(*  Title:      HOL/Library/Float.thy

     Author:     Steven Obua 2008

     Author:     Johannes Hoelzl, TU Muenchen <hoelzl@in.tum.de> 2008 / 2009

 *)

 header {* Floating-Point Numbers *}

 theory Float

 imports Complex_Main

 begin

 definition

   pow2 :: "int \<Rightarrow> real" where

   [simp]: "pow2 a = (if (0 <= a) then (2^(nat a)) else (inverse (2^(nat (-a)))))"

 datatype float = Float int int

 primrec of_float :: "float \<Rightarrow> real" where

   "of_float (Float a b) = real a * pow2 b"

 defs (overloaded)

   real_of_float_def [code_unfold]: "real == of_float"

 primrec mantissa :: "float \<Rightarrow> int" where

   "mantissa (Float a b) = a"

 primrec scale :: "float \<Rightarrow> int" where

   "scale (Float a b) = b"

 instantiation float :: zero begin

 definition zero_float where "0 = Float 0 0"

 instance ..

 end

 instantiation float :: one begin

 definition one_float where "1 = Float 1 0"

 instance ..

 end

 instantiation float :: number begin

 definition number_of_float where "number_of n = Float n 0"

 instance ..

 end

 lemma number_of_float_Float [code_inline, symmetric, code_post]:

   "number_of k = Float (number_of k) 0"

   by (simp add: number_of_float_def number_of_is_id)

 lemma real_of_float_simp[simp]: "real (Float a b) = real a * pow2 b"

   unfolding real_of_float_def using of_float.simps .

 lemma real_of_float_neg_exp: "e < 0 \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto

 lemma real_of_float_nge0_exp: "\<not> 0 \<le> e \<Longrightarrow> real (Float m e) = real m * inverse (2^nat (-e))" by auto

 lemma real_of_float_ge0_exp: "0 \<le> e \<Longrightarrow> real (Float m e) = real m * (2^nat e)" by auto

 lemma Float_num[simp]: shows

    "real (Float 1 0) = 1" and "real (Float 1 1) = 2" and "real (Float 1 2) = 4" and

    "real (Float 1 -1) = 1/2" and "real (Float 1 -2) = 1/4" and "real (Float 1 -3) = 1/8" and

    "real (Float -1 0) = -1" and "real (Float (number_of n) 0) = number_of n"

   by auto

 lemma float_number_of[simp]: "real (number_of x :: float) = number_of x"

   by (simp only:number_of_float_def Float_num[unfolded number_of_is_id])

 lemma float_number_of_int[simp]: "real (Float n 0) = real n"

   by (simp add: Float_num[unfolded number_of_is_id] real_of_float_simp pow2_def)

 lemma pow2_0[simp]: "pow2 0 = 1" by simp

 lemma pow2_1[simp]: "pow2 1 = 2" by simp

 lemma pow2_neg: "pow2 x = inverse (pow2 (-x))" by simp

 declare pow2_def[simp del]

 lemma pow2_add1: "pow2 (1 + a) = 2 * (pow2 a)"

 proof -

   have h: "! n. nat (2 + int n) - Suc 0 = nat (1 + int n)" by arith

   have g: "! a b. a - -1 = a + (1::int)" by arith

   have pos: "! n. pow2 (int n + 1) = 2 * pow2 (int n)"

     apply (auto, induct_tac n)

     apply (simp_all add: pow2_def)

     apply (rule_tac m1="2" and n1="nat (2 + int na)" in ssubst[OF realpow_num_eq_if])

     by (auto simp add: h)

   show ?thesis

   proof (induct a)

     case (1 n)

     from pos show ?case by (simp add: algebra_simps)

   next

     case (2 n)

     show ?case

       apply (auto)

       apply (subst pow2_neg[of "- int n"])

       apply (subst pow2_neg[of "-1 - int n"])

       apply (auto simp add: g pos)

       done

   qed

 qed

 lemma pow2_add: "pow2 (a+b) = (pow2 a) * (pow2 b)"

 proof (induct b)

   case (1 n)

   show ?case

   proof (induct n)

     case 0

     show ?case by simp

   next

     case (Suc m)

     show ?case by (auto simp add: algebra_simps pow2_add1 prems)

   qed

 next

   case (2 n)

   show ?case

   proof (induct n)

     case 0

     show ?case

       apply (auto)

       apply (subst pow2_neg[of "a + -1"])

       apply (subst pow2_neg[of "-1"])

       apply (simp)

       apply (insert pow2_add1[of "-a"])

       apply (simp add: algebra_simps)

       apply (subst pow2_neg[of "-a"])

       apply (simp)

       done

     case (Suc m)

     have a: "int m - (a + -2) =  1 + (int m - a + 1)" by arith

     have b: "int m - -2 = 1 + (int m + 1)" by arith

     show ?case

       apply (auto)

       apply (subst pow2_neg[of "a + (-2 - int m)"])

       apply (subst pow2_neg[of "-2 - int m"])

       apply (auto simp add: algebra_simps)

       apply (subst a)

       apply (subst b)

       apply (simp only: pow2_add1)

       apply (subst pow2_neg[of "int m - a + 1"])

       apply (subst pow2_neg[of "int m + 1"])

       apply auto

       apply (insert prems)

       apply (auto simp add: algebra_simps)

       done

   qed

 qed

 lemma float_components[simp]: "Float (mantissa f) (scale f) = f" by (cases f, auto)

 lemma float_split: "\<exists> a b. x = Float a b" by (cases x, auto)

 lemma float_split2: "(\<forall> a b. x \<noteq> Float a b) = False" by (auto simp add: float_split)

 lemma float_zero[simp]: "real (Float 0 e) = 0" by simp

 lemma abs_div_2_less: "a \<noteq> 0 \<Longrightarrow> a \<noteq> -1 \<Longrightarrow> abs((a::int) div 2) < abs a"

 by arith

 function normfloat :: "float \<Rightarrow> float" where

 "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)"

 by pat_completeness auto

 termination by (relation "measure (nat o abs o mantissa)") (auto intro: abs_div_2_less)

 declare normfloat.simps[simp del]

 theorem normfloat[symmetric, simp]: "real f = real (normfloat f)"

 proof (induct f rule: normfloat.induct)

   case (1 a b)

   have real2: "2 = real (2::int)"

     by auto

   show ?case

     apply (subst normfloat.simps)

     apply (auto simp add: float_zero)

     apply (subst 1[symmetric])

     apply (auto simp add: pow2_add even_def)

     done

 qed

 lemma pow2_neq_zero[simp]: "pow2 x \<noteq> 0"

   by (auto simp add: pow2_def)

 lemma pow2_int: "pow2 (int c) = 2^c"

 by (simp add: pow2_def)

 lemma zero_less_pow2[simp]:

   "0 < pow2 x"

 proof -

   {

     fix y

     have "0 <= y \<Longrightarrow> 0 < pow2 y"

       by (induct y, induct_tac n, simp_all add: pow2_add)

   }

   note helper=this

   show ?thesis

     apply (case_tac "0 <= x")

     apply (simp add: helper)

     apply (subst pow2_neg)

     apply (simp add: helper)

     done

 qed

 lemma normfloat_imp_odd_or_zero: "normfloat f = Float a b \<Longrightarrow> odd a \<or> (a = 0 \<and> b = 0)"

 proof (induct f rule: normfloat.induct)

   case (1 u v)

   from 1 have ab: "normfloat (Float u v) = Float a b" by auto

   {

     assume eu: "even u"

     assume z: "u \<noteq> 0"

     have "normfloat (Float u v) = normfloat (Float (u div 2) (v + 1))"

       apply (subst normfloat.simps)

       by (simp add: eu z)

     with ab have "normfloat (Float (u div 2) (v + 1)) = Float a b" by simp

     with 1 eu z have ?case by auto

   }

   note case1 = this

   {

     assume "odd u \<or> u = 0"

     then have ou: "\<not> (u \<noteq> 0 \<and> even u)" by auto

     have "normfloat (Float u v) = (if u = 0 then Float 0 0 else Float u v)"

       apply (subst normfloat.simps)

       apply (simp add: ou)

       done

     with ab have "Float a b = (if u = 0 then Float 0 0 else Float u v)" by auto

     then have ?case

       apply (case_tac "u=0")

       apply (auto)

       by (insert ou, auto)

