src/HOL/Matrix/Float.thy
author obua
Fri Sep 03 17:10:36 2004 +0200 (2004-09-03)
changeset 15178 5f621aa35c25
child 15236 f289e8ba2bb3
permissions -rw-r--r--
Matrix theory, linear programming
     1 theory Float = Hyperreal:
     2 
     3 constdefs  
     4   pow2 :: "int \<Rightarrow> real"
     5   "pow2 a == if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a))))" 
     6   float :: "int * int \<Rightarrow> real"
     7   "float x == (real (fst x)) * (pow2 (snd x))"
     8 
     9 lemma pow2_0[simp]: "pow2 0 = 1"
    10 by (simp add: pow2_def)
    11 
    12 lemma pow2_1[simp]: "pow2 1 = 2"
    13 by (simp add: pow2_def)
    14 
    15 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
    16 by (simp add: pow2_def)
    17 
    18 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)" 
    19 proof -
    20   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
    21   have g: "! a b. a - -1 = a + (1::int)" by arith
    22   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
    23     apply (auto, induct_tac n)
    24     apply (simp_all add: pow2_def)
    25     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
    26     apply (auto simp add: h)
    27     apply arith
    28     done  
    29   show ?thesis
    30   proof (induct a)
    31     case (1 n)
    32     from pos show ?case by (simp add: ring_eq_simps)
    33   next
    34     case (2 n)
    35     show ?case
    36       apply (auto)
    37       apply (subst pow2_neg[of "- int n"])
    38       apply (subst pow2_neg[of "-1 - int n"])
    39       apply (auto simp add: g pos)
    40       done
    41   qed  
    42 qed
    43   
    44 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
    45 proof (induct b)
    46   case (1 n) 
    47   show ?case
    48   proof (induct n)
    49     case 0
    50     show ?case by simp
    51   next
    52     case (Suc m)
    53     show ?case by (auto simp add: ring_eq_simps pow2_add1 prems)
    54   qed
    55 next
    56   case (2 n)
    57   show ?case 
    58   proof (induct n)
    59     case 0
    60     show ?case 
    61       apply (auto)
    62       apply (subst pow2_neg[of "a + -1"])
    63       apply (subst pow2_neg[of "-1"])
    64       apply (simp)
    65       apply (insert pow2_add1[of "-a"])
    66       apply (simp add: ring_eq_simps)
    67       apply (subst pow2_neg[of "-a"])
    68       apply (simp)
    69       done
    70     case (Suc m)
    71     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith	
    72     have b: "int m - -2 = 1 + (int m + 1)" by arith
    73     show ?case
    74       apply (auto)
    75       apply (subst pow2_neg[of "a + (-2 - int m)"])
    76       apply (subst pow2_neg[of "-2 - int m"])
    77       apply (auto simp add: ring_eq_simps)
    78       apply (subst a)
    79       apply (subst b)
    80       apply (simp only: pow2_add1)
    81       apply (subst pow2_neg[of "int m - a + 1"])
    82       apply (subst pow2_neg[of "int m + 1"])
    83       apply auto
    84       apply (insert prems)
    85       apply (auto simp add: ring_eq_simps)
    86       done
    87   qed
    88 qed
    89 
    90 lemma "float (a, e) + float (b, e) = float (a + b, e)"  
    91 by (simp add: float_def ring_eq_simps)
    92 
    93 constdefs 
    94   int_of_real :: "real \<Rightarrow> int"
    95   "int_of_real x == SOME y. real y = x"  
    96   real_is_int :: "real \<Rightarrow> bool"
    97   "real_is_int x == ? (u::int). x = real u" 
    98 
    99 lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
   100 by (auto simp add: real_is_int_def int_of_real_def)
   101 
   102 lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
   103 by (simp add: float_def real_is_int_def2 pow2_add[symmetric])
   104 
   105 lemma pow2_int: "pow2 (int c) = (2::real)^c"
   106 by (simp add: pow2_def)
   107 
   108 lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)" 
   109 by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])
   110 
   111 lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
   112 by (auto simp add: real_is_int_def int_of_real_def)
   113 
   114 lemma int_of_real_real[simp]: "int_of_real (real x) = x"
   115 by (simp add: int_of_real_def)
   116 
   117 lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
   118 by (auto simp add: int_of_real_def real_is_int_def)
   119 
   120 lemma real_is_int_add_int_of_real: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a+b)) = (int_of_real a) + (int_of_real b)"
   121 by (auto simp add: int_of_real_def real_is_int_def)
   122 
