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Proof step 326 from proof refine-link (previous step)

dirs0[ino2] = mkdir(# entries1, md, # links(ino2, dirs0), m3, entries1), files0[entries[str]] = mkfile(m0, md2, # links(entries[str], dirs0), content0), dirs0[ino1] = mkdir(# entries0, md0, # links(ino1, dirs0), m2, entries0), entries0[str] = entries[str], dirs0[ino3] = mkdir(# entries, md1, # links(ino3, dirs0), m4, entries), root.dir?, root[q].dir?, root[q0].dir?, dom of < ino0, abs(root, ROOT_INO)(dirs0), afs-cons(dirs, files0[entries[str], mkfile(m0, md2, # links(entries[str], dirs0) + 1, content0)]), afs-cons(dirs0, files0), fids-cons(root, abs-files(files0), abs-of(of)), ispath(q0, ROOT_INO, ino2, dirs0, files0), ispath(q, ROOT_INO, ino1, dirs0, files0), of-cons(of, dirs, files0[entries[str], mkfile(m0, md2, # links(entries[str], dirs0) + 1, content0)]), of-cons(of, dirs0, files0), pw(user, md), px(user, md0), px(user, md), px-parent(user, q0, ROOT_INO, dirs0, files0), px-parent(user, q, ROOT_INO, dirs0, files0), str ∈ entries0, entries[str] ∈ files0, ino1 ∈ dirs0, ino3 ∈ dirs0, str ∈ entries, ino2 ∈ dirs0, dirs ≡ dirs0[ino2, mkdir(# entries1 + 1, md, # links(ino2, dirs0), m3, entries1[str0, entries[str]])], psynced → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino2, dirs0) ≠ ∅ ∨ ino2 = ROOT_INO, links(ino3, dirs0) ≠ ∅ ∨ ino3 = ROOT_INO ⊦ root[q + str '].dir?, entries[str] ∈ dirs0, str0 ∈ entries1, q + str ' ∈ root ∧ ¬ root[q + str '].dir? ∧ ¬ ispath(q0 + str0 ', ROOT_INO, dirs0, files0) ∧ parent(q0 + str0 ') ∈ root ∧ root[parent(q0 + str0 ')].dir? ∧ px(user, parent(q + str '), root, abs-files(files0)) ∧ py(user, parent(q0 + str0 '), root, abs-files(files0))

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 327:
⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files))
Proof step 328:
dirs0[ino3] = mkdir(# entries, md1, # links(ino3, dirs0), m4, entries), entries0[str] = entries[str], dirs0[ino1] = mkdir(# entries0, md0, # links(ino1, dirs0), m2, entries0), files0[entries[str]] = mkfile(m0, md2, # links(entries[str], dirs0), content0), dirs0[ino2] = mkdir(# entries1, md, # links(ino2, dirs0), m3, entries1), root[q0].dir?, root[q].dir?, root.dir?, dom of < ino0, abs(root, ROOT_INO)(dirs0), afs-cons(dirs, files0[entries[str], mkfile(m0, md2, # links(entries[str], dirs0) + 1, content0)]), afs-cons(dirs0, files0), fids-cons(root, abs-files(files0), abs-of(of)), ispath(q, ROOT_INO, ino1, dirs0, files0), ispath(q0, ROOT_INO, ino2, dirs0, files0), of-cons(of, dirs0, files0), of-cons(of, dirs, files0[entries[str], mkfile(m0, md2, # links(entries[str], dirs0) + 1, content0)]), pw(user, md), px(user, md), px(user, md0), px-parent(user, q, ROOT_INO, dirs0, files0), px-parent(user, q0, ROOT_INO, dirs0, files0), ino2 ∈ dirs0, str ∈ entries, ino3 ∈ dirs0, ino1 ∈ dirs0, entries[str] ∈ files0, str ∈ entries0, dirs ≡ dirs0[ino2, mkdir(# entries1 + 1, md, # links(ino2, dirs0), m3, entries1[str0, entries[str]])], psynced → asynced0, links(ino3, dirs0) ≠ ∅ ∨ ino3 = ROOT_INO, links(ino2, dirs0) ≠ ∅ ∨ ino2 = ROOT_INO, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ root[q + str '].dir?, str0 ∈ entries1, entries[str] ∈ dirs0, ispath(q + str ', ROOT_INO, dirs0, files0) ∧ ¬ root[q + str '].dir? ∧ ¬ ispath(q0 + str0 ', ROOT_INO, dirs0, files0) ∧ parent(q0 + str0 ') ∈ root ∧ root[parent(q0 + str0 ')].dir? ∧ px(user, parent(q + str '), root, abs-files(files0)) ∧ py(user, parent(q0 + str0 '), root, abs-files(files0))

Rule argument

spec: <toplevel> name: path-in-eq seq: ⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files)) subst: [ino, dirs, files, ofh, tree, p] → [ROOT_INO, dirs0, files0, abs-of(of), root, q + str '] flags: <all paths>

Additional information

Used terms: ROOT_INO, dirs0, files0, abs-of(of), root, q + str '