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Proof step 9 from proof abs-orphans-diff-datakeys (previous step)

⊦ ino0 ∈ orphans(abs-dirs(fs, ri), abs-files(fs, ri \ datakeys(inodekey(ino), n, ri))) ↔ (inodekey(ino0) ∈ ri ∧ fs[ri[inodekey(ino0)]].directory ∨ inodekey(ino0) ∈ ri ∧ ¬ fs[ri[inodekey(ino0)]].directory) ∧ links(ino0, abs-dirs(fs, ri)) = ∅, ino0 = ROOT_INO

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 10:
⊦ ino ∈ orphans(dirs, files) ↔ valid-ino(ino, dirs, files) ∧ links(ino, dirs) = ∅ ∧ ino ≠ ROOT_INO
Proof step 11:
⊦ valid-ino(ino0, abs-dirs(fs, ri), abs-files(fs, ri \ datakeys(inodekey(ino), n, ri))) ∧ links(ino0, abs-dirs(fs, ri)) = ∅ ∧ ino0 ≠ ROOT_INO ↔ (inodekey(ino0) ∈ ri ∧ fs[ri[inodekey(ino0)]].directory ∨ inodekey(ino0) ∈ ri ∧ ¬ fs[ri[inodekey(ino0)]].directory) ∧ links(ino0, abs-dirs(fs, ri)) = ∅, ino0 = ROOT_INO

Rule argument

spec: afs-base name: orphans seq: ⊦ ino ∈ orphans(dirs, files) ↔ valid-ino(ino, dirs, files) ∧ links(ino, dirs) = ∅ ∧ ino ≠ ROOT_INO subst: [files, dirs, ino] → [abs-files(fs, ri \ datakeys(inodekey(ino), n, ri)), abs-dirs(fs, ri), ino0] flags: <all paths>

Additional information

Used terms: abs-files(fs, ri \ datakeys(inodekey(ino), n, ri)), abs-dirs(fs, ri), ino0