src/HOL/Real/Float.thy
 author wenzelm Fri Nov 17 02:20:03 2006 +0100 (2006-11-17) changeset 21404 eb85850d3eb7 parent 21256 47195501ecf7 child 22964 2284e0d02e7f permissions -rw-r--r--
more robust syntax for definition/abbreviation/notation;
```     1 (*  Title: HOL/Real/Float.thy
```
```     2     ID:    \$Id\$
```
```     3     Author: Steven Obua
```
```     4 *)
```
```     5
```
```     6 header {* Floating Point Representation of the Reals *}
```
```     7
```
```     8 theory Float
```
```     9 imports Real Parity
```
```    10 uses ("float.ML")
```
```    11 begin
```
```    12
```
```    13 definition
```
```    14   pow2 :: "int \<Rightarrow> real" where
```
```    15   "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
```
```    16
```
```    17 definition
```
```    18   float :: "int * int \<Rightarrow> real" where
```
```    19   "float x = real (fst x) * pow2 (snd x)"
```
```    20
```
```    21 lemma pow2_0[simp]: "pow2 0 = 1"
```
```    22 by (simp add: pow2_def)
```
```    23
```
```    24 lemma pow2_1[simp]: "pow2 1 = 2"
```
```    25 by (simp add: pow2_def)
```
```    26
```
```    27 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
```
```    28 by (simp add: pow2_def)
```
```    29
```
```    30 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
```
```    31 proof -
```
```    32   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
```
```    33   have g: "! a b. a - -1 = a + (1::int)" by arith
```
```    34   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
```
```    35     apply (auto, induct_tac n)
```
```    36     apply (simp_all add: pow2_def)
```
```    37     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
```
```    38     by (auto simp add: h)
```
```    39   show ?thesis
```
```    40   proof (induct a)
```
```    41     case (1 n)
```
```    42     from pos show ?case by (simp add: ring_eq_simps)
```
```    43   next
```
```    44     case (2 n)
```
```    45     show ?case
```
```    46       apply (auto)
```
```    47       apply (subst pow2_neg[of "- int n"])
```
```    48       apply (subst pow2_neg[of "-1 - int n"])
```
```    49       apply (auto simp add: g pos)
```
```    50       done
```
```    51   qed
```
```    52 qed
```
```    53
```
```    54 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
```
```    55 proof (induct b)
```
```    56   case (1 n)
```
```    57   show ?case
```
```    58   proof (induct n)
```
```    59     case 0
```
```    60     show ?case by simp
```
```    61   next
```
```    62     case (Suc m)
```
```    63     show ?case by (auto simp add: ring_eq_simps pow2_add1 prems)
```
```    64   qed
```
```    65 next
```
```    66   case (2 n)
```
```    67   show ?case
```
```    68   proof (induct n)
```
```    69     case 0
```
```    70     show ?case
```
```    71       apply (auto)
```
```    72       apply (subst pow2_neg[of "a + -1"])
```
```    73       apply (subst pow2_neg[of "-1"])
```
```    74       apply (simp)
```
```    75       apply (insert pow2_add1[of "-a"])
```
```    76       apply (simp add: ring_eq_simps)
```
```    77       apply (subst pow2_neg[of "-a"])
```
```    78       apply (simp)
```
```    79       done
```
```    80     case (Suc m)
```
```    81     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
```
```    82     have b: "int m - -2 = 1 + (int m + 1)" by arith
```
```    83     show ?case
```
```    84       apply (auto)
```
```    85       apply (subst pow2_neg[of "a + (-2 - int m)"])
```
```    86       apply (subst pow2_neg[of "-2 - int m"])
```
```    87       apply (auto simp add: ring_eq_simps)
```
```    88       apply (subst a)
```
```    89       apply (subst b)
```
```    90       apply (simp only: pow2_add1)
```
```    91       apply (subst pow2_neg[of "int m - a + 1"])
```
```    92       apply (subst pow2_neg[of "int m + 1"])
```
```    93       apply auto
```
```    94       apply (insert prems)
```
```    95       apply (auto simp add: ring_eq_simps)
```
```    96       done
```
```    97   qed
```
```    98 qed
```
```    99
```
```   100 lemma "float (a, e) + float (b, e) = float (a + b, e)"
```
```   101 by (simp add: float_def ring_eq_simps)
```
```   102
```
```   103 definition
```
```   104   int_of_real :: "real \<Rightarrow> int" where
```
```   105   "int_of_real x = (SOME y. real y = x)"
```
```   106
```
```   107 definition
```
```   108   real_is_int :: "real \<Rightarrow> bool" where
