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Proof step 72 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), bh = bin-heap(kar0, m2), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, m / 2 < # kar, m + 1 < # kar, kar0[m + 1].key < kar0[m / 2].key, n1 < m2, m + 1 < m2, 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), # ar ≤ kar0[m / 2].idx, m2 ≤ # kar, 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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 73:
# kar0 = # kar, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, m / 2 < # kar, m + 1 < # kar, kar0[m + 1].key < kar0[m / 2].key, n1 < m2, m + 1 < m2, 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), # ar ≤ kar0[m / 2].idx, m2 ≤ # kar, 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: ⊦ 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