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Proof step 2 from proof heap-order-pull-up-child (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

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

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