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Proof step 24 from proof rename-fids (previous step)

fid0 ∈ fs ↔ fid0 ∈ root0.fids ∨ fid0 ∈ ofh.fids, root0[path'] = fnode(fid), 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 ⊦ fid0 ≠ fid ∧ fid0 ∈ fs ↔ fid0 ∈ ((root0 -- path).fids \ (root0 -- path)[path'].fids) ∪ root0[path].fids ∨ fid0 ∈ ofh.fids, path' = ε, root0[path].dir?, fid ∈ ((root0 -- path).fids \ (root0 -- path)[path'].fids) ∪ root0[path].fids, fid ∈ ofh.fids, path ⊑ path'

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
fid0 ∈ fs ↔ fid0 ∈ root0.fids ∨ fid0 ∈ ofh.fids, root0[path'] = fnode(fid), (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 ⊦ fid0 ≠ fid ∧ fid0 ∈ fs ↔ fid0 ∈ ((root0.fids \ root0[path].fids) \ (path' ⊑ path ⊃ fnode(fid) -- restn(# path', path);fnode(fid)).fids) ∪ root0[path].fids ∨ fid0 ∈ ofh.fids, path' = ε, root0[path].dir?, fid ∈ ofh.fids, fid ∈ (root0.fids \ root0[path].fids) \ (path' ⊑ path ⊃ fnode(fid) -- restn(# path', path);fnode(fid)).fids, fid ∈ root0[path].fids, path ⊑ path'

Rule argument

Additional information

Used simplifier rules: union-com: ⊦ fidb ∪ fidb0 = fidb0 ∪ fidb; from specification bag-union[fidbag]%union-ass: ⊦ fidb ∪ fidb0 ∪ fidb1 = (fidb ∪ fidb0) ∪ fidb1; from specification bag-union[fidbag]% ⊦ fid ∈ fidb ∪ fidb0 ↔ fid ∈ fidb ∨ fid ∈ fidb0; (ws)% ⊦ p ≠ ε → (parent(p) = ε ↔ p = p.head '); (s)% ⊦ ¬ tree.dir? → (p ∈ tree ↔ p = ε); (s)% ⊦ p ⊑ ε ↔ p = ε; (s)% ⊦ tree[ε] = tree; (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)