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Proof step 137 from proof refine-read-buf (previous step)

# buf3 = n4 + n2, # avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, (n0 + # buf4) + n3 + n2 = LEB_SIZE, n1 + n5 = n4 + n2, avol0[m2] = mapped(buf4), avol1[m2] = mapped(buf4 + subarray(buf2, 0, n2 + n0)), n6 < n4, m2 < # avol1, n2 + n0 < EB_PAGE_SIZE, n0 < EB_PAGE_SIZE, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf4, EB_PAGE_SIZE), n1 ≤ n2 + n3, # buf4 ≤ LEB_SIZE, n2 ≤ n1 ⊦ EB_PAGE_SIZE = 1, (mkempbuf(n1) + subarray(buf3, n1, n5))[n2 + n6] = (n6 + n2 < n1 ⊃ (subarray(buf2, 0, n2 + n0) + mkempbuf(n3))[n0 + n6 + n2];subarray(buf3, n1, n5)[n6 + n2 - n1]), n0 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 138:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 139:
# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, # buf3 = n4 + n2, n1 + n5 = n4 + n2, (n0 + # buf4) + n3 + n2 = LEB_SIZE, avol1[m2] = mapped(buf4 + subarray(buf2, 0, n2 + n0)), avol0[m2] = mapped(buf4), n0 < EB_PAGE_SIZE, n2 + n0 < EB_PAGE_SIZE, m2 < # avol1, n6 < n4, n2 + n6 < # mkempbuf(n1) + # subarray(buf3, n1, n5), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf4, EB_PAGE_SIZE), n2 ≤ n1, # buf4 ≤ LEB_SIZE, n1 ≤ n2 + n3 ⊦ (n2 + n6 < # mkempbuf(n1) ⊃ mkempbuf(n1)[n2 + n6];subarray(buf3, n1, n5)[n2 + n6 - # mkempbuf(n1)]) = (n6 + n2 < n1 ⊃ (subarray(buf2, 0, n2 + n0) + mkempbuf(n3))[n0 + n6 + n2];subarray(buf3, n1, n5)[n6 + n2 - n1]), EB_PAGE_SIZE = 1, n0 = 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] → [n2 + n6, subarray(buf3, n1, n5), mkempbuf(n1)] flags: <all paths>

Additional information

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