Mercurial > hg > Papers > 2021 > soto-thesis
diff prepaper/src/stackTest.agda.replaced @ 0:3dba680da508
init-test
author | soto |
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date | Tue, 08 Dec 2020 19:06:49 +0900 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/prepaper/src/stackTest.agda.replaced Tue Dec 08 19:06:49 2020 +0900 @@ -0,0 +1,144 @@ +open import Level renaming (suc to succ ; zero to Zero ) +module stackTest where + +open import stack + +open import Relation.Binary.PropositionalEquality +open import Relation.Binary.Core +open import Data.Nat +open import Function + + +open SingleLinkedStack +open Stack + +---- +-- +-- proof of properties ( concrete cases ) +-- + +test01 : {n : Level } {a : Set n} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ Maybe a @$\rightarrow$@ Bool {n} +test01 stack _ with (top stack) +... | (Just _) = True +... | Nothing = False + + +test02 : {n : Level } {a : Set n} @$\rightarrow$@ SingleLinkedStack a @$\rightarrow$@ Bool +test02 stack = popSingleLinkedStack stack test01 + +test03 : {n : Level } {a : Set n} @$\rightarrow$@ a @$\rightarrow$@ Bool +test03 v = pushSingleLinkedStack emptySingleLinkedStack v test02 + +-- after a push and a pop, the stack is empty +lemma : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ test03 a @$\equiv$@ False +lemma = refl + +testStack01 : {n m : Level } {a : Set n} @$\rightarrow$@ a @$\rightarrow$@ Bool {m} +testStack01 v = pushStack createSingleLinkedStack v ( + \s @$\rightarrow$@ popStack s (\s1 d1 @$\rightarrow$@ True)) + +-- after push 1 and 2, pop2 get 1 and 2 + +testStack02 : {m : Level } @$\rightarrow$@ ( Stack @$\mathbb{N}$@ (SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ Bool {m} ) @$\rightarrow$@ Bool {m} +testStack02 cs = pushStack createSingleLinkedStack 1 ( + \s @$\rightarrow$@ pushStack s 2 cs) + + +testStack031 : (d1 d2 : @$\mathbb{N}$@ ) @$\rightarrow$@ Bool {Zero} +testStack031 2 1 = True +testStack031 _ _ = False + +testStack032 : (d1 d2 : Maybe @$\mathbb{N}$@) @$\rightarrow$@ Bool {Zero} +testStack032 (Just d1) (Just d2) = testStack031 d1 d2 +testStack032 _ _ = False + +testStack03 : {m : Level } @$\rightarrow$@ Stack @$\mathbb{N}$@ (SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ ((Maybe @$\mathbb{N}$@) @$\rightarrow$@ (Maybe @$\mathbb{N}$@) @$\rightarrow$@ Bool {m} ) @$\rightarrow$@ Bool {m} +testStack03 s cs = pop2Stack s ( + \s d1 d2 @$\rightarrow$@ cs d1 d2 ) + +testStack04 : Bool +testStack04 = testStack02 (\s @$\rightarrow$@ testStack03 s testStack032) + +testStack05 : testStack04 @$\equiv$@ True +testStack05 = refl + +testStack06 : {m : Level } @$\rightarrow$@ Maybe (Element @$\mathbb{N}$@) +testStack06 = pushStack createSingleLinkedStack 1 ( + \s @$\rightarrow$@ pushStack s 2 (\s @$\rightarrow$@ top (stack s))) + +testStack07 : {m : Level } @$\rightarrow$@ Maybe (Element @$\mathbb{N}$@) +testStack07 = pushSingleLinkedStack emptySingleLinkedStack 1 ( + \s @$\rightarrow$@ pushSingleLinkedStack s 2 (\s @$\rightarrow$@ top s)) + +testStack08 = pushSingleLinkedStack emptySingleLinkedStack 1 + $ \s @$\rightarrow$@ pushSingleLinkedStack s 2 + $ \s @$\rightarrow$@ pushSingleLinkedStack s 3 + $ \s @$\rightarrow$@ pushSingleLinkedStack s 4 + $ \s @$\rightarrow$@ pushSingleLinkedStack s 5 + $ \s @$\rightarrow$@ top s + +------ +-- +-- proof of properties with indefinite state of stack +-- +-- this should be proved by properties of the stack inteface, not only by the implementation, +-- and the implementation have to provides the properties. +-- +-- we cannot write "s @$\equiv$@ s3", since level of the Set does not fit , but use stack s @$\equiv$@ stack s3 is ok. +-- anyway some implementations may result s != s3 +-- + +stackInSomeState : {l m : Level } {D : Set l} {t : Set m } (s : SingleLinkedStack D ) @$\rightarrow$@ Stack {l} {m} D {t} ( SingleLinkedStack D ) +stackInSomeState s = record { stack = s ; stackMethods = singleLinkedStackSpec } + +push@$\rightarrow$@push@$\rightarrow$@pop2 : {l : Level } {D : Set l} (x y : D ) (s : SingleLinkedStack D ) @$\rightarrow$@ + 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 ) ) )) +push@$\rightarrow$@push@$\rightarrow$@pop2 {l} {D} x y s = record { pi1 = refl ; pi2 = refl } + + +-- id : {n : Level} {A : Set n} @$\rightarrow$@ A @$\rightarrow$@ A +-- id a = a + +-- push a, n times + +n-push : {n : Level} {A : Set n} {a : A} @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A +n-push zero s = s +n-push {l} {A} {a} (suc n) s = pushSingleLinkedStack (n-push {l} {A} {a} n s) a (\s @$\rightarrow$@ s ) + +n-pop : {n : Level}{A : Set n} {a : A} @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack A @$\rightarrow$@ SingleLinkedStack A +n-pop zero s = s +n-pop {_} {A} {a} (suc n) s = popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ @$\rightarrow$@ s ) + +open @$\equiv$@-Reasoning + +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 +push-pop-equiv s = refl + +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 +push-and-n-pop zero s = refl +push-and-n-pop {_} {A} {a} (suc n) s = begin + n-pop {_} {A} {a} (suc (suc n)) (pushSingleLinkedStack s a id) + @$\equiv$@@$\langle$@ refl @$\rangle$@ + popSingleLinkedStack (n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack s a id)) (\s _ @$\rightarrow$@ s) + @$\equiv$@@$\langle$@ cong (\s @$\rightarrow$@ popSingleLinkedStack s (\s _ @$\rightarrow$@ s )) (push-and-n-pop n s) @$\rangle$@ + popSingleLinkedStack (n-pop {_} {A} {a} n s) (\s _ @$\rightarrow$@ s) + @$\equiv$@@$\langle$@ refl @$\rangle$@ + n-pop {_} {A} {a} (suc n) s + @$\blacksquare$@ + + +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 +n-push-pop-equiv zero s = refl +n-push-pop-equiv {_} {A} {a} (suc n) s = begin + n-pop {_} {A} {a} (suc n) (n-push (suc n) s) + @$\equiv$@@$\langle$@ refl @$\rangle$@ + n-pop {_} {A} {a} (suc n) (pushSingleLinkedStack (n-push n s) a (\s @$\rightarrow$@ s)) + @$\equiv$@@$\langle$@ push-and-n-pop n (n-push n s) @$\rangle$@ + n-pop {_} {A} {a} n (n-push n s) + @$\equiv$@@$\langle$@ n-push-pop-equiv n s @$\rangle$@ + s + @$\blacksquare$@ + + +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 +n-push-pop-equiv-empty n = n-push-pop-equiv n emptySingleLinkedStack