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Proof step 499 from proof refine-unlink (previous step)

⟨if ¬ is-open(entries0[str], of) ∧ ¬ ⌜entries0[str]⌝ ∈ (root0 -- (q + str ')).fids then fs := fs -- ⌜entries0[str]⌝ ⟩ (fs = fs0), ⟨vfs_put_inode#(getinode(entries0[str], dirs1, files), of, asynced; err0, dirs, files)⟩ (err0 = ESUCCESS ∧ dirs = dirs1 ∧ files = files1), # entries0 = m0 + 1, # links(entries0[str], dirs0) = m + 1, files0[entries0[str]] = mkfile(m1, md1, m + 1, content0), dirs0[ino1] = mkdir(m0 + 1, md0, # links(ino1, dirs0), m3, entries0), dirs = dirs1, err0 = ESUCCESS, files = files0[entries0[str], mkfile(m1, md1, m, content0)], fs = abs-files(files0), root0[q].dir?, root0.dir?, dom of < ino0, abs(root0, ROOT_INO)(dirs0), afs-cons(dirs0, files0), afs-cons(dirs1, files1), fids-cons(root0, abs-files(files0), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files0), of-cons(of, dirs0, files0), of-cons(of, dirs1, files1), pw(user, md0), px(user, md0), px-parent(user, q, ROOT_INO, dirs0, files0), ino1 ∈ dirs0, str ∈ entries0, entries0[str] ∈ files0, dirs1 ≡ dirs0[ino1, mkdir(m0, md0, # links(ino1, dirs0), m3, entries0 -- str)], psynced → asynced0 ⊦ ⟨psynced := ?⟩ ( (root0 -- (q + str ')).dir? ∧ fids-cons(root0 -- (q + str '), fs0, abs-of(of)) → fs0 = abs-files(files1) ∧ abs(root0 -- (q + str '), ROOT_INO)(dirs1) ∧ (psynced → asynced)), # links(ino1, dirs0) = 0, entries0 = ∅, ino1 = ROOT_INO, links(ino1, dirs0) = ∅, root0[q + str '].dir?, entries0[str] ∈ dirs0, ino1 ∈ dirs1, entries0[str] ∈ dirs1

User applied rule cut formula

And obtained the following new goals

Proof step 500:
⟨if ¬ is-open(entries0[str], of) ∧ ¬ ⌜entries0[str]⌝ ∈ (root0 -- (q + str ')).fids then fs := fs -- ⌜entries0[str]⌝ ⟩ (fs = fs0), ⟨vfs_put_inode#(getinode(entries0[str], dirs1, files), of, asynced; err0, dirs, files)⟩ (err0 = ESUCCESS ∧ dirs = dirs1 ∧ files = files1), # links(entries0[str], dirs0) = m + 1, # entries0 = m0 + 1, dirs0[ino1] = mkdir(m0 + 1, md0, # links(ino1, dirs0), m3, entries0), files0[entries0[str]] = mkfile(m1, md1, m + 1, content0), dirs = dirs1, err0 = ESUCCESS, files = files0[entries0[str], mkfile(m1, md1, m, content0)], fs = abs-files(files0), root0.dir?, root0[q].dir?, dom of < ino0, abs(root0, ROOT_INO)(dirs0), abs(root0 -- (q + str '), ROOT_INO)(dirs), afs-cons(dirs1, files1), afs-cons(dirs0, files0), fids-cons(root0, abs-files(files0), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files0), of-cons(of, dirs1, files1), of-cons(of, dirs0, files0), pw(user, md0), px(user, md0), px-parent(user, q, ROOT_INO, dirs0, files0), entries0[str] ∈ files0, str ∈ entries0, ino1 ∈ dirs0, dirs1 ≡ dirs0[ino1, mkdir(m0, md0, # links(ino1, dirs0), m3, entries0 -- str)], psynced → asynced0 ⊦ ⟨psynced := ?⟩ ( (root0 -- (q + str ')).dir? ∧ fids-cons(root0 -- (q + str '), fs0, abs-of(of)) → fs0 = abs-files(files1) ∧ abs(root0 -- (q + str '), ROOT_INO)(dirs1) ∧ (psynced → asynced)), # links(ino1, dirs0) = 0, entries0 = ∅, ino1 = ROOT_INO, links(ino1, dirs0) = ∅, root0[q + str '].dir?, entries0[str] ∈ dirs1, ino1 ∈ dirs1, entries0[str] ∈ dirs0
Proof step 559:
⊦ ispath(p, ROOT_INO, ino, dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n, md, m0, m1, entries) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) ∧ ¬ abs(tree0, ROOT_INO)(dirs0) → ( abs(tree, ROOT_INO)(dirs) ↔ (∃ sdt. ((abs|(tree, ROOT_INO, p, ino) * ino => mkdir(n, md, m0, m1, entries) * abs(sdt, entries)))(dirs) ∧ p ∈ tree ∧ tree[p] = dnode(md, sdt)))

Rule argument

fma: abs(root0 -- (q + str '), ROOT_INO)(dirs)

Additional information

Used simplifier rules: abs-unfold: ⊦ ispath(p, ROOT_INO, ino, dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n, md, m0, m1, entries) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) ∧ ¬ abs(tree0, ROOT_INO)(dirs0) → ( abs(tree, ROOT_INO)(dirs) ↔ (∃ sdt. (( abs|(tree, ROOT_INO, p, ino) * ino => mkdir(n, md, m0, m1, entries) * abs(sdt, entries)) )(dirs) ∧ p ∈ tree ∧ tree[p] = dnode(md, sdt)));%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]% ⊦ ((ino => ad * D))(dirs) → (dirs[ino] = ad0 ↔ ad = ad0); (s)% ⊦ ((ino => ad * D))(dirs) → ino ∈ dirs; (s)%⊦ dirs ≡ dirs0[ino, ad] → ( ((ino => ad0 * D))(dirs0) ↔ ((ino => ad * D))(dirs) ∧ dirs0[ino] = ad0 ∧ ino ∈ dirs0); (ws) Used formulas: abs(root0 -- (q + str '), ROOT_INO)(dirs)