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Proof step 9 from proof heap-order-min (previous step)

kar[m] = key-index(key, m1), kar[0] = key-index(key0, m2), m = m3 + 1, m < # kar, key < key0, n0 < # kar, m ≤ n0, kar[m3 / 2].key < key0 ∧ m3 / 2 < n0 ∧ (m0 < n0 ∧ m0 ≠ 0 → ¬ kar[m0].key < kar[(m0 - 1) / 2].key) → m3 = 0 ∨ m3 = 1, ∀ m0. ¬ n0 < m0 ∧ m0 ≠ 0 → ¬ kar[m0].key < kar[(m0 - 1) / 2].key ⊦ m = 0

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
kar[m3 + 1] = key-index(key, m1), kar[0] = key-index(key0, m2), m3 < # kar, m3 < n0, key < key0, n0 < # kar, m3 + 1 < # kar, kar[m3 / 2].key < key0 ∧ m3 / 2 < n0 ∧ (m0 < n0 ∧ m0 ≠ 0 → ¬ kar[m0].key < kar[(m0 - 1) / 2].key) → m3 = 0 ∨ m3 = 1, ∀ m0. ¬ n0 < m0 ∧ m0 ≠ 0 → ¬ kar[m0].key < kar[(m0 - 1) / 2].key ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (s)% ⊦ m < n → n ≠ 0; (s)%trans: ⊦ n < n0 ∧ n0 < n1 → n < n1; (forward) from specification nat-basic%lf: ⊦ m < n ∧ n < n0 → m +1 < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m + 1 ≤ 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 + 1 ≠ 0; (s)