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Proof step 4 from proof abs-keys-orphans-add2 (previous step)

⊦ abs-keys(orphans(fs0, ri0)) = abs-keys(orphans(fs, ri)) ++ n ↔ (∀ key. key.inode? ∧ key ∈ ri0 ∧ fs0[ri0[key]].nlink = 0 ∧ key.ino ≠ ROOT_INO ↔ key = inodekey(n) ∨ key.inode? ∧ key ∈ ri ∧ fs[ri[key]].nlink = 0 ∧ key.ino ≠ ROOT_INO)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 5:
⊦ ns = ns0 ↔ (∀ ino. ino ∈ ns ↔ ino ∈ ns0)
Proof step 6:
⊦ (∀ ino. ino ∈ abs-keys(orphans(fs0, ri0)) ↔ ino ∈ abs-keys(orphans(fs, ri)) ++ n) ↔ (∀ key. key.inode? ∧ key ∈ ri0 ∧ fs0[ri0[key]].nlink = 0 ∧ key.ino ≠ ROOT_INO ↔ key = inodekey(n) ∨ key.inode? ∧ key ∈ ri ∧ fs[ri[key]].nlink = 0 ∧ key.ino ≠ 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)) ++ n, abs-keys(orphans(fs0, ri0))] flags: <all paths>

Additional information

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