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Proof step 21 from proof rename-root (previous step)

root0[parent(path')].dir?, root0.dir?, py(user, parent(path'), root0, fs), py(user, parent(path), root0, fs), parent(path') ∈ root0, path ∈ root0, root0[path].dir? → pw(user, path, root0, fs), path' ∈ root0 ∧ root0[path'].dir? → root0[path'].entries = ∅, path' ∈ root0 → (root0[path].dir? ↔ root0[path'].dir?) ⊦ path' = ε, (root0 -- path)[path', root0[path]].dir?, path ⊑ path', path' ∈ root0 ∧ ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ (root0 -- path)[path', root0[path]].fids

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 22:
p ≠ ε ⊦ tree.dir? ∧ parent(p) ∈ tree → tree[p, tree0].dir?
Proof step 23:
root0.dir?, root0[parent(path')].dir?, py(user, parent(path), root0, fs), py(user, parent(path'), root0, fs), path ∈ root0, parent(path') ∈ root0, path' ∈ root0 → (root0[path].dir? ↔ root0[path'].dir?), path' ∈ root0 ∧ root0[path'].dir? → root0[path'].entries = ∅, root0[path].dir? → pw(user, path, root0, fs) ⊦ path' = ε, (root0 -- path)[path', root0[path]].dir?, path ⊑ path', path' ∈ root0 ∧ ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ (root0 -- path)[path', root0[path]].fids, path' ≠ ε ∧ (root0 -- path).dir? ∧ parent(path') ∈ root0 -- path

Rule argument

spec: directory-tree name: dir-put3 seq: p ≠ ε ⊦ tree.dir? ∧ parent(p) ∈ tree → tree[p, tree0].dir? subst: [tree0, p, tree] → [root0[path], path', root0 -- path] flags: <all paths>

Additional information

Used terms: root0[path], path', root0 -- path