src/HOL/ex/NormalForm.thy
author haftmann
Thu Sep 25 09:28:08 2008 +0200 (2008-09-25)
changeset 28350 715163ec93c0
parent 28337 93964076e7b8
child 28351 abfc66969d1f
permissions -rw-r--r--
non left-linear equations for nbe
nipkow@19829
     1
(*  ID:         $Id$
nipkow@19829
     2
    Authors:    Klaus Aehlig, Tobias Nipkow
wenzelm@20807
     3
*)
nipkow@19829
     4
haftmann@21059
     5
header {* Test of normalization function *}
nipkow@19829
     6
nipkow@19829
     7
theory NormalForm
haftmann@25165
     8
imports Main "~~/src/HOL/Real/Rational"
nipkow@19829
     9
begin
nipkow@19829
    10
haftmann@28350
    11
lemma [code nbe]:
haftmann@28350
    12
  "x = x \<longleftrightarrow> True" by rule+
haftmann@28350
    13
haftmann@28350
    14
lemma [code nbe]:
haftmann@28350
    15
  "eq_class.eq (x::bool) x \<longleftrightarrow> True" unfolding eq by rule+
haftmann@28350
    16
haftmann@28350
    17
lemma [code nbe]:
haftmann@28350
    18
  "eq_class.eq (x::nat) x \<longleftrightarrow> True" unfolding eq by rule+
haftmann@28350
    19
haftmann@21117
    20
lemma "True" by normalization
nipkow@19971
    21
lemma "p \<longrightarrow> True" by normalization
haftmann@28350
    22
declare disj_assoc [code nbe]
haftmann@28350
    23
lemma "((P | Q) | R) = (P | (Q | R))" by normalization
haftmann@22845
    24
declare disj_assoc [code func del]
haftmann@28350
    25
lemma "0 + (n::nat) = n" by normalization
haftmann@28350
    26
lemma "0 + Suc n = Suc n" by normalization
haftmann@28350
    27
lemma "Suc n + Suc m = n + Suc (Suc m)" by normalization
nipkow@19971
    28
lemma "~((0::nat) < (0::nat))" by normalization
nipkow@19971
    29
nipkow@19829
    30
datatype n = Z | S n
haftmann@28350
    31
haftmann@28350
    32
lemma [code nbe]:
haftmann@28350
    33
  "eq_class.eq (x::n) x \<longleftrightarrow> True" unfolding eq by rule+
haftmann@28350
    34
nipkow@19829
    35
consts
haftmann@20842
    36
  add :: "n \<Rightarrow> n \<Rightarrow> n"
haftmann@20842
    37
  add2 :: "n \<Rightarrow> n \<Rightarrow> n"
haftmann@20842
    38
  mul :: "n \<Rightarrow> n \<Rightarrow> n"
haftmann@20842
    39
  mul2 :: "n \<Rightarrow> n \<Rightarrow> n"
haftmann@20842
    40
  exp :: "n \<Rightarrow> n \<Rightarrow> n"
nipkow@19829
    41
primrec
haftmann@20842
    42
  "add Z = id"
haftmann@20842
    43
  "add (S m) = S o add m"
nipkow@19829
    44
primrec
haftmann@20842
    45
  "add2 Z n = n"
haftmann@20842
    46
  "add2 (S m) n = S(add2 m n)"
nipkow@19829
    47
haftmann@28143
    48
declare add2.simps [code]
nipkow@19829
    49
lemma [code]: "add2 (add2 n m) k = add2 n (add2 m k)"
haftmann@28143
    50
  by (induct n) auto
haftmann@20842
    51
lemma [code]: "add2 n (S m) =  S (add2 n m)"
haftmann@20842
    52
  by(induct n) auto
nipkow@19829
    53
lemma [code]: "add2 n Z = n"
haftmann@20842
    54
  by(induct n) auto
nipkow@19971
    55
haftmann@28350
    56
lemma "add2 (add2 n m) k = add2 n (add2 m k)" by normalization
haftmann@28350
    57
