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Proof step 2 from proof heap-order-overwrite-children-ok (previous step)

m < n, heap-order(kar, n), n ≤ # kar, 2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key, 2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key ⊦ ¬ heap-order(kar[m, ki], n) ↔ false, ki.key < kar[m].key

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
m < n, kar[m, ki][m0].key < kar[m, ki][(m0 - 1) / 2].key, m0 < n, m < # kar, m0 < # kar, n ≤ # kar, 2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key, 2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ m0 = 0, ki.key < kar[m].key

Rule argument

Additional information

Used simplifier rules: heap-order: ⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key);% ⊦ m < n → n ≠ 0; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic