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Proof step 45 from proof prepost-bubble-up (previous step)

# kar0 = # kar, kar0[1, key-index(key1, m3)][0, key-index(key2, m4)][(0 - 1) / 2] = key-index(key3, m5), kar[(n1 - 1) / 2] = key-index(key0, m1), kar[n1] = key-index(key, m0), kar0[0] = key-index(key1, m3), kar0[1] = key-index(key2, m4), keyset(kar0, m2) = keyset(kar, m2), n1 < # kar, n1 < m2, m3 < # ar, key2 < key1, m4 < # ar, heap-order(kar0[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, heap-order(kar0[1, key-index(key1, m3)][0, 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: heap-order-overwrite0 seq: n ≤ # kar, n ≠ 0 ⊦ heap-order(kar, n) ∧ ¬ kar[0].key < ki.key → heap-order(kar[0, ki], n) subst: [n, ki, kar] → [m2, key-index(key2, m4), kar0[1, key-index(key1, m3)]] flags: <all paths>

Additional information

Used simplifier rules: ⊦ # kar[n, ki] = # kar; (s)% ⊦ m < n → n ≠ 0; (s)%m ≠ n ⊦ kar[(* m), ki][(* n)] = kar[n]; (s)% ⊦ key-index(key, m).key = key; (s)%transitivity: ⊦ key < key0 ∧ key0 < key1 → key < key1; (forward) from specification oelem[binheap-key]% ⊦ ¬ key < key; (s) Used terms: m2, key-index(key2, m4), kar0[1, key-index(key1, m3)]