Mercurial > hg > Gears > GearsAgda
annotate stackTest.agda @ 661:323533798054
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| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sun, 21 Nov 2021 19:31:44 +0900 |
| parents | 429ece770187 |
| children | 0b791ae19543 |
| rev | line source |
|---|---|
| 537 | 1 open import Level renaming (suc to succ ; zero to Zero ) |
| 2 module stackTest where | |
| 3 | |
| 4 open import stack | |
| 5 | |
| 6 open import Relation.Binary.PropositionalEquality | |
| 7 open import Relation.Binary.Core | |
| 8 open import Data.Nat | |
| 541 | 9 open import Function |
| 537 | 10 |
| 11 | |
| 12 open SingleLinkedStack | |
| 13 open Stack | |
| 14 | |
| 15 ---- | |
| 16 -- | |
| 17 -- proof of properties ( concrete cases ) | |
| 18 -- | |
| 19 | |
| 20 test01 : {n : Level } {a : Set n} -> SingleLinkedStack a -> Maybe a -> Bool {n} | |
| 21 test01 stack _ with (top stack) | |
| 22 ... | (Just _) = True | |
| 23 ... | Nothing = False | |
| 24 | |
| 25 | |
| 26 test02 : {n : Level } {a : Set n} -> SingleLinkedStack a -> Bool | |
| 27 test02 stack = popSingleLinkedStack stack test01 | |
| 28 | |
| 29 test03 : {n : Level } {a : Set n} -> a -> Bool | |
| 30 test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02 | |
| 31 | |
| 32 -- after a push and a pop, the stack is empty | |
| 33 lemma : {n : Level} {A : Set n} {a : A} -> test03 a ≡ False | |
| 34 lemma = refl | |
| 35 | |
| 36 testStack01 : {n m : Level } {a : Set n} -> a -> Bool {m} | |
| 37 testStack01 v = pushStack createSingleLinkedStack v ( | |
| 38 \s -> popStack s (\s1 d1 -> True)) | |
| 39 | |
| 40 -- after push 1 and 2, pop2 get 1 and 2 | |
| 41 | |
| 42 testStack02 : {m : Level } -> ( Stack ℕ (SingleLinkedStack ℕ) -> Bool {m} ) -> Bool {m} | |
| 43 testStack02 cs = pushStack createSingleLinkedStack 1 ( | |
| 44 \s -> pushStack s 2 cs) | |
| 45 | |
| 46 | |
| 47 testStack031 : (d1 d2 : ℕ ) -> Bool {Zero} | |
| 48 testStack031 2 1 = True | |
| 49 testStack031 _ _ = False | |
| 50 | |
| 51 testStack032 : (d1 d2 : Maybe ℕ) -> Bool {Zero} | |
| 52 testStack032 (Just d1) (Just d2) = testStack031 d1 d2 | |
| 53 testStack032 _ _ = False | |
| 54 | |
| 55 testStack03 : {m : Level } -> Stack ℕ (SingleLinkedStack ℕ) -> ((Maybe ℕ) -> (Maybe ℕ) -> Bool {m} ) -> Bool {m} | |
| 56 testStack03 s cs = pop2Stack s ( | |
| 57 \s d1 d2 -> cs d1 d2 ) | |
| 58 | |
| 59 testStack04 : Bool | |
| 60 testStack04 = testStack02 (\s -> testStack03 s testStack032) | |
| 61 | |
| 62 testStack05 : testStack04 ≡ True | |
| 63 testStack05 = refl | |
| 64 | |
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65 testStack06 : {m : Level } -> Maybe (Element ℕ) |
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66 testStack06 = pushStack createSingleLinkedStack 1 ( |
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67 \s -> pushStack s 2 (\s -> top (stack s))) |
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68 |
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69 testStack07 : {m : Level } -> Maybe (Element ℕ) |
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70 testStack07 = pushSingleLinkedStack emptySingleLinkedStack 1 ( |
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71 \s -> pushSingleLinkedStack s 2 (\s -> top s)) |
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72 |
