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Proof step 5 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 ⊦ 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)

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 6:
kar[m] = key-index(key0, m0), m1 < n, m1 < # kar, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ kar[m, key-index(key, m0)][m1].idx < # ar ∧ ar[kar[m, key-index(key, m0)][m1].idx].idx = m1

Rule argument

Additional information

Used simplifier rules: ⊦ m < n → n ≠ 0; (s)%fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat%f: ⊦ m < n → n ≠ 0; (forward) from specification nat-basic% ⊦ # kar[n, ki] = # kar; (s)