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Proof step 45 from proof afs-cons-rename-file-negdentry (previous step)

# links(entries0[str], dirs0) = files[entries0[str]].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(m, dirs0) = dirs0[m].nlink, entries0[str] = dirs0[m].entries[str], dirs0[m] = mkdir(dirs0[m].size, md4, dirs0[m].nlink, dirs0[m].nsubdirs, entries0), files[entries0[str]] = mkfile(m1, md2, files[entries0[str]].nlink, content), dirs0[m0] = mkdir(dirs0[m0].size, md3, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries), dirs = dirs0[m, (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1)] [m0, ( (m0 = m ⊃ (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1);dirs0[m0]) .entries:= (m0 = m ⊃ (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1);dirs0[m0]).entries[str0, entries0[str]]) .size:= ((m0 = m ⊃ (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= (dirs0[m].size - 1);dirs0[m0]).size + 1) ], dirs0[m].size ≠ 0, afs-cons(dirs0, files), str ∈ dirs0[m].entries, str ∈ entries0, m ∈ dirs0, entries0[str] ∈ files, m0 ∈ dirs0, 1 ≤ dirs0[m].size, ¬ str0 ∈ entries, ¬ entries0[str] ∈ dirs0, ¬ str0 ∈ dirs0[m0].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, 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)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 46:
# links(m, dirs0) = dirs0[m].nlink, # links(m0, dirs0) = dirs0[m0].nlink, # links(entries0[str], dirs0) = files[entries0[str]].nlink, dirs0[m].size = m2 + 1, dirs0[m0] = mkdir(dirs0[m0].size, md3, dirs0[m0].nlink, dirs0[m0].nsubdirs, entries), files[entries0[str]] = mkfile(m1, md2, files[entries0[str]].nlink, content), dirs0[m] = mkdir(dirs0[m].size, md4, dirs0[m].nlink, dirs0[m].nsubdirs, entries0), entries0[str] = dirs0[m].entries[str], dirs = dirs0[m, (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m2] [m0, ( (m0 = m ⊃ (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m2;dirs0[m0]) .entries:= (m0 = m ⊃ (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m2;dirs0[m0]).entries[str0, entries0[str]]) .size:= ((m0 = m ⊃ (dirs0[m].entries:= (dirs0[m].entries -- str)).size:= m2;dirs0[m0]).size + 1) ], afs-cons(dirs0, files), m0 ∈ dirs0, entries0[str] ∈ files, m ∈ dirs0, str ∈ entries0, str ∈ dirs0[m].entries, 1 ≤ dirs0[m].size, m0 = m → md1 = md0, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = 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), dirs0[m].size = 0, str0 ∈ entries, entries0[str] ∈ dirs0, str0 ∈ dirs0[m0].entries

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [dirs0[m].size, m2]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: dirs0[m].size, m2