```
```   109   "real_is_int x = (EX (u::int). x = real u)"
```
```   110
```
```   111 lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
```
```   112 by (auto simp add: real_is_int_def int_of_real_def)
```
```   113
```
```   114 lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
```
```   115 by (simp add: float_def real_is_int_def2 pow2_add[symmetric])
```
```   116
```
```   117 lemma pow2_int: "pow2 (int c) = (2::real)^c"
```
```   118 by (simp add: pow2_def)
```
```   119
```
```   120 lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
```
```   121 by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])
```
```   122
```
```   123 lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
```
```   124 by (auto simp add: real_is_int_def int_of_real_def)
```
```   125
```
```   126 lemma int_of_real_real[simp]: "int_of_real (real x) = x"
```
```   127 by (simp add: int_of_real_def)
```
```   128
```
```   129 lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
```
```   130 by (auto simp add: int_of_real_def real_is_int_def)
```
```   131
```
```   132 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)"
```
```   133 by (auto simp add: int_of_real_def real_is_int_def)
```
```   134
```
```   135 lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
```
```   136 apply (subst real_is_int_def2)
```
```   137 apply (simp add: real_is_int_add_int_of_real real_int_of_real)
```
```   138 done
```
```   139
```
```   140 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)"
```
```   141 by (auto simp add: int_of_real_def real_is_int_def)
```
```   142
```
```   143 lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
```
```   144 apply (subst real_is_int_def2)
```
```   145 apply (simp add: int_of_real_sub real_int_of_real)
```
```   146 done
```
```   147
```
```   148 lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
```
```   149 by (auto simp add: real_is_int_def)
```
```   150
```
```   151 lemma int_of_real_mult:
```
```   152   assumes "real_is_int a" "real_is_int b"
```
```   153   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
```
```   154 proof -
```
```   155   from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
```
```   156   from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
```
```   157   from a obtain a'::int where a':"a = real a'" by auto
```
```   158   from b obtain b'::int where b':"b = real b'" by auto
```
```   159   have r: "real a' * real b' = real (a' * b')" by auto
```
```   160   show ?thesis
```
```   161     apply (simp add: a' b')
```
```   162     apply (subst r)
```
```   163     apply (simp only: int_of_real_real)
```
```   164     done
```
```   165 qed
```
```   166
```
```   167 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
```
```   168 apply (subst real_is_int_def2)
```
```   169 apply (simp add: int_of_real_mult)
```
```   170 done
```
```   171
```
```   172 lemma real_is_int_0[simp]: "real_is_int (0::real)"
```
```   173 by (simp add: real_is_int_def int_of_real_def)
```
```   174
```
```   175 lemma real_is_int_1[simp]: "real_is_int (1::real)"
```
```   176 proof -
```
```   177   have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
```
```   178   also have "\<dots> = True" by (simp only: real_is_int_real)
```
```   179   ultimately show ?thesis by auto
```
```   180 qed
```
```   181
```
```   182 lemma real_is_int_n1: "real_is_int (-1::real)"
```
```   183 proof -
```
```   184   have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
```
```   185   also have "\<dots> = True" by (simp only: real_is_int_real)
```
```   186   ultimately show ?thesis by auto
```
```   187 qed
```
```   188
```
```   189 lemma real_is_int_number_of[simp]: "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
```
```   190 proof -
```
```   191   have neg1: "real_is_int (-1::real)"
```
```   192   proof -
```
```   193     have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
```
```   194     also have "\<dots> = True" by (simp only: real_is_int_real)
```
```   195     ultimately show ?thesis by auto
```
```   196   qed
```
```   197
```
```   198   {
```
```   199     fix x :: int
```
```   200     have "real_is_int ((number_of \<Colon> int \<Rightarrow> real) x)"
```
```   201       unfolding number_of_eq
```
```   202       apply (induct x)
```
```   203       apply (induct_tac n)
```
```   204       apply (simp)
```
```   205       apply (simp)
```
