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Proof step 14 from proof prepost-remove (previous step)

ar[m] = elemindex(a, m1), kar[m1] = key-index(key, m), key-index(key, m) = kar[m2], 2exp-size(# kar), m2 < m0, m2 < # kar, m2 < m1, heap-order(kar, m0), no-dups-idx(kar, m0), valid-idx(kar, ar, m0), m0 ≤ # kar, m0 ≤ m1 ⊦ # keyset(kar, m0) = 0, keyset(kar, m0) = ∅

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 15:
⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m)
Proof step 16:
kar[m1] = key-index(key, m), ar[m] = elemindex(a, m1), key-index(key, m) = kar[m2], 2exp-size(# kar), m2 < m1, m2 < # kar, m2 < m0, heap-order(kar, m0), no-dups-idx(kar, m0), m0 ≤ m1, m0 ≤ # kar, ∀ m. m < m0 → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ # keyset(kar, m0) = 0, keyset(kar, 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] → [m0, ar, kar] flags: <all paths>

Additional information

Used terms: m0, ar, kar