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Proof step 3 from proof valid-idx-move-last (previous step)

kar[m] = key-index(key, m1), m < # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ valid-idx(kar[m0, key-index(key, m1)], ar[m1, ar[m1].idx:= m0], m)

User applied rule insert rewrite lemma

And obtained the following new goals

Proof step 4:
⊦ valid-idx(kar, ar, n) ↔ n ≤ # kar ∧ (∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m)
Proof step 5:
kar[m] = key-index(key, m1), m < # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ m ≤ # kar[m0, key-index(key, m1)] ∧ (∀ m2. m2 < m → kar[m0, key-index(key, m1)][m2].idx < # ar[m1, ar[m1].idx:= m0] ∧ ar[m1, ar[m1].idx:= m0][kar[m0, key-index(key, m1)][m2].idx].idx = m2)

Rule argument

spec: <toplevel> 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[m1, ar[m1].idx:= m0], kar[m0, key-index(key, m1)]] flags: <all paths>

Additional information

Used terms: m, ar[m1, ar[m1].idx:= m0], kar[m0, key-index(key, m1)]