src/HOL/Library/Float.thy
author haftmann
Wed Apr 22 19:09:21 2009 +0200 (2009-04-22)
changeset 30960 fec1a04b7220
parent 30242 aea5d7fa7ef5
child 31021 53642251a04f
permissions -rw-r--r--
power operation defined generic
     1 (*  Title:      HOL/Library/Float.thy
     2     Author:     Steven Obua 2008
     3     Author:     Johannes Hoelzl, TU Muenchen <hoelzl@in.tum.de> 2008 / 2009
     4 *)
     5 
     6 header {* Floating-Point Numbers *}
     7 
     8 theory Float
     9 imports Complex_Main
    10 begin
    11 
    12 definition
    13   pow2 :: "int \<Rightarrow> real" where
    14   [simp]: "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"
    15 
    16 datatype float = Float int int
    17 
    18 primrec Ifloat :: "float \<Rightarrow> real" where
    19   "Ifloat (Float a b) = real a * pow2 b"
    20 
    21 instantiation float :: zero begin
    22 definition zero_float where "0 = Float 0 0" 
    23 instance ..
    24 end
    25 
    26 instantiation float :: one begin
    27 definition one_float where "1 = Float 1 0"
    28 instance ..
    29 end
    30 
    31 instantiation float :: number begin
    32 definition number_of_float where "number_of n = Float n 0"
    33 instance ..
    34 end
    35 
    36 primrec mantissa :: "float \<Rightarrow> int" where
    37   "mantissa (Float a b) = a"
    38 
    39 primrec scale :: "float \<Rightarrow> int" where
    40   "scale (Float a b) = b"
    41 
    42 lemma Ifloat_neg_exp: "e < 0 \<Longrightarrow> Ifloat (Float m e) = real m * inverse (2^nat (-e))" by auto
    43 lemma Ifloat_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> Ifloat (Float m e) = real m * inverse (2^nat (-e))" by auto
    44 lemma Ifloat_ge0_exp: "0 \<le> e \<Longrightarrow> Ifloat (Float m e) = real m * (2^nat e)" by auto
    45 
    46 lemma Float_num[simp]: shows
    47    "Ifloat (Float 1 0) = 1" and "Ifloat (Float 1 1) = 2" and "Ifloat (Float 1 2) = 4" and 
    48    "Ifloat (Float 1 -1) = 1/2" and "Ifloat (Float 1 -2) = 1/4" and "Ifloat (Float 1 -3) = 1/8" and
    49    "Ifloat (Float -1 0) = -1" and "Ifloat (Float (number_of n) 0) = number_of n"
    50   by auto
    51 
    52 lemma pow2_0[simp]: "pow2 0 = 1" by simp
    53 lemma pow2_1[simp]: "pow2 1 = 2" by simp
    54 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" by simp
    55 
    56 declare pow2_def[simp del]
    57 
    58 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"
    59 proof -
    60   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith
    61   have g: "! a b. a - -1 = a + (1::int)" by arith
    62   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"
    63     apply (auto, induct_tac n)
    64     apply (simp_all add: pow2_def)
    65     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])
    66     by (auto simp add: h)
    67   show ?thesis
    68   proof (induct a)
    69     case (1 n)
    70     from pos show ?case by (simp add: algebra_simps)
    71   next
    72     case (2 n)
    73     show ?case
    74       apply (auto)
    75       apply (subst pow2_neg[of "- int n"])
    76       apply (subst pow2_neg[of "-1 - int n"])
    77       apply (auto simp add: g pos)
    78       done
    79   qed
    80 qed
    81 
    82 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"
    83 proof (induct b)
    84   case (1 n)
    85   show ?case
    86   proof (induct n)
    87     case 0
    88     show ?case by simp
    89   next
    90     case (Suc m)
    91     show ?case by (auto simp add: algebra_simps pow2_add1 prems)
    92   qed
    93 next
    94   case (2 n)
    95   show ?case
    96   proof (induct n)
    97     case 0
    98     show ?case
    99       apply (auto)
   100       apply (subst pow2_neg[of "a + -1"])
   101       apply (subst pow2_neg[of "-1"])
   102       apply (simp)
   103       apply (insert pow2_add1[of "-a"])
   104       apply (simp add: algebra_simps)
   105       apply (subst pow2_neg[of "-a"])
   106       apply (simp)
   107       done
   108     case (Suc m)
   109     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith
   110     have b: "int m - -2 = 1 + (int m + 1)" by arith
   111     show ?case
   112       apply (auto)
   113       apply (subst pow2_neg[of "a + (-2 - int m)"])
   114       apply (subst pow2_neg[of "-2 - int m"])
   115       apply (auto simp add: algebra_simps)
   116       apply (subst a)
   117       apply (subst b)
   118       apply (simp only: pow2_add1)
   119       apply (subst pow2_neg[of "int m - a + 1"])
   120       apply (subst pow2_neg[of "int m + 1"])
   121       apply auto
   122       apply (insert prems)
   123       apply (auto simp add: algebra_simps)
   124       done
   125   qed
   126 qed
   127 
   128 lemma float_components[simp]: "Float (mantissa f) (scale f) = f" by (cases f, auto)
   129 
   130 lemma float_split: "\<exists> a b. x = Float a b" by (cases x, auto)
   131 
   132 lemma float_split2: "(\<forall> a b. x \<noteq> Float a b) = False" by (auto simp add: float_split)
   133 
   134 lemma float_zero[simp]: "Ifloat (Float 0 e) = 0" by simp
   135 
   136 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"
   137 by arith
   138 
   139 function normfloat :: "float \<Rightarrow> float" where
   140 "normfloat (Float a b) = (if a \<noteq> 0 \<and> even a then normfloat (Float (a div 2) (b+1)) else if a=0 then Float 0 0 else Float a b)"
   141 by pat_completeness auto
   142 termination by (relation "measure (nat o abs o mantissa)") (auto intro: abs_div_2_less)
   143 declare normfloat.simps[simp del]
   144 
   145 theorem normfloat[symmetric, simp]: "Ifloat f = Ifloat (normfloat f)"
   146 proof (induct f rule: normfloat.induct)
   147   case (1 a b)
   148   have real2: "2 = real (2::int)"
   149     by auto
   150   show ?case
   151     apply (subst normfloat.simps)
   152     apply (auto simp add: float_zero)
   153     apply (subst 1[symmetric])
   154     apply (auto simp add: pow2_add even_def)
   155     done
   156 qed
   157 
   158 lemma pow2_neq_zero[simp]: "pow2 x \<noteq> 0"
   159   by (auto simp add: pow2_def)
   160 
   161 lemma pow2_int: "pow2 (int c) = 2^c"
   162 by (simp add: pow2_def)
   163 
   164 lemma zero_less_pow2[simp]:
   165   "0 < pow2 x"
   166 proof -
   167   {
   168     fix y
   169     have "0 <= y \<Longrightarrow> 0 < pow2 y"
   170       by (induct y, induct_tac n, simp_all add: pow2_add)
   171   }
   172   note helper=this
   173   show ?thesis
   174     apply (case_tac "0 <= x")
   175     apply (simp add: helper)
   176     apply (subst pow2_neg)
   177     apply (simp add: helper)
   178     done
   179 qed
   180 
   181 lemma normfloat_imp_odd_or_zero: "normfloat f = Float a b \<Longrightarrow> odd a \<or> (a = 0 \<and> b = 0)"
   182 proof (induct f rule: normfloat.induct)
   183   case (1 u v)
   184   from 1 have ab: "normfloat (Float u v) = Float a b" by auto
   185   {
   186     assume eu: "even u"
   187     assume z: "u \<noteq> 0"
   188     have "normfloat (Float u v) = normfloat (Float (u div 2) (v + 1))"
   189       apply (subst normfloat.simps)
   190       by (simp add: eu z)
   191     with ab have "normfloat (Float (u div 2) (v + 1)) = Float a b" by simp
   192     with 1 eu z have ?case by auto
   193   }
   194   note case1 = this
   195   {
   196     assume "odd u \<or> u = 0"
   197     then have ou: "\<not> (u \<noteq> 0 \<and> even u)" by auto
   198     have "normfloat (Float u v) = (if u = 0 then Float 0 0 else Float u v)"
   199       apply (subst normfloat.simps)
   200       apply (simp add: ou)
   201       done
   202     with ab have "Float a b = (if u = 0 then Float 0 0 else Float u v)" by auto
   203     then have ?case
   204       apply (case_tac "u=0")
   205       apply (auto)
   206       by (insert ou, auto)
   207   }
   208   note case2 = this
   209   show ?case
   210     apply (case_tac "odd u \<or> u = 0")
   211     apply (rule case2)
   212     apply simp
   213     apply (rule case1)
   214     apply auto
   215     done
   216 qed
   217 
   218 lemma float_eq_odd_helper: 
   219   assumes odd: "odd a'"
   220   and floateq: "Ifloat (Float a b) = Ifloat (Float a' b')"
   221   shows "b \<le> b'"
   222 proof - 
   223   {
   224     assume bcmp: "b > b'"
   225     from floateq have eq: "real a * pow2 b = real a' * pow2 b'" by simp
   226     {
   227       fix x y z :: real
   228       assume "y \<noteq> 0"
