Last change
on this file since 1134 was
1134,
checked in by campbell, 8 years ago

Extra results for nonfailing map updates.

File size:
870 bytes

Line  

1  include "basics/types.ma". 

2  

3  definition option_map : ∀A,B:Type[0]. (A → B) → option A → option B ≝ 

4  λA,B,f,o. match o with [ None ⇒ None B  Some a ⇒ Some B (f a) ]. 

5  

6  lemma option_map_none : ∀A,B,f,x. 

7  option_map A B f x = None B → x = None A. 

8  #A #B #f * [ //  #a #E whd in E:(??%?); destruct ] 

9  qed. 

10  

11  lemma option_map_some : ∀A,B,f,x,v. 

12  option_map A B f x = Some B v → ∃y. x = Some ? y ∧ f y = v. 

13  #A #B #f * 

14  [ #v normalize #E destruct 

15   #y #v normalize #E %{y} destruct % // 

16  ] qed. 

17  

18  lemma refute_none_by_refl : ∀A,B:Type[0]. ∀P:A → B. ∀Q:B → Type[0]. ∀x:option A. ∀H:x = None ? → False. 

19  (∀v. x = Some ? v → Q (P v)) → 

20  Q (match x return λy.x = y → ? with [ Some v ⇒ λ_. P v  None ⇒ λE. match H E in False with [ ] ] (refl ? x)). 

21  #A #B #P #Q * 

22  [ #H cases (H (refl ??)) 

23   #a #H #p normalize @p @refl 

24  ] qed. 

25  

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