   }

   note case2 = this

   show ?case

     apply (case_tac "odd u \<or> u = 0")

     apply (rule case2)

     apply simp

     apply (rule case1)

     apply auto

     done

 qed

 lemma float_eq_odd_helper: 

   assumes odd: "odd a'"

   and floateq: "real (Float a b) = real (Float a' b')"

   shows "b \<le> b'"

 proof - 

   {

     assume bcmp: "b > b'"

     from floateq have eq: "real a * pow2 b = real a' * pow2 b'" by simp

     {

       fix x y z :: real

       assume "y \<noteq> 0"

       then have "(x * inverse y = z) = (x = z * y)"

 	by auto

     }

     note inverse = this

     have eq': "real a * (pow2 (b - b')) = real a'"

       apply (subst diff_int_def)

       apply (subst pow2_add)

       apply (subst pow2_neg[where x = "-b'"])

       apply simp

       apply (subst mult_assoc[symmetric])

       apply (subst inverse)

       apply (simp_all add: eq)

       done

     have "\<exists> z > 0. pow2 (b-b') = 2^z"

       apply (rule exI[where x="nat (b - b')"])

       apply (auto)

       apply (insert bcmp)

       apply simp

       apply (subst pow2_int[symmetric])

       apply auto

       done

     then obtain z where z: "z > 0 \<and> pow2 (b-b') = 2^z" by auto

     with eq' have "real a * 2^z = real a'"

       by auto

     then have "real a * real ((2::int)^z) = real a'"

       by auto

     then have "real (a * 2^z) = real a'"

       apply (subst real_of_int_mult)

       apply simp

       done

     then have a'_rep: "a * 2^z = a'" by arith

     then have "a' = a*2^z" by simp

     with z have "even a'" by simp

     with odd have False by auto

   }

   then show ?thesis by arith

 qed

 lemma float_eq_odd: 

   assumes odd1: "odd a"

   and odd2: "odd a'"

   and floateq: "real (Float a b) = real (Float a' b')"

   shows "a = a' \<and> b = b'"

 proof -

   from 

      float_eq_odd_helper[OF odd2 floateq] 

      float_eq_odd_helper[OF odd1 floateq[symmetric]]

   have beq: "b = b'"  by arith

   with floateq show ?thesis by auto

 qed

 theorem normfloat_unique:

   assumes real_of_float_eq: "real f = real g"

   shows "normfloat f = normfloat g"

 proof - 

   from float_split[of "normfloat f"] obtain a b where normf:"normfloat f = Float a b" by auto

   from float_split[of "normfloat g"] obtain a' b' where normg:"normfloat g = Float a' b'" by auto

   have "real (normfloat f) = real (normfloat g)"

     by (simp add: real_of_float_eq)

   then have float_eq: "real (Float a b) = real (Float a' b')"

     by (simp add: normf normg)

   have ab: "odd a \<or> (a = 0 \<and> b = 0)" by (rule normfloat_imp_odd_or_zero[OF normf])

   have ab': "odd a' \<or> (a' = 0 \<and> b' = 0)" by (rule normfloat_imp_odd_or_zero[OF normg])

   {

     assume odd: "odd a"

     then have "a \<noteq> 0" by (simp add: even_def, arith)

     with float_eq have "a' \<noteq> 0" by auto

     with ab' have "odd a'" by simp

     from odd this float_eq have "a = a' \<and> b = b'" by (rule float_eq_odd)

   }

   note odd_case = this

   {

     assume even: "even a"

     with ab have a0: "a = 0" by simp

     with float_eq have a0': "a' = 0" by auto 

     from a0 a0' ab ab' have "a = a' \<and> b = b'" by auto

   }

   note even_case = this

   from odd_case even_case show ?thesis

     apply (simp add: normf normg)

     apply (case_tac "even a")

     apply auto

     done

 qed

 instantiation float :: plus begin

 fun plus_float where

 [simp del]: "(Float a_m a_e) + (Float b_m b_e) = 

      (if a_e \<le> b_e then Float (a_m + b_m * 2^(nat(b_e - a_e))) a_e 

                    else Float (a_m * 2^(nat (a_e - b_e)) + b_m) b_e)"

 instance ..

 end

 instantiation float :: uminus begin

 primrec uminus_float where [simp del]: "uminus_float (Float m e) = Float (-m) e"

 instance ..

 end

 instantiation float :: minus begin

 definition minus_float where [simp del]: "(z::float) - w = z + (- w)"

 instance ..

 end

 instantiation float :: times begin

 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)"

 instance ..

 end

 primrec float_pprt :: "float \<Rightarrow> float" where

   "float_pprt (Float a e) = (if 0 <= a then (Float a e) else 0)"

 primrec float_nprt :: "float \<Rightarrow> float" where

   "float_nprt (Float a e) = (if 0 <= a then 0 else (Float a e))" 

 instantiation float :: ord begin

 definition le_float_def: "z \<le> (w :: float) \<equiv> real z \<le> real w"

 definition less_float_def: "z < (w :: float) \<equiv> real z < real w"

 instance ..

 end

 lemma real_of_float_add[simp]: "real (a + b) = real a + real (b :: float)"

   by (cases a, cases b, simp add: algebra_simps plus_float.simps, 

       auto simp add: pow2_int[symmetric] pow2_add[symmetric])

 lemma real_of_float_minus[simp]: "real (- a) = - real (a :: float)"

   by (cases a, simp add: uminus_float.simps)

 lemma real_of_float_sub[simp]: "real (a - b) = real a - real (b :: float)"

   by (cases a, cases b, simp add: minus_float_def)

 lemma real_of_float_mult[simp]: "real (a*b) = real a * real (b :: float)"

   by (cases a, cases b, simp add: times_float.simps pow2_add)

 lemma real_of_float_0[simp]: "real (0 :: float) = 0"

   by (auto simp add: zero_float_def float_zero)

 lemma real_of_float_1[simp]: "real (1 :: float) = 1"

   by (auto simp add: one_float_def)

 lemma zero_le_float:

   "(0 <= real (Float a b)) = (0 <= a)"

   apply auto

   apply (auto simp add: zero_le_mult_iff)

   apply (insert zero_less_pow2[of b])

   apply (simp_all)

   done

 lemma float_le_zero:

   "(real (Float a b) <= 0) = (a <= 0)"

   apply auto

   apply (auto simp add: mult_le_0_iff)

   apply (insert zero_less_pow2[of b])

   apply auto

   done

 declare real_of_float_simp[simp del]

 lemma real_of_float_pprt[simp]: "real (float_pprt a) = pprt (real a)"

   by (cases a, auto simp add: float_pprt.simps zero_le_float float_le_zero float_zero)

 lemma real_of_float_nprt[simp]: "real (float_nprt a) = nprt (real a)"

   by (cases a,  auto simp add: float_nprt.simps zero_le_float float_le_zero float_zero)

 instance float :: ab_semigroup_add

 proof (intro_classes)

   fix a b c :: float

   show "a + b + c = a + (b + c)"

     by (cases a, cases b, cases c, auto simp add: algebra_simps power_add[symmetric] plus_float.simps)

 next

   fix a b :: float

   show "a + b = b + a"

     by (cases a, cases b, simp add: plus_float.simps)

 qed

 instance float :: comm_monoid_mult

 proof (intro_classes)

   fix a b c :: float

   show "a * b * c = a * (b * c)"

     by (cases a, cases b, cases c, simp add: times_float.simps)

 next

   fix a b :: float

   show "a * b = b * a"

     by (cases a, cases b, simp add: times_float.simps)

 next

   fix a :: float

   show "1 * a = a"

     by (cases a, simp add: times_float.simps one_float_def)

 qed

 (* Floats do NOT form a cancel_semigroup_add: *)

 lemma "0 + Float 0 1 = 0 + Float 0 2"

   by (simp add: plus_float.simps zero_float_def)

 instance float :: comm_semiring

 proof (intro_classes)

