src/HOL/Matrix/ComputeFloat.thy
author wenzelm
Fri Dec 17 17:43:54 2010 +0100 (2010-12-17)
changeset 41229 d797baa3d57c
parent 38273 d31a34569542
child 41413 64cd30d6b0b8
permissions -rw-r--r--
replaced command 'nonterminals' by slightly modernized version 'nonterminal';
     1 (*  Title:  HOL/Tools/ComputeFloat.thy
     2     Author: Steven Obua
     3 *)
     4 
     5 header {* Floating Point Representation of the Reals *}
     6 
     7 theory ComputeFloat
     8 imports Complex_Main Lattice_Algebras
     9 uses "~~/src/Tools/float.ML" ("~~/src/HOL/Tools/float_arith.ML")
    10 begin
    11 
    12 definition pow2 :: "int \<Rightarrow> real"
    13   where "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
    14 
    15 definition float :: "int * int \<Rightarrow> real"
    16   where "float x = real (fst x) * pow2 (snd x)"
    17 
    18 lemma pow2_0[simp]: "pow2 0 = 1"
    19 by (simp add: pow2_def)
    20 
    21 lemma pow2_1[simp]: "pow2 1 = 2"
    22 by (simp add: pow2_def)
    23 
    24 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
    25 by (simp add: pow2_def)
    26 
    27 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
    28 proof -
    29   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
    30   have g: "! a b. a - -1 = a + (1::int)" by arith
    31   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
    32     apply (auto, induct_tac n)
    33     apply (simp_all add: pow2_def)
    34     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
    35     by (auto simp add: h)
    36   show ?thesis
    37   proof (induct a)
    38     case (1 n)
    39     from pos show ?case by (simp add: algebra_simps)
    40   next
    41     case (2 n)
    42     show ?case
    43       apply (auto)
    44       apply (subst pow2_neg[of "- int n"])
    45       apply (subst pow2_neg[of "-1 - int n"])
    46       apply (auto simp add: g pos)
    47       done
    48   qed
    49 qed
    50 
    51 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
    52 proof (induct b)
    53   case (1 n)
    54   show ?case
    55   proof (induct n)
    56     case 0
    57     show ?case by simp
    58   next
    59     case (Suc m)
    60     show ?case by (auto simp add: algebra_simps pow2_add1 prems)
    61   qed
    62 next
    63   case (2 n)
    64   show ?case
    65   proof (induct n)
    66     case 0
    67     show ?case
    68       apply (auto)
    69       apply (subst pow2_neg[of "a + -1"])
    70       apply (subst pow2_neg[of "-1"])
    71       apply (simp)
    72       apply (insert pow2_add1[of "-a"])
    73       apply (simp add: algebra_simps)
    74       apply (subst pow2_neg[of "-a"])
    75       apply (simp)
    76       done
    77     case (Suc m)
    78     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
    79     have b: "int m - -2 = 1 + (int m + 1)" by arith
    80     show ?case
    81       apply (auto)
    82       apply (subst pow2_neg[of "a + (-2 - int m)"])
    83       apply (subst pow2_neg[of "-2 - int m"])
    84       apply (auto simp add: algebra_simps)
    85       apply (subst a)
    86       apply (subst b)
    87       apply (simp only: pow2_add1)
    88       apply (subst pow2_neg[of "int m - a + 1"])
    89       apply (subst pow2_neg[of "int m + 1"])
    90       apply auto
    91       apply (insert prems)
    92       apply (auto simp add: algebra_simps)
    93       done
    94   qed
    95 qed
    96 
    97 lemma "float (a, e) + float (b, e) = float (a + b, e)"
    98 by (simp add: float_def algebra_simps)
    99 
   100 definition int_of_real :: "real \<Rightarrow> int"
   101   where "int_of_real x = (SOME y. real y = x)"
   102 
   103 definition real_is_int :: "real \<Rightarrow> bool"
   104   where "real_is_int x = (EX (u::int). x = real u)"
   105 
   106 lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
   107 by (auto simp add: real_is_int_def int_of_real_def)
   108 
   109 lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
   110 by (simp add: float_def real_is_int_def2 pow2_add[symmetric])
   111 
   112 lemma pow2_int: "pow2 (int c) = 2^c"
   113 by (simp add: pow2_def)
   114 
   115 lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
   116 by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])
   117 
   118 lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
   119 by (auto simp add: real_is_int_def int_of_real_def)
   120 
   121 lemma int_of_real_real[simp]: "int_of_real (real x) = x"
   122 by (simp add: int_of_real_def)
   123 
   124 lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
   125 by (auto simp add: int_of_real_def real_is_int_def)
   126 
   127 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)"
   128 by (auto simp add: int_of_real_def real_is_int_def)
   129 
   130 lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
   131 apply (subst real_is_int_def2)
