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Proof step 276 from proof refine-read-buf (previous step)

# buf3 = n5 + n1, # avol0 = # avol1, # lpt0 = # avol1, (# buf + # buf4) + n3 + n2 = LEB_SIZE, n1 + n5 = n4 + min(n1, n2), # buf + # buf0 = EB_PAGE_SIZE, avol0[m2] = mapped(buf4), avol1[m2] = mapped(buf4 + buf + subarray(buf0, 0, n2)), n2 < # buf0, m2 < # avol1, # buf < EB_PAGE_SIZE, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf4, EB_PAGE_SIZE), # buf4 ≤ LEB_SIZE, min(n1, n2) ≤ n1 + n5, n1 ≤ n2 + n3, min(n1, n2) ≤ # buf0 ⊦ # buf = 0, subarray(buf0, 0, min(n1, n2)) + subarray(mkempbuf(n1) + subarray(buf3, n1, n5), min(n1, n2), n4) = subarray(subarray(buf0, 0, n2) + mkempbuf(n3), 0, n1) + subarray(buf3, n1, n5), EB_PAGE_SIZE = 1

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 277:
⊦ # buf ≤ m ∧ m ≤ # buf + # buf0 → subarray(buf + buf0, 0, m) = buf + subarray(buf0, 0, m - # buf)
Proof step 278:
# lpt0 = # avol1, # avol0 = # avol1, # buf3 = n5 + n1, # buf + # buf0 = EB_PAGE_SIZE, n1 + n5 = n4 + min(n1, n2), (# buf + # buf4) + n3 + n2 = LEB_SIZE, avol1[m2] = mapped(buf4 + buf + subarray(buf0, 0, n2)), avol0[m2] = mapped(buf4), # buf < EB_PAGE_SIZE, m2 < # avol1, n2 < # buf0, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf4, EB_PAGE_SIZE), min(n1, n2) ≤ # buf0, n1 ≤ n2 + n3, min(n1, n2) ≤ n1 + n5, # buf4 ≤ LEB_SIZE, # subarray(buf0, 0, n2) ≤ n1, n1 ≤ # subarray(buf0, 0, n2) + # mkempbuf(n3) ⊦ # buf = 0, subarray(buf0, 0, min(n1, n2)) + subarray(mkempbuf(n1) + subarray(buf3, n1, n5), min(n1, n2), n4) = (subarray(buf0, 0, n2) + subarray(mkempbuf(n3), 0, n1 - # subarray(buf0, 0, n2))) + subarray(buf3, n1, n5), EB_PAGE_SIZE = 1

Rule argument

spec: array-splice, inst: buffer name: subarray-append-shift-right seq: ⊦ # buf ≤ m ∧ m ≤ # buf + # buf0 → subarray(buf + buf0, 0, m) = buf + subarray(buf0, 0, m - # buf) subst: [m, buf0, buf] → [n1, mkempbuf(n3), subarray(buf0, 0, n2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ # mkempbuf(n) = n; (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m ≤ m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m ≤ n ↔ ¬ n < m; (ss)% ⊦ min(n, m) < n ↔ min(n, m) = m ∧ m < n; (ws)% ⊦ n1 < n0 → min(n0, n1) = n1; (s)% ⊦ m < n → m ≤ n; (s)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ n + 0 = n; (s)%n + m ≤ # buf ⊦ subarray(buf + buf0, n, m) = subarray(buf, n, m); (s)%n2 ≤ n1 ⊦ n1 < # buf → subarray(subarray(buf, 0, n1), 0, n2) = subarray(buf, 0, n2); (s)%min-comm: ⊦ min(n0, n1) = min(n1, n0); from specification min-max% ⊦ m + n = m + n0 ↔ n = n0; (s)% ⊦ m - m = 0; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ buf + buf0 = buf + buf1 ↔ buf0 = buf1; (s) Used terms: n1, mkempbuf(n3), subarray(buf0, 0, n2)