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1 open import Level renaming (suc to succ ; zero to Zero )
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2 module stackTest where
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3
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4 open import stack
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5
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6 open import Relation.Binary.PropositionalEquality
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7 open import Relation.Binary.Core
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8 open import Data.Nat
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9 open import Function
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10
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11
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12 open SingleLinkedStack
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13 open Stack
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14
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15 ----
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16 --
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17 -- proof of properties ( concrete cases )
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18 --
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19
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20 test01 : {n : Level } {a : Set n} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ Maybe a @$\rightarrow$@ Bool {n}
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21 test01 stack _ with (top stack)
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22 ... | (Just _) = True
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23 ... | Nothing = False
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24
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25
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26 test02 : {n : Level } {a : Set n} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ Bool
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27 test02 stack = popSingleLinkedStack stack test01
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28
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29 test03 : {n : Level } {a : Set n} @$\rightarrow$@ a @$\rightarrow$@ Bool
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30 test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02
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31
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32 -- after a push and a pop, the stack is empty
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33 lemma : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ test03 a @$\equiv$@ False
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34 lemma = refl
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35
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36 testStack01 : {n m : Level } {a : Set n} @$\rightarrow$@ a @$\rightarrow$@ Bool {m}
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37 testStack01 v = pushStack createSingleLinkedStack v (
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38 \s @$\rightarrow$@ popStack s (\s1 d1 @$\rightarrow$@ True))
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39
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40 -- after push 1 and 2, pop2 get 1 and 2
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41
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42 testStack02 : {m : Level } @$\rightarrow$@ ( Stack @$\mathbb{N}$@ (SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ Bool {m} ) @$\rightarrow$@ Bool {m}
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43 testStack02 cs = pushStack createSingleLinkedStack 1 (
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44 \s @$\rightarrow$@ pushStack s 2 cs)
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45
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46
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47 testStack031 : (d1 d2 : @$\mathbb{N}$@ ) @$\rightarrow$@ Bool {Zero}
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48 testStack031 2 1 = True
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49 testStack031 _ _ = False
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50
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51 testStack032 : (d1 d2 : Maybe @$\mathbb{N}$@) @$\rightarrow$@ Bool {Zero}
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52 testStack032 (Just d1) (Just d2) = testStack031 d1 d2
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53 testStack032 _ _ = False
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54
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55 testStack03 : {m : Level } @$\rightarrow$@ Stack @$\mathbb{N}$@ (SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ ((Maybe @$\mathbb{N}$@) @$\rightarrow$@ (Maybe @$\mathbb{N}$@) @$\rightarrow$@ Bool {m} ) @$\rightarrow$@ Bool {m}
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56 testStack03 s cs = pop2Stack s (
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57 \s d1 d2 @$\rightarrow$@ cs d1 d2 )
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58
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59 testStack04 : Bool
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60 testStack04 = testStack02 (\s @$\rightarrow$@ testStack03 s testStack032)
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61
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62 testStack05 : testStack04 @$\equiv$@ True
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63 testStack05 = refl
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64
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65 testStack06 : {m : Level } @$\rightarrow$@ Maybe (Element @$\mathbb{N}$@)
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66 testStack06 = pushStack createSingleLinkedStack 1 (
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67 \s @$\rightarrow$@ pushStack s 2 (\s @$\rightarrow$@ top (stack s)))
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68
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69 testStack07 : {m : Level } @$\rightarrow$@ Maybe (Element @$\mathbb{N}$@)
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70 testStack07 = pushSingleLinkedStack emptySingleLinkedStack 1 (
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71 \s @$\rightarrow$@ pushSingleLinkedStack s 2 (\s @$\rightarrow$@ top s))
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72
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73 testStack08 = pushSingleLinkedStack emptySingleLinkedStack 1
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74 $ \s @$\rightarrow$@ pushSingleLinkedStack s 2
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75 $ \s @$\rightarrow$@ pushSingleLinkedStack s 3
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76 $ \s @$\rightarrow$@ pushSingleLinkedStack s 4
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77 $ \s @$\rightarrow$@ pushSingleLinkedStack s 5
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78 $ \s @$\rightarrow$@ top s
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79
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80 ------
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81 --
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82 -- proof of properties with indefinite state of stack
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83 --
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84 -- this should be proved by properties of the stack inteface, not only by the implementation,
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85 -- and the implementation have to provides the properties.