   123 lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
   124 apply (subst real_is_int_def2)
   125 apply (simp add: real_is_int_add_int_of_real real_int_of_real)
   126 done
   127 
   128 lemma int_of_real_sub: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> (int_of_real (a-b)) = (int_of_real a) - (int_of_real b)"
   129 by (auto simp add: int_of_real_def real_is_int_def)
   130 
   131 lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
   132 apply (subst real_is_int_def2)
   133 apply (simp add: int_of_real_sub real_int_of_real)
   134 done
   135 
   136 lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
   137 by (auto simp add: real_is_int_def)
   138 
   139 lemma int_of_real_mult: 
   140   assumes "real_is_int a" "real_is_int b"
   141   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
   142 proof -
   143   from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
   144   from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
   145   from a obtain a'::int where a':"a = real a'" by auto
   146   from b obtain b'::int where b':"b = real b'" by auto
   147   have r: "real a' * real b' = real (a' * b')" by auto
   148   show ?thesis
   149     apply (simp add: a' b')
   150     apply (subst r)
   151     apply (simp only: int_of_real_real)
   152     done
   153 qed
   154 
   155 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
   156 apply (subst real_is_int_def2)
   157 apply (simp add: int_of_real_mult)
   158 done
   159 
   160 lemma real_is_int_0[simp]: "real_is_int (0::real)"
   161 by (simp add: real_is_int_def int_of_real_def)
   162 
   163 lemma real_is_int_1[simp]: "real_is_int (1::real)"
   164 proof -
   165   have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
   166   also have "\<dots> = True" by (simp only: real_is_int_real)
   167   ultimately show ?thesis by auto
   168 qed
   169 
   170 lemma real_is_int_n1: "real_is_int (-1::real)"
   171 proof -
   172   have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
   173   also have "\<dots> = True" by (simp only: real_is_int_real)
   174   ultimately show ?thesis by auto
   175 qed
   176 
   177 lemma real_is_int_number_of[simp]: "real_is_int ((number_of::bin\<Rightarrow>real) x)"
   178 proof -
   179   have neg1: "real_is_int (-1::real)"
   180   proof -
   181     have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
   182     also have "\<dots> = True" by (simp only: real_is_int_real)
   183     ultimately show ?thesis by auto
   184   qed
   185   
   186   { 
   187     fix x::int
   188     have "!! y. real_is_int ((number_of::bin\<Rightarrow>real) (Abs_Bin x))"
   189       apply (simp add: number_of_eq)
   190       apply (subst Abs_Bin_inverse)
   191       apply (simp add: Bin_def)
   192       apply (induct x)
   193       apply (induct_tac n)
   194       apply (simp)
   195       apply (simp)
   196       apply (induct_tac n)
   197       apply (simp add: neg1)
   198     proof -
   199       fix n :: nat
   200       assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
   201       have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
   202       show "real_is_int (of_int (- (int (Suc (Suc n)))))"
   203 	apply (simp only: s of_int_add)
   204 	apply (rule real_is_int_add)
   205 	apply (simp add: neg1)
   206 	apply (simp only: rn)
   207 	done
   208     qed
   209   }
   210   note Abs_Bin = this
   211   {
   212     fix x :: bin
   213     have "? u. x = Abs_Bin u"
   214       apply (rule exI[where x = "Rep_Bin x"])
   215       apply (simp add: Rep_Bin_inverse)
   216       done
   217   }
   218   then obtain u::int where "x = Abs_Bin u" by auto
   219   with Abs_Bin show ?thesis by auto
   220 qed
   221 
   222 lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
   223 by (simp add: int_of_real_def)
   224 
   225 lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
   226 proof - 
   227   have 1: "(1::real) = real (1::int)" by auto
   228   show ?thesis by (simp only: 1 int_of_real_real)
   229 qed
   230 
   231 lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
   232 proof -
   233   have "real_is_int (number_of b)" by simp
   234   then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
   235   then obtain u::int where u:"number_of b = real u" by auto
   236   have "number_of b = real ((number_of b)::int)" 
   237     by (simp add: number_of_eq real_of_int_def)
   238   have ub: "number_of b = real ((number_of b)::int)" 
   239     by (simp add: number_of_eq real_of_int_def)
