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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 {l : Level} where
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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
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9 data _∨_ (A B : Set) : Set where
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10 or1 : A -> A ∨ B
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11 or2 : B -> A ∨ B
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12
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13 lemma01 : A -> A ∨ B
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14 lemma01 a = or1 a
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15
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16 lemma02 : B -> A ∨ B
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17 lemma02 b = or2 b
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18
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19 lemma03 : {C : Set} -> (A ∨ B) -> (A -> C) -> (B -> C) -> C
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20 lemma03 (or1 a) ac bc = ac a
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21 lemma03 (or2 b) ac bc = bc b
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22
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23 postulate U : Set l
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24 postulate V : Set l
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25
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26
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27 Bool = \{l : Level} -> {X : Set l} -> X -> X -> X
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28
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29 T : {l : Level} -> Bool
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30 T {l} = \{X : Set l} -> \(x y : X) -> x
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31
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32 F : {l : Level} -> Bool
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33 F {l} = \{X : Set l} -> \(x y : X) -> y
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34
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35 D : {l : Level} -> {U : Set l} -> U -> U -> Bool -> U
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36 D {l} {U} u v t = t {U} u v
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37
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38 lemma04 : {u v : U} -> D u v T ≡ u
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39 lemma04 = refl
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40
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41 lemma05 : {u v : U} -> D u v F ≡ v
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42 lemma05 = refl
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43
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44 _×_ : {l : Level} -> Set l -> Set l -> Set (suc l)
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45 _×_ {l} U V = {X : Set l} -> (U -> V -> X) -> X
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46
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47 <_,_> : {l : Level} {U V : Set l} -> U -> V -> (U × V)
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48 <_,_> {l} {U} {V} u v = \{X} -> \(x : U -> V -> X) -> x u v
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49
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50 π1 : {l : Level} {U V : Set l} -> (U × V) -> U
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51 π1 {l} {U} {V} t = t {U} (\(x : U) -> \(y : V) -> x)
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52
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53 π2 : {l : Level} {U V : Set l} -> (U × V) -> V
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54 π2 {l} {U} {V} t = t {V} (\(x : U) -> \(y : V) -> y)
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55
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56 lemma06 : {U V : Set l } -> {u : U } -> {v : V} -> π1 ( < u , v > ) ≡ u
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57 lemma06 = refl
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58
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59 lemma07 : {U V : Set l } -> {u : U } -> {v : V} -> π2 ( < u , v > ) ≡ v
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60 lemma07 = refl
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61
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62 hoge : {U V : Set l} -> U -> V -> (U × V)
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63 hoge u v = < u , v >
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64
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65 -- lemma08 : (t : U × V) -> < π1 t , π2 t > ≡ t
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66 -- lemma08 t = {!!}
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67
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68 -- Emp definision is still wrong...
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69
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70 Emp : ∀{l : Level} {X : Set l} -> Set l
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71 Emp {l} = \{X : Set l} -> X
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72
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73 -- ε : {l : Level} {U : Set l} -> Emp -> Emp
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74 -- ε {l} {U} t = t {l} {U}
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75
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76 -- lemma09 : {l : Level} {U : Set l} -> (t : Emp) -> ε U (ε Emp t) ≡ ε U t
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77 -- lemma09 t = refl
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78
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79 -- lemma10 : {l : Level} {U V X : Set l} -> (t : Emp ) -> (U × V)
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80 -- lemma10 {l} {U} {V} t = ε (U × V) t
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81
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82 -- lemma100 : {l : Level} {U V X : Set l} -> (t : Emp ) -> Emp
