src/HOL/ex/NormalForm.thy
 author haftmann Wed Mar 12 19:38:14 2008 +0100 (2008-03-12) changeset 26265 4b63b9e9b10d parent 25934 7b8f3a9efa03 child 26513 6f306c8c2c54 permissions -rw-r--r--
separated Random.thy from Quickcheck.thy
```     1 (*  ID:         \$Id\$
```
```     2     Authors:    Klaus Aehlig, Tobias Nipkow
```
```     3 *)
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```     4
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```     5 header {* Test of normalization function *}
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```     6
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```     7 theory NormalForm
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```     8 imports Main "~~/src/HOL/Real/Rational"
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```     9 begin
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```    10
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```    11 lemma "True" by normalization
```
```    12 lemma "p \<longrightarrow> True" by normalization
```
```    13 declare disj_assoc [code func]
```
```    14 lemma "((P | Q) | R) = (P | (Q | R))" by normalization rule
```
```    15 declare disj_assoc [code func del]
```
```    16 lemma "0 + (n::nat) = n" by normalization rule
```
```    17 lemma "0 + Suc n = Suc n" by normalization rule
```
```    18 lemma "Suc n + Suc m = n + Suc (Suc m)" by normalization rule
```
```    19 lemma "~((0::nat) < (0::nat))" by normalization
```
```    20
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```    21 datatype n = Z | S n
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```    22 consts
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```    23   add :: "n \<Rightarrow> n \<Rightarrow> n"
```
```    24   add2 :: "n \<Rightarrow> n \<Rightarrow> n"
```
```    25   mul :: "n \<Rightarrow> n \<Rightarrow> n"
```
```    26   mul2 :: "n \<Rightarrow> n \<Rightarrow> n"
```
```    27   exp :: "n \<Rightarrow> n \<Rightarrow> n"
```
```    28 primrec
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```    29   "add Z = id"
```
```    30   "add (S m) = S o add m"
```
```    31 primrec
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```    32   "add2 Z n = n"
```
```    33   "add2 (S m) n = S(add2 m n)"
```
```    34
```
```    35 lemma [code]: "add2 (add2 n m) k = add2 n (add2 m k)"
```
```    36   by(induct n) auto
```
```    37 lemma [code]: "add2 n (S m) =  S (add2 n m)"
```
```    38   by(induct n) auto
```
```    39 lemma [code]: "add2 n Z = n"
```
```    40   by(induct n) auto
```
```    41
```
```    42 lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization rule
```
```    43 lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization rule
```
```    44 lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization rule
```
```    45
```
```    46 primrec
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```    47   "mul Z = (%n. Z)"
```
```    48   "mul (S m) = (%n. add (mul m n) n)"
```
```    49 primrec
```
```    50   "mul2 Z n = Z"
```
```    51   "mul2 (S m) n = add2 n (mul2 m n)"
```
```    52 primrec
```
```    53   "exp m Z = S Z"
```
```    54   "exp m (S n) = mul (exp m n) m"
```
```    55
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```    56 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
```
```    57 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
```
```    58 lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization
```
```    59
```
```    60 lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization
```
```    61 lemma "split (%x y. x) (a, b) = a" by normalization rule
```
```    62 lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization
```
```    63
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```    64 lemma "case Z of Z \<Rightarrow> True | S x \<Rightarrow> False" by normalization
```
```    65
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```    66 lemma "[] @ [] = []" by normalization
```
```    67 lemma "map f [x,y,z::'x] = [f x, f y, f z]" by normalization rule+
```
```    68 lemma "[a, b, c] @ xs = a # b # c # xs" by normalization rule+
```
```    69 lemma "[] @ xs = xs" by normalization rule
```
```    70 lemma "map (%f. f True) [id, g, Not] = [True, g True, False]" by normalization rule+
```
```    71 lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs" by normalization rule+
```
```    72 lemma "rev [a, b, c] = [c, b, a]" by normalization rule+
```
```    73 normal_form "rev (a#b#cs) = rev cs @ [b, a]"
```
```    74 normal_form "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
```
```    75 normal_form "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
```
```    76 normal_form "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
```
```    77 lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]"
```
```    78   by normalization
```
```    79 normal_form "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
```
```    80 normal_form "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
```
```    81 lemma "let x = y in [x, x] = [y, y]" by normalization rule+
```
```    82 lemma "Let y (%x. [x,x]) = [y, y]" by normalization rule+
```
```    83 normal_form "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
```
```    84 lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization rule+
```
```    85 normal_form "filter (%x. x) ([True,False,x]@xs)"
```
```    86 normal_form "filter Not ([True,False,x]@xs)"
```
```    87
```
```    88 lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b ,c]" by normalization rule+
```
```    89 lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization rule+
```
```    90 lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" by normalization
```
```    91
```
```    92 lemma "last [a, b, c] = c" by normalization rule
```
```    93 lemma "last ([a, b, c] @ xs) = (if null xs then c else last xs)"
```
```    94   by normalization rule
```
```    95
```
```    96 lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization rule
```
```    97 lemma "(-4::int) * 2 = -8" by normalization
```
```    98 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
```
```    99 lemma "(2::int) + 3 = 5" by normalization
```
```   100 lemma "(2::int) + 3 * (- 4) * (- 1) = 14" by normalization
```
```   101 lemma "(2::int) + 3 * (- 4) * 1 + 0 = -10" by normalization
```
```   102 lemma "(2::int) < 3" by normalization
```
```   103 lemma "(2::int) <= 3" by normalization
```
```   104 lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
```
```   105 lemma "4 - 42 * abs (3 + (-7\<Colon>int)) = -164" by normalization
```
```   106 lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>nat else x) = 0" by normalization
```
```   107 lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization
```
```   108 lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization
```
```   109 lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization
```
```   110 lemma "max (Suc 0) 0 = Suc 0" by normalization
```
```   111 lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization
```
```   112 normal_form "Suc 0 \<in> set ms"
```
```   113
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```   114 lemma "f = f" by normalization rule+
```
```   115 lemma "f x = f x" by normalization rule+
```
```   116 lemma "(f o g) x = f (g x)" by normalization rule+
```
```   117 lemma "(f o id) x = f x" by normalization rule+
```
```   118 normal_form "(\<lambda>x. x)"
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```   119
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```   120 (* Church numerals: *)
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```   121
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```   122 normal_form "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
```
```   123 normal_form "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
```
```   124 normal_form "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
```
```   125
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```   126 end
```