Plain-language statement
Converse of slice_contains_reachable: every entity in the level-n slice (the reachable set computed by Entities.sliceAtLevel) is reachable in n steps. Together with reachable_then_mem_slice this gives euid ∈ slice ↔ ReachableIn, the bridge that makes Entities.closedAtLevel decide EntitiesClosedAtLevel.
Exact Lean statement
theorem reachable_of_mem_slice {n : Nat} {work : Set EntityUID} {euid : EntityUID} {entities : Entities}
(h : euid ∈ Entities.sliceAtLevel.sliceAtLevel entities work n) :
ReachableIn entities work euid nFormal artifact
Lean source
theorem reachable_of_mem_slice {n : Nat} {work : Set EntityUID} {euid : EntityUID} {entities : Entities} (h : euid ∈ Entities.sliceAtLevel.sliceAtLevel entities work n) : ReachableIn entities work euid n:= by cases n case zero => simp only [Entities.sliceAtLevel.sliceAtLevel] at h exact absurd h (Set.not_mem_empty euid) case succ m => simp only [Entities.sliceAtLevel.sliceAtLevel] at h rw [Set.mem_union] at h cases h case inl hw => exact ReachableIn.in_start hw case inr hs => simp only [List.mem_mapUnion_iff_mem_exists, List.mem_filterMap] at hs obtain ⟨ed, ⟨i, hi, hfind⟩, hmem⟩ := hs exact ReachableIn.step i ((Set.mem_elts_iff_mem_set i work).mp hi) hfind (reachable_of_mem_slice hmem)- Project
- Cedar Specification
- License
- Apache-2.0
- Commit
- 3f093947b8ae
- Source
- cedar-lean/Cedar/Thm/Validation/Slice/Reachable.lean:263-279
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Person-level attribution pending.