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Proof step 16 from proof heap-order-pull-up-child (previous step)

m1 / 2 = m, kar[m1 / 2] = key-index(key4, m6), kar[m, (2 * m + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3))][m1 / 2] = key-index(key0, m2), kar[m, (2 * m + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3))][m1 + 1] = key-index(key, m0), kar[2 * m + 1] = key-index(key1, m3), kar[2 * m + 2] = key-index(key2, m4), kar[m1 + 1] = key-index(key3, m5), m1 + 1 < # kar, 2 * m + 1 < # kar, m < # kar, m1 + 1 < n, key < key0, 2 * m + 1 < n, m < n, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key3 < key4

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 17:
(2 * m1 / 2 + 2 < n ∧ key2 < key1 ⊃ key-index(key2, m4);key-index(key1, m3)) = key-index(key0, m2), kar[2 * m1 / 2 + 2] = key-index(key2, m4), kar[2 * m1 / 2 + 1] = key-index(key1, m3), kar[m1 + 1] = key-index(key3, m5), kar[m1 / 2] = key-index(key4, m6), m1 / 2 < n, 2 * m1 / 2 + 1 < n, key3 < key0, m1 + 1 < n, m1 + 1 < # kar, m1 / 2 < # kar, 2 * m1 / 2 + 1 < # kar, key4 < key0, n ≤ # kar, ∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ key3 < key4

Rule argument

Additional information

Used simplifier rules: nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key]% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ m < # kar → kar[m, ki][m] = ki; (s)%m ≠ n ⊦ kar[m, ki][n] = kar[n]; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m / 2 ≠ m + 1; (s)