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Proof step 86 from proof recovery (previous step)

fs0[ri1[inodekey(ino)]].directory, fo0.inodes?, orphans(fs0, ri1).inodes?, afs-cons(abs-dirs(fs0, ri1), abs-files(fs0, ri1)), file-node-cons(fs0, ri1 ⊖ orphans(fs0, ri1)), file-node-cons(fs0, ri1), log-cons(log0, fs0), orphans-cons(fs0, ri1 ⊖ orphans(fs0, ri1), ∅), orphans-cons(fs0, ri1, orphans(fs0, ri1)), replay-safe(log0, fs0, fi0 ⊖ fo0), ri-inj(ri1 ⊖ orphans(fs0, ri1)), ri-inj(fi0), ri-inj(ri1), ubifs-cons(fs0, ri1 ⊖ orphans(fs0, ri1), ∅, fi0, fo0, log0), ubifs-cons(fs0, ri1, orphans(fs0, ri1), fi0, fo0, log0), inodekey(ino) ∈ ri1 ⊦ fs0[ri1[inodekey(ino)]].nlink = 0 ∧ ino ≠ ROOT_INO, (dentrykey(ino, str) ∈ ri1 ⊖ orphans(fs0, ri1) ↔ dentrykey(ino, str) ∈ ri1) ∧ (dentrykey(ino, str) ∈ ri1 ⊖ orphans(fs0, ri1) → fs0[(ri1 ⊖ orphans(fs0, ri1))[dentrykey(ino, str)]].ino = fs0[ri1[dentrykey(ino, str)]].ino)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-base name: dels-with-data seq: ⊦ ri ⊖ ks = (ri \ ks) \ datakeys(ks, ri) subst: [ks, ri] → [orphans(fs0, ri1), ri1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ls)% ⊦ ks.inodes? → ¬ dentrykey(ino, str) ∈ ks; (s)%⊦ key ∈ datakeys(ks, ri) ↔ key ∈ ri ∧ key.data? ∧ inodekey(key.ino) ∈ ks; (s)% ⊦ ¬ dentrykey(m, str).data?; (s)% ⊦ (ri \ ks)[key] = (key ∈ ks ⊃ (∅)[key];ri[key]); (s) Used terms: orphans(fs0, ri1), ri1