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1 module list-nat0 where
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2
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3 open import Level
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4
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5
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6 postulate a : Set
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7 postulate b : Set
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8 postulate c : Set
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9
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10
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11 infixr 40 _::_
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12 data List ∀ {a} (A : Set a) : Set a where
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13 [] : List A
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14 _::_ : A -> List A -> List A
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15
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16
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17 infixl 30 _++_
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18 -- _++_ : {a : Level } -> {A : Set a} -> List A -> List A -> List A
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19 _++_ : ∀ {a} {A : Set a} -> List A -> List A -> List A
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20 [] ++ ys = ys
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21 (x :: xs) ++ ys = x :: (xs ++ ys)
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22
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23
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24 l1 = a :: []
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25 l2 = a :: b :: a :: c :: []
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26
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27 l3 = l1 ++ l2
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28
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29 infixr 20 _==_
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30
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31 data _==_ {n} {A : Set n} : List A -> List A -> Set n where
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32 reflection : {x : List A} -> x == x
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33 eq-cons : {x y : List A} { a : A } -> x == y -> ( a :: x ) == ( a :: y )
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34 trans-list : {x y z : List A} { a : A } -> x == y -> y == z -> x == z
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35 -- eq-nil : [] == []
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36
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37 list-id-l : { A : Set } -> { x : List A} -> [] ++ x == x
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38 list-id-l = reflection
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39
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40 list-id-r : { A : Set } -> ( x : List A ) -> x ++ [] == x
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41 list-id-r [] = reflection
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42 list-id-r (x :: xs) = eq-cons ( list-id-r xs )
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43
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44
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45 -- listAssoc : { A : Set } -> { a b c : List A} -> ( ( a ++ b ) ++ c ) == ( a ++ ( b ++ c ) )
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46 -- listAssoc = eq-reflection
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47
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48 list-assoc : {A : Set } -> ( xs ys zs : List A ) ->
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49 ( ( xs ++ ys ) ++ zs ) == ( xs ++ ( ys ++ zs ) )
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50 list-assoc [] ys zs = reflection
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51 list-assoc (x :: xs) ys zs = eq-cons ( list-assoc xs ys zs )
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52
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53
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54
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55 open import Relation.Binary.PropositionalEquality
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56 open ≡-Reasoning
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57
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58 cong1 : ∀{a} {A : Set a } {b} { B : Set b } ->
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59 ( f : A -> B ) -> {x : A } -> {y : A} -> x ≡ y -> f x ≡ f y
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60 cong1 f refl = refl
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61
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62 lemma11 : ∀{n} -> ( Set n ) IsRelatedTo ( Set n )
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63 lemma11 = relTo refl
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64
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65 lemma12 : {L : Set} ( x : List L ) -> x ++ x ≡ x ++ x
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66 lemma12 x = begin x ++ x ∎
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67
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68
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69 ++-assoc : {L : Set} ( xs ys zs : List L ) -> (xs ++ ys) ++ zs ≡ xs ++ (ys ++ zs)
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70 ++-assoc [] ys zs = -- {A : Set} -> -- let open ==-Reasoning A in
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71 begin -- to prove ([] ++ ys) ++ zs ≡ [] ++ (ys ++ zs)
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72 ( [] ++ ys ) ++ zs
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73 ≡⟨ refl ⟩
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74 ys ++ zs
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75 ≡⟨ refl ⟩
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76 [] ++ ( ys ++ zs )
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77 ∎
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78
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79 ++-assoc (x :: xs) ys zs = -- {A : Set} -> -- let open ==-Reasoning A in
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80 begin -- to prove ((x :: xs) ++ ys) ++ zs ≡ (x :: xs) ++ (ys ++ zs)
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81 ((x :: xs) ++ ys) ++ zs
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82 ≡⟨ refl ⟩
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83 (x :: (xs ++ ys)) ++ zs
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84 ≡⟨ refl ⟩
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85 x :: ((xs ++ ys) ++ zs)
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86 ≡⟨ cong1 (_::_ x) (++-assoc xs ys zs) ⟩
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87 x :: (xs ++ (ys ++ zs))
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88 ≡⟨ refl ⟩
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89 (x :: xs) ++ (ys ++ zs)
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90 ∎
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91
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92
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93
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94
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95
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96 --data Bool : Set where
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97 -- true : Bool
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98 -- false : Bool
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99
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100
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101 --postulate Obj : Set
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102
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103 --postulate Hom : Obj -> Obj -> Set
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104
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105
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106 --postulate id : { a : Obj } -> Hom a a
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107
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108
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109 --infixr 80 _○_
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110 --postulate _○_ : { a b c : Obj } -> Hom b c -> Hom a b -> Hom a c
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111
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112 -- postulate axId1 : {a b : Obj} -> ( f : Hom a b ) -> f == id ○ f
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113 -- postulate axId2 : {a b : Obj} -> ( f : Hom a b ) -> f == f ○ id
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114
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115 --assoc : { a b c d : Obj } ->
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116 -- (f : Hom c d ) -> (g : Hom b c) -> (h : Hom a b) ->
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117 -- (f ○ g) ○ h == f ○ ( g ○ h)
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118
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119
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120 --a = Set
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121
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122 -- ListObj : {A : Set} -> List A
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123 -- ListObj = List Set
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124
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125 -- ListHom : ListObj -> ListObj -> Set
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126
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