Mercurial > hg > Papers > 2020 > soto-midterm
comparison src/stackTest.agda.replaced @ 1:73127e0ab57c
(none)
| 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 |
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| 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 | |
| 9 open import Function | |
| 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} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ Maybe a @$\rightarrow$@ Bool {n} | |
| 21 test01 stack _ with (top stack) | |
| 22 ... | (Just _) = True | |
| 23 ... | Nothing = False | |
| 24 | |
| 25 | |
| 26 test02 : {n : Level } {a : Set n} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ Bool | |
| 27 test02 stack = popSingleLinkedStack stack test01 | |
| 28 | |
| 29 test03 : {n : Level } {a : Set n} @$\rightarrow$@ a @$\rightarrow$@ 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} @$\rightarrow$@ test03 a @$\equiv$@ False | |
| 34 lemma = refl | |
| 35 | |
| 36 testStack01 : {n m : Level } {a : Set n} @$\rightarrow$@ a @$\rightarrow$@ Bool {m} | |
| 37 testStack01 v = pushStack createSingleLinkedStack v ( | |
| 38 \s @$\rightarrow$@ popStack s (\s1 d1 @$\rightarrow$@ True)) | |
| 39 | |
| 40 -- after push 1 and 2, pop2 get 1 and 2 | |
| 41 | |
| 42 testStack02 : {m : Level } @$\rightarrow$@ ( Stack @$\mathbb{N}$@ (SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ Bool {m} ) @$\rightarrow$@ Bool {m} | |
| 43 testStack02 cs = pushStack createSingleLinkedStack 1 ( | |
| 44 \s @$\rightarrow$@ pushStack s 2 cs) | |
| 45 | |
| 46 | |
| 47 testStack031 : (d1 d2 : @$\mathbb{N}$@ ) @$\rightarrow$@ Bool {Zero} | |
| 48 testStack031 2 1 = True | |
| 49 testStack031 _ _ = False | |
| 50 | |
| 51 testStack032 : (d1 d2 : Maybe @$\mathbb{N}$@) @$\rightarrow$@ Bool {Zero} | |
| 52 testStack032 (Just d1) (Just d2) = testStack031 d1 d2 | |
| 53 testStack032 _ _ = False | |
| 54 | |
| 55 testStack03 : {m : Level } @$\rightarrow$@ Stack @$\mathbb{N}$@ (SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ ((Maybe @$\mathbb{N}$@) @$\rightarrow$@ (Maybe @$\mathbb{N}$@) @$\rightarrow$@ Bool {m} ) @$\rightarrow$@ Bool {m} | |
| 56 testStack03 s cs = pop2Stack s ( | |
| 57 \s d1 d2 @$\rightarrow$@ cs d1 d2 ) | |
| 58 | |
| 59 testStack04 : Bool | |
| 60 testStack04 = testStack02 (\s @$\rightarrow$@ testStack03 s testStack032) | |
| 61 | |
| 62 testStack05 : testStack04 @$\equiv$@ True | |
| 63 testStack05 = refl | |
| 64 | |
| 65 testStack06 : {m : Level } @$\rightarrow$@ Maybe (Element @$\mathbb{N}$@) | |
| 66 testStack06 = pushStack createSingleLinkedStack 1 ( | |
| 67 \s @$\rightarrow$@ pushStack s 2 (\s @$\rightarrow$@ top (stack s))) | |
| 68 | |
| 69 testStack07 : {m : Level } @$\rightarrow$@ Maybe (Element @$\mathbb{N}$@) | |
| 70 testStack07 = pushSingleLinkedStack emptySingleLinkedStack 1 ( | |
| 71 \s @$\rightarrow$@ pushSingleLinkedStack s 2 (\s @$\rightarrow$@ top s)) | |
| 72 | |
| 73 testStack08 = pushSingleLinkedStack emptySingleLinkedStack 1 | |
| 74 $ \s @$\rightarrow$@ pushSingleLinkedStack s 2 | |
| 75 $ \s @$\rightarrow$@ pushSingleLinkedStack s 3 | |
| 76 $ \s @$\rightarrow$@ pushSingleLinkedStack s 4 | |
| 77 $ \s @$\rightarrow$@ pushSingleLinkedStack s 5 | |
| 78 $ \s @$\rightarrow$@ top s | |
| 79 | |
