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Proof step 9 from proof heap-order-pull-down-parent (previous step)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 10:
m2 / 2 = m1 + 1, kar[m1 + 1, kar[m1 / 2]][m2 / 2] = key-index(key0, m3), kar[m1 + 1, kar[m1 / 2]][m2 + 1] = key-index(key, m0), m2 + 1 < # kar, m1 + 1 < # kar, m2 + 1 < n, key < key0, m1 + 1 < n, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦
Proof step 17:
kar[m1 + 1, kar[m1 / 2]][m2 / 2] = key-index(key0, m3), kar[m1 + 1, kar[m1 / 2]][m2 + 1] = key-index(key, m0), m2 + 1 < # kar, m1 + 1 < # kar, m2 + 1 < n, key < key0, m1 + 1 < n, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ m2 / 2 = m1 + 1

Rule argument

fma: m2 / 2 = m1 + 1

Additional information

Used formulas: m2 / 2 = m1 + 1