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Proof step 116 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries0[str], dirs0) = m12 + 1, (ino = m ⊃ (m = m0 ⊃ mkdir(m8, md2, m10, m11, entries[str, entries2[str0]]);mkdir(m8, md4, m10, m11, entries[str, entries2[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m1, md2, # links(m0, dirs0), m2, entries2 -- str0) ;mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4) ) ) = mkdir(m20, md10, m21, m22, entries5), dirs0[m0] = mkdir(m1 + 1, md0, # links(m0, dirs0), m2 + 1, entries2), dirs0[m] = mkdir(m6, md5, # links(m, dirs0), m9, entries0), dirs0[entries0[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[entries2[str0]] = mkdir(m4, md3, # links(entries2[str0], dirs0), m7, entries1), files[ino] = mkfile(m14, md8, m16, content), dirs0[ino] = mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4), links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m8, md4, m10, m11, entries[str, entries2[str0]]);mkdir(m1, md0, # links(m0, dirs0), m2, entries2 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m8, md2, m10, m11, entries[str, entries2[str0]]);mkdir(m8, md4, m10, m11, entries[str, entries2[str0]])).meta:= md1 ]) = ∅, m = m0, mkdir(m1, md0, # links(m0, dirs0), m2, entries2 -- str0) = mkdir(m8, md4, m10, m11, entries), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries0[str] ∈ dirs0, m0 ∈ dirs0, str0 ∈ entries2, entries2[str0] ∈ dirs0, str ∈ entries0, m ∈ dirs0, ino ∈ dirs0, # links(ino, dirs0) ≤ 1, m = m0 → md1 = md2, name ∈ entries4 → entries4[name] ∈ dirs0 ∨ entries4[name] ∈ files, links(ino, dirs0) = ∅ → entries4 = ∅, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str] = ROOT_INO, entries0[str] = m, entries0[str] = m0, entries5 = ∅, ino = ROOT_INO, ino = entries0[str], m0 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries2[str0])

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 117:
# links(entries0[str], dirs0) = m12 + 1, (ino = m0 ⊃ mkdir(m8, md1, # links(m0, dirs0), m11, (entries0 -- str0)[str, entries0[str0]]) ;mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4) ) = mkdir(m20, md10, m21, m22, entries5), (entries0[str0] = ino ⊃ (links(entries0[str0], dirs0) -- mklink(m0, str0)) ++ mklink(m0, str);links(ino, dirs0)) = ∅, dirs0[ino] = mkdir(# entries4, md9, # links(ino, dirs0), # subdirs(ino, dirs0), entries4), files[ino] = mkfile(m14, md8, m16, content), dirs0[entries0[str0]] = mkdir(m4, md3, # links(entries0[str0], dirs0), m7, entries1), dirs0[entries0[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), dirs0[m0] = mkdir(m8 + 1, md5, # links(m0, dirs0), m11 + 1, entries0), afs-cons(dirs0, files), ino ∈ dirs0, str ∈ entries0, entries0[str0] ∈ dirs0, str0 ∈ entries0, m0 ∈ dirs0, entries0[str] ∈ dirs0, ROOT_INO ∈ dirs0, # links(ino, dirs0) ≤ 1, links(ino, dirs0) = ∅ → entries4 = ∅, name ∈ entries4 → entries4[name] ∈ dirs0 ∨ entries4[name] ∈ files, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries0[str] = m0, entries0[str] = ROOT_INO, entries5 = ∅, ino = entries0[str], ino = ROOT_INO, mkdentry(str, entries0[str]) = mkdentry(str0, entries0[str0])

Rule argument

Additional information

Used simplifier rules: Del-at-entries: ⊦ (entries -- str0)[str] = (str ≠ str0 ⊃ entries[str];(∅)[str]);%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(m0, md0, n1, n2, entries) → links(ino0, dirs[ino, mkdir(m, md, n, n0, entries)]) = links(ino0, dirs); (s)% ⊦ str ≠ str0 → (entries -- str0)[str] = entries[str]; (s)% ⊦ str ∈ entries -- str0 ↔ str ≠ str0 ∧ str ∈ entries; (s)%⊦ ino0 ∈ dirs → links(ino, dirs[ino0, mkdir(m, md, n0, n1, entries -- str)]) = (str ∈ entries ∧ entries[str] = ino ⊃ links(ino, dirs[ino0, mkdir(m, md, n0, n1, entries)]) -- mklink(ino0, str) ;links(ino, dirs[ino0, mkdir(m, md, n0, n1, entries)]) ); (s)%⊦ p_ino ∈ dirs → links (ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries[str, ino])]) = (ino = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ++ mklink(p_ino, str) ;(str ∈ entries ∧ entries[str] = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) -- mklink(p_ino, str) ;links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ) ); (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)%⊦ mkdir(m, md, m0, m1, entries).meta:= md0 = mkdir(m, md0, m0, m1, entries); (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)