author | nipkow |
Fri, 21 Jun 2013 09:00:26 +0200 | |
changeset 52402 | c2f30ba4bb98 |
parent 41037 | 6d6f23b3a879 |
child 56927 | 4044a7d1720f |
permissions | -rw-r--r-- |
30946 | 1 |
(* Authors: Klaus Aehlig, Tobias Nipkow *) |
19829 | 2 |
|
30946 | 3 |
header {* Testing implementation of normalization by evaluation *} |
19829 | 4 |
|
39395 | 5 |
theory Normalization_by_Evaluation |
35372 | 6 |
imports Complex_Main |
19829 | 7 |
begin |
8 |
||
21117 | 9 |
lemma "True" by normalization |
19971 | 10 |
lemma "p \<longrightarrow> True" by normalization |
28350 | 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 |
|
19971 | 16 |
lemma "~((0::nat) < (0::nat))" by normalization |
17 |
||
19829 | 18 |
datatype n = Z | S n |
28350 | 19 |
|
30946 | 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)" |
|
19829 | 27 |
|
28143 | 28 |
declare add2.simps [code] |
28709 | 29 |
lemma [code nbe]: "add2 (add2 n m) k = add2 n (add2 m k)" |
28143 | 30 |
by (induct n) auto |
20842 | 31 |
lemma [code]: "add2 n (S m) = S (add2 n m)" |
32 |
by(induct n) auto |
|
19829 | 33 |
lemma [code]: "add2 n Z = n" |
20842 | 34 |
by(induct n) auto |
19971 | 35 |
|
28350 | 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 |
|
19829 | 39 |
|
30946 | 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" |
|
19829 | 51 |
|
19971 | 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 |
|
28350 | 57 |
lemma "split (%x y. x) (a, b) = a" by normalization |
19971 | 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 |
|
19829 | 61 |
|
20842 | 62 |
lemma "[] @ [] = []" by normalization |
28350 | 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 |
||
28422 | 68 |
lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs" |
41037
6d6f23b3a879
removed experimental equality checking of closures; acknowledge underapproximation of equality in function name
haftmann
parents:
40730
diff
changeset
|
69 |
by normalization rule |
28350 | 70 |
lemma "rev [a, b, c] = [c, b, a]" by normalization |
39395 | 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])" |
|
25934 | 75 |
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" |
76 |
by normalization |
|
39395 | 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" |
|
28350 | 79 |
lemma "let x = y in [x, x] = [y, y]" by normalization |
80 |
lemma "Let y (%x. [x,x]) = [y, y]" by normalization |
|
39395 | 81 |
value [nbe] "case n of Z \<Rightarrow> True | S x \<Rightarrow> False" |
28350 | 82 |
lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization |
39395 | 83 |
value [nbe] "filter (%x. x) ([True,False,x]@xs)" |
84 |
value [nbe] "filter Not ([True,False,x]@xs)" |
|
19829 | 85 |
|
28350 | 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 |
|
25100 | 88 |
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" by normalization |
19829 | 89 |
|
28350 | 90 |
lemma "last [a, b, c] = c" by normalization |
91 |
lemma "last ([a, b, c] @ xs) = last (c # xs)" by normalization |
|
19829 | 92 |
|
28350 | 93 |
lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization |
20842 | 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 |
|
22394 | 104 |
lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization |
105 |
lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization |
|
25100 | 106 |
lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization |
107 |
lemma "max (Suc 0) 0 = Suc 0" by normalization |
|
25187 | 108 |
lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization |
39395 | 109 |
value [nbe] "Suc 0 \<in> set ms" |
20922 | 110 |
|
40730
2aa0390a2da7
nbe decides equality of abstractions by extensionality
haftmann
parents:
39395
diff
changeset
|
111 |
(* non-left-linear patterns, equality by extensionality *) |
2aa0390a2da7
nbe decides equality of abstractions by extensionality
haftmann
parents:
39395
diff
changeset
|
112 |
|
28350 | 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 |
|
40730
2aa0390a2da7
nbe decides equality of abstractions by extensionality
haftmann
parents:
39395
diff
changeset
|
117 |
lemma "(id :: bool \<Rightarrow> bool) = id" by normalization |
39395 | 118 |
value [nbe] "(\<lambda>x. x)" |
21987 | 119 |
|
23396 | 120 |
(* Church numerals: *) |
121 |
||
39395 | 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))))" |
|
23396 | 125 |
|
32544 | 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 |
||
19829 | 134 |
end |