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1 module stack-product where
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2
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3 open import product
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4 open import Data.Product
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5 open import Data.Nat
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6 open import Function using (id)
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7 open import Relation.Binary.PropositionalEquality
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8
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9 -- definition based from Gears(209:5708390a9d88) src/parallel_execution
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10 goto = executeCS
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11
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12 data Bool : Set where
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13 True : Bool
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14 False : Bool
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15
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16 data Maybe (a : Set) : Set where
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17 Nothing : Maybe a
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18 Just : a @$\rightarrow$@ Maybe a
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19
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20
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21 record Stack {a t : Set} (stackImpl : Set) : Set where
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22 field
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23 stack : stackImpl
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24 push : CodeSegment (stackImpl @$\times$@ a @$\times$@ (CodeSegment stackImpl t)) t
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25 pop : CodeSegment (stackImpl @$\times$@ (CodeSegment (stackImpl @$\times$@ Maybe a) t)) t
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26
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27
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28 data Element (a : Set) : Set where
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29 cons : a @$\rightarrow$@ Maybe (Element a) @$\rightarrow$@ Element a
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30
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31 datum : {a : Set} @$\rightarrow$@ Element a @$\rightarrow$@ a
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32 datum (cons a _) = a
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33
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34 next : {a : Set} @$\rightarrow$@ Element a @$\rightarrow$@ Maybe (Element a)
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35 next (cons _ n) = n
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36
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37 record SingleLinkedStack (a : Set) : Set where
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38 field
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39 top : Maybe (Element a)
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40 open SingleLinkedStack
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41
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42 emptySingleLinkedStack : {a : Set} @$\rightarrow$@ SingleLinkedStack a
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43 emptySingleLinkedStack = record {top = Nothing}
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44
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45
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46
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47
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48 pushSingleLinkedStack : {a t : Set} @$\rightarrow$@ CodeSegment ((SingleLinkedStack a) @$\times$@ a @$\times$@ (CodeSegment (SingleLinkedStack a) t)) t
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49 pushSingleLinkedStack = cs push
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50 where
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51 push : {a t : Set} @$\rightarrow$@ ((SingleLinkedStack a) @$\times$@ a @$\times$@ (CodeSegment (SingleLinkedStack a) t)) @$\rightarrow$@ t
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52 push (stack , datum , next) = goto next stack1
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53 where
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54 element = cons datum (top stack)
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55 stack1 = record {top = Just element}
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56
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57 popSingleLinkedStack : {a t : Set} @$\rightarrow$@ CodeSegment (SingleLinkedStack a @$\times$@ (CodeSegment (SingleLinkedStack a @$\times$@ Maybe a) t)) t
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58 popSingleLinkedStack = cs pop
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59 where
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60 pop : {a t : Set} @$\rightarrow$@ (SingleLinkedStack a @$\times$@ (CodeSegment (SingleLinkedStack a @$\times$@ Maybe a) t)) @$\rightarrow$@ t
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61 pop (record { top = Nothing } , nextCS) = goto nextCS (emptySingleLinkedStack , Nothing)
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62 pop (record { top = Just x } , nextCS) = goto nextCS (stack1 , (Just datum1))
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63 where
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64 datum1 = datum x
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65 stack1 = record { top = (next x) }
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66
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67
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68
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69
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70
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71 createSingleLinkedStack : {a b : Set} @$\rightarrow$@ Stack {a} {b} (SingleLinkedStack a)
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72 createSingleLinkedStack = record { stack = emptySingleLinkedStack
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73 ; push = pushSingleLinkedStack
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74 ; pop = popSingleLinkedStack
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75 }
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76
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77
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78
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79
