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Proof step 6 from proof valid-idx-update-key (previous step)

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

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 7:
kar[m] = key-index(key0, m0), kar[m, key-index(key, m0)][m1] = key-index(key1, m2), m1 < # kar, m1 < n, n ≤ # kar, ∀ m. m < n → kar[m].idx < # ar ∧ ar[kar[m].idx].idx = m ⊦ m2 < # ar ∧ ar[m2].idx = m1

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar[m, key-index(key, m0)][m1], key1, m2]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m, key-index(key, m0)][m1], key1, m2