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1 open import Level
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2 open import Relation.Binary.PropositionalEquality
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3
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4 module system-f where
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5
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6 Bool : {l : Level} (X : Set l) -> Set l
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7 Bool = \{l : Level} -> \(X : Set l) -> X -> X -> X
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8
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9 T : {l : Level} (X : Set l) -> Bool X
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10 T X = \(x y : X) -> x
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11
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12 F : {l : Level} (X : Set l) -> Bool X
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13 F X = \(x y : X) -> y
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14
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15 D : {l : Level} -> {U : Set l} -> U -> U -> Bool U -> U
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16 D u v t = t u v
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17
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18 lemma04 : {l : Level} { U : Set l} {u v : U} -> D {_} {U} u v (T U ) ≡ u
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19 lemma04 = refl
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20
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21 lemma05 : {l : Level} { U : Set l} {u v : U} -> D {_} {U} u v (F U ) ≡ v
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22 lemma05 = refl
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23
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24 _×_ : {l : Level} -> Set l -> Set l -> Set (suc l)
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25 _×_ {l} U V = {X : Set l} -> (U -> V -> X) -> X
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26
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27 <_,_> : {l : Level} {U V : Set l} -> U -> V -> (U × V)
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28 <_,_> {l} {U} {V} u v = \{X} -> \(x : U -> V -> X) -> x u v
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29
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30 π1 : {l : Level} {U V : Set l} -> (U × V) -> U
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31 π1 {l} {U} {V} t = t {U} (\(x : U) -> \(y : V) -> x)
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32
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33 π2 : {l : Level} {U V : Set l} -> (U × V) -> V
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34 π2 {l} {U} {V} t = t {V} (\(x : U) -> \(y : V) -> y)
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35
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36 lemma06 : {l : Level} {U V : Set l } -> {u : U } -> {v : V} -> π1 ( < u , v > ) ≡ u
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37 lemma06 = refl
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38
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39 lemma07 : {l : Level} {U V : Set l } -> {u : U } -> {v : V} -> π2 ( < u , v > ) ≡ v
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40 lemma07 = refl
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41
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42 hoge : {l : Level} {U V : Set l} -> U -> V -> (U × V)
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43 hoge u v = < u , v >
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44
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45 -- lemma08 : (t : U × V) -> < π1 t , π2 t > ≡ t
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46 -- lemma08 t = {!!}
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47
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48 -- Emp definision is still wrong...
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49
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50 Emp : {l : Level} {X : Set l} -> Set l
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51 Emp {l} = \{X : Set l} -> X
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52
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53 -- ε : {l : Level} (U : Set l) {l' : Level} {U' : Set l'} -> Emp -> Emp
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54 -- ε {l} U t = t
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55
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56 -- lemma09 : {l : Level} {U : Set l} {l' : Level} {U' : Set l} -> (t : Emp {l} {U} ) -> ε U (ε Emp t) ≡ ε U t
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57 -- lemma09 t = refl
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58
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59 -- lemma10 : {l : Level} {U V X : Set l} -> (t : Emp {_} {U × V}) -> U × V
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60 -- lemma10 {l} {U} {V} t = ε (U × V) t
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61
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62 -- lemma10' : {l : Level} {U V X : Set l} -> (t : Emp {_} {U × V}) -> Emp
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63 -- lemma10' {l} {U} {V} {X} t = ε (U × V) t
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64
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65 -- lemma100 : {l : Level} {U V X : Set l} -> (t : Emp {_} {U}) -> Emp
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66 -- lemma100 {l} {U} {V} t = ε U t
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67
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68 -- lemma101 : {l k : Level} {U V : Set l} -> (t : Emp {_} {U × V}) -> π1 (ε (U × V) t) ≡ ε U t
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69 -- lemma101 t = refl
