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Proof step 48 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)) + n2 ≤ # buf, (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(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, # buf - (n + NODE_HEADER_SIZE + node-size-headerless(nd)) + n2), 2 * NODE_HEADER_SIZE + n2 - NODE_HEADER_SIZE - n2, 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))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 49:
# buf = n0 + n + 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd), # buf0 = # buf, # buf2 = # buf, 2 * NODE_HEADER_SIZE + n2 - NODE_HEADER_SIZE = n1 + n2, 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, n2 ≤ 2 * NODE_HEADER_SIZE + n2 - NODE_HEADER_SIZE ⊦ 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, # buf - (n + NODE_HEADER_SIZE + node-size-headerless(nd)) + n2), n1, 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))

Rule argument

spec: nat name: elim-sub seq: n ≤ m ⊦ n0 = m - n ↔ m = n0 + n subst: [n, m, n0] → [n2, 2 * NODE_HEADER_SIZE + n2 - NODE_HEADER_SIZE, n1]

Additional information

Used simplifier rules: ⊦ m ≤ n ↔ ¬ n < m; (ss)%m2 ≠ 0 ⊦ ¬ (* m2) * m + n < m; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%n0 ≤ n ⊦ m ≤ n - n0 ↔ m + n0 ≤ n; (s)% ⊦ m + n ≤ m + n0 ↔ n ≤ n0; (s)% ⊦ ¬ 2 * n < n; (s)%n ≤ m ⊦ n0 = m - n ↔ m = n0 + n; (elim) Used terms: n2, 2 * NODE_HEADER_SIZE + n2 - NODE_HEADER_SIZE, n1