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Proof step 32 from proof prepost-insert (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [ki, key1, m]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: ki, key1, m