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Proof step 8 from proof abs-keys-orphans2 (previous step)

abs-keys(orphans(fs0, ri0)) = abs-keys(orphans(fs, ri)) ⊦ inodekey(m) ∈ ri0 ∧ fs0[ri0[inodekey(m)]].nlink = 0 ↔ inodekey(m) ∈ ri ∧ fs[ri[inodekey(m)]].nlink = 0, m = ROOT_INO

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ ns = ns0 ↔ (∀ ino. ino ∈ ns ↔ ino ∈ ns0)
Proof step 10:
∀ ino. ino ∈ abs-keys(orphans(fs0, ri0)) ↔ ino ∈ abs-keys(orphans(fs, ri)) ⊦ inodekey(m) ∈ ri0 ∧ fs0[ri0[inodekey(m)]].nlink = 0 ↔ inodekey(m) ∈ ri ∧ fs[ri[inodekey(m)]].nlink = 0, m = ROOT_INO

Rule argument

spec: set-basic, inst: dir-heap-refset name: Extension seq: ⊦ ns = ns0 ↔ (∀ ino. ino ∈ ns ↔ ino ∈ ns0) subst: [ns0, ns] → [abs-keys(orphans(fs, ri)), abs-keys(orphans(fs0, ri0))] flags: <all paths>

Additional information

Used terms: abs-keys(orphans(fs, ri)), abs-keys(orphans(fs0, ri0))