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Proof step 45 from proof encode-node (previous step)

# buf2 = # buf, # buf0 = # buf, # buf = n0 + n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd), align↑(node-size-headerless(nd), 2 * NODE_HEADER_SIZE) = n2 + node-size-headerless(nd), subarray(buf2, 0, n + NODE_HEADER_SIZE) = subarray(buf0, 0, n + NODE_HEADER_SIZE), subarray(buf2, (n + NODE_HEADER_SIZE) + node-size-headerless(nd), NODE_HEADER_SIZE + n0 + n2) = subarray(buf0, (n + NODE_HEADER_SIZE) + node-size-headerless(nd), NODE_HEADER_SIZE + n0 + n2), subarray(buf0, 0, n) = subarray(buf, 0, n), subarray(buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n0 + n2) = subarray(buf, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n0 + n2), is-encoded(nodeheader(node-size-headerless(nd), false), subarray(buf0, n, NODE_HEADER_SIZE)), is-encoded(nd, subarray(buf2, n + NODE_HEADER_SIZE, node-size-headerless(nd))), (n + NODE_HEADER_SIZE) + node-size-headerless(nd) ≤ # buf, n + NODE_HEADER_SIZE ≤ # buf, n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd) ≤ # buf, (n2 + node-size-headerless(nd)) + n + NODE_HEADER_SIZE ≤ # buf, 2 * NODE_HEADER_SIZE + n + n2 + node-size-headerless(nd) ≤ # buf ⊦ subarray(fill(buf2, empty, n + NODE_HEADER_SIZE + node-size-headerless(nd), n2), 0, n) = subarray(buf, 0, n) ∧ subarray(fill(buf2, empty, n + NODE_HEADER_SIZE + node-size-headerless(nd), n2), n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd), n0) = subarray(buf, n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd), n0) ∧ (∃ buf1, buf3. subarray ( subarray (fill(buf2, empty, n + NODE_HEADER_SIZE + node-size-headerless(nd), n2), 0, n + NODE_HEADER_SIZE + n2 + node-size-headerless(nd)) + validtrailer, n, 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd)) = buf1 + buf3 + validtrailer ∧ is-encoded(nodeheader(node-size-headerless(nd), false), buf1) ∧ is-encoded(nd, subarray(buf3, 0, node-size-headerless(nd))) ∧ # buf3 = n2 + node-size-headerless(nd))

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: array-splice, inst: buffer name: fill2append seq: ⊦ n0 + n1 ≤ # buf → fill(buf, d, n0, n1) = subarray(buf, 0, n0) + fill(mkbuf(n1), d) + subarray(buf, n0 + n1, # buf - n0 + n1) subst: [n1, n0, d, buf] → [n2, n + NODE_HEADER_SIZE + node-size-headerless(nd), empty, buf2] flags: <all paths>

Additional information

Used simplifier rules: buf-size-zero: ⊦ # buf = 0 ↔ buf = mkbuf(0);% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)% ⊦ m ≤ n ↔ ¬ n < m; (ss)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ NODE_HEADER_SIZE + n ≠ 0; (s)% ⊦ m ≤ n → (n < m ↔ false); (s)%# buf = n ⊦ subarray(buf, 0, n) = buf; (s)% ⊦ 2 * m = 0 ↔ m = 0; (s)% ⊦ NODE_HEADER_SIZE ≠ 0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%n ≠ 0, m % n = 0 ⊦ align↑(m, n) = m; (s)% ⊦ m = m + n ↔ n = 0; (s)% ⊦ fill(buf, d, m, 0) = buf; (s)% ⊦ m + n = 0 ↔ m = 0 ∧ n = 0; (ws)% ⊦ # fill(buf, d, m, n) = # buf; (s)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ # validtrailer = NODE_HEADER_SIZE; (s)%m2 ≠ 0 ⊦ (* m2) * m + n ≤ m + n0 ↔ (m2 - 1) * m + n ≤ n0; (s)% ⊦ m0 * 1 = m0; (s)%# buf ≤ n0, n0 + m ≤ # buf + # buf0 ⊦ subarray(buf + buf0, n0, m) = subarray(buf0, n0 - # buf, m); (s)%append-a: ⊦ (buf + buf0) + buf1 = buf + buf0 + buf1; from specification array-splice[buffer]%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ n + 0 = n; (s) Used terms: n2, n + NODE_HEADER_SIZE + node-size-headerless(nd), empty, buf2