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Proof step 5 from proof heap-order-add-parent-key (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 6:
kar[m1 / 2] = key-index(key, m), m0 = m3 + 1, m1 + 1 < # kar, kar[m1 + 1, key-index(key, m)][m0].key < kar[m1 + 1, key-index(key, m)][m3 / 2].key, m0 < # kar, m0 ≤ m1 + 1, ∀ m2. ¬ m1 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m0 = 0

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [m0, m3]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m0, m3