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Proof step 211 from proof inv-rename-dir-negdentry (previous step)

(m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])) = mkdir(m13, md6, m14, m15, entries5), (m9 = m ⊃ mkdir(m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str)) = mkdir(m0, md1, m11, m12, entries1), (m = m9 ⊃ mkdir(m1, md, # links(m9, dirs0), m2, entries0 -- str);mkdir(m19, md8, # links(m, dirs0), m21, entries6)) = mkdir(m4, md3, m6, m7, entries2), dirs0[m9] = mkdir(m1 + 1, md, # links(m9, dirs0), m2 + 1, entries0), dirs0[m] = mkdir(m19, md8, # links(m, dirs0), m21, entries6), dirs0[entries0[str]] = mkdir(m3, md2, # links(entries0[str], dirs0), m5, entries), m = m9, afs-cons (dirs0 [m9, (m9 = m ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md, # links(m9, dirs0), m2, entries0 -- str)).meta:= md0 ] [m, mkdir(m13, md4, m14, m15, entries5) ], files), afs-cons(dirs0, files), synced(dirs0, icache0, pcache), m ∈ dirs0, m9 ∈ dirs0, str ∈ entries0, entries0[str] ∈ dirs0, dentrykey(m, str0) ∈ dcache0 → str0 = dcache0[dentrykey(m, str0)].name ∧ (dcache0[dentrykey(m, str0)].ino ∈ dirs0 ∨ dcache0[dentrykey(m, str0)].ino ∈ files) ∧ dcache0[dentrykey(m, str0)].name ∈ entries6 ∧ dcache0[dentrykey(m, str0)].ino = entries6[dcache0[dentrykey(m, str0)].name] ∧ dcache0[dentrykey(m, str0)].dentry?, asynced2 → synced(files, icache0, pcache), m = m9 → md4 = md0, ∀ 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(m9, dirs0) ≠ ∅ ∨ m9 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ str0 ∈ entries6, ( links (m, dirs0 [m9, (m9 = m ⊃ mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md, # links(m9, dirs0), m2, entries0 -- str)) .meta:= md0 ] [m, mkdir(m13, md4, m14, m15, entries5) ]) ≠ ∅ ∨ m = ROOT_INO) ∧ str0 ∈ entries5 ∧ entries0[str] = entries5[str0]

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 212:
(entries6[str] = m9 ⊃ (links(entries6[str], dirs0) -- mklink(entries6[str], str)) ++ mklink(entries6[str], str0);links(m9, dirs0)) = ∅, dirs0[m9] = mkdir(m4 + 1, md8, # links(m9, dirs0), m7 + 1, entries6), dirs0[entries6[str]] = mkdir(m3, md2, # links(entries6[str], dirs0), m5, entries), afs-cons(dirs0, files), afs-cons(dirs0[m9, mkdir(m4 + 1, md0, # links(m9, dirs0), m7 + 1, (entries6 -- str)[str0, entries6[str]])], files), synced(dirs0, icache0, pcache), entries6[str] ∈ dirs0, str ∈ entries6, m9 ∈ dirs0, asynced2 → synced(files, icache0, pcache), dentrykey(m9, str0) ∈ dcache0 → str0 = dcache0[dentrykey(m9, str0)].name ∧ (dcache0[dentrykey(m9, str0)].ino ∈ dirs0 ∨ dcache0[dentrykey(m9, str0)].ino ∈ files) ∧ dcache0[dentrykey(m9, str0)].name ∈ entries6 ∧ dcache0[dentrykey(m9, str0)].ino = entries6[dcache0[dentrykey(m9, str0)].name] ∧ dcache0[dentrykey(m9, str0)].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(m9, dirs0) = 0, links(m9, dirs0) = ∅, m9 = ROOT_INO, str0 ∈ entries6

Rule argument

Additional information

Used simplifier rules: size-zero: ⊦ # ls ≠ 0 → ls ≠ ∅; (forward) from specification set-basic[alink-set]%⊦ ino ∈ dirs ∧ dirs[ino] = mkdir(m0, md0, n1, n2, entries) → links(ino0, dirs[ino, mkdir(m, md, n, n0, entries)]) = links(ino0, dirs); (s)%size-zero-01: ⊦ ls ≠ ∅ → # ls ≠ 0; (forward) from specification set-basic[alink-set]% ⊦ entries[str, ino][str] = ino; (s)% ⊦ str ∈ entries[str, ino]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ str ≠ str0 → (entries -- str0)[str] = entries[str]; (s)% ⊦ str ∈ entries -- str0 ↔ str ≠ str0 ∧ str ∈ entries; (s)%⊦ ino0 ∈ dirs → links(ino, dirs[ino0, mkdir(m, md, n0, n1, entries -- str)]) = (str ∈ entries ∧ entries[str] = ino ⊃ links(ino, dirs[ino0, mkdir(m, md, n0, n1, entries)]) -- mklink(ino0, str) ;links(ino, dirs[ino0, mkdir(m, md, n0, n1, entries)]) ); (s)%⊦ p_ino ∈ dirs → links (ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries[str, ino])]) = (ino = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ++ mklink(p_ino, str) ;(str ∈ entries ∧ entries[str] = ino0 ⊃ links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) -- mklink(p_ino, str) ;links(ino0, dirs[p_ino, mkdir(m0, md, n, n0, entries)]) ) ); (s)%⊦ p_ino ∈ dirs → links(ino, dirs[p_ino, mkdir(m0, md, n0, n1 + 1, entries)]) = links(ino, dirs[p_ino, mkdir(m0, md, n0, n1, entries)]); (s)% ⊦ dirs[ino, ad][ino, ad0] = dirs[ino, ad0]; (s)%⊦ mkdir(m, md, m0, m1, entries).meta:= md0 = mkdir(m, md0, m0, m1, entries); (s)%⊦ mkdir(m, md, m0, m1, entries) = mkdir(m2, md0, m3, m4, entries0) ↔ m = m2 ∧ md = md0 ∧ m0 = m3 ∧ m1 = m4 ∧ entries = entries0; (ws)