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Proof step 54 from proof encode-node (previous step)

# buf = n0 + n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd), # buf0 = # buf, # buf2 = # buf, align↑(node-size-headerless(nd), 2 * NODE_HEADER_SIZE) = n2 + node-size-headerless(nd), 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), subarray(buf0, 0, n) = subarray(buf, 0, n), 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(buf2, 0, n + NODE_HEADER_SIZE) = subarray(buf0, 0, n + NODE_HEADER_SIZE), is-encoded(nd, subarray(buf2, n + NODE_HEADER_SIZE, node-size-headerless(nd))), is-encoded(nodeheader(node-size-headerless(nd), false), subarray(buf0, n, NODE_HEADER_SIZE)), 2 * NODE_HEADER_SIZE + n + n2 + node-size-headerless(nd) ≤ # buf, (n2 + node-size-headerless(nd)) + n + NODE_HEADER_SIZE ≤ # buf, n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd) ≤ # buf, n + NODE_HEADER_SIZE ≤ # buf, (n + NODE_HEADER_SIZE) + node-size-headerless(nd) ≤ # buf, (n + NODE_HEADER_SIZE + node-size-headerless(nd)) + n2 ≤ # buf ⊦ subarray(subarray(buf2, 0, n + NODE_HEADER_SIZE + node-size-headerless(nd)), 0, n) = subarray(buf, 0, n) ∧ subarray(subarray(buf2, (n + NODE_HEADER_SIZE + node-size-headerless(nd)) + n2, NODE_HEADER_SIZE + n0), NODE_HEADER_SIZE, n0) = subarray(buf, n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd), n0) ∧ (∃ buf1, buf3. subarray ((subarray(buf2, 0, n + NODE_HEADER_SIZE + node-size-headerless(nd)) + fill(mkbuf(n2), empty)) + 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-copy, inst: buffer name: subarray-subarray-elim seq: ⊦ n0 + n1 ≤ # buf ∧ m0 + m1 ≤ n1 → subarray(subarray(buf, n0, n1), m0, m1) = subarray(buf, n0 + m0, m1) subst: [m1, m0, n1, n0, buf] → [n, 0, n + NODE_HEADER_SIZE + node-size-headerless(nd), 0, buf2] flags: <all paths>

Additional information

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