openAI-math-dataset / lean /OAI /SetTheory /PartitionPrinciple /Definability /DecodedFunctionGraphs.lean
| 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 | |