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Proof step 3 from proof lebs-write-last (previous step)

# buf = LEB_SIZE, n + m < # avol ⊦ lebs(avol[n + m, mapped(buf)], n, m + 1) = lebs(avol, n, m) + buf

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ n + m < # avol → lebs(avol, n, m + 1) = lebs(avol, n, m) + (avol[n + m].mapped? ⊃ avol[n + m].data + mkempbuf(LEB_SIZE - # avol[n + m].data);mkempbuf(LEB_SIZE))

Rule argument

spec: <toplevel> name: lebs-rec seq: ⊦ n + m < # avol → lebs(avol, n, m + 1) = lebs(avol, n, m) + (avol[n + m].mapped? ⊃ avol[n + m].data + mkempbuf(LEB_SIZE - # avol[n + m].data) ;mkempbuf(LEB_SIZE) ) subst: [m, n, avol] → [m, n, avol[n + m, mapped(buf)]] flags: <all paths>

Additional information

Used simplifier rules: lebs-write-above: n + m ≤ n0, n + m ≤ # avol ⊦ lebs(avol[n0, aleb], n, m) = lebs(avol, n, m);% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ m ≤ m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ m < n → m ≤ n; (s)% ⊦ m < # avol → avol[m, aleb][m] = aleb; (s)% ⊦ mapped(buf).mapped?; (s)% ⊦ mapped(buf).data = buf; (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]% ⊦ buf + buf0 = buf + buf1 ↔ buf0 = buf1; (s)% ⊦ buf + buf0 = buf ↔ # buf0 = 0; (s)% ⊦ # mkempbuf(n) = n; (s)% ⊦ # avol[n, aleb] = # avol; (s)% ⊦ m - m = 0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s) Used terms: m, n, avol[n + m, mapped(buf)]