include "basics/list.ma".
include "basics/types.ma".
include "arithmetics/nat.ma".
  
lemma eq_rect_Type0_r :
  ∀A: Type[0].
  ∀a:A.
  ∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x: A.∀p:eq ? x a. P x p.
  #A #a #P #H #x #p
  generalize in match H
  generalize in match P
  cases p
  //
qed.
  
let rec safe_nth (A: Type[0]) (n: nat) (l: list A) (p: n < length A l) on n: A ≝
  match n return λo. o < length A l → A with
  [ O ⇒
    match l return λm. 0 < length A m → A with
    [ nil ⇒ λabsd1. ?
    | cons hd tl ⇒ λprf1. hd
    ]
  | S n' ⇒
    match l return λm. S n' < length A m → A with
    [ nil ⇒ λabsd2. ?
    | cons hd tl ⇒ λprf2. safe_nth A n' tl ?
    ]
  ] ?.
  [ 1:
    @ p
  | 4:
    normalize in prf2
    normalize
    @ le_S_S_to_le
    assumption
  | 2:
    normalize in absd1;
    cases (not_le_Sn_O O)
    # H
    elim (H absd1)
  | 3:
    normalize in absd2;
    cases (not_le_Sn_O (S n'))
    # H
    elim (H absd2)
  ]
qed.
  
let rec nub_by_internal (A: Type[0]) (f: A → A → bool) (l: list A) (n: nat) on n ≝
  match n with
  [ O ⇒
    match l with
    [ nil ⇒ [ ]
    | cons hd tl ⇒ l
    ]
  | S n ⇒
    match l with
    [ nil ⇒ [ ]
    | cons hd tl ⇒
      hd :: nub_by_internal A f (filter ? (λy. notb (f y hd)) tl) n
    ]
  ].
  
definition nub_by ≝
  λA: Type[0].
  λf: A → A → bool.
  λl: list A.
    nub_by_internal A f l (length ? l).
  
let rec member (A: Type[0]) (eq: A → A → bool) (a: A) (l: list A) on l ≝
  match l with
  [ nil ⇒ false
  | cons hd tl ⇒ orb (eq a hd) (member A eq a tl)
  ].
  
let rec take (A: Type[0]) (n: nat) (l: list A) on n: list A ≝
  match n with
  [ O ⇒ [ ]
  | S n ⇒
    match l with
    [ nil ⇒ [ ]
    | cons hd tl ⇒ hd :: take A n tl
    ]
  ].
  
let rec drop (A: Type[0]) (n: nat) (l: list A) on n ≝
  match n with
  [ O ⇒ l
  | S n ⇒
    match l with
    [ nil ⇒ [ ]
    | cons hd tl ⇒ drop A n tl
    ]
  ].
  
definition list_split ≝
  λA: Type[0].
  λn: nat.
  λl: list A.
    〈take A n l, drop A n l〉.
  
let rec mapi_internal (A: Type[0]) (B: Type[0]) (n: nat) (f: nat → A → B)
                      (l: list A) on l: list B ≝
  match l with
  [ nil ⇒ nil ?
  | cons hd tl ⇒ (f n hd) :: (mapi_internal A B (n + 1) f tl)
  ].  

definition mapi ≝
  λA, B: Type[0].
  λf: nat → A → B.
  λl: list A.
    mapi_internal A B 0 f l.

let rec zip (A: Type[0]) (B: Type[0]) (l: list A) (r: list B) on l: option (list (A × B)) ≝
  match l with
  [ nil ⇒ Some ? (nil (A × B))
  | cons hd tl ⇒
    match r with
    [ nil ⇒ None ?
    | cons hd' tl' ⇒
      match zip ? ? tl tl' with
      [ None ⇒ None ?
      | Some tail ⇒ Some ? (〈hd, hd'〉 :: tail)
      ]
    ]
  ].

let rec foldl (A: Type[0]) (B: Type[0]) (f: A → B → A) (a: A) (l: list B) on l ≝
  match l with
  [ nil ⇒ a
  | cons hd tl ⇒ foldl A B f (f a hd) tl
  ].

definition flatten ≝
  λA: Type[0].
  λl: list (list A).
    foldl ? ? (append ?) [ ] l.

definition if_then_else ≝
  λA: Type[0].
  λb: bool.
  λt: A.
  λf: A.
  match b with
  [ true ⇒ t
  | false ⇒ f
  ].

let rec rev (A: Type[0]) (l: list A) on l ≝ 
  match l with
  [ nil ⇒ nil A
  | cons hd tl ⇒ (rev A tl) @ [ hd ]
  ]. 
    
(*
notation "hvbox('if' b 'then' t 'else' f)"
  non associative with precedence 83
  for @{ 'if_then_else $b $t $f }.
*)
notation > "'if' term 19 e 'then' term 19 t 'else' term 48 f" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.
notation < "'if' \nbsp term 19 e \nbsp 'then' \nbsp term 19 t \nbsp 'else' \nbsp term 48 f \nbsp" non associative with precedence 19 for @{ 'if_then_else $e $t $f }.

interpretation "Bool if_then_else" 'if_then_else b t f = (if_then_else ? b t f).

let rec fold_left_i_aux (A: Type[0]) (B: Type[0])
                        (f: nat → A → B → A) (x: A) (i: nat) (l: list B) on l ≝
  match l with
    [ nil ⇒ x
    | cons hd tl ⇒ fold_left_i_aux A B f (f i x hd) (S i) tl
    ].

definition fold_left_i ≝ λA,B,f,x. fold_left_i_aux A B f x O.

notation "hvbox(t⌈o ↦ h⌉)"
  with precedence 45
  for @{ match (? : $o=$h) with [ refl ⇒ $t ] }.

definition function_apply ≝
  λA, B: Type[0].
  λf: A → B.
  λa: A.
    f a.
    
notation "f break $ x"
  left associative with precedence 99
  for @{ 'function_apply $f $x }.
  
interpretation "Function application" 'function_apply f x = (function_apply ? ? f x).

let rec iterate (A: Type[0]) (f: A → A) (a: A) (n: nat) on n ≝
  match n with
    [ O ⇒ a
    | S o ⇒ f (iterate A f a o)
    ].
    
notation > "hvbox('let' 〈ident x,ident y〉 ≝ t 'in' s)"
 with precedence 10
for @{ match $t with [ pair ${ident x} ${ident y} ⇒ $s ] }.