lemma "add2 (add2 (S n) (S m)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization
haftmann@28350
    58
lemma "add2 (add2 (S n) (add2 (S m) Z)) (S k) = S(S(S(add2 n (add2 m k))))" by normalization
nipkow@19829
    59
nipkow@19829
    60
primrec
haftmann@20842
    61
  "mul Z = (%n. Z)"
haftmann@20842
    62
  "mul (S m) = (%n. add (mul m n) n)"
nipkow@19829
    63
primrec
haftmann@20842
    64
  "mul2 Z n = Z"
haftmann@20842
    65
  "mul2 (S m) n = add2 n (mul2 m n)"
nipkow@19829
    66
primrec
haftmann@20842
    67
  "exp m Z = S Z"
haftmann@20842
    68
  "exp m (S n) = mul (exp m n) m"
nipkow@19829
    69
nipkow@19971
    70
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
nipkow@19971
    71
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
nipkow@19971
    72
lemma "exp (S(S Z)) (S(S(S(S Z)))) = exp (S(S(S(S Z)))) (S(S Z))" by normalization
nipkow@19971
    73
nipkow@19971
    74
lemma "(let ((x,y),(u,v)) = ((Z,Z),(Z,Z)) in add (add x y) (add u v)) = Z" by normalization
haftmann@28350
    75
lemma "split (%x y. x) (a, b) = a" by normalization
nipkow@19971
    76
lemma "(%((x,y),(u,v)). add (add x y) (add u v)) ((Z,Z),(Z,Z)) = Z" by normalization
nipkow@19971
    77
nipkow@19971
    78
lemma "case Z of Z \<Rightarrow> True | S x \<Rightarrow> False" by normalization
nipkow@19829
    79
haftmann@20842
    80
lemma "[] @ [] = []" by normalization
haftmann@28350
    81
lemma "map f [x,y,z::'x] = [f x, f y, f z]" by normalization
haftmann@28350
    82
lemma "[a, b, c] @ xs = a # b # c # xs" by normalization
haftmann@28350
    83
lemma "[] @ xs = xs" by normalization
haftmann@28350
    84
lemma "map (%f. f True) [id, g, Not] = [True, g True, False]" by normalization
haftmann@28350
    85
haftmann@28350
    86
lemma [code nbe]:
haftmann@28350
    87
  "eq_class.eq (x :: 'a\<Colon>eq list) x \<longleftrightarrow> True" unfolding eq by rule+
haftmann@28350
    88
haftmann@25934
    89
lemma "map (%f. f True) ([id, g, Not] @ fs) = [True, g True, False] @ map (%f. f True) fs" by normalization rule+
haftmann@28350
    90
lemma "rev [a, b, c] = [c, b, a]" by normalization
haftmann@26739
    91
normal_form "rev (a#b#cs) = rev cs @ [b, a]"
nipkow@19829
    92
normal_form "map (%F. F [a,b,c::'x]) (map map [f,g,h])"
nipkow@19829
    93
normal_form "map (%F. F ([a,b,c] @ ds)) (map map ([f,g,h]@fs))"
nipkow@19829
    94
normal_form "map (%F. F [Z,S Z,S(S Z)]) (map map [S,add (S Z),mul (S(S Z)),id])"
haftmann@25934
    95
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" 
haftmann@25934
    96
  by normalization
nipkow@19829
    97
normal_form "case xs of [] \<Rightarrow> True | x#xs \<Rightarrow> False"
haftmann@25934
    98
normal_form "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) xs = P"
haftmann@28350
    99
lemma "let x = y in [x, x] = [y, y]" by normalization
haftmann@28350
   100
lemma "Let y (%x. [x,x]) = [y, y]" by normalization
nipkow@19829
   101
normal_form "case n of Z \<Rightarrow> True | S x \<Rightarrow> False"