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83 -- lemma100 {l} {U} {V} t = ε U t
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84
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85 -- lemma101 : {l k : Level} {U V : Set l} -> (t : Emp ) -> π1 (ε (U × V) t) ≡ ε U t
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86 -- lemma101 t = refl
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87
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88 -- lemma102 : {l k : Level} {U V : Set l} -> (t : Emp ) -> π2 (ε (U × V) t) ≡ ε V t
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89 -- lemma102 t = refl
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90
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91 -- lemma103 : {l : Level} {U V : Set l} -> (u : U) -> (t : Emp) -> ε (U -> V) u ≡ ε V t
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92 -- lemma103 u t = refl
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93
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94 _+_ : Set l -> Set l -> Set (suc l)
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95 U + V = {X : Set l} -> ( U -> X ) -> (V -> X) -> X
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96
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97 ι1 : {U V : Set l} -> U -> U + V
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98 ι1 {U} {V} u = \{X} -> \(x : U -> X) -> \(y : V -> X ) -> x u
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99
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100 ι2 : {U V : Set l} -> V -> U + V
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101 ι2 {U} {V} v = \{X} -> \(x : U -> X) -> \(y : V -> X ) -> y v
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102
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103 δ : { U V R S : Set l } -> (R -> U) -> (S -> U) -> ( R + S ) -> U
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104 δ {U} {V} {R} {S} u v t = t {U} (\(x : R) -> u x) ( \(y : S) -> v y)
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105
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106 lemma11 : { U V R S : Set l } -> (u : R -> U ) (v : S -> U ) -> (r : R) -> δ {U} {V} {R} {S} u v ( ι1 r ) ≡ u r
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107 lemma11 u v r = refl
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108
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109 lemma12 : { U V R S : Set l } -> (u : R -> U ) (v : S -> U ) -> (s : S) -> δ {U} {V} {R} {S} u v ( ι2 s ) ≡ v s
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110 lemma12 u v s = refl
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111
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112
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113 _××_ : {l : Level} -> Set (suc l) -> Set l -> Set (suc l)
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114 _××_ {l} U V = {X : Set l} -> (U -> V -> X) -> X
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115
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116 <<_,_>> : {l : Level} {U : Set (suc l) } {V : Set l} -> U -> V -> (U ×× V)
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117 <<_,_>> {l} {U} {V} u v = \{X} -> \(x : U -> V -> X) -> x u v
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118
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120 Int = \{l : Level } -> \{ X : Set l } -> X -> ( X -> X ) -> X
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121
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122 Zero : {l : Level } -> { X : Set l } -> Int
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123 Zero {l} {X} = \(x : X ) -> \(y : X -> X ) -> x
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124
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125 S : {l : Level } -> { X : Set l } -> Int -> Int
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126 S {l} {X} t = \(x : X) -> \(y : X -> X ) -> y ( t x y )
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127
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128 n0 : {l : Level} {X : Set l} -> Int {l} {X}
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129 n0 = Zero
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130
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131 n1 : {l : Level } -> { X : Set l } -> Int {l} {X}
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132 n1 {l} {X} = \(x : X ) -> \(y : X -> X ) -> y x
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133
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134 n2 : {l : Level } -> { X : Set l } -> Int {l} {X}
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135 n2 {l} {X} = \(x : X ) -> \(y : X -> X ) -> y (y x)
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136
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137 n3 : {l : Level } -> { X : Set l } -> Int {l} {X}
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138 n3 {l} {X} = \(x : X ) -> \(y : X -> X ) -> y (y (y x))
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139
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140 lemma13 : {l : Level } -> { X : Set l } -> S ( S ( Zero {l} {X}) ) ≡ n2
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141 lemma13 {l} {X} = refl
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142
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143 It : {l : Level} {U : Set l} -> U -> ( U -> U ) -> Int -> U
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144 It {l} {U} u f t = t u f
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145
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147 R : {l : Level} { U X : Set l} -> U -> ( U -> Int {l} {X} -> U ) -> Int -> U
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148 R {l} {U} u v t = π1 ( It {suc l} {U × Int} (< u , Zero >) (λ (x : U × Int) → < v (π1 x) (π2 x) , S (π2 x) > ) t )
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149
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150 sum : {l : Level} {X : Set l} -> Int -> Int {l} {X} -> Int
