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Proof step 86 from proof afs-cons-rename-file-negdentry (previous step)

(m0 = m ⊃ mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str);mkdir(m7, md5, # links(m0, dirs0), m9, entries2)) = mkdir(m1, md4, m4, m6, entries0), dirs0[ino] = mkdir(m11, md6, m12, m13, entries), files[ino] = mkfile(m8, md3, m10, content), dirs0[m0] = mkdir(m7, md5, # links(m0, dirs0), m9, entries2), files[entries1[str]] = mkfile(m3, md, # links(entries1[str], dirs0), content0), dirs0[m] = mkdir(m2 + 1, md2, # links(m, dirs0), m5, entries1), afs-cons(dirs0, files), m ∈ dirs0, entries1[str] ∈ files, str ∈ entries1, m0 ∈ dirs0, ROOT_INO ∈ dirs0, m0 = m → md1 = md0, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅, ino ∈ files → ¬ ino ∈ dirs0 ∧ m10 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files, ino ∈ dirs0 → m12 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m11 = # entries ∧ m13 = # subdirs(ino, dirs0), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ str0 ∈ entries2, entries1[str] ∈ dirs0, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ]) = ∅ → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .nlink = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ]) ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .nlink ≤ 1 ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .size = # (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .entries ∧ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .nsubdirs = # subdirs (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ m10 = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .entries → (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .entries [name ] ∈ dirs0 ∨ (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) .entries [name ] ∈ files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 87:
(m0 = m ⊃ mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str);mkdir(m7, md5, # links(m0, dirs0), m9, entries2)) = mkdir(m1, md4, m4, m6, entries0), (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str);mkdir(m11, md6, m12, m13, entries)) ) = mkdir(m14, md7, m15, m16, entries3), dirs0[m] = mkdir(m2 + 1, md2, # links(m, dirs0), m5, entries1), files[entries1[str]] = mkfile(m3, md, # links(entries1[str], dirs0), content0), dirs0[m0] = mkdir(m7, md5, # links(m0, dirs0), m9, entries2), files[ino] = mkfile(m8, md3, m10, content), dirs0[ino] = mkdir(m11, md6, m12, m13, entries), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, m0 ∈ dirs0, str ∈ entries1, entries1[str] ∈ files, m ∈ dirs0, ino ∈ dirs0 → m12 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m11 = # entries ∧ m13 = # subdirs(ino, dirs0), ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files, ino ∈ files → ¬ ino ∈ dirs0 ∧ m10 = # links(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅, m0 = m → md1 = md0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries1[str] ∈ dirs0, str0 ∈ entries2, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ]) = ∅ → entries3 = ∅) ∧ ( ino ∈ dirs0 → m15 = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ]) ∧ m15 ≤ 1 ∧ m14 = # entries3 ∧ m16 = # subdirs (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ m10 = # links (ino, dirs0 [m, (m = m0 ⊃ mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]);mkdir(m2, md2, # links(m, dirs0), m5, entries1 -- str)) .meta:= md0 ] [m0, (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]);mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]])).meta:= md1 ])) ∧ (ino ∈ dirs0 ∧ name ∈ entries3 → entries3[name] ∈ dirs0 ∨ entries3[name] ∈ files)

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] → [(ino = m0 ⊃ (m0 = m ⊃ mkdir (m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]) ;mkdir (m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir (m2, md0, # links(m, dirs0), m5, entries1 -- str) ;mkdir(m11, md6, m12, m13, entries) ) ), m14, md7, m15, m16, entries3]

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: (ino = m0 ⊃ (m0 = m ⊃ mkdir(m1 + 1, md0, m4, m6, entries0[str0, entries1[str]]) ;mkdir(m1 + 1, md4, m4, m6, entries0[str0, entries1[str]]) ) .meta:= md1 ;(ino = m ⊃ mkdir(m2, md0, # links(m, dirs0), m5, entries1 -- str) ;mkdir(m11, md6, m12, m13, entries) ) ), m14, md7, m15, m16, entries3