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Proof step 92 from proof prepost-bubble-down (previous step)

# bh.ar = # kar, kar[n1] = key-index(key1, m2), kar[2 * n1 + 2] = key-index(key0, m1), kar[2 * n1 + 1] = key-index(key, m0), keyset(bh.ar, bh.size) = keyset(kar, bh.size), 2 * n + 2 < # kar, 2 * n + 1 < # kar, n < # kar, 2 * n + 1 < bh.size, bh.ar[2 * n + 2].key < bh.ar[2 * n + 1].key, 2 * n + 2 < bh.size, heap-order(bh.ar[n, bh.ar[2 * n + 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[2 * n + 2].idx, bh.size ≤ # kar, ¬ bh.ar[2 * n + 1].key < bh.ar[n].key ∧ ¬ bh.ar[2 * n + 2].key < bh.ar[n].key → heap-order(bh.ar, bh.size), 2 * n1 + 1 < bh.size ∧ key < key1 ∨ 2 * n1 + 2 < bh.size ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < bh.size ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], bh.size), (2 * n1 + 1 < bh.size → ¬ key < key1) ∧ (2 * n1 + 2 < bh.size → ¬ key0 < key1) → heap-order(kar, bh.size), bh.ar[2 * n + 1].key < bh.ar[n].key ∨ bh.ar[2 * n + 2].key < bh.ar[n].key ⊦

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 93:
# kar0 = # kar, kar[2 * n1 + 1] = key-index(key, m0), kar[2 * n1 + 2] = key-index(key0, m1), kar[n1] = key-index(key1, m2), bh = bin-heap(kar0, m), keyset(kar0, m) = keyset(kar, m), 2 * n + 2 < m, kar0[2 * n + 2].key < kar0[2 * n + 1].key, 2 * n + 1 < m, n < # kar, 2 * n + 1 < # kar, 2 * n + 2 < # kar, heap-order(kar0[n, kar0[2 * n + 2]], m), id-idx(ar1, ar), no-dups-idx(kar, m), no-dups-idx(kar0, m), valid-idx(kar, ar1, m), valid-idx(kar0, ar, m), m ≤ # kar, # ar ≤ kar0[2 * n + 2].idx, (2 * n1 + 1 < m → ¬ key < key1) ∧ (2 * n1 + 2 < m → ¬ key0 < key1) → heap-order(kar, m), 2 * n1 + 1 < m ∧ key < key1 ∨ 2 * n1 + 2 < m ∧ key0 < key1 → heap-order(kar[n1, (2 * n1 + 2 < m ∧ key0 < key ⊃ key-index(key0, m1);key-index(key, m0))], m), ¬ kar0[2 * n + 1].key < kar0[n].key ∧ ¬ kar0[2 * n + 2].key < kar0[n].key → heap-order(kar0, m), kar0[2 * n + 1].key < kar0[n].key ∨ kar0[2 * n + 2].key < kar0[n].key ⊦

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, m]

Additional information

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