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Proof step 43 from proof prepost-remove (previous step)

# 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), keyset(kar[m2 + 1, key-index(key0, m0)], m1) = keyset(kar0, m1), 2exp-size(# kar), m2 + 1 < m1, m2 / 2 < # kar, key0 < key1, m2 + 1 < # kar, m1 < # kar, m0 < # ar1, m2 < m1, m2 < # kar, 2 + m2 < # kar, 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

User applied rule apply lemma

And obtained the following new goals

Proof step 44:
⊦ ar[m] = elemindex(a, m0) ∧ valid-idx(kar, ar, n) ∧ key-index(key, m) ∈ keyset(kar, n) → kar[m0] = key-index(key, m)
Proof step 45:
# kar0 = # kar, kar[m2 / 2] = key-index(key1, m3), kar[m2 + 1] = key-index(key, m), ar1[m] = elemindex(a, m2 + 1), kar[m1] = key-index(key0, m0), keyset(kar[m2 + 1, key-index(key0, m0)], m1) = keyset(kar0, m1), 2exp-size(# kar), 2 + m2 < # kar, m2 < # kar, m2 < m1, m0 < # ar1, m1 < # kar, m2 + 1 < # kar, key0 < key1, m2 / 2 < # kar, m2 + 1 < m1, heap-order(kar[m2 + 1, key-index(key1, m3)], m1), heap-order(kar, m1 + 1), heap-order(kar0, m1), id-idx(ar1, ar0), no-dups-idx(kar, m1 + 1), no-dups-idx(kar0, m1), valid-idx(kar, ar1, m1 + 1), valid-idx(kar0, ar0, m1), ar1[m] = elemindex(a, m0 + 1) ∧ valid-idx(kar, ar1, m1 + 1) ∧ key-index(key, m) ∈ keyset(kar, m1 + 1) → kar[m0 + 1] = key-index(key, m), key-index(key, m) = key-index(key0, m0) ∨ key-index(key, m) ∈ keyset(kar, m1) ⊦ #(keyset(kar, m1) ++ key-index(key0, m0)) = 0, # kar = 1, keyset(kar0, m1) = (keyset(kar, m1) ++ key-index(key0, m0)) -- key-index(key, m), m1 = m2 + 1

Rule argument

spec: bin-min-heap-inv name: key-index-in-keyset-array-extract seq: ⊦ ar[m] = elemindex(a, m0) ∧ valid-idx(kar, ar, n) ∧ key-index(key, m) ∈ keyset(kar, n) → kar[m0] = key-index(key, m) subst: [ar, m, a, m0, kar, n, key] → [ar1, m, a, m0 + 1, kar, m1 + 1, key]

Additional information

Used terms: ar1, m, a, m0 + 1, kar, m1 + 1, key