[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 376 from proof afs-cons-rename-dir-dentry (previous step)

files[ino] = mkfile(m0, md0, m2, content), dirs0[entries0[str0]] = mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅), dirs0[m] = mkdir(m1, md1, # links(m, dirs0), m4, entries0), afs-cons(dirs0, files), m ∈ dirs0, str0 ∈ entries0, entries0[str0] ∈ dirs0, ROOT_INO ∈ dirs0, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅, ino ∈ dirs0 → dirs0[ino].nlink = # links(ino, dirs0) ∧ dirs0[ino].nlink ≤ 1 ∧ dirs0[ino].size = # dirs0[ino].entries ∧ dirs0[ino].nsubdirs = # subdirs(ino, dirs0), ino ∈ files → ¬ ino ∈ dirs0 ∧ m2 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str0] = ROOT_INO, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ]) = ∅ → (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .nlink = # links (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ]) ∧ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .nlink ≤ 1 ∧ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .size = # (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries ∧ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .nsubdirs = # subdirs (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ m2 = # links (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries → (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries [name ] ∈ dirs0 ∨ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries [name ] ∈ files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 377:
dirs0[m] = mkdir(m1, md1, # links(m, dirs0), m4, entries0), dirs0[entries0[str0]] = mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅), files[ino] = mkfile(m0, md0, m2, content), dirs0[ino] = mkdir(m5, md4, m7, m8, entries), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries0[str0] ∈ dirs0, str0 ∈ entries0, m ∈ dirs0, ino ∈ dirs0 ∧ name ∈ entries → entries[name] ∈ dirs0 ∨ entries[name] ∈ files, ino ∈ files → ¬ ino ∈ dirs0 ∧ m2 = # links(ino, dirs0), ino ∈ dirs0 → m7 = # links(ino, dirs0) ∧ m7 ≤ 1 ∧ m5 = # entries ∧ m8 = # subdirs(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries = ∅, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str0] = ROOT_INO, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ]) = ∅ → (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .nlink = # links (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ]) ∧ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .nlink ≤ 1 ∧ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .size = # (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries ∧ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .nsubdirs = # subdirs (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ m2 = # links (ino, dirs0[m, mkdir(m1, md2, # links(m, dirs0), m4, entries0)] [entries0[str0], (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries → (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries [name ] ∈ dirs0 ∨ (ino = entries0[str0] ⊃ (entries0[str0] = m ⊃ mkdir(m1, md2, # links(entries0[str0], dirs0), m4, entries0) ;mkdir(m3, md3, # links(entries0[str0], dirs0), m6, ∅) ) .meta:= md ;(ino = m ⊃ mkdir(m1, md2, # links(m, dirs0), m4, entries0);dirs0[ino]) ) .entries [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] → [dirs0[ino], m5, md4, m7, m8, entries]

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[ino], m5, md4, m7, m8, entries