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Proof step 228 from proof refine-read-buf (previous step)

# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, # buf3 = n5 + n1, n1 + n5 = n4 + min(n1, n2), LEB_SIZE = n0 + # buf0 + # buf, avol1[m2] = mapped(buf0 + buf), avol0[m2] = mapped(buf0 + buf), subarray(buf2, 0, n2 + n3) = subarray(buf, 0, n2 + n3), n6 < n4, n3 + n2 < # buf, # buf0 + 1 < LEB_SIZE, # buf0 < LEB_SIZE, n3 < # buf, n2 + n3 < EB_PAGE_SIZE, m2 < # avol1, n3 < EB_PAGE_SIZE, (n3 + # buf0) + n2 < LEB_SIZE, n3 < n0 + # buf, min(n1, n2) < n1 + n5, min(n1, n2) < n5 + n1, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0, EB_PAGE_SIZE), n3 + min(n1, n2) ≤ EB_PAGE_SIZE, n3 + n1 ≤ # buf + n0 ⊦ # buf = 1, EB_PAGE_SIZE = 1, (subarray(buf + mkempbuf(n0), n3, n1) + subarray(buf3, n1, n5))[min(n1, n2) + n6] = (n6 + min(n1, n2) < n1 ⊃ (buf + mkempbuf(n0))[n3 + n6 + min(n1, n2)];subarray(buf3, n1, n5)[n6 + min(n1, n2) - n1]), n3 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 229:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])

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] → [min(n1, n2) + n6, subarray(buf3, n1, n5), subarray(buf + mkempbuf(n0), n3, n1)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ n0 < n → subarray(buf, m, n)[n0] = buf[m + n0]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ m = m0 + n0 ∧ n < n0 → m0 + n < m; (s) Used terms: min(n1, n2) + n6, subarray(buf3, n1, n5), subarray(buf + mkempbuf(n0), n3, n1)