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Proof step 34 from proof valid-peb-write-valid-peb (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 35:
⊦ m ≤ # buf0 + # buf1 ∧ (m ≥ # buf0 → datasize(buf1, m - # buf0) = 0) → datasize(buf0 + buf1, m) = datasize(buf0, min(m, # buf0))
Proof step 36:
LEB_SIZE = n0 + m, LEB_SIZE = n1 + m0, apebs[n] = mkpeb(aehdr, avidhdr(v, m2, m4, m3, checksum(subarray(buf, 0, m0), m3)), subarray(buf, 0, m0) + mkempbuf(n1), m, false), subarray(subarray(buf, 0, m0) + mkempbuf(n1), m, n0) = mkempbuf(n0), aehdr.valid?, n < # apebs, is-aligned(m, EB_PAGE_SIZE), m3 ≤ LEB_SIZE, m3 ≤ m0, m ≤ m0, m0 ≤ LEB_SIZE, m ≤ LEB_SIZE, m3 ≤ datasize(subarray(buf, 0, m0)), m0 ≤ # subarray(buf, 0, m0) + # mkempbuf(n1), m0 ≥ # subarray(buf, 0, m0) → datasize(mkempbuf(n1), m0 - # subarray(buf, 0, m0)) = 0 ⊦ aehdr = empty, m3 = 0, m3 ≤ (datasize(buf1) = 0 ⊃ datasize(subarray(buf, 0, m0), min(m0, # subarray(buf, 0, m0)));m0 + datasize(buf1)) ∧ checksum(subarray(buf, 0, m0) + mkempbuf(n1), m3) = checksum(subarray(buf, 0, m0) + buf1, m3)

Rule argument

spec: datasize name: datasize-append-empty-h seq: ⊦ m ≤ # buf0 + # buf1 ∧ (m ≥ # buf0 → datasize(buf1, m - # buf0) = 0) → datasize(buf0 + buf1, m) = datasize(buf0, min(m, # buf0)) subst: [m, buf1, buf0] → [m0, mkempbuf(n1), subarray(buf, 0, m0)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ # mkempbuf(n) = n; (s)% ⊦ m ≤ m + n; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m - m = 0; (s)% ⊦ datasize(buf, 0) = 0; (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ n ≥ m ↔ m ≤ n; (s) Used terms: m0, mkempbuf(n1), subarray(buf, 0, m0)