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Proof step 184 from proof inv-link (previous step)

(key.ino = m11 ⊃ mkdir(m1 + 1, md0, # links(key.ino, dirs0), m9, entries3[str0, entries2[str]]);dirs0[key.ino]) = mkdir(m, md7, m3, m13, entries4), dirs0[ino3] = mkdir(m7, md4, # links(ino3, dirs0), m12, entries2), entries1[str] = entries2[str], dirs0[ino2] = mkdir(m4, md3, # links(ino2, dirs0), m6, entries0), dirs0[m11] = mkdir(m1, md5, # links(m11, dirs0), m9, entries3), files0[entries2[str]] = mkfile(m0, md6, # links(entries2[str], dirs0), content0), entries0[str] = entries2[str], dirs0[ino0] = mkdir(m2, md1, # links(ino0, dirs0), m8, entries), entries[str] = entries2[str], dirs0[ino1] = mkdir(m5, md2, # links(ino1, dirs0), m10, entries1), afs-cons (dirs0[m11, mkdir(m1 + 1, md0, # links(m11, dirs0), m9, entries3[str0, entries2[str]])], files0[entries2[str], mkfile(m15, md, # links(entries2[str], dirs0) + 1, content0)]), afs-cons(dirs0, files0), synced(dirs0, icache0, pcache), key ∈ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])], entries2[str] ∈ files0, m11 ∈ dirs0, str ∈ entries2, ino3 ∈ dirs0, str ∈ entries0, ino2 ∈ dirs0, ino0 ∈ dirs0, str ∈ entries, ino1 ∈ dirs0, str ∈ entries1, asynced1 → synced(files0, icache0, pcache), key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs0 ∧ (links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs0 ∨ dcache[key].ino ∈ files0) ∧ dcache[key].name ∈ dirs0[key.ino].entries ∧ dcache[key].ino = dirs0[key.ino].entries[dcache[key].name] ∧ dcache[key].dentry?, ∀ key0. key0 ∈ dcache → key0.dentry? ∧ key0.name = dcache[key0].name ∧ key0.ino ∈ dirs0 ∧ (links(key0.ino, dirs0) ≠ ∅ ∨ key0.ino = ROOT_INO) ∧ (dcache[key0].ino ∈ dirs0 ∨ dcache[key0].ino ∈ files0) ∧ dcache[key0].name ∈ dirs0[key0.ino].entries ∧ dcache[key0].ino = dirs0[key0.ino].entries[dcache[key0].name] ∧ dcache[key0].dentry?, links(ino3, dirs0) ≠ ∅ ∨ ino3 = ROOT_INO, links(m11, dirs0) ≠ ∅ ∨ m11 = ROOT_INO, links(ino2, dirs0) ≠ ∅ ∨ ino2 = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO ⊦ str0 ∈ entries3, key.dentry? ∧ key.name = dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].name ∧ key.ino ∈ dirs0 ∧ ((entries2[str] = key.ino ⊃ links(key.ino, dirs0) ++ mklink(m11, str0);links(key.ino, dirs0)) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].ino ∈ dirs0 ∨ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].ino ∈ files0) ∧ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].name ∈ entries4 ∧ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].ino = entries4[dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].name] ∧ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].dentry?

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 185:
(key.ino = m11 ⊃ mkdir(m1 + 1, md0, # links(key.ino, dirs0), m9, entries3[str0, entries2[str]]);dirs0[key.ino]) = mkdir(m, md7, m3, m13, entries4), dirs0[ino1] = mkdir(m5, md2, # links(ino1, dirs0), m10, entries1), entries[str] = entries2[str], dirs0[ino0] = mkdir(m2, md1, # links(ino0, dirs0), m8, entries), entries0[str] = entries2[str], files0[entries2[str]] = mkfile(m0, md6, # links(entries2[str], dirs0), content0), dirs0[m11] = mkdir(m1, md5, # links(m11, dirs0), m9, entries3), dirs0[ino2] = mkdir(m4, md3, # links(ino2, dirs0), m6, entries0), entries1[str] = entries2[str], dirs0[ino3] = mkdir(m7, md4, # links(ino3, dirs0), m12, entries2), dirs0[key.ino] = mkdir(m14, md8, m16, m17, entries5), afs-cons(dirs0, files0), afs-cons (dirs0[m11, mkdir(m1 + 1, md0, # links(m11, dirs0), m9, entries3[str0, entries2[str]])], files0[entries2[str], mkfile(m15, md, # links(entries2[str], dirs0) + 1, content0)]), synced(dirs0, icache0, pcache), str ∈ entries1, ino1 ∈ dirs0, str ∈ entries, ino0 ∈ dirs0, ino2 ∈ dirs0, str ∈ entries0, ino3 ∈ dirs0, str ∈ entries2, m11 ∈ dirs0, entries2[str] ∈ files0, key ∈ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])], key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ key.ino ∈ dirs0 ∧ (links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache[key].ino ∈ dirs0 ∨ dcache[key].ino ∈ files0) ∧ dcache[key].name ∈ entries5 ∧ dcache[key].ino = entries5[dcache[key].name] ∧ dcache[key].dentry?, asynced1 → synced(files0, icache0, pcache), ∀ key0. key0 ∈ dcache → key0.dentry? ∧ key0.name = dcache[key0].name ∧ key0.ino ∈ dirs0 ∧ (links(key0.ino, dirs0) ≠ ∅ ∨ key0.ino = ROOT_INO) ∧ (dcache[key0].ino ∈ dirs0 ∨ dcache[key0].ino ∈ files0) ∧ dcache[key0].name ∈ dirs0[key0.ino].entries ∧ dcache[key0].ino = dirs0[key0.ino].entries[dcache[key0].name] ∧ dcache[key0].dentry?, links(ino1, dirs0) ≠ ∅ ∨ ino1 = ROOT_INO, links(ino0, dirs0) ≠ ∅ ∨ ino0 = ROOT_INO, links(ino2, dirs0) ≠ ∅ ∨ ino2 = ROOT_INO, links(m11, dirs0) ≠ ∅ ∨ m11 = ROOT_INO, links(ino3, dirs0) ≠ ∅ ∨ ino3 = ROOT_INO ⊦ str0 ∈ entries3, key.dentry? ∧ key.name = dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].name ∧ key.ino ∈ dirs0 ∧ ((entries2[str] = key.ino ⊃ links(key.ino, dirs0) ++ mklink(m11, str0);links(key.ino, dirs0)) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].ino ∈ dirs0 ∨ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].ino ∈ files0) ∧ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].name ∈ entries4 ∧ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].ino = entries4[dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][key].name] ∧ dcache[dentrykey(m11, str0), mkdentry(str0, entries2[str])][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] → [dirs0[key.ino], m14, md8, m16, m17, entries5]

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: dirs0[key.ino], m14, md8, m16, m17, entries5