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Proof step 8 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 = ∅, ((abs(tree[p, dnode(md, sdt -- str)], ROOT_INO) * D))(dirs)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 9:
⊦ p ∈ tree → abs(tree, ino0) = _∃(λ ino1. abs|(tree, ino0, p, ino1) * abs(tree[p], ino1))
Proof step 10:
# 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, p ∈ tree[p, dnode(md, sdt -- str)] ⊦ entries = ∅, ((_∃(λ ino1. abs|(tree[p, dnode(md, sdt -- str)], ROOT_INO, p, ino1) * abs(tree[p, dnode(md, sdt -- str)][p], ino1)) * D))(dirs)

Rule argument

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

Additional information

Used simplifier rules: ⊦ p ∈ tree → p ∈ tree[p, tree0]; (s) Used terms: p, ROOT_INO, tree[p, dnode(md, sdt -- str)]