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86 --
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87 -- we cannot write "s @$\equiv$@ s3", since level of the Set does not fit , but use stack s @$\equiv$@ stack s3 is ok.
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88 -- anyway some implementations may result s != s3
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89 --
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90
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91 stackInSomeState : {l m : Level } {D : Set l} {t : Set m } (s : SingleLinkedStack D ) @$\rightarrow$@ Stack {l} {m} D {t} ( SingleLinkedStack D )
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92 stackInSomeState s = record { stack = s ; stackMethods = singleLinkedStackSpec }
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93
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94 push@$\rightarrow$@push@$\rightarrow$@pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) @$\rightarrow$@
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95 pushStack ( stackInSomeState s ) x ( \s1 @$\rightarrow$@ pushStack s1 y ( \s2 @$\rightarrow$@ pop2Stack s2 ( \s3 y1 x1 @$\rightarrow$@ (Just x @$\equiv$@ x1 ) @$\wedge$@ (Just y @$\equiv$@ y1 ) ) ))
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96 push@$\rightarrow$@push@$\rightarrow$@pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl }
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97
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98
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99 -- id : {n : Level} {A : Set n} @$\rightarrow$@ A @$\rightarrow$@ A
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100 -- id a = a
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101
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102 -- push a, n times
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103
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104 n-push : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A
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105 n-push zero s = s
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106 n-push {l} {A} {a} (suc n) s = pushSingleLinkedStack (n-push {l} {A} {a} n s) a (\s @$\rightarrow$@ s )
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107
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108 n-pop : {n : Level}{A : Set n} {a : A} @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A
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109 n-pop zero s = s
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110 n-pop {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ @$\rightarrow$@ s )
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111
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112 open @$\equiv$@-Reasoning
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113
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114 push-pop-equiv : {n : Level} {A : Set n} {a : A} (s : SingleLinkedStack A) @$\rightarrow$@ (popSingleLinkedStack (pushSingleLinkedStack s a (\s @$\rightarrow$@ s)) (\s _ @$\rightarrow$@ s) ) @$\equiv$@ s
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115 push-pop-equiv s = refl
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116
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117 push-and-n-pop : {n : Level} {A : Set n} {a : A} (n : @$\mathbb{N}$@) (s : SingleLinkedStack A) @$\rightarrow$@ n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id) @$\equiv$@ n-pop {_} {A} {a} n s
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118 push-and-n-pop zero s = refl
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119 push-and-n-pop {_} {A} {a} (suc n) s = begin
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120 n-pop {_} {A} {a} (suc (suc n)) (pushSingleLinkedStack s a id)
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121 @$\equiv$@@$\langle$@ refl @$\rangle$@
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122 popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ @$\rightarrow$@ s)
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123 @$\equiv$@@$\langle$@ cong (\s @$\rightarrow$@ popSingleLinkedStack s (\s _ @$\rightarrow$@ s )) (push-and-n-pop n s) @$\rangle$@
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124 popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ @$\rightarrow$@ s)
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125 @$\equiv$@@$\langle$@ refl @$\rangle$@
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126 n-pop {_} {A} {a} (suc n) s
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127 @$\blacksquare$@
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128
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129
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130 n-push-pop-equiv : {n : Level} {A : Set n} {a : A} (n : @$\mathbb{N}$@) (s : SingleLinkedStack A) @$\rightarrow$@ (n-pop {_} {A} {a} n (n-push {_} {A} {a} n s)) @$\equiv$@ s
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131 n-push-pop-equiv zero s = refl
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132 n-push-pop-equiv {_} {A} {a} (suc n) s = begin
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133 n-pop {_} {A} {a} (suc n) (n-push (suc n) s)
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134 @$\equiv$@@$\langle$@ refl @$\rangle$@
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135 n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s @$\rightarrow$@ s))
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136 @$\equiv$@@$\langle$@ push-and-n-pop n (n-push n s) @$\rangle$@
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137 n-pop {_} {A} {a} n (n-push n s)
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138 @$\equiv$@@$\langle$@ n-push-pop-equiv n s @$\rangle$@
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139 s
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140 @$\blacksquare$@
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141
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142
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143 n-push-pop-equiv-empty : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ (n : @$\mathbb{N}$@) @$\rightarrow$@ n-pop {_} {A} {a} n (n-push {_} {A} {a} n emptySingleLinkedStack) @$\equiv$@ emptySingleLinkedStack
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144 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
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