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Proof step 42 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))), 2 * NODE_HEADER_SIZE + n + n2 + node-size-headerless(nd) ≤ # 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 ⊦ 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) ∧ (∃ buf, buf1. subarray (copy (validtrailer, 0, fill(buf2, empty, n + NODE_HEADER_SIZE + node-size-headerless(nd), n2), n + NODE_HEADER_SIZE + n2 + node-size-headerless(nd), NODE_HEADER_SIZE), n, 2 * NODE_HEADER_SIZE + n2 + node-size-headerless(nd)) = buf + buf1 + validtrailer ∧ is-encoded(nodeheader(node-size-headerless(nd), false), buf) ∧ is-encoded(nd, subarray(buf1, 0, node-size-headerless(nd))) ∧ # buf1 = 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: copy2append0 seq: ⊦ n1 ≤ # buf ∧ n0 + n1 ≤ # buf0 → copy(buf, 0, buf0, n0, n1) = subarray(buf0, 0, n0) + subarray(buf, 0, n1) + subarray(buf0, n0 + n1, # buf0 - n0 - n1) subst: [n1, n0, buf0, buf] → [NODE_HEADER_SIZE, n + NODE_HEADER_SIZE + n2 + node-size-headerless(nd), fill (buf2, empty, n + NODE_HEADER_SIZE + node-size-headerless(nd), n2), validtrailer] flags: <all paths>

Additional information

Used simplifier rules: buf-size-zero: ⊦ # buf = 0 ↔ buf = mkbuf(0);% ⊦ # validtrailer = NODE_HEADER_SIZE; (s)% ⊦ m ≤ m; (s)% ⊦ m + m = 2 * m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic% ⊦ # fill(buf, d, m, n) = # buf; (s)% ⊦ 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%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic% ⊦ 2 * NODE_HEADER_SIZE + n ≠ 0; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ 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)%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)% ⊦ m < n → (n ≤ m ↔ false); (s)%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ n + 0 = n; (s)% ⊦ m + n < m + n0 ↔ n < n0; (s)% ⊦ ¬ m + n < m; (s) Used terms: NODE_HEADER_SIZE, n + NODE_HEADER_SIZE + n2 + node-size-headerless(nd), fill(buf2, empty, n + NODE_HEADER_SIZE + node-size-headerless(nd), n2), validtrailer