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Proof step 74 from proof prepost-remove (previous step)

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), 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), 2 * m1 + 1 < m2 → ¬ key3 < key0, 2 * m1 + 2 < m2 → ¬ key2 < key0, m1 ≠ 0 → (m1 - 1) / 2 < # kar ∧ 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), 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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 75:
n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ (m ≠ 0 → ¬ ki.key < kar[(m - 1) / 2].key) ∧ (2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key) ∧ (2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key) → (¬ heap-order(kar[m, ki], n) ↔ false)
Proof step 76:
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)

Rule argument

spec: bin-min-heap-inv name: heap-order-overwrite-children-at seq: n ≤ # kar, m < n ⊦ heap-order(kar, n) ∧ (m ≠ 0 → ¬ ki.key < kar[(m - 1) / 2].key) ∧ (2 * m + 1 < n → ¬ kar[2 * m + 1].key < ki.key) ∧ (2 * m + 2 < n → ¬ kar[2 * m + 2].key < ki.key) → (¬ heap-order(kar[m, ki], n) ↔ false) subst: [n, ki, m, kar] → [m2, key-index(key0, m0), m1, kar] flags: <all paths>

Additional information

Used terms: m2, key-index(key0, m0), m1, kar