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Proof step 80 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[m + 1] = key-index(key1, m3), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, m / 2 < # kar, m + 1 < # kar, key1 < key2, n1 < m2, m + 1 < m2, heap-order(kar0[m + 1, key-index(key2, m4)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar, ar1, m2), # ar ≤ m4, m2 ≤ # kar, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), m / 2 < m2 → kar0[m / 2].idx < # ar ∧ ar[kar0[m / 2].idx].idx = m / 2 ⊦ # kar = 1, m2 = 1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 81:
# kar0 = # kar, kar0[m + 1] = key-index(key1, m3), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, m / 2 < # kar, m + 1 < # kar, key1 < key2, n1 < m2, m + 1 < m2, heap-order(kar0[m + 1, key-index(key2, m4)], m2), id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar, ar1, m2), # ar ≤ m4, m2 ≤ # kar, m2 * 2 ≤ m, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2) ⊦ # kar = 1, m2 = 1

Rule argument

Additional information

Used simplifier rules: m0 ≠ 0 ⊦ n ≤ m / m0 ↔ n * m0 ≤ m; (s)% ⊦ m < n ↔ ¬ n ≤ m; (ss)% ⊦ m ≤ n → (n < m ↔ false); (s)% ⊦ key-index(key, m).idx = m; (s)