   fix a b c :: float

   show "(a + b) * c = a * c + b * c"

     by (cases a, cases b, cases c, simp, simp add: plus_float.simps times_float.simps algebra_simps)

 qed

 (* Floats do NOT form an order, because "(x < y) = (x <= y & x <> y)" does NOT hold *)

 instance float :: zero_neq_one

 proof (intro_classes)

   show "(0::float) \<noteq> 1"

     by (simp add: zero_float_def one_float_def)

 qed

 lemma float_le_simp: "((x::float) \<le> y) = (0 \<le> y - x)"

   by (auto simp add: le_float_def)

 lemma float_less_simp: "((x::float) < y) = (0 < y - x)"

   by (auto simp add: less_float_def)

 lemma real_of_float_min: "real (min x y :: float) = min (real x) (real y)" unfolding min_def le_float_def by auto

 lemma real_of_float_max: "real (max a b :: float) = max (real a) (real b)" unfolding max_def le_float_def by auto

 lemma float_power: "real (x ^ n :: float) = real x ^ n"

   by (induct n) simp_all

 lemma zero_le_pow2[simp]: "0 \<le> pow2 s"

   apply (subgoal_tac "0 < pow2 s")

   apply (auto simp only:)

   apply auto

   done

 lemma pow2_less_0_eq_False[simp]: "(pow2 s < 0) = False"

   apply auto

   apply (subgoal_tac "0 \<le> pow2 s")

   apply simp

   apply simp

   done

 lemma pow2_le_0_eq_False[simp]: "(pow2 s \<le> 0) = False"

   apply auto

   apply (subgoal_tac "0 < pow2 s")

   apply simp

   apply simp

   done

 lemma float_pos_m_pos: "0 < Float m e \<Longrightarrow> 0 < m"

   unfolding less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff

   by auto

 lemma float_pos_less1_e_neg: assumes "0 < Float m e" and "Float m e < 1" shows "e < 0"

 proof -

   have "0 < m" using float_pos_m_pos `0 < Float m e` by auto

   hence "0 \<le> real m" and "1 \<le> real m" by auto

   show "e < 0"

   proof (rule ccontr)

     assume "\<not> e < 0" hence "0 \<le> e" by auto

     hence "1 \<le> pow2 e" unfolding pow2_def by auto

     from mult_mono[OF `1 \<le> real m` this `0 \<le> real m`]

     have "1 \<le> Float m e" by (simp add: le_float_def real_of_float_simp)

     thus False using `Float m e < 1` unfolding less_float_def le_float_def by auto

   qed

 qed

 lemma float_less1_mantissa_bound: assumes "0 < Float m e" "Float m e < 1" shows "m < 2^(nat (-e))"

 proof -

   have "e < 0" using float_pos_less1_e_neg assms by auto

   have "\<And>x. (0::real) < 2^x" by auto

   have "real m < 2^(nat (-e))" using `Float m e < 1`

     unfolding less_float_def real_of_float_neg_exp[OF `e < 0`] real_of_float_1

           real_mult_less_iff1[of _ _ 1, OF `0 < 2^(nat (-e))`, symmetric] 

           real_mult_assoc by auto

   thus ?thesis unfolding real_of_int_less_iff[symmetric] by auto

 qed

 function bitlen :: "int \<Rightarrow> int" where

 "bitlen 0 = 0" | 

 "bitlen -1 = 1" | 

 "0 < x \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))" | 

 "x < -1 \<Longrightarrow> bitlen x = 1 + (bitlen (x div 2))"

   apply (case_tac "x = 0 \<or> x = -1 \<or> x < -1 \<or> x > 0")

   apply auto

   done

 termination by (relation "measure (nat o abs)", auto)

 lemma bitlen_ge0: "0 \<le> bitlen x" by (induct x rule: bitlen.induct, auto)

 lemma bitlen_ge1: "x \<noteq> 0 \<Longrightarrow> 1 \<le> bitlen x" by (induct x rule: bitlen.induct, auto simp add: bitlen_ge0)

 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")

   using `0 < x`

 proof (induct x rule: bitlen.induct)

   fix x

   assume "0 < x" and hyp: "0 < x div 2 \<Longrightarrow> ?P (x div 2)" hence "0 \<le> x" and "x \<noteq> 0" by auto

   { fix x have "0 \<le> 1 + bitlen x" using bitlen_ge0[of x] by auto } note gt0_pls1 = this

   have "0 < (2::int)" by auto

   show "?P x"

   proof (cases "x = 1")

     case True show "?P x" unfolding True by auto

   next

     case False hence "2 \<le> x" using `0 < x` `x \<noteq> 1` by auto

     hence "2 div 2 \<le> x div 2" by (rule zdiv_mono1, auto)

     hence "0 < x div 2" and "x div 2 \<noteq> 0" by auto

     hence bitlen_s1_ge0: "0 \<le> bitlen (x div 2) - 1" using bitlen_ge1[OF `x div 2 \<noteq> 0`] by auto

     { from hyp[OF `0 < x div 2`]

       have "2 ^ nat (bitlen (x div 2) - 1) \<le> x div 2" by auto

       hence "2 ^ nat (bitlen (x div 2) - 1) * 2 \<le> x div 2 * 2" by (rule mult_right_mono, auto)

       also have "\<dots> \<le> x" using `0 < x` by auto

       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

     } moreover

     { have "x + 1 \<le> x - x mod 2 + 2"

       proof -

 	have "x mod 2 < 2" using `0 < x` by auto

  	hence "x < x - x mod 2 +  2" unfolding algebra_simps by auto

 	thus ?thesis by auto

       qed

       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

       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)

       also have "\<dots> = 2^(1 + nat (bitlen (x div 2)))" unfolding power_Suc2[symmetric] by auto

       finally have "x + 1 \<le> 2^(1 + nat (bitlen (x div 2)))" .

     }

     ultimately show ?thesis

       unfolding bitlen.simps(3)[OF `0 < x`] nat_add_distrib[OF zero_le_one bitlen_ge0]

       unfolding add_commute nat_add_distrib[OF zero_le_one gt0_pls1]

       by auto

   qed

 next

   fix x :: int assume "x < -1" and "0 < x" hence False by auto

   thus "?P x" by auto

 qed auto

 lemma bitlen_bounds: assumes "0 < x" shows "2^nat (bitlen x - 1) \<le> x \<and> x < 2^nat (bitlen x)"

   using bitlen_bounds'[OF `0<x`] by auto

 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"

 proof -

   let ?B = "2^nat(bitlen m - 1)"

   have "?B \<le> m" using bitlen_bounds[OF `0 <m`] ..

   hence "1 * ?B \<le> real m" unfolding real_of_int_le_iff[symmetric] by auto

   thus "1 \<le> real m / ?B" by auto

   have "m \<noteq> 0" using assms by auto

   have "0 \<le> bitlen m - 1" using bitlen_ge1[OF `m \<noteq> 0`] by auto

   have "m < 2^nat(bitlen m)" using bitlen_bounds[OF `0 <m`] ..

