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Proof step 18 from proof heap-order-overwrite-children-at (previous step)

kar[2 * m1 / 2 + 1] = key-index(key3, m5), kar[m1 + 1] = key-index(key5, m7), kar[2 * m1 / 2 + 2] = key-index(key2, m4), kar[(m1 / 2 - 1) / 2] = key-index(key4, m6), kar[m1 / 2] = key-index(key6, m8), m1 + 1 < # kar, m1 + 1 < n, key5 < key1, m1 / 2 < n, m1 / 2 < # kar, key6 < key1, n ≤ # kar, 2 * m1 / 2 + 1 < n → ¬ key3 < key1, 2 * m1 / 2 + 2 < n → ¬ key2 < key1, m1 ≠ 0 ∧ m1 ≠ 1 → ¬ key1 < key4 ⊦ key5 < key6

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 19:
⊦ m % 2 = 1 → 2 * m / 2 = m - 1
Proof step 20:
m1 % 2 = 1, kar[m1 / 2] = key-index(key6, m8), kar[(m1 / 2 - 1) / 2] = key-index(key4, m6), kar[(m1 - 1) + 2] = key-index(key2, m4), kar[m1 + 1] = key-index(key5, m7), kar[(m1 - 1) + 1] = key-index(key3, m5), key6 < key1, m1 / 2 < # kar, m1 / 2 < n, key5 < key1, m1 + 1 < n, m1 + 1 < # kar, n ≤ # kar, m1 ≠ 0 ∧ m1 ≠ 1 → ¬ key1 < key4, (m1 - 1) + 2 < n → ¬ key2 < key1, (m1 - 1) + 1 < n → ¬ key3 < key1 ⊦ key5 < key6

Rule argument

spec: nat-div name: div-two-mult seq: ⊦ m % 2 = 1 → 2 * m / 2 = m - 1 subst: [m] → [m1] flags: <all paths>

Additional information

Used simplifier rules: ⊦ m % 2 = 1 ↔ m % 2 ≠ 0; (ss)%n ≠ 0 ⊦ m % n = 0 → m / n * n = m; (s)%mult-a: ⊦ m0 * m * n = (m0 * m) * n; from specification nat-mult%mult-c: ⊦ m * n = n * m; from specification nat-mult% ⊦ m % 2 = 0 → (m / 2 < n ↔ m < n * 2); (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws) Used terms: m1