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Proof step 143 from proof prepost-bubble-down (previous step)

# kar0 = # kar, kar0[2 * n0 + 2] = key-index(key2, m), kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), kar0[n0] = key-index(key3, m3), kar0[4 + 4 * n0] = key-index(key8, m4), kar0[3 + 4 * n0] = key-index(key7, m7), kar0[2 * n0 + 1] = key-index(key6, m6), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), key8 < key6, key7 < key6, 2 * n0 + 2 < # kar, n0 + 2 < n2, n0 + 1 < n2, m6 < # ar, m3 < # ar, 2 * n0 + 1 < # kar, n0 < # kar, key8 < key7, 3 * n0 + 3 < n2, key8 < key3, key7 < key3, heap-order(kar0[n0, key-index(key6, m6)], n2 + n0), id-idx(ar1, ar), no-dups-idx(kar0, n2 + n0), no-dups-idx(kar, n2 + n0), valid-idx(kar0, ar, n2 + n0), valid-idx(kar, ar1, n2 + n0), n2 + n0 ≤ # kar, n2 ≤ 3 * n0 + 4, ¬ key6 < key3 ∧ ¬ key2 < key3 → heap-order(kar0, n2 + n0), 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), key6 < key3 ∨ key2 < key3 ⊦ key2 < key6, heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, key-index(key7, m7)], n2 + n0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 144:
⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key)
Proof step 145:
# kar0 = # kar, kar0[2 * n0 + 1] = key-index(key6, m6), kar0[3 + 4 * n0] = key-index(key7, m7), kar0[4 + 4 * n0] = key-index(key8, m4), kar0[n0] = key-index(key3, m3), kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), kar0[2 * n0 + 2] = key-index(key2, m), keyset(kar0, n2 + n0) = keyset(kar, n2 + n0), key7 < key3, key8 < key3, 3 * n0 + 3 < n2, key8 < key7, n0 < # kar, 2 * n0 + 1 < # kar, m3 < # ar, m6 < # ar, n0 + 1 < n2, n0 + 2 < n2, 2 * n0 + 2 < # kar, key7 < key6, key8 < key6, id-idx(ar1, ar), no-dups-idx(kar, n2 + n0), no-dups-idx(kar0, n2 + n0), valid-idx(kar, ar1, n2 + n0), valid-idx(kar0, ar, n2 + n0), n2 ≤ 3 * n0 + 4, n2 + n0 ≤ # kar, (2 * n1 + 1 < n2 + n0 → ¬ key < key1) ∧ (2 * n1 + 2 < n2 + n0 → ¬ key0 < key1) → heap-order(kar, n2 + n0), 2 * n1 + 1 < n2 + n0 ∧ key < key1 ∨ 2 * n1 + 2 < n2 + n0 ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < n2 + n0 ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], n2 + n0), ¬ key6 < key3 ∧ ¬ key2 < key3 → heap-order(kar0, n2 + n0), ∀ m. m < n2 + n0 ∧ m ≠ 0 → ¬ kar0[n0, key-index(key6, m6)][m].key < kar0[n0, key-index(key6, m6)][(m - 1) / 2].key, key6 < key3 ∨ key2 < key3 ⊦ key2 < key6, heap-order(kar0[n0, key-index(key6, m6)][2 * n0 + 1, key-index(key7, m7)], n2 + n0)

Rule argument

spec: bin-min-heap-inv name: heap-order seq: ⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key) subst: [n, kar] → [n2 + n0, kar0[n0, key-index(key6, m6)]] flags: <all paths>

Additional information

Used terms: n2 + n0, kar0[n0, key-index(key6, m6)]