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Proof step 315 from proof refine-rmdir (previous step)

cache_dirs[m] = mkdir(m0 + 1, md2, # links(m, cache_dirs), m1 + 1, entries), cache_dirs[entries[str]] = mkdir(m2, md1, # links(entries[str], cache_dirs), m4, ∅), afs_files = (cache_files ⊕ cache_pcache) ⊕ cache_icache, dirs = cache_dirs[m, mkdir(m0, md2, # links(m, cache_dirs), m1, entries -- str)], files = afs_files, ino = mkdentry(str, entries[str]).ino, mkdentry(str, entries[str]).dentry?, 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), entries[str] ∈ cache_dirs, m ∈ cache_dirs, str ∈ entries, ⌊err0⌋ , cache_sync → synced(cache_files, cache_icache, cache_pcache), links(m, cache_dirs) ≠ ∅ ∨ m = ROOT_INO ⊦ ⟨fail#(; ; err); asynced := ?⟩ ((asynced ↔ asynced0) ∧ err0 = err), entries[str] = ROOT_INO, dirs[ino].nlink ≥ 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 316:
(entries[str] = m ⊃ mkdir(m0, md2, # links(entries[str], cache_dirs), m1, entries -- str) ;mkdir(m2, md1, # links(entries[str], cache_dirs), m4, ∅) ) .nlink = 0, cache_dirs[entries[str]] = mkdir(m2, md1, # links(entries[str], cache_dirs), m4, ∅), cache_dirs[m] = mkdir(m0 + 1, md2, # links(m, cache_dirs), m1 + 1, entries), # links(m, cache_dirs) ≠ 0, entries[str] ≠ ROOT_INO, links(m, cache_dirs) ≠ ∅, m ≠ ROOT_INO, 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), str ∈ entries, m ∈ cache_dirs, entries[str] ∈ cache_dirs, ⌊err0⌋ , cache_sync → synced(cache_files, cache_icache, cache_pcache) ⊦ ⟨fail#(; ; err); asynced := ?⟩ ((asynced ↔ asynced0) ∧ err0 = err)

Rule argument

Additional information

Used simplifier rules: size-zero: ⊦ # ls ≠ 0 → ls ≠ ∅; (forward) from specification set-basic[alink-set]%size-zero-01: ⊦ ls ≠ ∅ → # ls ≠ 0; (forward) from specification set-basic[alink-set]%⊦ p_ino ∈ dirs ∧ dirs[p_ino] = mkdir(m, md, n, n0, entries) ∧ str ∈ entries → links(entries[str], dirs) ≠ ∅; (s)%size-zero-02: ⊦ # ls = 0 → ls = ∅; (forward) from specification set-basic[alink-set]% ⊦ mkdir(m, md, m0, m1, entries).nlink = m0; (s)% ⊦ m < 1 ↔ m = 0; (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ n ≥ m ↔ m ≤ n; (s)% ⊦ dirs[ino, ad][ino0] = (ino0 = ino ∧ ino ∈ dirs ⊃ ad;dirs[ino0]); (s)% ⊦ mkdentry(str, m).ino = m; (s)% ⊦ mkdentry(str, m).dentry?; (s)