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Proof step 44 from proof inv-rmdir (previous step)

(entries0[str] = m ⊃ mkdir(m3, md3, # links(entries0[str], dirs0), m0, entries0 -- str) ;mkdir(m1, md2, # links(entries0[str], dirs0), m4, ∅) ) = mkdir(m5, md1, m2 + 1, m7, entries), dirs0[entries0[str]] = mkdir(m1, md2, # links(entries0[str], dirs0), m4, ∅), dirs0[m] = mkdir(m3 + 1, md3, # links(m, dirs0), m0 + 1, entries0), afs-cons(dirs0, files), entries0[str] ∈ dirs0, str ∈ entries0, m ∈ 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 ∧ files[ino].nlink = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files, ino ∈ files ∧ n ∈ files[ino].content → # files[ino].content[n] = VFS_PAGE_SIZE, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str] = ROOT_INO, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)]) = ∅ → (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .nlink = # links(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)]) ∧ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .nlink ≤ 1 ∧ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .size = # (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries ∧ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .nsubdirs = # subdirs(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ files[ino].nlink = # links(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries → (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries [name ] ∈ dirs0 ∨ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries [name ] ∈ files)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 45:
(entries0[str] = m ⊃ mkdir(m3, md3, # links(entries0[str], dirs0), m0, entries0 -- str) ;mkdir(m1, md2, # links(entries0[str], dirs0), m4, ∅) ) = mkdir(m5, md1, m2 + 1, m7, entries), dirs0[m] = mkdir(m3 + 1, md3, # links(m, dirs0), m0 + 1, entries0), dirs0[entries0[str]] = mkdir(m1, md2, # links(entries0[str], dirs0), m4, ∅), files[ino] = mkfile(m6, md, m8, content), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, m ∈ dirs0, str ∈ entries0, entries0[str] ∈ dirs0, ino ∈ files ∧ n ∈ content → # content[n] = VFS_PAGE_SIZE, ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files, ino ∈ files → ¬ ino ∈ dirs0 ∧ m8 = # links(ino, dirs0), 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 ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dirs0[ino].entries = ∅, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str] = ROOT_INO, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)]) = ∅ → (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries = ∅) ∧ ( ino ∈ dirs0 → (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .nlink = # links(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)]) ∧ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .nlink ≤ 1 ∧ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .size = # (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries ∧ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .nsubdirs = # subdirs(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)])) ∧ ( ino ∈ files → ¬ ino ∈ dirs0 ∧ m8 = # links(ino, dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)])) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries → (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries [name ] ∈ dirs0 ∨ (ino = entries0[str] ⊃ mkdir(m5, md1, m2, m7, entries);(ino = m ⊃ mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str);dirs0[ino])) .entries [name ] ∈ files)

Rule argument

spec: abstract-file name: elim seq: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content) subst: [af, m, md, m0, content] → [files[ino], m6, md, m8, content]

Additional information

Used simplifier rules: ⊦ ((m = af.size ∧ md = af.meta) ∧ m0 = af.nlink) ∧ content = af.content ↔ af = mkfile(m, md, m0, content); (elim) Used terms: files[ino], m6, md, m8, content