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151 sum x y = It y ( λ z -> S z ) x
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152
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153 mul : {l : Level } {X : Set l} -> Int {l} {_} -> Int -> Int {l} {X}
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154 mul x y = It Zero ( λ (z : Int) -> sum y z ) x
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155
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156 copyInt : {l : Level } {X : Set l} -> Int {l} {_} -> Int {l} {X}
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157 copyInt x = It Zero ( λ (z : Int) -> S z ) x
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159 -- fact : {l : Level} {X X' : Set l} -> Int -> Int {l} {_}
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160 -- fact {l} {X} {X'} n = R (S Zero) (λ ( z w : Int) -> mul {l} {_} z (S w) ) n
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161
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162 -- lemma13' : fact n3 ≡ mul n1 ( mul n2 n3)
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163 -- lemma13' = refl
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164
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165 -- lemma14 : (x y : Int) -> mul x y ≡ mul y x
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166 -- lemma14 x y = It {!!} {!!} {!!}
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167
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168 lemma15 : {l : Level} {X : Set l} (x y : Int {l} {X}) -> mul {l} {X} n2 n3 ≡ mul {l} {X} n3 n2
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169 lemma15 x y = refl
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170
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171 lemma16 : {l : Level} {X U : Set l} -> (u : U ) -> (v : U -> Int {l} {X} -> U ) -> R u v Zero ≡ u
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172 lemma16 u v = refl
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173
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174 -- 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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175 -- lemma17 u v t = refl
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176
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177 -- 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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180 List = \{l : Level} -> \{ U X : Set l} -> X -> ( U -> X -> X ) -> X
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182 Nil : {l : Level} {U : Set l} {X : Set l} -> List {l} {U} {X}
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183 Nil {l} {U} {X} = \(x : X) -> \(y : U -> X -> X) -> x
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184
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185 Cons : {l : Level} {U : Set l} {X : Set l} -> U -> List {l} {U} {X} -> List {l} {U} {X}
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186 Cons {l} {U} {X} u t = \(x : X) -> \(y : U -> X -> X) -> y u (t x y )
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187
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188 l0 : {l : Level} {X : Set l} -> List {l} {Int {l} {X}} {X}
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189 l0 = Nil
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190
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191 l1 : {l : Level} {X : Set l} -> List {l} {Int {l} {X}} {X}
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192 l1 = Cons n1 Nil
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193
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194 l2 : {l : Level} {X : Set l} -> List {l} {Int {l} {X}} {X}
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195 l2 = Cons n2 l1
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197 ListIt : {l : Level} {U W X : Set l} -> W -> ( U -> W -> W ) -> List {l} {U} {W} -> W
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198 ListIt {l} {U} {W} {X} w f t = t w f
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199
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200 Nullp : {l : Level} {U : Set (suc l)} { X : Set (suc l)} -> List {suc l} {U} {Bool {l}} -> Bool
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201 Nullp {l} {U} {X} list = ListIt {suc l} {U} {Bool} {X} T (\u w -> F) list
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202
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203 Append : {l : Level} {U : Set l} {X : Set l} -> List {l} {U} {_} -> List {l} {U} {X} -> List {l} {U} {_}
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204 Append {l} {U} {X} x y = ListIt {l} {U} {_} {X} y (\u w -> Cons u w) x
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205
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206 -- lemma18 : Append l1 l2 ≡ Cons n1 ( Cons n2 Nil )
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207 -- lemma18 = refl
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208
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209 Tree = \{l : Level} -> \{ U X : Set l} -> X -> ( (U -> X) -> X ) -> X
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210
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211 NilTree : {l : Level} {U : Set l} {X : Set l} -> Tree {l} {U} {X}
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212 NilTree {l} {U} {X} = \(x : X) -> \(y : (U -> X) -> X) -> x
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213
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214 Collect : {l : Level} {U : Set l} {X : Set l} -> (U -> X -> ((U -> X) -> X) -> X ) -> Tree {l} {U} {X}
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215 Collect {l} {U} {X} f = \(x : X) -> \(y : (U -> X) -> X) -> y (\(z : U) -> f z x y )
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216
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217 TreeIt : {l : Level} {U W X : Set l} -> W -> ( (U -> W) -> W ) -> Tree {l} {U} {W} -> W
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218 TreeIt {l} {U} {W} {X} w h t = t w h
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