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Proof step 57 from proof rename-fids (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 58:
fid ∈ fs ↔ fid ∈ root0.fids ∨ fid ∈ ofh.fids, root0[path].dir? ↔ root0[path'].dir?, (root0 -- path)[parent(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.fids \ root0[path].fids) \ (path' ⊑ path ⊃ root0[path'] -- restn(# path', path);root0[path']).fids) ∪ root0[path].fids ∨ fid ∈ ofh.fids, path' = ε, path ⊑ path', ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ ((root0.fids \ root0[path].fids) \ (path' ⊑ path ⊃ root0[path'] -- restn(# path', path);root0[path']).fids) ∪ root0[path].fids

Rule argument

Additional information

Used simplifier rules: ⊦ p ≠ ε → (parent(p) = ε ↔ p = p.head '); (s)% ⊦ ¬ tree.dir? → (p ∈ tree ↔ p = ε); (s)% ⊦ p ⊑ ε ↔ p = ε; (s)%p ≠ ε, q ∈ tree -- p ⊦ p ∈ tree → (tree -- p)[q] = (q ⊑ p ⊃ tree[q] -- restn(# q, p);tree[q]); (s)%p ≠ ε ⊦ p ∈ tree → (q ∈ tree -- p ↔ ¬ p ⊑ q ∧ q ∈ tree); (s)%p ≠ ε ⊦ p ∈ tree → (tree -- p).fids = tree.fids \ tree[p].fids; (s)% ⊦ ε ⊑ p; (s)% ⊦ ε ∈ tree; (s)% ⊦ tree[ε] = tree; (s)