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Proof step 2 from proof heap-order-add-smaller (previous step)

heap-order(kar, n), n + 1 ≤ # kar ⊦ n = 0, ki.key < kar[(n - 1) / 2].key, heap-order(kar[n, ki], n + 1)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 3:
m < # kar, kar[n, ki][m].key < kar[n, ki][(m - 1) / 2].key, n < # kar, m ≤ n, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ m = 0, n = 0, ki.key < kar[(n - 1) / 2].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)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m < n + 1 ↔ ¬ n < m; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≤ n ↔ m < n; (s)