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ReWrite stack.agda using product type definition
author atton <atton@cr.ie.u-ryukyu.ac.jp>
date Fri, 23 Dec 2016 10:20:05 +0000
parents
children 67978ba63a6f
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27
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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1 module stack-product where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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2
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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3 open import product
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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4 open import Data.Product
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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5 open import Relation.Binary.PropositionalEquality
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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6
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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7 -- definition based from Gears(209:5708390a9d88) src/parallel_execution
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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8 goto = executeCS
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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9
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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10 data Bool : Set where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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11 True : Bool
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12 False : Bool
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13
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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14 data Maybe (a : Set) : Set where
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15 Nothing : Maybe a
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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16 Just : a -> Maybe a
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17
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18
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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19 record Stack {a t : Set} (stackImpl : Set) : Set where
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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20 field
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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21 stack : stackImpl
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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22 push : CodeSegment (stackImpl × a × (CodeSegment stackImpl t)) t
892f8b3aa57e ReWrite stack.agda using product type definition
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23 pop : CodeSegment (stackImpl × (CodeSegment (stackImpl × Maybe a) t)) t
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24
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25
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26 data Element (a : Set) : Set where
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27 cons : a -> Maybe (Element a) -> Element a
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28
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29 datum : {a : Set} -> Element a -> a
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30 datum (cons a _) = a
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31
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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32 next : {a : Set} -> Element a -> Maybe (Element a)
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33 next (cons _ n) = n
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34
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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35 record SingleLinkedStack (a : Set) : Set where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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36 field
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37 top : Maybe (Element a)
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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38 open SingleLinkedStack
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39
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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40 emptySingleLinkedStack : {a : Set} -> SingleLinkedStack a
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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41 emptySingleLinkedStack = record {top = Nothing}
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42
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43
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44
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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45
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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46 pushSingleLinkedStack : {a t : Set} -> CodeSegment ((SingleLinkedStack a) × a × (CodeSegment (SingleLinkedStack a) t)) t
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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47 pushSingleLinkedStack = cs push
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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48 where
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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49 push : {a t : Set} -> ((SingleLinkedStack a) × a × (CodeSegment (SingleLinkedStack a) t)) -> t
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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50 push (stack , datum , next) = goto next stack1
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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51 where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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52 element = cons datum (top stack)
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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53 stack1 = record {top = Just element}
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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54
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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55 popSingleLinkedStack : {a t : Set} -> CodeSegment (SingleLinkedStack a × (CodeSegment (SingleLinkedStack a × Maybe a) t)) t
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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56 popSingleLinkedStack = cs pop
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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57 where
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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58 pop : {a t : Set} -> (SingleLinkedStack a × (CodeSegment (SingleLinkedStack a × Maybe a) t)) -> t
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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59 pop (record { top = Nothing } , nextCS) = goto nextCS (emptySingleLinkedStack , Nothing)
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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60 pop (record { top = Just x } , nextCS) = goto nextCS (stack1 , (Just datum1))
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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61 where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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62 datum1 = datum x
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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63 stack1 = record { top = (next x) }
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64
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65
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66
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67
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68
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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69 createSingleLinkedStack : {a b : Set} -> Stack {a} {b} (SingleLinkedStack a)
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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70 createSingleLinkedStack = record { stack = emptySingleLinkedStack
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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71 ; push = pushSingleLinkedStack
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72 ; pop = popSingleLinkedStack
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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73 }
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74
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75
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76
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77
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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78 test01 : {a : Set} -> CodeSegment (SingleLinkedStack a × Maybe a) Bool
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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79 test01 = cs test01'
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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80 where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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81 test01' : {a : Set} -> (SingleLinkedStack a × Maybe a) -> Bool
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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82 test01' (record { top = Nothing } , _) = False
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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83 test01' (record { top = Just x } , _) = True
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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84
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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85
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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86 test02 : {a : Set} -> CodeSegment (SingleLinkedStack a) Bool
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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87 test02 = cs test02'
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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88 where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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89 test02' : {a : Set} -> SingleLinkedStack a -> Bool
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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90 test02' stack = goto popSingleLinkedStack (stack , test01)
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91
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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92
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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93 test03 : {a : Set} -> CodeSegment a Bool
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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94 test03 = cs test03'
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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95 where
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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96 test03' : {a : Set} -> a -> Bool
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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97 test03' a = goto pushSingleLinkedStack (emptySingleLinkedStack , a , test02)
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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98
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99
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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100 lemma : {A : Set} {a : A} -> goto test03 a ≡ False
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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101 lemma = refl
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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102
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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103 id : {A : Set} -> A -> A
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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104 id a = a
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
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105
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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106
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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107 {-
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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108
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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109 n-push : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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110 n-push zero s = s
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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111 n-push {A} {a} (suc n) s = pushSingleLinkedStack (n-push {A} {a} n s) a (\s -> s)
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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112
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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113 n-pop : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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114 n-pop zero s = s
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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115 n-pop {A} {a} (suc n) s = popSingleLinkedStack (n-pop {A} {a} n s) (\s _ -> s)
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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116
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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117 open ≡-Reasoning
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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118
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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119 push-pop-equiv : {A : Set} {a : A} (s : SingleLinkedStack A) -> popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ≡ s
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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120 push-pop-equiv s = refl
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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121
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atton <atton@cr.ie.u-ryukyu.ac.jp>
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122 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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atton <atton@cr.ie.u-ryukyu.ac.jp>
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123 push-and-n-pop zero s = refl
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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124 push-and-n-pop {A} {a} (suc n) s = begin
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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125 n-pop (suc (suc n)) (pushSingleLinkedStack s a id)
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
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126 ≡⟨ refl ⟩
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atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
127 popSingleLinkedStack (n-pop (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s)
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
128 ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s)) (push-and-n-pop n s) ⟩
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
129 popSingleLinkedStack (n-pop n s) (\s _ -> s)
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
130 ≡⟨ refl ⟩
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
131 n-pop (suc n) s
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
132 ∎
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
133
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
134
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
135 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
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
136 n-push-pop-equiv zero s = refl
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
137 n-push-pop-equiv {A} {a} (suc n) s = begin
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
138 n-pop (suc n) (n-push (suc n) s)
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
139 ≡⟨ refl ⟩
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
140 n-pop (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s))
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
141 ≡⟨ push-and-n-pop n (n-push n s) ⟩
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
142 n-pop n (n-push n s)
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
143 ≡⟨ n-push-pop-equiv n s ⟩
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
144 s
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
145 ∎
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
146
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
147
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
148 n-push-pop-equiv-empty : {A : Set} {a : A} -> (n : Nat) -> n-pop {A} {a} n (n-push {A} {a} n emptySingleLinkedStack) ≡ emptySingleLinkedStack
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
149 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
150 -}
892f8b3aa57e ReWrite stack.agda using product type definition
atton <atton@cr.ie.u-ryukyu.ac.jp>
parents:
diff changeset
151