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Proof step 3 from proof valid-idx-rec (previous step)

n < # kar, m < # ar, ∀ m0, m1. m0 < # kar ∧ ¬ n < m0 ∧ m1 < # kar ∧ ¬ n < m1 ∧ m0 ≠ m1 → kar[n, key-index(key, m)][m0].idx ≠ kar[n, key-index(key, m)][m1].idx, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ valid-idx(kar[n, key-index(key, m)], ar[m, ar[m].idx:= n], n + 1)

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:
m < # ar, n < # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m, ∀ m0, m1. m0 < # kar ∧ ¬ n < m0 ∧ m1 < # kar ∧ ¬ n < m1 ∧ m0 ≠ m1 → kar[n, key-index(key, m)][m0].idx ≠ kar[n, key-index(key, m)][m1].idx ⊦ n + 1 ≤ # kar[n, key-index(key, m)] ∧ (∀ m0. m0 < n + 1 → kar[n, key-index(key, m)][m0].idx < # ar[m, ar[m].idx:= n] ∧ ar[m, ar[m].idx:= n][kar[n, key-index(key, m)][m0].idx].idx = m0)

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 + 1, ar[m, ar[m].idx:= n], kar[n, key-index(key, m)]] flags: <all paths>

Additional information

Used terms: n + 1, ar[m, ar[m].idx:= n], kar[n, key-index(key, m)]