simp only [push_cons_eq_cons_push, nodup_cons, mem_push, mem_cons, IH]
case cons
α : Type u
inst✝ : DecidableEq α
a b : α
x : Array α
IH : Nodup (push x a) ↔ Nodup x ∧ ¬a ∈ x
⊢ Nodup (push (cons b x) a) ↔ Nodup (cons b x) ∧ ¬a ∈ cons b x
case cons
α : Type u
inst✝ : DecidableEq α
a b : α
x : Array α
IH : Nodup (push x a) ↔ Nodup x ∧ ¬a ∈ x
⊢ ¬(b ∈ x ∨ b = a) ∧ Nodup x ∧ ¬a ∈ x ↔ (¬b ∈ x ∧ Nodup x) ∧ ¬(a = b ∨ a ∈ x)