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Proof step 17 from proof heap-order-add-parent-key (previous step)

m3 / 2 = (m4 * 2 + n0) + 1, kar[m3 + 1] = key-index(key0, m0), kar[m4] = key-index(key1, m), m3 / 2 < # kar, m3 + 1 < # kar, key0 < key1, m3 < m4 * 2 + n0, n0 < 2, ∀ m2. ¬ m4 * 2 + n0 < m2 ∧ m2 ≠ 0 → ¬ kar[m2].key < kar[(m2 - 1) / 2].key ⊦ m3 = m4 * 2 + n0

User applied rule insert elim lemma

And closed the goal!

Rule argument

spec: nat-div name: div-elim seq: n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n subst: [m, n, m0, n0] → [m3, 2, m1, n]

Additional information

Used simplifier rules: ⊦ m * (n + 1) = m * n + m; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult%m1 ≠ 0 ⊦ m = (* m1) * m + n0 ↔ 0 = (m1 - 1) * m + n0; (s)% ⊦ m0 * 1 = m0; (s)%n < n0 ⊦ (* n) ≠ m0 + (* n0); (s)% ⊦ m * n + n0 < n ↔ m = 0 ∧ n0 < n; (ws)%n ≠ 0 ⊦ m0 = m / n ∧ n0 = m % n ↔ m = m0 * n + n0 ∧ n0 < n; (elim) Used terms: m3, 2, m1, n