File size: 1,465 Bytes
2f887db | 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 | 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
|