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1
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1 pop-n-push-type : @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack @$\mathbb{N}$@ @$\rightarrow$@ Set@$\_{1}$@
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2 pop-n-push-type n cn ce s = M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) meta
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3 @$\equiv$@ M.exec (n-push n) meta
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4 where
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5 meta = id-meta cn ce s
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6
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7 pop-n-push : (n cn ce : @$\mathbb{N}$@) @$\rightarrow$@ (s : SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ pop-n-push-type n cn ce s
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8 pop-n-push zero cn ce s = refl
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9 pop-n-push (suc n) cn ce s = begin
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10 M.exec (M.csComp (M.cs popOnce) (n-push (suc (suc n)))) (id-meta cn ce s)
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11 @$\equiv$@@$\langle$@ refl @$\rangle$@
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12 M.exec (M.csComp (M.cs popOnce) (M.csComp (n-push (suc n)) (M.cs pushOnce))) (id-meta cn ce s)
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13 @$\equiv$@@$\langle$@ exec-comp (M.cs popOnce) (M.csComp (n-push (suc n)) (M.cs pushOnce)) (id-meta cn ce s) @$\rangle$@
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14 M.exec (M.cs popOnce) (M.exec (M.csComp (n-push (suc n)) (M.cs pushOnce)) (id-meta cn ce s))
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15 @$\equiv$@@$\langle$@ cong (\x @$\rightarrow$@ M.exec (M.cs popOnce) x) (exec-comp (n-push (suc n)) (M.cs pushOnce) (id-meta cn ce s)) @$\rangle$@
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16 M.exec (M.cs popOnce) (M.exec (n-push (suc n))(M.exec (M.cs pushOnce) (id-meta cn ce s)))
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17 @$\equiv$@@$\langle$@ refl @$\rangle$@
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18 M.exec (M.cs popOnce) (M.exec (n-push (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))})))
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19 @$\equiv$@@$\langle$@ sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))) @$\rangle$@
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20 M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
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21 @$\equiv$@@$\langle$@ pop-n-push n cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}) @$\rangle$@
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22 M.exec (n-push n) (id-meta cn ce (record {top = just (cons ce (SingleLinkedStack.top s))}))
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23 @$\equiv$@@$\langle$@ refl @$\rangle$@
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24 M.exec (n-push n) (pushOnce (id-meta cn ce s))
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25 @$\equiv$@@$\langle$@ refl @$\rangle$@
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26 M.exec (n-push n) (M.exec (M.cs pushOnce) (id-meta cn ce s))
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27 @$\equiv$@@$\langle$@ refl @$\rangle$@
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28 M.exec (n-push (suc n)) (id-meta cn ce s)
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29 @$\blacksquare$@
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30
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31
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32
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33 n-push-pop-type : @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ @$\mathbb{N}$@ @$\rightarrow$@ SingleLinkedStack @$\mathbb{N}$@ @$\rightarrow$@ Set@$\_{1}$@
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34 n-push-pop-type n cn ce st = M.exec (M.csComp (n-pop n) (n-push n)) meta @$\equiv$@ meta
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35 where
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36 meta = id-meta cn ce st
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37
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38 n-push-pop : (n cn ce : @$\mathbb{N}$@) @$\rightarrow$@ (s : SingleLinkedStack @$\mathbb{N}$@) @$\rightarrow$@ n-push-pop-type n cn ce s
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39 n-push-pop zero cn ce s = refl
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40 n-push-pop (suc n) cn ce s = begin
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41 M.exec (M.csComp (n-pop (suc n)) (n-push (suc n))) (id-meta cn ce s)
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42 @$\equiv$@@$\langle$@ refl @$\rangle$@
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43 M.exec (M.csComp (M.cs (\m @$\rightarrow$@ M.exec (n-pop n) (popOnce m))) (n-push (suc n))) (id-meta cn ce s)
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44 @$\equiv$@@$\langle$@ exec-comp (M.cs (\m @$\rightarrow$@ M.exec (n-pop n) (popOnce m))) (n-push (suc n)) (id-meta cn ce s) @$\rangle$@
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45 M.exec (M.cs (\m @$\rightarrow$@ M.exec (n-pop n) (popOnce m))) (M.exec (n-push (suc n)) (id-meta cn ce s))
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46 @$\equiv$@@$\langle$@ refl @$\rangle$@
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47 M.exec (n-pop n) (popOnce (M.exec (n-push (suc n)) (id-meta cn ce s)))
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48 @$\equiv$@@$\langle$@ refl @$\rangle$@
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49 M.exec (n-pop n) (M.exec (M.cs popOnce) (M.exec (n-push (suc n)) (id-meta cn ce s)))
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50 @$\equiv$@@$\langle$@ cong (\x @$\rightarrow$@ M.exec (n-pop n) x) (sym (exec-comp (M.cs popOnce) (n-push (suc n)) (id-meta cn ce s))) @$\rangle$@
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51 M.exec (n-pop n) (M.exec (M.csComp (M.cs popOnce) (n-push (suc n))) (id-meta cn ce s))
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52 @$\equiv$@@$\langle$@ cong (\x @$\rightarrow$@ M.exec (n-pop n) x) (pop-n-push n cn ce s) @$\rangle$@
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53 M.exec (n-pop n) (M.exec (n-push n) (id-meta cn ce s))
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54 @$\equiv$@@$\langle$@ sym (exec-comp (n-pop n) (n-push n) (id-meta cn ce s)) @$\rangle$@
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55 M.exec (M.csComp (n-pop n) (n-push n)) (id-meta cn ce s)
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56 @$\equiv$@@$\langle$@ n-push-pop n cn ce s @$\rangle$@
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57 id-meta cn ce s
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58 @$\blacksquare$@
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