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Proof step 174 from proof inv-rmdir (previous step)

dcache[key].ino = dirs0[key.ino].entries[key.name], key.name = dcache[key].name, (entries0[str] = m5 ⊃ mkdir(m0 - 1, md1, # links(entries0[str], dirs0), m2 - 1, entries0 -- str) ;mkdir(m1, md3, # links(entries0[str], dirs0), m4, ∅) ) = mkdir(m3, md2, m7 + 1, m6, entries), (entries0[str] = m5 ⊃ mkdir(m3, md, m7, m6, entries);mkdir(m3, md2, m7, m6, entries)) = mkdir(m, md4, m8, m9, entries1), (m5 = entries0[str] ⊃ mkdir(m3, md0, m7, m6, entries);mkdir(m0 - 1, md, # links(m5, dirs0), m2 - 1, entries0 -- str)) = mkdir(m10, md5, m11, m12, entries2), (key.ino = entries0[str] ⊃ mkdir(m, md0, m8, m9, entries1) ;(key.ino = m5 ⊃ mkdir(m0 - 1, md, # links(key.ino, dirs0), m2 - 1, entries0 -- str);dirs0[key.ino]) ) = mkdir(m13, md6, m14, m15, entries3), dirs0[entries0[str]] = mkdir(m1, md3, # links(entries0[str], dirs0), m4, ∅), dirs0[m5] = mkdir(m0, md1, # links(m5, dirs0), m2, entries0), key.dentry?, dcache[key].dentry?, afs-cons(dirs0, files), afs-cons (dirs0[m5, (m5 = entries0[str] ⊃ mkdir(m3, md2, m7, m6, entries);mkdir(m0 - 1, md1, # links(m5, dirs0), m2 - 1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], files), synced(dirs0, icache0, pcache), key ∈ dcache, entries0[str] ∈ dirs0, str ∈ entries0, m5 ∈ dirs0, key.ino ∈ dirs0, key.name ∈ dirs0[key.ino].entries, 1 ≤ m2, 1 ≤ m0, asynced1 → synced(files, 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 ∈ files) ∧ dcache[key0].name ∈ dirs0[key0.ino].entries ∧ dcache[key0].ino = dirs0[key0.ino].entries[dcache[key0].name] ∧ dcache[key0].dentry?, links(m5, dirs0) ≠ ∅ ∨ m5 = ROOT_INO, links(key.ino, dirs0) ≠ ∅ ∨ key.ino = ROOT_INO, dcache[key].ino ∈ dirs0 ∨ dcache[key].ino ∈ files ⊦ entries0[str] = ROOT_INO, key = dentrykey(m5, str), m0 = 0, m2 = 0, ( links (key.ino, dirs0 [m5, (m5 = entries0[str] ⊃ mkdir(m3, md2, m7, m6, entries);mkdir(m0 - 1, md1, # links(m5, dirs0), m2 - 1, entries0 -- str)).meta:= md ] [entries0[str], mkdir(m, md0, m8, m9, entries1) ]) ≠ ∅ ∨ key.ino = ROOT_INO) ∧ key.name ∈ entries3 ∧ dcache[key].ino = entries3[key.name]

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 175:
dcache[key].ino = dirs0[m16].entries[str0], (m16 = entries0[str] ⊃ mkdir(m, md0, m8, m9, entries1);(m16 = m5 ⊃ mkdir(m0 - 1, md, # links(m16, dirs0), m2 - 1, entries0 -- str);dirs0[m16])) = mkdir(m13, md6, m14, m15, entries3), (m5 = entries0[str] ⊃ mkdir(m3, md0, m7, m6, entries);mkdir(m0 - 1, md, # links(m5, dirs0), m2 - 1, entries0 -- str)) = mkdir(m10, md5, m11, m12, entries2), (entries0[str] = m5 ⊃ mkdir(m3, md, m7, m6, entries);mkdir(m3, md2, m7, m6, entries)) = mkdir(m, md4, m8, m9, entries1), (entries0[str] = m5 ⊃ mkdir(m0 - 1, md1, # links(entries0[str], dirs0), m2 - 1, entries0 -- str) ;mkdir(m1, md3, # links(entries0[str], dirs0), m4, ∅) ) = mkdir(m3, md2, m7 + 1, m6, entries), dirs0[m5] = mkdir(m0, md1, # links(m5, dirs0), m2, entries0), dirs0[entries0[str]] = mkdir(m1, md3, # links(entries0[str], dirs0), m4, ∅), key = dentrykey(m16, str0), str0 = dcache[key].name, dcache[key].dentry?, key.dentry?, afs-cons (dirs0[m5, (m5 = entries0[str] ⊃ mkdir(m3, md2, m7, m6, entries);mkdir(m0 - 1, md1, # links(m5, dirs0), m2 - 1, entries0 -- str)).meta:= md] [entries0[str], mkdir(m, md0, m8, m9, entries1) ], files), afs-cons(dirs0, files), synced(dirs0, icache0, pcache), str0 ∈ dirs0[m16].entries, m16 ∈ dirs0, m5 ∈ dirs0, str ∈ entries0, entries0[str] ∈ dirs0, key ∈ dcache, 1 ≤ m0, 1 ≤ m2, asynced1 → synced(files, 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 ∈ files) ∧ dcache[key0].name ∈ dirs0[key0.ino].entries ∧ dcache[key0].ino = dirs0[key0.ino].entries[dcache[key0].name] ∧ dcache[key0].dentry?, dcache[key].ino ∈ dirs0 ∨ dcache[key].ino ∈ files, links(m16, dirs0) ≠ ∅ ∨ m16 = ROOT_INO, links(m5, dirs0) ≠ ∅ ∨ m5 = ROOT_INO ⊦ entries0[str] = ROOT_INO, key = dentrykey(m5, str), m0 = 0, m2 = 0, ( links (m16, dirs0 [m5, (m5 = entries0[str] ⊃ mkdir(m3, md2, m7, m6, entries);mkdir(m0 - 1, md1, # links(m5, dirs0), m2 - 1, entries0 -- str)).meta:= md ] [entries0[str], mkdir(m, md0, m8, m9, entries1) ]) ≠ ∅ ∨ m16 = ROOT_INO) ∧ str0 ∈ entries3 ∧ dcache[key].ino = entries3[str0]

Rule argument

spec: key name: elim-03 seq: ⊦ key.dentry? → (m = key.ino ∧ str = key.name ↔ key = dentrykey(m, str)) subst: [key, m, str] → [key, m16, str0]

Additional information

Used simplifier rules: ⊦ key.dentry? → (m = key.ino ∧ str = key.name ↔ key = dentrykey(m, str)); (elim) Used terms: key, m16, str0