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Proof step 250 from proof refine-read-buf (previous step)

# avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, # buf + n4 + n2 = # buf3, n1 + n5 = n4 + n2, LEB_SIZE = n3 + n2 + # buf0 + # buf, avol0[m2] = mapped(buf0 + buf), avol1[m2] = mapped((buf0 + buf) + subarray(buf2, 0, n2)), n < n4 + n2, n < n2 + n4, m2 < # avol1, n2 < EB_PAGE_SIZE, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0 + # buf, EB_PAGE_SIZE), # buf0 ≤ LEB_SIZE, n1 ≤ n2 + n3, # buf ≤ # buf3, n2 ≤ n1 ⊦ (subarray(buf2, 0, n2) + subarray(mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4))[n] = (subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1) + subarray(buf3, n1 + # buf, n5))[n]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 251:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 252:
# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, n1 + n5 = n4 + n2, # buf + n4 + n2 = # buf3, LEB_SIZE = n3 + n2 + # buf0 + # buf, avol1[m2] = mapped((buf0 + buf) + subarray(buf2, 0, n2)), avol0[m2] = mapped(buf0 + buf), n2 < EB_PAGE_SIZE, m2 < # avol1, n < n2 + n4, n < n4 + n2, n < # subarray(buf2, 0, n2) + # subarray(mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0 + # buf, EB_PAGE_SIZE), n2 ≤ n1, # buf ≤ # buf3, n1 ≤ n2 + n3, # buf0 ≤ LEB_SIZE ⊦ (n < # subarray(buf2, 0, n2) ⊃ subarray(buf2, 0, n2)[n] ;subarray(mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4)[n - # subarray(buf2, 0, n2)] ) = (subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1) + subarray(buf3, n1 + # buf, n5))[n]

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 (mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4), subarray(buf2, 0, n2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ # mkempbuf(n) = n; (s)% ⊦ m ≤ n → (n ≤ m ↔ m = n); (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)%n = # buf ⊦ subarray(buf + buf0, 0, n) = buf; (s)% ⊦ EB_PAGE_SIZE ≠ 0; (s)%m ≠ 0, is-aligned(n1, m) ⊦ is-aligned(n0 + n1, m) ↔ is-aligned(n0, m); (s) Used terms: n, subarray(mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4), subarray(buf2, 0, n2)