include "basics/types.ma".

definition option_map : ∀A,B:Type[0]. (A → B) → option A → option B ≝
λA,B,f,o. match o with [ None ⇒ None B | Some a ⇒ Some B (f a) ].

lemma option_map_none : ∀A,B,f,x.
  option_map A B f x = None B → x = None A.
#A #B #f * [ // | #a #E whd in E:(??%?); destruct ]
qed.

lemma option_map_some : ∀A,B,f,x,v.
  option_map A B f x = Some B v → ∃y. x = Some ? y ∧ f y = v.
#A #B #f *
[ #v normalize #E destruct
| #y #v normalize #E %{y} destruct % //
] qed.

lemma refute_none_by_refl : ∀A,B:Type[0]. ∀P:A → B. ∀Q:B → Type[0]. ∀x:option A. ∀H:x = None ? → False.
  (∀v. x = Some ? v → Q (P v)) →
  Q (match x return λy.x = y → ? with [ Some v ⇒ λ_. P v | None ⇒ λE. match H E in False with [ ] ] (refl ? x)).
#A #B #P #Q *
[ #H cases (H (refl ??))
| #a #H #p normalize @p @refl
] qed.