   240   from uu u ub have unb: "u = number_of b"
   241     by blast
   242   have "int_of_real (number_of b) = u" by (simp add: u)
   243   with unb show ?thesis by simp
   244 qed
   245 
   246 lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"
   247   apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
   248   apply (simp_all add: pow2_def even_def real_is_int_def ring_eq_simps)
   249   apply (auto)
   250 proof -
   251   fix q::int
   252   have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith
   253   show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))" 
   254     by (simp add: a)
   255 qed
   256     
   257 consts
   258   norm_float :: "int*int \<Rightarrow> int*int"
   259 
   260 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
   261 apply (subst split_div, auto)
   262 apply (subst split_zdiv, auto)
   263 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
   264 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
   265 done
   266 
   267 lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
   268 apply (subst split_mod, auto)
   269 apply (subst split_zmod, auto)
   270 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
   271 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
   272 done
   273 
   274 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
   275 by arith
   276 
   277 lemma terminating_norm_float: "\<forall>a. (a::int) \<noteq> 0 \<and> even a \<longrightarrow> a \<noteq> 0 \<and> \<bar>a div 2\<bar> < \<bar>a\<bar>"
   278 apply (auto)
   279 apply (rule abs_div_2_less)
   280 apply (auto)
   281 done
   282 
   283 ML {* simp_depth_limit := 2 *} 
   284 recdef norm_float "measure (% (a,b). nat (abs a))"
   285   "norm_float (a,b) = (if (a \<noteq> 0) & (even a) then norm_float (a div 2, b+1) else (if a=0 then (0,0) else (a,b)))"
   286 (hints simp: terminating_norm_float)
   287 ML {* simp_depth_limit := 1000 *}
   288 
   289 
   290 lemma norm_float: "float x = float (norm_float x)"
   291 proof -
   292   {
   293     fix a b :: int 
   294     have norm_float_pair: "float (a,b) = float (norm_float (a,b))" 
   295     proof (induct a b rule: norm_float.induct)
   296       case (1 u v)
   297       show ?case 
   298       proof cases
   299 	assume u: "u \<noteq> 0 \<and> even u"
   300 	with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
   301 	with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even) 
   302 	then show ?thesis
   303 	  apply (subst norm_float.simps)
   304 	  apply (simp add: ind)
   305 	  done
   306       next
   307 	assume "~(u \<noteq> 0 \<and> even u)"
   308 	then show ?thesis
   309 	  by (simp add: prems float_def)
   310       qed
   311     qed
   312   }
   313   note helper = this
   314   have "? a b. x = (a,b)" by auto
   315   then obtain a b where "x = (a, b)" by blast
   316   then show ?thesis by (simp only: helper)
   317 qed
   318 
   319 lemma pow2_int: "pow2 (int n) = 2^n"
   320   by (simp add: pow2_def)
   321 
   322 lemma float_add: 
   323   "float (a1, e1) + float (a2, e2) = 
   324   (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1) 
   325   else float (a1*2^(nat (e1-e2))+a2, e2))"
   326   apply (simp add: float_def ring_eq_simps)
   327   apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
   328   done
   329 
   330 lemma float_mult:
   331   "float (a1, e1) * float (a2, e2) = 
   332   (float (a1 * a2, e1 + e2))"
   333   by (simp add: float_def pow2_add)
   334 
   335 lemma float_minus:
   336   "- (float (a,b)) = float (-a, b)"
   337   by (simp add: float_def)
   338 
   339 lemma zero_less_pow2:
   340   "0 < pow2 x"
   341 proof -
   342   {
   343     fix y
   344     have "0 <= y \<Longrightarrow> 0 < pow2 y"    
   345       by (induct y, induct_tac n, simp_all add: pow2_add)
   346   }
   347   note helper=this
   348   show ?thesis
   349     apply (case_tac "0 <= x")
   350     apply (simp add: helper)
   351     apply (subst pow2_neg)
   352     apply (simp add: helper)
   353     done
   354 qed
   355 
   356 lemma zero_le_float:
   357   "(0 <= float (a,b)) = (0 <= a)"
   358   apply (auto simp add: float_def)
   359   apply (auto simp add: zero_le_mult_iff zero_less_pow2) 
   360   apply (insert zero_less_pow2[of b])
   361   apply (simp_all)
   362   done
   363 
   364 lemma float_abs:
   365   "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
   366   apply (auto simp add: abs_if)
   367   apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
   368   done
   369 
   370 lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
