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Proof step 60 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)][(m / 2 - 1) / 2] = key-index(key3, m5), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[m / 2] = key-index(key1, m3), kar0[m + 1] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), m + 1 < # kar, m + 1 < m2, n1 < # kar, n1 < m2, m3 < # ar, key2 < key1, m4 < # ar, m / 2 < m2, m / 2 < # kar, 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, m2 = 1, no-dups-idx(kar0[m + 1, key-index(key1, m3)][m / 2, key-index(key2, m4)], m2)

User applied rule insert rewrite lemma

And obtained the following new goals

Rule argument

spec: bin-min-heap-inv name: no-dups-idx-xchg seq: ⊦ no-dups-idx(kar, n) ∧ kar[m0] = ki0 ∧ kar[m1] = ki1 ∧ m0 < n ∧ m1 < n ∧ m0 < # kar ∧ m1 < # kar → no-dups-idx(kar[m0, ki1][m1, ki0], n) subst: [n, ki0, m1, ki1, m0, kar] → [m2, key-index(key2, m4), m / 2, key-index(key1, m3), m + 1, kar0] flags: <all paths>

Additional information

Used terms: m2, key-index(key2, m4), m / 2, key-index(key1, m3), m + 1, kar0