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Proof step 163 from proof refine-rmdir (previous step)

(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, # links(entries3[str], cache_dirs), m1, entries3 -- str) ;mkdir(m2, md1, # links(entries3[str], cache_dirs), m13, ∅) ) = 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(m4, md3, m14, m15, entries0), cache_dirs[entries3[str]] = mkdir(m2, md1, # links(entries3[str], cache_dirs), m13, ∅), cache_dirs[m6] = mkdir(m0 + 1, md6, # links(m6, cache_dirs), m1 + 1, entries3), 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, md3, true, m14, m15, m4))], 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, md3, true, m14, m15, m4))], cache_pcache), m6 ∈ cache_dirs, entries3[str] ∈ cache_dirs, str ∈ entries3, ino ∈ cache_files, cache_sync → synced (cache_files, cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ], cache_pcache), cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ entries3[str] = ROOT_INO, ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = ( (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] )[ino ]

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 164:
⊦ ino ∈ icache ∧ ¬ icache[ino].inode.directory → (files ⊕ icache)[ino] = mkfile(icache[ino].inode.size, icache[ino].inode.meta, icache[ino].inode.nlink, files[ino].content)
Proof step 165:
(m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)) = mkdir(m4, md3, m14, m15, entries0), (entries3[str] = m6 ⊃ mkdir(m0, md6, # links(entries3[str], cache_dirs), m1, entries3 -- str) ;mkdir(m2, md1, # links(entries3[str], cache_dirs), m13, ∅) ) = 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, md1, # links(entries3[str], cache_dirs), m13, ∅), 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, md3, true, m14, m15, m4))], 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, md3, true, m14, m15, m4))], cache_pcache), synced(cache_dirs, cache_icache, cache_pcache), ino ∈ cache_files, 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, md3, true, m14, m15, m4)) ], cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = ( (cache_files ⊕ cache_pcache) ⊕ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] )[ino ], entries3[str] = ROOT_INO, ino ∈ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] ∧ ¬ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] [ino ] .inode .directory
Proof step 195:
(m6 = entries3[str] ⊃ mkdir(m5, md0, m3, m7, entries);mkdir(m0, md, # links(m6, cache_dirs), m1, entries3 -- str)) = mkdir(m4, md3, m14, m15, entries0), (entries3[str] = m6 ⊃ mkdir(m0, md6, # links(entries3[str], cache_dirs), m1, entries3 -- str) ;mkdir(m2, md1, # links(entries3[str], cache_dirs), m13, ∅) ) = 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, md1, # links(entries3[str], cache_dirs), m13, ∅), 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, md3, true, m14, m15, m4))], 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, md3, true, m14, m15, m4))], cache_pcache), synced(cache_dirs, cache_icache, cache_pcache), ino ∈ cache_files, str ∈ entries3, entries3[str] ∈ cache_dirs, m6 ∈ cache_dirs, ino ∈ cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ], 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, md3, true, m14, m15, m4)) ], cache_pcache), links(m6, cache_dirs) ≠ ∅ ∨ m6 = ROOT_INO ⊦ ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[ino] = mkfile (cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] [ino ] .inode .size, cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] [ino ] .inode .meta, cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] [ino ] .inode .nlink, (cache_files ⊕ cache_pcache)[ino].content), entries3[str] = ROOT_INO, cache_icache[entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m))] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ] [ino ] .inode .directory

Rule argument

spec: cache-base name: at-icache-overwrite seq: ⊦ ino ∈ icache ∧ ¬ icache[ino].inode.directory → (files ⊕ icache)[ino] = mkfile (icache[ino].inode.size, icache[ino].inode.meta, icache[ino].inode.nlink, files[ino].content) subst: [ino, icache, files] → [ino, cache_icache [entries3[str], mkientry (false, mkinode(entries3[str], md0, true, m8, m9, m)) ] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ], cache_files ⊕ cache_pcache] flags: <all paths>

Additional information

Used terms: ino, cache_icache [entries3[str], mkientry(false, mkinode(entries3[str], md0, true, m8, m9, m)) ] [m6, mkientry(false, mkinode(m6, md3, true, m14, m15, m4)) ], cache_files ⊕ cache_pcache