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Proof step 40 from proof inv-link (previous step)

dirs0[ino0] = mkdir(m2, md3, # links(ino0, dirs0), m6, entries0), files0[entries[str]] = mkfile(m1, md4, # links(entries[str], dirs0), content0), dirs0[m] = mkdir(m0, md1, # links(m, dirs0), m5, entries1), entries0[str] = entries[str], dirs0[ino1] = mkdir(m4, md2, # links(ino1, dirs0), m8, entries), dirs = dirs0[m, mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]])], files = files0[entries[str], mkfile(m1, md4, # links(entries[str], dirs0) + 1, content0)], afs-cons(dirs0, files0), entries[str] ∈ files0, m ∈ dirs0, str ∈ entries0, ino0 ∈ dirs0, ino1 ∈ dirs0, str ∈ entries, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ afs-cons(dirs, files), str0 ∈ entries1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 41:
dirs0[ino1] = mkdir(m4, md2, # links(ino1, dirs0), m8, entries), entries0[str] = entries[str], dirs0[m] = mkdir(m0, md1, # links(m, dirs0), m5, entries1), files0[entries[str]] = mkfile(m1, md4, # links(entries[str], dirs0), content0), dirs0[ino0] = mkdir(m2, md3, # links(ino0, dirs0), m6, entries0), afs-cons(dirs0, files0), str ∈ entries, ino1 ∈ dirs0, ino0 ∈ dirs0, str ∈ entries0, m ∈ dirs0, entries[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 ∧ files0[ino].nlink = # links(ino, dirs0), ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → dirs0[ino].entries[name] ∈ dirs0 ∨ dirs0[ino].entries[name] ∈ files0, ino ∈ files0 ∧ n ∈ files0[ino].content → # files0[ino].content[n] = VFS_PAGE_SIZE, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ str0 ∈ entries1, ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0)) = ∅ → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries = ∅) ∧ ( ino ∈ dirs0 → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nlink = # (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0)) ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nlink ≤ 1 ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).size = # (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries ∧ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).nsubdirs = # subdirs(ino, dirs0[m, mkdir(m0, md1, # links(m, dirs0), m5, entries1[str0, entries[str]])])) ∧ ( ino ∈ files0 → ¬ ino ∈ dirs0 ∧ (ino = entries[str] ⊃ mkfile(m1, md4, # links(entries[str], dirs0) + 1, content0);files0[ino]).nlink = # (entries[str] = ino ⊃ links(entries[str], dirs0) ++ mklink(m, str0);links(ino, dirs0))) ∧ ( ino ∈ dirs0 ∧ name ∈ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries → (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries[name] ∈ dirs0 ∨ (ino = m ⊃ mkdir(m0 + 1, md1, # links(m, dirs0), m5, entries1[str0, entries[str]]);dirs0[ino]).entries[name] ∈ files0) ∧ ( ino ∈ files0 ∧ n ∈ (ino = entries[str] ⊃ mkfile(m1, md4, # links(entries[str], dirs0) + 1, content0);files0[ino]).content → # (ino = entries[str] ⊃ mkfile(m1, md4, # links(entries[str], dirs0) + 1, content0);files0[ino]).content[n] = VFS_PAGE_SIZE)

Rule argument

Additional information

Used simplifier rules: afs-cons-reduce: ⊦ afs-cons(dirs0, files0) → ( afs-cons(dirs1, files1) ↔ (∀ ino, name, n. ROOT_INO ∈ dirs0 ∧ ( ino ∈ dirs0 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs0) = ∅ → dir-empty(ino, dirs0, files0)) ∧ ( ino ∈ dirs0 → ¬ ino ∈ files0 ∧ 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 ∧ files0[ino].nlink = # links(ino, dirs0)) ∧ ( ino ∈ dirs0 ∧ name ∈ dirs0[ino].entries → valid-ino(dirs0[ino].entries[name], dirs0, files0)) ∧ ( ino ∈ files0 ∧ n ∈ files0[ino].content → valid-pagesize(ino, n, files0)) → ROOT_INO ∈ dirs1 ∧ ( ino ∈ dirs1 ∧ ino ≠ ROOT_INO ∧ links(ino, dirs1) = ∅ → dir-empty(ino, dirs1, files1)) ∧ ( ino ∈ dirs1 → ¬ ino ∈ files1 ∧ dirs1[ino].nlink = # links(ino, dirs1) ∧ dirs1[ino].nlink ≤ 1 ∧ dirs1[ino].size = # dirs1[ino].entries ∧ dirs1[ino].nsubdirs = # subdirs(ino, dirs1)) ∧ ( ino ∈ files1 → ¬ ino ∈ dirs1 ∧ files1[ino].nlink = # links(ino, dirs1)) ∧ ( ino ∈ dirs1 ∧ name ∈ dirs1[ino].entries → valid-ino(dirs1[ino].entries[name], dirs1, files1)) ∧ ( ino ∈ files1 ∧ n ∈ files1[ino].content → valid-pagesize(ino, n, files1))));% ⊦ files[ino, af][ino0] = (ino0 = ino ∧ ino ∈ files ⊃ af;files[ino0]); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%⊦ p_ino ∈ dirs → subdirs(ino, dirs[p_ino, mkdir(m0 + 1, md, n0, n1, entries)]) = subdirs(ino, dirs[p_ino, mkdir(m0, md, n0, n1, entries)]); (s)% ⊦ ino ∈ files[ino0, af] ↔ ino ∈ files; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(m0, md0, n1, n2, entries) → links(ino0, dirs[ino, mkdir(m, md, n, n0, entries)]) = links(ino0, dirs); (s)%⊦ p_ino ∈ dirs → links (ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries[str, ino])]) = (ino = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ++ mklink(p_ino, str) ;(str ∈ entries ∧ entries[str] = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) -- mklink(p_ino, str) ;links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ) ); (s)% ⊦ afs-cons(dirs, files) → ROOT_INO ∈ dirs; (ss)% ⊦ ino ∈ dirs[ino0, ad] ↔ ino ∈ dirs; (s)%⊦ valid-pagesize(ino, n, files) ↔ # files[ino].content[n] = VFS_PAGE_SIZE; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs → ¬ ino ∈ files; (s)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)