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Proof step 39 from proof inv-unlink (previous step)

# links(entries0[str], dirs0) = m1 + 1, files0[ino] = mkfile(m4, md, m5, content0), files0[entries0[str]] = mkfile(m0, md1, m1 + 1, content), dirs0[m] = mkdir(m2 + 1, md2, # links(m, dirs0), m3, entries0), afs-cons(dirs0, files0), m ∈ dirs0, str ∈ entries0, entries0[str] ∈ files0, 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 ∈ files0 → ¬ ino ∈ dirs0 ∧ m5 = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ (entries0[str] = ino ⊃ links(entries0[str], dirs0) -- mklink(m, str);links(ino, dirs0)) = ∅ → (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).nlink = # (entries0[str] = ino ⊃ links(entries0[str], dirs0) -- mklink(m, str);links(ino, dirs0)) ∧ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).nlink ≤ 1 ∧ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).size = # (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries ∧ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).nsubdirs = # subdirs(ino, dirs0[m, mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str)])) ∧ ( ino ∈ files0 → ¬ ino ∈ dirs0 ∧ (ino = entries0[str] ⊃ mkfile(m0, md1, m1, content);mkfile(m4, md, m5, content0)).nlink = # (entries0[str] = ino ⊃ links(entries0[str], dirs0) -- mklink(m, str);links(ino, dirs0))) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries → (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries[name] ∈ dirs0 ∨ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries[name] ∈ files0) ∧ ( ino ∈ files0 ∧ n ∈ (ino = entries0[str] ⊃ mkfile(m0, md1, m1, content);mkfile(m4, md, m5, content0)).content → # (ino = entries0[str] ⊃ mkfile(m0, md1, m1, content);mkfile(m4, md, m5, content0)).content[n] = VFS_PAGE_SIZE)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 40:
# links(entries0[str], dirs0) = m1 + 1, (ino = entries0[str] ⊃ mkfile(m0, md1, m1, content);mkfile(m4, md, m5, content0)) = mkfile(m6, md0, m7, content1), dirs0[m] = mkdir(m2 + 1, md2, # links(m, dirs0), m3, entries0), files0[entries0[str]] = mkfile(m0, md1, m1 + 1, content), files0[ino] = mkfile(m4, md, m5, content0), afs-cons(dirs0, files0), ROOT_INO ∈ dirs0, entries0[str] ∈ files0, str ∈ entries0, m ∈ dirs0, ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files0, ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m5 = # 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 ⊦ ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ (entries0[str] = ino ⊃ links(entries0[str], dirs0) -- mklink(m, str);links(ino, dirs0)) = ∅ → (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).nlink = # (entries0[str] = ino ⊃ links(entries0[str], dirs0) -- mklink(m, str);links(ino, dirs0)) ∧ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).nlink ≤ 1 ∧ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).size = # (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries ∧ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).nsubdirs = # subdirs(ino, dirs0[m, mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str)])) ∧ (ino ∈ files0 → ¬ ino ∈ dirs0 ∧ m7 = # (entries0[str] = ino ⊃ links(entries0[str], dirs0) -- mklink(m, str);links(ino, dirs0))) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries → (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries[name] ∈ dirs0 ∨ (ino = m ⊃ mkdir(m2, md2, # links(m, dirs0), m3, entries0 -- str);dirs0[ino]).entries[name] ∈ files0) ∧ (ino ∈ files0 ∧ n ∈ content1 → # content1[n] = VFS_PAGE_SIZE)

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] → [(ino = entries0[str] ⊃ mkfile(m0, md1, m1, content) ;mkfile(m4, md, m5, content0) ), m6, md0, m7, content1]

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: (ino = entries0[str] ⊃ mkfile(m0, md1, m1, content) ;mkfile(m4, md, m5, content0) ), m6, md0, m7, content1