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Proof step 52 from proof valid-peb-write-valid-peb (previous step)

# buf = LEB_SIZE, LEB_SIZE = n0 + m, LEB_SIZE = n1 + m0, apebs[n] = mkpeb(aehdr, avidhdr(v, m2, m4, m3, checksum(buf, m3)), buf, m, false), aehdr.valid?, n2 < LEB_SIZE, n < # apebs, is-aligned(m, EB_PAGE_SIZE), m3 ≤ LEB_SIZE, m3 ≤ datasize(buf), m ≤ m0, m0 ≤ LEB_SIZE, m ≤ LEB_SIZE, ∀ n3. n3 < n0 → buf[m + n3] = empty ⊦ buf[n2] = (subarray(buf, 0, m0) + mkempbuf(n1))[n2], aehdr = empty, m3 = 0, m3 ≤ datasize(subarray(buf, 0, m0) + buf1) ∧ checksum(buf, m3) = checksum(subarray(buf, 0, m0) + buf1, m3)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 53:
⊦ n < # buf → (buf + buf0)[n] = buf[n]
Proof step 54:
# buf = LEB_SIZE, LEB_SIZE = n1 + m0, LEB_SIZE = n0 + m, apebs[n] = mkpeb(aehdr, avidhdr(v, m2, m4, m3, checksum(buf, m3)), buf, m, false), aehdr.valid?, n < # apebs, n2 < LEB_SIZE, is-aligned(m, EB_PAGE_SIZE), m ≤ LEB_SIZE, m0 ≤ LEB_SIZE, m ≤ m0, m3 ≤ datasize(buf), m3 ≤ LEB_SIZE, ∀ n3. n3 < n0 → buf[m + n3] = empty ⊦ buf[n2] = (subarray(buf, 0, m0) + mkempbuf(n1))[n2], aehdr = empty, m3 = 0, n2 < # subarray(buf, 0, m0), m3 ≤ datasize(subarray(buf, 0, m0) + buf1) ∧ checksum(buf, m3) = checksum(subarray(buf, 0, m0) + buf1, m3)

Rule argument

spec: array-splice, inst: buffer name: append-at1 seq: ⊦ n < # buf → (buf + buf0)[n] = buf[n] subst: [n, buf0, buf] → [n2, mkempbuf(n1), subarray(buf, 0, m0)] 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)% ⊦ n + 0 = n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: n2, mkempbuf(n1), subarray(buf, 0, m0)