src/HOL/ex/Normalization_by_Evaluation.thy
author bulwahn
Fri Oct 21 11:17:14 2011 +0200 (2011-10-21)
changeset 45231 d85a2fdc586c
parent 41037 6d6f23b3a879
child 56927 4044a7d1720f
permissions -rw-r--r--
replacing code_inline by code_unfold, removing obsolete code_unfold, code_inline del now that the ancient code generator is removed
     1 (*  Authors:  Klaus Aehlig, Tobias Nipkow *)
     2 
     3 header {* Testing implementation of normalization by evaluation *}
     4 
     5 theory Normalization_by_Evaluation
     6 imports Complex_Main
     7 begin
     8 
     9 lemma "True" by normalization
    10 lemma "p \<longrightarrow> True" by normalization
    11 declare disj_assoc [code nbe]
    12 lemma "((P | Q) | R) = (P | (Q | R))" by normalization
    13 lemma "0 + (n::nat) = n" by normalization
    14 lemma "0 + Suc n = Suc n" by normalization
    15 lemma "Suc n + Suc m = n + Suc (Suc m)" by normalization
    16 lemma "~((0::nat) < (0::nat))" by normalization
    17 
    18 datatype n = Z | S n
    19 
    20 primrec add :: "n \<Rightarrow> n \<Rightarrow> n" where
    21    "add Z = id"
    22  | "add (S m) = S o add m"
    23 
    24 primrec add2 :: "n \<Rightarrow> n \<Rightarrow> n" where
    25    "add2 Z n = n"
    26  | "add2 (S m) n = S(add2 m n)"
    27 
    28 declare add2.simps [code]
    29 lemma [code nbe]: "add2 (add2 n m) k = add2 n (add2 m k)"
    30   by (induct n) auto
    31 lemma [code]: "add2 n (S m) =  S (add2 n m)"
    32   by(induct n) auto
    33 lemma [code]: "add2 n Z = n"
    34   by(induct n) auto
    35 
    36 lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization
    37 lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization
    38 lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization
    39 
    40 primrec mul :: "n \<Rightarrow> n \<Rightarrow> n" where
    41    "mul Z = (%n. Z)"
    42  | "mul (S m) = (%n. add (mul m n) n)"
    43 
    44 primrec mul2 :: "n \<Rightarrow> n \<Rightarrow> n" where
    45    "mul2 Z n = Z"
    46  | "mul2 (S m) n = add2 n (mul2 m n)"
    47 
    48 primrec exp :: "n \<Rightarrow> n \<Rightarrow> n" where
    49    "exp m Z = S Z"
    50  | "exp m (S n) = mul (exp m n) m"
    51 
    52 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
    53 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
    54 lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization
    55 
    56 lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization
    57 lemma "split (%x y. x) (a, b) = a" by normalization
    58 lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization
    59 
    60 lemma "case Z of Z \<Rightarrow> True | S x \<Rightarrow> False" by normalization
    61 
    62 lemma "[] @ [] = []" by normalization
    63 lemma "map f [x,y,z::'x] = [f x, f y, f z]" by normalization
    64 lemma "[a, b, c] @ xs = a # b # c # xs" by normalization
    65 lemma "[] @ xs = xs" by normalization
    66 lemma "map (%f. f True) [id, g, Not] = [True, g True, False]" by normalization
    67 
    68 lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs"
    69   by normalization rule
    70 lemma "rev [a, b, c] = [c, b, a]" by normalization
    71 value [nbe] "rev (a#b#cs) = rev cs @ [b, a]"
    72 value [nbe] "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
    73 value [nbe] "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
    74 value [nbe] "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
    75 lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" 
    76   by normalization
    77 value [nbe] "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
    78 value [nbe] "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
    79 lemma "let x = y in [x, x] = [y, y]" by normalization
    80 lemma "Let y (%x. [x,x]) = [y, y]" by normalization
    81 value [nbe] "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
    82 lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization
    83 value [nbe] "filter (%x. x) ([True,False,x]@xs)"
    84 value [nbe] "filter Not ([True,False,x]@xs)"
    85 
    86 lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b, c]" by normalization
    87 lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization
    88 lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" by normalization
    89 
    90 lemma "last [a, b, c] = c" by normalization
    91 lemma "last ([a, b, c] @ xs) = last (c # xs)" by normalization
    92 
    93 lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization
    94 lemma "(-4::int) * 2 = -8" by normalization
    95 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
    96 lemma "(2::int) + 3 = 5" by normalization
    97 lemma "(2::int) + 3 * (- 4) * (- 1) = 14" by normalization
    98 lemma "(2::int) + 3 * (- 4) * 1 + 0 = -10" by normalization
    99 lemma "(2::int) < 3" by normalization
   100 lemma "(2::int) <= 3" by normalization
   101 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
   102 lemma "4 - 42 * abs (3 + (-7\<Colon>int)) = -164" by normalization
   103 lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>nat else x) = 0" by normalization
   104 lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization
   105 lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization
   106 lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization
   107 lemma "max (Suc 0) 0 = Suc 0" by normalization
   108 lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization
   109 value [nbe] "Suc 0 \<in> set ms"
   110 
   111 (* non-left-linear patterns, equality by extensionality *)
   112 
   113 lemma "f = f" by normalization
   114 lemma "f x = f x" by normalization
   115 lemma "(f o g) x = f (g x)" by normalization
   116 lemma "(f o id) x = f x" by normalization
   117 lemma "(id :: bool \<Rightarrow> bool) = id" by normalization
   118 value [nbe] "(\<lambda>x. x)"
   119 
   120 (* Church numerals: *)
   121 
   122 value [nbe] "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
   123 value [nbe] "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
   124 value [nbe] "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
   125 
   126 (* handling of type classes in connection with equality *)
   127 
   128 lemma "map f [x, y] = [f x, f y]" by normalization
   129 lemma "(map f [x, y], w) = ([f x, f y], w)" by normalization
   130 lemma "map f [x, y] = [f x \<Colon> 'a\<Colon>semigroup_add, f y]" by normalization
   131 lemma "map f [x \<Colon> 'a\<Colon>semigroup_add, y] = [f x, f y]" by normalization
   132 lemma "(map f [x \<Colon> 'a\<Colon>semigroup_add, y], w \<Colon> 'b\<Colon>finite) = ([f x, f y], w)" by normalization
   133 
   134 end