```   206       apply (induct_tac n)
```
```   207       apply (simp add: neg1)
```
```   208     proof -
```
```   209       fix n :: nat
```
```   210       assume rn: "(real_is_int (of_int (- (int (Suc n)))))"
```
```   211       have s: "-(int (Suc (Suc n))) = -1 + - (int (Suc n))" by simp
```
```   212       show "real_is_int (of_int (- (int (Suc (Suc n)))))"
```
```   213         apply (simp only: s of_int_add)
```
```   214         apply (rule real_is_int_add)
```
```   215         apply (simp add: neg1)
```
```   216         apply (simp only: rn)
```
```   217         done
```
```   218     qed
```
```   219   }
```
```   220   note Abs_Bin = this
```
```   221   {
```
```   222     fix x :: int
```
```   223     have "? u. x = u"
```
```   224       apply (rule exI[where x = "x"])
```
```   225       apply (simp)
```
```   226       done
```
```   227   }
```
```   228   then obtain u::int where "x = u" by auto
```
```   229   with Abs_Bin show ?thesis by auto
```
```   230 qed
```
```   231
```
```   232 lemma int_of_real_0[simp]: "int_of_real (0::real) = (0::int)"
```
```   233 by (simp add: int_of_real_def)
```
```   234
```
```   235 lemma int_of_real_1[simp]: "int_of_real (1::real) = (1::int)"
```
```   236 proof -
```
```   237   have 1: "(1::real) = real (1::int)" by auto
```
```   238   show ?thesis by (simp only: 1 int_of_real_real)
```
```   239 qed
```
```   240
```
```   241 lemma int_of_real_number_of[simp]: "int_of_real (number_of b) = number_of b"
```
```   242 proof -
```
```   243   have "real_is_int (number_of b)" by simp
```
```   244   then have uu: "?! u::int. number_of b = real u" by (auto simp add: real_is_int_rep)
```
```   245   then obtain u::int where u:"number_of b = real u" by auto
```
```   246   have "number_of b = real ((number_of b)::int)"
```
```   247     by (simp add: number_of_eq real_of_int_def)
```
```   248   have ub: "number_of b = real ((number_of b)::int)"
```
```   249     by (simp add: number_of_eq real_of_int_def)
```
```   250   from uu u ub have unb: "u = number_of b"
```
```   251     by blast
```
```   252   have "int_of_real (number_of b) = u" by (simp add: u)
```
```   253   with unb show ?thesis by simp
```
```   254 qed
```
```   255
```
```   256 lemma float_transfer_even: "even a \<Longrightarrow> float (a, b) = float (a div 2, b+1)"
```
```   257   apply (subst float_transfer[where a="a" and b="b" and c="-1", simplified])
```
```   258   apply (simp_all add: pow2_def even_def real_is_int_def ring_eq_simps)
```
```   259   apply (auto)
```
```   260 proof -
```
```   261   fix q::int
```
```   262   have a:"b - (-1\<Colon>int) = (1\<Colon>int) + b" by arith
```
```   263   show "(float (q, (b - (-1\<Colon>int)))) = (float (q, ((1\<Colon>int) + b)))"
```
```   264     by (simp add: a)
```
```   265 qed
```
```   266
```
```   267 consts
```
```   268   norm_float :: "int*int \<Rightarrow> int*int"
```
```   269
```
```   270 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
```
```   271 apply (subst split_div, auto)
```
```   272 apply (subst split_zdiv, auto)
```
```   273 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
```
```   274 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
```
```   275 done
```
```   276
```
```   277 lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
```
```   278 apply (subst split_mod, auto)
```
```   279 apply (subst split_zmod, auto)
```
```   280 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
```
```   281 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
```
```   282 done
```
```   283
```
```   284 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
```
```   285 by arith
```
```   286
```
```   287 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>"
```
```   288 apply (auto)
```
```   289 apply (rule abs_div_2_less)
```
```   290 apply (auto)
```
```   291 done
```
```   292
```
```   293 ML {* simp_depth_limit := 2 *}
```
```   294 recdef norm_float "measure (% (a,b). nat (abs a))"
```
```   295   "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)))"
```
```   296 (hints simp: terminating_norm_float)
```
```   297 ML {* simp_depth_limit := 1000 *}
```
```   298
```
```   299 lemma norm_float: "float x = float (norm_float x)"
```
```   300 proof -
```
```   301   {
```
```   302     fix a b :: int
```
```   303     have norm_float_pair: "float (a,b) = float (norm_float (a,b))"
```
```   304     proof (induct a b rule: norm_float.induct)
