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Proof step 76 from proof prepost-remove (previous step)

kar[2 * m1 + 1] = key-index(key3, m5), kar[2 * m1 + 2] = key-index(key2, m4), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), ar1[m] = elemindex(a, m1), kar[m1] = key-index(key, m), kar[m2] = key-index(key0, m0), 2exp-size(# kar), m1 + 1 < # kar, m1 < # kar, m2 < # kar, m0 < # ar1, m1 < m2, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar1, m2 + 1), elemindex(a, m1) = elemindex(a, m3) ∧ valid-idx(kar, ar1, m1) ∧ key-index(key, m) ∈ keyset(kar, m1) → kar[m3] = key-index(key, m), m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 1 ≤ m1, 2 * m1 + 2 < m2 → ¬ key2 < key0, 2 * m1 + 1 < m2 → ¬ key3 < key0, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, m2 = m1, heap-order(kar[m1, key-index(key0, m0)], m2), m1 ≠ 0 ∧ key0 < key1, m2 ≤ # kar ∧ m1 < m2 ∧ heap-order(kar, m2) ∧ (m1 ≠ 0 → ¬ key-index(key0, m0).key < kar[(m1 - 1) / 2].key) ∧ (2 * m1 + 1 < m2 → ¬ kar[2 * m1 + 1].key < key-index(key0, m0).key) ∧ (2 * m1 + 2 < m2 → ¬ kar[2 * m1 + 2].key < key-index(key0, m0).key)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 77:
kar[m2] = key-index(key0, m0), kar[m1] = key-index(key, m), ar1[m] = elemindex(a, m1), kar[m1, key-index(key0, m0)][(m1 - 1) / 2] = key-index(key1, m3), kar[2 * m1 + 2] = key-index(key2, m4), kar[2 * m1 + 1] = key-index(key3, m5), 2exp-size(# kar), key1 < kar[(m1 - 1) / 2].key, (m1 - 1) / 2 < # kar, key0 < kar[(m1 - 1) / 2].key, m1 < m2, m0 < # ar1, m2 < # kar, m1 < # kar, m1 + 1 < # kar, heap-order(kar, m2 + 1), no-dups-idx(kar, m2 + 1), valid-idx(kar, ar1, m2 + 1), 1 ≤ m1, elemindex(a, m1) = elemindex(a, m3) ∧ valid-idx(kar, ar1, m1) ∧ key-index(key, m) ∈ keyset(kar, m1) → kar[m3] = key-index(key, m), 2 * m1 + 1 < m2 → ¬ key3 < key0, 2 * m1 + 2 < m2 → ¬ key2 < key0, key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m2) ⊦ #(keyset(kar, m2) ++ key-index(key0, m0)) = 0, # kar = 1, m1 = 0, m2 = m1, m2 = 1, key0 < key1, heap-order(kar[m1, key-index(key0, m0)], m2)

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)% ⊦ 1 < m ↔ ¬(m = 0 ∨ m = 1); (s)%nll: ⊦ ¬ key < key0 ∧ key < key1 → key0 < key1; (forward) from specification oelem[binheap-key]%fle-01: ⊦ m ≤ n ∧ n < n0 → m < n0; (forward) from specification nat%not-zero: 0 < n ⊦ (* n) ≤ m → m ≠ 0; (forward) from specification nat% ⊦ key-index(key, m).key = key; (s)%n < # kar ⊦ heap-order(kar, n + 1) → heap-order(kar, n); (ss)% ⊦ m < n → (n ≤ m ↔ false); (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m < n → m ≤ n; (s)