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Proof step 126 from proof refine-rmdir (previous step)

# links(entries3[str], cache_dirs) = cache_dirs[entries3[str]].nlink, cache_dirs[entries3[str]].entries = ∅, (entries3[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)) = mkdir(m, md4, m8, m9, entries1), (entries3[str] = m6 ⊃ mkdir(m0, md6, cache_dirs[entries3[str]].nlink, m1, entries3 -- str);cache_dirs[entries3[str]]) = mkdir(m5, md2, m3 + 1, m7, entries), (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)) = mkdir(m10, md5, m11, m12, entries2), cache_dirs[entries3[str]] = mkdir(cache_dirs[entries3[str]].size, md3, cache_dirs[entries3[str]].nlink, cache_dirs[entries3[str]].nsubdirs, ∅), cache_dirs[m6] = mkdir(m0 + 1, md6, # links(m6, cache_dirs), m1 + 1, entries3), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m6, mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)], entries3[str] ≠ ROOT_INO, afs-cons (cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons (cache_dcache -- dentrykey(m6, str), cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons (cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), synced (cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), m6 ∈ cache_dirs, entries3[str] ∈ cache_dirs, str ∈ entries3, cache_sync → synced (cache_files, cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ⟨afs_drop_nlink#(mkdentry(str, entries3[str]).ino; dirs, afs_files); dent := negdentry(mkdentry(str, entries3[str]).name); afs_inode_sync#(mkinode(m6, md, true, # links(m6, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync# (mkinode(entries3[str], md0, true, # links(entries3[str], cache_dirs), cache_dirs[entries3[str]].nsubdirs, cache_dirs[entries3[str]].size) ; dirs, afs_files); p_inode := (m6 ∈ dirs ⊃ mkinode(m6, dirs[m6].meta, true, dirs[m6].nlink, dirs[m6].nsubdirs, dirs[m6].size) ;mkinode(m6, afs_files[m6].meta, false, afs_files[m6].nlink, 0, afs_files[m6].size) ); c_inode := getinode(entries3[str], dirs, afs_files); err := ESUCCESS; asynced := ?⟩ ( dirs = cache_dirs [m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md ] [entries3[str], (entries3[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ] ∧ (asynced ↔ asynced0) ∧ mkinode (m6, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).meta, true, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).nlink, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).nsubdirs, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).size) = p_inode ∧ mkinode(entries3[str], md0, true, m8, m9, m) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 127:
# links(entries3[str], cache_dirs) = m4, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)) = mkdir(m10, md5, m11, m12, entries2), (entries3[str] = m6 ⊃ mkdir(m0, md6, m4, m1, entries3 -- str);cache_dirs[entries3[str]]) = mkdir(m5, md2, m3 + 1, m7, entries), (entries3[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)) = mkdir(m, md4, m8, m9, entries1), cache_dirs[m6] = mkdir(m0 + 1, md6, # links(m6, cache_dirs), m1 + 1, entries3), cache_dirs[entries3[str]] = mkdir(m2, md3, m4, m13, ∅), cache_dirs[entries3[str]] = mkdir(m2, md1, m4, m13, entries0), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m6, mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)], entries0 = ∅, afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), afs-cons (cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), dcache-cons (cache_dcache -- dentrykey(m6, str), cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), icache-cons(cache_icache, cache_dirs, cache_files), icache-cons (cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), pcache-cons (cache_pcache, cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_files), synced (cache_dirs[m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md] [entries3[str], mkdir(m, md0, m8, m9, entries1) ], cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), synced(cache_dirs, cache_icache, cache_pcache), str ∈ entries3, entries3[str] ∈ cache_dirs, m6 ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), cache_sync → synced (cache_files, cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ], cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ⟨afs_drop_nlink#(mkdentry(str, entries3[str]).ino; dirs, afs_files); dent := negdentry(mkdentry(str, entries3[str]).name); afs_inode_sync#(mkinode(m6, md, true, # links(m6, cache_dirs), m1 + 1, m0 + 1); dirs, afs_files); afs_inode_sync#(mkinode(entries3[str], md0, true, # links(entries3[str], cache_dirs), m13, m2); dirs, afs_files); p_inode := (m6 ∈ dirs ⊃ mkinode(m6, dirs[m6].meta, true, dirs[m6].nlink, dirs[m6].nsubdirs, dirs[m6].size) ;mkinode(m6, afs_files[m6].meta, false, afs_files[m6].nlink, 0, afs_files[m6].size) ); c_inode := getinode(entries3[str], dirs, afs_files); err := ESUCCESS; asynced := ?⟩ ( dirs = cache_dirs [m6, (m6 = entries3[str] ⊃ mkdir(m5, md2, m3, m7, entries);mkdir(m0, md6, # links(m6, cache_dirs), m1, entries3 -- str)).meta:= md ] [entries3[str], (entries3[str] = m6 ⊃ mkdir(m5, md, m3, m7, entries);mkdir(m5, md2, m3, m7, entries)).meta:= md0 ] ∧ afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md5, true, m11, m12, m10)) ] ∧ (asynced ↔ asynced0) ∧ mkinode (m6, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).meta, true, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).nlink, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).nsubdirs, (m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)).size) = p_inode ∧ mkinode(entries3[str], md0, true, m8, m9, m) = c_inode ∧ negdentry(str) = dent ∧ ESUCCESS = err), entries3[str] = ROOT_INO

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] → [cache_dirs[entries3[str]], m2, md1, m4, m13, entries0]

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: cache_dirs[entries3[str]], m2, md1, m4, m13, entries0