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Proof step 170 from proof encode-node (previous step)

# buf2 = # buf, # buf0 = # buf, # buf = n0 + n + 2 * NODE_HEADER_SIZE + n3 + node-size-headerless(nd), buf2[2 * NODE_HEADER_SIZE + n + node-size-headerless(nd) + n2 + n3] = buf0[2 * NODE_HEADER_SIZE + n + node-size-headerless(nd) + n2 + n3], align↑(node-size-headerless(nd), 2 * NODE_HEADER_SIZE) = n3 + node-size-headerless(nd), subarray(buf2, 0, n + NODE_HEADER_SIZE) = subarray(buf0, 0, n + NODE_HEADER_SIZE), subarray(buf0, 0, n) = subarray(buf, 0, n), subarray(buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n0 + n3) = subarray(buf, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n0 + n3), n2 < n0, is-encoded(nodeheader(node-size-headerless(nd), false), subarray(buf0, n, NODE_HEADER_SIZE)), (n + NODE_HEADER_SIZE) + node-size-headerless(nd) ≤ # buf, n + NODE_HEADER_SIZE ≤ # buf, n + 2 * NODE_HEADER_SIZE + n3 + node-size-headerless(nd) ≤ # buf, ⌊err⌋ ⊦ buf0[2 * NODE_HEADER_SIZE + n + node-size-headerless(nd) + n2 + n3] = buf[(n + 2 * NODE_HEADER_SIZE + n3 + node-size-headerless(nd)) + n2]

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 171:
⊦ buf = buf0 ↔ # buf = # buf0 ∧ (∀ n. n < # buf → buf[n] = buf0[n])
Proof step 172:
# subarray(buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0) = # subarray(buf, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0), # buf = n0 + n + 2 * NODE_HEADER_SIZE + n3 + node-size-headerless(nd), # buf0 = # buf, # buf2 = # buf, buf2[2 * NODE_HEADER_SIZE + n + node-size-headerless(nd) + n2 + n3] = buf0[2 * NODE_HEADER_SIZE + n + node-size-headerless(nd) + n2 + n3], align↑(node-size-headerless(nd), 2 * NODE_HEADER_SIZE) = n3 + node-size-headerless(nd), subarray(buf0, 0, n) = subarray(buf, 0, n), subarray(buf2, 0, n + NODE_HEADER_SIZE) = subarray(buf0, 0, n + NODE_HEADER_SIZE), n2 < n0, is-encoded(nodeheader(node-size-headerless(nd), false), subarray(buf0, n, NODE_HEADER_SIZE)), n + 2 * NODE_HEADER_SIZE + n3 + node-size-headerless(nd) ≤ # buf, n + NODE_HEADER_SIZE ≤ # buf, (n + NODE_HEADER_SIZE) + node-size-headerless(nd) ≤ # buf, ⌊err⌋ , ∀ n1. n1 < # subarray(buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0) → subarray(buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0)[n1] = subarray(buf, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0)[n1] ⊦ buf0[2 * NODE_HEADER_SIZE + n + node-size-headerless(nd) + n2 + n3] = buf[(n + 2 * NODE_HEADER_SIZE + n3 + node-size-headerless(nd)) + n2]

Rule argument

spec: array, inst: buffer name: Extension seq: ⊦ buf = buf0 ↔ # buf = # buf0 ∧ (∀ n. n < # buf → buf[n] = buf0[n]) subst: [buf0, buf] → [subarray (buf, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0), subarray (buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0)] flags: <all paths>

Additional information

Used simplifier rules: a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic Used terms: subarray (buf, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0), subarray (buf0, n + NODE_HEADER_SIZE, node-size-headerless(nd) + NODE_HEADER_SIZE + n3 + n0)