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Proof step 31 from proof refine-rename-file-dentry (previous step)

cache_files[entries[str]] = mkfile(m4, md2, # links(entries[str], cache_dirs), content2), cache_dirs[m] = mkdir(m11, md5, # links(m, cache_dirs), m17, entries), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str0]] = mkfile(m8, md4, m15, content0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m12, md3, m14, content), cache_dirs[m3] = mkdir(m0, md6, # links(m3, cache_dirs), m2, entries0), cache_files[entries0[str0]] = mkfile(m1, md1, # links(entries0[str0], cache_dirs), content1), 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), str ∈ entries, str0 ∈ entries0, m ∈ cache_dirs, m3 ∈ cache_dirs, entries[str] ∈ cache_files, entries0[str0] ∈ cache_files, m = m3 → md = md0, cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO, links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO ⊦ m15 = # links(entries0[str0], cache_dirs)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 32:
⊦ icache-cons(icache, dirs, files) ↔ (∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ valid-file-inode(icache[ino].inode, dirs, files) ;valid-ino(ino, dirs, files) ∧ valid-inode(icache[ino].inode, dirs, files) ))
Proof step 33:
cache_files[entries0[str0]] = mkfile(m1, md1, # links(entries0[str0], cache_dirs), content1), cache_dirs[m3] = mkdir(m0, md6, # links(m3, cache_dirs), m2, entries0), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries[str]] = mkfile(m12, md3, m14, content), ((cache_files ⊕ cache_pcache) ⊕ cache_icache)[entries0[str0]] = mkfile(m8, md4, m15, content0), cache_dirs[m] = mkdir(m11, md5, # links(m, cache_dirs), m17, entries), cache_files[entries[str]] = mkfile(m4, md2, # links(entries[str], cache_dirs), content2), afs-cons(cache_dirs, cache_files), afs-cons(cache_dirs, (cache_files ⊕ cache_pcache) ⊕ cache_icache), dcache-cons(cache_dcache, cache_dirs, cache_files), pcache-cons(cache_pcache, cache_dirs, cache_files), synced(cache_dirs, cache_icache, cache_pcache), entries0[str0] ∈ cache_files, entries[str] ∈ cache_files, m3 ∈ cache_dirs, m ∈ cache_dirs, str0 ∈ entries0, str ∈ entries, cache_sync → synced(cache_files, cache_icache, cache_pcache), m = m3 → md = md0, ∀ ino. ino ∈ cache_icache → cache_icache[ino].inode.ino = ino ∧ (cache_icache[ino].dirty ⊃ valid-file-inode(cache_icache[ino].inode, cache_dirs, cache_files) ;valid-ino(ino, cache_dirs, cache_files) ∧ valid-inode(cache_icache[ino].inode, cache_dirs, cache_files) ), links(m3, cache_dirs) ≠ ∅ ∨ m3 = ROOT_INO, links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ m15 = # links(entries0[str0], cache_dirs)

Rule argument

spec: cache-base name: icache-cons seq: ⊦ icache-cons(icache, dirs, files) ↔ (∀ ino. ino ∈ icache → icache[ino].inode.ino = ino ∧ (icache[ino].dirty ⊃ valid-file-inode(icache[ino].inode, dirs, files) ; valid-ino(ino, dirs, files) ∧ valid-inode(icache[ino].inode, dirs, files) )) subst: [files, dirs, icache] → [cache_files, cache_dirs, cache_icache] flags: <all paths>

Additional information

Used terms: cache_files, cache_dirs, cache_icache