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comparison src/stack-product.agda.replaced @ 1:73127e0ab57c

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author soto@cr.ie.u-ryukyu.ac.jp
date Tue, 08 Sep 2020 18:38:08 +0900
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0:b919985837a3 1:73127e0ab57c
1 module stack-product where
2
3 open import product
4 open import Data.Product
5 open import Data.Nat
6 open import Function using (id)
7 open import Relation.Binary.PropositionalEquality
8
9 -- definition based from Gears(209:5708390a9d88) src/parallel_execution
10 goto = executeCS
11
12 data Bool : Set where
13 True : Bool
14 False : Bool
15
16 data Maybe (a : Set) : Set where
17 Nothing : Maybe a
18 Just : a @$\rightarrow$@ Maybe a
19
20
21 record Stack {a t : Set} (stackImpl : Set) : Set where
22 field
23 stack : stackImpl
24 push : CodeSegment (stackImpl @$\times$@ a @$\times$@ (CodeSegment stackImpl t)) t
25 pop : CodeSegment (stackImpl @$\times$@ (CodeSegment (stackImpl @$\times$@ Maybe a) t)) t
26
27
28 data Element (a : Set) : Set where
29 cons : a @$\rightarrow$@ Maybe (Element a) @$\rightarrow$@ Element a
30
31 datum : {a : Set} @$\rightarrow$@ Element a @$\rightarrow$@ a
32 datum (cons a _) = a
33
34 next : {a : Set} @$\rightarrow$@ Element a @$\rightarrow$@ Maybe (Element a)
35 next (cons _ n) = n
36
37 record SingleLinkedStack (a : Set) : Set where
38 field
39 top : Maybe (Element a)
40 open SingleLinkedStack
41
42 emptySingleLinkedStack : {a : Set} @$\rightarrow$@ SingleLinkedStack a
43 emptySingleLinkedStack = record {top = Nothing}
44
45
46
47
48 pushSingleLinkedStack : {a t : Set} @$\rightarrow$@ CodeSegment ((SingleLinkedStack a) @$\times$@ a @$\times$@ (CodeSegment (SingleLinkedStack a) t)) t
49 pushSingleLinkedStack = cs push
50 where
51 push : {a t : Set} @$\rightarrow$@ ((SingleLinkedStack a) @$\times$@ a @$\times$@ (CodeSegment (SingleLinkedStack a) t)) @$\rightarrow$@ t
52 push (stack , datum , next) = goto next stack1
53 where
54 element = cons datum (top stack)
55 stack1 = record {top = Just element}
56
57 popSingleLinkedStack : {a t : Set} @$\rightarrow$@ CodeSegment (SingleLinkedStack a @$\times$@ (CodeSegment (SingleLinkedStack a @$\times$@ Maybe a) t)) t
58 popSingleLinkedStack = cs pop
59 where
60 pop : {a t : Set} @$\rightarrow$@ (SingleLinkedStack a @$\times$@ (CodeSegment (SingleLinkedStack a @$\times$@ Maybe a) t)) @$\rightarrow$@ t
61 pop (record { top = Nothing } , nextCS) = goto nextCS (emptySingleLinkedStack , Nothing)
62 pop (record { top = Just x } , nextCS) = goto nextCS (stack1 , (Just datum1))
63 where
64 datum1 = datum x
65 stack1 = record { top = (next x) }
66
67
68
69
70
71 createSingleLinkedStack : {a b : Set} @$\rightarrow$@ Stack {a} {b} (SingleLinkedStack a)
72 createSingleLinkedStack = record { stack = emptySingleLinkedStack
73 ; push = pushSingleLinkedStack
74 ; pop = popSingleLinkedStack
75 }
76
77
78
79
80 test01 : {a : Set} @$\rightarrow$@ CodeSegment (SingleLinkedStack a @$\times$@ Maybe a) Bool
81 test01 = cs test01'
82 where
83 test01' : {a : Set} @$\rightarrow$@ (SingleLinkedStack a @$\times$@ Maybe a) @$\rightarrow$@ Bool
84 test01' (record { top = Nothing } , _) = False
85 test01' (record { top = Just x } , _) = True