   229       then have "(x * inverse y = z) = (x = z * y)"
   230 	by auto
   231     }
   232     note inverse = this
   233     have eq': "real a * (pow2 (b - b')) = real a'"
   234       apply (subst diff_int_def)
   235       apply (subst pow2_add)
   236       apply (subst pow2_neg[where x = "-b'"])
   237       apply simp
   238       apply (subst mult_assoc[symmetric])
   239       apply (subst inverse)
   240       apply (simp_all add: eq)
   241       done
   242     have "\<exists> z > 0. pow2 (b-b') = 2^z"
   243       apply (rule exI[where x="nat (b - b')"])
   244       apply (auto)
   245       apply (insert bcmp)
   246       apply simp
   247       apply (subst pow2_int[symmetric])
   248       apply auto
   249       done
   250     then obtain z where z: "z > 0 \<and> pow2 (b-b') = 2^z" by auto
   251     with eq' have "real a * 2^z = real a'"
   252       by auto
   253     then have "real a * real ((2::int)^z) = real a'"
   254       by auto
   255     then have "real (a * 2^z) = real a'"
   256       apply (subst real_of_int_mult)
   257       apply simp
   258       done
   259     then have a'_rep: "a * 2^z = a'" by arith
   260     then have "a' = a*2^z" by simp
   261     with z have "even a'" by simp
   262     with odd have False by auto
   263   }
   264   then show ?thesis by arith
   265 qed
   266 
   267 lemma float_eq_odd: 
   268   assumes odd1: "odd a"
   269   and odd2: "odd a'"
   270   and floateq: "Ifloat (Float a b) = Ifloat (Float a' b')"
   271   shows "a = a' \<and> b = b'"
   272 proof -
   273   from 
   274      float_eq_odd_helper[OF odd2 floateq] 
   275      float_eq_odd_helper[OF odd1 floateq[symmetric]]
   276   have beq: "b = b'"  by arith
   277   with floateq show ?thesis by auto
   278 qed
   279 
   280 theorem normfloat_unique:
   281   assumes Ifloat_eq: "Ifloat f = Ifloat g"
   282   shows "normfloat f = normfloat g"
   283 proof - 
   284   from float_split[of "normfloat f"] obtain a b where normf:"normfloat f = Float a b" by auto
   285   from float_split[of "normfloat g"] obtain a' b' where normg:"normfloat g = Float a' b'" by auto
   286   have "Ifloat (normfloat f) = Ifloat (normfloat g)"
   287     by (simp add: Ifloat_eq)
   288   then have float_eq: "Ifloat (Float a b) = Ifloat (Float a' b')"
   289     by (simp add: normf normg)
   290   have ab: "odd a \<or> (a = 0 \<and> b = 0)" by (rule normfloat_imp_odd_or_zero[OF normf])
   291   have ab': "odd a' \<or> (a' = 0 \<and> b' = 0)" by (rule normfloat_imp_odd_or_zero[OF normg])
   292   {
   293     assume odd: "odd a"
   294     then have "a \<noteq> 0" by (simp add: even_def, arith)
   295     with float_eq have "a' \<noteq> 0" by auto
   296     with ab' have "odd a'" by simp
   297     from odd this float_eq have "a = a' \<and> b = b'" by (rule float_eq_odd)
   298   }
   299   note odd_case = this
   300   {
   301     assume even: "even a"
   302     with ab have a0: "a = 0" by simp
   303     with float_eq have a0': "a' = 0" by auto 
   304     from a0 a0' ab ab' have "a = a' \<and> b = b'" by auto
   305   }
   306   note even_case = this
   307   from odd_case even_case show ?thesis
   308     apply (simp add: normf normg)
   309     apply (case_tac "even a")
   310     apply auto
   311     done
   312 qed
   313 
   314 instantiation float :: plus begin
   315 fun plus_float where
   316 [simp del]: "(Float a_m a_e) + (Float b_m b_e) = 
   317      (if a_e \<le> b_e then Float (a_m + b_m * 2^(nat(b_e - a_e))) a_e 
   318                    else Float (a_m * 2^(nat (a_e - b_e)) + b_m) b_e)"
   319 instance ..
   320 end
   321 
   322 instantiation float :: uminus begin
   323 primrec uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"
   324 instance ..
   325 end
   326 
   327 instantiation float :: minus begin
   328 definition minus_float where [simp del]: "(z::float) - w = z + (- w)"
   329 instance ..
   330 end
   331 
   332 instantiation float :: times begin
   333 fun times_float where [simp del]: "(Float a_m a_e) * (Float b_m b_e) = Float (a_m * b_m) (a_e + b_e)"
   334 instance ..
   335 end
   336 
   337 primrec float_pprt :: "float \<Rightarrow> float" where
   338   "float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"
   339 
   340 primrec float_nprt :: "float \<Rightarrow> float" where
   341   "float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))" 
   342 
   343 instantiation float :: ord begin
   344 definition le_float_def: "z \<le> w \<equiv> Ifloat z \<le> Ifloat w"
   345 definition less_float_def: "z < w \<equiv> Ifloat z < Ifloat w"
   346 instance ..
   347 end
   348 
   349 lemma Ifloat_add[simp]: "Ifloat (a + b) = Ifloat a + Ifloat b"
   350   by (cases a, cases b, simp add: algebra_simps plus_float.simps, 
   351       auto simp add: pow2_int[symmetric] pow2_add[symmetric])
   352 
   353 lemma Ifloat_minus[simp]: "Ifloat (- a) = - Ifloat a"
   354   by (cases a, simp add: uminus_float.simps)
   355 
   356 lemma Ifloat_sub[simp]: "Ifloat (a - b) = Ifloat a - Ifloat b" 
   357   by (cases a, cases b, simp add: minus_float_def)
   358 
   359 lemma Ifloat_mult[simp]: "Ifloat (a*b) = Ifloat a * Ifloat b"
   360   by (cases a, cases b, simp add: times_float.simps pow2_add)
   361 
   362 lemma Ifloat_0[simp]: "Ifloat 0 = 0"
   363   by (auto simp add: zero_float_def float_zero)
   364 
   365 lemma Ifloat_1[simp]: "Ifloat 1 = 1"
   366   by (auto simp add: one_float_def)
   367 
   368 lemma zero_le_float:
   369   "(0 <= Ifloat (Float a b)) = (0 <= a)"
   370   apply auto
   371   apply (auto simp add: zero_le_mult_iff)
   372   apply (insert zero_less_pow2[of b])
   373   apply (simp_all)
   374   done
   375 
   376 lemma float_le_zero:
   377   "(Ifloat (Float a b) <= 0) = (a <= 0)"
   378   apply auto
   379   apply (auto simp add: mult_le_0_iff)
   380   apply (insert zero_less_pow2[of b])
   381   apply auto
   382   done
   383 
   384 declare Ifloat.simps[simp del]
   385 
   386 lemma Ifloat_pprt[simp]: "Ifloat (float_pprt a) = pprt (Ifloat a)"
   387   by (cases a, auto simp add: float_pprt.simps zero_le_float float_le_zero float_zero)
   388 
   389 lemma Ifloat_nprt[simp]: "Ifloat (float_nprt a) = nprt (Ifloat a)"
   390   by (cases a,  auto simp add: float_nprt.simps zero_le_float float_le_zero float_zero)
   391 
   392 instance float :: ab_semigroup_add
   393 proof (intro_classes)
   394   fix a b c :: float
   395   show "a + b + c = a + (b + c)"
   396     by (cases a, cases b, cases c, auto simp add: algebra_simps power_add[symmetric] plus_float.simps)
   397 next
   398   fix a b :: float
   399   show "a + b = b + a"
   400     by (cases a, cases b, simp add: plus_float.simps)
   401 qed
   402 
   403 instance float :: comm_monoid_mult
   404 proof (intro_classes)
   405   fix a b c :: float
   406   show "a * b * c = a * (b * c)"
   407     by (cases a, cases b, cases c, simp add: times_float.simps)
   408 next
   409   fix a b :: float
   410   show "a * b = b * a"
   411     by (cases a, cases b, simp add: times_float.simps)
   412 next
   413   fix a :: float
   414   show "1 * a = a"
   415     by (cases a, simp add: times_float.simps one_float_def)
   416 qed
   417 
   418 (* Floats do NOT form a cancel_semigroup_add: *)
   419 lemma "0 + Float 0 1 = 0 + Float 0 2"
   420   by (simp add: plus_float.simps zero_float_def)
   421 
   422 instance float :: comm_semiring
   423 proof (intro_classes)
   424   fix a b c :: float
   425   show "(a + b) * c = a * c + b * c"
   426     by (cases a, cases b, cases c, simp, simp add: plus_float.simps times_float.simps algebra_simps)
   427 qed
   428 
   429 (* Floats do NOT form an order, because "(x < y) = (x <= y & x <> y)" does NOT hold *)
   430 
   431 instance float :: zero_neq_one
   432 proof (intro_classes)
   433   show "(0::float) \<noteq> 1"
   434     by (simp add: zero_float_def one_float_def)
   435 qed
   436 
   437 lemma float_le_simp: "((x::float) \<le> y) = (0 \<le> y - x)"
   438   by (auto simp add: le_float_def)
   439 
   440 lemma float_less_simp: "((x::float) < y) = (0 < y - x)"
   441   by (auto simp add: less_float_def)
   442 
   443 lemma Ifloat_min: "Ifloat (min x y) = min (Ifloat x) (Ifloat y)" unfolding min_def le_float_def by auto
   444 lemma Ifloat_max: "Ifloat (max a b) = max (Ifloat a) (Ifloat b)" unfolding max_def le_float_def by auto
   445 
   446 instance float :: recpower ..
