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Proof step 201 from proof refine-mkdir (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

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, files, abs-of(of), root, q + str '] flags: <all paths>

Additional information

Used simplifier rules: path-in-eq: ⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files));%py: ⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → ( py(user, p, tree, abs-files(files)) ↔ py(user, p, ino, dirs, files));%py-strong: ⊦ ispath(q, ino, m, dirs, files) ∧ px-parent(user, q, ino, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → ( py(user, q, ino, dirs, files) ↔ pw(user, meta(ε, m, dirs, files)) ∧ px(user, meta(ε, m, dirs, files)));%⊦ ispath(p, n, m, dirs, files) → (ispath(p + q, n, dirs, files) ↔ ispath(q, m, dirs, files)); (s)%⊦ ispath(str ', n, dirs, files) ↔ n ∈ dirs ∧ str ∈ dirs[n].entries ∧ (dirs[n].entries[str] ∈ dirs ∨ dirs[n].entries[str] ∈ files); (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ parent(p + str ') = p; (s)% ⊦ ispath(p, n, m, dirs, files) → ispath(p, n, dirs, files); (s)% ⊦ mkdir(m, md, m0, m1, entries).entries = entries; (s)% ⊦ ino ∈ dirs → meta(ε, ino, dirs, files) = dirs[ino].meta; (s)% ⊦ mkdir(m, md, m0, m1, entries).meta = md; (s) Used terms: ROOT_INO, dirs0, files, abs-of(of), root, q + str '