openAI-math-dataset / lean /OAI /Computability /PerfectCompleteness /Foundations /CutSlotOutside.lean
| import OAI.Computability.PerfectCompleteness.Foundations.CutSlotAssembly | |
| namespace OAI | |
| section | |
| namespace PerfectCompleteness.CutSlotAssembly | |
| noncomputable section | |
| open scoped Classical | |
| open RecursiveSpaces DescendantSpaces | |
| variable {branch : Nat → Nat} {n m : Nat} {V : Type*} | |
| theorem fill_outside (p : Path branch n m) | |
| (outside : Slots branch n → V) (inside : Slots branch m → V) | |
| (s : Slots branch n) (hs : s ∉ Set.range p.slotEmbedding) : | |
| fill p outside inside s = outside s := by | |
| revert outside inside s hs | |
| induction p with | |
| | refl _ => | |
| intro outside inside s hs | |
| exact False.elim (hs ⟨s, rfl⟩) | |
| | step i p ih => | |
| intro outside inside s hs | |
| obtain ⟨j, s⟩ := s | |
| by_cases hji : j = i | |
| · subst j | |
| have htail : s ∉ Set.range p.slotEmbedding := by | |
| rintro ⟨a, ha⟩ | |
| apply hs | |
| exact ⟨a, congrArg (fun b => (i, b)) ha⟩ | |
| simpa only [fill, ite_eq_left rfl, ite_true] using | |
| ih (fun b => outside (i, b)) inside s htail | |
| · simp only [fill, ite_eq_right hji] | |
| theorem fill_outside_eq (p : Path branch n m) | |
| (outside : Slots branch n → V) (left right : Slots branch m → V) | |
| (s : Slots branch n) (hs : s ∉ Set.range p.slotEmbedding) : | |
| fill p outside left s = fill p outside right s := by | |
| rw [fill_outside p outside left s hs, fill_outside p outside right s hs] | |
| end | |
| end PerfectCompleteness.CutSlotAssembly | |
| end | |
| end OAI | |