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Proof step 12 from proof heap-order-pull-up-child (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

Rule argument

Additional information

Used simplifier rules: ⊦ n + 0 = n; (s)% ⊦ n ≤ m → m + n0 - n = (m - n) + n0; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)