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Proof step 109 from proof refine-create (previous step)

cache_dirs[m1] = mkdir(m2, md1, # links(m1, cache_dirs), m5, entries0), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), m1 ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m1, cache_dirs) ≠ ∅ ∨ m1 = ROOT_INO ⊦ (((cache_files ⊕ cache_pcache) ⊕ cache_icache) ++ m)[m, mkfile(0, md0, 1, ∅)] = ((cache_files ++ m)[m, mkfile(0, md0, 1, ∅)] ⊕ cache_pcache) ⊕ cache_icache[m1, mkientry(false, mkinode(m1, md2, true, # links(m1, cache_dirs), m5, m2 + 1))] [m, mkientry(false, mkinode(m, md0, false, 1, 0, 0)) ], m = 0, str ∈ entries0, m ∈ cache_files, m ∈ cache_dirs

Heuristic use patterns applied rule insert rewrite lemma

And obtained the following new goals

Proof step 110:
⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino]))
Proof step 111:
cache_dirs[m1] = mkdir(m2, md1, # links(m1, cache_dirs), m5, entries0), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), afs-cons(cache_dirs, cache_files), dcache-cons(cache_dcache, cache_dirs, cache_files), icache-cons(cache_icache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), m1 ∈ cache_dirs, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m1, cache_dirs) ≠ ∅ ∨ m1 = ROOT_INO ⊦ m = 0, m ∈ cache_dirs, m ∈ cache_files, str ∈ entries0, ∀ ino. ( ino ∈ (((cache_files ⊕ cache_pcache) ⊕ cache_icache) ++ m)[m, mkfile(0, md0, 1, ∅)] ↔ ino ∈ ((cache_files ++ m)[m, mkfile(0, md0, 1, ∅)] ⊕ cache_pcache) ⊕ cache_icache[m1, mkientry(false, mkinode(m1, md2, true, # links(m1, cache_dirs), m5, m2 + 1))] [m, mkientry(false, mkinode(m, md0, false, 1, 0, 0)) ]) ∧ ( ino ∈ (((cache_files ⊕ cache_pcache) ⊕ cache_icache) ++ m)[m, mkfile(0, md0, 1, ∅)] → (((cache_files ⊕ cache_pcache) ⊕ cache_icache) ++ m)[m, mkfile(0, md0, 1, ∅)][ino] = ( ((cache_files ++ m)[m, mkfile(0, md0, 1, ∅)] ⊕ cache_pcache) ⊕ cache_icache[m1, mkientry(false, mkinode(m1, md2, true, # links(m1, cache_dirs), m5, m2 + 1))] [m, mkientry(false, mkinode(m, md0, false, 1, 0, 0)) ] )[ino ])

Rule argument

spec: heap, inst: file-heap name: ext seq: ⊦ files = files0 ↔ (∀ ino. (ino ∈ files ↔ ino ∈ files0) ∧ (ino ∈ files → files[ino] = files0[ino])) subst: [files, files0] → [(((cache_files ⊕ cache_pcache) ⊕ cache_icache) ++ m )[m, mkfile(0, md0, 1, ∅) ], ( (cache_files ++ m)[m, mkfile(0, md0, 1, ∅)] ⊕ cache_pcache) ⊕ cache_icache [m1, mkientry (false, mkinode (m1, md2, true, # links(m1, cache_dirs), m5, m2 + 1)) ] [m, mkientry(false, mkinode(m, md0, false, 1, 0, 0)) ]] flags: <all paths>

Additional information

Used terms: (((cache_files ⊕ cache_pcache) ⊕ cache_icache) ++ m )[m, mkfile(0, md0, 1, ∅) ], ((cache_files ++ m)[m, mkfile(0, md0, 1, ∅)] ⊕ cache_pcache) ⊕ cache_icache [m1, mkientry (false, mkinode(m1, md2, true, # links(m1, cache_dirs), m5, m2 + 1)) ] [m, mkientry(false, mkinode(m, md0, false, 1, 0, 0)) ]