```
```   305       case (1 u v)
```
```   306       show ?case
```
```   307       proof cases
```
```   308         assume u: "u \<noteq> 0 \<and> even u"
```
```   309         with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
```
```   310         with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
```
```   311         then show ?thesis
```
```   312           apply (subst norm_float.simps)
```
```   313           apply (simp add: ind)
```
```   314           done
```
```   315       next
```
```   316         assume "~(u \<noteq> 0 \<and> even u)"
```
```   317         then show ?thesis
```
```   318           by (simp add: prems float_def)
```
```   319       qed
```
```   320     qed
```
```   321   }
```
```   322   note helper = this
```
```   323   have "? a b. x = (a,b)" by auto
```
```   324   then obtain a b where "x = (a, b)" by blast
```
```   325   then show ?thesis by (simp only: helper)
```
```   326 qed
```
```   327
```
```   328 lemma pow2_int: "pow2 (int n) = 2^n"
```
```   329   by (simp add: pow2_def)
```
```   330
```
```   331 lemma float_add:
```
```   332   "float (a1, e1) + float (a2, e2) =
```
```   333   (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
```
```   334   else float (a1*2^(nat (e1-e2))+a2, e2))"
```
```   335   apply (simp add: float_def ring_eq_simps)
```
```   336   apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
```
```   337   done
```
```   338
```
```   339 lemma float_mult:
```
```   340   "float (a1, e1) * float (a2, e2) =
```
```   341   (float (a1 * a2, e1 + e2))"
```
```   342   by (simp add: float_def pow2_add)
```
```   343
```
```   344 lemma float_minus:
```
```   345   "- (float (a,b)) = float (-a, b)"
```
```   346   by (simp add: float_def)
```
```   347
```
```   348 lemma zero_less_pow2:
```
```   349   "0 < pow2 x"
```
```   350 proof -
```
```   351   {
```
```   352     fix y
```
```   353     have "0 <= y \<Longrightarrow> 0 < pow2 y"
```
```   354       by (induct y, induct_tac n, simp_all add: pow2_add)
```
```   355   }
```
```   356   note helper=this
```
```   357   show ?thesis
```
```   358     apply (case_tac "0 <= x")
```
```   359     apply (simp add: helper)
```
```   360     apply (subst pow2_neg)
```
```   361     apply (simp add: helper)
```
```   362     done
```
```   363 qed
```
```   364
```
```   365 lemma zero_le_float:
```
```   366   "(0 <= float (a,b)) = (0 <= a)"
```
```   367   apply (auto simp add: float_def)
```
```   368   apply (auto simp add: zero_le_mult_iff zero_less_pow2)
```
```   369   apply (insert zero_less_pow2[of b])
```
```   370   apply (simp_all)
```
```   371   done
```
```   372
```
```   373 lemma float_le_zero:
```
```   374   "(float (a,b) <= 0) = (a <= 0)"
```
```   375   apply (auto simp add: float_def)
```
```   376   apply (auto simp add: mult_le_0_iff)
```
```   377   apply (insert zero_less_pow2[of b])
```
```   378   apply auto
```
```   379   done
```
```   380
```
```   381 lemma float_abs:
```
```   382   "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
```
```   383   apply (auto simp add: abs_if)
```
```   384   apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
```
```   385   done
```
```   386
```
```   387 lemma float_zero:
```
```   388   "float (0, b) = 0"
```
```   389   by (simp add: float_def)
```
```   390
```
```   391 lemma float_pprt:
```
```   392   "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
```
```   393   by (auto simp add: zero_le_float float_le_zero float_zero)
```
```   394
```
```   395 lemma float_nprt:
```
```   396   "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
```
```   397   by (auto simp add: zero_le_float float_le_zero float_zero)
```
```   398
```
```   399 lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
```
```   400   by auto
```
```   401
```
```   402 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
```
```   403   by simp
```
```   404
```
```   405 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
```
```   406   by simp
```
```   407
```
```   408 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
```
```   409   by simp
```
```   410
```
```   411 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
```
```   412   by simp
```
```   413
```
```   414 lemma int_pow_0: "(a::int)^(Numeral0) = 1"
```
```   415   by simp
```
```   416
```
```   417 lemma int_pow_1: "(a::int)^(Numeral1) = a"
