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Proof step 41 from proof prepost-remove (previous step)

# bh0.ar = # kar, kar[m2 / 2] = key-index(key1, m3), kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[bh0.size] = key-index(key0, m0), keyset(kar[m2 + 1, key-index(key0, m0)], bh0.size) = keyset(bh0.ar, bh0.size), 2exp-size(# kar), m2 + 1 < bh0.size, 2 + m2 < # kar, m2 < # kar, m2 < bh0.size, m0 < # ar1, bh0.size < # kar, m2 + 1 < # kar, key0 < key1, m2 / 2 < # kar, heap-order(kar[m2 + 1, key-index(key1, m3)], bh0.size), heap-order(kar, bh0.size + 1), heap-order(bh0.ar, bh0.size), id-idx(ar1, ar0), no-dups-idx(kar, bh0.size + 1), no-dups-idx(bh0.ar, bh0.size), valid-idx(kar, ar1, bh0.size + 1), valid-idx(bh0.ar, ar0, bh0.size), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, bh0.size) ⊦ #(keyset(kar, bh0.size) ++ key-index(key0, m0)) = 0, # kar = 1, bh0.size = m2 + 1, keyset(bh0.ar, bh0.size) = (keyset(kar, bh0.size) ++ key-index(key0, m0)) -- key-index(key, m)

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 42:
# kar0 = # kar, kar[m1] = key-index(key0, m0), ar1[m] = elemindex(a, m2 + 1), kar[m2 + 1] = key-index(key, m), kar[m2 / 2] = key-index(key1, m3), bh0 = bin-heap(kar0, m1), keyset(kar[m2 + 1, key-index(key0, m0)], m1) = keyset(kar0, m1), 2exp-size(# kar), m2 / 2 < # kar, key0 < key1, m2 + 1 < # kar, m1 < # kar, m0 < # ar1, m2 < m1, m2 < # kar, 2 + m2 < # kar, m2 + 1 < m1, heap-order(kar0, m1), heap-order(kar, m1 + 1), heap-order(kar[m2 + 1, key-index(key1, m3)], m1), id-idx(ar1, ar0), no-dups-idx(kar0, m1), no-dups-idx(kar, m1 + 1), valid-idx(kar0, ar0, m1), valid-idx(kar, ar1, m1 + 1), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ # kar = 1, #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, keyset(kar0, m1) = (keyset(kar, m1) ++ key-index(key0, m0)) -- key-index(key, m), m1 = m2 + 1

Rule argument

spec: bin-min-heap name: elim seq: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m) subst: [bh, kar, m] → [bh0, kar0, m1]

Additional information

Used simplifier rules: ⊦ kar = bh.ar ∧ m = bh.size ↔ bh = bin-heap(kar, m); (elim) Used terms: bh0, kar0, m1