   132 apply (simp add: real_is_int_add_int_of_real real_int_of_real)
   133 done
   134 
   135 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)"
   136 by (auto simp add: int_of_real_def real_is_int_def)
   137 
   138 lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
   139 apply (subst real_is_int_def2)
   140 apply (simp add: int_of_real_sub real_int_of_real)
   141 done
   142 
   143 lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
   144 by (auto simp add: real_is_int_def)
   145 
   146 lemma int_of_real_mult:
   147   assumes "real_is_int a" "real_is_int b"
   148   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
   149 proof -
   150   from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
   151   from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
   152   from a obtain a'::int where a':"a = real a'" by auto
   153   from b obtain b'::int where b':"b = real b'" by auto
   154   have r: "real a' * real b' = real (a' * b')" by auto
   155   show ?thesis
   156     apply (simp add: a' b')
   157     apply (subst r)
   158     apply (simp only: int_of_real_real)
   159     done
   160 qed
   161 
   162 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
   163 apply (subst real_is_int_def2)
   164 apply (simp add: int_of_real_mult)
   165 done
   166 
   167 lemma real_is_int_0[simp]: "real_is_int (0::real)"
   168 by (simp add: real_is_int_def int_of_real_def)
   169 
   170 lemma real_is_int_1[simp]: "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_n1: "real_is_int (-1::real)"
   178 proof -
   179   have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
   180   also have "\<dots> = True" by (simp only: real_is_int_real)
   181   ultimately show ?thesis by auto
   182 qed
   183 
   184 lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
   185 proof -
   186   have neg1: "real_is_int (-1::real)"
   187   proof -
   188     have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
   189     also have "\<dots> = True" by (simp only: real_is_int_real)
   190     ultimately show ?thesis by auto
   191   qed
   192 
   193   {
   194     fix x :: int
   195     have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
   196       unfolding number_of_eq
   197       apply (induct x)
   198       apply (induct_tac n)
   199       apply (simp)
   200       apply (simp)
   201       apply (induct_tac n)
   202       apply (simp add: neg1)
   203     proof -
   204       fix n :: nat
   205       assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
   206       have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
   207       show "real_is_int (of_int (- (int (Suc (Suc n)))))"
   208         apply (simp only: s of_int_add)
   209         apply (rule real_is_int_add)
   210         apply (simp add: neg1)
   211         apply (simp only: rn)
   212         done
   213     qed
   214   }
   215   note Abs_Bin = this
   216   {
   217     fix x :: int
   218     have "? u. x = u"
   219       apply (rule exI[where x = "x"])
   220       apply (simp)
   221       done
   222   }
   223   then obtain u::int where "x = u" by auto
   224   with Abs_Bin show ?thesis by auto
   225 qed
   226 
   227 lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
   228 by (simp add: int_of_real_def)
   229 
   230 lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
   231 proof -
   232   have 1: "(1::real) = real (1::int)" by auto
   233   show ?thesis by (simp only: 1 int_of_real_real)
   234 qed
   235 
   236 lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
   237 proof -
   238   have "real_is_int (number_of b)" by simp
   239   then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
   240   then obtain u::int where u:"number_of b = real u" by auto
   241   have "number_of b = real ((number_of b)::int)"
   242     by (simp add: number_of_eq real_of_int_def)
   243   have ub: "number_of b = real ((number_of b)::int)"
   244     by (simp add: number_of_eq real_of_int_def)
   245   from uu u ub have unb: "u = number_of b"
   246     by blast
   247   have "int_of_real (number_of b) = u" by (simp add: u)
   248   with unb show ?thesis by simp
   249 qed
   250 
   251 lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"
   252   apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
   253   apply (simp_all add: pow2_def even_def real_is_int_def algebra_simps)
   254   apply (auto)
   255 proof -
   256   fix q::int
   257   have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith
   258   show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))"
   259     by (simp add: a)
   260 qed
   261 
   262 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
   263 by (rule zdiv_int)
   264 
   265 lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
   266 by (rule zmod_int)
   267 
   268 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
   269 by arith
   270 