   447 
   448 lemma float_power: "Ifloat (x ^ n) = Ifloat x ^ n"
   449   by (induct n) simp_all
   450 
   451 lemma zero_le_pow2[simp]: "0 \<le> pow2 s"
   452   apply (subgoal_tac "0 < pow2 s")
   453   apply (auto simp only:)
   454   apply auto
   455   done
   456 
   457 lemma pow2_less_0_eq_False[simp]: "(pow2 s < 0) = False"
   458   apply auto
   459   apply (subgoal_tac "0 \<le> pow2 s")
   460   apply simp
   461   apply simp
   462   done
   463 
   464 lemma pow2_le_0_eq_False[simp]: "(pow2 s \<le> 0) = False"
   465   apply auto
   466   apply (subgoal_tac "0 < pow2 s")
   467   apply simp
   468   apply simp
   469   done
   470 
   471 lemma float_pos_m_pos: "0 < Float m e \<Longrightarrow> 0 < m"
   472   unfolding less_float_def Ifloat.simps Ifloat_0 zero_less_mult_iff
   473   by auto
   474 
   475 lemma float_pos_less1_e_neg: assumes "0 < Float m e" and "Float m e < 1" shows "e < 0"
   476 proof -
   477   have "0 < m" using float_pos_m_pos `0 < Float m e` by auto
   478   hence "0 \<le> real m" and "1 \<le> real m" by auto
   479   
   480   show "e < 0"
   481   proof (rule ccontr)
   482     assume "\<not> e < 0" hence "0 \<le> e" by auto
   483     hence "1 \<le> pow2 e" unfolding pow2_def by auto
   484     from mult_mono[OF `1 \<le> real m` this `0 \<le> real m`]
   485     have "1 \<le> Float m e" by (simp add: le_float_def Ifloat.simps)
   486     thus False using `Float m e < 1` unfolding less_float_def le_float_def by auto
   487   qed
   488 qed
   489 
   490 lemma float_less1_mantissa_bound: assumes "0 < Float m e" "Float m e < 1" shows "m < 2^(nat (-e))"
   491 proof -
   492   have "e < 0" using float_pos_less1_e_neg assms by auto
   493   have "\<And>x. (0::real) < 2^x" by auto
   494   have "real m < 2^(nat (-e))" using `Float m e < 1`
   495     unfolding less_float_def Ifloat_neg_exp[OF `e < 0`] Ifloat_1
   496           real_mult_less_iff1[of _ _ 1, OF `0 < 2^(nat (-e))`, symmetric] 
   497           real_mult_assoc by auto
   498   thus ?thesis unfolding real_of_int_less_iff[symmetric] by auto
   499 qed
   500 
   501 function bitlen :: "int \<Rightarrow> int" where
   502 "bitlen 0 = 0" | 
   503 "bitlen -1 = 1" | 
   504 "0 < x \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))" | 
   505 "x < -1 \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))"
   506   apply (case_tac "x = 0 \<or> x = -1 \<or> x < -1 \<or> x > 0")
   507   apply auto
   508   done
   509 termination by (relation "measure (nat o abs)", auto)
   510 
   511 lemma bitlen_ge0: "0 \<le> bitlen x" by (induct x rule: bitlen.induct, auto)
   512 lemma bitlen_ge1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> bitlen x" by (induct x rule: bitlen.induct, auto simp add: bitlen_ge0)
   513 
   514 lemma bitlen_bounds': assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x + 1 \<le> 2^nat (bitlen x)" (is "?P x")
   515   using `0 < x`
   516 proof (induct x rule: bitlen.induct)
   517   fix x
   518   assume "0 < x" and hyp: "0 < x div 2 \<Longrightarrow> ?P (x div 2)" hence "0 \<le> x" and "x \<noteq> 0" by auto
   519   { fix x have "0 \<le> 1 + bitlen x" using bitlen_ge0[of x] by auto } note gt0_pls1 = this
   520 
   521   have "0 < (2::int)" by auto
   522 
   523   show "?P x"
   524   proof (cases "x = 1")
   525     case True show "?P x" unfolding True by auto
   526   next
   527     case False hence "2 \<le> x" using `0 < x` `x \<noteq> 1` by auto
   528     hence "2 div 2 \<le> x div 2" by (rule zdiv_mono1, auto)
   529     hence "0 < x div 2" and "x div 2 \<noteq> 0" by auto
   530     hence bitlen_s1_ge0: "0 \<le> bitlen (x div 2) - 1" using bitlen_ge1[OF `x div 2 \<noteq> 0`] by auto
   531 
   532     { from hyp[OF `0 < x div 2`]
   533       have "2 ^ nat (bitlen (x div 2) - 1) \<le> x div 2" by auto
   534       hence "2 ^ nat (bitlen (x div 2) - 1) * 2 \<le> x div 2 * 2" by (rule mult_right_mono, auto)
   535       also have "\<dots> \<le> x" using `0 < x` by auto
   536       finally have "2^nat (1 + bitlen (x div 2) - 1) \<le> x" unfolding power_Suc2[symmetric] Suc_nat_eq_nat_zadd1[OF bitlen_s1_ge0] by auto
   537     } moreover
   538     { have "x + 1 \<le> x - x mod 2 + 2"
   539       proof -
   540 	have "x mod 2 < 2" using `0 < x` by auto
   541  	hence "x < x - x mod 2 +  2" unfolding algebra_simps by auto
   542 	thus ?thesis by auto
   543       qed
   544       also have "x - x mod 2 + 2 = (x div 2 + 1) * 2" unfolding algebra_simps using `0 < x` zdiv_zmod_equality2[of x 2 0] by auto
   545       also have "\<dots> \<le> 2^nat (bitlen (x div 2)) * 2" using hyp[OF `0 < x div 2`, THEN conjunct2] by (rule mult_right_mono, auto)
   546       also have "\<dots> = 2^(1 + nat (bitlen (x div 2)))" unfolding power_Suc2[symmetric] by auto
   547       finally have "x + 1 \<le> 2^(1 + nat (bitlen (x div 2)))" .
   548     }
   549     ultimately show ?thesis
   550       unfolding bitlen.simps(3)[OF `0 < x`] nat_add_distrib[OF zero_le_one bitlen_ge0]
   551       unfolding add_commute nat_add_distrib[OF zero_le_one gt0_pls1]
   552       by auto
   553   qed
   554 next
   555   fix x :: int assume "x < -1" and "0 < x" hence False by auto
   556   thus "?P x" by auto
   557 qed auto
   558 
   559 lemma bitlen_bounds: assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x < 2^nat (bitlen x)"
   560   using bitlen_bounds'[OF `0<x`] by auto
   561 
   562 lemma bitlen_div: assumes "0 < m" shows "1 \<le> real m / 2^nat (bitlen m - 1)" and "real m / 2^nat (bitlen m - 1) < 2"
   563 proof -
   564   let ?B = "2^nat(bitlen m - 1)"
   565 
   566   have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..
   567   hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto
   568   thus "1 \<le> real m / ?B" by auto
   569 
   570   have "m \<noteq> 0" using assms by auto
   571   have "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto
   572 
   573   have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..