(* Yeah, I probably ought to do something more general... *)
notation "hvbox(\langle term 19 a, break term 19 b, break term 19 c\rangle)" 
with precedence 90 for @{ 'triple $a $b $c}.
interpretation "Triple construction" 'triple x y z = (pair ? ? (pair ? ? x y) z).

notation > "hvbox('let' 〈ident x,ident y,ident z〉 ≝ t 'in' s)"
 with precedence 10
for @{ match $t with [ pair ${fresh xy} ${ident z} ⇒ match ${fresh xy} with [ pair ${ident x} ${ident y} ⇒ $s ] ] }.

notation "⊥" with precedence 90
  for @{ match ? in False with [ ] }.

let rec exclusive_disjunction (b: bool) (c: bool) on b ≝
  match b with
    [ true ⇒
      match c with
        [ false ⇒ true
        | true ⇒ false
        ]
    | false ⇒
      match c with
        [ false ⇒ false
        | true ⇒ true
        ]
    ].
    
definition ltb ≝
  λm, n: nat.
    leb (S m) n.
    
definition geb ≝
  λm, n: nat.
    ltb n m.

definition gtb ≝
  λm, n: nat.
    leb n m.
    
(* dpm: unless I'm being stupid, this isn't defined in the stdlib? *)
let rec eq_nat (n: nat) (m: nat) on n: bool ≝
  match n with
  [ O ⇒ match m with [ O ⇒ true | _ ⇒ false ]
  | S n' ⇒ match m with [ S m' ⇒ eq_nat n' m' | _ ⇒ false ]
  ].

(* dpm: conflicts with library definitions
interpretation "Nat less than" 'lt m n = (ltb m n).
interpretation "Nat greater than" 'gt m n = (gtb m n).
interpretation "Nat greater than eq" 'geq m n = (geb m n).
*)

let rec division_aux (m: nat) (n : nat) (p: nat) ≝
  match ltb n (S p) with
    [ true ⇒ O
    | false ⇒
      match m with
        [ O ⇒ O
        | (S q) ⇒ S (division_aux q (n - (S p)) p)
        ]
    ].
    
definition division ≝
  λm, n: nat.
    match n with
      [ O ⇒ S m
      | S o ⇒ division_aux m m o
      ].
      
notation "hvbox(n break ÷ m)"
  right associative with precedence 47
  for @{ 'division $n $m }.
  
interpretation "Nat division" 'division n m = (division n m).

let rec modulus_aux (m: nat) (n: nat) (p: nat) ≝
  match leb n p with
    [ true ⇒ n
    | false ⇒
      match m with
        [ O ⇒ n
        | S o ⇒ modulus_aux o (n - (S p)) p
        ]
    ].
    
definition modulus ≝
  λm, n: nat.
    match n with
      [ O ⇒ m
      | S o ⇒ modulus_aux m m o
      ].
    
notation "hvbox(n break 'mod' m)"
  right associative with precedence 47
  for @{ 'modulus $n $m }.
  
interpretation "Nat modulus" 'modulus m n = (modulus m n).

definition divide_with_remainder ≝
  λm, n: nat.
    pair ? ? (m ÷ n) (modulus m n).
    
let rec exponential (m: nat) (n: nat) on n ≝
  match n with
    [ O ⇒ S O
    | S o ⇒ m * exponential m o
    ].

interpretation "Nat exponential" 'exp n m = (exponential n m).
    
notation "hvbox(a break ⊎ b)"
 left associative with precedence 50
for @{ 'disjoint_union $a $b }.
interpretation "sum" 'disjoint_union A B = (Sum A B).

theorem less_than_or_equal_monotone:
  ∀m, n: nat.
    m ≤ n → (S m) ≤ (S n).
 #m #n #H
 elim H
 /2/
qed.

theorem less_than_or_equal_b_complete:
  ∀m, n: nat.
    leb m n = false → ¬(m ≤ n).
 #m;
 elim m;
 normalize
 [ #n #H
   destruct
 | #y #H1 #z
   cases z
   normalize
   [ #H
     /2/
   | /3/
   ]
 ]
qed.

theorem less_than_or_equal_b_correct:
  ∀m, n: nat.
    leb m n = true → m ≤ n.
 #m
 elim m
 //
 #y #H1 #z
 cases z
 normalize
 [ #H
   destruct
 | #n #H lapply (H1 … H) /2/
 ]
qed.

definition less_than_or_equal_b_elim:
 ∀m, n: nat.
 ∀P: bool → Type[0].
   (m ≤ n → P true) → (¬(m ≤ n) → P false) → P (leb m n).
 #m #n #P #H1 #H2;
 lapply (less_than_or_equal_b_correct m n)
 lapply (less_than_or_equal_b_complete m n)
 cases (leb m n)
 /3/
qed.