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Proof step 46 from proof inv-write (previous step)

LEB_SIZE = n1 + m1 + n, m2 < # avols[v], n0 < n1, n0 < LEB_SIZE, ebm-cons(avols), is-aligned(n, EB_PAGE_SIZE), is-aligned(m1, EB_PAGE_SIZE), v ∈ avols, n + n2 ≤ # buf, m1 + n ≤ LEB_SIZE, n2 + n ≤ # buf, m1 ≤ LEB_SIZE, n + m1 ≤ LEB_SIZE, n1 ≤ LEB_SIZE ⊦ (mkempbuf(m1) + subarray(buf, n2, n) + mkempbuf(n1))[(m1 + n) + n0] = empty, avols[v][m2].mapped?

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 47:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 48:
LEB_SIZE = n1 + m1 + n, n0 < LEB_SIZE, n0 < n1, m2 < # avols[v], (m1 + n) + n0 < # mkempbuf(m1) + #(subarray(buf, n2, n) + mkempbuf(n1)), ebm-cons(avols), is-aligned(m1, EB_PAGE_SIZE), is-aligned(n, EB_PAGE_SIZE), v ∈ avols, n1 ≤ LEB_SIZE, n + m1 ≤ LEB_SIZE, m1 ≤ LEB_SIZE, n2 + n ≤ # buf, m1 + n ≤ LEB_SIZE, n + n2 ≤ # buf ⊦ ((m1 + n) + n0 < # mkempbuf(m1) ⊃ mkempbuf(m1)[(m1 + n) + n0];(subarray(buf, n2, n) + mkempbuf(n1))[(m1 + n) + n0 - # mkempbuf(m1)]) = empty, avols[v][m2].mapped?

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] → [(m1 + n) + n0, subarray(buf, n2, n) + mkempbuf(n1), mkempbuf(m1)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # mkempbuf(n) = n; (s)% ⊦ #(buf + buf0) = # buf + # buf0; (s)% ⊦ # subarray(buf, m, 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% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%c: ⊦ m + n = n + m; from specification nat-basic Used terms: (m1 + n) + n0, subarray(buf, n2, n) + mkempbuf(n1), mkempbuf(m1)