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Proof step 10 from proof heap-order-pull-down-parent (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 11:
m2 / 2 = m1 + 1, kar[m2 + 1] = key-index(key, m0), kar[m1 / 2] = key-index(key0, m3), m2 + 1 < # kar, m1 + 1 < # kar, m2 + 1 < n, key < key0, m1 + 1 < n, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦

Rule argument

Additional information

Used simplifier rules: ⊦ m < # kar → kar[m, ki][m] = ki; (s)%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)% ⊦ m / 2 ≠ m + 1; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + n = m + n0 ↔ n = n0; (s)