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Proof step 7 from proof abs-dirs-orphans (previous step)

⊦ ( inodekey(ino) ∈ ri → fs[ri[inodekey(ino)]].nlink ≠ 0 ∨ ino = ROOT_INO → (fs[ri[inodekey(ino)]].directory ↔ fs[(ri \ orphans(fs, ri))[inodekey(ino)]].directory)) ∧ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ∧ (fs[ri[inodekey(ino)]].nlink ≠ 0 ∨ ino = ROOT_INO) → (abs-dirs(fs, ri) \ abs-keys(orphans(fs, ri)))[ino] = mkdir (fs[(ri \ orphans(fs, ri))[inodekey(ino)]].size, fs[(ri \ orphans(fs, ri))[inodekey(ino)]].meta, fs[(ri \ orphans(fs, ri))[inodekey(ino)]].nlink, fs[(ri \ orphans(fs, ri))[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri \ orphans(fs, ri)))), inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: store+set, inst: node-index name: Diff-get seq: ⊦ ¬ key ∈ ks → (ri \ ks)[key] = ri[key] subst: [key, ks, ri] → [inodekey(ino), orphans(fs, ri), ri] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key ∈ orphans(fs, ri) ↔ key.inode? ∧ key ∈ ri ∧ fs[ri[key]].nlink = 0 ∧ key.ino ≠ ROOT_INO; (s)% ⊦ inodekey(m).inode?; (s)% ⊦ inodekey(m).ino = m; (s) Used terms: inodekey(ino), orphans(fs, ri), ri