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Proof step 107 from proof write-inv-rec (previous step)

n0 + n = n3 + m0 * VFS_PAGE_SIZE, files1 -- ino = files0 -- ino, VFS_PAGE_SIZE = n4 + n3 + m, files1[ino] = mkfile(m2, md0, # links(ino, dirs), content0), files0[ino] = mkfile(m2, md0, m4, content), afs-cons(dirs, files1), ino ∈ files1, ino ∈ files0, m0 * VFS_PAGE_SIZE ≤ n1, n + m ≤ # buf, n ≤ # buf, n1 ≤ # buf + n0, n0 + n ≤ n1, m0 * VFS_PAGE_SIZE ≤ # buf + n0, m0 * VFS_PAGE_SIZE ≤ n0 + n, m0 * VFS_PAGE_SIZE ≤ n + n0 ⊦ n4 + m = 0, (⌜buf⌝ to n) + ((⌜buf⌝ from n) to m) = ⌜buf⌝ to n + m

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 108:
⊦ ¬ # σ0 < ⌜n + n1⌝ → (σ0 to n) + ((σ0 from n) to n1) = σ0 to n + n1

Rule argument

spec: stream, inst: buffer-inf name: from-append-tofrom seq: ⊦ ¬ # σ0 < ⌜n + n1⌝ → (σ0 to n) + ((σ0 from n) to n1) = σ0 to n + n1 subst: [n1, n, σ0] → [m, n, ⌜buf⌝ ] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ⌜n⌝ < ⌜m⌝ + ⌜n0⌝ ↔ n < m + n0; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%add-assoc: ⊦ (n∞ + n∞1) + n∞2 = n∞ + n∞1 + n∞2; from specification nat-inf%add-comm: ⊦ n∞1 + n∞2 = n∞2 + n∞1; from specification nat-inf Used terms: m, n, ⌜buf⌝