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70
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71 -- lemma102 : {l k : Level} {U V : Set l} -> (t : Emp {_} {U × V}) -> π2 (ε (U × V) t) ≡ ε V t
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72 -- lemma102 t = refl
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73
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74 -- lemma103 : {l : Level} {U V : Set l} -> (u : U) -> (t : Emp {l} {_} ) -> (ε (U -> V) t) u ≡ ε V t
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75 -- lemma103 u t = refl
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76
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77 _+_ : {l : Level} -> Set l -> Set l -> Set (suc l)
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78 _+_ {l} U V = {X : Set l} -> ( U -> X ) -> (V -> X) -> X
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79
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80 ι1 : {l : Level } {U V : Set l} -> U -> U + V
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81 ι1 {l} {U} {V} u = \{X} -> \(x : U -> X) -> \(y : V -> X ) -> x u
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82
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83 ι2 : {l : Level } {U V : Set l} -> V -> U + V
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84 ι2 {l} {U} {V} v = \{X} -> \(x : U -> X) -> \(y : V -> X ) -> y v
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85
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86 δ : {l : Level} { U V R S : Set l } -> (R -> U) -> (S -> U) -> ( R + S ) -> U
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87 δ {l} {U} {V} {R} {S} u v t = t {U} (\(x : R) -> u x) ( \(y : S) -> v y)
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88
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89 lemma11 : {l : Level} { U V R S : Set _ } -> (u : R -> U ) (v : S -> U ) -> (r : R) -> δ {l} {U} {V} {R} {S} u v ( ι1 r ) ≡ u r
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90 lemma11 u v r = refl
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92 lemma12 : {l : Level} { U V R S : Set _ } -> (u : R -> U ) (v : S -> U ) -> (s : S) -> δ {l} {U} {V} {R} {S} u v ( ι2 s ) ≡ v s
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93 lemma12 u v s = refl
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96 _××_ : {l : Level} -> Set (suc l) -> Set l -> Set (suc l)
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97 _××_ {l} U V = {X : Set l} -> (U -> V -> X) -> X
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98
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99 <<_,_>> : {l : Level} {U : Set (suc l) } {V : Set l} -> U -> V -> (U ×× V)
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100 <<_,_>> {l} {U} {V} u v = \{X} -> \(x : U -> V -> X) -> x u v
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101
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103 Int : {l : Level } ( X : Set l ) -> Set l
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104 Int {l} X = X -> ( X -> X ) -> X
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105
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106 Zero : {l : Level } -> { X : Set l } -> Int X
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107 Zero {l} {X} = \(x : X ) -> \(y : X -> X ) -> x
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108
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109 S : {l : Level } -> { X : Set l } -> Int X -> Int X
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110 S {l} {X} t = \(x : X) -> \(y : X -> X ) -> y ( t x y )
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112 n0 : {l : Level} {X : Set l} -> Int X
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113 n0 = Zero
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114
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115 n1 : {l : Level } -> { X : Set l } -> Int X
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116 n1 {_} {X} = \(x : X ) -> \(y : X -> X ) -> y x
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117
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118 n2 : {l : Level } -> { X : Set l } -> Int X
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119 n2 {_} {X} = \(x : X ) -> \(y : X -> X ) -> y (y x)
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120
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121 n3 : {l : Level } -> { X : Set l } -> Int X
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122 n3 {_} {X} = \(x : X ) -> \(y : X -> X ) -> y (y (y x))
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123
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124 lemma13 : {l : Level } -> { X : Set l } -> S (S (Zero {_} {X})) ≡ n2
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125 lemma13 = refl
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126
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127 It : {l : Level} {U : Set l} -> U -> ( U -> U ) -> Int U -> U
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128 It u f t = t u f
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130 R : {l : Level} { U X : Set l} -> U -> ( U -> Int X -> U ) -> Int _ -> U
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131 R {l} {U} {X} u v t = π1 ( It {suc l} {U × Int X} (< u , Zero >) (λ (x : U × Int X) → < v (π1 x) (π2 x) , S (π2 x) > ) t )
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133 add : {l : Level} {X : Set l} -> Int (Int X) -> Int X -> Int X
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134 add x y = It y S x
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135
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136 mul : {l : Level } {X : Set l} -> Int (Int X) -> Int (Int X) -> Int X
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137 mul {l} {X} x y = It Zero (add x) y
