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Proof step 71 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), m + 1 < bh.size, n1 < bh.size, bh.ar[m + 1].key < bh.ar[m / 2].key, m + 1 < # kar, m / 2 < # kar, n1 < # kar, heap-order(bh.ar[m + 1, bh.ar[m / 2]], bh.size), id-idx(ar1, ar), no-dups-idx(kar, bh.size), no-dups-idx(bh.ar, bh.size), valid-idx(kar, ar1, bh.size), valid-idx(bh.ar, ar, bh.size), bh.size ≤ # kar, # ar ≤ bh.ar[m / 2].idx, 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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 72:
# 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

Rule argument

spec: bin-min-heap name: elim seq: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m) subst: [bh, kar, m] → [bh, kar0, m2]

Additional information

Used simplifier rules: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m); (elim) Used terms: bh, kar0, m2