File size: 3,090 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 | import OAI.SetTheory.PartitionPrinciple.Transfinite.GroundSmallFunctionSpaces
namespace OAI
section
namespace PartitionPilot.Forcing.InjectionCode
universe u
theorem restrict {D N f Q : ZFSet.{u}} (h : InjectionCode D N f) (subset : Q ⊆ D) :
InjectionCode Q N (ZFSet.sep (fun z => z ∈ ZFSet.prod Q N) f) := by
refine ⟨⟨fun _ hz => (ZFSet.mem_sep.1 hz).2, ?_⟩, ?_⟩
· intro x hx
obtain ⟨y, rel, unique⟩ := h.function.2 x (subset hx)
have hy := (ZFSet.pair_mem_prod.1 (h.function.1 rel)).2
exact ⟨y, ZFSet.mem_sep.2 ⟨rel, ZFSet.pair_mem_prod.2 ⟨hx, hy⟩⟩,
fun z hz => unique z (ZFSet.mem_sep.1 hz).1⟩
· intro x y z hx hy
exact h.injective x y z (ZFSet.mem_sep.1 hx).1 (ZFSet.mem_sep.1 hy).1
end PartitionPilot.Forcing.InjectionCode
namespace PartitionPilot.Forcing.TransitiveGround
open SetFormula
universe u
variable (M : TransitiveGround.{u})
theorem injection_restriction_mem (separation : SeparationSchema M.sets)
{Q N f : ZFSet.{u}} (hQ : Q ∈ M.sets) (hN : N ∈ M.sets) (hf : f ∈ M.sets) :
ZFSet.sep (fun z => z ∈ ZFSet.prod Q N) f ∈ M.sets := by
let params := extend Q (extend N Fin.elim0)
have selected := M.separation_mem separation (BoundedFormula.productElementFormula 0 1 2).toFormula
params (Fin.cases hQ (Fin.cases hN (fun j => Fin.elim0 j))) hf
have eq : ZFSet.sep (fun z => Realize M.sets
(BoundedFormula.productElementFormula 0 1 2).toFormula (extend z params)) f =
ZFSet.sep (fun z => z ∈ ZFSet.prod Q N) f := by
apply ZFSet.ext
intro z
rw [ZFSet.mem_sep, ZFSet.mem_sep]
constructor
· rintro ⟨hz, rel⟩
exact ⟨hz, (BoundedFormula.truth_productElementFormula 0 1 2 _).1
((BoundedFormula.realize_toFormula M _ _ (Fin.cases (M.transitive hf hz)
(Fin.cases hQ (Fin.cases hN (fun j => Fin.elim0 j))))).1 rel)⟩
· rintro ⟨hz, rel⟩
exact ⟨hz, (BoundedFormula.realize_toFormula M _ _ (Fin.cases (M.transitive hf hz)
(Fin.cases hQ (Fin.cases hN (fun j => Fin.elim0 j))))).2
((BoundedFormula.truth_productElementFormula 0 1 2 _).2 rel)⟩
exact eq ▸ selected
theorem groundEmbeddable_restrict (separation : SeparationSchema M.sets)
{D Q N : ZFSet.{u}} (hQ : Q ∈ M.sets) (hN : N ∈ M.sets) (subset : Q ⊆ D)
(small : M.groundEmbeddable D N) : M.groundEmbeddable Q N := by
obtain ⟨f, hf, code⟩ := small
exact ⟨ZFSet.sep (fun z => z ∈ ZFSet.prod Q N) f,
M.injection_restriction_mem separation hQ hN hf, code.restrict subset⟩
theorem smallPartialFunctions_restrict (separation : SeparationSchema M.sets)
{X Y N p q : ZFSet.{u}} (hp : p ∈ M.smallPartialFunctions X Y N)
(hq : q ∈ M.sets) (hN : N ∈ M.sets) (subset : q ⊆ p) : q ∈ M.smallPartialFunctions X Y N := by
obtain ⟨_, code, small⟩ := (M.mem_smallPartialFunctions X Y N p).1 hp
exact (M.mem_smallPartialFunctions X Y N q).2
⟨hq, code.mono subset, M.groundEmbeddable_restrict separation hq hN subset small⟩
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
|