   574   also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
   575   also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto
   576   finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto
   577   hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)
   578   thus "real m / ?B < 2" by auto
   579 qed
   580 
   581 lemma float_gt1_scale: assumes "1 \<le> Float m e"
   582   shows "0 \<le> e + (bitlen m - 1)"
   583 proof (cases "0 \<le> e")
   584   have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
   585   hence "0 < m" using float_pos_m_pos by auto
   586   hence "m \<noteq> 0" by auto
   587   case True with bitlen_ge1[OF `m \<noteq> 0`] show ?thesis by auto
   588 next
   589   have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto
   590   hence "0 < m" using float_pos_m_pos by auto
   591   hence "m \<noteq> 0" and "1 < (2::int)" by auto
   592   case False let ?S = "2^(nat (-e))"
   593   have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def Ifloat_nge0_exp[OF False] by auto
   594   hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
   595   hence "?S \<le> real m" unfolding mult_assoc by auto
   596   hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto
   597   from this bitlen_bounds[OF `0 < m`, THEN conjunct2]
   598   have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)
   599   hence "-e < bitlen m" using False bitlen_ge0 by auto
   600   thus ?thesis by auto
   601 qed
   602 
   603 lemma normalized_float: assumes "m \<noteq> 0" shows "Ifloat (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"
   604 proof (cases "- (bitlen m - 1) = 0")
   605   case True show ?thesis unfolding Ifloat.simps pow2_def using True by auto
   606 next
   607   case False hence P: "\<not> 0 \<le> - (bitlen m - 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto
   608   show ?thesis unfolding Ifloat_nge0_exp[OF P] real_divide_def by auto
   609 qed
   610 
   611 lemma bitlen_Pls: "bitlen (Int.Pls) = Int.Pls" by (subst Pls_def, subst Pls_def, simp)
   612 
   613 lemma bitlen_Min: "bitlen (Int.Min) = Int.Bit1 Int.Pls" by (subst Min_def, simp add: Bit1_def) 
   614 
   615 lemma bitlen_B0: "bitlen (Int.Bit0 b) = (if iszero b then Int.Pls else Int.succ (bitlen b))"
   616   apply (auto simp add: iszero_def succ_def)
   617   apply (simp add: Bit0_def Pls_def)
   618   apply (subst Bit0_def)
   619   apply simp
   620   apply (subgoal_tac "0 < 2 * b \<or> 2 * b < -1")
   621   apply auto
   622   done
   623 
   624 lemma bitlen_B1: "bitlen (Int.Bit1 b) = (if iszero (Int.succ b) then Int.Bit1 Int.Pls else Int.succ (bitlen b))"
   625 proof -
   626   have h: "! x. (2*x + 1) div 2 = (x::int)"
   627     by arith    
   628   show ?thesis
   629     apply (auto simp add: iszero_def succ_def)
   630     apply (subst Bit1_def)+
   631     apply simp
   632     apply (subgoal_tac "2 * b + 1 = -1")
   633     apply (simp only:)
   634     apply simp_all
   635     apply (subst Bit1_def)
   636     apply simp
   637     apply (subgoal_tac "0 < 2 * b + 1 \<or> 2 * b + 1 < -1")
   638     apply (auto simp add: h)
   639     done
   640 qed
   641 
   642 lemma bitlen_number_of: "bitlen (number_of w) = number_of (bitlen w)"
   643   by (simp add: number_of_is_id)
   644 
   645 lemma [code]: "bitlen x = 
   646      (if x = 0  then 0 
   647  else if x = -1 then 1 
   648                 else (1 + (bitlen (x div 2))))"
   649   by (cases "x = 0 \<or> x = -1 \<or> 0 < x") auto
   650 
   651 definition lapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
   652 where
   653   "lapprox_posrat prec x y = 
   654    (let 
   655        l = nat (int prec + bitlen y - bitlen x) ;
   656        d = (x * 2^l) div y
   657     in normfloat (Float d (- (int l))))"
   658 
   659 lemma pow2_minus: "pow2 (-x) = inverse (pow2 x)"
   660   unfolding pow2_neg[of "-x"] by auto
   661 
   662 lemma lapprox_posrat: 
   663   assumes x: "0 \<le> x"
   664   and y: "0 < y"
   665   shows "Ifloat (lapprox_posrat prec x y) \<le> real x / real y"
   666 proof -
   667   let ?l = "nat (int prec + bitlen y - bitlen x)"
   668   
   669   have "real (x * 2^?l div y) * inverse (2^?l) \<le> (real (x * 2^?l) / real y) * inverse (2^?l)" 
   670     by (rule mult_right_mono, fact real_of_int_div4, simp)
   671   also have "\<dots> \<le> (real x / real y) * 2^?l * inverse (2^?l)" by auto
   672   finally have "real (x * 2^?l div y) * inverse (2^?l) \<le> real x / real y" unfolding real_mult_assoc by auto
   673   thus ?thesis unfolding lapprox_posrat_def Let_def normfloat Ifloat.simps
   674     unfolding pow2_minus pow2_int minus_minus .
   675 qed
   676 
   677 lemma real_of_int_div_mult: 
   678   fixes x y c :: int assumes "0 < y" and "0 < c"
   679   shows "real (x div y) \<le> real (x * c div y) * inverse (real c)"
   680 proof -
   681   have "c * (x div y) + 0 \<le> c * x div y" unfolding zdiv_zmult1_eq[of c x y]
   682     by (rule zadd_left_mono, 
   683         auto intro!: mult_nonneg_nonneg 
   684              simp add: pos_imp_zdiv_nonneg_iff[OF `0 < y`] `0 < c`[THEN less_imp_le] pos_mod_sign[OF `0 < y`])
   685   hence "real (x div y) * real c \<le> real (x * c div y)" 
   686     unfolding real_of_int_mult[symmetric] real_of_int_le_iff zmult_commute by auto
   687   hence "real (x div y) * real c * inverse (real c) \<le> real (x * c div y) * inverse (real c)"
   688     using `0 < c` by auto
   689   thus ?thesis unfolding real_mult_assoc using `0 < c` by auto
   690 qed
   691 
   692 lemma lapprox_posrat_bottom: assumes "0 < y"
   693   shows "real (x div y) \<le> Ifloat (lapprox_posrat n x y)" 
   694 proof -
   695   have pow: "\<And>x. (0::int) < 2^x" by auto
   696   show ?thesis
   697     unfolding lapprox_posrat_def Let_def Ifloat_add normfloat Ifloat.simps pow2_minus pow2_int
   698     using real_of_int_div_mult[OF `0 < y` pow] by auto
   699 qed
   700 
   701 lemma lapprox_posrat_nonneg: assumes "0 \<le> x" and "0 < y"
   702   shows "0 \<le> Ifloat (lapprox_posrat n x y)" 
   703 proof -
   704   show ?thesis
   705     unfolding lapprox_posrat_def Let_def Ifloat_add normfloat Ifloat.simps pow2_minus pow2_int
   706     using pos_imp_zdiv_nonneg_iff[OF `0 < y`] assms by (auto intro!: mult_nonneg_nonneg)
   707 qed
   708 
   709 definition rapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
   710 where
   711   "rapprox_posrat prec x y = (let
   712      l = nat (int prec + bitlen y - bitlen x) ;
   713      X = x * 2^l ;
   714      d = X div y ;
   715      m = X mod y
   716    in normfloat (Float (d + (if m = 0 then 0 else 1)) (- (int l))))"
   717 
   718 lemma rapprox_posrat:
   719   assumes x: "0 \<le> x"
   720   and y: "0 < y"
   721   shows "real x / real y \<le> Ifloat (rapprox_posrat prec x y)"
   722 proof -
   723   let ?l = "nat (int prec + bitlen y - bitlen x)" let ?X = "x * 2^?l"
   724   show ?thesis 
   725   proof (cases "?X mod y = 0")
   726     case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
   727     from real_of_int_div[OF this]
   728     have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
   729     also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
   730     finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
   731     thus ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True] 
   732       unfolding Ifloat.simps pow2_minus pow2_int minus_minus by auto
   733   next
   734     case False
   735     have "0 \<le> real y" and "real y \<noteq> 0" using `0 < y` by auto
   736     have "0 \<le> real y * 2^?l" by (rule mult_nonneg_nonneg, rule `0 \<le> real y`, auto)
   737 
   738     have "?X = y * (?X div y) + ?X mod y" by auto
   739     also have "\<dots> \<le> y * (?X div y) + y" by (rule add_mono, auto simp add: pos_mod_bound[OF `0 < y`, THEN less_imp_le])
   740     also have "\<dots> = y * (?X div y + 1)" unfolding zadd_zmult_distrib2 by auto
   741     finally have "real ?X \<le> real y * real (?X div y + 1)" unfolding real_of_int_le_iff real_of_int_mult[symmetric] .
   742     hence "real ?X / (real y * 2^?l) \<le> real y * real (?X div y + 1) / (real y * 2^?l)" 
   743       by (rule divide_right_mono, simp only: `0 \<le> real y * 2^?l`)
   744     also have "\<dots> = real y * real (?X div y + 1) / real y / 2^?l" by auto
   745     also have "\<dots> = real (?X div y + 1) * inverse (2^?l)" unfolding nonzero_mult_divide_cancel_left[OF `real y \<noteq> 0`] 
   746       unfolding real_divide_def ..
   747     finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_not_P[OF False]
   748       unfolding pow2_minus pow2_int minus_minus by auto
   749   qed
   750 qed
   751 
   752 lemma rapprox_posrat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
   753   shows "Ifloat (rapprox_posrat n x y) \<le> 1"
   754 proof -
   755   let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
   756   show ?thesis
   757   proof (cases "?X mod y = 0")
   758     case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
   759     from real_of_int_div[OF this]
   760     have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
   761     also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
   762     finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
   763     also have "real x / real y \<le> 1" using `0 \<le> x` and `0 < y` and `x \<le> y` by auto
   764     finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]
   765       unfolding Ifloat.simps pow2_minus pow2_int minus_minus by auto
   766   next
   767     case False
   768     have "x \<noteq> y"
   769     proof (rule ccontr)
   770       assume "\<not> x \<noteq> y" hence "x = y" by auto
   771       have "?X mod y = 0" unfolding `x = y` using mod_mult_self1_is_0 by auto
   772       thus False using False by auto
   773     qed
   774     hence "x < y" using `x \<le> y` by auto
   775     hence "real x / real y < 1" using `0 < y` and `0 \<le> x` by auto
   776 
   777     from real_of_int_div4[of "?X" y]
   778     have "real (?X div y) \<le> (real x / real y) * 2^?l" unfolding real_of_int_mult times_divide_eq_left real_of_int_power[symmetric] real_number_of .
