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Proof step 3 from proof abs-delorphans (previous step)

abs(tree, ROOT_INO)(dirs), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files ⊦ dirs \ orphans(dirs, files) = dirs

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino]))
Proof step 5:
abs(tree, ROOT_INO)(dirs), ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ ∀ ino. (ino ∈ dirs \ orphans(dirs, files) ↔ ino ∈ dirs) ∧ (ino ∈ dirs \ orphans(dirs, files) → (dirs \ orphans(dirs, files))[ino] = dirs[ino])

Rule argument

spec: heap, inst: dir-heap name: ext seq: ⊦ dirs = dirs0 ↔ (∀ ino. (ino ∈ dirs ↔ ino ∈ dirs0) ∧ (ino ∈ dirs → dirs[ino] = dirs0[ino])) subst: [dirs0, dirs] → [dirs, dirs \ orphans(dirs, files)] flags: <all paths>

Additional information

Used terms: dirs, dirs \ orphans(dirs, files)