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Proof step 79 from proof refine-writemeta (previous step)

root[path] = dnode(md0, sdt), dirs0[m] = mkdir(# entries, md0, # links(m, dirs0), m2, entries), ((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs|(root, ROOT_INO, path, m) * abs(sdt, entries)))(dirs), root.dir?, dom of < ino0, afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs0, files), of-cons(of, dirs0, files), pw(user, md0), px-parent(user, path, ROOT_INO, dirs0, files), path ∈ root, m ∈ dirs0, dirs ≡ dirs0[m, mkdir(# entries, md, # links(m, dirs0), m2, entries)], psynced0 → asynced0 ⊦ (((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs(sdt, entries)) * abs|(root[path, dnode(md, sdt)], ROOT_INO, path, m)) )(dirs), path ≠ ε ∧ parent(path) ∈ root ∧ root[parent(path)].dir?

Heuristic nonsplitting cut applied rule cut formula

And obtained the following new goals

Proof step 80:
dirs0[m] = mkdir(# entries, md0, # links(m, dirs0), m2, entries), root[path] = dnode(md0, sdt), ((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs|(root, ROOT_INO, path, m) * abs(sdt, entries)))(dirs), root.dir?, dom of < ino0, afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(path, ROOT_INO, m, dirs0, files), of-cons(of, dirs0, files), pw(user, md0), px-parent(user, path, ROOT_INO, dirs0, files), m ∈ dirs0, path ∈ root, dirs ≡ dirs0[m, mkdir(# entries, md, # links(m, dirs0), m2, entries)], psynced0 → asynced0 ⊦ path = ε, (((m => mkdir(# entries, md, # links(m, dirs0), m2, entries) * abs(sdt, entries)) * abs|(root[path, dnode(md, sdt)], ROOT_INO, path, m)) )(dirs), path ≠ ε ∧ parent(path) ∈ root ∧ root[parent(path)].dir?

Rule argument

fma: path = ε

Additional information

Used simplifier rules: ⊦ tree[ε] = tree; (s)% ⊦ ε ∈ tree; (s)% ⊦ px-parent(user, ε, ino, dirs, files); (s)% ⊦ ispath(ε, n, m, dirs, files) ↔ valid-ino(n, dirs, files) ∧ m = n; (s)% ⊦ valid-ino(ino, dirs, files) ↔ ino ∈ dirs ∨ ino ∈ files; (s)% ⊦ abs|(tree, ino, ε, ino0) = (emp ∧ ⌜ino = ino0⌝ ); (s)% ⊦ ⌜true⌝ = true; (s)% ⊦ (true ∧ D) = D; (s)%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]% ⊦ emp * D = D; (s)%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]% ⊦ tree[ε, tree0] = tree0; (s) Used formulas: path = ε