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Proof step 99 from proof write-inv-rec (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 100:
⊦ ¬ # σ1 < ⌜n⌝ ∧ ¬ # σ2 < ⌜n1⌝ → (σ1 to n) + σ2 to n1 + n = (σ1 to n) + (σ2 to n1)
Proof step 101:
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, ((content.stream to n0) + (⌜buf⌝ to n) to m0 * VFS_PAGE_SIZE) + ((content.stream to n0) + (⌜buf⌝ to n to n) from m0 * VFS_PAGE_SIZE) + ((⌜buf⌝ from n) to m) = (content.stream to n0) + (⌜buf⌝ to n + m), #(⌜buf⌝ to n) < ⌜n⌝ , # content.stream < ⌜n0⌝

Rule argument

spec: stream, inst: buffer-inf name: to-append-add seq: ⊦ ¬ # σ1 < ⌜n⌝ ∧ ¬ # σ2 < ⌜n1⌝ → (σ1 to n) + σ2 to n1 + n = (σ1 to n) + (σ2 to n1) subst: [n1, σ2, n, σ1] → [n, ⌜buf⌝ to n, n0, content.stream] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # content.stream = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)% ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ ¬ n < n; (s)%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n, ⌜buf⌝ to n, n0, content.stream