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Proof step 69 from proof prepost-bubble-up (previous step)

# bh.ar = # kar, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), keyset(bh.ar, bh.size) = keyset(kar, bh.size), n1 < # kar, (n - 1) / 2 < # kar, n < # kar, bh.ar[n].key < bh.ar[(n - 1) / 2].key, n1 < bh.size, n < bh.size, heap-order(bh.ar[n, bh.ar[(n - 1) / 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 - 1) / 2].idx, bh.size ≤ # kar, 1 ≤ n, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], bh.size), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, bh.size) ⊦ # kar = 1, bh.size = 1, n = 0

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 70:
# 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, n < bh.size, n1 < bh.size, bh.ar[n].key < bh.ar[m / 2].key, n < # kar, m / 2 < # kar, n1 < # kar, heap-order(bh.ar[n, 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), 1 ≤ n, 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, n = 0

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [n, m]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: n, m