   271 function norm_float :: "int \<Rightarrow> int \<Rightarrow> int \<times> int" where
   272   "norm_float a b = (if a \<noteq> 0 \<and> even a then norm_float (a div 2) (b + 1)
   273     else if a = 0 then (0, 0) else (a, b))"
   274 by auto
   275 
   276 termination by (relation "measure (nat o abs o fst)")
   277   (auto intro: abs_div_2_less)
   278 
   279 lemma norm_float: "float x = float (split norm_float x)"
   280 proof -
   281   {
   282     fix a b :: int
   283     have norm_float_pair: "float (a, b) = float (norm_float a b)"
   284     proof (induct a b rule: norm_float.induct)
   285       case (1 u v)
   286       show ?case
   287       proof cases
   288         assume u: "u \<noteq> 0 \<and> even u"
   289         with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2) (v + 1))" by auto
   290         with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
   291         then show ?thesis
   292           apply (subst norm_float.simps)
   293           apply (simp add: ind)
   294           done
   295       next
   296         assume "~(u \<noteq> 0 \<and> even u)"
   297         then show ?thesis
   298           by (simp add: prems float_def)
   299       qed
   300     qed
   301   }
   302   note helper = this
   303   have "? a b. x = (a,b)" by auto
   304   then obtain a b where "x = (a, b)" by blast
   305   then show ?thesis by (simp add: helper)
   306 qed
   307 
   308 lemma float_add_l0: "float (0, e) + x = x"
   309   by (simp add: float_def)
   310 
   311 lemma float_add_r0: "x + float (0, e) = x"
   312   by (simp add: float_def)
   313 
   314 lemma float_add:
   315   "float (a1, e1) + float (a2, e2) =
   316   (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
   317   else float (a1*2^(nat (e1-e2))+a2, e2))"
   318   apply (simp add: float_def algebra_simps)
   319   apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
   320   done
   321 
   322 lemma float_add_assoc1:
   323   "(x + float (y1, e1)) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x"
   324   by simp
   325 
   326 lemma float_add_assoc2:
   327   "(float (y1, e1) + x) + float (y2, e2) = (float (y1, e1) + float (y2, e2)) + x"
   328   by simp
   329 
   330 lemma float_add_assoc3:
   331   "float (y1, e1) + (x + float (y2, e2)) = (float (y1, e1) + float (y2, e2)) + x"
   332   by simp
   333 
   334 lemma float_add_assoc4:
   335   "float (y1, e1) + (float (y2, e2) + x) = (float (y1, e1) + float (y2, e2)) + x"
   336   by simp
   337 
   338 lemma float_mult_l0: "float (0, e) * x = float (0, 0)"
   339   by (simp add: float_def)
   340 
   341 lemma float_mult_r0: "x * float (0, e) = float (0, 0)"
   342   by (simp add: float_def)
   343 
   344 definition lbound :: "real \<Rightarrow> real"
   345   where "lbound x = min 0 x"
   346 
   347 definition ubound :: "real \<Rightarrow> real"
   348   where "ubound x = max 0 x"
   349 
   350 lemma lbound: "lbound x \<le> x"   
   351   by (simp add: lbound_def)
   352 
   353 lemma ubound: "x \<le> ubound x"
   354   by (simp add: ubound_def)
   355 
   356 lemma float_mult:
   357   "float (a1, e1) * float (a2, e2) =
   358   (float (a1 * a2, e1 + e2))"
   359   by (simp add: float_def pow2_add)
   360 
   361 lemma float_minus:
   362   "- (float (a,b)) = float (-a, b)"
   363   by (simp add: float_def)
   364 
   365 lemma zero_less_pow2:
   366   "0 < pow2 x"
   367 proof -
   368   {
   369     fix y
   370     have "0 <= y \<Longrightarrow> 0 < pow2 y"
   371       by (induct y, induct_tac n, simp_all add: pow2_add)
   372   }
   373   note helper=this
   374   show ?thesis
   375     apply (case_tac "0 <= x")
   376     apply (simp add: helper)
   377     apply (subst pow2_neg)
   378     apply (simp add: helper)
   379     done
   380 qed
   381 
   382 lemma zero_le_float:
   383   "(0 <= float (a,b)) = (0 <= a)"
   384   apply (auto simp add: float_def)
   385   apply (auto simp add: zero_le_mult_iff zero_less_pow2)
   386   apply (insert zero_less_pow2[of b])
   387   apply (simp_all)
   388   done
   389 
   390 lemma float_le_zero:
   391   "(float (a,b) <= 0) = (a <= 0)"
   392   apply (auto simp add: float_def)
   393   apply (auto simp add: mult_le_0_iff)
   394   apply (insert zero_less_pow2[of b])
   395   apply auto
   396   done
   397 
   398 lemma float_abs:
   399   "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
   400   apply (auto simp add: abs_if)
   401   apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
   402   done
   403 
   404 lemma float_zero:
   405   "float (0, b) = 0"
   406   by (simp add: float_def)
   407 
   408 lemma float_pprt:
   409   "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
   410   by (auto simp add: zero_le_float float_le_zero float_zero)
   411 
   412 lemma pprt_lbound: "pprt (lbound x) = float (0, 0)"
   413   apply (simp add: float_def)
   414   apply (rule pprt_eq_0)
