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Proof step 64 from proof prepost-bubble-up (previous step)

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

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 65:
# bh.ar = # kar, kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), keyset(bh.ar, bh.size) = keyset(kar, bh.size), bh.ar[m / 2].idx < bh.ar[m + 1].idx, n1 < bh.size, n1 < # kar, m / 2 < # kar, m + 1 < bh.size, bh.ar[m + 1].key < bh.ar[m / 2].key, m + 1 < # kar, bh.ar[m / 2].idx < # ar, heap-order(bh.ar[m + 1, bh.ar[m / 2]], bh.size), id-idx(ar1, ar), no-dups-idx(bh.ar, bh.size), no-dups-idx(kar, bh.size), valid-idx(bh.ar, ar, bh.size), valid-idx(kar, ar1, bh.size), # ar ≤ bh.ar[m + 1].idx, bh.size ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, bh.size), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], bh.size) ⊦ # kar = 1, bh.size = 1

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%lf-01: ⊦ m < n ∧ n = n0 → m < n0; (forward) from specification nat-basic%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ m ≤ m + n; (s)%c: ⊦ m + n = n + m; from specification nat-basic%a: ⊦ (m + n0) + n1 = m + n0 + n1; from specification nat-basic% ⊦ m + 1 ≠ 0; (s)