|
1
|
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 -> Maybe a
|
|
|
19
|
|
|
20
|
|
|
21 record Stack {a t : Set} (stackImpl : Set) : Set where
|
|
|
22 field
|
|
|
23 stack : stackImpl
|
|
|
24 push : CodeSegment (stackImpl × a × (CodeSegment stackImpl t)) t
|
|
|
25 pop : CodeSegment (stackImpl × (CodeSegment (stackImpl × Maybe a) t)) t
|
|
|
26
|
|
|
27
|
|
|
28 data Element (a : Set) : Set where
|
|
|
29 cons : a -> Maybe (Element a) -> Element a
|
|
|
30
|
|
|
31 datum : {a : Set} -> Element a -> a
|
|
|
32 datum (cons a _) = a
|
|
|
33
|
|
|
34 next : {a : Set} -> Element a -> 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} -> SingleLinkedStack a
|
|
|
43 emptySingleLinkedStack = record {top = Nothing}
|
|
|
44
|
|
|
45
|
|
|
46
|
|
|
47
|
|
|
48 pushSingleLinkedStack : {a t : Set} -> CodeSegment ((SingleLinkedStack a) × a × (CodeSegment (SingleLinkedStack a) t)) t
|
|
|
49 pushSingleLinkedStack = cs push
|
|
|
50 where
|
|
|
51 push : {a t : Set} -> ((SingleLinkedStack a) × a × (CodeSegment (SingleLinkedStack a) t)) -> 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} -> CodeSegment (SingleLinkedStack a × (CodeSegment (SingleLinkedStack a × Maybe a) t)) t
|
|
|
58 popSingleLinkedStack = cs pop
|
|
|
59 where
|
|
|
60 pop : {a t : Set} -> (SingleLinkedStack a × (CodeSegment (SingleLinkedStack a × Maybe a) t)) -> 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} -> 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} -> CodeSegment (SingleLinkedStack a × Maybe a) Bool
|
|
|
81 test01 = cs test01'
|
|
|
82 where
|
|
|
83 test01' : {a : Set} -> (SingleLinkedStack a × Maybe a) -> Bool
|
|
|
84 test01' (record { top = Nothing } , _) = False
|
|
|
85 test01' (record { top = Just x } , _) = True
|
|
|
86
|
|
|
87
|
|
|
88 test02 : {a : Set} -> CodeSegment (SingleLinkedStack a) (SingleLinkedStack a × Maybe a)
|
|
|
89 test02 = cs test02'
|
|
|
90 where
|
|
|
91 test02' : {a : Set} -> SingleLinkedStack a -> (SingleLinkedStack a × Maybe a)
|
|
|
92 test02' stack = goto popSingleLinkedStack (stack , (cs id))
|
|
|
93
|
|
|
94
|
|
|
95 test03 : {a : Set} -> CodeSegment a (SingleLinkedStack a)
|
|
|
96 test03 = cs test03'
|
|
|
97 where
|
|
|
98 test03' : {a : Set} -> a -> SingleLinkedStack a
|
|
|
99 test03' a = goto pushSingleLinkedStack (emptySingleLinkedStack , a , (cs id))
|
|
|
100
|
|
|
101
|
|
|
102 lemma : {A : Set} {a : A} -> goto (test03 ◎ test02 ◎ test01) a ≡ False
|
|
|
103 lemma = refl
|
|
|
104
|
|
|
105
|
|
|
106 n-push : {A : Set} {a : A} -> CodeSegment (ℕ × SingleLinkedStack A) (ℕ × SingleLinkedStack A)
|
|
|
107 n-push {A} {a} = cs (push {A} {a})
|
|
|
108 where
|
|
|
109 push : {A : Set} {a : A} -> (ℕ × SingleLinkedStack A) -> (ℕ × 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} -> Nat -> SingleLinkedStack A -> 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 -> s)
|
|
|
119
|
|
|
120 n-pop : {A : Set} {a : A} -> Nat -> SingleLinkedStack A -> SingleLinkedStack A
|
|
|
121 n-pop zero s = s
|
|
|
122 n-pop {A} {a} (suc n) s = popSingleLinkedStack (n-pop {A} {a} n s) (\s _ -> s)
|
|
|
123
|
|
|
124 open ≡-Reasoning
|
|
|
125
|
|
|
126 push-pop-equiv : {A : Set} {a : A} (s : SingleLinkedStack A) -> popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ≡ s
|
|
|
127 push-pop-equiv s = refl
|
|
|
128
|
|
|
129 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
|
|
|
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 ≡⟨ refl ⟩
|
|
|
134 popSingleLinkedStack (n-pop (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s)
|
|
|
135 ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s)) (push-and-n-pop n s) ⟩
|
|
|
136 popSingleLinkedStack (n-pop n s) (\s _ -> s)
|
|
|
137 ≡⟨ refl ⟩
|
|
|
138 n-pop (suc n) s
|
|
|
139 ∎
|
|
|
140
|
|
|
141
|
|
|
142 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
|
|
|
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 ≡⟨ refl ⟩
|
|
|
147 n-pop (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s))
|
|
|
148 ≡⟨ push-and-n-pop n (n-push n s) ⟩
|
|
|
149 n-pop n (n-push n s)
|
|
|
150 ≡⟨ n-push-pop-equiv n s ⟩
|
|
|
151 s
|
|
|
152 ∎
|
|
|
153
|
|
|
154
|
|
|
155 n-push-pop-equiv-empty : {A : Set} {a : A} -> (n : Nat) -> n-pop {A} {a} n (n-push {A} {a} n emptySingleLinkedStack) ≡ emptySingleLinkedStack
|
|
|
156 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
|
|
|
157 -}
|
|
|
158
|
