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Proof step 13 from proof ls-datasize-zero (previous step)

# buf = n1 + n, n0 < n1 + n, n ≤ n0, ∀ n0. n0 < n → buf[n0] = empty ⊦ buf[n0] = (mkempbuf(n) + subarray(buf, n, n1))[n0], n1 = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 14:
⊦ # buf ≤ n ∧ n < # buf + # buf0 → (buf + buf0)[n] = buf0[n - # buf]
Proof step 15:
# buf = n1 + n, n0 < n1 + n, n0 < # mkempbuf(n) + # subarray(buf, n, n1), n ≤ n0, # mkempbuf(n) ≤ n0, ∀ n0. n0 < n → buf[n0] = empty ⊦ buf[n0] = subarray(buf, n, n1)[n0 - # mkempbuf(n)], n1 = 0

Rule argument

spec: array-splice, inst: buffer name: append-at2 seq: ⊦ # buf ≤ n ∧ n < # buf + # buf0 → (buf + buf0)[n] = buf0[n - # buf] subst: [n, buf0, buf] → [n0, subarray(buf, n, n1), mkempbuf(n)] flags: <all paths>

Additional information

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