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Proof step 22 from proof abs-nlinks-h (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 23:
⊦ p ∈ tree → abs(tree, ino0) = _∃(λ ino1. abs|(tree, ino0, p, ino1) * abs(tree[p], ino1))
Proof step 24:
dirs[m0] = mkdir(m, md, m1, m2, entries), tree[p0 + str '] = fnode(⌜ino1⌝ ), links(ino1, dirs) = ∅, (_∃(λ ino1. abs|(tree, ino0, p0, ino1) * abs(tree[p0], ino1)))(dirs), ispath(p0, ino0, m0, dirs, files), str ∈ entries, m0 ∈ dirs, ino0 ∈ dirs, str ' ∈ tree[p0], ino1 ∈ files, p0 ∈ tree, ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs, entries[str] ∈ dirs ∨ entries[str] ∈ files ⊦

Rule argument

spec: <toplevel> name: path-unfold seq: ⊦ p ∈ tree → abs(tree, ino0) = _∃(λ ino1. abs|(tree, ino0, p, ino1) * abs(tree[p], ino1)) subst: [p, ino0, tree] → [p0, ino0, tree] flags: <all paths>

Additional information

Used terms: p0, ino0, tree