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Proof step 204 from proof inv-rename-dir-negdentry (previous step)

(key.ino = m ⊃ mkdir(m13, md4, m14, m15, entries4);(key.ino = m9 ⊃ mkdir(m1, md0, # links(key.ino, dirs0), m2, entries0 -- str);dirs0[key.ino])) = mkdir(m16, md7, m17, m18, entries5), (m = m9 ⊃ mkdir(m1, md, # links(m9, dirs0), m2, entries0 -- str);mkdir(m8, md5, # links(m, dirs0), m10, entries3)) = mkdir(m4, md3, m6, m7, entries2), (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(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, entries4), dirs0[entries0[str]] = mkdir(m3, md2, # links(entries0[str], dirs0), m5, entries), dirs0[m] = mkdir(m8, md5, # links(m, dirs0), m10, entries3), dirs0[m9] = mkdir(m1 + 1, md, # links(m9, dirs0), m2 + 1, entries0), afs-cons(dirs0, files), 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, entries4) ], files), synced(dirs0, icache0, pcache), key ∈ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])], entries0[str] ∈ dirs0, m ∈ dirs0, str ∈ entries0, m9 ∈ dirs0, m = m9 → md4 = md0, asynced2 → synced(files, icache0, pcache), 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(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m9, dirs0) ≠ ∅ ∨ m9 = ROOT_INO ⊦ str0 ∈ entries3, key.dentry? ∧ key.name = (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].name ∧ key.ino ∈ dirs0 ∧ ( links (key.ino, 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, entries4) ]) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].ino ∈ dirs0 ∨ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].ino ∈ files) ∧ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].name ∈ entries5 ∧ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].ino = entries5[(dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].name] ∧ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].dentry?

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 205:
(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, entries4), (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(m8, md5, # links(m, dirs0), m10, entries3)) = mkdir(m4, md3, m6, m7, entries2), (key.ino = m ⊃ mkdir(m13, md4, m14, m15, entries4);(key.ino = m9 ⊃ mkdir(m1, md0, # links(key.ino, dirs0), m2, entries0 -- str);dirs0[key.ino])) = mkdir(m16, md7, m17, m18, entries5), dirs0[m9] = mkdir(m1 + 1, md, # links(m9, dirs0), m2 + 1, entries0), dirs0[m] = mkdir(m8, md5, # links(m, dirs0), m10, entries3), dirs0[entries0[str]] = mkdir(m3, md2, # links(entries0[str], dirs0), m5, entries), dirs0[key.ino] = mkdir(m19, md8, m20, m21, entries6), 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, entries4) ], files), afs-cons(dirs0, files), synced(dirs0, icache0, pcache), m9 ∈ dirs0, str ∈ entries0, m ∈ dirs0, entries0[str] ∈ dirs0, key ∈ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[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 ∈ entries6 ∧ dcache0[key].ino = entries6[dcache0[key].name] ∧ dcache0[key].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 ∈ entries3, key.dentry? ∧ key.name = (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].name ∧ key.ino ∈ dirs0 ∧ ( links (key.ino, 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, entries4) ]) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ ( (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].ino ∈ dirs0 ∨ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].ino ∈ files) ∧ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].name ∈ entries5 ∧ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].ino = entries5[(dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[str])][key].name] ∧ (dcache0 -- dentrykey(m9, str))[dentrykey(m, str0), mkdentry(str0, entries0[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], m19, md8, m20, m21, entries6]

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], m19, md8, m20, m21, entries6