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Proof step 49 from proof refine-write (previous step)

LEB_SIZE = (n4 + n) + n0 + n2, avols[default-volid][m2] = mapped(buf0, n2), m2 < # avols[default-volid], n1 < n2 + n0, n1 < LEB_SIZE, ebm-cons(avols), ebm-vol-inv(abs(avols[default-volid])), is-aligned(n, EB_PAGE_SIZE), is-aligned(n0, EB_PAGE_SIZE), default-volid ∈ avols, n + n0 + n2 ≤ LEB_SIZE, n0 + n2 ≤ LEB_SIZE, n2 ≤ LEB_SIZE, n3 + n ≤ # buf, (n0 + n2) + n ≤ LEB_SIZE ⊦ (subarray(buf0, 0, n2) + mkempbuf(n0))[n1] = buf0[n1], n = 0

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 50:
⊦ n < # buf + # buf0 → (buf + buf0)[n] = (n < # buf ⊃ buf[n];buf0[n - # buf])
Proof step 51:
LEB_SIZE = (n4 + n) + n0 + n2, avols[default-volid][m2] = mapped(buf0, n2), n1 < LEB_SIZE, n1 < n2 + n0, m2 < # avols[default-volid], n1 < # subarray(buf0, 0, n2) + # mkempbuf(n0), ebm-cons(avols), ebm-vol-inv(abs(avols[default-volid])), is-aligned(n0, EB_PAGE_SIZE), is-aligned(n, EB_PAGE_SIZE), default-volid ∈ avols, (n0 + n2) + n ≤ LEB_SIZE, n3 + n ≤ # buf, n2 ≤ LEB_SIZE, n0 + n2 ≤ LEB_SIZE, n + n0 + n2 ≤ LEB_SIZE ⊦ (n1 < # subarray(buf0, 0, n2) ⊃ subarray(buf0, 0, n2)[n1];mkempbuf(n0)[n1 - # subarray(buf0, 0, n2)]) = buf0[n1], n = 0

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] → [n1, mkempbuf(n0), subarray(buf0, 0, n2)] flags: <all paths>

Additional information

Used simplifier rules: ⊦ v ∈ avols ∧ m < # avols[v] ∧ avols[v][m] = mapped(buf, m0) ∧ ebm-cons(avols) → # buf = LEB_SIZE; (s)% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ # subarray(buf, m, n) = n; (s)% ⊦ # mkempbuf(n) = n; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s) Used terms: n1, mkempbuf(n0), subarray(buf0, 0, n2)