| 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 @$\equiv$@ s3", since level of the Set does not fit , but use stack s @$\equiv$@ 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 ) @$\rightarrow$@ Stack {l} {m} D {t} ( SingleLinkedStack D ) | |
| 92 stackInSomeState s = record { stack = s ; stackMethods = singleLinkedStackSpec } | |
| 93 | |
| 94 push@$\rightarrow$@push@$\rightarrow$@pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) @$\rightarrow$@ | |
| 95 pushStack ( stackInSomeState s ) x ( \s1 @$\rightarrow$@ pushStack s1 y ( \s2 @$\rightarrow$@ pop2Stack s2 ( \s3 y1 x1 @$\rightarrow$@ (Just x @$\equiv$@ x1 ) @$\wedge$@ (Just y @$\equiv$@ y1 ) ) )) | |
| 96 push@$\rightarrow$@push@$\rightarrow$@pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl } | |
| 97 | |
| 98 | |
| 99 -- id : {n : Level} {A : Set n} @$\rightarrow$@ A @$\rightarrow$@ A | |
| 100 -- id a = a | |
| 101 | |
| 102 -- push a, n times | |
| 103 | |
| 104 n-push : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ 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 @$\rightarrow$@ s ) | |
| 107 | |
| 108 n-pop : {n : Level}{A : Set n} {a : A} @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A | |
| 109 n-pop zero s = s | |
| 110 n-pop {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ @$\rightarrow$@ s ) | |
| 111 | |
| 112 open @$\equiv$@-Reasoning | |
| 113 | |
| 114 push-pop-equiv : {n : Level} {A : Set n} {a : A} (s : SingleLinkedStack A) @$\rightarrow$@ (popSingleLinkedStack (pushSingleLinkedStack s a (\s @$\rightarrow$@ s)) (\s _ @$\rightarrow$@ s) ) @$\equiv$@ s | |
| 115 push-pop-equiv s = refl | |
| 116 | |
| 117 push-and-n-pop : {n : Level} {A : Set n} {a : A} (n : @$\mathbb{N}$@) (s : SingleLinkedStack A) @$\rightarrow$@ n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id) @$\equiv$@ 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 @$\equiv$@@$\langle$@ refl @$\rangle$@ | |
| 122 popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ @$\rightarrow$@ s) | |
| 123 @$\equiv$@@$\langle$@ cong (\s @$\rightarrow$@ popSingleLinkedStack s (\s _ @$\rightarrow$@ s )) (push-and-n-pop n s) @$\rangle$@ | |
| 124 popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ @$\rightarrow$@ s) | |
| 125 @$\equiv$@@$\langle$@ refl @$\rangle$@ | |
| 126 n-pop {_} {A} {a} (suc n) s | |
| 127 @$\blacksquare$@ | |
| 128 | |
| 129 | |
| 130 n-push-pop-equiv : {n : Level} {A : Set n} {a : A} (n : @$\mathbb{N}$@) (s : SingleLinkedStack A) @$\rightarrow$@ (n-pop {_} {A} {a} n (n-push {_} {A} {a} n s)) @$\equiv$@ 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 @$\equiv$@@$\langle$@ refl @$\rangle$@ | |
| 135 n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s @$\rightarrow$@ s)) | |
| 136 @$\equiv$@@$\langle$@ push-and-n-pop n (n-push n s) @$\rangle$@ | |
| 137 n-pop {_} {A} {a} n (n-push n s) | |
| 138 @$\equiv$@@$\langle$@ n-push-pop-equiv n s @$\rangle$@ | |
| 139 s | |
| 140 @$\blacksquare$@ | |
| 141 | |
| 142 | |
| 143 n-push-pop-equiv-empty : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ (n : @$\mathbb{N}$@) @$\rightarrow$@ n-pop {_} {A} {a} n (n-push {_} {A} {a} n emptySingleLinkedStack) @$\equiv$@ emptySingleLinkedStack | |
| 144 n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack |