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Proof step 93 from proof refine-read-buf (previous step)

# buf2 = EB_PAGE_SIZE, # lpt0 = # avol1, # avol0 = # avol1, # buf + n4 + n2 = # buf3, n1 + n5 = n4 + n2, LEB_SIZE = n3 + n2 + # buf0 + # buf, avol1[m2] = mapped((buf0 + buf) + subarray(buf2, 0, n2)), avol0[m2] = mapped(buf0 + buf), n < n4 + n2, m2 < # avol1, n2 < EB_PAGE_SIZE, n < n2 + n4, awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0 + # buf, EB_PAGE_SIZE), # buf ≤ # buf3, n1 ≤ n2 + n3, # buf0 ≤ LEB_SIZE, n2 ≤ n1 ⊦ (n < n2 ⊃ buf2[n];subarray(mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4)[n - n2]) = (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 94:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 95:
# avol0 = # avol1, # lpt0 = # avol1, # buf2 = EB_PAGE_SIZE, n1 + n5 = n4 + n2, # buf + n4 + n2 = # buf3, LEB_SIZE = n3 + n2 + # buf0 + # buf, avol0[m2] = mapped(buf0 + buf), avol1[m2] = mapped((buf0 + buf) + subarray(buf2, 0, n2)), n < n2 + n4, n2 < EB_PAGE_SIZE, m2 < # avol1, n < n4 + n2, n < # subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1) + # subarray(buf3, n1 + # buf, n5), awbuf-inv(avol1, buffered(m2)), ebm-vol-inv(avol0), is-aligned(# buf0 + # buf, EB_PAGE_SIZE), n2 ≤ n1, # buf0 ≤ LEB_SIZE, n1 ≤ n2 + n3, # buf ≤ # buf3 ⊦ (n < n2 ⊃ buf2[n];subarray(mkempbuf(n1) + subarray(buf3, n1 + # buf, n5), n2, n4)[n - n2]) = (n < # subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1) ⊃ subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1)[n] ;subarray(buf3, n1 + # buf, n5)[n - # subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1)] )

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, n1 + # buf, n5), subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%c: ⊦ m + n = n + m; from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ n < n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%n = # buf ⊦ subarray(buf + buf0, 0, n) = buf; (s) Used terms: n, subarray(buf3, n1 + # buf, n5), subarray(subarray(buf2, 0, n2) + mkempbuf(n3), 0, n1)