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Proof step 44 from proof prepost-resize (previous step)

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

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 45:
⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m)
Proof step 46:
kar[m0] = key-index(key, m1), ar[m1] = elemindex(a, m2), 2exp-size(# kar), 2exp-size(n0), m0 < n0, m0 < # kar, m0 < minsize, m0 < m, m < n0, m < minsize, m0 + 1 < minsize, m0 + 1 < n0, m1 < # ar, heap-order(kar, m), no-dups-idx(kar, m), minsize ≤ n0, m ≤ # kar, ∀ m0. m0 < m → kar[m0].idx < # ar ∧ ar[kar[m0].idx].idx = m0 ⊦ m2 = m0

Rule argument

spec: bin-min-heap-inv name: valid-idx seq: ⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m) subst: [n, ar, kar] → [m, ar, kar] flags: <all paths>

Additional information

Used terms: m, ar, kar