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Proof step 62 from proof inv-evict (previous step)

replay-log(log, fs, fi ⊖ fo) = ri1 ⊖ ro0, ro0.inodes?, fo.inodes?, file-node-cons(fs, (ri1 \ datakeys(inodekey(m), ri1)) -- inodekey(m)), file-node-cons(fs, ri1), log-cons(log, fs), orphans-cons(fs, (ri1 \ datakeys(inodekey(m), ri1)) -- inodekey(m), ro0 -- inodekey(m)), orphans-cons(fs, ri1, ro0), replay-safe(log, fs, fi ⊖ fo), ri-inj(ri1), ri-inj(fi), ubifs-cons(fs, ri1, ro0, fi, fo, log), key ∈ ri1, inodekey(m) ∈ ro0, inodekey(m) ∈ ri1, ∀ ino. inodekey(ino) ∈ ro0 → dentrykeys(inodekey(ino), ri1) = ∅, ∀ ino. inodekey(ino) ∈ ro0 → links(ino, fs, ri1) = ∅ ⊦ # ro0 = 0, ri1[key] = (((ri1 \ datakeys(inodekey(m), ri1)) -- inodekey(m)) ⊖ (ro0 -- inodekey(m)))[key], m = ROOT_INO, ro0 = ∅, key.data?, key ∈ ro0

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: aubifs-base name: at-dels-ro-nodata seq: ⊦ key ∈ ri ∧ ¬ key ∈ ks ∧ ¬ key.data? → (ri ⊖ ks)[key] = ri[key] subst: [key, ks, ri] → [key, ro0 -- inodekey(m), (ri1 \ datakeys(inodekey(m), ri1)) -- inodekey(m)] flags: <all paths>

Additional information

Used simplifier rules: at-del-ri: ⊦ (ri -- key0)[key1] = (key0 = key1 ⊃ (∅)[key0];ri[key1]);%at-diff-ri: ⊦ (ri \ ks)[key] = (key ∈ ks ⊃ (∅)[key];ri[key]);%⊦ key ∈ datakeys(key0, ri) ↔ key ∈ ri ∧ key.data? ∧ key0.inode? ∧ key.ino = key0.ino; (s)% ⊦ key ∈ ri -- key0 ↔ key ≠ key0 ∧ key ∈ ri; (s)% ⊦ key ∈ ri \ ks ↔ key ∈ ri ∧ ¬ key ∈ ks; (ls)% ⊦ key ∈ ks -- key0 ↔ key ≠ key0 ∧ key ∈ ks; (s) Used terms: key, ro0 -- inodekey(m), (ri1 \ datakeys(inodekey(m), ri1)) -- inodekey(m)