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Proof step 12 from proof heap-order-add-smaller (previous step)

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

Heuristic weak cut applied rule cut formula

And obtained the following new goals

Proof step 13:
kar[m1 / 2] = key-index(key2, m5), kar[m1 + 1, key-index(key, m2)][m0 + 1] = key-index(key0, m3), kar[m0 / 2] = key-index(key1, m4), m1 + 1 < # kar, m0 + 1 < # kar, m0 < # kar, key0 < key1, m0 ≤ m1, ∀ m. ¬ m1 < m ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key ⊦ m0 + 1 = m1 + 1, key < key2

Rule argument

fma: m0 + 1 = m1 + 1

Additional information

Used simplifier rules: ⊦ m + n = m + n0 ↔ n = n0; (s)%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic%c: ⊦ m + n = n + m; from specification nat-basic%lf-02: ⊦ m < n ∧ m = n0 → n0 < n; (forward) from specification nat-basic% ⊦ m < # kar → kar[m, ki][m] = ki; (s)% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ m ≤ m; (s)%lel: ⊦ key < key0 ∧ key = key1 → key1 < key0; (forward) from specification oelem[binheap-key] Used formulas: m0 + 1 = m1 + 1