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Theorem bubble-down-inv

⊦ bubble-down-inv(n, bh, ar, bh0, ar0) ↔ bh.size ≤ # bh.ar ∧ # bh.ar = # bh0.ar ∧ bh.size = bh0.size ∧ id-idx(ar0, ar) ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ ( (2 * n + 1 < bh.size → ¬ bh.ar[2 * n + 1].key < bh.ar[n].key) ∧ (2 * n + 2 < bh.size → ¬ bh.ar[2 * n + 2].key < bh.ar[n].key) → heap-order(bh.ar, bh.size)) ∧ ( 2 * n + 1 < bh.size ∧ bh.ar[2 * n + 1].key < bh.ar[n].key ∨ 2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[n].key → heap-order(bh.ar[n, (2 * n + 2 < bh.size ∧ bh.ar[2 * n + 2].key < bh.ar[2 * n + 1].key ⊃ bh.ar[2 * n + 2];bh.ar[2 * n + 1])], bh.size)) ∧ keyset(bh.ar, bh.size) = keyset(bh0.ar, bh0.size)