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80 test01 : {a : Set} @$\rightarrow$@ CodeSegment (SingleLinkedStack a @$\times$@ Maybe a) Bool
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81 test01 = cs test01'
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82 where
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83 test01' : {a : Set} @$\rightarrow$@ (SingleLinkedStack a @$\times$@ Maybe a) @$\rightarrow$@ Bool
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84 test01' (record { top = Nothing } , _) = False
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85 test01' (record { top = Just x } , _) = True
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86
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87
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88 test02 : {a : Set} @$\rightarrow$@ CodeSegment (SingleLinkedStack a) (SingleLinkedStack a @$\times$@ Maybe a)
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89 test02 = cs test02'
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90 where
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91 test02' : {a : Set} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ (SingleLinkedStack a @$\times$@ Maybe a)
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92 test02' stack = goto popSingleLinkedStack (stack , (cs id))
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93
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94
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95 test03 : {a : Set} @$\rightarrow$@ CodeSegment a (SingleLinkedStack a)
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96 test03 = cs test03'
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97 where
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98 test03' : {a : Set} @$\rightarrow$@ a @$\rightarrow$@ SingleLinkedStack a
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99 test03' a = goto pushSingleLinkedStack (emptySingleLinkedStack , a , (cs id))
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100
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101
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102 lemma : {A : Set} {a : A} @$\rightarrow$@ goto (test03 ◎ test02 ◎ test01) a @$\equiv$@ False
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103 lemma = refl
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104
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105
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106 n-push : {A : Set} {a : A} @$\rightarrow$@ CodeSegment (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A) (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A)
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107 n-push {A} {a} = cs (push {A} {a})
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108 where
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109 push : {A : Set} {a : A} @$\rightarrow$@ (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A) @$\rightarrow$@ (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A)
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110 push {A} {a} (zero , s) = (zero , s)
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111 push {A} {a} (suc n , s) = goto pushSingleLinkedStack (s , a , {!!} {- n-push -}) -- needs subtype
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112
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113
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114 {-
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115
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116 n-push : {A : Set} {a : A} @$\rightarrow$@ Nat @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ 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 @$\rightarrow$@ s)
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119
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120 n-pop : {A : Set} {a : A} @$\rightarrow$@ Nat @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ 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 _ @$\rightarrow$@ s)
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123
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124 open @$\equiv$@-Reasoning
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125
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126 push-pop-equiv : {A : Set} {a : A} (s : SingleLinkedStack A) @$\rightarrow$@ popSingleLinkedStack (pushSingleLinkedStack s a (\s @$\rightarrow$@ s)) (\s _ @$\rightarrow$@ s) @$\equiv$@ 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) @$\rightarrow$@ n-pop {A} {a} (suc n) (pushSingleLinkedStack s a id) @$\equiv$@ 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 @$\equiv$@@$\langle$@ refl @$\rangle$@
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134 popSingleLinkedStack (n-pop (suc n) (pushSingleLinkedStack s a id)) (\s _ @$\rightarrow$@ s)
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135 @$\equiv$@@$\langle$@ cong (\s @$\rightarrow$@ popSingleLinkedStack s (\s _ @$\rightarrow$@ s)) (push-and-n-pop n s) @$\rangle$@
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136 popSingleLinkedStack (n-pop n s) (\s _ @$\rightarrow$@ s)
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137 @$\equiv$@@$\langle$@ refl @$\rangle$@
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138 n-pop (suc n) s
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139 @$\blacksquare$@
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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) @$\rightarrow$@ (n-pop {A} {a} n (n-push {A} {a} n s)) @$\equiv$@ 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 @$\equiv$@@$\langle$@ refl @$\rangle$@
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147 n-pop (suc n) (pushSingleLinkedStack (n-push n s) a (\s @$\rightarrow$@ s))
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148 @$\equiv$@@$\langle$@ push-and-n-pop n (n-push n s) @$\rangle$@
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149 n-pop n (n-push n s)
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150 @$\equiv$@@$\langle$@ n-push-pop-equiv n s @$\rangle$@
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151 s
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152 @$\blacksquare$@
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153
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154
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155 n-push-pop-equiv-empty : {A : Set} {a : A} @$\rightarrow$@ (n : Nat) @$\rightarrow$@ n-pop {A} {a} n (n-push {A} {a} n emptySingleLinkedStack) @$\equiv$@ emptySingleLinkedStack
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156 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
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157 -}
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158
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