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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
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10
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11 open SingleLinkedStack
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12 open Stack
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13
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14 ----
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15 --
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16 -- proof of properties ( concrete cases )
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17 --
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18
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19 test01 : {n : Level } {a : Set n} -> SingleLinkedStack a -> Maybe a -> Bool {n}
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20 test01 stack _ with (top stack)
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21 ... | (Just _) = True
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22 ... | Nothing = False
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23
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24
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25 test02 : {n : Level } {a : Set n} -> SingleLinkedStack a -> Bool
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26 test02 stack = popSingleLinkedStack stack test01
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27
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28 test03 : {n : Level } {a : Set n} -> a -> Bool
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29 test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02
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30
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31 -- after a push and a pop, the stack is empty
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32 lemma : {n : Level} {A : Set n} {a : A} -> test03 a ≡ False
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33 lemma = refl
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34
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35 testStack01 : {n m : Level } {a : Set n} -> a -> Bool {m}
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36 testStack01 v = pushStack createSingleLinkedStack v (
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37 \s -> popStack s (\s1 d1 -> True))
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38
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39 -- after push 1 and 2, pop2 get 1 and 2
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40
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41 testStack02 : {m : Level } -> ( Stack ℕ (SingleLinkedStack ℕ) -> Bool {m} ) -> Bool {m}
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42 testStack02 cs = pushStack createSingleLinkedStack 1 (
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43 \s -> pushStack s 2 cs)
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44
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45
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46 testStack031 : (d1 d2 : ℕ ) -> Bool {Zero}
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47 testStack031 2 1 = True
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48 testStack031 _ _ = False
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49
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50 testStack032 : (d1 d2 : Maybe ℕ) -> Bool {Zero}
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51 testStack032 (Just d1) (Just d2) = testStack031 d1 d2
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52 testStack032 _ _ = False
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53
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54 testStack03 : {m : Level } -> Stack ℕ (SingleLinkedStack ℕ) -> ((Maybe ℕ) -> (Maybe ℕ) -> Bool {m} ) -> Bool {m}
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55 testStack03 s cs = pop2Stack s (
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56 \s d1 d2 -> cs d1 d2 )
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57
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58 testStack04 : Bool
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59 testStack04 = testStack02 (\s -> testStack03 s testStack032)
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60
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61 testStack05 : testStack04 ≡ True
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62 testStack05 = refl
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63
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64 ------
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65 --
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66 -- proof of properties with indefinite state of stack
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67 --
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68 -- this should be proved by properties of the stack inteface, not only by the implementation,
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69 -- and the implementation have to provides the properties.
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70 --
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71 -- we cannot write "s ≡ s3", since level of the Set does not fit , but use stack s ≡ stack s3 is ok.
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72 -- anyway some implementations may result s != s3
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73 --
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74
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75 stackInSomeState : {l m : Level } {D : Set l} {t : Set m } (s : SingleLinkedStack D ) -> Stack {l} {m} D {t} ( SingleLinkedStack D )
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76 stackInSomeState s = record { stack = s ; stackMethods = singleLinkedStackSpec }
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77
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78 push->push->pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) ->
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79 pushStack ( stackInSomeState s ) x ( \s1 -> pushStack s1 y ( \s2 -> pop2Stack s2 ( \s3 y1 x1 -> (Just x ≡ x1 ) ∧ (Just y ≡ y1 ) ) ))
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80 push->push->pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl }
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81
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82
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83 id : {n : Level} {A : Set n} -> A -> A
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84 id a = a
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85
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86 -- push a, n times
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87
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88 n-push : {n : Level} {A : Set n} {a : A} -> ℕ -> SingleLinkedStack A -> SingleLinkedStack A
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89 n-push zero s = s
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90 n-push {l} {A} {a} (suc n) s = pushSingleLinkedStack (n-push {l} {A} {a} n s) a (\s -> s )
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91
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92 n-pop : {n : Level}{A : Set n} {a : A} -> ℕ -> SingleLinkedStack A -> SingleLinkedStack A
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93 n-pop zero s = s
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94 n-pop {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ -> s )
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95
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96 open ≡-Reasoning
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97
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98 push-pop-equiv : {n : Level} {A : Set n} {a : A} (s : SingleLinkedStack A) -> (popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ) ≡ s
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99 push-pop-equiv s = refl
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100
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101 push-and-n-pop : {n : Level} {A : Set n} {a : A} (n : ℕ) (s : SingleLinkedStack A) -> n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id) ≡ n-pop {_} {A} {a} n s
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102 push-and-n-pop zero s = refl
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103 push-and-n-pop {_} {A} {a} (suc n) s = begin
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104 n-pop {_} {A} {a} (suc (suc n)) (pushSingleLinkedStack s a id)
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105 ≡⟨ refl ⟩
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106 popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s)
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107 ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s )) (push-and-n-pop n s) ⟩
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108 popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ -> s)
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109 ≡⟨ refl ⟩
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110 n-pop {_} {A} {a} (suc n) s
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111 ∎
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112
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113
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114 n-push-pop-equiv : {n : Level} {A : Set n} {a : A} (n : ℕ) (s : SingleLinkedStack A) -> (n-pop {_} {A} {a} n (n-push {_} {A} {a} n s)) ≡ s
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115 n-push-pop-equiv zero s = refl
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116 n-push-pop-equiv {_} {A} {a} (suc n) s = begin
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117 n-pop {_} {A} {a} (suc n) (n-push (suc n) s)
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118 ≡⟨ refl ⟩
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119 n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s))
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120 ≡⟨ push-and-n-pop n (n-push n s) ⟩
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121 n-pop {_} {A} {a} n (n-push n s)
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122 ≡⟨ n-push-pop-equiv n s ⟩
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123 s
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124 ∎
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125
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126
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127 n-push-pop-equiv-empty : {n : Level} {A : Set n} {a : A} -> (n : ℕ) -> n-pop {_} {A} {a} n (n-push {_} {A} {a} n emptySingleLinkedStack) ≡ emptySingleLinkedStack
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128 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack
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