(*  Title:      HOL/Library/Float.thy

     Author:     Johannes Hölzl, Fabian Immler

     Copyright   2012  TU München

 *)

 header {* Floating-Point Numbers *}

 theory Float

 imports Complex_Main Lattice_Algebras

 begin

 definition "float = {m * 2 powr e | (m :: int) (e :: int). True}"

 typedef float = float

   morphisms real_of_float float_of

   unfolding float_def by auto

 defs (overloaded)

   real_of_float_def[code_unfold]: "real \<equiv> real_of_float"

 lemma type_definition_float': "type_definition real float_of float"

   using type_definition_float unfolding real_of_float_def .

 setup_lifting (no_code) type_definition_float'

 lemmas float_of_inject[simp]

 declare [[coercion "real :: float \<Rightarrow> real"]]

 lemma real_of_float_eq:

   fixes f1 f2 :: float shows "f1 = f2 \<longleftrightarrow> real f1 = real f2"

   unfolding real_of_float_def real_of_float_inject ..

 lemma float_of_real[simp]: "float_of (real x) = x"

   unfolding real_of_float_def by (rule real_of_float_inverse)

 lemma real_float[simp]: "x \<in> float \<Longrightarrow> real (float_of x) = x"

   unfolding real_of_float_def by (rule float_of_inverse)

 subsection {* Real operations preserving the representation as floating point number *}

 lemma floatI: fixes m e :: int shows "m * 2 powr e = x \<Longrightarrow> x \<in> float"

   by (auto simp: float_def)

 lemma zero_float[simp]: "0 \<in> float" by (auto simp: float_def)

 lemma one_float[simp]: "1 \<in> float" by (intro floatI[of 1 0]) simp

 lemma numeral_float[simp]: "numeral i \<in> float" by (intro floatI[of "numeral i" 0]) simp

 lemma neg_numeral_float[simp]: "- numeral i \<in> float" by (intro floatI[of "- numeral i" 0]) simp

 lemma real_of_int_float[simp]: "real (x :: int) \<in> float" by (intro floatI[of x 0]) simp

 lemma real_of_nat_float[simp]: "real (x :: nat) \<in> float" by (intro floatI[of x 0]) simp

 lemma two_powr_int_float[simp]: "2 powr (real (i::int)) \<in> float" by (intro floatI[of 1 i]) simp

 lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) \<in> float" by (intro floatI[of 1 i]) simp

 lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) \<in> float" by (intro floatI[of 1 "-i"]) simp

 lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) \<in> float" by (intro floatI[of 1 "-i"]) simp

 lemma two_powr_numeral_float[simp]: "2 powr numeral i \<in> float" by (intro floatI[of 1 "numeral i"]) simp

 lemma two_powr_neg_numeral_float[simp]: "2 powr - numeral i \<in> float" by (intro floatI[of 1 "- numeral i"]) simp

 lemma two_pow_float[simp]: "2 ^ n \<in> float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)

 lemma real_of_float_float[simp]: "real (f::float) \<in> float" by (cases f) simp

 lemma plus_float[simp]: "r \<in> float \<Longrightarrow> p \<in> float \<Longrightarrow> r + p \<in> float"

   unfolding float_def

 proof (safe, simp)

   fix e1 m1 e2 m2 :: int

   { fix e1 m1 e2 m2 :: int assume "e1 \<le> e2"

     then have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"

       by (simp add: powr_realpow[symmetric] powr_divide2[symmetric] field_simps)

     then have "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"

       by blast }

   note * = this

   show "\<exists>(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"

   proof (cases e1 e2 rule: linorder_le_cases)

     assume "e2 \<le> e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)

   qed (rule *)

 qed

 lemma uminus_float[simp]: "x \<in> float \<Longrightarrow> -x \<in> float"

   apply (auto simp: float_def)

   apply hypsubst_thin

   apply (rule_tac x="-x" in exI)

   apply (rule_tac x="xa" in exI)

   apply (simp add: field_simps)

   done

 lemma times_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x * y \<in> float"

   apply (auto simp: float_def)

   apply hypsubst_thin

   apply (rule_tac x="x * xa" in exI)

   apply (rule_tac x="xb + xc" in exI)

   apply (simp add: powr_add)

   done

 lemma minus_float[simp]: "x \<in> float \<Longrightarrow> y \<in> float \<Longrightarrow> x - y \<in> float"

   using plus_float [of x "- y"] by simp

 lemma abs_float[simp]: "x \<in> float \<Longrightarrow> abs x \<in> float"

   by (cases x rule: linorder_cases[of 0]) auto

 lemma sgn_of_float[simp]: "x \<in> float \<Longrightarrow> sgn x \<in> float"

   by (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)

 lemma div_power_2_float[simp]: "x \<in> float \<Longrightarrow> x / 2^d \<in> float"

   apply (auto simp add: float_def)

   apply hypsubst_thin

   apply (rule_tac x="x" in exI)

   apply (rule_tac x="xa - d" in exI)

   apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])

   done

 lemma div_power_2_int_float[simp]: "x \<in> float \<Longrightarrow> x / (2::int)^d \<in> float"

   apply (auto simp add: float_def)

   apply hypsubst_thin

   apply (rule_tac x="x" in exI)

   apply (rule_tac x="xa - d" in exI)

   apply (simp add: powr_realpow[symmetric] field_simps powr_add[symmetric])

   done

 lemma div_numeral_Bit0_float[simp]:

   assumes x: "x / numeral n \<in> float" shows "x / (numeral (Num.Bit0 n)) \<in> float"

 proof -

   have "(x / numeral n) / 2^1 \<in> float"

     by (intro x div_power_2_float)

   also have "(x / numeral n) / 2^1 = x / (numeral (Num.Bit0 n))"

     by (induct n) auto

   finally show ?thesis .

