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Proof step 69 from proof prepost-resize (previous step)

kar[1 + m1] = key-index(key, m0), kar[m1 / 2] = key-index(key0, m2), 2exp-size(n0), 2exp-size(# kar), m1 < m * 2, m1 / 2 < n0, m1 + 1 < n0, 2 + m1 < minsize, 2 + m1 < n0, key < key0, m1 + 1 < m, m < minsize, m < n0, m1 + 1 < minsize, m1 + 1 < # kar, heap-order(kar, m), no-dups-idx(kar, m), valid-idx(kar, ar, m), minsize ≤ n0, m ≤ # kar ⊦

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 70:
⊦ heap-order(kar, n) ↔ (∀ m. m < n ∧ m ≠ 0 → ¬ kar[m].key < kar[(m - 1) / 2].key)
Proof step 71:
kar[m1 / 2] = key-index(key0, m2), kar[1 + m1] = key-index(key, m0), 2exp-size(# kar), 2exp-size(n0), m1 + 1 < # kar, m1 + 1 < minsize, m < n0, m < minsize, m1 + 1 < m, key < key0, 2 + m1 < n0, 2 + m1 < minsize, m1 + 1 < n0, m1 / 2 < n0, m1 < m * 2, no-dups-idx(kar, m), valid-idx(kar, ar, m), m ≤ # kar, minsize ≤ n0, ∀ m0. m0 < m ∧ m0 ≠ 0 → ¬ kar[m0].key < kar[(m0 - 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] → [m, kar] flags: <all paths>

Additional information

Used terms: m, kar