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Proof step 2 from proof abs-dirs-orphans (previous step)

⊦ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ∧ (fs[ri[inodekey(ino)]].nlink ≠ 0 ∨ ino = ROOT_INO) ↔ inodekey(ino) ∈ ri \ orphans(fs, ri) ∧ 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))))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 3:
⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks
Proof step 4:
⊦ ( inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory ∧ (fs[ri[inodekey(ino)]].nlink ≠ 0 ∨ ino = ROOT_INO) ↔ (inodekey(ino) ∈ ri ∧ ¬ inodekey(ino) ∈ orphans(fs, ri)) ∧ 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))))

Rule argument

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

Additional information

Used terms: orphans(fs, ri), ri, inodekey(ino)