unions powers separation infinite hG hP0 k x.property.2 | |
| exact ⟨(w, r), owned, covered⟩ | |
| let pick (x : Slice) : Copies := Classical.choose (witnesses x) | |
| have selected (x : Slice) : | |
| ZFSet.pair x.val ((vpres).encode (.inr (pick x))) ∈ owner ∧ | |
| x.val ∈ relationImage U B (e (pick x)) := Classical.choose_spec (witnesses x) | |
| have geometry := data.externalGeometryData group oldAction pairing unions powers separation infinite hG hP0 | |
| have levelCode := (data.externalOwnerLevels_ground_function group oldAction pairing unions powers separation infinite hG hP0).2 | |
| have sc := completion.externalFamilyGroupPresentation_carrier group oldAction data.original | |
| pairing unions powers separation infinite hG data.words.letters_ground | |
| have oldSubset : P ⊆ B := fun _ hx => ((mem_relationImage _ _ _ _).1 hx).1 | |
| have times : oldTimes ⊆ births.limit := fun _ ht => | |
| births.times.union.subset 0 (compressionStamp_old_mem (Or.inl ht) 0) | |
| have oldBefore : ∀ x a n t, ZFSet.pair x a ∈ completion.externalSeedBirth P0 r0 μ0 A0 oldTimes oldBirth → | |
| ZFSet.pair (natSetCode n) t ∈ births.stamps → a ∈ t := by | |
| intro x a n t entry stamp | |
| rw [(births.times.entries n t).1 stamp] | |
| exact births.old_before entry n | |
| refine ownerBirthGraph_separated births.seed_function oldSubset times levelCode births.times.ownerStampCode | |
| oldBefore ((births.times.entries k _).2 rfl) Copies (fun _ => oldTimes) e | |
| (fun _ => compressionStamp_base_mem oldTimes k) ?_ ?_ pick ?_ ?_ | |
| · intro v | |
| exact geometry.copies_ground _ (sc ▸ (s).encode_mem _) | |
| · intro v | |
| rw [atSpace, atOrder, atMeet] | |
| exact geometry.copies _ (sc ▸ (s).encode_mem _) | |
| · intro x _ | |
| rw [atSpace] | |
| exact (selected x).2 | |
| · intro x y _ _ different | |
| have differentOwners : (vpres).encode (.inr (pick x)) ≠ (vpres).encode (.inr (pick y)) := by | |
| intro equal | |
| exact different (Sum.inr.inj ((vpres).injective equal)) | |
| have hx := ((mem_ownerFresh P B x.val).1 x.property.1).1 | |
| have hy := ((mem_ownerFresh P B y.val).1 y.property.1).1 | |
| obtain ⟨z, meetEntry, _⟩ := geometry.meet.operation.2 (ZFSet.pair x.val y.val) (ZFSet.pair_mem_prod.2 ⟨hx, hy⟩) | |
| refine ⟨z, (ZFSet.pair_mem_prod.1 (geometry.meet.operation.1 meetEntry)).2, meetEntry, ?_⟩ | |
| exact data.externalOwnerLevels_meet_old_or_lower group oldAction pairing unions powers separation infinite hG hP0 | |
| (selected x).1 (selected y).1 differentOwners x.property.2 y.property.2 meetEntry | |
| end PartitionPilot.Forcing.GroundExternalGeometryInputs | |
| end OAI | |