[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 28 from proof abs-paths-eq (previous step)

dirs[m] = mkdir(m0, md, m1, m2, entries), tree[p3 + str0 ' + p1].dir?, tree[p3 + str ' + p0].dir?, abs(tree, ino0)(dirs), ispath(p0, entries[str], ino1, dirs, files), ispath(p3, ino0, m, dirs, files), ispath(p3 + str0 ' + p1, ino0, ino1, dirs, files), str ∈ entries, m ∈ dirs, ino0 ∈ dirs, p1 ∈ tree[p3][str0 '], str0 ' ∈ tree[p3], p0 ∈ tree[p3][str '], str ' ∈ tree[p3], p3 ∈ tree, ino1 ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ str = str0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 29:
⊦ ispath(p + q, n, m, dirs, files) ↔ (∃ m0. ispath(p, n, m0, dirs, files) ∧ ispath(q, m0, m, dirs, files))
Proof step 30:
dirs[m] = mkdir(m0, md, m1, m2, entries), tree[p3 + str ' + p0].dir?, tree[p3 + str0 ' + p1].dir?, abs(tree, ino0)(dirs), ispath(p3, ino0, m, dirs, files), ispath(p0, entries[str], ino1, dirs, files), ino1 ∈ dirs, p3 ∈ tree, str ' ∈ tree[p3], p0 ∈ tree[p3][str '], str0 ' ∈ tree[p3], p1 ∈ tree[p3][str0 '], ino0 ∈ dirs, m ∈ dirs, str ∈ entries, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∃ m0. ispath(p3, ino0, m0, dirs, files) ∧ ispath(str0 ' + p1, m0, ino1, dirs, files) ⊦ str = str0

Rule argument

spec: afs-base name: path-split seq: ⊦ ispath(p + q, n, m, dirs, files) ↔ (∃ m0. ispath(p, n, m0, dirs, files) ∧ ispath(q, m0, m, dirs, files)) subst: [files, dirs, m, n, q, p] → [files, dirs, ino1, ino0, str0 ' + p1, p3] flags: <all paths>

Additional information

Used terms: files, dirs, ino1, ino0, str0 ' + p1, p3