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Proof step 8 from proof content-trunc-exact (previous step)

n1 < VFS_PAGE_SIZE ⊦ (m0 < n0 ∧ m0 ∈ content ⊃ content[m0];zeropage)[n1] = (⌜(λ n. (n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE];zeropage)[n % VFS_PAGE_SIZE]) to n0 * VFS_PAGE_SIZE⌝ + σ⊥ )[m0 * VFS_PAGE_SIZE + n1 ]

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: stream, inst: buffer-inf name: get-append-ls seq: ⊦ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ0[m] subst: [m, σ2, σ0] → [m0 * VFS_PAGE_SIZE + n1, σ⊥, ⌜ (λ n. (n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE] ;zeropage ) [n % VFS_PAGE_SIZE ]) to n0 * VFS_PAGE_SIZE ⌝ ] flags: <all paths>

Additional information

Used simplifier rules: ⊦ ⌜dx⌝ [n] = dx[n]; (s)% ⊦ n0 < m0 → (m0 * m + n0 < m0 * n ↔ m < n); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ ⌜m + n⌝ = ⌜m⌝ + ⌜n⌝ ; (s)% ⊦ # ⌜nf0 to n⌝ = ⌜n⌝ ; (s)% ⊦ ⌜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%m < n ⊦ (nf0 to n)[m] = nf0(m); (s)% ⊦ VFS_PAGE_SIZE ≠ 0; (s)%n ≠ 0 ⊦ (n0 + m * n) / n = n0 / n + m; (s)% ⊦ m < n → m / n = 0; (s)% ⊦ n + 0 = n; (s)%n ≠ 0 ⊦ (m + n0 * n) % n = m % n; (s)% ⊦ m < n → m % n = m; (s)% ⊦ ¬ ⌜m⌝ < # σ0 → (σ0 + σ2)[m] = σ2[m - (# σ0).n]; (s)% ⊦ ⌜m⌝ .n = m; (s)% ⊦ (σ⊥)[n] = zero; (s)%append-assoc: ⊦ (σ0 + σ3) + σ4 = σ0 + σ3 + σ4; from specification stream[buffer-inf]% ⊦ n < VFS_PAGE_SIZE → (zeropage)[n] = zero; (s) Used terms: m0 * VFS_PAGE_SIZE + n1, σ⊥, ⌜ (λ n. (n / VFS_PAGE_SIZE ∈ content ⊃ content[n / VFS_PAGE_SIZE] ;zeropage ) [n % VFS_PAGE_SIZE ]) to n0 * VFS_PAGE_SIZE ⌝