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Proof step 46 from proof afs-cons-rename-dir-negdentry (previous step)

# links(entries1[str], dirs0) = dirs0[entries1[str]].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(m, dirs0) = dirs0[m].nlink, dirs0[m].nsubdirs = m2 + 1, dirs0[m].size = m1 + 1, entries1[str] = dirs0[m].entries[str], dirs0[m] = mkdir(m1 + 1, md4, dirs0[m].nlink, m2 + 1, entries1), dirs0[m0] = mkdir(dirs0[m0].size, md2, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries), dirs0[entries1[str]] = mkdir(dirs0[entries1[str]].size, md3, dirs0[entries1[str]].nlink, dirs0[entries1[str]].nsubdirs, entries0), 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, entries1[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), str ∈ dirs0[m].entries, str ∈ entries1, m ∈ dirs0, m0 ∈ dirs0, entries1[str] ∈ dirs0, ¬ str0 ∈ dirs0[m0].entries, ¬ str0 ∈ entries, 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)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 47:
# links(m, dirs0) = m4, # links(m0, dirs0) = dirs0[m0].nlink, # links(entries1[str], dirs0) = dirs0[entries1[str]].nlink, dirs0[entries1[str]] = mkdir(dirs0[entries1[str]].size, md3, dirs0[entries1[str]].nlink, dirs0[entries1[str]].nsubdirs, entries0), dirs0[m0] = mkdir(dirs0[m0].size, md2, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries), dirs0[m] = mkdir(m1 + 1, md4, m4, m2 + 1, entries1), entries1[str] = entries2[str], dirs0[m] = mkdir(m3, md, m4, m5, entries2), dirs = dirs0[m, ((dirs0[m].entries:= (entries2 -- str)).size:= m1).nsubdirs:= m2] [m0, ( ( (m0 = m ⊃ ((dirs0[m].entries:= (entries2 -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]) .entries:= (m0 = m ⊃ ((dirs0[m].entries:= (entries2 -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).entries[str0, entries1[str]]) .size:= ((m0 = m ⊃ ((dirs0[m].entries:= (entries2 -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).size + 1)) .nsubdirs:= ((m0 = m ⊃ ((dirs0[m].entries:= (entries2 -- str)).size:= m1).nsubdirs:= m2;dirs0[m0]).nsubdirs + 1) ], m3 = m1 + 1, m5 = m2 + 1, afs-cons(dirs0, files), entries1[str] ∈ dirs0, m0 ∈ dirs0, m ∈ dirs0, str ∈ entries1, str ∈ entries2, m0 = m → md1 = md0, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪afs_inode_sync#(mkinode(m, md0, true, m4, 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), str0 ∈ dirs0[m0].entries, str0 ∈ entries

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[m], m3, md, 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[m], m3, md, m4, m5, entries2