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(* ID: $Id$


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Authors: Klaus Aehlig, Tobias Nipkow


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Test of normalization function


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*)


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theory NormalForm


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imports Main


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begin


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lemma "p \<longrightarrow> True" by normalization

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(* FIXME Eventually the code generator should be able to handle this


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by regenerating the existing code for "or":


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declare disj_assoc[code]


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normal_form "(P  Q)  R"


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*)


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lemma "0 + (n::nat) = n" by normalization


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lemma "0 + Suc(n) = Suc n" by normalization


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lemma "Suc(n) + Suc m = n + Suc(Suc m)" by normalization


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lemma "~((0::nat) < (0::nat))" by normalization


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datatype n = Z  S n


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consts


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add :: "n \<Rightarrow> n \<Rightarrow> n"


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add2 :: "n \<Rightarrow> n \<Rightarrow> n"


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mul :: "n \<Rightarrow> n \<Rightarrow> n"


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mul2 :: "n \<Rightarrow> n \<Rightarrow> n"


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exp :: "n \<Rightarrow> n \<Rightarrow> n"


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primrec


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"add Z = id"


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"add (S m) = S o add m"


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primrec


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"add2 Z n = n"


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"add2 (S m) n = S(add2 m n)"


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lemma [code]: "add2 (add2 n m) k = add2 n (add2 m k)"


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by(induct n, auto)


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lemma [code]: "add2 n (S m) = S(add2 n m)"


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by(induct n, auto)


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lemma [code]: "add2 n Z = n"


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by(induct n, auto)

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lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization


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lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization


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lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization

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primrec


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"mul Z = (%n. Z)"


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"mul (S m) = (%n. add (mul m n) n)"


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primrec


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"mul2 Z n = Z"


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"mul2 (S m) n = add2 n (mul2 m n)"


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primrec


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"exp m Z = S Z"


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"exp m (S n) = mul (exp m n) m"


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lemma "mul2 (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization


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lemma "mul (S(S(S(S(S Z))))) (S(S(S Z))) = S(S(S(S(S(S(S(S(S(S(S(S(S(S(S Z))))))))))))))" by normalization


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lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization


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lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization


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lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization


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lemma "case Z of Z \<Rightarrow> True  S x \<Rightarrow> False" by normalization

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normal_form "[] @ []"


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normal_form "[] @ xs"


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normal_form "[a::'d,b,c] @ xs"


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normal_form "[%a::'x. a, %b. b, c] @ xs"


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normal_form "[%a::'x. a, %b. b, c] @ [u,v]"


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normal_form "map f (xs::'c list)"


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normal_form "map f [x,y,z::'x]"


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normal_form "map (%f. f True) [id,g,Not]"


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normal_form "map (%f. f True) ([id,g,Not] @ fs)"


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normal_form "rev[a,b,c]"


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normal_form "rev(a#b#cs)"


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normal_form "map map [f,g,h]"


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normal_form "map (%F. F [a,b,c::'x]) (map map [f,g,h])"


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normal_form "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"


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normal_form "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"


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normal_form "map (%x. case x of None \<Rightarrow> False  Some y \<Rightarrow> True) [None, Some ()]"


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normal_form "case xs of [] \<Rightarrow> True  x#xs \<Rightarrow> False"


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normal_form "map (%x. case x of None \<Rightarrow> False  Some y \<Rightarrow> True) xs"


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normal_form "let x = y::'x in [x,x]"


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normal_form "Let y (%x. [x,x])"


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normal_form "case n of Z \<Rightarrow> True  S x \<Rightarrow> False"


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normal_form "(%(x,y). add x y) (S z,S z)"


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normal_form "filter (%x. x) ([True,False,x]@xs)"


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normal_form "filter Not ([True,False,x]@xs)"


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normal_form "[x,y,z] @ [a,b,c]"


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normal_form "%(xs, ys). xs @ ys"


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normal_form "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f])"


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normal_form "%x. case x of None \<Rightarrow> False  Some y \<Rightarrow> True"


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normal_form "map (%x. case x of None \<Rightarrow> False  Some y \<Rightarrow> True) [None, Some ()]"


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normal_form "last[a,b,c]"


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normal_form "last([a,b,c]@xs)"


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(* FIXME


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won't work since it relies on


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polymorphically used adhoc overloaded function:


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normal_form "max 0 (0::nat)"


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*)


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text {*


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Numerals still take their time\<dots>


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*}


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end
