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Proof step 14 from proof abs-orphans (previous step)

inodekey(ino) ∈ ri, ∀ ino. ROOT_INO ∈ abs-dirs(fs, ri) ∧ (ino ∈ abs-dirs(fs, ri) ∧ ino ≠ ROOT_INO ∧ links(ino, abs-dirs(fs, ri)) = ∅ → dir-empty(ino, abs-dirs(fs, ri), abs-files(fs, ri))) ∧ ( ino ∈ abs-dirs(fs, ri) → ¬ ino ∈ abs-files(fs, ri) ∧ abs-dirs(fs, ri)[ino].nlink = # links(ino, abs-dirs(fs, ri)) ∧ abs-dirs(fs, ri)[ino].nlink ≤ 1 ∧ abs-dirs(fs, ri)[ino].size = # abs-dirs(fs, ri)[ino].entries ∧ abs-dirs(fs, ri)[ino].nsubdirs = # subdirs(ino, abs-dirs(fs, ri))) ∧ (ino ∈ abs-files(fs, ri) → ¬ ino ∈ abs-dirs(fs, ri) ∧ abs-files(fs, ri)[ino].nlink = # links(ino, abs-dirs(fs, ri))) ∧ (∀ name. ino ∈ abs-dirs(fs, ri) ∧ name ∈ abs-dirs(fs, ri)[ino].entries → valid-ino(abs-dirs(fs, ri)[ino].entries[name], abs-dirs(fs, ri), abs-files(fs, ri))) ∧ (∀ n. ino ∈ abs-files(fs, ri) ∧ n ∈ abs-files(fs, ri)[ino].content → valid-pagesize(ino, n, abs-files(fs, ri))) ⊦ links(ino, abs-dirs(fs, ri)) = ∅ ↔ fs[ri[inodekey(ino)]].nlink = 0, ino = ROOT_INO

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 15:
fs[ri[inodekey(ROOT_INO)]].directory, inodekey(ino) ∈ ri, inodekey(ROOT_INO) ∈ ri, ∀ ino0. (inodekey(ino0) ∈ ri ∧ fs[ri[inodekey(ino0)]].directory ∧ ino0 ≠ ROOT_INO ∧ links(ino0, abs-dirs(fs, ri)) = ∅ → abs-children(ino0, fs, ri) = ∅) ∧ ( inodekey(ino0) ∈ ri ∧ fs[ri[inodekey(ino0)]].directory → fs[ri[inodekey(ino0)]].nlink = # links(ino0, abs-dirs(fs, ri)) ∧ fs[ri[inodekey(ino0)]].nlink ≤ 1 ∧ fs[ri[inodekey(ino0)]].size = # abs-children(ino0, fs, ri) ∧ fs[ri[inodekey(ino0)]].nsubdirs = # subdirs(ino0, abs-dirs(fs, ri))) ∧ (inodekey(ino0) ∈ ri ∧ ¬ fs[ri[inodekey(ino0)]].directory → fs[ri[inodekey(ino0)]].nlink = # links(ino0, abs-dirs(fs, ri))) ∧ (∀ name. inodekey(ino0) ∈ ri ∧ fs[ri[inodekey(ino0)]].directory ∧ dentrykey(ino0, name) ∈ ri → inodekey(fs[ri[dentrykey(ino0, name)]].ino) ∈ ri ∧ fs[ri[inodekey(fs[ri[dentrykey(ino0, name)]].ino)]].directory ∨ inodekey(fs[ri[dentrykey(ino0, name)]].ino) ∈ ri ∧ ¬ fs[ri[inodekey(fs[ri[dentrykey(ino0, name)]].ino)]].directory) ∧ (∀ n. inodekey(ino0) ∈ ri ∧ ¬ fs[ri[inodekey(ino0)]].directory ∧ datakey(ino0, n) ∈ ri → # fs[ri[datakey(ino0, n)]].data = VFS_PAGE_SIZE) ⊦ links(ino, abs-dirs(fs, ri)) = ∅ ↔ fs[ri[inodekey(ino)]].nlink = 0, ino = ROOT_INO

Rule argument

Additional information

Used simplifier rules: in-dirs: ⊦ ino ∈ abs-dirs(fs, ri) ↔ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory;%at-dirs: ⊦ inodekey(ino) ∈ ri ∧ fs[ri[inodekey(ino)]].directory → abs-dirs(fs, ri)[ino] = abs-dir(ino, fs, ri);%dir: ⊦ abs-dir(ino, fs, ri) = mkdir (fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, fs[ri[inodekey(ino)]].nsubdirs, abs-children(ino, fs, ri));%in-files: ⊦ ino ∈ abs-files(fs, ri) ↔ inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory;%at-files: ⊦ inodekey(ino) ∈ ri ∧ ¬ fs[ri[inodekey(ino)]].directory → abs-files(fs, ri)[ino] = abs-file(ino, fs, ri);%file: ⊦ abs-file(ino, fs, ri) = mkfile (fs[ri[inodekey(ino)]].size, fs[ri[inodekey(ino)]].meta, fs[ri[inodekey(ino)]].nlink, abs-content(ino, fs, ri));%in-children: ⊦ str ∈ abs-children(ino, fs, ri) ↔ dentrykey(ino, str) ∈ ri;%at-children: ⊦ dentrykey(ino, str) ∈ ri → abs-children(ino, fs, ri)[str] = fs[ri[dentrykey(ino, str)]].ino;%in-content: ⊦ n ∈ abs-content(ino, fs, ri) ↔ datakey(ino, n) ∈ ri;%at-content: ⊦ datakey(ino, n) ∈ ri → abs-content(ino, fs, ri)[n] = fs[ri[datakey(ino, n)]].data;%⊦ valid-pagesize(ino, n, files) ↔ # files[ino].content[n] = VFS_PAGE_SIZE; (s)% ⊦ mkfile(m, md, m0, content).content = content; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ mkfile(m, md, m0, content).nlink = m0; (s)% ⊦ mkdir(m, md, m0, m1, entries).nsubdirs = m1; (s)% ⊦ mkdir(m, md, m0, m1, entries).size = m; (s)% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ dir-empty(ino, dirs, files) ↔ dirs[ino].entries = ∅; (s)