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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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