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Proof step 3 from proof valid-idx-update-key (previous step)

kar[m] = key-index(key0, m0), n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ valid-idx(kar[m, key-index(key, m0)], ar, n)

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(key0, m0), n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ n ≤ # kar[m, key-index(key, m0)] ∧ (∀ m1. m1 < n → kar[m, key-index(key, m0)][m1].idx < # ar ∧ ar[kar[m, key-index(key, m0)][m1].idx].idx = m1)

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] → [n, ar, kar[m, key-index(key, m0)]] flags: <all paths>

Additional information

Used terms: n, ar, kar[m, key-index(key, m0)]