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Proof step 23 from proof prepost-bubble-up (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic