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Proof step 12 from proof heap-order-overwrite-children-ok (previous step)

kar[2 * m + 2] = key-index(key3, m5), kar[m, key-index(key, m0)][m1 / 2] = key-index(key1, m3), kar[m, key-index(key, m0)][m1 + 1] = key-index(key0, m2), kar[m] = key-index(key2, m4), kar[2 * m + 1] = key-index(key4, m6), m1 + 1 < # kar, m < # kar, m1 + 1 < n, key0 < key1, m < n, n ≤ # kar, 2 * m + 1 < n → ¬ key4 < key, 2 * m + 2 < n → ¬ key3 < key, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key < key2

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 13:
⊦ m ≠ n → kar[n, ki][m] = kar[m]
Proof step 14:
m1 / 2 = m, kar[2 * m + 1] = key-index(key4, m6), kar[m] = key-index(key2, m4), kar[m, key-index(key, m0)][m1 + 1] = key-index(key0, m2), kar[m, key-index(key, m0)][m1 / 2] = key-index(key1, m3), kar[2 * m + 2] = key-index(key3, m5), m < n, key0 < key1, m1 + 1 < n, m < # kar, m1 + 1 < # kar, n ≤ # kar, 2 * m + 2 < n → ¬ key3 < key, 2 * m + 1 < n → ¬ key4 < key, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key < key2
Proof step 18:
kar[2 * m + 1] = key-index(key4, m6), kar[m] = key-index(key2, m4), kar[m, key-index(key, m0)][m1 + 1] = key-index(key0, m2), kar[m1 / 2] = key-index(key1, m3), kar[2 * m + 2] = key-index(key3, m5), m < n, key0 < key1, m1 + 1 < n, m < # kar, m1 + 1 < # kar, n ≤ # kar, 2 * m + 2 < n → ¬ key3 < key, 2 * m + 1 < n → ¬ key4 < key, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ m1 / 2 = m, key < key2

Rule argument

spec: array, inst: key-index-array name: Put-other seq: ⊦ m ≠ n → kar[n, ki][m] = kar[m] subst: [m, ki, n, kar] → [m1 / 2, key-index(key, m0), m, kar] flags: <all paths>

Additional information

Used terms: m1 / 2, key-index(key, m0), m, kar