include "basics/lists/list.ma".

lemma All_append : ∀A,P,l,r. All A P l → All A P r → All A P (l@r).
#A #P #l elim l
[ //
| #hd #tl #IH #r * #H1 #H2 #H3 % // @IH //
] qed.

lemma All_append_l : ∀A,P,l,r. All A P (l@r) → All A P l.
#A #P #l elim l
[ //
| #hd #tl #IH #r * #H1 #H2 % /2/
] qed.

lemma All_append_r : ∀A,P,l,r. All A P (l@r) → All A P r.
#A #P #l elim l
[ //
| #h #t #IH #r * /2/
] qed.

(* An alternative form of All that can be easier to use sometimes. *)
lemma All_alt : ∀A,P,l.
  (∀a,pre,post. l = pre@a::post → P a) →
  All A P l.
#A #P #l #H lapply (refl ? l) change with ([ ] @ l) in ⊢ (???% → ?);
generalize in ⊢ (???(??%?) → ?); elim l in ⊢ (? → ???(???%) → %);
[ #pre #E %
| #a #tl #IH #pre #E %
  [ @(H a pre tl E)
  | @(IH (pre@[a])) >associative_append @E
  ]
] qed.

let rec All2 (A,B:Type[0]) (P:A → B → Prop) (la:list A) (lb:list B) on la : Prop ≝
match la with
[ nil ⇒ match lb with [ nil ⇒ True | _ ⇒ False ]
| cons ha ta ⇒
    match lb with [ nil ⇒ False | cons hb tb ⇒ P ha hb ∧ All2 A B P ta tb ]
].

lemma All2_length : ∀A,B,P,la,lb. All2 A B P la lb → |la| = |lb|.
#A #B #P #la elim la
[ * [ // | #x #y * ]
| #ha #ta #IH * [ * | #hb #tb * #H1 #H2 whd in ⊢ (??%%); >(IH … H2) @refl
] qed.

lemma All2_mp : ∀A,B,P,Q,la,lb. (∀a,b. P a b → Q a b) → All2 A B P la lb → All2 A B Q la lb.
#A #B #P #Q #la elim la
[ * [ // | #h #t #_ * ]
| #ha #ta #IH * [ // | #hb #tb #H * #H1 #H2 % [ @H @H1 | @(IH … H2) @H ] ]
] qed.

let rec map_All (A,B:Type[0]) (P:A → Prop) (f:∀a. P a → B) (l:list A) (H:All A P l) on l : list B ≝
match l return λl. All A P l → ? with
[ nil ⇒ λ_. nil B
| cons hd tl ⇒ λH. cons B (f hd (proj1 … H)) (map_All A B P f tl (proj2 … H))
] H.

include "utilities/monad.ma".

definition Append : ∀A.Aop ? (nil A) ≝ λA.mk_Aop ? ? (append ?) ? ? ?.
// qed.

definition List ≝ MakeMonadProps
  list
  (λX,x.[x])
  (λX,Y,l,f.\fold [append ?, [ ]]_{x ∈ l} (f x))
  ???. normalize
[ / by /
| #X#m elim m normalize //
| #X#Y#Z #m #f#g
  elim m normalize [//]
  #x#l' #Hi
  <(fold_sum ?? (f x) ? [ ] (Append ?))
  >Hi //
] qed.

unification hint 0 ≔ X ;
N ≟ max_def List, M ≟ m_def N
(*---------------------------*)⊢
list X ≡ monad M X.