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Proof step 25 from proof prepost-insert (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 26:
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), m0 < # ar, m < # kar0, heap-order(kar0, m), heap-order(kar, m), no-dups-idx(kar0, m), no-dups-idx(kar, m), valid-idx(kar0, ar, m), valid-idx(kar, ar, m), m ≤ # kar, ∀ ki0. ki0 ∈ keyset(kar, m) → ki0.idx ≠ m0 ⊦ no-dups-idx(kar0[m, key-index(key, m0)], m + 1) ∧ valid-idx(kar0[m, key-index(key, m0)], ar[m0, ar[m0].idx:= m], m + 1) ∧ (m = 0 ∨ ¬ key < key0 → heap-order(kar0[m, key-index(key, m0)], m + 1)) ∧ (m ≠ 0 ∧ key < key0 → heap-order(kar0[m, kar0[m, key-index(key, m0)][(m - 1) / 2]], m + 1))

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar0[m, key-index(key, m0)][(m - 1) / 2], key0, m1]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar0[m, key-index(key, m0)][(m - 1) / 2], key0, m1