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139 fact : {l : Level} {X : Set l} -> Int _ -> Int (Int X)
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140 fact {l} {X} n = R (S Zero) (λ (z : Int (Int X)) -> λ w -> mul w (S w) ) n
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142 lemma13' : {l : Level} {X : Set l} -> fact {l} {X} n3 ≡ mul n1 ( mul n2 n3)
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143 lemma13' = refl
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144
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145 -- lemma14 : (x y : Int) -> mul x y ≡ mul y x
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146 -- lemma14 x y = It {!!} {!!} {!!}
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148 lemma15 : {l : Level} {X : Set l} (x y : Int X) -> mul n2 n3 ≡ mul {l} {X} n3 n2
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149 lemma15 x y = refl
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151 lemma16 : {l : Level} {X U : Set l} -> (u : U ) -> (v : U -> Int X -> U ) -> R u v Zero ≡ u
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152 lemma16 u v = refl
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154 -- lemma17 : {l : Level} {X U : Set l} -> (u : U ) -> (v : U -> Int -> U ) -> (t : Int ) -> R u v (S t) ≡ v ( R u v t ) t
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155 -- lemma17 u v t = refl
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156
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157 -- postulate lemma17 : {l : Level} {X U : Set l} -> (u : U ) -> (v : U -> Int -> U ) -> (t : Int ) -> R u v (S t) ≡ v ( R u v t ) t
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159 List : {l : Level} (U X : Set l) -> Set l
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160 List {l} = \( U X : Set l) -> X -> ( U -> X -> X ) -> X
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162 Nil : {l : Level} {U : Set l} {X : Set l} -> List U X
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163 Nil {l} {U} {X} = \(x : X) -> \(y : U -> X -> X) -> x
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165 Cons : {l : Level} {U : Set l} {X : Set l} -> U -> List U X -> List U X
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166 Cons {l} {U} {X} u t = \(x : X) -> \(y : U -> X -> X) -> y u (t x y )
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168 l0 : {l : Level} {X X' : Set l} -> List (Int X) (X')
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169 l0 = Nil
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171 l1 : {l : Level} {X X' : Set l} -> List (Int X) (X')
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172 l1 = Cons n1 Nil
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174 l2 : {l : Level} {X X' : Set l} -> List (Int X) (X')
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175 l2 = Cons n2 l1
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177 l3 : {l : Level} {X X' : Set l} -> List (Int X) (X')
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178 l3 = Cons n3 l2
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180 ListIt : {l : Level} {U W X : Set l} -> W -> ( U -> W -> W ) -> List U W -> W
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181 ListIt w f t = t w f
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183 Nullp : {l : Level} {U : Set (suc l)} { X : Set (suc l)} -> List U (Bool X) -> Bool _
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184 Nullp {l} {U} {X} list = ListIt {suc l} {U} {Bool _} {X} (T X) (\u w -> (F X)) list
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185
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186 Append : {l : Level} {U : Set l} {X : Set l} -> List U _ -> List U X -> List U _
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187 Append x y = \s t -> x (y s t) t
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188
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189 lemma18 :{l : Level} {U : Set l} {X : Set l} -> Append {_} {Int U} {X} l1 l2 ≡ Cons n1 (Cons n2 (Cons n1 Nil))
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190 lemma18 = refl
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192 Reverse : {l : Level} {U : Set l} {X : Set l} -> List U _ -> List U X
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193 Reverse {l} {U} {X} x = ListIt {_} {U} {List U _} {X} Nil ( \u w -> Append w (Cons u Nil) ) x
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194
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195 lemma19 :{l : Level} {U : Set l} {X : Set l} -> Reverse {_} {Int U} {X} l3 ≡ Cons n1 (Cons n2 (Cons n3 Nil))
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196 lemma19 = refl
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197
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198 Tree : {l : Level} -> Set l -> Set l -> Set l
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199 Tree {l} = \( U X : Set l) -> X -> ( (U -> X) -> X ) -> X
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200
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201 NilTree : {l : Level} (U : Set l) (X : Set l) -> Tree U X
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202 NilTree U X = \(x : X) -> \(y : (U -> X) -> X) -> x
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203
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204 Collect : {l : Level} (U : Set l) (X : Set l) -> (U -> X -> ((U -> X) -> X) -> X ) -> Tree U X
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205 Collect U X f = \(x : X) -> \(y : (U -> X) -> X) -> y (\(z : U) -> f z x y )
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207 TreeIt : {l : Level} {U W X : Set l} -> W -> ( (U -> W) -> W ) -> Tree U W -> W
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208 TreeIt w h t = t w h
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