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Proof step 11 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), m1 + 1 < # kar, key0 < key1, m3 + 1 < # kar, m3 ≤ m1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m3 + 1 = m1 + 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 12:
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

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≠ n → (m ≤ n ↔ m < n); (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)