   779     also have "\<dots> < 1 * 2^?l" using `real x / real y < 1` by (rule mult_strict_right_mono, auto)
   780     finally have "?X div y < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
   781     hence "?X div y + 1 \<le> 2^?l" by auto
   782     hence "real (?X div y + 1) * inverse (2^?l) \<le> 2^?l * inverse (2^?l)"
   783       unfolding real_of_int_le_iff[of _ "2^?l", symmetric] real_of_int_power[symmetric] real_number_of
   784       by (rule mult_right_mono, auto)
   785     hence "real (?X div y + 1) * inverse (2^?l) \<le> 1" by auto
   786     thus ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_not_P[OF False]
   787       unfolding pow2_minus pow2_int minus_minus by auto
   788   qed
   789 qed
   790 
   791 lemma zdiv_greater_zero: fixes a b :: int assumes "0 < a" and "a \<le> b"
   792   shows "0 < b div a"
   793 proof (rule ccontr)
   794   have "0 \<le> b" using assms by auto
   795   assume "\<not> 0 < b div a" hence "b div a = 0" using `0 \<le> b`[unfolded pos_imp_zdiv_nonneg_iff[OF `0<a`, of b, symmetric]] by auto
   796   have "b = a * (b div a) + b mod a" by auto
   797   hence "b = b mod a" unfolding `b div a = 0` by auto
   798   hence "b < a" using `0 < a`[THEN pos_mod_bound, of b] by auto
   799   thus False using `a \<le> b` by auto
   800 qed
   801 
   802 lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"
   803   shows "Ifloat (rapprox_posrat n x y) < 1"
   804 proof (cases "x = 0")
   805   case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat Ifloat.simps by auto
   806 next
   807   case False hence "0 < x" using `0 \<le> x` by auto
   808   hence "x < y" using assms by auto
   809   
   810   let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"
   811   show ?thesis
   812   proof (cases "?X mod y = 0")
   813     case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto
   814     from real_of_int_div[OF this]
   815     have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto
   816     also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto
   817     finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto
   818     also have "real x / real y < 1" using `0 \<le> x` and `0 < y` and `x < y` by auto
   819     finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_P[OF True]
   820       unfolding pow2_minus pow2_int minus_minus by auto
   821   next
   822     case False
   823     hence "(real x / real y) < 1 / 2" using `0 < y` and `0 \<le> x` `2 * x < y` by auto
   824 
   825     have "0 < ?X div y"
   826     proof -
   827       have "2^nat (bitlen x - 1) \<le> y" and "y < 2^nat (bitlen y)"
   828 	using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto
   829       hence "(2::int)^nat (bitlen x - 1) < 2^nat (bitlen y)" by (rule order_le_less_trans)
   830       hence "bitlen x \<le> bitlen y" by auto
   831       hence len_less: "nat (bitlen x - 1) \<le> nat (int (n - 1) + bitlen y)" by auto
   832 
   833       have "x \<noteq> 0" and "y \<noteq> 0" using `0 < x` `0 < y` by auto
   834 
   835       have exp_eq: "nat (int (n - 1) + bitlen y) - nat (bitlen x - 1) = ?l"
   836 	using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto
   837 
   838       have "y * 2^nat (bitlen x - 1) \<le> y * x" 
   839 	using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)
   840       also have "\<dots> \<le> 2^nat (bitlen y) * x" using bitlen_bounds[OF `0 < y`, THEN conjunct2, THEN less_imp_le] `0 \<le> x` by (rule mult_right_mono)
   841       also have "\<dots> \<le> x * 2^nat (int (n - 1) + bitlen y)" unfolding mult_commute[of x] by (rule mult_right_mono, auto simp add: `0 \<le> x`)
   842       finally have "real y * 2^nat (bitlen x - 1) * inverse (2^nat (bitlen x - 1)) \<le> real x * 2^nat (int (n - 1) + bitlen y) * inverse (2^nat (bitlen x - 1))"
   843 	unfolding real_of_int_le_iff[symmetric] by auto
   844       hence "real y \<le> real x * (2^nat (int (n - 1) + bitlen y) / (2^nat (bitlen x - 1)))" 
   845 	unfolding real_mult_assoc real_divide_def by auto
   846       also have "\<dots> = real x * (2^(nat (int (n - 1) + bitlen y) - nat (bitlen x - 1)))" using power_diff[of "2::real", OF _ len_less] by auto
   847       finally have "y \<le> x * 2^?l" unfolding exp_eq unfolding real_of_int_le_iff[symmetric] by auto
   848       thus ?thesis using zdiv_greater_zero[OF `0 < y`] by auto
   849     qed
   850 
   851     from real_of_int_div4[of "?X" y]
   852     have "real (?X div y) \<le> (real x / real y) * 2^?l" unfolding real_of_int_mult times_divide_eq_left real_of_int_power[symmetric] real_number_of .
   853     also have "\<dots> < 1/2 * 2^?l" using `real x / real y < 1/2` by (rule mult_strict_right_mono, auto)
   854     finally have "?X div y * 2 < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto
   855     hence "?X div y + 1 < 2^?l" using `0 < ?X div y` by auto
   856     hence "real (?X div y + 1) * inverse (2^?l) < 2^?l * inverse (2^?l)"
   857       unfolding real_of_int_less_iff[of _ "2^?l", symmetric] real_of_int_power[symmetric] real_number_of
   858       by (rule mult_strict_right_mono, auto)
   859     hence "real (?X div y + 1) * inverse (2^?l) < 1" by auto
   860     thus ?thesis unfolding rapprox_posrat_def Let_def normfloat Ifloat.simps if_not_P[OF False]
   861       unfolding pow2_minus pow2_int minus_minus by auto
   862   qed
   863 qed
   864 
   865 lemma approx_rat_pattern: fixes P and ps :: "nat * int * int"
   866   assumes Y: "\<And>y prec x. \<lbrakk>y = 0; ps = (prec, x, 0)\<rbrakk> \<Longrightarrow> P" 
   867   and A: "\<And>x y prec. \<lbrakk>0 \<le> x; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
   868   and B: "\<And>x y prec. \<lbrakk>x < 0; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
   869   and C: "\<And>x y prec. \<lbrakk>x < 0; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
   870   and D: "\<And>x y prec. \<lbrakk>0 \<le> x; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"
   871   shows P
   872 proof -
   873   obtain prec x y where [simp]: "ps = (prec, x, y)" by (cases ps, auto)
   874   from Y have "y = 0 \<Longrightarrow> P" by auto
   875   moreover { assume "0 < y" have P proof (cases "0 \<le> x") case True with A and `0 < y` show P by auto next case False with B and `0 < y` show P by auto qed } 
   876   moreover { assume "y < 0" have P proof (cases "0 \<le> x") case True with D and `y < 0` show P by auto next case False with C and `y < 0` show P by auto qed }
   877   ultimately show P by (cases "y = 0 \<or> 0 < y \<or> y < 0", auto)
   878 qed
   879 
   880 function lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
   881 where
   882   "y = 0 \<Longrightarrow> lapprox_rat prec x y = 0"
   883 | "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec x y"
   884 | "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec (-x) y)"
   885 | "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec (-x) (-y)"
   886 | "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec x (-y))"
   887 apply simp_all by (rule approx_rat_pattern)
   888 termination by lexicographic_order
   889 
   890 lemma compute_lapprox_rat[code]:
   891       "lapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then (if 0 < y then lapprox_posrat prec x y else - (rapprox_posrat prec x (-y))) 
   892                                                              else (if 0 < y then - (rapprox_posrat prec (-x) y) else lapprox_posrat prec (-x) (-y)))"
   893   by auto
   894             
   895 lemma lapprox_rat: "Ifloat (lapprox_rat prec x y) \<le> real x / real y"
   896 proof -      
   897   have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
   898   show ?thesis
   899     apply (case_tac "y = 0")
   900     apply simp
   901     apply (case_tac "0 \<le> x \<and> 0 < y")
   902     apply (simp add: lapprox_posrat)
   903     apply (case_tac "x < 0 \<and> 0 < y")
   904     apply simp
   905     apply (subst minus_le_iff)   
   906     apply (rule h[OF rapprox_posrat])
   907     apply (simp_all)
   908     apply (case_tac "x < 0 \<and> y < 0")
   909     apply simp
   910     apply (rule h[OF _ lapprox_posrat])
   911     apply (simp_all)
   912     apply (case_tac "0 \<le> x \<and> y < 0")
   913     apply (simp)
   914     apply (subst minus_le_iff)   
   915     apply (rule h[OF rapprox_posrat])
   916     apply simp_all
   917     apply arith
   918     done
   919 qed
   920 
   921 lemma lapprox_rat_bottom: assumes "0 \<le> x" and "0 < y"
   922   shows "real (x div y) \<le> Ifloat (lapprox_rat n x y)" 
   923   unfolding lapprox_rat.simps(2)[OF assms]  using lapprox_posrat_bottom[OF `0<y`] .
   924 
   925 function rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"
   926 where
   927   "y = 0 \<Longrightarrow> rapprox_rat prec x y = 0"
   928 | "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec x y"
   929 | "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec (-x) y)"
   930 | "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec (-x) (-y)"
   931 | "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec x (-y))"
   932 apply simp_all by (rule approx_rat_pattern)
   933 termination by lexicographic_order
   934 
   935 lemma compute_rapprox_rat[code]:
   936       "rapprox_rat prec x y = (if y = 0 then 0 else if 0 \<le> x then (if 0 < y then rapprox_posrat prec x y else - (lapprox_posrat prec x (-y))) else 
   937                                                                   (if 0 < y then - (lapprox_posrat prec (-x) y) else rapprox_posrat prec (-x) (-y)))"
   938   by auto
   939 
   940 lemma rapprox_rat: "real x / real y \<le> Ifloat (rapprox_rat prec x y)"
   941 proof -      
   942   have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto
   943   show ?thesis
   944     apply (case_tac "y = 0")
   945     apply simp
   946     apply (case_tac "0 \<le> x \<and> 0 < y")
   947     apply (simp add: rapprox_posrat)
   948     apply (case_tac "x < 0 \<and> 0 < y")
   949     apply simp
   950     apply (subst le_minus_iff)   
   951     apply (rule h[OF _ lapprox_posrat])
   952     apply (simp_all)
   953     apply (case_tac "x < 0 \<and> y < 0")
   954     apply simp
   955     apply (rule h[OF rapprox_posrat])
   956     apply (simp_all)
   957     apply (case_tac "0 \<le> x \<and> y < 0")
   958     apply (simp)
   959     apply (subst le_minus_iff)   
   960     apply (rule h[OF _ lapprox_posrat])
   961     apply simp_all
   962     apply arith
   963     done
   964 qed
   965 
   966 lemma rapprox_rat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"
   967   shows "Ifloat (rapprox_rat n x y) \<le> 1"
   968   unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`] using rapprox_posrat_le1[OF assms] .
   969 
   970 lemma rapprox_rat_neg: assumes "x < 0" and "0 < y"
   971   shows "Ifloat (rapprox_rat n x y) \<le> 0"
   972   unfolding rapprox_rat.simps(3)[OF assms] using lapprox_posrat_nonneg[of "-x" y n] assms by auto
   973 
   974 lemma rapprox_rat_nonneg_neg: assumes "0 \<le> x" and "y < 0"
   975   shows "Ifloat (rapprox_rat n x y) \<le> 0"
   976   unfolding rapprox_rat.simps(5)[OF assms] using lapprox_posrat_nonneg[of x "-y" n] assms by auto
   977 
   978 lemma rapprox_rat_nonpos_pos: assumes "x \<le> 0" and "0 < y"
   979   shows "Ifloat (rapprox_rat n x y) \<le> 0"
   980 proof (cases "x = 0") 
   981   case True hence "0 \<le> x" by auto show ?thesis unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`]
   982     unfolding True rapprox_posrat_def Let_def by auto
   983 next
   984   case False hence "x < 0" using assms by auto
   985   show ?thesis using rapprox_rat_neg[OF `x < 0` `0 < y`] .
   986 qed
   987 
   988 fun float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
   989 where
   990   "float_divl prec (Float m1 s1) (Float m2 s2) = 
   991     (let
   992        l = lapprox_rat prec m1 m2;
   993        f = Float 1 (s1 - s2)
   994      in
   995        f * l)"     
   996 
   997 lemma float_divl: "Ifloat (float_divl prec x y) \<le> Ifloat x / Ifloat y"
   998 proof - 
   999   from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto
  1000   from float_split[of y] obtain my sy where y: "y = Float my sy" by auto
  1001   have "real mx / real my \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
  1002     apply (case_tac "my = 0")
  1003     apply simp
  1004     apply (case_tac "my > 0")       
  1005     apply (subst pos_le_divide_eq)
  1006     apply simp
  1007     apply (subst pos_le_divide_eq)
  1008     apply (simp add: mult_pos_pos)
  1009     apply simp
  1010     apply (subst pow2_add[symmetric])
  1011     apply simp
  1012     apply (subgoal_tac "my < 0")
  1013     apply auto
  1014     apply (simp add: field_simps)
  1015     apply (subst pow2_add[symmetric])
  1016     apply (simp add: field_simps)
  1017     done
  1018   then have "Ifloat (lapprox_rat prec mx my) \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
  1019     by (rule order_trans[OF lapprox_rat])
  1020   then have "Ifloat (lapprox_rat prec mx my) * pow2 (sx - sy) \<le> real mx * pow2 sx / (real my * pow2 sy)"
  1021     apply (subst pos_le_divide_eq[symmetric])
  1022     apply simp_all
  1023     done
  1024   then have "pow2 (sx - sy) * Ifloat (lapprox_rat prec mx my) \<le> real mx * pow2 sx / (real my * pow2 sy)"
  1025     by (simp add: algebra_simps)
  1026   then show ?thesis
  1027     by (simp add: x y Let_def Ifloat.simps)
  1028 qed
  1029 
  1030 lemma float_divl_lower_bound: assumes "0 \<le> x" and "0 < y" shows "0 \<le> float_divl prec x y"
  1031 proof (cases x, cases y)
  1032   fix xm xe ym ye :: int
  1033   assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
  1034   have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def Ifloat.simps Ifloat_0 zero_le_mult_iff] by auto
  1035   have "0 < ym" using `0 < y`[unfolded y_eq less_float_def Ifloat.simps Ifloat_0 zero_less_mult_iff] by auto
  1036 
  1037   have "\<And>n. 0 \<le> Ifloat (Float 1 n)" unfolding Ifloat.simps using zero_le_pow2 by auto
  1038   moreover have "0 \<le> Ifloat (lapprox_rat prec xm ym)" by (rule order_trans[OF _ lapprox_rat_bottom[OF `0 \<le> xm` `0 < ym`]], auto simp add: `0 \<le> xm` pos_imp_zdiv_nonneg_iff[OF `0 < ym`])
  1039   ultimately show "0 \<le> float_divl prec x y"
  1040     unfolding x_eq y_eq float_divl.simps Let_def le_float_def Ifloat_0 by (auto intro!: mult_nonneg_nonneg)
  1041 qed
  1042 
  1043 lemma float_divl_pos_less1_bound: assumes "0 < x" and "x < 1" and "0 < prec" shows "1 \<le> float_divl prec 1 x"
  1044 proof (cases x)
  1045   case (Float m e)
  1046   from `0 < x` `x < 1` have "0 < m" "e < 0" using float_pos_m_pos float_pos_less1_e_neg unfolding Float by auto
  1047   let ?b = "nat (bitlen m)" and ?e = "nat (-e)"
  1048   have "1 \<le> m" and "m \<noteq> 0" using `0 < m` by auto
  1049   with bitlen_bounds[OF `0 < m`] have "m < 2^?b" and "(2::int) \<le> 2^?b" by auto
  1050   hence "1 \<le> bitlen m" using power_le_imp_le_exp[of "2::int" 1 ?b] by auto
  1051   hence pow_split: "nat (int prec + bitlen m - 1) = (prec - 1) + ?b" using `0 < prec` by auto
  1052   
  1053   have pow_not0: "\<And>x. (2::real)^x \<noteq> 0" by auto
  1054 
  1055   from float_less1_mantissa_bound `0 < x` `x < 1` Float 
  1056   have "m < 2^?e" by auto
  1057   with bitlen_bounds[OF `0 < m`, THEN conjunct1]
  1058   have "(2::int)^nat (bitlen m - 1) < 2^?e" by (rule order_le_less_trans)
  1059   from power_less_imp_less_exp[OF _ this]
  1060   have "bitlen m \<le> - e" by auto
  1061   hence "(2::real)^?b \<le> 2^?e" by auto
  1062   hence "(2::real)^?b * inverse (2^?b) \<le> 2^?e * inverse (2^?b)" by (rule mult_right_mono, auto)
  1063   hence "(1::real) \<le> 2^?e * inverse (2^?b)" by auto
  1064   also
  1065   let ?d = "real (2 ^ nat (int prec + bitlen m - 1) div m) * inverse (2 ^ nat (int prec + bitlen m - 1))"
  1066   { have "2^(prec - 1) * m \<le> 2^(prec - 1) * 2^?b" using `m < 2^?b`[THEN less_imp_le] by (rule mult_left_mono, auto)
  1067     also have "\<dots> = 2 ^ nat (int prec + bitlen m - 1)" unfolding pow_split zpower_zadd_distrib by auto
  1068     finally have "2^(prec - 1) * m div m \<le> 2 ^ nat (int prec + bitlen m - 1) div m" using `0 < m` by (rule zdiv_mono1)
  1069     hence "2^(prec - 1) \<le> 2 ^ nat (int prec + bitlen m - 1) div m" unfolding div_mult_self2_is_id[OF `m \<noteq> 0`] .
  1070     hence "2^(prec - 1) * inverse (2 ^ nat (int prec + bitlen m - 1)) \<le> ?d"
  1071       unfolding real_of_int_le_iff[of "2^(prec - 1)", symmetric] by auto }
  1072   from mult_left_mono[OF this[unfolded pow_split power_add inverse_mult_distrib real_mult_assoc[symmetric] right_inverse[OF pow_not0] real_mult_1], of "2^?e"]
  1073   have "2^?e * inverse (2^?b) \<le> 2^?e * ?d" unfolding pow_split power_add by auto
  1074   finally have "1 \<le> 2^?e * ?d" .
