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Proof step 45 from proof prepost-insert (previous step)

kar0[m, key-index(key, m0)][(m - 1) / 2] = key-index(key0, m1), subarray(kar0, 0, m) = subarray(kar, 0, m), 2exp-size(# kar0), 2exp-size(# kar), m < # kar0, m0 < # ar, heap-order(kar0, m), heap-order(kar, m), no-dups-idx(kar0, m), no-dups-idx(kar, m), no-dups-idx(kar0[m, key-index(key, m0)], m + 1), valid-idx(kar, ar, m), valid-idx(kar0, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ m = 0, key < key0, heap-order(kar0[m, key-index(key, m0)], m + 1)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 46:
kar0[m, key-index(key, m0)][m2 / 2] = key-index(key0, m1), m = m2 + 1, subarray(kar0, 0, m) = subarray(kar, 0, m), 2exp-size(# kar), 2exp-size(# kar0), m0 < # ar, m < # kar0, heap-order(kar, m), heap-order(kar0, m), no-dups-idx(kar0[m, key-index(key, m0)], m + 1), no-dups-idx(kar, m), no-dups-idx(kar0, m), valid-idx(kar0, ar, m), valid-idx(kar, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ m = 0, key < key0, heap-order(kar0[m, key-index(key, m0)], m + 1)

Rule argument

spec: nat name: elim-pred-c seq: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1 subst: [n, m] → [m, m2]

Additional information

Used simplifier rules: n ≠ 0 ⊦ m = n - 1 ↔ n = m + 1; (elim) Used terms: m, m2