```
```   418   by simp
```
```   419
```
```   420 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
```
```   421   by simp
```
```   422
```
```   423 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
```
```   424   by simp
```
```   425
```
```   426 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
```
```   427   by simp
```
```   428
```
```   429 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
```
```   430   by simp
```
```   431
```
```   432 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
```
```   433   by simp
```
```   434
```
```   435 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
```
```   436 proof -
```
```   437   have 1:"((-1)::nat) = 0"
```
```   438     by simp
```
```   439   show ?thesis by (simp add: 1)
```
```   440 qed
```
```   441
```
```   442 lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
```
```   443   by simp
```
```   444
```
```   445 lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
```
```   446   by simp
```
```   447
```
```   448 lemma lift_bool: "x \<Longrightarrow> x=True"
```
```   449   by simp
```
```   450
```
```   451 lemma nlift_bool: "~x \<Longrightarrow> x=False"
```
```   452   by simp
```
```   453
```
```   454 lemma not_false_eq_true: "(~ False) = True" by simp
```
```   455
```
```   456 lemma not_true_eq_false: "(~ True) = False" by simp
```
```   457
```
```   458 lemmas binarith =
```
```   459   Pls_0_eq Min_1_eq
```
```   460   pred_Pls pred_Min pred_1 pred_0
```
```   461   succ_Pls succ_Min succ_1 succ_0
```
```   462   add_Pls add_Min add_BIT_0 add_BIT_10
```
```   463   add_BIT_11 minus_Pls minus_Min minus_1
```
```   464   minus_0 mult_Pls mult_Min mult_num1 mult_num0
```
```   465   add_Pls_right add_Min_right
```
```   466
```
```   467 lemma int_eq_number_of_eq:
```
```   468   "(((number_of v)::int)=(number_of w)) = iszero ((number_of (v + uminus w))::int)"
```
```   469   by simp
```
```   470
```
```   471 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
```
```   472   by (simp only: iszero_number_of_Pls)
```
```   473
```
```   474 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
```
```   475   by simp
```
```   476
```
```   477 lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
```
```   478   by simp
```
```   479
```
```   480 lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
```
```   481   by simp
```
```   482
```
```   483 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (x + (uminus y)))::int)"
```
```   484   by simp
```
```   485
```
```   486 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
```
```   487   by simp
```
```   488
```
```   489 lemma int_neg_number_of_Min: "neg (-1::int)"
```
```   490   by simp
```
```   491
```
```   492 lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
```
```   493   by simp
```
```   494
```
```   495 lemma int_le_number_of_eq: "(((number_of x)::int) \<le> number_of y) = (\<not> neg ((number_of (y + (uminus x)))::int))"
```
```   496   by simp
```
```   497
```
```   498 lemmas intarithrel =
```
```   499   int_eq_number_of_eq
```
```   500   lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0
```
```   501   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]
```
```   502   int_neg_number_of_BIT int_le_number_of_eq
```
```   503
```
```   504 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (v + w)"
```
```   505   by simp
```
```   506
```
```   507 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (v + (uminus w))"
```
```   508   by simp
```
```   509
```
```   510 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (v * w)"
```
```   511   by simp
```
```   512
```
```   513 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (uminus v)"
```
```   514   by simp
```
```   515
```
```   516 lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
```
```   517
```
```   518 lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
```
```   519
```
```   520 lemmas powerarith = nat_number_of zpower_number_of_even
```
```   521   zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
```
```   522   zpower_Pls zpower_Min
```
```   523
```
```   524 lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float float_pprt float_nprt
```
```   525
```
```   526 (* for use with the compute oracle *)
```
```   527 lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
```
```   528
```
```   529 use "float.ML";
```
```   530
```
```   531 end
```