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Proof step 205 from proof inv-rename-file-negdentry (previous step)

# links (entries[str], dirs0 [m8, (m8 = m ⊃ mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str)).meta:= md0 ] [m, mkdir(m12, md5, m13, m14, entries3) ]) = # links(entries[str], dirs0), (m = m8 ⊃ mkdir(m1 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]])) = mkdir(m12, md6, m13, m14, entries3), (m8 = m ⊃ mkdir(m1 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md0, # links(m8, dirs0), m5, entries -- str)) = mkdir(m0, md1, m10, m11, entries0), (m = m8 ⊃ mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str);mkdir(m7, md4, # links(m, dirs0), m9, entries2)) = mkdir(m1, md2, m4, m6, entries1), (key.ino = m ⊃ mkdir(m12, md5, m13, m14, entries3);(key.ino = m8 ⊃ mkdir(m2, md0, # links(key.ino, dirs0), m5, entries -- str);dirs0[key.ino])) = mkdir(m15, md7, m16, m17, entries4), dirs0[m] = mkdir(m7, md4, # links(m, dirs0), m9, entries2), files[entries[str]] = mkfile(m3, md, # links(entries[str], dirs0), content0), dirs0[m8] = mkdir(m2 + 1, md3, # links(m8, dirs0), m5, entries), afs-cons (dirs0[m8, (m8 = m ⊃ mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str)).meta:= md0] [m, mkdir(m12, md5, m13, m14, entries3) ], files), afs-cons(dirs0, files), dcache-cons(dcache0, dirs0, files), synced(dirs0, icache0, pcache), m8 ∈ dirs0, m ∈ dirs0, entries[str] ∈ files, str ∈ entries, key ∈ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])], m = m8 → md5 = md0, asynced2 → synced(files, icache0, pcache), links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m8, dirs0) ≠ ∅ ∨ m8 = ROOT_INO ⊦ str0 ∈ entries2, entries[str] ∈ dirs0, key.dentry? ∧ key.name = (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].name ∧ key.ino ∈ dirs0 ∧ ( links (key.ino, dirs0 [m8, (m8 = m ⊃ mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str)).meta:= md0 ] [m, mkdir(m12, md5, m13, m14, entries3) ]) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].ino ∈ dirs0 ∨ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].ino ∈ files) ∧ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].name ∈ entries4 ∧ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].ino = entries4[(dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].name] ∧ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].dentry?

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 206:
⊦ dcache-cons(dcache, dirs, files) ↔ (∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files))
Proof step 207:
# links (entries[str], dirs0 [m8, (m8 = m ⊃ mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str)).meta:= md0 ] [m, mkdir(m12, md5, m13, m14, entries3) ]) = # links(entries[str], dirs0), (key.ino = m ⊃ mkdir(m12, md5, m13, m14, entries3);(key.ino = m8 ⊃ mkdir(m2, md0, # links(key.ino, dirs0), m5, entries -- str);dirs0[key.ino])) = mkdir(m15, md7, m16, m17, entries4), (m = m8 ⊃ mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str);mkdir(m7, md4, # links(m, dirs0), m9, entries2)) = mkdir(m1, md2, m4, m6, entries1), (m8 = m ⊃ mkdir(m1 + 1, md5, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md0, # links(m8, dirs0), m5, entries -- str)) = mkdir(m0, md1, m10, m11, entries0), (m = m8 ⊃ mkdir(m1 + 1, md0, m4, m6, entries1[str0, entries[str]]);mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]])) = mkdir(m12, md6, m13, m14, entries3), dirs0[m8] = mkdir(m2 + 1, md3, # links(m8, dirs0), m5, entries), files[entries[str]] = mkfile(m3, md, # links(entries[str], dirs0), content0), dirs0[m] = mkdir(m7, md4, # links(m, dirs0), m9, entries2), afs-cons(dirs0, files), afs-cons (dirs0[m8, (m8 = m ⊃ mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str)).meta:= md0] [m, mkdir(m12, md5, m13, m14, entries3) ], files), synced(dirs0, icache0, pcache), key ∈ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])], str ∈ entries, entries[str] ∈ files, m ∈ dirs0, m8 ∈ dirs0, asynced2 → synced(files, icache0, pcache), m = m8 → md5 = md0, ∀ key. key ∈ dcache0 → key.dentry? ∧ key.name = dcache0[key].name ∧ valid-dentry(key.ino, dcache0[key], dirs0, files), links(m8, dirs0) ≠ ∅ ∨ m8 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ entries[str] ∈ dirs0, str0 ∈ entries2, key.dentry? ∧ key.name = (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].name ∧ key.ino ∈ dirs0 ∧ ( links (key.ino, dirs0 [m8, (m8 = m ⊃ mkdir(m1 + 1, md2, m4, m6, entries1[str0, entries[str]]);mkdir(m2, md3, # links(m8, dirs0), m5, entries -- str)).meta:= md0 ] [m, mkdir(m12, md5, m13, m14, entries3) ]) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].ino ∈ dirs0 ∨ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].ino ∈ files) ∧ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].name ∈ entries4 ∧ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].ino = entries4[(dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].name] ∧ (dcache0 -- dentrykey(m8, str))[dentrykey(m, str0), mkdentry(str0, entries[str])][key].dentry?

Rule argument

spec: cache-base name: dcache-cons seq: ⊦ dcache-cons(dcache, dirs, files) ↔ (∀ key. key ∈ dcache → key.dentry? ∧ key.name = dcache[key].name ∧ valid-dentry(key.ino, dcache[key], dirs, files)) subst: [dcache, dirs, files] → [dcache0, dirs0, files] flags: <all paths>

Additional information

Used terms: dcache0, dirs0, files