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Proof step 8 from proof align-down-app-same (previous step)

# buf0 % EB_PAGE_SIZE < # buf0, n < # buf0 + # buf1, # buf0 % EB_PAGE_SIZE + # buf1 < EB_PAGE_SIZE, n < align↓(# buf0 + # buf1, EB_PAGE_SIZE), align↓(# buf0, EB_PAGE_SIZE) < # buf0 + # buf1, align↓(# buf0, EB_PAGE_SIZE) < align↓(# buf0 + # buf1, EB_PAGE_SIZE), EB_PAGE_SIZE ≤ # buf0 + # buf1, align↓(# buf0, EB_PAGE_SIZE) ≤ n ⊦ (buf0 + buf1)[n] = subarray(buf0, 0, align↓(# buf0, EB_PAGE_SIZE))[n]

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-copy, inst: buffer name: subarray-at seq: ⊦ n0 < n → subarray(buf, m, n)[n0] = buf[m + n0] subst: [n0, n, m, buf] → [n, align↓(# buf0, EB_PAGE_SIZE), 0, buf0] flags: <all paths>

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ EB_PAGE_SIZE ≠ 0; (s)%aligndown-is-smaller: m ≠ 0 ⊦ n0 < align↓(n, m) → n0 < n; (forward) from specification align%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)% ⊦ n < # buf → (buf + buf0)[n] = buf[n]; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%n ≠ 0, m % n = 0 ⊦ align↓(m, n) = m; (s)% ⊦ # buf ≤ n ∧ n < # buf + # buf0 → (buf + buf0)[n] = buf0[n - # buf]; (s)%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 < m ↔ false); (s)% ⊦ m < m + n ↔ n ≠ 0; (s) Used terms: n, align↓(# buf0, EB_PAGE_SIZE), 0, buf0