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Proof step 180 from proof refine-mkdir (previous step)

dirs0[ino1] = mkdir(# entries0, md1, # links(ino1, dirs0), m1, entries0), err0 = ESUCCESS, ((ino2 => mkdir(0, md, 1, 0, ∅) * (λ dirs1. dirs3 = dirs1)))(dirs1), root[q].dir?, root.dir?, dom of < ino0, abs(root, ROOT_INO)(dirs0), afs-cons(dirs, files), afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files), of-cons(of, dirs, files), of-cons(of, dirs0, files), pw(user, md1), px(user, md1), px-parent(user, q, ROOT_INO, dirs0, files), ino1 ∈ dirs1, ino1 ∈ dirs0, ino2 ∈ dirs4, dirs ≡ dirs1[ino1, dirs1[ino1].meta:= md1], dirs3 ≡ dirs0[ino1, mkdir(# entries0 + 1, md1, # links(ino1, dirs0), m1 + 1, entries0[str, ino2])], dirs4 ≡ dirs3 ++ ino2, dirs1 ≡ dirs4[ino2, mkdir(0, md, 1, 0, ∅)], psynced0 → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ ino2 ∈ dirs3, str ∈ entries0, ino2 ∈ files, ino2 ∈ dirs0, ∃ psynced. abs(root[q + str ', dnode(md, ∅)], ROOT_INO)(dirs) ∧ (psynced → asynced) ∧ ESUCCESS = err0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 181:
⊦ 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)))
Proof step 182:
dirs1[ino1] = mkdir(# entries0 + 1, md1, # links(ino1, dirs0), m1 + 1, entries0[str, ino2]), dirs0[ino1] = mkdir(# entries0, md1, # links(ino1, dirs0), m1, entries0), root[q] = dnode(md1, sdt), ((ino2 => mkdir(0, md, 1, 0, ∅) * (λ dirs1. dirs3 = dirs1)))(dirs1), (( ino1 => mkdir(# entries0 + 1, md1, # links(ino1, dirs0), m1 + 1, entries0[str, ino2]) * ino2 => mkdir(0, md, 1, 0, ∅) * abs|(root, ROOT_INO, q, ino1) * abs(sdt, entries0)) )(dirs), root.dir?, dom of < ino0, afs-cons(dirs, files), afs-cons(dirs0, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files), of-cons(of, dirs, files), of-cons(of, dirs0, files), pw(user, md1), px(user, md1), px-parent(user, q, ROOT_INO, dirs0, files), q ∈ root, ino1 ∈ dirs0, ino2 ∈ dirs4, ino1 ∈ dirs1, dirs3 ≡ dirs0[ino1, mkdir(# entries0 + 1, md1, # links(ino1, dirs0), m1 + 1, entries0[str, ino2])], dirs4 ≡ dirs3 ++ ino2, dirs1 ≡ dirs4[ino2, mkdir(0, md, 1, 0, ∅)], dirs ≡ dirs1, psynced0 → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ abs(root[q + str ', dnode(md, ∅)], ROOT_INO)(dirs), ino2 ∈ dirs3, str ∈ entries0, ino2 ∈ files, ino2 ∈ dirs0

Rule argument

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)));% ⊦ dnode(md, sdt).dir?; (s)%ino ∈ dirs ⊦ dirs[ino] = ad → dirs[ino, ad] = dirs; (s)%⊦ mkdir(m, md, m0, m1, entries).meta:= md0 = mkdir(m, md0, m0, m1, entries); (s)%⊦ dirs ≡ dirs0[ino, ad] → (((ino =>ad * D))(dirs0) ↔ ((ino => ad * D))(dirs) ∧ ino ∈ dirs0); (ws)% ⊦ ((ino =>ad * D))(dirs) → ino ∈ dirs; (s)%ino ≠ 0, ¬ ino ∈ dirs ⊦ dirs0 ≡ dirs ++ ino → (((D * D0))(dirs) ↔ ((ino =>ad * D * D0))(dirs0)); (s)% ⊦ ino ∈ dirs → ino ≠ 0; (s)%⊦ dirs ≡ dirs0[ino, ad] → ( ((ino => ad0 * D))(dirs0) ↔ ((ino => ad * D))(dirs) ∧ dirs0[ino] = ad0 ∧ ino ∈ dirs0); (ws)% ⊦ ((ino => ad * D))(dirs) → ino ∈ dirs; (s)% ⊦ ((ino => ad * D))(dirs) → (dirs[ino] = ad0 ↔ ad = ad0); (s)%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]