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Proof step 63 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, dirs0[m].nsubdirs = m2 + 1, dirs0[m].size = m1 + 1, entries[str] = dirs0[m].entries[str], dirs0[m] = mkdir(m1 + 1, 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:= m1).nsubdirs:= m2] [m0, ( ( (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]) .entries:= (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).entries [str0, entries[str] ]) .size:= ((m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).size + 1)) .nsubdirs:= ((m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;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].nsubdirs, 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, m1 + 1); 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)))), dirs0[m].nsubdirs = 0, str0 ∈ dirs0[m0].entries, str0 ∈ entries1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 64:
# links(entries[str], dirs0) = dirs0[entries[str]].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(m, dirs0) = dirs0[m].nlink, dirs0[m].nsubdirs = m2 + 1, dirs0[m].size = m1 + 1, entries[str] = dirs0[m].entries[str], dirs0[m] = mkdir(m1 + 1, md2, dirs0[m].nlink, m2 + 1, 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:= m1).nsubdirs:= m2] [m0, ( ( (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]) .entries:= (m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).entries [str0, entries[str] ]) .size:= ((m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).size + 1)) .nsubdirs:= ((m0 = m ⊃ ((dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m1).nsubdirs:= m2;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, ¬ 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, m2 + 1, m1 + 1); 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))))

Rule argument

Additional information

Used simplifier rules: ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat