src/HOL/Real/Float.thy
 author wenzelm Tue Jul 19 17:28:37 2005 +0200 (2005-07-19) changeset 16890 c4e5afaba440 parent 16782 b214f21ae396 child 19765 dfe940911617 permissions -rw-r--r--
```     1 (*  Title: HOL/Real/Float.thy
```
```     2     ID:    \$Id\$
```
```     3     Author: Steven Obua
```
```     4 *)
```
```     5
```
```     6 theory Float imports Real begin
```
```     7
```
```     8 constdefs
```
```     9   pow2 :: "int \<Rightarrow> real"
```
```    10   "pow2 a == if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a))))"
```
```    11   float :: "int * int \<Rightarrow> real"
```
```    12   "float x == (real (fst x)) * (pow2 (snd x))"
```
```    13
```
```    14 lemma pow2_0[simp]: "pow2 0 = 1"
```
```    15 by (simp add: pow2_def)
```
```    16
```
```    17 lemma pow2_1[simp]: "pow2 1 = 2"
```
```    18 by (simp add: pow2_def)
```
```    19
```
```    20 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))"
```
```    21 by (simp add: pow2_def)
```
```    22
```
```    23 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
```
```    24 proof -
```
```    25   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
```
```    26   have g: "! a b. a - -1 = a + (1::int)" by arith
```
```    27   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
```
```    28     apply (auto, induct_tac n)
```
```    29     apply (simp_all add: pow2_def)
```
```    30     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
```
```    31     apply (auto simp add: h)
```
```    32     apply arith
```
```    33     done
```
```    34   show ?thesis
```
```    35   proof (induct a)
```
```    36     case (1 n)
```
```    37     from pos show ?case by (simp add: ring_eq_simps)
```
```    38   next
```
```    39     case (2 n)
```
```    40     show ?case
```
```    41       apply (auto)
```
```    42       apply (subst pow2_neg[of "- int n"])
```
```    43       apply (subst pow2_neg[of "-1 - int n"])
```
```    44       apply (auto simp add: g pos)
```
```    45       done
```
```    46   qed
```
```    47 qed
```
```    48
```
```    49 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
```
```    50 proof (induct b)
```
```    51   case (1 n)
```
```    52   show ?case
```
```    53   proof (induct n)
```
```    54     case 0
```
```    55     show ?case by simp
```
```    56   next
```
```    57     case (Suc m)
```
```    58     show ?case by (auto simp add: ring_eq_simps pow2_add1 prems)
```
```    59   qed
```
```    60 next
```
```    61   case (2 n)
```
```    62   show ?case
```
```    63   proof (induct n)
```
```    64     case 0
```
```    65     show ?case
```
```    66       apply (auto)
```
```    67       apply (subst pow2_neg[of "a + -1"])
```
```    68       apply (subst pow2_neg[of "-1"])
```
```    69       apply (simp)
```
```    70       apply (insert pow2_add1[of "-a"])
```
```    71       apply (simp add: ring_eq_simps)
```
```    72       apply (subst pow2_neg[of "-a"])
```
```    73       apply (simp)
```
```    74       done
```
```    75     case (Suc m)
```
```    76     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
```
```    77     have b: "int m - -2 = 1 + (int m + 1)" by arith
```
```    78     show ?case
```
```    79       apply (auto)
```
```    80       apply (subst pow2_neg[of "a + (-2 - int m)"])
```
```    81       apply (subst pow2_neg[of "-2 - int m"])
```
```    82       apply (auto simp add: ring_eq_simps)
```
```    83       apply (subst a)
```
```    84       apply (subst b)
```
```    85       apply (simp only: pow2_add1)
```
```    86       apply (subst pow2_neg[of "int m - a + 1"])
```
```    87       apply (subst pow2_neg[of "int m + 1"])
```
```    88       apply auto
```
```    89       apply (insert prems)
```
```    90       apply (auto simp add: ring_eq_simps)
```
```    91       done
```
```    92   qed
```
```    93 qed
```
```    94
```
```    95 lemma "float (a, e) + float (b, e) = float (a + b, e)"
```
```    96 by (simp add: float_def ring_eq_simps)
```
```    97
```
```    98 constdefs
```
```    99   int_of_real :: "real \<Rightarrow> int"
```
```   100   "int_of_real x == SOME y. real y = x"
```
```   101   real_is_int :: "real \<Rightarrow> bool"
```
```   102   "real_is_int x == ? (u::int). x = real u"
```
```   103
```
```   104 lemma real_is_int_def2: "real_is_int x = (x = real (int_of_real x))"
```
```   105 by (auto simp add: real_is_int_def int_of_real_def)
```
```   106
```
```   107 lemma float_transfer: "real_is_int ((real a)*(pow2 c)) \<Longrightarrow> float (a, b) = float (int_of_real ((real a)*(pow2 c)), b - c)"
```
```   108 by (simp add: float_def real_is_int_def2 pow2_add[symmetric])
```
```   109
```
```   110 lemma pow2_int: "pow2 (int c) = (2::real)^c"
```
```   111 by (simp add: pow2_def)
```
```   112
```
```   113 lemma float_transfer_nat: "float (a, b) = float (a * 2^c, b - int c)"
```
```   114 by (simp add: float_def pow2_int[symmetric] pow2_add[symmetric])
```
```   115
```
```   116 lemma real_is_int_real[simp]: "real_is_int (real (x::int))"
```
```   117 by (auto simp add: real_is_int_def int_of_real_def)
```
```   118
```
```   119 lemma int_of_real_real[simp]: "int_of_real (real x) = x"
```
```   120 by (simp add: int_of_real_def)
```
```   121
```
```   122 lemma real_int_of_real[simp]: "real_is_int x \<Longrightarrow> real (int_of_real x) = x"
```
```   123 by (auto simp add: int_of_real_def real_is_int_def)
```
```   124
```
```   125 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)"
```
```   126 by (auto simp add: int_of_real_def real_is_int_def)
```
```   127
```
```   128 lemma real_is_int_add[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a+b)"
```
```   129 apply (subst real_is_int_def2)
```
```   130 apply (simp add: real_is_int_add_int_of_real real_int_of_real)
```
```   131 done
```
```   132
```
```   133 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)"
```
```   134 by (auto simp add: int_of_real_def real_is_int_def)
```
```   135
```
```   136 lemma real_is_int_sub[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a-b)"
```
```   137 apply (subst real_is_int_def2)
```
```   138 apply (simp add: int_of_real_sub real_int_of_real)
```
```   139 done
```
```   140
```
```   141 lemma real_is_int_rep: "real_is_int x \<Longrightarrow> ?! (a::int). real a = x"
```
```   142 by (auto simp add: real_is_int_def)
```
```   143
```
```   144 lemma int_of_real_mult:
```
```   145   assumes "real_is_int a" "real_is_int b"
```
```   146   shows "(int_of_real (a*b)) = (int_of_real a) * (int_of_real b)"
```
```   147 proof -
```
```   148   from prems have a: "?! (a'::int). real a' = a" by (rule_tac real_is_int_rep, auto)
```
```   149   from prems have b: "?! (b'::int). real b' = b" by (rule_tac real_is_int_rep, auto)
```
```   150   from a obtain a'::int where a':"a = real a'" by auto
```
```   151   from b obtain b'::int where b':"b = real b'" by auto
```
```   152   have r: "real a' * real b' = real (a' * b')" by auto
```
```   153   show ?thesis
```
```   154     apply (simp add: a' b')
```
```   155     apply (subst r)
```
```   156     apply (simp only: int_of_real_real)
```
```   157     done
```
```   158 qed
```
```   159
```
```   160 lemma real_is_int_mult[simp]: "real_is_int a \<Longrightarrow> real_is_int b \<Longrightarrow> real_is_int (a*b)"
```
```   161 apply (subst real_is_int_def2)
```
```   162 apply (simp add: int_of_real_mult)
```
```   163 done
```
```   164
```
```   165 lemma real_is_int_0[simp]: "real_is_int (0::real)"
```
```   166 by (simp add: real_is_int_def int_of_real_def)
```
```   167
```
```   168 lemma real_is_int_1[simp]: "real_is_int (1::real)"
```
```   169 proof -
```
```   170   have "real_is_int (1::real) = real_is_int(real (1::int))" by auto
```
```   171   also have "\<dots> = True" by (simp only: real_is_int_real)
```
```   172   ultimately show ?thesis by auto
```
```   173 qed
```
```   174
```
```   175 lemma real_is_int_n1: "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_number_of[simp]: "real_is_int ((number_of::bin\<Rightarrow>real) x)"
```
```   183 proof -
```
```   184   have neg1: "real_is_int (-1::real)"
```
```   185   proof -
```
```   186     have "real_is_int (-1::real) = real_is_int(real (-1::int))" by auto
```
```   187     also have "\<dots> = True" by (simp only: real_is_int_real)
```
```   188     ultimately show ?thesis by auto
```
```   189   qed
```
```   190
```
```   191   {
```
```   192     fix x::int
```
```   193     have "!! y. real_is_int ((number_of::bin\<Rightarrow>real) (Abs_Bin x))"
```
```   194       apply (simp add: number_of_eq)
```
```   195       apply (subst Abs_Bin_inverse)
```
```   196       apply (simp add: Bin_def)
```
```   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 :: bin
```
```   218     have "? u. x = Abs_Bin u"
```
```   219       apply (rule exI[where x = "Rep_Bin x"])
```
```   220       apply (simp add: Rep_Bin_inverse)
```
```   221       done
```
```   222   }
```
```   223   then obtain u::int where "x = Abs_Bin 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 ring_eq_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 consts
```
```   263   norm_float :: "int*int \<Rightarrow> int*int"
```
```   264
```
```   265 lemma int_div_zdiv: "int (a div b) = (int a) div (int b)"
```
```   266 apply (subst split_div, auto)
```
```   267 apply (subst split_zdiv, auto)
```
```   268 apply (rule_tac a="int (b * i) + int j" and b="int b" and r="int j" and r'=ja in IntDiv.unique_quotient)
```
```   269 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
```
```   270 done
```
```   271
```
```   272 lemma int_mod_zmod: "int (a mod b) = (int a) mod (int b)"
```
```   273 apply (subst split_mod, auto)
```
```   274 apply (subst split_zmod, auto)
```
```   275 apply (rule_tac a="int (b * i) + int j" and b="int b" and q="int i" and q'=ia in IntDiv.unique_remainder)
```
```   276 apply (auto simp add: IntDiv.quorem_def int_eq_of_nat)
```
```   277 done
```
```   278
```
```   279 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
```
```   280 by arith
```
```   281
```
```   282 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>"
```
```   283 apply (auto)
```
```   284 apply (rule abs_div_2_less)
```
```   285 apply (auto)
```
```   286 done
```
```   287
```
```   288 ML {* simp_depth_limit := 2 *}
```
```   289 recdef norm_float "measure (% (a,b). nat (abs a))"
```
```   290   "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)))"
```
```   291 (hints simp: terminating_norm_float)
```
```   292 ML {* simp_depth_limit := 1000 *}
```
```   293
```
```   294 lemma norm_float: "float x = float (norm_float x)"
```
```   295 proof -
```
```   296   {
```
```   297     fix a b :: int
```
```   298     have norm_float_pair: "float (a,b) = float (norm_float (a,b))"
```
```   299     proof (induct a b rule: norm_float.induct)
```
```   300       case (1 u v)
```
```   301       show ?case
```
```   302       proof cases
```
```   303 	assume u: "u \<noteq> 0 \<and> even u"
```
```   304 	with prems have ind: "float (u div 2, v + 1) = float (norm_float (u div 2, v + 1))" by auto
```
```   305 	with u have "float (u,v) = float (u div 2, v+1)" by (simp add: float_transfer_even)
```
```   306 	then show ?thesis
```
```   307 	  apply (subst norm_float.simps)
```
```   308 	  apply (simp add: ind)
```
```   309 	  done
```
```   310       next
```
```   311 	assume "~(u \<noteq> 0 \<and> even u)"
```
```   312 	then show ?thesis
```
```   313 	  by (simp add: prems float_def)
```
```   314       qed
```
```   315     qed
```
```   316   }
```
```   317   note helper = this
```
```   318   have "? a b. x = (a,b)" by auto
```
```   319   then obtain a b where "x = (a, b)" by blast
```
```   320   then show ?thesis by (simp only: helper)
```
```   321 qed
```
```   322
```
```   323 lemma pow2_int: "pow2 (int n) = 2^n"
```
```   324   by (simp add: pow2_def)
```
```   325
```
```   326 lemma float_add:
```
```   327   "float (a1, e1) + float (a2, e2) =
```
```   328   (if e1<=e2 then float (a1+a2*2^(nat(e2-e1)), e1)
```
```   329   else float (a1*2^(nat (e1-e2))+a2, e2))"
```
```   330   apply (simp add: float_def ring_eq_simps)
```
```   331   apply (auto simp add: pow2_int[symmetric] pow2_add[symmetric])
```
```   332   done
```
```   333
```
```   334 lemma float_mult:
```
```   335   "float (a1, e1) * float (a2, e2) =
```
```   336   (float (a1 * a2, e1 + e2))"
```
```   337   by (simp add: float_def pow2_add)
```
```   338
```
```   339 lemma float_minus:
```
```   340   "- (float (a,b)) = float (-a, b)"
```
```   341   by (simp add: float_def)
```
```   342
```
```   343 lemma zero_less_pow2:
```
```   344   "0 < pow2 x"
```
```   345 proof -
```
```   346   {
```
```   347     fix y
```
```   348     have "0 <= y \<Longrightarrow> 0 < pow2 y"
```
```   349       by (induct y, induct_tac n, simp_all add: pow2_add)
```
```   350   }
```
```   351   note helper=this
```
```   352   show ?thesis
```
```   353     apply (case_tac "0 <= x")
```
```   354     apply (simp add: helper)
```
```   355     apply (subst pow2_neg)
```
```   356     apply (simp add: helper)
```
```   357     done
```
```   358 qed
```
```   359
```
```   360 lemma zero_le_float:
```
```   361   "(0 <= float (a,b)) = (0 <= a)"
```
```   362   apply (auto simp add: float_def)
```
```   363   apply (auto simp add: zero_le_mult_iff zero_less_pow2)
```
```   364   apply (insert zero_less_pow2[of b])
```
```   365   apply (simp_all)
```
```   366   done
```
```   367
```
```   368 lemma float_le_zero:
```
```   369   "(float (a,b) <= 0) = (a <= 0)"
```
```   370   apply (auto simp add: float_def)
```
```   371   apply (auto simp add: mult_le_0_iff)
```
```   372   apply (insert zero_less_pow2[of b])
```
```   373   apply auto
```
```   374   done
```
```   375
```
```   376 lemma float_abs:
```
```   377   "abs (float (a,b)) = (if 0 <= a then (float (a,b)) else (float (-a,b)))"
```
```   378   apply (auto simp add: abs_if)
```
```   379   apply (simp_all add: zero_le_float[symmetric, of a b] float_minus)
```
```   380   done
```
```   381
```
```   382 lemma float_zero:
```
```   383   "float (0, b) = 0"
```
```   384   by (simp add: float_def)
```
```   385
```
```   386 lemma float_pprt:
```
```   387   "pprt (float (a, b)) = (if 0 <= a then (float (a,b)) else (float (0, b)))"
```
```   388   by (auto simp add: zero_le_float float_le_zero float_zero)
```
```   389
```
```   390 lemma float_nprt:
```
```   391   "nprt (float (a, b)) = (if 0 <= a then (float (0,b)) else (float (a, b)))"
```
```   392   by (auto simp add: zero_le_float float_le_zero float_zero)
```
```   393
```
```   394 lemma norm_0_1: "(0::_::number_ring) = Numeral0 & (1::_::number_ring) = Numeral1"
```
```   395   by auto
```
```   396
```
```   397 lemma add_left_zero: "0 + a = (a::'a::comm_monoid_add)"
```
```   398   by simp
```
```   399
```
```   400 lemma add_right_zero: "a + 0 = (a::'a::comm_monoid_add)"
```
```   401   by simp
```
```   402
```
```   403 lemma mult_left_one: "1 * a = (a::'a::semiring_1)"
```
```   404   by simp
```
```   405
```
```   406 lemma mult_right_one: "a * 1 = (a::'a::semiring_1)"
```
```   407   by simp
```
```   408
```
```   409 lemma int_pow_0: "(a::int)^(Numeral0) = 1"
```
```   410   by simp
```
```   411
```
```   412 lemma int_pow_1: "(a::int)^(Numeral1) = a"
```
```   413   by simp
```
```   414
```
```   415 lemma zero_eq_Numeral0_nring: "(0::'a::number_ring) = Numeral0"
```
```   416   by simp
```
```   417
```
```   418 lemma one_eq_Numeral1_nring: "(1::'a::number_ring) = Numeral1"
```
```   419   by simp
```
```   420
```
```   421 lemma zero_eq_Numeral0_nat: "(0::nat) = Numeral0"
```
```   422   by simp
```
```   423
```
```   424 lemma one_eq_Numeral1_nat: "(1::nat) = Numeral1"
```
```   425   by simp
```
```   426
```
```   427 lemma zpower_Pls: "(z::int)^Numeral0 = Numeral1"
```
```   428   by simp
```
```   429
```
```   430 lemma zpower_Min: "(z::int)^((-1)::nat) = Numeral1"
```
```   431 proof -
```
```   432   have 1:"((-1)::nat) = 0"
```
```   433     by simp
```
```   434   show ?thesis by (simp add: 1)
```
```   435 qed
```
```   436
```
```   437 lemma fst_cong: "a=a' \<Longrightarrow> fst (a,b) = fst (a',b)"
```
```   438   by simp
```
```   439
```
```   440 lemma snd_cong: "b=b' \<Longrightarrow> snd (a,b) = snd (a,b')"
```
```   441   by simp
```
```   442
```
```   443 lemma lift_bool: "x \<Longrightarrow> x=True"
```
```   444   by simp
```
```   445
```
```   446 lemma nlift_bool: "~x \<Longrightarrow> x=False"
```
```   447   by simp
```
```   448
```
```   449 lemma not_false_eq_true: "(~ False) = True" by simp
```
```   450
```
```   451 lemma not_true_eq_false: "(~ True) = False" by simp
```
```   452
```
```   453
```
```   454 lemmas binarith =
```
```   455   Pls_0_eq Min_1_eq
```
```   456   bin_pred_Pls bin_pred_Min bin_pred_1 bin_pred_0
```
```   457   bin_succ_Pls bin_succ_Min bin_succ_1 bin_succ_0
```
```   458   bin_add_Pls bin_add_Min bin_add_BIT_0 bin_add_BIT_10
```
```   459   bin_add_BIT_11 bin_minus_Pls bin_minus_Min bin_minus_1
```
```   460   bin_minus_0 bin_mult_Pls bin_mult_Min bin_mult_1 bin_mult_0
```
```   461   bin_add_Pls_right bin_add_Min_right
```
```   462
```
```   463 lemma int_eq_number_of_eq: "(((number_of v)::int)=(number_of w)) = iszero ((number_of (bin_add v (bin_minus w)))::int)"
```
```   464   by simp
```
```   465
```
```   466 lemma int_iszero_number_of_Pls: "iszero (Numeral0::int)"
```
```   467   by (simp only: iszero_number_of_Pls)
```
```   468
```
```   469 lemma int_nonzero_number_of_Min: "~(iszero ((-1)::int))"
```
```   470   by simp
```
```   471
```
```   472 lemma int_iszero_number_of_0: "iszero ((number_of (w BIT bit.B0))::int) = iszero ((number_of w)::int)"
```
```   473   by simp
```
```   474
```
```   475 lemma int_iszero_number_of_1: "\<not> iszero ((number_of (w BIT bit.B1))::int)"
```
```   476   by simp
```
```   477
```
```   478 lemma int_less_number_of_eq_neg: "(((number_of x)::int) < number_of y) = neg ((number_of (bin_add x (bin_minus y)))::int)"
```
```   479   by simp
```
```   480
```
```   481 lemma int_not_neg_number_of_Pls: "\<not> (neg (Numeral0::int))"
```
```   482   by simp
```
```   483
```
```   484 lemma int_neg_number_of_Min: "neg (-1::int)"
```
```   485   by simp
```
```   486
```
```   487 lemma int_neg_number_of_BIT: "neg ((number_of (w BIT x))::int) = neg ((number_of w)::int)"
```
```   488   by simp
```
```   489
```
```   490 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))"
```
```   491   by simp
```
```   492
```
```   493 lemmas intarithrel =
```
```   494   int_eq_number_of_eq
```
```   495   lift_bool[OF int_iszero_number_of_Pls] nlift_bool[OF int_nonzero_number_of_Min] int_iszero_number_of_0
```
```   496   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]
```
```   497   int_neg_number_of_BIT int_le_number_of_eq
```
```   498
```
```   499 lemma int_number_of_add_sym: "((number_of v)::int) + number_of w = number_of (bin_add v w)"
```
```   500   by simp
```
```   501
```
```   502 lemma int_number_of_diff_sym: "((number_of v)::int) - number_of w = number_of (bin_add v (bin_minus w))"
```
```   503   by simp
```
```   504
```
```   505 lemma int_number_of_mult_sym: "((number_of v)::int) * number_of w = number_of (bin_mult v w)"
```
```   506   by simp
```
```   507
```
```   508 lemma int_number_of_minus_sym: "- ((number_of v)::int) = number_of (bin_minus v)"
```
```   509   by simp
```
```   510
```
```   511 lemmas intarith = int_number_of_add_sym int_number_of_minus_sym int_number_of_diff_sym int_number_of_mult_sym
```
```   512
```
```   513 lemmas natarith = add_nat_number_of diff_nat_number_of mult_nat_number_of eq_nat_number_of less_nat_number_of
```
```   514
```
```   515 lemmas powerarith = nat_number_of zpower_number_of_even
```
```   516   zpower_number_of_odd[simplified zero_eq_Numeral0_nring one_eq_Numeral1_nring]
```
```   517   zpower_Pls zpower_Min
```
```   518
```
```   519 lemmas floatarith[simplified norm_0_1] = float_add float_mult float_minus float_abs zero_le_float float_pprt float_nprt
```
```   520
```
```   521 (* for use with the compute oracle *)
```
```   522 lemmas arith = binarith intarith intarithrel natarith powerarith floatarith not_false_eq_true not_true_eq_false
```
```   523
```
```   524 end
```
```   525
```