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Proof step 96 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 ⊦ n4 + m = 0, ((content.stream to n0) + (⌜buf⌝ to n) to m0 * VFS_PAGE_SIZE) + (((content.stream to n0) + (⌜buf⌝ to n) from m0 * VFS_PAGE_SIZE) to n3) + ((⌜buf⌝ from n) to m) = (content.stream to n0) + (⌜buf⌝ to n + m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 97:
⊦ ¬ # σ0 < ⌜m⌝ + ⌜n⌝ → (σ0 from m) to n = (σ0 to m + n) from m
Proof step 98:
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 ≤ 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 m0 * VFS_PAGE_SIZE + n3) from m0 * VFS_PAGE_SIZE) + ((⌜buf⌝ from n) to m) = (content.stream to n0) + (⌜buf⌝ to n + m), n4 + m = 0, #((content.stream to n0) + (⌜buf⌝ to n)) < ⌜m0 * VFS_PAGE_SIZE⌝ + ⌜n3⌝

Rule argument

spec: stream, inst: buffer-inf name: swap-from-to seq: ⊦ ¬ # σ0 < ⌜m⌝ + ⌜n⌝ → (σ0 from m) to n = (σ0 to m + n) from m subst: [n, m, σ0] → [n3, m0 * VFS_PAGE_SIZE, (content.stream to n0) + (⌜buf⌝ to n)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(σ0 + σ2) = # σ0 + # σ2; (s)% ⊦ # content.stream = ∞; (s)% ⊦ ¬ ∞ < n∞; (s)%¬ # σ0 < ⌜n⌝ ⊦ #(σ0 to n) = ⌜n⌝ ; (s)% ⊦ # ⌜buf⌝ = ⌜# buf⌝ ; (s)% ⊦ ⌜m⌝ < ⌜n⌝ ↔ m < n; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]% ⊦ ⌜n⌝ + ⌜m⌝ < ⌜n0⌝ + ⌜n1⌝ ↔ n + m < n0 + n1; (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%c: ⊦ m + n = n + m; from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (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 → ¬ n < m; (s) Used terms: n3, m0 * VFS_PAGE_SIZE, (content.stream to n0) + (⌜buf⌝ to n)