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