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Proof step 4 from proof heap-order-add-smaller (previous step)

m = m0 + 1, n < # kar, kar[n, ki][m].key < kar[n, ki][m0 / 2].key, m < # 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 5:
m0 < # kar, kar[n, ki][m0 + 1].key < kar[m0 / 2].key, m0 < n, n < # kar, m0 + 1 < # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ ki.key < kar[(n - 1) / 2].key

Rule argument

Additional information

Used simplifier rules: ⊦ m +1 = m + 1; (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 ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ ¬ m < m / 2; (s)% ⊦ m < n → n ≠ 0; (s)% ⊦ 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)