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Proof step 215 from proof refine-readmeta (previous step)

dirs[m1] = mkdir(m5, md0, m6, m7, entries0), files[m] = mkfile(m0, md1, # links(m, dirs), content), dirs[lookup(parent(p), ROOT_INO, dirs, files)] = mkdir(m2, md, m3, m4, entries), root[q] = fnode(⌜m⌝ ), root[q0 + str '] = fnode(⌜m⌝ ), q = q1 + str ', root.dir?, root[q0].dir?, dom of < ino6, abs(root, ROOT_INO)(dirs), afs-cons(dirs, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q0, ROOT_INO, lookup(q1, ROOT_INO, dirs, files), dirs, files), ispath(path, ROOT_INO, m, dirs, files), ispath(q, ROOT_INO, m8, dirs, files), ispath(q0 + str ', ROOT_INO, m9, dirs, files), of-cons(of, dirs, files), pr(user, md1), px-parent(user, path, ROOT_INO, dirs, files), ROOT_INO ∈ dirs, m ∈ files, str ' ∈ root[q0], lookup(q1, ROOT_INO, dirs, files) ∈ dirs, psynced → asynced ⊦ # links(m, dirs) = count(⌜m⌝ , root.fids), # links(⌜m⌝ , root) = 0, q0 + str ' = q, links(⌜m⌝ , root) = ∅, q = ε, root[path].dir?, m ∈ dirs, m8 ∈ dirs, m9 ∈ dirs

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 216:
⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → p ∈ tree ∧ (tree[p].dir? ↔ m ∈ dirs)
Proof step 217:
⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files))
Proof step 218:
⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p + str ' ∈ tree ↔ m ∈ dirs ∧ str ∈ dirs[m].entries)
Proof step 219:
⊦ ispath(p, ino, dirs, files) ↔ (∃ m. ispath(p, ino, m, dirs, files))
Proof step 220:
root[q0 + str '] = fnode(⌜m⌝ ), root[q1 + str '] = fnode(⌜m⌝ ), dirs[lookup(parent(p), ROOT_INO, dirs, files)] = mkdir(m2, md, m3, m4, entries), files[m] = mkfile(m0, md1, # links(m, dirs), content), dirs[m1] = mkdir(m5, md0, m6, m7, entries0), root[q0].dir?, root.dir?, root[q1].dir?, dom of < ino6, abs(root, ROOT_INO)(dirs), afs-cons(dirs, files), fids-cons(root, abs-files(files), abs-of(of)), ispath(q0 + str ', ROOT_INO, m9, dirs, files), ispath(q1 + str ', ROOT_INO, m8, dirs, files), ispath(path, ROOT_INO, m, dirs, files), ispath(q0, ROOT_INO, m10, dirs, files), ispath(q1, ROOT_INO, m10, dirs, files), of-cons(of, dirs, files), pr(user, md1), px-parent(user, path, ROOT_INO, dirs, files), m10 ∈ dirs, str ' ∈ root[q0], m ∈ files, ROOT_INO ∈ dirs, str ∈ dirs[m10].entries, str ' ∈ root[q1], psynced → asynced ⊦ # links(⌜m⌝ , root) = 0, # links(m, dirs) = count(⌜m⌝ , root.fids), links(⌜m⌝ , root) = ∅, q0 = q1, root[path].dir?, m9 ∈ dirs, m8 ∈ dirs, m ∈ dirs

Rule argument

Additional information

Used simplifier rules: path-in: ⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → p ∈ tree ∧ (tree[p].dir? ↔ m ∈ dirs);%path-in-eq: ⊦ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p ∈ tree ↔ ispath(p, ino, dirs, files));%path-in-last: ⊦ ispath(p, ino, m, dirs, files) ∧ abs(tree, ino)(dirs) ∧ fids-cons(tree, abs-files(files), ofh) ∧ afs-cons(dirs, files) → (p + str ' ∈ tree ↔ m ∈ dirs ∧ str ∈ dirs[m].entries);%ispath-target: ⊦ ispath(p, ino, dirs, files) ↔ (∃ m. ispath(p, ino, m, dirs, files));% ⊦ ¬ fnode(fid).dir?; (s)% ⊦ ispath(p, n, m, dirs, files) → lookup(p, n, dirs, files) = m; (s)% ⊦ p + q ∈ tree ↔ p ∈ tree ∧ q ∈ tree[p]; (s)% ⊦ ispath(p, n, m, dirs, files) → ispath(p, n, dirs, files); (s)% ⊦ p + p0 = p1 + p0 ↔ p = p1; (s)%a: ⊦ (p + p0) + p1 = p + p0 + p1; from specification list[path]% ⊦ p + str ' ≠ ε; (s)