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Proof step 30 from proof prepost-update (previous step)

kar[m1, key-index(key, m)][(m1 - 1) / 2] = key-index(key0, m2), ar[m] = elemindex(a, m1), kar[m1] = key-index(key1, m), 2exp-size(# kar), m1 < m0, m1 < # kar, key < key1, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, m1 = 0 ∨ ¬ key < key0 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, heap-order(kar[m1, key-index(key, m)], m0)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 31:
⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key)
Proof step 32:
kar[m1] = key-index(key1, m), ar[m] = elemindex(a, m1), kar[m1, key-index(key, m)][(m1 - 1) / 2] = key-index(key0, m2), 2exp-size(# kar), key < key1, m1 < # kar, m1 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), key-index(key1, m) ∈ keyset(kar, m0), m0 ≤ # kar, m1 = 0 ∨ ¬ key < key0 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅, ∀ m2. m2 < m0 ∧ m2 ≠ 0 → ¬ kar[m1, key-index(key, m)][m2].key < kar[m1, key-index(key, m)][(m2 - 1) / 2].key

Rule argument

spec: bin-min-heap-inv name: heap-order seq: ⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key) subst: [n, kar] → [m0, kar[m1, key-index(key, m)]] flags: <all paths>

Additional information

Used terms: m0, kar[m1, key-index(key, m)]