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Proof step 6 from proof valid-idx-move-last (previous step)

kar[m] = key-index(key, m1), m2 < # kar, m2 < m, m < # kar, m2 + 1 < # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ kar[m0, key-index(key, m1)][m2].idx < # ar ∧ ar[m1, ar[m1].idx:= m0][kar[m0, key-index(key, m1)][m2].idx].idx = m2

Heuristic elimination applied rule insert elim lemma

And obtained the following new goals

Proof step 7:
kar[m] = key-index(key, m1), kar[m0, key-index(key, m1)][m2] = key-index(key0, m3), m2 + 1 < # kar, m < # kar, m2 < m, m2 < # kar, ∀ m2. ¬ m < m2 → kar[m2].idx < # ar ∧ ar[kar[m2].idx].idx = m2 ⊦ m3 < # ar ∧ ar[m1, ar[m1].idx:= m0][m3].idx = m2

Rule argument

spec: key-index name: elim seq: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m) subst: [ki, key, m] → [kar[m0, key-index(key, m1)][m2], key0, m3]

Additional information

Used simplifier rules: ⊦ key = ki.key ∧ m = ki.idx ↔ ki = key-index(key, m); (elim) Used terms: kar[m0, key-index(key, m1)][m2], key0, m3