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Proof step 52 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, (root0 -- path)[parent(path')].dir?, path ⊑ path', ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ (root0 -- path)[path', root0[path]].fids

User applied rule insert elim lemma

And obtained the following new goals

Proof step 53:
root0[path].dir? ↔ root0[path'].dir?, fid ∈ fs ↔ fid ∈ root0.fids ∨ fid ∈ ofh.fids, path' = p0 + str ', root0[p0].dir?, fids-cons(root0, fs, ofh), py(user, p0, root0, fs), py(user, parent(path), root0, fs), p0 ∈ 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, (root0 -- path)[p0].dir?, path ⊑ path', ¬ root0[path'].dir? ∧ ¬ root0[path'].fid ∈ ofh.fids ∧ ¬ root0[path'].fid ∈ (root0 -- path)[path', root0[path]].fids

Rule argument

spec: list-last, inst: path name: last-elim seq: p ≠ ε ⊦ str = p.last ∧ p0 = parent(p) ↔ p = p0 + str ' subst: [p, str, p0] → [path', str, p]

Additional information

Used simplifier rules: p ≠ ε ⊦ str = p.last ∧ p0 = parent(p) ↔ p = p0 + str '; (elim) Used terms: path', str, p