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Proof step 60 from proof inv-rename-dir-negdentry (previous step)

# links(entries[str], dirs0) = dirs0[entries[str]].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(m, dirs0) = dirs0[m].nlink, entries[str] = dirs0[m].entries[str], dirs0[m] = mkdir(dirs0[m].size, md2, dirs0[m].nlink, dirs0[m].nsubdirs, entries), dirs0[m0] = mkdir(dirs0[m0].size, md4, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries1), dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md5, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, entries2), dirs = dirs0[m, ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1)] [m0, ( ( (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1) ;dirs0[m0] ) .entries:= (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1) ;dirs0[m0] ) .entries [str0, entries[str] ]) .size:= ( (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1) ;dirs0[m0] ) .size + 1)) .nsubdirs:= ( (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)).nsubdirs:= (dirs0[m].nsubdirs - 1);dirs0[m0]) .nsubdirs + 1) ], afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), icache-cons(icache, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache, pcache), str ∈ dirs0[m].entries, str ∈ entries, m ∈ dirs0, m0 ∈ dirs0, entries[str] ∈ dirs0, 1 ≤ dirs0[m].size, 1 ≤ dirs0[m].nsubdirs, ¬ str0 ∈ dirs0[m0].entries, ¬ str0 ∈ entries1, asynced2 → synced(files, icache, pcache), m0 = m → md1 = md0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ [afs_inode_sync#(mkinode(m, md0, true, dirs0[m].nlink, dirs0[m].nsubdirs, dirs0[m].size); dirs, files); afs_inode_sync#(mkinode(m0, md1, true, # links(m0, dirs0), dirs0[m0].nsubdirs, dirs0[m0].size); dirs, files); skip; old_parent_inode := (m ∈ dirs ⊃ mkinode(m, dirs[m].meta, true, dirs[m].nlink, dirs[m].nsubdirs, dirs[m].size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ); new_parent_inode := (m0 ∈ dirs ⊃ mkinode(m0, dirs[m0].meta, true, dirs[m0].nlink, dirs[m0].nsubdirs, dirs[m0].size) ;mkinode(m0, files[m0].meta, false, files[m0].nlink, 0, files[m0].size) ); err := ESUCCESS; asynced := ?] ( afs-cons(dirs, files) → ⟪if err = ESUCCESS then let asynced = false in {icache_set#(old_parent_inode, asynced; err, icache); icache_set#(new_parent_inode, asynced; err, icache); skip; dcache_delete#(old_parent_inode.ino, mkdentry(str, entries[str]).name; err, dcache); dcache_set# (new_parent_inode.ino, mkdentry(str0, mkdentry(str, entries[str]).ino).name, mkdentry(str0, mkdentry(str, entries[str]).ino) ; err, dcache)}⟫ ( icache-cons(icache, dirs, files) ∧ dcache-cons(dcache, dirs, files) ∧ pcache-cons(pcache, dirs, files) ∧ synced(dirs, icache, pcache) ∧ (asynced2 → synced(files, icache, pcache))))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [dirs0[m].size, m1]

Additional information

Used simplifier rules: afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: dirs0[m].size, m1