   415   apply (simp add: lbound_def)
   416   done
   417 
   418 lemma nprt_ubound: "nprt (ubound x) = float (0, 0)"
   419   apply (simp add: float_def)
   420   apply (rule nprt_eq_0)
   421   apply (simp add: ubound_def)
   422   done
   423 
   424 lemma float_nprt:
   425   "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
   426   by (auto simp add: zero_le_float float_le_zero float_zero)
   427 
   428 lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
   429   by auto
   430 
   431 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
   432   by simp
   433 
   434 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
   435   by simp
   436 
   437 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
   438   by simp
   439 
   440 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
   441   by simp
   442 
   443 lemma int_pow_0: "(a::int)^(Numeral0) = 1"
   444   by simp
   445 
   446 lemma int_pow_1: "(a::int)^(Numeral1) = a"
   447   by simp
   448 
   449 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
   450   by simp
   451 
   452 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
   453   by simp
   454 
   455 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
   456   by simp
   457 
   458 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
   459   by simp
   460 
   461 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
   462   by simp
   463 
   464 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
   465 proof -
   466   have 1:"((-1)::nat) = 0"
   467     by simp
   468   show ?thesis by (simp add: 1)
   469 qed
   470 
   471 lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
   472   by simp
   473 
   474 lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
   475   by simp
   476 
   477 lemma lift_bool: "x \<Longrightarrow> x=True"
   478   by simp
   479 
   480 lemma nlift_bool: "~x \<Longrightarrow> x=False"
   481   by simp
   482 
   483 lemma not_false_eq_true: "(~ False) = True" by simp
   484 
   485 lemma not_true_eq_false: "(~ True) = False" by simp
   486 
   487 lemmas binarith =
   488   normalize_bin_simps
   489   pred_bin_simps succ_bin_simps
   490   add_bin_simps minus_bin_simps mult_bin_simps
   491 
   492 lemma int_eq_number_of_eq:
   493   "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)"
   494   by (rule eq_number_of_eq)
   495 
   496 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
   497   by (simp only: iszero_number_of_Pls)
   498 
   499 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
   500   by simp
   501 
   502 lemma int_iszero_number_of_Bit0: "iszero ((number_of (Int.Bit0 w))::int) = iszero ((number_of w)::int)"
   503   by simp
   504 
   505 lemma int_iszero_number_of_Bit1: "\<not> iszero ((number_of (Int.Bit1 w))::int)"
   506   by simp
   507 
   508 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
   509   unfolding neg_def number_of_is_id by simp
   510 
   511 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
   512   by simp
   513 
   514 lemma int_neg_number_of_Min: "neg (-1::int)"
   515   by simp
   516 
   517 lemma int_neg_number_of_Bit0: "neg ((number_of (Int.Bit0 w))::int) = neg ((number_of w)::int)"
   518   by simp
   519 
   520 lemma int_neg_number_of_Bit1: "neg ((number_of (Int.Bit1 w))::int) = neg ((number_of w)::int)"
   521   by simp
   522 
   523 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
   524   unfolding neg_def number_of_is_id by (simp add: not_less)
   525 
   526 lemmas intarithrel =
   527   int_eq_number_of_eq
   528   lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_Bit0
   529   lift_bool[OF int_iszero_number_of_Bit1] int_less_number_of_eq_neg nlift_bool[OF int_not_neg_number_of_Pls] lift_bool[OF int_neg_number_of_Min]
   530   int_neg_number_of_Bit0 int_neg_number_of_Bit1 int_le_number_of_eq
   531 
   532 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
   533   by simp
   534 
   535 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
   536   by simp
   537 
   538 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)"
   539   by simp
   540 
   541 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
   542   by simp
   543 
   544 lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
   545 
   546 lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
   547 
   548 lemmas powerarith = nat_number_of zpower_number_of_even
   549   zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
   550   zpower_Pls zpower_Min
   551 
   552 lemmas floatarith[simplified norm_0_1] = float_add float_add_l0 float_add_r0 float_mult float_mult_l0 float_mult_r0 
   553           float_minus float_abs zero_le_float float_pprt float_nprt pprt_lbound nprt_ubound
   554 
   555 (* for use with the compute oracle *)
   556 lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
   557 
   558 use "~~/src/HOL/Tools/float_arith.ML"
   559 
   560 end