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 *)
3 header {* Testing implementation of normalization by evaluation *}
5 theory Normalization_by_Evaluation
6 imports Complex_Main
7 begin
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
18 datatype n = Z | S n
20 primrec add :: "n \<Rightarrow> n \<Rightarrow> n" where
21    "add Z = id"
22  | "add (S m) = S o add m"
24 primrec add2 :: "n \<Rightarrow> n \<Rightarrow> n" where
25    "add2 Z n = n"
26  | "add2 (S m) n = S(add2 m n)"
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
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
40 primrec mul :: "n \<Rightarrow> n \<Rightarrow> n" where
41    "mul Z = (%n. Z)"
42  | "mul (S m) = (%n. add (mul m n) n)"
44 primrec mul2 :: "n \<Rightarrow> n \<Rightarrow> n" where
45    "mul2 Z n = Z"
46  | "mul2 (S m) n = add2 n (mul2 m n)"
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"
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
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
60 lemma "case Z of Z \<Rightarrow> True | S x \<Rightarrow> False" by normalization
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
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)"
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
90 lemma "last [a, b, c] = c" by normalization
91 lemma "last ([a, b, c] @ xs) = last (c # xs)" by normalization
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"
111 (* non-left-linear patterns, equality by extensionality *)
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)"
120 (* Church numerals: *)
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))))"
126 (* handling of type classes in connection with equality *)
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
134 end