   also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto

   also have "\<dots> = ?B * 2" unfolding nat_add_distrib[OF `0 \<le> bitlen m - 1` zero_le_one] by auto

   finally have "real m < 2 * ?B" unfolding real_of_int_less_iff[symmetric] by auto

   hence "real m / ?B < 2 * ?B / ?B" by (rule divide_strict_right_mono, auto)

   thus "real m / ?B < 2" by auto

 qed

 lemma float_gt1_scale: assumes "1 \<le> Float m e"

   shows "0 \<le> e + (bitlen m - 1)"

 proof (cases "0 \<le> e")

   have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto

   hence "0 < m" using float_pos_m_pos by auto

   hence "m \<noteq> 0" by auto

   case True with bitlen_ge1[OF `m \<noteq> 0`] show ?thesis by auto

 next

   have "0 < Float m e" using assms unfolding less_float_def le_float_def by auto

   hence "0 < m" using float_pos_m_pos by auto

   hence "m \<noteq> 0" and "1 < (2::int)" by auto

   case False let ?S = "2^(nat (-e))"

   have "1 \<le> real m * inverse ?S" using assms unfolding le_float_def real_of_float_nge0_exp[OF False] by auto

   hence "1 * ?S \<le> real m * inverse ?S * ?S" by (rule mult_right_mono, auto)

   hence "?S \<le> real m" unfolding mult_assoc by auto

   hence "?S \<le> m" unfolding real_of_int_le_iff[symmetric] by auto

   from this bitlen_bounds[OF `0 < m`, THEN conjunct2]

   have "nat (-e) < (nat (bitlen m))" unfolding power_strict_increasing_iff[OF `1 < 2`, symmetric] by (rule order_le_less_trans)

   hence "-e < bitlen m" using False bitlen_ge0 by auto

   thus ?thesis by auto

 qed

 lemma normalized_float: assumes "m \<noteq> 0" shows "real (Float m (- (bitlen m - 1))) = real m / 2^nat (bitlen m - 1)"

 proof (cases "- (bitlen m - 1) = 0")

   case True show ?thesis unfolding real_of_float_simp pow2_def using True by auto

 next

   case False hence P: "\<not> 0 \<le> - (bitlen m - 1)" using bitlen_ge1[OF `m \<noteq> 0`] by auto

   show ?thesis unfolding real_of_float_nge0_exp[OF P] real_divide_def by auto

 qed

 lemma bitlen_Pls: "bitlen (Int.Pls) = Int.Pls" by (subst Pls_def, subst Pls_def, simp)

 lemma bitlen_Min: "bitlen (Int.Min) = Int.Bit1 Int.Pls" by (subst Min_def, simp add: Bit1_def) 

 lemma bitlen_B0: "bitlen (Int.Bit0 b) = (if iszero b then Int.Pls else Int.succ (bitlen b))"

   apply (auto simp add: iszero_def succ_def)

   apply (simp add: Bit0_def Pls_def)

   apply (subst Bit0_def)

   apply simp

   apply (subgoal_tac "0 < 2 * b \<or> 2 * b < -1")

   apply auto

   done

 lemma bitlen_B1: "bitlen (Int.Bit1 b) = (if iszero (Int.succ b) then Int.Bit1 Int.Pls else Int.succ (bitlen b))"

 proof -

   have h: "! x. (2*x + 1) div 2 = (x::int)"

     by arith    

   show ?thesis

     apply (auto simp add: iszero_def succ_def)

     apply (subst Bit1_def)+

     apply simp

     apply (subgoal_tac "2 * b + 1 = -1")

     apply (simp only:)

     apply simp_all

     apply (subst Bit1_def)

     apply simp

     apply (subgoal_tac "0 < 2 * b + 1 \<or> 2 * b + 1 < -1")

     apply (auto simp add: h)

     done

 qed

 lemma bitlen_number_of: "bitlen (number_of w) = number_of (bitlen w)"

   by (simp add: number_of_is_id)

 lemma [code]: "bitlen x = 

      (if x = 0  then 0 

  else if x = -1 then 1 

                 else (1 + (bitlen (x div 2))))"

   by (cases "x = 0 \<or> x = -1 \<or> 0 < x") auto

 definition lapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"

 where

   "lapprox_posrat prec x y = 

    (let 

        l = nat (int prec + bitlen y - bitlen x) ;

        d = (x * 2^l) div y

     in normfloat (Float d (- (int l))))"

 lemma pow2_minus: "pow2 (-x) = inverse (pow2 x)"

   unfolding pow2_neg[of "-x"] by auto

 lemma lapprox_posrat: 

   assumes x: "0 \<le> x"

   and y: "0 < y"

   shows "real (lapprox_posrat prec x y) \<le> real x / real y"

 proof -

   let ?l = "nat (int prec + bitlen y - bitlen x)"

   have "real (x * 2^?l div y) * inverse (2^?l) \<le> (real (x * 2^?l) / real y) * inverse (2^?l)" 

     by (rule mult_right_mono, fact real_of_int_div4, simp)

   also have "\<dots> \<le> (real x / real y) * 2^?l * inverse (2^?l)" by auto

   finally have "real (x * 2^?l div y) * inverse (2^?l) \<le> real x / real y" unfolding real_mult_assoc by auto

   thus ?thesis unfolding lapprox_posrat_def Let_def normfloat real_of_float_simp

     unfolding pow2_minus pow2_int minus_minus .

 qed

 lemma real_of_int_div_mult: 

   fixes x y c :: int assumes "0 < y" and "0 < c"

   shows "real (x div y) \<le> real (x * c div y) * inverse (real c)"

 proof -

   have "c * (x div y) + 0 \<le> c * x div y" unfolding zdiv_zmult1_eq[of c x y]

     by (rule zadd_left_mono, 

         auto intro!: mult_nonneg_nonneg 

              simp add: pos_imp_zdiv_nonneg_iff[OF `0 < y`] `0 < c`[THEN less_imp_le] pos_mod_sign[OF `0 < y`])

   hence "real (x div y) * real c \<le> real (x * c div y)" 

     unfolding real_of_int_mult[symmetric] real_of_int_le_iff zmult_commute by auto

   hence "real (x div y) * real c * inverse (real c) \<le> real (x * c div y) * inverse (real c)"

     using `0 < c` by auto

   thus ?thesis unfolding real_mult_assoc using `0 < c` by auto

 qed

 lemma lapprox_posrat_bottom: assumes "0 < y"

   shows "real (x div y) \<le> real (lapprox_posrat n x y)" 

 proof -

   have pow: "\<And>x. (0::int) < 2^x" by auto

   show ?thesis

     unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int

     using real_of_int_div_mult[OF `0 < y` pow] by auto

 qed

 lemma lapprox_posrat_nonneg: assumes "0 \<le> x" and "0 < y"

   shows "0 \<le> real (lapprox_posrat n x y)" 

 proof -

   show ?thesis

     unfolding lapprox_posrat_def Let_def real_of_float_add normfloat real_of_float_simp pow2_minus pow2_int

     using pos_imp_zdiv_nonneg_iff[OF `0 < y`] assms by (auto intro!: mult_nonneg_nonneg)

 qed

 definition rapprox_posrat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"

 where

   "rapprox_posrat prec x y = (let

      l = nat (int prec + bitlen y - bitlen x) ;

      X = x * 2^l ;

      d = X div y ;

      m = X mod y

    in normfloat (Float (d + (if m = 0 then 0 else 1)) (- (int l))))"

 lemma rapprox_posrat:

   assumes x: "0 \<le> x"

   and y: "0 < y"

   shows "real x / real y \<le> real (rapprox_posrat prec x y)"

 proof -

   let ?l = "nat (int prec + bitlen y - bitlen x)" let ?X = "x * 2^?l"

   show ?thesis 

   proof (cases "?X mod y = 0")

     case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto

     from real_of_int_div[OF this]

     have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto

     also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto

     finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto

     thus ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True] 

       unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto

   next

     case False

     have "0 \<le> real y" and "real y \<noteq> 0" using `0 < y` by auto

     have "0 \<le> real y * 2^?l" by (rule mult_nonneg_nonneg, rule `0 \<le> real y`, auto)

     have "?X = y * (?X div y) + ?X mod y" by auto

     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])

     also have "\<dots> = y * (?X div y + 1)" unfolding zadd_zmult_distrib2 by auto

     finally have "real ?X \<le> real y * real (?X div y + 1)" unfolding real_of_int_le_iff real_of_int_mult[symmetric] .

     hence "real ?X / (real y * 2^?l) \<le> real y * real (?X div y + 1) / (real y * 2^?l)" 

       by (rule divide_right_mono, simp only: `0 \<le> real y * 2^?l`)

     also have "\<dots> = real y * real (?X div y + 1) / real y / 2^?l" by auto

     also have "\<dots> = real (?X div y + 1) * inverse (2^?l)" unfolding nonzero_mult_divide_cancel_left[OF `real y \<noteq> 0`] 

       unfolding real_divide_def ..

     finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]

       unfolding pow2_minus pow2_int minus_minus by auto

   qed

 qed

 lemma rapprox_posrat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"

   shows "real (rapprox_posrat n x y) \<le> 1"

 proof -

   let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"

   show ?thesis

   proof (cases "?X mod y = 0")

     case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto

     from real_of_int_div[OF this]

     have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto

     also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto

     finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto

     also have "real x / real y \<le> 1" using `0 \<le> x` and `0 < y` and `x \<le> y` by auto

     finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat if_P[OF True]

       unfolding real_of_float_simp pow2_minus pow2_int minus_minus by auto

   next

     case False

     have "x \<noteq> y"

     proof (rule ccontr)

       assume "\<not> x \<noteq> y" hence "x = y" by auto

       have "?X mod y = 0" unfolding `x = y` using mod_mult_self1_is_0 by auto

       thus False using False by auto

     qed

     hence "x < y" using `x \<le> y` by auto

     hence "real x / real y < 1" using `0 < y` and `0 \<le> x` by auto

     from real_of_int_div4[of "?X" y]

     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 .

     also have "\<dots> < 1 * 2^?l" using `real x / real y < 1` by (rule mult_strict_right_mono, auto)

     finally have "?X div y < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto

     hence "?X div y + 1 \<le> 2^?l" by auto

     hence "real (?X div y + 1) * inverse (2^?l) \<le> 2^?l * inverse (2^?l)"

       unfolding real_of_int_le_iff[of _ "2^?l", symmetric] real_of_int_power[symmetric] real_number_of

       by (rule mult_right_mono, auto)

     hence "real (?X div y + 1) * inverse (2^?l) \<le> 1" by auto

     thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]

       unfolding pow2_minus pow2_int minus_minus by auto

   qed

 qed

 lemma zdiv_greater_zero: fixes a b :: int assumes "0 < a" and "a \<le> b"

   shows "0 < b div a"

 proof (rule ccontr)

   have "0 \<le> b" using assms by auto

   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

   have "b = a * (b div a) + b mod a" by auto

   hence "b = b mod a" unfolding `b div a = 0` by auto

   hence "b < a" using `0 < a`[THEN pos_mod_bound, of b] by auto

   thus False using `a \<le> b` by auto

 qed

 lemma rapprox_posrat_less1: assumes "0 \<le> x" and "0 < y" and "2 * x < y" and "0 < n"

   shows "real (rapprox_posrat n x y) < 1"

 proof (cases "x = 0")

   case True thus ?thesis unfolding rapprox_posrat_def True Let_def normfloat real_of_float_simp by auto

 next

   case False hence "0 < x" using `0 \<le> x` by auto

   hence "x < y" using assms by auto

   let ?l = "nat (int n + bitlen y - bitlen x)" let ?X = "x * 2^?l"

   show ?thesis

   proof (cases "?X mod y = 0")

     case True hence "y \<noteq> 0" and "y dvd ?X" using `0 < y` by auto

     from real_of_int_div[OF this]

     have "real (?X div y) * inverse (2 ^ ?l) = real ?X / real y * inverse (2 ^ ?l)" by auto

     also have "\<dots> = real x / real y * (2^?l * inverse (2^?l))" by auto

     finally have "real (?X div y) * inverse (2^?l) = real x / real y" by auto

     also have "real x / real y < 1" using `0 \<le> x` and `0 < y` and `x < y` by auto

     finally show ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_P[OF True]

       unfolding pow2_minus pow2_int minus_minus by auto

   next

     case False

     hence "(real x / real y) < 1 / 2" using `0 < y` and `0 \<le> x` `2 * x < y` by auto

     have "0 < ?X div y"

     proof -

       have "2^nat (bitlen x - 1) \<le> y" and "y < 2^nat (bitlen y)"

 	using bitlen_bounds[OF `0 < x`, THEN conjunct1] bitlen_bounds[OF `0 < y`, THEN conjunct2] `x < y` by auto

       hence "(2::int)^nat (bitlen x - 1) < 2^nat (bitlen y)" by (rule order_le_less_trans)

       hence "bitlen x \<le> bitlen y" by auto

       hence len_less: "nat (bitlen x - 1) \<le> nat (int (n - 1) + bitlen y)" by auto

       have "x \<noteq> 0" and "y \<noteq> 0" using `0 < x` `0 < y` by auto

       have exp_eq: "nat (int (n - 1) + bitlen y) - nat (bitlen x - 1) = ?l"

 	using `bitlen x \<le> bitlen y` bitlen_ge1[OF `x \<noteq> 0`] bitlen_ge1[OF `y \<noteq> 0`] `0 < n` by auto

       have "y * 2^nat (bitlen x - 1) \<le> y * x" 

 	using bitlen_bounds[OF `0 < x`, THEN conjunct1] `0 < y`[THEN less_imp_le] by (rule mult_left_mono)

       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)

       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`)

       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))"

 	unfolding real_of_int_le_iff[symmetric] by auto

       hence "real y \<le> real x * (2^nat (int (n - 1) + bitlen y) / (2^nat (bitlen x - 1)))" 

 	unfolding real_mult_assoc real_divide_def by auto

       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

       finally have "y \<le> x * 2^?l" unfolding exp_eq unfolding real_of_int_le_iff[symmetric] by auto

       thus ?thesis using zdiv_greater_zero[OF `0 < y`] by auto

     qed

     from real_of_int_div4[of "?X" y]

     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 .

     also have "\<dots> < 1/2 * 2^?l" using `real x / real y < 1/2` by (rule mult_strict_right_mono, auto)

     finally have "?X div y * 2 < 2^?l" unfolding real_of_int_less_iff[of _ "2^?l", symmetric] by auto

     hence "?X div y + 1 < 2^?l" using `0 < ?X div y` by auto

     hence "real (?X div y + 1) * inverse (2^?l) < 2^?l * inverse (2^?l)"

       unfolding real_of_int_less_iff[of _ "2^?l", symmetric] real_of_int_power[symmetric] real_number_of

       by (rule mult_strict_right_mono, auto)

     hence "real (?X div y + 1) * inverse (2^?l) < 1" by auto

     thus ?thesis unfolding rapprox_posrat_def Let_def normfloat real_of_float_simp if_not_P[OF False]

       unfolding pow2_minus pow2_int minus_minus by auto

   qed

 qed

 lemma approx_rat_pattern: fixes P and ps :: "nat * int * int"

   assumes Y: "\<And>y prec x. \<lbrakk>y = 0; ps = (prec, x, 0)\<rbrakk> \<Longrightarrow> P" 

   and A: "\<And>x y prec. \<lbrakk>0 \<le> x; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"

   and B: "\<And>x y prec. \<lbrakk>x < 0; 0 < y; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"

   and C: "\<And>x y prec. \<lbrakk>x < 0; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"

   and D: "\<And>x y prec. \<lbrakk>0 \<le> x; y < 0; ps = (prec, x, y)\<rbrakk> \<Longrightarrow> P"

   shows P

 proof -

   obtain prec x y where [simp]: "ps = (prec, x, y)" by (cases ps, auto)

   from Y have "y = 0 \<Longrightarrow> P" by auto

   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 } 

   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 }

   ultimately show P by (cases "y = 0 \<or> 0 < y \<or> y < 0", auto)

 qed

 function lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"

 where

   "y = 0 \<Longrightarrow> lapprox_rat prec x y = 0"

 | "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec x y"

 | "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec (-x) y)"

 | "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = lapprox_posrat prec (-x) (-y)"

 | "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> lapprox_rat prec x y = - (rapprox_posrat prec x (-y))"

 apply simp_all by (rule approx_rat_pattern)

 termination by lexicographic_order

 lemma compute_lapprox_rat[code]:

       "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))) 

