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Proof step 100 from proof afs-cons-rename-file-dentry (previous step)

# links(entries0[str], dirs0) = m11 + 1, (m = m0 ⊃ mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m4, md6, # links(m, dirs0), m8, entries0)) = mkdir(m6, md5, m9, m10, entries), (ino = entries0[str] ⊃ mkfile(m2, md, m11, content1);mkfile(m12, md7, m13, content0)) = mkfile(m14, md8, m15, content2), dirs0[ino] = mkdir(m16, md9, m17, m18, entries2), dirs0[m0] = mkdir(m3 + 1, md4, # links(m0, dirs0), m7, entries1), files0[entries1[str0]] = mkfile(m1, md3, # links(entries1[str0], dirs0), content), files0[entries0[str]] = mkfile(m5, md0, m11 + 1, content1), dirs0[m] = mkdir(m4, md6, # links(m, dirs0), m8, entries0), files0[ino] = mkfile(m12, md7, m13, content0), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, entries0[str] ∈ files0, m0 ∈ dirs0, str ∈ entries0, entries1[str0] ∈ files0, str0 ∈ entries1, m ∈ dirs0, ino ∈ dirs0 ∧ name ∈ entries2 → entries2[name] ∈ dirs0 ∨ entries2[name] ∈ files0, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m13 = # links(ino, dirs0), ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries2 = ∅, m = m0 → md1 = md2, ino ∈ dirs0 → m17 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m16 = # entries2 ∧ m18 = # subdirs(ino, dirs0), links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str] ∈ dirs0, entries1[str0] ∈ dirs0, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ]) = ∅ → (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .nlink = # links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ]) ∧ (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .nlink ≤ 1 ∧ (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .size = # (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .entries ∧ (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .nsubdirs = # subdirs (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ])) ∧ ( ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m15 = # links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .entries → (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .entries [name ] ∈ dirs0 ∨ (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) .entries [name ] ∈ files0) ∧ (ino ∈ files0 ∧ n ∈ content2 → # content2[n] = VFS_PAGE_SIZE), m0 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries1[str0])

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 101:
# links(entries0[str], dirs0) = m11 + 1, (ino = entries0[str] ⊃ mkfile(m2, md, m11, content1);mkfile(m12, md7, m13, content0)) = mkfile(m14, md8, m15, content2), (m = m0 ⊃ mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m4, md6, # links(m, dirs0), m8, entries0)) = mkdir(m6, md5, m9, m10, entries), (ino = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0);mkdir(m16, md9, m17, m18, entries2)) ) = mkdir(m19, md10, m20, m21, entries3), files0[ino] = mkfile(m12, md7, m13, content0), dirs0[m] = mkdir(m4, md6, # links(m, dirs0), m8, entries0), files0[entries0[str]] = mkfile(m5, md0, m11 + 1, content1), files0[entries1[str0]] = mkfile(m1, md3, # links(entries1[str0], dirs0), content), dirs0[m0] = mkdir(m3 + 1, md4, # links(m0, dirs0), m7, entries1), dirs0[ino] = mkdir(m16, md9, m17, m18, entries2), afs-cons(dirs0, files0), m ∈ dirs0, str0 ∈ entries1, entries1[str0] ∈ files0, str ∈ entries0, m0 ∈ dirs0, entries0[str] ∈ files0, ROOT_INO ∈ dirs0, ino ∈ dirs0 → m17 = # links(ino, dirs0) ∧ # links(ino, dirs0) ≤ 1 ∧ m16 = # entries2 ∧ m18 = # subdirs(ino, dirs0), m = m0 → md1 = md2, ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → entries2 = ∅, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m13 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ entries2 → entries2[name] ∈ dirs0 ∨ entries2[name] ∈ files0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m0, dirs0) ≠ ∅ ∨ m0 = ROOT_INO ⊦ entries1[str0] ∈ dirs0, entries0[str] ∈ dirs0, m0 = m ∧ mkdentry(str, entries0[str]) = mkdentry(str0, entries1[str0]), ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ]) = ∅ → entries3 = ∅) ∧ ( ino ∈ dirs0 → m20 = # links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ]) ∧ m20 ≤ 1 ∧ m19 = # entries3 ∧ m21 = # subdirs (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ])) ∧ ( ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m15 = # links (ino, dirs0 [m0, (m0 = m ⊃ mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]);mkdir(m3, md4, # links(m0, dirs0), m7, entries1 -- str0)).meta:= md2 ] [m, (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]);mkdir(m6, md5, m9, m10, entries[str, entries1[str0]])).meta:= md1 ])) ∧ (ino ∈ dirs0 ∧ name ∈ entries3 → entries3[name] ∈ dirs0 ∨ entries3[name] ∈ files0) ∧ (ino ∈ files0 ∧ n ∈ content2 → # content2[n] = VFS_PAGE_SIZE)

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 = m ⊃ (m = m0 ⊃ mkdir (m6, md2, m9, m10, entries[str, entries1[str0]]) ;mkdir (m6, md5, m9, m10, entries[str, entries1[str0]]) ) .meta:= md1 ;(ino = m0 ⊃ mkdir (m3, md2, # links(m0, dirs0), m7, entries1 -- str0) ;mkdir(m16, md9, m17, m18, entries2) ) ), m19, md10, m20, m21, 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 = m ⊃ (m = m0 ⊃ mkdir(m6, md2, m9, m10, entries[str, entries1[str0]]) ;mkdir(m6, md5, m9, m10, entries[str, entries1[str0]]) ) .meta:= md1 ;(ino = m0 ⊃ mkdir(m3, md2, # links(m0, dirs0), m7, entries1 -- str0) ;mkdir(m16, md9, m17, m18, entries2) ) ), m19, md10, m20, m21, entries3