File size: 2,415 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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.StageOneDataBoundLaws
namespace OAI
section
namespace PartitionPilot.Forcing
open SetRealization
universe u
variable (M : TransitiveGround.{u}) {R P : Type u}
variable (c : Presentation R) (s : Presentation P)
variable (I T levels alpha : ZFSet.{u}) (tests : ZFSet.{u} → P → Prop)
structure FlatSuccessorCondition where
row : R
assignments : ZFSet.{u}
assignments_mem : assignments ∈
M.groundSmallSubsets (stageOneAssignments I T) ZFSet.omega
lower : P
lower_code : s.encode lower = ZFSet.pair (c.encode row)
(assignmentCut levels alpha assignments)
tests : tests assignments lower
namespace FlatSuccessorCondition
local notation "Condition" => FlatSuccessorCondition M c s I T levels alpha tests
@[ext] theorem ext {p q : Condition}
(rows : p.row = q.row) (data : p.assignments = q.assignments) : p = q := by
cases p with
| mk rp Ap smallp lp codep testp =>
cases q with
| mk rq Aq smallq lq codeq testq =>
dsimp only at rows data
subst rq
subst Aq
have equal : lp = lq := s.injective (codep.trans codeq.symm)
subst lq
rfl
noncomputable def code (p : Condition) : ZFSet.{u} := ZFSet.pair (c.encode p.row) p.assignments
theorem code_eq_iff (p q : Condition) :
code M c s I T levels alpha tests p =
code M c s I T levels alpha tests q ↔ p.row = q.row ∧ p.assignments = q.assignments := by
rw [code, code, ZFSet.pair_inj]
exact and_congr c.injective.eq_iff Iff.rfl
theorem code_injective : Function.Injective (code M c s I T levels alpha tests) := by
intro p q equal
obtain ⟨rows, data⟩ := (code_eq_iff M c s I T levels alpha tests p q).1 equal
exact ext M c s I T levels alpha tests rows data
theorem code_mem_candidateBound (p : Condition) :
code M c s I T levels alpha tests p ∈ M.stageOneCandidateBound c.carrier I T :=
ZFSet.pair_mem_prod.2 ⟨c.encode_mem p.row, p.assignments_mem⟩
instance small : Small.{u} Condition := by
let bound := M.stageOneCandidateBound c.carrier I T
let f : Condition → bound := fun p =>
⟨code M c s I T levels alpha tests p,
code_mem_candidateBound M c s I T levels alpha tests p⟩
apply small_of_injective (f := f)
intro p q equal
exact code_injective M c s I T levels alpha tests (congrArg Subtype.val equal)
end FlatSuccessorCondition
end PartitionPilot.Forcing
end
end OAI
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