                                                              else (if 0 < y then - (rapprox_posrat prec (-x) y) else lapprox_posrat prec (-x) (-y)))"

   by auto

 lemma lapprox_rat: "real (lapprox_rat prec x y) \<le> real x / real y"

 proof -      

   have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto

   show ?thesis

     apply (case_tac "y = 0")

     apply simp

     apply (case_tac "0 \<le> x \<and> 0 < y")

     apply (simp add: lapprox_posrat)

     apply (case_tac "x < 0 \<and> 0 < y")

     apply simp

     apply (subst minus_le_iff)   

     apply (rule h[OF rapprox_posrat])

     apply (simp_all)

     apply (case_tac "x < 0 \<and> y < 0")

     apply simp

     apply (rule h[OF _ lapprox_posrat])

     apply (simp_all)

     apply (case_tac "0 \<le> x \<and> y < 0")

     apply (simp)

     apply (subst minus_le_iff)   

     apply (rule h[OF rapprox_posrat])

     apply simp_all

     apply arith

     done

 qed

 lemma lapprox_rat_bottom: assumes "0 \<le> x" and "0 < y"

   shows "real (x div y) \<le> real (lapprox_rat n x y)" 

   unfolding lapprox_rat.simps(2)[OF assms]  using lapprox_posrat_bottom[OF `0<y`] .

 function rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float"

 where

   "y = 0 \<Longrightarrow> rapprox_rat prec x y = 0"

 | "0 \<le> x \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec x y"

 | "x < 0 \<Longrightarrow> 0 < y \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec (-x) y)"

 | "x < 0 \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = rapprox_posrat prec (-x) (-y)"

 | "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> rapprox_rat prec x y = - (lapprox_posrat prec x (-y))"

 apply simp_all by (rule approx_rat_pattern)

 termination by lexicographic_order

 lemma compute_rapprox_rat[code]:

       "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 

                                                                   (if 0 < y then - (lapprox_posrat prec (-x) y) else rapprox_posrat prec (-x) (-y)))"

   by auto

 lemma rapprox_rat: "real x / real y \<le> real (rapprox_rat prec x y)"

 proof -      

   have h[rule_format]: "! a b b'. b' \<le> b \<longrightarrow> a \<le> b' \<longrightarrow> a \<le> (b::real)" by auto

   show ?thesis

     apply (case_tac "y = 0")

     apply simp

     apply (case_tac "0 \<le> x \<and> 0 < y")

     apply (simp add: rapprox_posrat)

     apply (case_tac "x < 0 \<and> 0 < y")

     apply simp

     apply (subst le_minus_iff)   

     apply (rule h[OF _ lapprox_posrat])

     apply (simp_all)

     apply (case_tac "x < 0 \<and> y < 0")

     apply simp

     apply (rule h[OF rapprox_posrat])

     apply (simp_all)

     apply (case_tac "0 \<le> x \<and> y < 0")

     apply (simp)

     apply (subst le_minus_iff)   

     apply (rule h[OF _ lapprox_posrat])

     apply simp_all

     apply arith

     done

 qed

 lemma rapprox_rat_le1: assumes "0 \<le> x" and "0 < y" and "x \<le> y"

   shows "real (rapprox_rat n x y) \<le> 1"

   unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`] using rapprox_posrat_le1[OF assms] .

 lemma rapprox_rat_neg: assumes "x < 0" and "0 < y"

   shows "real (rapprox_rat n x y) \<le> 0"

   unfolding rapprox_rat.simps(3)[OF assms] using lapprox_posrat_nonneg[of "-x" y n] assms by auto

 lemma rapprox_rat_nonneg_neg: assumes "0 \<le> x" and "y < 0"

   shows "real (rapprox_rat n x y) \<le> 0"

   unfolding rapprox_rat.simps(5)[OF assms] using lapprox_posrat_nonneg[of x "-y" n] assms by auto

 lemma rapprox_rat_nonpos_pos: assumes "x \<le> 0" and "0 < y"

   shows "real (rapprox_rat n x y) \<le> 0"

 proof (cases "x = 0") 

   case True hence "0 \<le> x" by auto show ?thesis unfolding rapprox_rat.simps(2)[OF `0 \<le> x` `0 < y`]

     unfolding True rapprox_posrat_def Let_def by auto

 next

   case False hence "x < 0" using assms by auto

   show ?thesis using rapprox_rat_neg[OF `x < 0` `0 < y`] .

 qed

 fun float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"

 where

   "float_divl prec (Float m1 s1) (Float m2 s2) = 

     (let

        l = lapprox_rat prec m1 m2;

        f = Float 1 (s1 - s2)

      in

        f * l)"     

 lemma float_divl: "real (float_divl prec x y) \<le> real x / real y"

 proof - 

   from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto

   from float_split[of y] obtain my sy where y: "y = Float my sy" by auto

   have "real mx / real my \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"

     apply (case_tac "my = 0")

     apply simp

     apply (case_tac "my > 0")       

     apply (subst pos_le_divide_eq)

     apply simp

     apply (subst pos_le_divide_eq)

     apply (simp add: mult_pos_pos)

     apply simp

     apply (subst pow2_add[symmetric])

     apply simp

     apply (subgoal_tac "my < 0")

     apply auto

     apply (simp add: field_simps)

     apply (subst pow2_add[symmetric])

     apply (simp add: field_simps)

     done

   then have "real (lapprox_rat prec mx my) \<le> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"

     by (rule order_trans[OF lapprox_rat])

   then have "real (lapprox_rat prec mx my) * pow2 (sx - sy) \<le> real mx * pow2 sx / (real my * pow2 sy)"

     apply (subst pos_le_divide_eq[symmetric])

     apply simp_all

     done

   then have "pow2 (sx - sy) * real (lapprox_rat prec mx my) \<le> real mx * pow2 sx / (real my * pow2 sy)"

     by (simp add: algebra_simps)

   then show ?thesis

     by (simp add: x y Let_def real_of_float_simp)

 qed

 lemma float_divl_lower_bound: assumes "0 \<le> x" and "0 < y" shows "0 \<le> float_divl prec x y"

 proof (cases x, cases y)

   fix xm xe ym ye :: int

   assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"

   have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto

   have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto

   have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto

   moreover have "0 \<le> real (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`])

   ultimately show "0 \<le> float_divl prec x y"

     unfolding x_eq y_eq float_divl.simps Let_def le_float_def real_of_float_0 by (auto intro!: mult_nonneg_nonneg)

 qed

 lemma float_divl_pos_less1_bound: assumes "0 < x" and "x < 1" and "0 < prec" shows "1 \<le> float_divl prec 1 x"

 proof (cases x)

   case (Float m e)

   from `0 < x` `x < 1` have "0 < m" "e < 0" using float_pos_m_pos float_pos_less1_e_neg unfolding Float by auto

   let ?b = "nat (bitlen m)" and ?e = "nat (-e)"

   have "1 \<le> m" and "m \<noteq> 0" using `0 < m` by auto

   with bitlen_bounds[OF `0 < m`] have "m < 2^?b" and "(2::int) \<le> 2^?b" by auto

   hence "1 \<le> bitlen m" using power_le_imp_le_exp[of "2::int" 1 ?b] by auto

   hence pow_split: "nat (int prec + bitlen m - 1) = (prec - 1) + ?b" using `0 < prec` by auto

   have pow_not0: "\<And>x. (2::real)^x \<noteq> 0" by auto

   from float_less1_mantissa_bound `0 < x` `x < 1` Float 

   have "m < 2^?e" by auto

   with bitlen_bounds[OF `0 < m`, THEN conjunct1]

   have "(2::int)^nat (bitlen m - 1) < 2^?e" by (rule order_le_less_trans)

   from power_less_imp_less_exp[OF _ this]

   have "bitlen m \<le> - e" by auto

   hence "(2::real)^?b \<le> 2^?e" by auto

   hence "(2::real)^?b * inverse (2^?b) \<le> 2^?e * inverse (2^?b)" by (rule mult_right_mono, auto)

   hence "(1::real) \<le> 2^?e * inverse (2^?b)" by auto

   also

   let ?d = "real (2 ^ nat (int prec + bitlen m - 1) div m) * inverse (2 ^ nat (int prec + bitlen m - 1))"

   { 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)

     also have "\<dots> = 2 ^ nat (int prec + bitlen m - 1)" unfolding pow_split zpower_zadd_distrib by auto

     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)

     hence "2^(prec - 1) \<le> 2 ^ nat (int prec + bitlen m - 1) div m" unfolding div_mult_self2_is_id[OF `m \<noteq> 0`] .

