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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 85 | import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundSerializedCompressionChronology
namespace OAI
section
namespace PartitionPilot.Forcing.GroundCompressionOrigin
open SetFormula
universe u
variable {M : TransitiveGround.{u}} {initial : RecursiveStateData.{u}}
variable {P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters C Q qlt original Targets Separators base old T0 : ZFSet.{u}}
variable {history : GroundChronologicalHistory M T0 C Q qlt}
variable (origin : GroundCompressionOrigin M initial P0 r0 μ0 A0 τ I Sources Reps G m inv one T Filters)
variable (compression : GroundChronologicalCompression M initial I Sources Reps G m one T τ
P0 C Q qlt original Targets Separators Filters base old T0 history)
variable (prior : GroundSerializedChronology M T0 C Q qlt)
include origin in
theorem exists_serializedCompression
(pairing : Realize M.sets SetFormula.pairing Fin.elim0)
(unions : Realize M.sets SetFormula.union Fin.elim0)
(powers : Realize M.sets powerSet Fin.elim0) (separation : SeparationSchema M.sets)
(collection : CollectionSchema M.sets) (infinite : Realize M.sets infinity Fin.elim0)
(choice : Realize M.sets SetFormula.choice Fin.elim0)
(hI : I ∈ M.sets) (hSources : Sources ∈ M.sets) (hReps : Reps ∈ M.sets)
(hG : G ∈ M.sets) (hT : T ∈ M.sets) (hBase : base ∈ T0)
(constant : RecursiveTemplateConstancy P0 r0 μ0 initial)
(initialDomain : initial.D = P0) (initialOrder : initial.rD = r0) (initialMeet : initial.μD = μ0)
(priorSpace : history.space base = P0) (priorOrder : history.order base = r0)
(priorMeet : history.meetGraph base = μ0) (priorBirth : history.birth base = old)
(agreement : ∀ j s, s ∈ T0 → prior.fields.value j s = history.fieldValue j s) :
∃ serial : GroundSerializedChronology M
(insert compression.born.births.limit compression.born.births.limit) C Q qlt,
serial.code ∈ M.sets ∧ ∀ j s, s ∈ insert compression.born.births.limit compression.born.births.limit →
serial.fields.value j s = compression.history.fieldValue j s := by
obtain ⟨seed, state, equal⟩ := origin
cases equal
let data := compression.born.profiled.heighted.completion.algebra.union
let births := compression.born.births
let heights := compression.born.profiled.heighted.heights
obtain ⟨serial, ground, values⟩ := state.exists_serializedPrefix data births heights prior
pairing unions powers separation collection infinite choice hI hSources hReps hG hT hBase
constant initialDomain initialOrder initialMeet
((agreement 0 base hBase).trans priorSpace) ((agreement 1 base hBase).trans priorOrder)
((agreement 2 base hBase).trans priorMeet) ((agreement 3 base hBase).trans priorBirth)
refine ⟨serial, ground, ?_⟩
intro j s hs
rw [values j s hs]
have oldValue (a : ZFSet.{u}) (before : a ∈ base ∨ a = base) :
(state.prefixFields data births heights prior.toHistory).fieldValue j a =
compression.history.fieldValue j a := by
have ha : a ∈ T0 := before.elim (fun h => prior.ordinal.1 base hBase a h) (fun h => h.symm ▸ hBase)
have left := state.prefixFields_old data births heights prior.toHistory before
have right := compression.old_values a before
have eLeft : (state.prefixFields data births heights prior.toHistory).fieldValue j a =
prior.fields.value j a :=
Fin.cases left.1 (Fin.cases left.2.1 (Fin.cases left.2.2.1
(Fin.cases left.2.2.2.1 (Fin.cases left.2.2.2.2 (fun i => Fin.elim0 i))))) j
have eRight : compression.history.fieldValue j a = history.fieldValue j a :=
Fin.cases right.1 (Fin.cases right.2.1 (Fin.cases right.2.2.1
(Fin.cases right.2.2.2.1 (Fin.cases right.2.2.2.2 (fun i => Fin.elim0 i))))) j
exact eLeft.trans ((agreement j a ha).trans eRight.symm)
rcases ZFSet.mem_insert_iff.1 hs with rfl | hs
· have left := state.prefixFields_limit data births heights prior.toHistory
have right := compression.limit_values
exact Fin.cases (left.1.trans right.1.symm)
(Fin.cases (left.2.1.trans right.2.1.symm) (Fin.cases (left.2.2.1.trans right.2.2.1.symm)
(Fin.cases (left.2.2.2.1.trans right.2.2.2.1.symm)
(Fin.cases (left.2.2.2.2.trans right.2.2.2.2.symm) (fun i => Fin.elim0 i))))) j
· rcases (births.times.mem_limit_iff s).1 hs with before | equal | ⟨n, rfl⟩
· exact oldValue s (Or.inl before)
· exact oldValue s (Or.inr equal)
· have left := state.prefixFields_stamp data births heights prior.toHistory n
have right := compression.stamp_values n
exact Fin.cases (left.1.trans right.1.symm)
(Fin.cases (left.2.1.trans right.2.1.symm) (Fin.cases (left.2.2.1.trans right.2.2.1.symm)
(Fin.cases (left.2.2.2.1.trans right.2.2.2.1.symm)
(Fin.cases (left.2.2.2.2.trans right.2.2.2.2.symm) (fun i => Fin.elim0 i))))) j
end PartitionPilot.Forcing.GroundCompressionOrigin
end
end OAI
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