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e3e0a0d | 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 | import OAI.SetTheory.PartitionPrinciple.Definability.PresentedGroupCodes
namespace OAI
section
namespace PartitionPilot.SetRealization.Presentation
universe u
variable {A B : Type u} (s : Presentation A) (t : Presentation B)
theorem exists_graph_value (f : ZFSet.{u}) (h : ZFSet.IsFunc s.carrier t.carrier f) (a : A) :
∃ b, ZFSet.pair (s.encode a) (t.encode b) ∈ f := by
obtain ⟨y, rel, _⟩ := h.2 (s.encode a) (s.encode_mem a)
have hy := (ZFSet.pair_mem_prod.1 (h.1 rel)).2
obtain ⟨b, rfl⟩ := (t.mem_carrier y).1 hy
exact ⟨b, rel⟩
noncomputable def decodeFunction (f : ZFSet.{u}) (h : ZFSet.IsFunc s.carrier t.carrier f) : A → B :=
fun a => Classical.choose (s.exists_graph_value t f h a)
theorem decodeFunction_mem (f : ZFSet.{u}) (h : ZFSet.IsFunc s.carrier t.carrier f) (a : A) :
ZFSet.pair (s.encode a) (t.encode (s.decodeFunction t f h a)) ∈ f :=
Classical.choose_spec (s.exists_graph_value t f h a)
theorem decodeFunction_iff (f : ZFSet.{u}) (h : ZFSet.IsFunc s.carrier t.carrier f) (a : A) (b : B) :
s.decodeFunction t f h a = b ↔ ZFSet.pair (s.encode a) (t.encode b) ∈ f := by
constructor
· intro eq
simpa only [eq] using s.decodeFunction_mem t f h a
· intro rel
obtain ⟨y, _, unique⟩ := h.2 (s.encode a) (s.encode_mem a)
apply t.injective
exact (unique _ (s.decodeFunction_mem t f h a)).trans (unique _ rel).symm
theorem graph_decodeFunction (f : ZFSet.{u}) (h : ZFSet.IsFunc s.carrier t.carrier f) :
s.graph t (s.decodeFunction t f h) = f := by
apply ZFSet.ext
intro z
constructor
· intro hz
obtain ⟨a, eq⟩ := (s.mem_graph t (s.decodeFunction t f h) z).1 hz
exact eq ▸ s.decodeFunction_mem t f h a
· intro hz
obtain ⟨x, hx, y, hy, eq⟩ := ZFSet.mem_prod.1 (h.1 hz)
obtain ⟨a, rfl⟩ := (s.mem_carrier x).1 hx
obtain ⟨b, rfl⟩ := (t.mem_carrier y).1 hy
rw [eq, s.pair_mem_graph]
exact (s.decodeFunction_iff t f h a b).2 (eq ▸ hz)
end PartitionPilot.SetRealization.Presentation
namespace PartitionPilot.Forcing.TransitiveGround
open SetRealization
universe u
variable (M : TransitiveGround.{u}) {A B : Type u}
theorem decoded_function_graph_mem (s : Presentation A) (t : Presentation B) {f : ZFSet.{u}}
(hf : f ∈ M.sets) (h : ZFSet.IsFunc s.carrier t.carrier f) :
s.graph t (s.decodeFunction t f h) ∈ M.sets := by
rw [s.graph_decodeFunction]
exact hf
end PartitionPilot.Forcing.TransitiveGround
end
end OAI
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