   371   by auto
   372   
   373 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
   374   by simp
   375 
   376 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
   377   by simp
   378 
   379 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
   380   by simp
   381 
   382 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
   383   by simp
   384 
   385 lemma int_pow_0: "(a::int)^(Numeral0) = 1"
   386   by simp
   387 
   388 lemma int_pow_1: "(a::int)^(Numeral1) = a"
   389   by simp
   390 
   391 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
   392   by simp
   393 
   394 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
   395   by simp
   396 
   397 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
   398   by simp
   399 
   400 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
   401   by simp
   402 
   403 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
   404   by simp
   405 
   406 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
   407 proof -
   408   have 1:"((-1)::nat) = 0"
   409     by simp
   410   show ?thesis by (simp add: 1)
   411 qed
   412 
   413 lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
   414   by simp
   415 
   416 lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
   417   by simp
   418 
   419 lemma lift_bool: "x \<Longrightarrow> x=True"
   420   by simp
   421 
   422 lemma nlift_bool: "~x \<Longrightarrow> x=False"
   423   by simp
   424 
   425 lemma not_false_eq_true: "(~ False) = True" by simp
   426 
   427 lemma not_true_eq_false: "(~ True) = False" by simp
   428 
   429 
   430 lemmas binarith = 
   431   Pls_0_eq Min_1_eq
   432   bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0     
   433   bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0
   434   bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10
   435   bin_add_BIT_11 bin_minus_Pls bin_minus_Min bin_minus_1 
   436   bin_minus_0 bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0 
   437   bin_add_Pls_right bin_add_Min_right
   438 
   439 thm eq_number_of_eq
   440 
   441 lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)"
   442   by simp
   443 
   444 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)" 
   445   by (simp only: iszero_number_of_Pls)
   446 
   447 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
   448   by simp
   449 thm iszero_number_of_1
   450 
   451 lemma int_iszero_number_of_0: "iszero ((number_of (w BIT False))::int) = iszero ((number_of w)::int)"
   452   by simp
   453 
   454 lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT True))::int)" 
   455   by simp
   456 
   457 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)"
   458   by simp
   459 
   460 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))" 
   461   by simp
   462 
   463 lemma int_neg_number_of_Min: "neg (-1::int)"
   464   by simp
   465 
   466 lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
   467   by simp
   468 
   469 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (bin_add y (bin_minus x)))::int))"
   470   by simp
   471 
   472 lemmas intarithrel = 
   473   (*abs_zero abs_one*)
   474   int_eq_number_of_eq 
   475   lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0 
   476   lift_bool[OF int_iszero_number_of_1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]
   477   int_neg_number_of_BIT int_le_number_of_eq
   478 
   479 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (bin_add v w)"
   480   by simp
   481 
   482 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (bin_add v (bin_minus w))"
   483   by simp
   484 
   485 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (bin_mult v w)"
   486   by simp
   487 
   488 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (bin_minus v)"
   489   by simp
   490 
   491 lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
   492 
   493 lemmas natarith = (*zero_eq_Numeral0_nat one_eq_Numeral1_nat*) add_nat_number_of 
   494   diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
   495 
   496 lemmas powerarith = nat_number_of zpower_number_of_even 
   497   zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]   
   498   zpower_Pls zpower_Min
   499 
   500 lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float
   501 
   502 lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
   503 
   504 (* needed for the verifying code generator *)
   505 lemmas arith_no_let = arith[simplified Let_def]
   506 
   507 end
   508