case nil
α : Type u_1
p✝ : α → Prop
inst✝ : DecidablePred p✝
l : List α
a : α
p : α → Bool
⊢ any [] p = true ↔ ∃ a, a ∈ [] ∧ p a = true

case cons
α : Type u_1
p✝ : α → Prop
inst✝ : DecidablePred p✝
l✝ : List α
a✝ : α
p : α → Bool
a : α
l : List α
ih : any l p = true ↔ ∃ a, a ∈ l ∧ p a = true
⊢ any (a :: l) p = true ↔ ∃ a_1, a_1 ∈ a :: l ∧ p a_1 = true
α : Type u_1
p✝ : α → Prop
inst✝ : DecidablePred p✝
l : List α
a : α
p : α → Bool
⊢ any l p = true ↔ ∃ a, a ∈ l ∧ p a = true
induction' l with a l ih
no goals
case cons
α : Type u_1
p✝ : α → Prop
inst✝ : DecidablePred p✝
l✝ : List α
a✝ : α
p : α → Bool
a : α
l : List α
ih : any l p = true ↔ ∃ a, a ∈ l ∧ p a = true
⊢ any (a :: l) p = true ↔ ∃ a_1, a_1 ∈ a :: l ∧ p a_1 = true
simp only [any_cons, Bool.or_coe_iff, ih, exists_mem_cons_iff]
no goals
case nil
α : Type u_1
p✝ : α → Prop
inst✝ : DecidablePred p✝
l : List α
a : α
p : α → Bool
⊢ any [] p = true ↔ ∃ a, a ∈ [] ∧ p a = true
exact iff_of_false Bool.not_false' (not_exists_mem_nil _)