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1 module stack where
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2
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3 open import Relation.Binary.PropositionalEquality
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4 open import Relation.Binary.Core
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5
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6 data Nat : Set where
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7 zero : Nat
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8 suc : Nat -> Nat
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9
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10 data Bool : Set where
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11 True : Bool
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12 False : Bool
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13
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14 -- equal : {a : Set} -> a -> a -> Bool
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15 -- equal x y with (x ≡)y)
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16 -- equal x .x | refl = True
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17 -- equal _ _ | _ = False
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18
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19
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20 data Maybe (a : Set) : Set where
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21 Nothing : Maybe a
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22 Just : a -> Maybe a
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23
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24 record Stack {a t : Set} (stackImpl : Set) : Set where
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25 field
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26 stack : stackImpl
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27 push : stackImpl -> a -> (stackImpl -> t) -> t
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28 pop : stackImpl -> (stackImpl -> Maybe a -> t) -> t
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29
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30 open Stack
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31
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32 pushStack : {a t si : Set} -> Stack si -> a -> (Stack si -> t) -> t
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33 pushStack {a} {t} s0 d next = push s0 (stack s0) d (\s1 -> next (record s0 {stack = s1} ))
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34
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35 popStack : {a t si : Set} -> Stack si -> (Stack si -> Maybe a -> t) -> t
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36 popStack {a} {t} s0 next = pop s0 (stack s0) (\s1 d1 -> next (record s0 {stack = s1}) d1 )
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37
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38
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39 data Element (a : Set) : Set where
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40 cons : a -> Maybe (Element a) -> Element a
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41
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42 datum : {a : Set} -> Element a -> a
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43 datum (cons a _) = a
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44
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45 next : {a : Set} -> Element a -> Maybe (Element a)
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46 next (cons _ n) = n
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47
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48
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49 {-
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50 -- cannot define recrusive record definition. so use linked list with maybe.
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51 record Element {l : Level} (a : Set l) : Set (suc l) where
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52 field
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53 datum : a -- `data` is reserved by Agda.
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54 next : Maybe (Element a)
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55 -}
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56
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57
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58
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59 record SingleLinkedStack (a : Set) : Set where
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60 field
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61 top : Maybe (Element a)
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62 open SingleLinkedStack
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63
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64 pushSingleLinkedStack : {Data t : Set} -> SingleLinkedStack Data -> Data -> (Code : SingleLinkedStack Data -> t) -> t
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65 pushSingleLinkedStack stack datum next = next stack1
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66 where
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67 element = cons datum (top stack)
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68 stack1 = record {top = Just element}
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69
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70
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71 popSingleLinkedStack : {a t : Set} -> SingleLinkedStack a -> (Code : SingleLinkedStack a -> (Maybe a) -> t) -> t
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72 popSingleLinkedStack stack cs with (top stack)
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73 ... | Nothing = cs stack Nothing
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74 ... | Just d = cs stack1 (Just data1)
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75 where
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76 data1 = datum d
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77 stack1 = record { top = (next d) }
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78
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79
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80 emptySingleLinkedStack : {a : Set} -> SingleLinkedStack a
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81 emptySingleLinkedStack = record {top = Nothing}
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82
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83 createSingleLinkedStack : {a b : Set} -> Stack {a} {b} (SingleLinkedStack a)
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84 createSingleLinkedStack = record { stack = emptySingleLinkedStack
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85 ; push = pushSingleLinkedStack
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86 ; pop = popSingleLinkedStack
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87 }
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88
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89
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90
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91 test01 : {a : Set} -> SingleLinkedStack a -> Maybe a -> Bool
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92 test01 stack _ with (top stack)
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93 ... | (Just _) = True
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94 ... | Nothing = False
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95
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96
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97 test02 : {a : Set} -> SingleLinkedStack a -> Bool
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98 test02 stack = (popSingleLinkedStack stack) test01
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99
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100 test03 : {a : Set} -> a -> Bool
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101 test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02
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102
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103 testStack01 : {a : Set} -> a -> Bool
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104 testStack01 v = pushStack createSingleLinkedStack v (
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105 \s -> popStack s (\s1 d1 -> True))
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106
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107
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108
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109 lemma : {A : Set} {a : A} -> test03 a ≡ False
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110 lemma = refl
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111
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112 id : {A : Set} -> A -> A
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113 id a = a
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114
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115
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116 n-push : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
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117 n-push zero s = s
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118 n-push {A} {a} (suc n) s = pushSingleLinkedStack (n-push {A} {a} n s) a (\s -> s)
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119
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120 n-pop : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
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121 n-pop zero s = s
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122 n-pop {A} {a} (suc n) s = popSingleLinkedStack (n-pop {A} {a} n s) (\s _ -> s)
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123
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124 open ≡-Reasoning
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125
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126 push-pop-equiv : {A : Set} {a : A} (s : SingleLinkedStack A) -> popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ≡ s
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127 push-pop-equiv s = refl
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128
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129 push-and-n-pop : {A : Set} {a : A} (n : Nat) (s : SingleLinkedStack A) -> n-pop {A} {a} (suc n) (pushSingleLinkedStack s a id) ≡ n-pop {A} {a} n s
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130 push-and-n-pop zero s = refl
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131 push-and-n-pop {A} {a} (suc n) s = begin
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132 n-pop (suc (suc n)) (pushSingleLinkedStack s a id)
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133 ≡⟨ refl ⟩
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134 popSingleLinkedStack (n-pop (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s)
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135 ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s)) (push-and-n-pop n s) ⟩
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136 popSingleLinkedStack (n-pop n s) (\s _ -> s)
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137 ≡⟨ refl ⟩
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138 n-pop (suc n) s
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139 ∎
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140
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141
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142 n-push-pop-equiv : {A : Set} {a : A} (n : Nat) (s : SingleLinkedStack A) -> (n-pop {A} {a} n (n-push {A} {a} n s)) ≡ s
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143 n-push-pop-equiv zero s = refl
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144 n-push-pop-equiv {A} {a} (suc n) s = begin
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145 n-pop (suc n) (n-push (suc n) s)
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146 ≡⟨ refl ⟩
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147 n-pop (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s))
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148 ≡⟨ push-and-n-pop n (n-push n s) ⟩
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149 n-pop n (n-push n s)
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150 ≡⟨ n-push-pop-equiv n s ⟩
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151 s
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152 ∎
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181
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153
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154
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155 n-push-pop-equiv-empty : {A : Set} {a : A} -> (n : Nat) -> n-pop {A} {a} n (n-push {A} {a} n emptySingleLinkedStack) ≡ emptySingleLinkedStack
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156 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
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