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Proof step 40 from proof prepost-io-write-data (previous step)

# apebs = # wlar, LEB_SIZE = (n1 + n2) + m0, wlar[n] = wl-entry(m3, used), apebs[n] = mkpeb(aehdr, avhdr, buf0, m, false), avhdr.valid?, aehdr.valid?, n < # wlar, n2 < m2, bflipset-cons(bflipset, wlar), is-aligned(m0, EB_PAGE_SIZE), is-aligned(m2, EB_PAGE_SIZE), is-aligned(n2, EB_PAGE_SIZE), ubi-io-inv(apebs[n, mkpeb(aehdr, avhdr, subarray(buf0, 0, m0) + subarray(buf, m1, n2) + mkempbuf(n1), m0 + n2, false)]), ubi-io-inv(apebs), ubi-io-peb-inv(mkpeb(aehdr, avhdr, buf0, m, false)), unique-sqns(apebs), valid-ebatbl(apebs, wlar, vols, n3), valid-eraseq(q), valid-eraseq(wlar, q), valid-eraseq(apebs, q), valid-global-sqn(apebs, n0), valid-peb(mkpeb(aehdr, avhdr, buf0, m, false)), valid-vtbl(apebs, vols, wlar, n3), valid-wl(apebs, wlar), n2 + m0 ≤ LEB_SIZE, m0 ≤ LEB_SIZE, m0 + m2 ≤ LEB_SIZE, m1 + m2 ≤ # buf, m ≤ m0, m1 + n2 ≤ # buf, m ≤ LEB_SIZE, ⌊err0⌋ ⊦ n2 = 0, n3 = n, valid-peb(mkpeb(aehdr, avhdr, subarray(buf0, 0, m0) + subarray(buf, m1, n2) + mkempbuf(n1), m0 + n2, false)), n ∈ q, VTBL_VOLID ∈ vols, n3 ∈ q, unique-sqns(apebs[n, mkpeb(aehdr, avhdr, subarray(buf0, 0, m0) + subarray(buf, m1, n2) + mkempbuf(n1), m0 + n2, false)]) ∧ valid-wl(apebs[n, mkpeb(aehdr, avhdr, subarray(buf0, 0, m0) + subarray(buf, m1, n2) + mkempbuf(n1), m0 + n2, false)], wlar) ∧ valid-ebatbl(apebs[n, mkpeb(aehdr, avhdr, subarray(buf0, 0, m0) + subarray(buf, m1, n2) + mkempbuf(n1), m0 + n2, false)], wlar, vols, n3)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: valid-peb name: valid-peb0-write-append seq: ⊦ valid-peb(mkpeb(aehdr, avhdr, buf, m, false)) ∧ m ≤ m0 ∧ m0 + # buf0 = LEB_SIZE ∧ ubi-io-peb-inv(mkpeb(aehdr, avhdr, buf, m, false)) → valid-peb (mkpeb (aehdr, avhdr, subarray(buf, 0, m0) + buf0, m0 + m1, false)) subst: [m, m1, buf0, m0, buf, avhdr, aehdr] → [m, n2, subarray(buf, m1, n2) + mkempbuf(n1), m0, buf0, avhdr, aehdr] flags: <all paths>

Additional information

Used simplifier rules: ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ # mkempbuf(n) = n; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: m, n2, subarray(buf, m1, n2) + mkempbuf(n1), m0, buf0, avhdr, aehdr