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Proof step 15 from proof inv-change (previous step)

LEB_SIZE = n0 + n, m < # avols[v], n1 < n0, n1 < LEB_SIZE, ebm-cons(avols), is-aligned(n, EB_PAGE_SIZE), v ∈ avols, n ≤ LEB_SIZE, n ≤ # buf, n0 ≤ LEB_SIZE, avols[v][m].mapped? → # avols[v][m].data = LEB_SIZE ∧ avols[v][m].fill ≤ LEB_SIZE ∧ is-aligned(avols[v][m].fill, EB_PAGE_SIZE) ∧ range=(avols[v][m].data, avols[v][m].fill, emptyleb, 0, LEB_SIZE - avols[v][m].fill) ⊦ (subarray(buf, 0, n) + mkempbuf(n0))[n + n1] = empty

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 16:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])

Rule argument

spec: array-splice, inst: buffer name: append-at seq: ⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf]) subst: [n, buf0, buf] → [n + n1, mkempbuf(n0), subarray(buf, 0, n)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # subarray(buf, m, n) = n; (s)% ⊦ ¬ m + n < 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 < n → mkempbuf(n)[m] = empty; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ # mkempbuf(n) = n; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n + n1, mkempbuf(n0), subarray(buf, 0, n)