File size: 4,688 Bytes
3277efa | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.SetTheory.PartitionPrinciple.Geometry.CompressionExternalTileOverlaps
import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundSerializedChronologyTimeEquality
namespace OAI
section
namespace PartitionPilot.Forcing
universe u
structure GroundChronologicalCompression (M : TransitiveGround.{u}) (initial : RecursiveStateData.{u})
(I Sources Reps G m one T τ P0 C Q qlt original Targets Separators Filters base old T0 : ZFSet.{u})
(prior : GroundChronologicalHistory M T0 C Q qlt) where
born : GroundBornCompression M initial I Sources Reps G m one T τ P0 C Q qlt original
Targets Separators Filters base old
history : GroundChronologicalHistory M (insert born.births.limit born.births.limit) C Q qlt
original_height : prior.height base = original
old_values : ∀ s, s ∈ base ∨ s = base →
history.space s = prior.space s ∧ history.order s = prior.order s ∧
history.meetGraph s = prior.meetGraph s ∧ history.birth s = prior.birth s ∧
history.height s = prior.height s
stamp_values : ∀ n,
history.space (compressionStamp base n) = (initial.iterate M I Sources Reps G T (n + 1)).D ∧
history.order (compressionStamp base n) = (initial.iterate M I Sources Reps G T (n + 1)).rD ∧
history.meetGraph (compressionStamp base n) = (initial.iterate M I Sources Reps G T (n + 1)).μD ∧
history.birth (compressionStamp base n) =
graphRestriction (initial.iterate M I Sources Reps G T (n + 1)).D born.births.birth ∧
history.height (compressionStamp base n) = born.profiled.heighted.heights.function.natValue (n + 1)
limit_values :
history.space born.births.limit = born.profiled.heighted.completion.algebra.union.fields 0 ∧
history.order born.births.limit = born.profiled.heighted.completion.algebra.union.fields 1 ∧
history.meetGraph born.births.limit = born.profiled.heighted.completion.algebra.union.fields 2 ∧
history.birth born.births.limit = born.births.birth ∧
history.height born.births.limit = born.profiled.heighted.heights.height
end PartitionPilot.Forcing
namespace PartitionPilot.Forcing.GroundChronologicalCompression
open SetFormula
universe u
variable {M N : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {I Sources Reps G m one T τ P0 C Q qlt original Targets Separators Filters base old T0 : ZFSet.{u}}
variable {prior : GroundChronologicalHistory M T0 C Q qlt}
variable (compression : GroundChronologicalCompression M initial I Sources Reps G m one T τ
P0 C Q qlt original Targets Separators Filters base old T0 prior)
theorem exists_extension_basis
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(includes : M.sets ⊆ N.sets)
(pairingN : Realize N.sets SetFormula.pairing Fin.elim0)
(unionsN : Realize N.sets SetFormula.union Fin.elim0)
(powersN : Realize N.sets powerSet Fin.elim0) (separationN : SeparationSchema N.sets)
(infiniteN : Realize N.sets SetFormula.infinity Fin.elim0)
(choiceN : Realize N.sets SetFormula.choice Fin.elim0)
(hC : C ∈ M.sets) (hQ : Q ∈ M.sets)
(smallQ : M.groundEmbeddable Q ZFSet.omega)
(irreflexive : ∀ q ∈ Q, ZFSet.pair q q ∉ qlt)
{F : ZFSet.{u}} (hF : F ∈ N.sets)
(filter : MeetFilterCode (compression.born.profiled.heighted.completion.algebra.union.fields 0)
(compression.born.profiled.heighted.completion.algebra.union.fields 1)
(compression.born.profiled.heighted.completion.algebra.union.fields 2) F) :
∃ B ∈ N.sets, ∃ f ∈ N.sets, InjectionCode B ZFSet.omega f ∧ B ⊆ F ∧
∀ x ∈ F, ∃ b ∈ B,
ZFSet.pair b x ∈ compression.born.profiled.heighted.completion.algebra.union.fields 1 := by
have filterAtLimit : MeetFilterCode (compression.history.space compression.born.births.limit)
(compression.history.order compression.born.births.limit)
(compression.history.meetGraph compression.born.births.limit) F := by
simpa only [compression.limit_values.1, compression.limit_values.2.1, compression.limit_values.2.2.1] using filter
have basis := compression.history.exists_extension_basis includes pairingN unionsN powersN
separationN infiniteN choiceN
(M.insert_mem pairing unions compression.born.births.limit_ground compression.born.births.limit_ground)
hC hQ (ordinalSetProperty_succ compression.born.births.times.ordinal) smallQ irreflexive
(ZFSet.mem_insert compression.born.births.limit compression.born.births.limit) hF filterAtLimit
simpa only [compression.limit_values.2.1] using basis
end PartitionPilot.Forcing.GroundChronologicalCompression
end
end OAI
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