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Proof step 38 from proof inv-lookup (previous step)

dirs[p_ino] = mkdir(m, md, # links(p_ino, dirs), m1, entries), afs-cons(dirs, files), dcache-cons(dcache, dirs, files), synced(dirs, icache, pcache), key ∈ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])], p_ino ∈ dirs, dent0.name ∈ entries, asynced1 → synced(files, icache, pcache), links(p_ino, dirs) ≠ ∅ ∨ p_ino = ROOT_INO ⊦ dentrykey(p_ino, dent0.name) ∈ dcache, key.dentry? ∧ key.name = dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].name ∧ key.ino ∈ dirs ∧ (links(key.ino, dirs) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].ino ∈ dirs ∨ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].ino ∈ files) ∧ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].name ∈ dirs[key.ino].entries ∧ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].ino = dirs[key.ino].entries[dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].name] ∧ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].dentry?

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 39:
dirs[p_ino] = mkdir(m, md, # links(p_ino, dirs), m1, entries), dirs[key.ino] = mkdir(m0, md0, m2, m3, entries0), afs-cons(dirs, files), dcache-cons(dcache, dirs, files), synced(dirs, icache, pcache), dent0.name ∈ entries, p_ino ∈ dirs, key ∈ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])], asynced1 → synced(files, icache, pcache), links(p_ino, dirs) ≠ ∅ ∨ p_ino = ROOT_INO ⊦ dentrykey(p_ino, dent0.name) ∈ dcache, key.dentry? ∧ key.name = dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].name ∧ key.ino ∈ dirs ∧ (links(key.ino, dirs) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].ino ∈ dirs ∨ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].ino ∈ files) ∧ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].name ∈ entries0 ∧ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].ino = entries0[dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].name] ∧ dcache[dentrykey(p_ino, dent0.name), mkdentry(dent0.name, entries[dent0.name])][key].dentry?

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[key.ino], m0, md0, m2, m3, entries0]

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[key.ino], m0, md0, m2, m3, entries0