     hence "2^(prec - 1) * inverse (2 ^ nat (int prec + bitlen m - 1)) \<le> ?d"

       unfolding real_of_int_le_iff[of "2^(prec - 1)", symmetric] by auto }

   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"]

   have "2^?e * inverse (2^?b) \<le> 2^?e * ?d" unfolding pow_split power_add by auto

   finally have "1 \<le> 2^?e * ?d" .

   have e_nat: "0 - e = int (nat (-e))" using `e < 0` by auto

   have "bitlen 1 = 1" using bitlen.simps by auto

   show ?thesis 

     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`

     unfolding le_float_def real_of_float_mult normfloat real_of_float_simp pow2_minus pow2_int e_nat

     using `1 \<le> 2^?e * ?d` by (auto simp add: pow2_def)

 qed

 fun float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float"

 where

   "float_divr prec (Float m1 s1) (Float m2 s2) = 

     (let

        r = rapprox_rat prec m1 m2;

        f = Float 1 (s1 - s2)

      in

        f * r)"  

 lemma float_divr: "real x / real y \<le> real (float_divr prec x y)"

 proof - 

   from float_split[of x] obtain mx sx where x: "x = Float mx sx" by auto

   from float_split[of y] obtain my sy where y: "y = Float my sy" by auto

   have "real mx / real my \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"

     apply (case_tac "my = 0")

     apply simp

     apply (case_tac "my > 0")

     apply auto

     apply (subst pos_divide_le_eq)

     apply (rule mult_pos_pos)+

     apply simp_all

     apply (subst pow2_add[symmetric])

     apply simp

     apply (subgoal_tac "my < 0")

     apply auto

     apply (simp add: field_simps)

     apply (subst pow2_add[symmetric])

     apply (simp add: field_simps)

     done

   then have "real (rapprox_rat prec mx my) \<ge> (real mx * pow2 sx / (real my * pow2 sy)) / (pow2 (sx - sy))"

     by (rule order_trans[OF _ rapprox_rat])

   then have "real (rapprox_rat prec mx my) * pow2 (sx - sy) \<ge> real mx * pow2 sx / (real my * pow2 sy)"

     apply (subst pos_divide_le_eq[symmetric])

     apply simp_all

     done

   then show ?thesis

     by (simp add: x y Let_def algebra_simps real_of_float_simp)

 qed

 lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> float_divr prec 1 x"

 proof -

   have "1 \<le> 1 / real x" using `0 < x` and `x < 1` unfolding less_float_def by auto

   also have "\<dots> \<le> real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto

   finally show ?thesis unfolding le_float_def by auto

 qed

 lemma float_divr_nonpos_pos_upper_bound: assumes "x \<le> 0" and "0 < y" shows "float_divr prec x y \<le> 0"

 proof (cases x, cases y)

   fix xm xe ym ye :: int

   assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"

   have "xm \<le> 0" using `x \<le> 0`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 mult_le_0_iff] by auto

   have "0 < ym" using `0 < y`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 zero_less_mult_iff] by auto

   have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto

   moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonpos_pos[OF `xm \<le> 0` `0 < ym`] .

   ultimately show "float_divr prec x y \<le> 0"

     unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)

 qed

 lemma float_divr_nonneg_neg_upper_bound: assumes "0 \<le> x" and "y < 0" shows "float_divr prec x y \<le> 0"

 proof (cases x, cases y)

   fix xm xe ym ye :: int

   assume x_eq: "x = Float xm xe" and y_eq: "y = Float ym ye"

   have "0 \<le> xm" using `0 \<le> x`[unfolded x_eq le_float_def real_of_float_simp real_of_float_0 zero_le_mult_iff] by auto

   have "ym < 0" using `y < 0`[unfolded y_eq less_float_def real_of_float_simp real_of_float_0 mult_less_0_iff] by auto

   hence "0 < - ym" by auto

   have "\<And>n. 0 \<le> real (Float 1 n)" unfolding real_of_float_simp using zero_le_pow2 by auto

   moreover have "real (rapprox_rat prec xm ym) \<le> 0" using rapprox_rat_nonneg_neg[OF `0 \<le> xm` `ym < 0`] .

   ultimately show "float_divr prec x y \<le> 0"

     unfolding x_eq y_eq float_divr.simps Let_def le_float_def real_of_float_0 real_of_float_mult by (auto intro!: mult_nonneg_nonpos)

 qed

 primrec round_down :: "nat \<Rightarrow> float \<Rightarrow> float" where

 "round_down prec (Float m e) = (let d = bitlen m - int prec in

      if 0 < d then let P = 2^nat d ; n = m div P in Float n (e + d)

               else Float m e)"

 primrec round_up :: "nat \<Rightarrow> float \<Rightarrow> float" where

 "round_up prec (Float m e) = (let d = bitlen m - int prec in

   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) 

            else Float m e)"

 lemma round_up: "real x \<le> real (round_up prec x)"

 proof (cases x)

   case (Float m e)

   let ?d = "bitlen m - int prec"

   let ?p = "(2::int)^nat ?d"

   have "0 < ?p" by auto

   show "?thesis"

   proof (cases "0 < ?d")

     case True

     hence pow_d: "pow2 ?d = real ?p" unfolding pow2_int[symmetric] power_real_number_of[symmetric] by auto

     show ?thesis

     proof (cases "m mod ?p = 0")

       case True

       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] .

       have "real (Float m e) = real (Float (m div ?p) (e + ?d))" unfolding real_of_float_simp arg_cong[OF m, of real]

 	by (auto simp add: pow2_add `0 < ?d` pow_d)

       thus ?thesis

 	unfolding Float round_up.simps Let_def if_P[OF `m mod ?p = 0`] if_P[OF `0 < ?d`]

 	by auto

     next

       case False

       have "m = m div ?p * ?p + m mod ?p" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..

       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])

       finally have "real (Float m e) \<le> real (Float (m div ?p + 1) (e + ?d))" unfolding real_of_float_simp add_commute[of e]

 	unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of m, symmetric]

 	by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)

       thus ?thesis

 	unfolding Float round_up.simps Let_def if_not_P[OF `\<not> m mod ?p = 0`] if_P[OF `0 < ?d`] .

     qed

   next

     case False

     show ?thesis

       unfolding Float round_up.simps Let_def if_not_P[OF False] .. 

   qed

 qed

 lemma round_down: "real (round_down prec x) \<le> real x"

 proof (cases x)

   case (Float m e)

   let ?d = "bitlen m - int prec"

   let ?p = "(2::int)^nat ?d"

   have "0 < ?p" by auto

   show "?thesis"

   proof (cases "0 < ?d")

     case True

     hence pow_d: "pow2 ?d = real ?p" unfolding pow2_int[symmetric] power_real_number_of[symmetric] by auto

     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])

     also have "\<dots> \<le> m" unfolding zdiv_zmod_equality2[where k=0, unfolded monoid_add_class.add_0_right] ..

     finally have "real (Float (m div ?p) (e + ?d)) \<le> real (Float m e)" unfolding real_of_float_simp add_commute[of e]

       unfolding pow2_add mult_assoc[symmetric] real_of_int_le_iff[of _ m, symmetric]

       by (auto intro!: mult_mono simp add: pow2_add `0 < ?d` pow_d)

     thus ?thesis

       unfolding Float round_down.simps Let_def if_P[OF `0 < ?d`] .

   next

     case False

     show ?thesis

       unfolding Float round_down.simps Let_def if_not_P[OF False] .. 

   qed

 qed

 definition lb_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where

 "lb_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let

     l = bitlen m - int prec

   in if l > 0 then Float (m div (2^nat l)) (e + l)

               else Float m e)"

 definition ub_mult :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where

 "ub_mult prec x y = (case normfloat (x * y) of Float m e \<Rightarrow> let

     l = bitlen m - int prec

   in if l > 0 then Float (m div (2^nat l) + 1) (e + l)

               else Float m e)"

 lemma lb_mult: "real (lb_mult prec x y) \<le> real (x * y)"

 proof (cases "normfloat (x * y)")

   case (Float m e)

   hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)

   let ?l = "bitlen m - int prec"

   have "real (lb_mult prec x y) \<le> real (normfloat (x * y))"

   proof (cases "?l > 0")

     case False thus ?thesis unfolding lb_mult_def Float Let_def float.cases by auto

   next

     case True

     have "real (m div 2^(nat ?l)) * pow2 ?l \<le> real m"

     proof -

       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] 

 	using `?l > 0` by auto

       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

       also have "\<dots> = real m" unfolding zmod_zdiv_equality[symmetric] ..