haftmann@28350
   102
lemma "(%(x,y). add x y) (S z,S z) = S (add z (S z))" by normalization
nipkow@19829
   103
normal_form "filter (%x. x) ([True,False,x]@xs)"
nipkow@19829
   104
normal_form "filter Not ([True,False,x]@xs)"
nipkow@19829
   105
haftmann@28350
   106
lemma "[x,y,z] @ [a,b,c] = [x, y, z, a, b, c]" by normalization
haftmann@28350
   107
lemma "(%(xs, ys). xs @ ys) ([a, b, c], [d, e, f]) = [a, b, c, d, e, f]" by normalization
haftmann@25100
   108
lemma "map (%x. case x of None \<Rightarrow> False | Some y \<Rightarrow> True) [None, Some ()] = [False, True]" by normalization
nipkow@19829
   109
haftmann@28350
   110
lemma "last [a, b, c] = c" by normalization
haftmann@28350
   111
lemma "last ([a, b, c] @ xs) = last (c # xs)" by normalization
nipkow@19829
   112
haftmann@28350
   113
lemma [code nbe]:
haftmann@28350
   114
  "eq_class.eq (x :: int) x \<longleftrightarrow> True" unfolding eq by rule+
haftmann@28350
   115
haftmann@28350
   116
lemma "(2::int) + 3 - 1 + (- k) * 2 = 4 + - k * 2" by normalization
haftmann@20842
   117
lemma "(-4::int) * 2 = -8" by normalization
haftmann@20842
   118
lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
haftmann@20842
   119
lemma "(2::int) + 3 = 5" by normalization
haftmann@20842
   120
lemma "(2::int) + 3 * (- 4) * (- 1) = 14" by normalization
haftmann@20842
   121
lemma "(2::int) + 3 * (- 4) * 1 + 0 = -10" by normalization
haftmann@20842
   122
lemma "(2::int) < 3" by normalization
haftmann@20842
   123
lemma "(2::int) <= 3" by normalization
haftmann@20842
   124
lemma "abs ((-4::int) + 2 * 1) = 2" by normalization
haftmann@20842
   125
lemma "4 - 42 * abs (3 + (-7\<Colon>int)) = -164" by normalization
haftmann@20842
   126
lemma "(if (0\<Colon>nat) \<le> (x\<Colon>nat) then 0\<Colon>nat else x) = 0" by normalization
haftmann@22394
   127
lemma "4 = Suc (Suc (Suc (Suc 0)))" by normalization
haftmann@22394
   128
lemma "nat 4 = Suc (Suc (Suc (Suc 0)))" by normalization
haftmann@25100
   129
lemma "[Suc 0, 0] = [Suc 0, 0]" by normalization
haftmann@25100
   130
lemma "max (Suc 0) 0 = Suc 0" by normalization
haftmann@25187
   131
lemma "(42::rat) / 1704 = 1 / 284 + 3 / 142" by normalization
haftmann@21059
   132
normal_form "Suc 0 \<in> set ms"
nipkow@20922
   133
haftmann@28350
   134
lemma "f = f" by normalization
haftmann@28350
   135
lemma "f x = f x" by normalization
haftmann@28350
   136
lemma "(f o g) x = f (g x)" by normalization
haftmann@28350
   137
lemma "(f o id) x = f x" by normalization
haftmann@25934
   138
normal_form "(\<lambda>x. x)"
haftmann@21987
   139
nipkow@23396
   140
(* Church numerals: *)
nipkow@23396
   141
nipkow@23396
   142
normal_form "(%m n f x. m f (n f x)) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
nipkow@23396
   143
normal_form "(%m n f x. m (n f) x) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
nipkow@23396
   144
normal_form "(%m n. n m) (%f x. f(f(f(x)))) (%f x. f(f(f(x))))"
nipkow@23396
   145
nipkow@19829
   146
end