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Proof step 43 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[m] = mkdir(m3 + 1, md3, # links(m, dirs0), m0 + 1, entries0), dirs0[entries0[str]] = mkdir(m1, md2, # links(entries0[str], dirs0), m4, ∅), dirs = dirs0[m, mkdir(m3, md3, # links(m, dirs0), m0, entries0 -- str)][entries0[str], mkdir(m5, md1, m2, m7, entries)], afs-cons(dirs0, files), m ∈ dirs0, str ∈ entries0, entries0[str] ∈ dirs0, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries0[str] = ROOT_INO, afs-cons(dirs, files)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 44:
(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)

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))));% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (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)