86
87
88 test02 : {a : Set} @$\rightarrow$@ CodeSegment (SingleLinkedStack a) (SingleLinkedStack a @$\times$@ Maybe a)
89 test02 = cs test02'
90 where
91 test02' : {a : Set} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ (SingleLinkedStack a @$\times$@ Maybe a)
92 test02' stack = goto popSingleLinkedStack (stack , (cs id))
93
94
95 test03 : {a : Set} @$\rightarrow$@ CodeSegment a (SingleLinkedStack a)
96 test03 = cs test03'
97 where
98 test03' : {a : Set} @$\rightarrow$@ a @$\rightarrow$@ SingleLinkedStack a
99 test03' a = goto pushSingleLinkedStack (emptySingleLinkedStack , a , (cs id))
100
101
102 lemma : {A : Set} {a : A} @$\rightarrow$@ goto (test03 ◎ test02 ◎ test01) a @$\equiv$@ False
103 lemma = refl
104
105
106 n-push : {A : Set} {a : A} @$\rightarrow$@ CodeSegment (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A) (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A)
107 n-push {A} {a} = cs (push {A} {a})
108 where
109 push : {A : Set} {a : A} @$\rightarrow$@ (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A) @$\rightarrow$@ (@$\mathbb{N}$@ @$\times$@ SingleLinkedStack A)
110 push {A} {a} (zero , s) = (zero , s)
111 push {A} {a} (suc n , s) = goto pushSingleLinkedStack (s , a , {!!} {- n-push -}) -- needs subtype
112
113
114 {-
115
116 n-push : {A : Set} {a : A} @$\rightarrow$@ Nat @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A
117 n-push zero s = s
118 n-push {A} {a} (suc n) s = pushSingleLinkedStack (n-push {A} {a} n s) a (\s @$\rightarrow$@ s)
119
120 n-pop : {A : Set} {a : A} @$\rightarrow$@ Nat @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A
121 n-pop zero s = s
122 n-pop {A} {a} (suc n) s = popSingleLinkedStack (n-pop {A} {a} n s) (\s _ @$\rightarrow$@ s)
123
124 open @$\equiv$@-Reasoning
125
126 push-pop-equiv : {A : Set} {a : A} (s : SingleLinkedStack A) @$\rightarrow$@ popSingleLinkedStack (pushSingleLinkedStack s a (\s @$\rightarrow$@ s)) (\s _ @$\rightarrow$@ s) @$\equiv$@ s
127 push-pop-equiv s = refl
128
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
130 push-and-n-pop zero s = refl
131 push-and-n-pop {A} {a} (suc n) s = begin
132 n-pop (suc (suc n)) (pushSingleLinkedStack s a id)
133 @$\equiv$@@$\langle$@ refl @$\rangle$@
134 popSingleLinkedStack (n-pop (suc n) (pushSingleLinkedStack s a id)) (\s _ @$\rightarrow$@ s)
135 @$\equiv$@@$\langle$@ cong (\s @$\rightarrow$@ popSingleLinkedStack s (\s _ @$\rightarrow$@ s)) (push-and-n-pop n s) @$\rangle$@
136 popSingleLinkedStack (n-pop n s) (\s _ @$\rightarrow$@ s)
137 @$\equiv$@@$\langle$@ refl @$\rangle$@
138 n-pop (suc n) s
139 @$\blacksquare$@
140
141
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
143 n-push-pop-equiv zero s = refl
144 n-push-pop-equiv {A} {a} (suc n) s = begin
145 n-pop (suc n) (n-push (suc n) s)
146 @$\equiv$@@$\langle$@ refl @$\rangle$@
147 n-pop (suc n) (pushSingleLinkedStack (n-push n s) a (\s @$\rightarrow$@ s))
148 @$\equiv$@@$\langle$@ push-and-n-pop n (n-push n s) @$\rangle$@
149 n-pop n (n-push n s)
150 @$\equiv$@@$\langle$@ n-push-pop-equiv n s @$\rangle$@
151 s
152 @$\blacksquare$@
153
154
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
156 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
157 -}
158