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Proof step 182 from proof refine-mkdir (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 183:
D ⊆ abs(tree0, ino1) ⊦ ¬ str ∈ entries ∧ p ∈ tree ∧ tree[p] = dnode(md, sdt) ∧ ((abs|(tree, ROOT_INO, p, ino0) * ino0 => mkdir(# entries + 1, md, m0, m1, entries[str, ino1]) * abs(sdt, entries) * D))(dirs) → abs(tree[p + str ', tree0], ROOT_INO)(dirs)
Proof step 184:
root[q] = dnode(md1, sdt), dirs0[ino1] = mkdir(# entries0, md1, # links(ino1, dirs0), m1, entries0), dirs1[ino1] = mkdir(# entries0 + 1, md1, # links(ino1, dirs0), m1 + 1, entries0[str, ino2]), (( 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), ((ino2 => mkdir(0, md, 1, 0, ∅) * (λ dirs1. dirs3 = dirs1)))(dirs1), root.dir?, dom of < ino0, 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), ino1 ∈ dirs1, ino2 ∈ dirs4, ino1 ∈ dirs0, q ∈ root, dirs ≡ dirs1, dirs1 ≡ 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])], psynced0 → asynced0, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ abs(root[q + str ', dnode(md, ∅)], ROOT_INO)(dirs), ino2 ∈ dirs0, ino2 ∈ files, str ∈ entries0, ino2 ∈ dirs3, ino2 => mkdir(0, md, 1, 0, ∅) ⊆ abs(dnode(md, ∅), ino2) ∧ ¬ str ∈ entries0 ∧ q ∈ root ∧ root[q] = dnode(md1, sdt) ∧ (( abs|(root, ROOT_INO, q, ino1) * ino1 => mkdir(# entries0 + 1, md1, # links(ino1, dirs0), m1 + 1, entries0[str, ino2]) * abs(sdt, entries0) * ino2 => mkdir(0, md, 1, 0, ∅)) )(dirs)

Rule argument

spec: <toplevel> name: abs-fold-new-p seq: D ⊆ abs(tree0, ino1) ⊦ ¬ str ∈ entries ∧ p ∈ tree ∧ tree[p] = dnode(md, sdt) ∧ (( abs|(tree, ROOT_INO, p, ino0) * ino0 => mkdir (# entries + 1, md, m0, m1, entries[str, ino1]) * abs(sdt, entries) * D) )(dirs) → abs(tree[p + str ', tree0], ROOT_INO)(dirs) subst: [D, ino1, entries, md, sdt, ino0, m0, m1, dirs, tree0, str, p, tree] → [ino2 => mkdir(0, md, 1, 0, ∅), ino2, entries0, md1, sdt, ino1, # links(ino1, dirs0), m1 + 1, dirs, dnode(md, ∅), str, q, root] flags: <all paths>

Additional information

Used terms: ino2 => mkdir(0, md, 1, 0, ∅), ino2, entries0, md1, sdt, ino1, # links(ino1, dirs0), m1 + 1, dirs, dnode(md, ∅), str, q, root