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Proof step 172 from proof afs-cons-rename-dir-dentry (previous step)

# links(entries2[str], dirs0) = m12 + 1, (entries2[str0] = m ⊃ mkdir(m8, md1, # links(entries2[str0], dirs0), m11, (entries2 -- str0)[str, entries2[str0]]) ;mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1) ) = mkdir(m20, md10, m21, m22, entries5), files[entries2[str0]] = mkfile(m14, md8, m16, content), dirs0[entries2[str0]] = mkdir(# entries1, md3, # links(entries2[str0], dirs0), # subdirs(entries2[str0], dirs0), entries1), dirs0[m] = mkdir(m8 + 1, md0, # links(m, dirs0), m11 + 1, entries2), dirs0[entries2[str]] = mkdir(m13, md6, m12 + 1, m15, ∅), afs-cons(dirs0, files), ROOT_INO ∈ dirs0, entries2[str] ∈ dirs0, str ∈ entries2, m ∈ dirs0, entries2[str0] ∈ dirs0, str0 ∈ entries2, # links(entries2[str0], dirs0) ≤ 1, name ∈ entries1 → entries1[name] ∈ dirs0 ∨ entries1[name] ∈ files, links(m, dirs0) ≠ ∅ ∨ m = ROOT_INO ⊦ # links(entries2[str0], dirs0) - 1 = 0, entries2[str] = ROOT_INO, entries2[str] = m, entries2[str0] = entries2[str], links(entries2[str0], dirs0) = ∅ ++ mklink(m, str0), mkdentry(str, entries2[str]) = mkdentry(str0, entries2[str0])

Heuristic elimination applied rule insert elim lemma

And closed the goal!

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [# links(entries2[str0], dirs0), m0]

Additional information

Used simplifier rules: lastlink-rename: ⊦ # links(entries[str0], dirs) = 1 ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(m, md, n, n0, entries) ∧ str0 ∈ entries ∧ str1 ∈ entries → (entries[str0] = entries[str1] ↔ str0 = str1);% ⊦ m ≠ 0 → (m ≤ 1 ↔ m = 1); (s)%afs-cons-dir-nlink: ⊦ afs-cons(dirs, files) ∧ ino ∈ dirs ∧ dirs[ino] = mkdir(n0, md, n, n1, entries) → # links(ino, dirs) = n; (forward) from specification afs-base% ⊦ n ≠ 0 → (m + n = 1 ↔ n = 1 ∧ m = 0); (ws)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%size-zero-02: ⊦ # ls = 0 → ls = ∅; (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)%n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: # links(entries2[str0], dirs0), m0