[Imprint][Data Privacy Statement][E-Mail]

View long theorem info. View the proof. Back to main index. View the specification. View theorem base summary, View theorem base listing, the used properties by spec, the dependency list, the symbol table,

Proof step 15 from proof abs-fold-new-p (previous step)

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), p ∈ tree, D ⊆ abs(tree0, ino1) ⊦ (( (ino0 => mkdir(# entries + 1, md, m0, m1, entries[str, ino1]) * abs(sdt[str, tree0], entries[str, ino1])) * abs|(tree[p, dnode(md, sdt[str, tree0])], ROOT_INO, p, ino0)) )(dirs), str ∈ entries

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ ¬ name ∈ sdt → abs(sdt[name, tree], entries) = (⌜name ∈ entries⌝ ∧ abs(tree, entries[name]) * abs(sdt, entries -- name))
Proof step 17:
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), p ∈ tree, D ⊆ abs(tree0, ino1) ⊦ (( ( ino0 => mkdir(# entries + 1, md, m0, m1, entries[str, ino1]) * (⌜str ∈ entries[str, ino1]⌝ ∧ abs(tree0, entries[str, ino1][str]) * abs(sdt, entries[str, ino1] -- str))) * abs|(tree[p, dnode(md, sdt[str, tree0])], ROOT_INO, p, ino0)) )(dirs), str ∈ entries, str ∈ sdt

Rule argument

spec: vfs-abs name: abs-rec seq: ⊦ ¬ name ∈ sdt → abs(sdt[name, tree], entries) = ( ⌜name ∈ entries⌝ ∧ abs(tree, entries[name]) * abs(sdt, entries -- name)) subst: [entries, tree, name, sdt] → [entries[str, ino1], tree0, str, sdt] flags: <all paths>

Additional information

Used simplifier rules: 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]%entries-dom2: ⊦ str ∈ sdt ∧ ((abs(sdt, sdi) * D))(dirs) → str ∈ sdi; (forward) from specification vfs-abs Used terms: entries[str, ino1], tree0, str, sdt