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Proof step 209 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), (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 ∈ dcache0 → key.dentry? ∧ key.name = dcache0[key].name ∧ key.ino ∈ dirs0 ∧ (links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache0[key].ino ∈ dirs0 ∨ dcache0[key].ino ∈ files) ∧ dcache0[key].name ∈ dirs0[key.ino].entries ∧ dcache0[key].ino = dirs0[key.ino].entries[dcache0[key].name] ∧ dcache0[key].dentry?, ∀ key0. key0 ∈ dcache0 → key0.dentry? ∧ key0.name = dcache0[key0].name ∧ key0.ino ∈ dirs0 ∧ (links(key0.ino, dirs0) ≠ ∅ ∨ key0.ino = ROOT_INO) ∧ (dcache0[key0].ino ∈ dirs0 ∨ dcache0[key0].ino ∈ files) ∧ dcache0[key0].name ∈ dirs0[key0.ino].entries ∧ dcache0[key0].ino = dirs0[key0.ino].entries[dcache0[key0].name] ∧ dcache0[key0].dentry?, 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?

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 210:
# 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), dirs0[key.ino] = mkdir(m18, md8, m19, m20, entries5), 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), 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])], key ∈ dcache0 → key.dentry? ∧ key.name = dcache0[key].name ∧ key.ino ∈ dirs0 ∧ (links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ (dcache0[key].ino ∈ dirs0 ∨ dcache0[key].ino ∈ files) ∧ dcache0[key].name ∈ entries5 ∧ dcache0[key].ino = entries5[dcache0[key].name] ∧ dcache0[key].dentry?, m = m8 → md5 = md0, asynced2 → synced(files, icache0, pcache), ∀ key0. key0 ∈ dcache0 → key0.dentry? ∧ key0.name = dcache0[key0].name ∧ key0.ino ∈ dirs0 ∧ (links(key0.ino, dirs0) ≠ ∅ ∨ key0.ino = ROOT_INO) ∧ (dcache0[key0].ino ∈ dirs0 ∨ dcache0[key0].ino ∈ files) ∧ dcache0[key0].name ∈ dirs0[key0.ino].entries ∧ dcache0[key0].ino = dirs0[key0.ino].entries[dcache0[key0].name] ∧ dcache0[key0].dentry?, 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?

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], m18, md8, m19, m20, 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], m18, md8, m19, m20, entries5