  1075   
  1076   have e_nat: "0 - e = int (nat (-e))" using `e < 0` by auto
  1077   have "bitlen 1 = 1" using bitlen.simps by auto
  1078   
  1079   show ?thesis 
  1080     unfolding one_float_def Float float_divl.simps Let_def lapprox_rat.simps(2)[OF zero_le_one `0 < m`] lapprox_posrat_def `bitlen 1 = 1`
  1081     unfolding le_float_def Ifloat_mult normfloat Ifloat.simps pow2_minus pow2_int e_nat
  1082     using `1 \<le> 2^?e * ?d` by (auto simp add: pow2_def)
  1083 qed
  1084 
  1085 fun float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"
  1086 where
  1087   "float_divr prec (Float m1 s1) (Float m2 s2) = 
  1088     (let
  1089        r = rapprox_rat prec m1 m2;
  1090        f = Float 1 (s1 - s2)
  1091      in
  1092        f * r)"  
  1093 
  1094 lemma float_divr: "Ifloat x / Ifloat y \<le> Ifloat (float_divr prec x y)"
  1095 proof - 
  1096   from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto
  1097   from float_split[of y] obtain my sy where y: "y = Float my sy" by auto
  1098   have "real mx / real my \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
  1099     apply (case_tac "my = 0")
  1100     apply simp
  1101     apply (case_tac "my > 0")
  1102     apply auto
  1103     apply (subst pos_divide_le_eq)
  1104     apply (rule mult_pos_pos)+
  1105     apply simp_all
  1106     apply (subst pow2_add[symmetric])
  1107     apply simp
  1108     apply (subgoal_tac "my < 0")
  1109     apply auto
  1110     apply (simp add: field_simps)
  1111     apply (subst pow2_add[symmetric])
  1112     apply (simp add: field_simps)
  1113     done
  1114   then have "Ifloat (rapprox_rat prec mx my) \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"
  1115     by (rule order_trans[OF _ rapprox_rat])
  1116   then have "Ifloat (rapprox_rat prec mx my) * pow2 (sx - sy) \<ge> real mx * pow2 sx / (real my * pow2 sy)"
  1117     apply (subst pos_divide_le_eq[symmetric])
  1118     apply simp_all
  1119     done
  1120   then show ?thesis
  1121     by (simp add: x y Let_def algebra_simps Ifloat.simps)
  1122 qed
  1123 
  1124 lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"
  1125 proof -
  1126   have "1 \<le> 1 / Ifloat x" using `0 < x` and `x < 1` unfolding less_float_def by auto
  1127   also have "\<dots> \<le> Ifloat (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
  1128   finally show ?thesis unfolding le_float_def by auto
  1129 qed
  1130 
  1131 lemma float_divr_nonpos_pos_upper_bound: assumes "x \<le> 0" and "0 < y" shows "float_divr prec x y \<le> 0"
  1132 proof (cases x, cases y)
  1133   fix xm xe ym ye :: int
  1134   assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
  1135   have "xm \<le> 0" using `x \<le> 0`[unfolded x_eq le_float_def Ifloat.simps Ifloat_0 mult_le_0_iff] by auto
  1136   have "0 < ym" using `0 < y`[unfolded y_eq less_float_def Ifloat.simps Ifloat_0 zero_less_mult_iff] by auto
  1137 
  1138   have "\<And>n. 0 \<le> Ifloat (Float 1 n)" unfolding Ifloat.simps using zero_le_pow2 by auto
  1139   moreover have "Ifloat (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .
  1140   ultimately show "float_divr prec x y \<le> 0"
  1141     unfolding x_eq y_eq float_divr.simps Let_def le_float_def Ifloat_0 Ifloat_mult by (auto intro!: mult_nonneg_nonpos)
  1142 qed
  1143 
  1144 lemma float_divr_nonneg_neg_upper_bound: assumes "0 \<le> x" and "y < 0" shows "float_divr prec x y \<le> 0"
  1145 proof (cases x, cases y)
  1146   fix xm xe ym ye :: int
  1147   assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"
  1148   have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def Ifloat.simps Ifloat_0 zero_le_mult_iff] by auto
  1149   have "ym < 0" using `y < 0`[unfolded y_eq less_float_def Ifloat.simps Ifloat_0 mult_less_0_iff] by auto
  1150   hence "0 < - ym" by auto
  1151 
  1152   have "\<And>n. 0 \<le> Ifloat (Float 1 n)" unfolding Ifloat.simps using zero_le_pow2 by auto
  1153   moreover have "Ifloat (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .
  1154   ultimately show "float_divr prec x y \<le> 0"
  1155     unfolding x_eq y_eq float_divr.simps Let_def le_float_def Ifloat_0 Ifloat_mult by (auto intro!: mult_nonneg_nonpos)
  1156 qed
  1157 
  1158 primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where
  1159 "round_down prec (Float m e) = (let d = bitlen m - int prec in
  1160      if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)
  1161               else Float m e)"
  1162 
  1163 primrec round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where
  1164 "round_up prec (Float m e) = (let d = bitlen m - int prec in
  1165   if 0 < d then let P = 2^nat d ; n = m div P ; r = m mod P in Float (n + (if r = 0 then 0 else 1)) (e + d) 
  1166            else Float m e)"
  1167 
  1168 lemma round_up: "Ifloat x \<le> Ifloat (round_up prec x)"
  1169 proof (cases x)
  1170   case (Float m e)
  1171   let ?d = "bitlen m - int prec"
  1172   let ?p = "(2::int)^nat ?d"
  1173   have "0 < ?p" by auto
  1174   show "?thesis"
  1175   proof (cases "0 < ?d")
  1176     case True
  1177     hence pow_d: "pow2 ?d = real ?p" unfolding pow2_int[symmetric] power_real_number_of[symmetric] by auto
  1178     show ?thesis
  1179     proof (cases "m mod ?p = 0")
  1180       case True
  1181       have m: "m = m div ?p * ?p + 0" unfolding True[symmetric] using zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right, symmetric] .
  1182       have "Ifloat (Float m e) = Ifloat (Float (m div ?p) (e + ?d))" unfolding Ifloat.simps arg_cong[OF m, of real]
  1183 	by (auto simp add: pow2_add `0 < ?d` pow_d)
  1184       thus ?thesis
  1185 	unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]
  1186 	by auto
  1187     next
  1188       case False
  1189       have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
  1190       also have "\<dots> \<le> (m div ?p + 1) * ?p" unfolding left_distrib zmult_1 by (rule add_left_mono, rule pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
  1191       finally have "Ifloat (Float m e) \<le> Ifloat (Float (m div ?p + 1) (e + ?d))" unfolding Ifloat.simps add_commute[of e]
  1192 	unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]
  1193 	by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
  1194       thus ?thesis
  1195 	unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .
  1196     qed
  1197   next
  1198     case False
  1199     show ?thesis
  1200       unfolding Float round_up.simps Let_def if_not_P[OF False] .. 
  1201   qed
  1202 qed
  1203 
  1204 lemma round_down: "Ifloat (round_down prec x) \<le> Ifloat x"
  1205 proof (cases x)
  1206   case (Float m e)
  1207   let ?d = "bitlen m - int prec"
  1208   let ?p = "(2::int)^nat ?d"
  1209   have "0 < ?p" by auto
  1210   show "?thesis"
  1211   proof (cases "0 < ?d")
  1212     case True
  1213     hence pow_d: "pow2 ?d = real ?p" unfolding pow2_int[symmetric] power_real_number_of[symmetric] by auto
  1214     have "m div ?p * ?p \<le> m div ?p * ?p + m mod ?p" by (auto simp add: pos_mod_bound[OF `0 < ?p`, THEN less_imp_le])
  1215     also have "\<dots> \<le> m" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..
  1216     finally have "Ifloat (Float (m div ?p) (e + ?d)) \<le> Ifloat (Float m e)" unfolding Ifloat.simps add_commute[of e]
  1217       unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of _ m, symmetric]
  1218       by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)
  1219     thus ?thesis
  1220       unfolding Float round_down.simps Let_def if_P[OF `0 < ?d`] .
  1221   next
  1222     case False
  1223     show ?thesis
  1224       unfolding Float round_down.simps Let_def if_not_P[OF False] .. 
  1225   qed
  1226 qed
  1227 
  1228 definition lb_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
  1229 "lb_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let
  1230     l = bitlen m - int prec
  1231   in if l > 0 then Float (m div (2^nat l)) (e + l)
  1232               else Float m e)"
  1233 
  1234 definition ub_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
  1235 "ub_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let
  1236     l = bitlen m - int prec
  1237   in if l > 0 then Float (m div (2^nat l) + 1) (e + l)
  1238               else Float m e)"
  1239 
  1240 lemma lb_mult: "Ifloat (lb_mult prec x y) \<le> Ifloat (x * y)"
  1241 proof (cases "normfloat (x * y)")
  1242   case (Float m e)
  1243   hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
  1244   let ?l = "bitlen m - int prec"
  1245   have "Ifloat (lb_mult prec x y) \<le> Ifloat (normfloat (x * y))"
  1246   proof (cases "?l > 0")
  1247     case False thus ?thesis unfolding lb_mult_def Float Let_def float.cases by auto
  1248   next
  1249     case True
  1250     have "real (m div 2^(nat ?l)) * pow2 ?l \<le> real m"
  1251     proof -
  1252       have "real (m div 2^(nat ?l)) * pow2 ?l = real (2^(nat ?l) * (m div 2^(nat ?l)))" unfolding real_of_int_mult real_of_int_power[symmetric] real_number_of unfolding pow2_int[symmetric] 
  1253 	using `?l > 0` by auto
  1254       also have "\<dots> \<le> real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding real_of_int_add by auto
  1255       also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..
  1256       finally show ?thesis by auto
  1257     qed
  1258     thus ?thesis unfolding lb_mult_def Float Let_def float.cases if_P[OF True] Ifloat.simps pow2_add real_mult_commute real_mult_assoc by auto
  1259   qed
  1260   also have "\<dots> = Ifloat (x * y)" unfolding normfloat ..
  1261   finally show ?thesis .
  1262 qed
  1263 
  1264 lemma ub_mult: "Ifloat (x * y) \<le> Ifloat (ub_mult prec x y)"
  1265 proof (cases "normfloat (x * y)")
  1266   case (Float m e)
  1267   hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)
  1268   let ?l = "bitlen m - int prec"
  1269   have "Ifloat (x * y) = Ifloat (normfloat (x * y))" unfolding normfloat ..
  1270   also have "\<dots> \<le> Ifloat (ub_mult prec x y)"
  1271   proof (cases "?l > 0")
  1272     case False thus ?thesis unfolding ub_mult_def Float Let_def float.cases by auto
  1273   next
  1274     case True
  1275     have "real m \<le> real (m div 2^(nat ?l) + 1) * pow2 ?l"
  1276     proof -
  1277       have "m mod 2^(nat ?l) < 2^(nat ?l)" by (rule pos_mod_bound) auto
  1278       hence mod_uneq: "real (m mod 2^(nat ?l)) \<le> 1 * 2^(nat ?l)" unfolding zmult_1 real_of_int_less_iff[symmetric] by auto
  1279       
  1280       have "real m = real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding zmod_zdiv_equality[symmetric] ..
  1281       also have "\<dots> = real (m div 2^(nat ?l)) * 2^(nat ?l) + real (m mod 2^(nat ?l))" unfolding real_of_int_add by auto
  1282       also have "\<dots> \<le> (real (m div 2^(nat ?l)) + 1) * 2^(nat ?l)" unfolding real_add_mult_distrib using mod_uneq by auto
  1283       finally show ?thesis unfolding pow2_int[symmetric] using True by auto
  1284     qed
  1285     thus ?thesis unfolding ub_mult_def Float Let_def float.cases if_P[OF True] Ifloat.simps pow2_add real_mult_commute real_mult_assoc by auto
  1286   qed
  1287   finally show ?thesis .
  1288 qed
  1289 
  1290 primrec float_abs :: "float \<Rightarrow> float" where
  1291   "float_abs (Float m e) = Float \<bar>m\<bar> e"
  1292 
  1293 instantiation float :: abs begin
  1294 definition abs_float_def: "\<bar>x\<bar> = float_abs x"
  1295 instance ..
  1296 end
  1297 
  1298 lemma Ifloat_abs: "Ifloat \<bar>x\<bar> = \<bar>Ifloat x\<bar>" 
  1299 proof (cases x)
  1300   case (Float m e)
  1301   have "\<bar>real m\<bar> * pow2 e = \<bar>real m * pow2 e\<bar>" unfolding abs_mult by auto
  1302   thus ?thesis unfolding Float abs_float_def float_abs.simps Ifloat.simps by auto
  1303 qed
  1304 
  1305 primrec floor_fl :: "float \<Rightarrow> float" where
  1306   "floor_fl (Float m e) = (if 0 \<le> e then Float m e
  1307                                   else Float (m div (2 ^ (nat (-e)))) 0)"
  1308 
  1309 lemma floor_fl: "Ifloat (floor_fl x) \<le> Ifloat x"
  1310 proof (cases x)
  1311   case (Float m e)
  1312   show ?thesis
  1313   proof (cases "0 \<le> e")
  1314     case False
  1315     hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
  1316     have "Ifloat (Float (m div (2 ^ (nat (-e)))) 0) = real (m div 2 ^ (nat (-e)))" unfolding Ifloat.simps by auto
  1317     also have "\<dots> \<le> real m / real ((2::int) ^ (nat (-e)))" using real_of_int_div4 .
  1318     also have "\<dots> = real m * inverse (2 ^ (nat (-e)))" unfolding power_real_number_of[symmetric] real_divide_def ..
  1319     also have "\<dots> = Ifloat (Float m e)" unfolding Ifloat.simps me_eq pow2_int pow2_neg[of e] ..
  1320     finally show ?thesis unfolding Float floor_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
  1321   next
  1322     case True thus ?thesis unfolding Float by auto
  1323   qed
  1324 qed
  1325 
  1326 lemma floor_pos_exp: assumes floor: "Float m e = floor_fl x" shows "0 \<le> e"
  1327 proof (cases x)
  1328   case (Float mx me)
  1329   from floor[unfolded Float floor_fl.simps] show ?thesis by (cases "0 \<le> me", auto)
  1330 qed
  1331 
  1332 declare floor_fl.simps[simp del]
  1333 
  1334 primrec ceiling_fl :: "float \<Rightarrow> float" where
  1335   "ceiling_fl (Float m e) = (if 0 \<le> e then Float m e
  1336                                     else Float (m div (2 ^ (nat (-e))) + 1) 0)"
  1337 
  1338 lemma ceiling_fl: "Ifloat x \<le> Ifloat (ceiling_fl x)"
  1339 proof (cases x)
  1340   case (Float m e)
  1341   show ?thesis
  1342   proof (cases "0 \<le> e")
  1343     case False
  1344     hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto
  1345     have "Ifloat (Float m e) = real m * inverse (2 ^ (nat (-e)))" unfolding Ifloat.simps me_eq pow2_int pow2_neg[of e] ..
  1346     also have "\<dots> = real m / real ((2::int) ^ (nat (-e)))" unfolding power_real_number_of[symmetric] real_divide_def ..
  1347     also have "\<dots> \<le> 1 + real (m div 2 ^ (nat (-e)))" using real_of_int_div3[unfolded diff_le_eq] .
  1348     also have "\<dots> = Ifloat (Float (m div (2 ^ (nat (-e))) + 1) 0)" unfolding Ifloat.simps by auto
  1349     finally show ?thesis unfolding Float ceiling_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .
  1350   next
  1351     case True thus ?thesis unfolding Float by auto
  1352   qed
  1353 qed
  1354 
  1355 declare ceiling_fl.simps[simp del]
  1356 
  1357 definition lb_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
  1358 "lb_mod prec x ub lb = x - ceiling_fl (float_divr prec x lb) * ub"
  1359 
  1360 definition ub_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where
  1361 "ub_mod prec x ub lb = x - floor_fl (float_divl prec x ub) * lb"
  1362 
  1363 lemma lb_mod: fixes k :: int assumes "0 \<le> Ifloat x" and "real k * y \<le> Ifloat x" (is "?k * y \<le> ?x")
  1364   assumes "0 < Ifloat lb" "Ifloat lb \<le> y" (is "?lb \<le> y") "y \<le> Ifloat ub" (is "y \<le> ?ub")
  1365   shows "Ifloat (lb_mod prec x ub lb) \<le> ?x - ?k * y"
  1366 proof -
  1367   have "?lb \<le> ?ub" by (auto!)
  1368   have "0 \<le> ?lb" and "?lb \<noteq> 0" by (auto!)
  1369   have "?k * y \<le> ?x" using assms by auto
  1370   also have "\<dots> \<le> ?x / ?lb * ?ub" by (metis mult_left_mono[OF `?lb \<le> ?ub` `0 \<le> ?x`] divide_right_mono[OF _ `0 \<le> ?lb` ] times_divide_eq_left nonzero_mult_divide_cancel_right[OF `?lb \<noteq> 0`])
  1371   also have "\<dots> \<le> Ifloat (ceiling_fl (float_divr prec x lb)) * ?ub" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divr ceiling_fl)
  1372   finally show ?thesis unfolding lb_mod_def Ifloat_sub Ifloat_mult by auto
  1373 qed
  1374 
  1375 lemma ub_mod: fixes k :: int assumes "0 \<le> Ifloat x" and "Ifloat x \<le> real k * y" (is "?x \<le> ?k * y")
  1376   assumes "0 < Ifloat lb" "Ifloat lb \<le> y" (is "?lb \<le> y") "y \<le> Ifloat ub" (is "y \<le> ?ub")
  1377   shows "?x - ?k * y \<le> Ifloat (ub_mod prec x ub lb)"
  1378 proof -
  1379   have "?lb \<le> ?ub" by (auto!)
  1380   hence "0 \<le> ?lb" and "0 \<le> ?ub" and "?ub \<noteq> 0" by (auto!)
  1381   have "Ifloat (floor_fl (float_divl prec x ub)) * ?lb \<le> ?x / ?ub * ?lb" by (metis mult_right_mono order_trans `0 \<le> ?lb` `?lb \<le> ?ub` float_divl floor_fl)
  1382   also have "\<dots> \<le> ?x" by (metis mult_left_mono[OF `?lb \<le> ?ub` `0 \<le> ?x`] divide_right_mono[OF _ `0 \<le> ?ub` ] times_divide_eq_left nonzero_mult_divide_cancel_right[OF `?ub \<noteq> 0`])
  1383   also have "\<dots> \<le> ?k * y" using assms by auto
  1384   finally show ?thesis unfolding ub_mod_def Ifloat_sub Ifloat_mult by auto
  1385 qed
  1386 
  1387 lemma le_float_def': "f \<le> g = (case f - g of Float a b \<Rightarrow> a \<le> 0)"
  1388 proof -
  1389   have le_transfer: "(f \<le> g) = (Ifloat (f - g) \<le> 0)" by (auto simp add: le_float_def)
  1390   from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
  1391   with le_transfer have le_transfer': "f \<le> g = (Ifloat (Float a b) \<le> 0)" by simp
  1392   show ?thesis by (simp add: le_transfer' f_diff_g float_le_zero)
  1393 qed
  1394 
  1395 lemma float_less_zero:
  1396   "(Ifloat (Float a b) < 0) = (a < 0)"
  1397   apply (auto simp add: mult_less_0_iff Ifloat.simps)
  1398   done
  1399 
  1400 lemma less_float_def': "f < g = (case f - g of Float a b \<Rightarrow> a < 0)"
  1401 proof -
  1402   have less_transfer: "(f < g) = (Ifloat (f - g) < 0)" by (auto simp add: less_float_def)
  1403   from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto
  1404   with less_transfer have less_transfer': "f < g = (Ifloat (Float a b) < 0)" by simp
  1405   show ?thesis by (simp add: less_transfer' f_diff_g float_less_zero)
  1406 qed
  1407 
  1408 end