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Proof step 24 from proof abs-dir-h (previous step)

dirs[ino] = mkdir(m, md, m0, m1, entries), tree = fnode(fid), ((abs(tree, ino) * D))(dirs), ispath(p0, entries[str], ino0, dirs, files), ino ∈ dirs, str ∈ entries, ∀ tree, ino, D, dirs, ino0, files. ((abs(tree, ino) * D))(dirs) ∧ ispath(p0, ino, ino0, dirs, files) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (tree[p0].dir? ↔ ino0 ∈ dirs), ∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files, ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ tree[str ' + p0].dir? ↔ ino0 ∈ dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 25:
⊦ abs(fnode(fid), ino) = (emp ∧ ⌜ino = fid.ino⌝ )
Proof step 26:
dirs[fid.ino] = mkdir(m, md, m0, m1, entries), D(dirs), ispath(p0, entries[str], ino0, dirs, files), fid.ino ∈ dirs, str ∈ entries, ∀ ino1. ⌜ino1⌝ = fid → ino1 ∈ files, ∀ tree, ino, D, dirs, ino0, files. ((abs(tree, ino) * D))(dirs) ∧ ispath(p0, ino, ino0, dirs, files) ∧ (∀ ino. ⌜ino⌝ ∈ tree.fids → ino ∈ files) ∧ (∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs) → (tree[p0].dir? ↔ ino0 ∈ dirs), ∀ ino. ¬ ino ∈ files ∨ ¬ ino ∈ dirs ⊦ fnode(fid)[str ' + p0].dir? ↔ ino0 ∈ dirs

Rule argument

Additional information

Used simplifier rules: abs-fnode: ⊦ abs(fnode(fid), ino) = (emp ∧ ⌜ino = fid.ino⌝ );% ⊦ fid ∈ {fid0} ↔ fid = fid0; (s)% ⊦ fnode(fid).fids = {fid} ; (s)% ⊦ ⌜bv⌝ (dirs) ↔ bv; (s)% ⊦ ((D ∧ D0))(dirs) ↔ D(dirs) ∧ D0(dirs); (s)%and-com: ⊦ (D ∧ D0) = (D0 ∧ D); from specification heap-sep[dir-heap]%and-ass: ⊦ ((D ∧ D0) ∧ D1) = (D ∧ D0 ∧ D1); from specification heap-sep[dir-heap]%sep-com: ⊦ D * D0 = D0 * D; from specification heap-sep[dir-heap]%sep-ass: ⊦ (D * D0) * D1 = D * D0 * D1; from specification heap-sep[dir-heap]% ⊦ emp * D = D; (s)% ⊦ (⌜bv⌝ ∧ D) * D0 = (⌜bv⌝ ∧ D * D0); (s)