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Proof step 5 from proof prepost-update (previous step)

ar0 = ar, bh = bin-heap(kar, m0), # keyset(kar, m0) ≠ 0, keyset(kar, m0) ≠ ∅, 2exp-size(# kar), heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key0, m) ∈ keyset(kar, m0), m0 ≤ # kar ⊦ ⟪bin_min_heap_update#(ar[m].idx, key; bh, ar)⟫ ( bh.size ≤ # bh.ar ∧ no-dups-idx(bh.ar, bh.size) ∧ valid-idx(bh.ar, ar, bh.size) ∧ 2exp-size(# bh.ar) ∧ heap-order(bh.ar, bh.size) ∧ id-idx(ar0, ar) ∧ keyset(bh.ar, bh.size) = (keyset(kar, m0) -- key-index(key0, m)) ++ key-index(key, m))

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Rule argument

spec: elem-index name: elim seq: ⊦ a = ei.elem ∧ m = ei.idx ↔ ei = elemindex(a, m) subst: [ei, a, m] → [ar[m], a, m1]

Additional information

Used simplifier rules: ⊦ a = ei.elem ∧ m = ei.idx ↔ ei = elemindex(a, m); (elim) Used terms: ar[m], a, m1