File size: 3,042 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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundStageOneOrder
namespace OAI
section
namespace PartitionPilot.Forcing
open SetRealization SetFormula
universe u
variable {R Q : Type u} [PartialOrder R] [PartialOrder Q]
structure FlatConditionSystem (M : TransitiveGround.{u})
(c : Presentation R) (s : Presentation Q) (J : ZFSet.{u}) where
row : Q → R
assignments : Q → ZFSet.{u}
encode : ∀ q, s.encode q = ZFSet.pair (c.encode (row q)) (assignments q)
le_iff : ∀ q p, q ≤ p ↔ row q ≤ row p ∧ assignments p ⊆ assignments q
assignments_mem : ∀ q, assignments q ∈ M.groundSmallSubsets J ZFSet.omega
namespace FlatConditionSystem
variable {M : TransitiveGround.{u}} {c : Presentation R} {s : Presentation Q} {J : ZFSet.{u}}
variable (F : FlatConditionSystem M c s J)
include F
theorem conditionOrder_eq : s.conditionOrder =
rowAssignmentOrder s.carrier c.carrier (M.groundSmallSubsets J ZFSet.omega) c.conditionOrder := by
apply ZFSet.ext
intro z
rw [Presentation.conditionOrder, Presentation.mem_subset]
constructor
· rintro ⟨⟨q, p⟩, below, rfl⟩
change ZFSet.pair (s.encode q) (s.encode p) ∈ _
rw [F.encode q, F.encode p]
exact (pair_mem_rowAssignmentOrder _ _ _ _ _ _ _ _).2
⟨F.encode q ▸ s.encode_mem q, F.encode p ▸ s.encode_mem p,
c.encode_mem (F.row q), F.assignments_mem q,
c.encode_mem (F.row p), F.assignments_mem p,
(c.pair_mem_conditionOrder _ _).2 ((F.le_iff q p).1 below).1,
((F.le_iff q p).1 below).2⟩
· intro member
obtain ⟨rawQ, hq, rawP, hp, equal⟩ := ZFSet.mem_prod.1
((mem_rowAssignmentOrder _ _ _ _ z).1 member).1
obtain ⟨q, eqQ⟩ := (s.mem_carrier rawQ).1 hq
obtain ⟨p, eqP⟩ := (s.mem_carrier rawP).1 hp
have entries : ZFSet.pair (s.encode q) (s.encode p) ∈
rowAssignmentOrder s.carrier c.carrier (M.groundSmallSubsets J ZFSet.omega) c.conditionOrder := by
rw [eqQ, eqP, ← equal]
exact member
rw [F.encode q, F.encode p] at entries
obtain ⟨_, _, _, _, _, _, rows, data⟩ :=
(pair_mem_rowAssignmentOrder _ _ _ _ _ _ _ _).1 entries
refine ⟨(q, p), (F.le_iff q p).2 ⟨(c.pair_mem_conditionOrder _ _).1 rows, data⟩, ?_⟩
change ZFSet.pair (s.encode q) (s.encode p) = z
rw [eqQ, eqP]
exact equal.symm
theorem conditionOrder_mem
(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)
(rows : c.carrier ∈ M.sets) (conditions : s.carrier ∈ M.sets)
(assignments : J ∈ M.sets) (omega : ZFSet.omega ∈ M.sets)
(order : c.conditionOrder ∈ M.sets) : s.conditionOrder ∈ M.sets := by
rw [F.conditionOrder_eq]
exact M.rowAssignmentOrder_mem pairing unions powers separation conditions rows
(M.groundSmallSubsets_mem powers separation assignments omega) order
end FlatConditionSystem
end PartitionPilot.Forcing
end
end OAI
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