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Proof step 17 from proof abs-fold-del-p (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 18:
⊦ p ≠ ε ∧ q ⊑ p ∧ parent(p) ∈ tree ∧ tree[parent(p)].dir? → abs|(tree[p, tree0], ino0, q, ino1) = abs|(tree, ino0, q, ino1)
Proof step 19:
# entries = m + 1, tree[p] = dnode(md, sdt), ((abs|(tree, ROOT_INO, p, ino0) * ino0 => mkdir(m, md, m0, m1, entries -- str) * abs(sdt -- str, entries -- str) * D))(dirs), p ∈ tree, str ∈ entries ⊦ entries = ∅, (((ino0 => mkdir(m, md, m0, m1, entries -- str) * abs(sdt -- str, entries -- str) * abs|(tree[p, dnode(md, sdt -- str)], ROOT_INO, p, ino0)) * D) )(dirs), p ≠ ε ∧ p ⊑ p ∧ parent(p) ∈ tree ∧ tree[parent(p)].dir?

Rule argument

spec: vfs-abs name: absc-drop seq: ⊦ p ≠ ε ∧ q ⊑ p ∧ parent(p) ∈ tree ∧ tree[parent(p)].dir? → abs|(tree[p, tree0], ino0, q, ino1) = abs|(tree, ino0, q, ino1) subst: [ino1, q, ino0, tree0, p, tree] → [ino0, p, ROOT_INO, dnode(md, sdt -- str), p, tree] 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] Used terms: ino0, p, ROOT_INO, dnode(md, sdt -- str), p, tree