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Proof step 166 from proof refine-write-buf (previous step)

# avol0 = # avol1, # lpt0 = # avol1, # buf1 = EB_PAGE_SIZE, avol0[m4] = mapped(buf2), avol1[m4] = mapped(buf2 + subarray(buf1, 0, m2)), align↓(m2 + n3, EB_PAGE_SIZE) = n0 + m2, m4 < # avol1, m2 < EB_PAGE_SIZE, n1 < n0, n1 < # buf3, n < m2 + n1, awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(n1 + m2, EB_PAGE_SIZE), is-aligned(n0 + m2, EB_PAGE_SIZE), is-aligned(# buf2, EB_PAGE_SIZE), n0 ≤ # buf3, n3 ≤ # buf3, EB_PAGE_SIZE ≤ m2 + n3, # buf2 + m2 + n3 ≤ LEB_SIZE, # buf2 + n0 + m2 ≤ LEB_SIZE, (# buf2 + m2) + n3 ≤ LEB_SIZE, ⌊err1⌋ ⊦ (n < m2 ⊃ buf1[n];subarray(buf3, 0, n1)[n - m2]) = (subarray(buf1, 0, m2) + subarray(buf3, 0, n0))[n], n1 = 0, n3 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 167:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 168:
# buf1 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, avol1[m4] = mapped(buf2 + subarray(buf1, 0, m2)), avol0[m4] = mapped(buf2), align↓(m2 + n3, EB_PAGE_SIZE) = n0 + m2, n < m2 + n1, n1 < # buf3, n1 < n0, m2 < EB_PAGE_SIZE, m4 < # avol1, n < # subarray(buf1, 0, m2) + # subarray(buf3, 0, n0), awbuf-inv(avol1, buffered(m4)), ebm-vol-inv(avol0), is-aligned(# buf2, EB_PAGE_SIZE), is-aligned(n0 + m2, EB_PAGE_SIZE), is-aligned(n1 + m2, EB_PAGE_SIZE), (# buf2 + m2) + n3 ≤ LEB_SIZE, # buf2 + n0 + m2 ≤ LEB_SIZE, # buf2 + m2 + n3 ≤ LEB_SIZE, EB_PAGE_SIZE ≤ m2 + n3, n3 ≤ # buf3, n0 ≤ # buf3, ⌊err1⌋ ⊦ (n < m2 ⊃ buf1[n];subarray(buf3, 0, n1)[n - m2]) = (n < # subarray(buf1, 0, m2) ⊃ subarray(buf1, 0, m2)[n];subarray(buf3, 0, n0)[n - # subarray(buf1, 0, m2)]), n1 = 0, n3 = 0

Rule argument

spec: array-splice, inst: buffer name: append-at seq: ⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf]) subst: [n, buf0, buf] → [n, subarray(buf3, 0, n0), subarray(buf1, 0, m2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ EB_PAGE_SIZE ≠ 0; (s)% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m < n → ¬ n < m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n ≠ 0, m % n = 0 ⊦ align↓(m, n) = m; (s)% ⊦ m + n = m + n0 ↔ n = n0; (s)%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ m < n → n ≠ 0; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%m ≠ 0, is-aligned(n1, m) ⊦ is-aligned(n0 + n1, m) ↔ is-aligned(n0, m); (s) Used terms: n, subarray(buf3, 0, n0), subarray(buf1, 0, m2)