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Proof step 19 from proof read-inv-rec (previous step)

# buf0 = # buf1, # buf1 = n4 + n + min(n3, n2, n1), VFS_PAGE_SIZE = n2 + (n0 + n) % VFS_PAGE_SIZE, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), afs-cons(dirs, files), ino ∈ files, (n0 + n) % VFS_PAGE_SIZE + min(n3, n2, n1) ≤ VFS_PAGE_SIZE, n + min(n3, n2, n1) ≤ # buf1, n3 + n ≤ # buf1 ⊦ (((content.stream from n0) to n) + (⌜buf0⌝ from n) to n) + ((((content.stream from (n0 + n) / VFS_PAGE_SIZE * VFS_PAGE_SIZE) to VFS_PAGE_SIZE) from (n0 + n) % VFS_PAGE_SIZE) to min(n3, n2, n1)) + (⌜buf0⌝ from n from min(n3, n2, n1)) = ((content.stream from n0) to min(n3, n2, n1) + n) + (⌜buf0⌝ from min(n3, n2, n1) + n)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 20:
⊦ ¬ # σ0 < ⌜m + n⌝ → σ0 from m from n = σ0 from m + n
Proof step 21:
# buf1 = n4 + n + min(n3, n2, n1), # buf0 = # buf1, VFS_PAGE_SIZE = n2 + (n0 + n) % VFS_PAGE_SIZE, files[ino] = mkfile(n1 + n0 + n, md, # links(ino, dirs), content), afs-cons(dirs, files), ino ∈ files, n3 + n ≤ # buf1, n + min(n3, n2, n1) ≤ # buf1, (n0 + n) % VFS_PAGE_SIZE + min(n3, n2, n1) ≤ VFS_PAGE_SIZE ⊦ (((content.stream from n0) to n) + (⌜buf0⌝ from n) to n) + ((((content.stream from (n0 + n) / VFS_PAGE_SIZE * VFS_PAGE_SIZE) to VFS_PAGE_SIZE) from (n0 + n) % VFS_PAGE_SIZE) to min(n3, n2, n1)) + (⌜buf0⌝ from n + min(n3, n2, n1)) = ((content.stream from n0) to min(n3, n2, n1) + n) + (⌜buf0⌝ from min(n3, n2, n1) + n), # ⌜buf0⌝ < ⌜n + min(n3, n2, n1)⌝

Rule argument

spec: stream, inst: buffer-inf name: from-from seq: ⊦ ¬ # σ0 < ⌜m + n⌝ → σ0 from m from n = σ0 from m + n subst: [n, m, σ0] → [min(n3, n2, n1), n, ⌜buf0⌝ ] 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)%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%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n → n ≠ 0; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ m + n < m; (s) Used terms: min(n3, n2, n1), n, ⌜buf0⌝