       finally show ?thesis by auto

     qed

     thus ?thesis unfolding lb_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add real_mult_commute real_mult_assoc by auto

   qed

   also have "\<dots> = real (x * y)" unfolding normfloat ..

   finally show ?thesis .

 qed

 lemma ub_mult: "real (x * y) \<le> real (ub_mult prec x y)"

 proof (cases "normfloat (x * y)")

   case (Float m e)

   hence "odd m \<or> (m = 0 \<and> e = 0)" by (rule normfloat_imp_odd_or_zero)

   let ?l = "bitlen m - int prec"

   have "real (x * y) = real (normfloat (x * y))" unfolding normfloat ..

   also have "\<dots> \<le> real (ub_mult prec x y)"

   proof (cases "?l > 0")

     case False thus ?thesis unfolding ub_mult_def Float Let_def float.cases by auto

   next

     case True

     have "real m \<le> real (m div 2^(nat ?l) + 1) * pow2 ?l"

     proof -

       have "m mod 2^(nat ?l) < 2^(nat ?l)" by (rule pos_mod_bound) auto

       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

       have "real m = real (2^(nat ?l) * (m div 2^(nat ?l)) + m mod 2^(nat ?l))" unfolding zmod_zdiv_equality[symmetric] ..

       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

       also have "\<dots> \<le> (real (m div 2^(nat ?l)) + 1) * 2^(nat ?l)" unfolding real_add_mult_distrib using mod_uneq by auto

       finally show ?thesis unfolding pow2_int[symmetric] using True by auto

     qed

     thus ?thesis unfolding ub_mult_def Float Let_def float.cases if_P[OF True] real_of_float_simp pow2_add real_mult_commute real_mult_assoc by auto

   qed

   finally show ?thesis .

 qed

 primrec float_abs :: "float \<Rightarrow> float" where

   "float_abs (Float m e) = Float \<bar>m\<bar> e"

 instantiation float :: abs begin

 definition abs_float_def: "\<bar>x\<bar> = float_abs x"

 instance ..

 end

 lemma real_of_float_abs: "real \<bar>x :: float\<bar> = \<bar>real x\<bar>" 

 proof (cases x)

   case (Float m e)

   have "\<bar>real m\<bar> * pow2 e = \<bar>real m * pow2 e\<bar>" unfolding abs_mult by auto

   thus ?thesis unfolding Float abs_float_def float_abs.simps real_of_float_simp by auto

 qed

 primrec floor_fl :: "float \<Rightarrow> float" where

   "floor_fl (Float m e) = (if 0 \<le> e then Float m e

                                   else Float (m div (2 ^ (nat (-e)))) 0)"

 lemma floor_fl: "real (floor_fl x) \<le> real x"

 proof (cases x)

   case (Float m e)

   show ?thesis

   proof (cases "0 \<le> e")

     case False

     hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto

     have "real (Float (m div (2 ^ (nat (-e)))) 0) = real (m div 2 ^ (nat (-e)))" unfolding real_of_float_simp by auto

     also have "\<dots> \<le> real m / real ((2::int) ^ (nat (-e)))" using real_of_int_div4 .

     also have "\<dots> = real m * inverse (2 ^ (nat (-e)))" unfolding power_real_number_of[symmetric] real_divide_def ..

     also have "\<dots> = real (Float m e)" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..

     finally show ?thesis unfolding Float floor_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .

   next

     case True thus ?thesis unfolding Float by auto

   qed

 qed

 lemma floor_pos_exp: assumes floor: "Float m e = floor_fl x" shows "0 \<le> e"

 proof (cases x)

   case (Float mx me)

   from floor[unfolded Float floor_fl.simps] show ?thesis by (cases "0 \<le> me", auto)

 qed

 declare floor_fl.simps[simp del]

 primrec ceiling_fl :: "float \<Rightarrow> float" where

   "ceiling_fl (Float m e) = (if 0 \<le> e then Float m e

                                     else Float (m div (2 ^ (nat (-e))) + 1) 0)"

 lemma ceiling_fl: "real x \<le> real (ceiling_fl x)"

 proof (cases x)

   case (Float m e)

   show ?thesis

   proof (cases "0 \<le> e")

     case False

     hence me_eq: "pow2 (-e) = pow2 (int (nat (-e)))" by auto

     have "real (Float m e) = real m * inverse (2 ^ (nat (-e)))" unfolding real_of_float_simp me_eq pow2_int pow2_neg[of e] ..

     also have "\<dots> = real m / real ((2::int) ^ (nat (-e)))" unfolding power_real_number_of[symmetric] real_divide_def ..

     also have "\<dots> \<le> 1 + real (m div 2 ^ (nat (-e)))" using real_of_int_div3[unfolded diff_le_eq] .

     also have "\<dots> = real (Float (m div (2 ^ (nat (-e))) + 1) 0)" unfolding real_of_float_simp by auto

     finally show ?thesis unfolding Float ceiling_fl.simps if_not_P[OF `\<not> 0 \<le> e`] .

   next

     case True thus ?thesis unfolding Float by auto

   qed

 qed

 declare ceiling_fl.simps[simp del]

 definition lb_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where

 "lb_mod prec x ub lb = x - ceiling_fl (float_divr prec x lb) * ub"

 definition ub_mod :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" where

 "ub_mod prec x ub lb = x - floor_fl (float_divl prec x ub) * lb"

 lemma lb_mod: fixes k :: int assumes "0 \<le> real x" and "real k * y \<le> real x" (is "?k * y \<le> ?x")

   assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")

   shows "real (lb_mod prec x ub lb) \<le> ?x - ?k * y"

 proof -

   have "?lb \<le> ?ub" by (auto!)

   have "0 \<le> ?lb" and "?lb \<noteq> 0" by (auto!)

   have "?k * y \<le> ?x" using assms by auto

   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`])

   also have "\<dots> \<le> real (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)

   finally show ?thesis unfolding lb_mod_def real_of_float_sub real_of_float_mult by auto

 qed

 lemma ub_mod: fixes k :: int and x :: float assumes "0 \<le> real x" and "real x \<le> real k * y" (is "?x \<le> ?k * y")

   assumes "0 < real lb" "real lb \<le> y" (is "?lb \<le> y") "y \<le> real ub" (is "y \<le> ?ub")

   shows "?x - ?k * y \<le> real (ub_mod prec x ub lb)"

 proof -

   have "?lb \<le> ?ub" by (auto!)

   hence "0 \<le> ?lb" and "0 \<le> ?ub" and "?ub \<noteq> 0" by (auto!)

   have "real (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)

   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`])

   also have "\<dots> \<le> ?k * y" using assms by auto

   finally show ?thesis unfolding ub_mod_def real_of_float_sub real_of_float_mult by auto

 qed

 lemma le_float_def': "f \<le> g = (case f - g of Float a b \<Rightarrow> a \<le> 0)"

 proof -

   have le_transfer: "(f \<le> g) = (real (f - g) \<le> 0)" by (auto simp add: le_float_def)

   from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto

   with le_transfer have le_transfer': "f \<le> g = (real (Float a b) \<le> 0)" by simp

   show ?thesis by (simp add: le_transfer' f_diff_g float_le_zero)

 qed

 lemma float_less_zero:

   "(real (Float a b) < 0) = (a < 0)"

   apply (auto simp add: mult_less_0_iff real_of_float_simp)

   done

 lemma less_float_def': "f < g = (case f - g of Float a b \<Rightarrow> a < 0)"

 proof -

   have less_transfer: "(f < g) = (real (f - g) < 0)" by (auto simp add: less_float_def)

   from float_split[of "f - g"] obtain a b where f_diff_g: "f - g = Float a b" by auto

   with less_transfer have less_transfer': "f < g = (real (Float a b) < 0)" by simp

   show ?thesis by (simp add: less_transfer' f_diff_g float_less_zero)

 qed

 end