| 541 | 73 testStack08 = pushSingleLinkedStack emptySingleLinkedStack 1 |
| 74 $ \s -> pushSingleLinkedStack s 2 | |
| 75 $ \s -> pushSingleLinkedStack s 3 | |
| 76 $ \s -> pushSingleLinkedStack s 4 | |
| 77 $ \s -> pushSingleLinkedStack s 5 | |
| 78 $ \s -> top s | |
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79 |
| 537 | 80 ------ |
| 81 -- | |
| 82 -- proof of properties with indefinite state of stack | |
| 83 -- | |
| 84 -- this should be proved by properties of the stack inteface, not only by the implementation, | |
| 85 -- and the implementation have to provides the properties. | |
| 86 -- | |
| 87 -- we cannot write "s ≡ s3", since level of the Set does not fit , but use stack s ≡ stack s3 is ok. | |
| 88 -- anyway some implementations may result s != s3 | |
| 89 -- | |
| 90 | |
| 91 stackInSomeState : {l m : Level } {D : Set l} {t : Set m } (s : SingleLinkedStack D ) -> Stack {l} {m} D {t} ( SingleLinkedStack D ) | |
| 92 stackInSomeState s = record { stack = s ; stackMethods = singleLinkedStackSpec } | |
| 93 | |
| 94 push->push->pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) -> | |
| 95 pushStack ( stackInSomeState s ) x ( \s1 -> pushStack s1 y ( \s2 -> pop2Stack s2 ( \s3 y1 x1 -> (Just x ≡ x1 ) ∧ (Just y ≡ y1 ) ) )) | |
| 96 push->push->pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl } | |
| 97 | |
| 98 | |
| 541 | 99 -- id : {n : Level} {A : Set n} -> A -> A |
| 100 -- id a = a | |
| 537 | 101 |
| 102 -- push a, n times | |
| 103 | |
| 104 n-push : {n : Level} {A : Set n} {a : A} -> ℕ -> SingleLinkedStack A -> SingleLinkedStack A | |
| 105 n-push zero s = s | |
| 106 n-push {l} {A} {a} (suc n) s = pushSingleLinkedStack (n-push {l} {A} {a} n s) a (\s -> s ) | |
| 107 | |
| 108 n-pop : {n : Level}{A : Set n} {a : A} -> ℕ -> SingleLinkedStack A -> SingleLinkedStack A | |
| 109 n-pop zero s = s | |
| 110 n-pop {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ -> s ) | |
| 111 | |
| 112 open ≡-Reasoning | |
| 113 | |
| 114 push-pop-equiv : {n : Level} {A : Set n} {a : A} (s : SingleLinkedStack A) -> (popSingleLinkedStack (pushSingleLinkedStack s a (\s -> s)) (\s _ -> s) ) ≡ s | |
| 115 push-pop-equiv s = refl | |
| 116 | |
| 117 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 | |
| 118 push-and-n-pop zero s = refl | |
| 119 push-and-n-pop {_} {A} {a} (suc n) s = begin | |
| 120 n-pop {_} {A} {a} (suc (suc n)) (pushSingleLinkedStack s a id) | |
| 121 ≡⟨ refl ⟩ | |
| 122 popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ -> s) | |
| 123 ≡⟨ cong (\s -> popSingleLinkedStack s (\s _ -> s )) (push-and-n-pop n s) ⟩ | |
| 124 popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ -> s) | |
| 125 ≡⟨ refl ⟩ | |
| 126 n-pop {_} {A} {a} (suc n) s | |
| 127 ∎ | |
| 128 | |
| 129 | |
| 130 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 | |
| 131 n-push-pop-equiv zero s = refl | |
| 132 n-push-pop-equiv {_} {A} {a} (suc n) s = begin | |
| 133 n-pop {_} {A} {a} (suc n) (n-push (suc n) s) | |
| 134 ≡⟨ refl ⟩ | |
| 135 n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s -> s)) | |
| 136 ≡⟨ push-and-n-pop n (n-push n s) ⟩ | |
| 137 n-pop {_} {A} {a} n (n-push n s) | |
| 138 ≡⟨ n-push-pop-equiv n s ⟩ | |
| 139 s | |
| 140 ∎ | |
| 141 | |
| 142 | |
| 143 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 | |
| 144 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack |