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Proof step 12 from proof heap-order-add-parent-key (previous step)

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

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:
m3 / 2 = m1 + 1, kar[m3 + 1] = key-index(key0, m0), kar[m1 / 2] = key-index(key, m), kar[m1 + 1, key-index(key, m)][m3 / 2] = key-index(key1, m4), m3 + 1 < # kar, key0 < key1, m1 + 1 < # kar, m3 < m1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m3 = m1
Proof step 18:
kar[m3 + 1] = key-index(key0, m0), kar[m1 / 2] = key-index(key, m), kar[m3 / 2] = key-index(key1, m4), m3 + 1 < # kar, key0 < key1, m1 + 1 < # kar, m3 < m1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m3 / 2 = m1 + 1, m3 = m1

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] → [m3 / 2, key-index(key, m), m1 + 1, kar] flags: <all paths>

Additional information

Used terms: m3 / 2, key-index(key, m), m1 + 1, kar