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Proof step 39 from proof prepost-bubble-up (previous step)

# kar0 = # kar, m / 2 = m6 + 1, kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m6 + 1] = key-index(key1, m3), kar0[m + 1] = key-index(key2, m4), kar0[m6 / 2] = key-index(key3, m5), keyset(kar0, m2) = keyset(kar, m2), m + 1 < # kar, m + 1 < m2, n1 < # kar, n1 < m2, m3 < # ar, key2 < key1, m4 < # ar, m6 + 1 < m2, m6 + 1 < # kar, key1 < key3, key2 < key3, heap-order(kar0[m + 1, key-index(key1, m3)], m2), id-idx(ar1, ar), no-dups-idx(kar, m2), no-dups-idx(kar0, m2), valid-idx(kar, ar1, m2), valid-idx(kar0, ar, m2), m2 ≤ # kar, n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2) ⊦ # kar = 1, m6 / 2 = m + 1, m = 0, m = 1, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key3, m5)], m2)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 40:
⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key)
Proof step 41:
# kar0 = # kar, m / 2 = m6 + 1, kar0[m6 / 2] = key-index(key3, m5), kar0[m + 1] = key-index(key2, m4), kar0[m6 + 1] = key-index(key1, m3), kar[n1] = key-index(key, m0), kar[(n1 - 1) / 2] = key-index(key0, m1), keyset(kar0, m2) = keyset(kar, m2), key2 < key3, key1 < key3, m6 + 1 < # kar, m6 + 1 < m2, m4 < # ar, key2 < key1, m3 < # ar, n1 < m2, n1 < # kar, m + 1 < m2, m + 1 < # kar, id-idx(ar1, ar), no-dups-idx(kar0, m2), no-dups-idx(kar, m2), valid-idx(kar0, ar, m2), valid-idx(kar, ar1, m2), m2 ≤ # kar, n1 = 0 ∨ ¬ key < key0 → heap-order(kar, m2), n1 ≠ 0 ∧ key < key0 → heap-order(kar[n1, key-index(key0, m1)], m2), ∀ m0. m0 < m2 ∧ m0 ≠ 0 → ¬ kar0[m + 1, key-index(key1, m3)][m0].key < kar0[m + 1, key-index(key1, m3)][(m0 - 1) / 2].key ⊦ # kar = 1, m6 / 2 = m + 1, m = 1, m = 0, m2 = 1, heap-order(kar0[m + 1, key-index(key1, m3)][m6 + 1, key-index(key3, m5)], m2)

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] → [m2, kar0[m + 1, key-index(key1, m3)]] flags: <all paths>

Additional information

Used terms: m2, kar0[m + 1, key-index(key1, m3)]