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Proof step 41 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries1[str0], dirs0) = dirs0[entries1[str0]].nlink, # links(m, dirs0) = dirs0[m].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(entries[str], dirs0) = dirs0[entries[str]].nlink, dirs0[entries[str]].entries = ∅, dirs0[m0].nsubdirs = m2 + 1, dirs0[m0].size = m1 + 1, entries1[str0] = dirs0[m0].entries[str0], entries[str] = dirs0[m].entries[str], dirs0[m0] = mkdir(m1 + 1, md6, dirs0[m0].nlink, m2 + 1, entries1), dirs0[m] = mkdir(dirs0[m].size, md4, dirs0[m].nlink, dirs0[m].nsubdirs, entries), dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md3, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), dirs0[entries1[str0]] = mkdir(dirs0[entries1[str0]].size, md5, dirs0[entries1[str0]].nlink, dirs0[entries1[str0]].nsubdirs, entries0), dirs = dirs0[m0, ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= m2] [m, (m = m0 ⊃ ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= m2;dirs0[m]) .entries:= (m = m0 ⊃ ((dirs0[m0].entries:= (dirs0[m0].entries -- str0)).size:= m1).nsubdirs:= m2;dirs0[m]).entries[str, entries1[str0]] ], entries[str] ≠ ROOT_INO, afs-cons(dirs0, files), str0 ∈ dirs0[m0].entries, str ∈ dirs0[m].entries, m0 ∈ dirs0, m ∈ dirs0, entries1[str0] ∈ dirs0, entries[str] ∈ dirs0, str0 ∈ entries1, str ∈ entries, m = m0 → md1 = md2, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); old_dent := mkdentry(str0, entries1[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m0, md2, true, dirs0[m0].nlink, m2 + 1, m1 + 1); dirs, files); afs_inode_sync#(mkinode(m, md1, true, # links(m, dirs0), dirs0[m].nsubdirs, dirs0[m].size); dirs, files); if asynced1 then {afs_inode_sync# (mkinode(entries[str], md, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, dirs0[entries[str]].size); dirs, files); new_child_inode := getinode(entries[str], dirs, files)}; old_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) ); new_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) ); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files), m0 = m ∧ mkdentry(str, entries[str]) = mkdentry(str0, entries1[str0])

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 42:
# links(entries[str], dirs0) = dirs0[entries[str]].nlink, # links(m0, dirs0) = m4, # links(m, dirs0) = dirs0[m].nlink, # links(entries1[str0], dirs0) = dirs0[entries1[str0]].nlink, dirs0[entries[str]].entries = ∅, dirs0[entries1[str0]] = mkdir(dirs0[entries1[str0]].size, md5, dirs0[entries1[str0]].nlink, dirs0[entries1[str0]].nsubdirs, entries0), dirs0[entries[str]] = mkdir(dirs0[entries[str]].size, md3, dirs0[entries[str]].nlink, dirs0[entries[str]].nsubdirs, ∅), dirs0[m] = mkdir(dirs0[m].size, md4, dirs0[m].nlink, dirs0[m].nsubdirs, entries), dirs0[m0] = mkdir(m1 + 1, md6, m4, m2 + 1, entries1), entries[str] = dirs0[m].entries[str], entries1[str0] = entries2[str0], dirs0[m0] = mkdir(m3, md0, m4, m5, entries2), dirs = dirs0[m0, ((dirs0[m0].entries:= (entries2 -- str0)).size:= m1).nsubdirs:= m2] [m, (m = m0 ⊃ ((dirs0[m0].entries:= (entries2 -- str0)).size:= m1).nsubdirs:= m2;dirs0[m]) .entries:= (m = m0 ⊃ ((dirs0[m0].entries:= (entries2 -- str0)).size:= m1).nsubdirs:= m2;dirs0[m]).entries[str, entries1[str0]] ], m3 = m1 + 1, m5 = m2 + 1, afs-cons(dirs0, files), str ∈ entries, str0 ∈ entries1, entries[str] ∈ dirs0, entries1[str0] ∈ dirs0, m ∈ dirs0, m0 ∈ dirs0, str ∈ dirs0[m].entries, str0 ∈ entries2, m = m0 → md1 = md2, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ ⟪afs_drop_nlink#(mkdentry(str, entries[str]).ino; dirs, files); old_dent := mkdentry(str0, entries1[str0]); asynced1 := true; if ¬ asynced1 then {dirs[m].size := dirs[m].size + 1; if old_dent.ino ∈ dirs then dirs[m].nsubdirs := dirs[m].nsubdirs + 1}; new_dent := mkdentry(mkdentry(str, entries[str]).name, old_dent.ino); old_dent := negdentry(old_dent.name); afs_inode_sync#(mkinode(m0, md2, true, m4, m2 + 1, m1 + 1); dirs, files); afs_inode_sync#(mkinode(m, md1, true, # links(m, dirs0), dirs0[m].nsubdirs, dirs0[m].size); dirs, files); if asynced1 then {afs_inode_sync# (mkinode(entries[str], md, true, # links(entries[str], dirs0), dirs0[entries[str]].nsubdirs, dirs0[entries[str]].size); dirs, files); new_child_inode := getinode(entries[str], dirs, files)}; old_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) ); new_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) ); err := ESUCCESS; asynced := ?⟫ afs-cons(dirs, files), entries[str] = ROOT_INO, m0 = m ∧ mkdentry(str, entries[str]) = mkdentry(str0, entries1[str0])

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [dirs0[m0], m3, md0, m4, m5, entries2]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: dirs0[m0], m3, md0, m4, m5, entries2