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Proof step 109 from proof inv-rename-dir-negdentry (previous step)

(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), 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), err = ESUCCESS, icache = icache0 [m9, mkientry (false, mkinode (m9, (m9 = m ⊃ mkdir(m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str)).meta, true, (m9 = m ⊃ mkdir(m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str)).nlink, (m9 = m ⊃ mkdir(m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str)).nsubdirs, (m9 = m ⊃ mkdir(m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str)).size)) ], 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])).meta:= md4 ], files), afs-cons(dirs0, files), dcache-cons(dcache, dirs0, files), icache-cons(icache0, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache0, pcache), m9 ∈ dirs0, str ∈ entries0, m ∈ dirs0, entries0[str] ∈ dirs0, ¬ str0 ∈ entries3, asynced2 → synced(files, icache0, pcache), m = m9 → md4 = md0, links(m9, dirs0) ≠ ∅ ∨ m9 = ROOT_INO, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟪icache_set# (( 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 ] ⊃ mkinode (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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .meta, true, 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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .nlink, 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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .nsubdirs, 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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ), false ; err, icache); skip; dcache_delete#(m9, mkdentry(str, entries0[str]).name; err, dcache); dcache_set#(m, mkdentry(str0, mkdentry(str, entries0[str]).ino).name, mkdentry(str0, mkdentry(str, entries0[str]).ino); err, dcache)⟫ ( icache-cons (icache, 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], files) ∧ dcache-cons (dcache, 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], files) ∧ pcache-cons (pcache, 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], files) ∧ synced (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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], icache, pcache) ∧ (asynced2 → synced(files, icache, pcache)))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 110:
(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), 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), err = ESUCCESS, icache = icache0[m9, mkientry(false, mkinode(m9, md1, true, m11, m12, m0))], 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]);mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])).meta:= md4 ], files), dcache-cons(dcache, dirs0, files), icache-cons(icache0, dirs0, files), pcache-cons(pcache, dirs0, files), synced(dirs0, icache0, pcache), entries0[str] ∈ dirs0, m ∈ dirs0, str ∈ entries0, m9 ∈ dirs0, m = m9 → md4 = md0, asynced2 → synced(files, icache0, pcache), links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO, links(m9, dirs0) ≠ ∅ ∨ m9 = ROOT_INO ⊦ ⟪icache_set# (( 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 ] ⊃ mkinode (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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .meta, true, 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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .nlink, 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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .nsubdirs, 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, dirs0[m, mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]])] [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 ] .meta:= md4 ] [m ] .size) ;mkinode(m, files[m].meta, false, files[m].nlink, 0, files[m].size) ), false ; err, icache); skip; dcache_delete#(m9, mkdentry(str, entries0[str]).name; err, dcache); dcache_set#(m, mkdentry(str0, mkdentry(str, entries0[str]).ino).name, mkdentry(str0, mkdentry(str, entries0[str]).ino); err, dcache)⟫ ( icache-cons (icache, 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], files) ∧ dcache-cons (dcache, 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], files) ∧ pcache-cons (pcache, 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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], files) ∧ synced (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, (m = m9 ⊃ mkdir(m4 + 1, md0, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m4 + 1, md3, m6, m7 + 1, entries2[str0, entries0[str]]) ) .meta:= md4 ], icache, pcache) ∧ (asynced2 → synced(files, icache, pcache))), str0 ∈ entries3

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] → [(m9 = m ⊃ mkdir (m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str) ), m0, md1, m11, m12, entries1]

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: (m9 = m ⊃ mkdir(m4 + 1, md4, m6, m7 + 1, entries2[str0, entries0[str]]) ;mkdir(m1, md0, # links(m9, dirs0), m2, entries0 -- str) ), m0, md1, m11, m12, entries1