 qed

 lemma div_neg_numeral_Bit0_float[simp]:

   assumes x: "x / numeral n \<in> float" shows "x / (- numeral (Num.Bit0 n)) \<in> float"

 proof -

   have "- (x / numeral (Num.Bit0 n)) \<in> float" using x by simp

   also have "- (x / numeral (Num.Bit0 n)) = x / - numeral (Num.Bit0 n)"

     by simp

   finally show ?thesis .

 qed

 lift_definition Float :: "int \<Rightarrow> int \<Rightarrow> float" is "\<lambda>(m::int) (e::int). m * 2 powr e" by simp

 declare Float.rep_eq[simp]

 lemma compute_real_of_float[code]:

   "real_of_float (Float m e) = (if e \<ge> 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))"

 by (simp add: real_of_float_def[symmetric] powr_int)

 code_datatype Float

 subsection {* Arithmetic operations on floating point numbers *}

 instantiation float :: "{ring_1, linorder, linordered_ring, linordered_idom, numeral, equal}"

 begin

 lift_definition zero_float :: float is 0 by simp

 declare zero_float.rep_eq[simp]

 lift_definition one_float :: float is 1 by simp

 declare one_float.rep_eq[simp]

 lift_definition plus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op +" by simp

 declare plus_float.rep_eq[simp]

 lift_definition times_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op *" by simp

 declare times_float.rep_eq[simp]

 lift_definition minus_float :: "float \<Rightarrow> float \<Rightarrow> float" is "op -" by simp

 declare minus_float.rep_eq[simp]

 lift_definition uminus_float :: "float \<Rightarrow> float" is "uminus" by simp

 declare uminus_float.rep_eq[simp]

 lift_definition abs_float :: "float \<Rightarrow> float" is abs by simp

 declare abs_float.rep_eq[simp]

 lift_definition sgn_float :: "float \<Rightarrow> float" is sgn by simp

 declare sgn_float.rep_eq[simp]

 lift_definition equal_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op = :: real \<Rightarrow> real \<Rightarrow> bool" .

 lift_definition less_eq_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op \<le>" .

 declare less_eq_float.rep_eq[simp]

 lift_definition less_float :: "float \<Rightarrow> float \<Rightarrow> bool" is "op <" .

 declare less_float.rep_eq[simp]

 instance

   proof qed (transfer, fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+

 end

 lemma real_of_float_power[simp]: fixes f::float shows "real (f^n) = real f^n"

   by (induct n) simp_all

 lemma fixes x y::float

   shows real_of_float_min: "real (min x y) = min (real x) (real y)"

     and real_of_float_max: "real (max x y) = max (real x) (real y)"

   by (simp_all add: min_def max_def)

 instance float :: unbounded_dense_linorder

 proof

   fix a b :: float

   show "\<exists>c. a < c"

     apply (intro exI[of _ "a + 1"])

     apply transfer

     apply simp

     done

   show "\<exists>c. c < a"

     apply (intro exI[of _ "a - 1"])

     apply transfer

     apply simp

     done

   assume "a < b"

   then show "\<exists>c. a < c \<and> c < b"

     apply (intro exI[of _ "(a + b) * Float 1 -1"])

     apply transfer

     apply (simp add: powr_minus)

     done

 qed

 instantiation float :: lattice_ab_group_add

 begin

 definition inf_float::"float\<Rightarrow>float\<Rightarrow>float"

 where "inf_float a b = min a b"

 definition sup_float::"float\<Rightarrow>float\<Rightarrow>float"

 where "sup_float a b = max a b"

 instance

   by default

      (transfer, simp_all add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+

 end

 lemma float_numeral[simp]: "real (numeral x :: float) = numeral x"

   apply (induct x)

   apply simp

   apply (simp_all only: numeral_Bit0 numeral_Bit1 real_of_float_eq real_float

                   plus_float.rep_eq one_float.rep_eq plus_float numeral_float one_float)

   done

 lemma transfer_numeral [transfer_rule]:

   "rel_fun (op =) pcr_float (numeral :: _ \<Rightarrow> real) (numeral :: _ \<Rightarrow> float)"

   unfolding rel_fun_def float.pcr_cr_eq  cr_float_def by simp

 lemma float_neg_numeral[simp]: "real (- numeral x :: float) = - numeral x"

   by simp

 lemma transfer_neg_numeral [transfer_rule]:

