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Proof step 6 from proof path-fid (previous step)

((abs(tree, ino0) * D))(dirs), ispath(p0, dirs[ino0].entries[str], ino1, dirs, files), str ' ∈ tree, p0 ∈ tree[str '], ino1 ∈ files, ino0 ∈ dirs, str ∈ dirs[ino0].entries, ∀ tree, ino0, D, dirs, ino1, files. ((abs(tree, ino0) * D))(dirs) ∧ ispath(p0, ino0, ino1, dirs, files) ∧ p0 ∈ tree ∧ ino1 ∈ files → tree[p0] = fnode(⌜ino1⌝ ) ∨ ino1 ∈ dirs ⊦ tree[str ' + p0] = fnode(⌜ino1⌝ ), ino1 ∈ dirs

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 7:
dirs[ino0] = mkdir(m, md, m0, m1, entries), ((abs(tree, ino0) * D))(dirs), ispath(p0, entries[str], ino1, dirs, files), str ∈ entries, ino0 ∈ dirs, ino1 ∈ files, p0 ∈ tree[str '], str ' ∈ tree, ∀ tree, ino0, D, dirs, ino1, files. ((abs(tree, ino0) * D))(dirs) ∧ ispath(p0, ino0, ino1, dirs, files) ∧ p0 ∈ tree ∧ ino1 ∈ files → tree[p0] = fnode(⌜ino1⌝ ) ∨ ino1 ∈ dirs ⊦ tree[str ' + p0] = fnode(⌜ino1⌝ ), ino1 ∈ dirs

Rule argument

spec: abstract-dir name: elim seq: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries) subst: [ad, m, md, m0, m1, entries] → [dirs[ino0], m, md, m0, m1, entries]

Additional information

Used simplifier rules: ⊦ ( ((m = ad.size ∧ md = ad.meta) ∧ m0 = ad.nlink) ∧ m1 = ad.nsubdirs) ∧ entries = ad.entries ↔ ad = mkdir(m, md, m0, m1, entries); (elim) Used terms: dirs[ino0], m, md, m0, m1, entries