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Proof step 49 from proof rename-fids (previous step)

fid ∈ fs ↔ fid ∈ root0.fids ∨ fid ∈ ofh.fids, root0[path].dir? ↔ root0[path'].dir?, root0[parent(path')].dir?, fids-cons(root0, fs, ofh), py(user, parent(path), root0, fs), py(user, parent(path'), root0, fs), path' ∈ root0, path ∈ root0, parent(path') ∈ root0, root0[path'].dir? → pw(user, path, root0, fs), root0[path'].dir? → root0[path'].entries = ∅ ⊦ fid ∈ fs ↔ fid ∈ (root0 -- path)[path', root0[path]].fids ∨ fid ∈ ofh.fids, path' = ε, path ⊑ path', ¬ 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 50:
p ≠ ε ⊦ p ∈ tree0 ∧ tree0[parent(p)].dir? → tree0[p, tree1].fids = (tree0.fids \ tree0[p].fids) ∪ tree1.fids
Proof step 51:
root0[path].dir? ↔ root0[path'].dir?, fid ∈ fs ↔ fid ∈ root0.fids ∨ fid ∈ ofh.fids, root0[parent(path')].dir?, fids-cons(root0, fs, ofh), py(user, parent(path'), root0, fs), py(user, parent(path), root0, fs), parent(path') ∈ root0, path ∈ root0, path' ∈ root0, root0[path'].dir? → root0[path'].entries = ∅, root0[path'].dir? → pw(user, path, root0, fs) ⊦ fid ∈ fs ↔ fid ∈ (root0 -- path)[path', root0[path]].fids ∨ fid ∈ ofh.fids, path' = ε, path ⊑ path', ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ (root0 -- path)[path', root0[path]].fids, path' ≠ ε ∧ path' ∈ root0 -- path ∧ (root0 -- path)[parent(path')].dir?
Proof step 57:
root0[path].dir? ↔ root0[path'].dir?, fid ∈ fs ↔ fid ∈ root0.fids ∨ fid ∈ ofh.fids, root0[parent(path')].dir?, (root0 -- path)[parent(path')].dir?, fids-cons(root0, fs, ofh), py(user, parent(path'), root0, fs), py(user, parent(path), root0, fs), parent(path') ∈ root0, path ∈ root0, path' ∈ root0, path' ∈ root0 -- path, root0[path'].dir? → root0[path'].entries = ∅, root0[path'].dir? → pw(user, path, root0, fs) ⊦ fid ∈ fs ↔ fid ∈ ((root0 -- path).fids \ (root0 -- path)[path'].fids) ∪ root0[path].fids ∨ fid ∈ ofh.fids, path' = ε, path ⊑ path', ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ ((root0 -- path).fids \ (root0 -- path)[path'].fids) ∪ root0[path].fids

Rule argument

spec: directory-tree-fids name: fids-put1 seq: p ≠ ε ⊦ p ∈ tree0 ∧ tree0[parent(p)].dir? → tree0[p, tree1].fids = (tree0.fids \ tree0[p].fids) ∪ tree1.fids subst: [tree1, p, tree0] → [root0[path], path', root0 -- path] flags: <all paths>

Additional information

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