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Proof step 9 from proof valid-idx-update-key (previous step)

ar[m2] = elemindex(a, m3), kar[m] = key-index(key0, m0), kar[m, key-index(key, m0)][m1] = key-index(key1, m2), m1 = m, m1 < # kar, m1 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m1

Heuristic simplifier applied rule simplifier

And obtained the following new goals

Proof step 10:
ar[m2] = elemindex(a, m3), kar[m] = key-index(key0, m2), m < # kar, m < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ m3 = m

Rule argument

Additional information

Used simplifier rules: fle: ⊦ m < n ∧ n ≤ n0 → m < n0; (forward) from specification nat% ⊦ key-index(key, m) = key-index(key0, m0) ↔ key = key0 ∧ m = m0; (ws)% ⊦ m < # kar → kar[m, ki][m] = ki; (s)