   "rel_fun (op =) pcr_float (- numeral :: _ \<Rightarrow> real) (- numeral :: _ \<Rightarrow> float)"

   unfolding rel_fun_def float.pcr_cr_eq cr_float_def by simp

 lemma

   shows float_of_numeral[simp]: "numeral k = float_of (numeral k)"

     and float_of_neg_numeral[simp]: "- numeral k = float_of (- numeral k)"

   unfolding real_of_float_eq by simp_all

 subsection {* Represent floats as unique mantissa and exponent *}

 lemma int_induct_abs[case_names less]:

   fixes j :: int

   assumes H: "\<And>n. (\<And>i. \<bar>i\<bar> < \<bar>n\<bar> \<Longrightarrow> P i) \<Longrightarrow> P n"

   shows "P j"

 proof (induct "nat \<bar>j\<bar>" arbitrary: j rule: less_induct)

   case less show ?case by (rule H[OF less]) simp

 qed

 lemma int_cancel_factors:

   fixes n :: int assumes "1 < r" shows "n = 0 \<or> (\<exists>k i. n = k * r ^ i \<and> \<not> r dvd k)"

 proof (induct n rule: int_induct_abs)

   case (less n)

   { fix m assume n: "n \<noteq> 0" "n = m * r"

     then have "\<bar>m \<bar> < \<bar>n\<bar>"

       by (metis abs_dvd_iff abs_ge_self assms comm_semiring_1_class.normalizing_semiring_rules(7)

                 dvd_imp_le_int dvd_refl dvd_triv_right linorder_neq_iff linorder_not_le

                 mult_eq_0_iff zdvd_mult_cancel1)

     from less[OF this] n have "\<exists>k i. n = k * r ^ Suc i \<and> \<not> r dvd k" by auto }

   then show ?case

     by (metis comm_semiring_1_class.normalizing_semiring_rules(12,7) dvdE power_0)

 qed

 lemma mult_powr_eq_mult_powr_iff_asym:

   fixes m1 m2 e1 e2 :: int

   assumes m1: "\<not> 2 dvd m1" and "e1 \<le> e2"

   shows "m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"

 proof

   have "m1 \<noteq> 0" using m1 unfolding dvd_def by auto

   assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"

   with `e1 \<le> e2` have "m1 = m2 * 2 powr nat (e2 - e1)"

     by (simp add: powr_divide2[symmetric] field_simps)

   also have "\<dots> = m2 * 2^nat (e2 - e1)"

     by (simp add: powr_realpow)

   finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"

     unfolding real_of_int_inject .

   with m1 have "m1 = m2"

     by (cases "nat (e2 - e1)") (auto simp add: dvd_def)

   then show "m1 = m2 \<and> e1 = e2"

     using eq `m1 \<noteq> 0` by (simp add: powr_inj)

 qed simp

 lemma mult_powr_eq_mult_powr_iff:

   fixes m1 m2 e1 e2 :: int

   shows "\<not> 2 dvd m1 \<Longrightarrow> \<not> 2 dvd m2 \<Longrightarrow> m1 * 2 powr e1 = m2 * 2 powr e2 \<longleftrightarrow> m1 = m2 \<and> e1 = e2"

   using mult_powr_eq_mult_powr_iff_asym[of m1 e1 e2 m2]

   using mult_powr_eq_mult_powr_iff_asym[of m2 e2 e1 m1]

   by (cases e1 e2 rule: linorder_le_cases) auto

 lemma floatE_normed:

   assumes x: "x \<in> float"

   obtains (zero) "x = 0"

    | (powr) m e :: int where "x = m * 2 powr e" "\<not> 2 dvd m" "x \<noteq> 0"

 proof atomize_elim

   { assume "x \<noteq> 0"

     from x obtain m e :: int where x: "x = m * 2 powr e" by (auto simp: float_def)

     with `x \<noteq> 0` int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "\<not> 2 dvd k"

       by auto

     with `\<not> 2 dvd k` x have "\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m"

       by (rule_tac exI[of _ "k"], rule_tac exI[of _ "e + int i"])

          (simp add: powr_add powr_realpow) }

   then show "x = 0 \<or> (\<exists>(m::int) (e::int). x = m * 2 powr e \<and> \<not> (2::int) dvd m \<and> x \<noteq> 0)"

     by blast

 qed

 lemma float_normed_cases:

   fixes f :: float

   obtains (zero) "f = 0"

    | (powr) m e :: int where "real f = m * 2 powr e" "\<not> 2 dvd m" "f \<noteq> 0"

 proof (atomize_elim, induct f)

   case (float_of y) then show ?case

     by (cases rule: floatE_normed) (auto simp: zero_float_def)

 qed

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

   "mantissa f = fst (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)

    \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"

 definition exponent :: "float \<Rightarrow> int" where

   "exponent f = snd (SOME p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)

    \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p))"

 lemma

   shows exponent_0[simp]: "exponent (float_of 0) = 0" (is ?E)

     and mantissa_0[simp]: "mantissa (float_of 0) = 0" (is ?M)

 proof -

   have "\<And>p::int \<times> int. fst p = 0 \<and> snd p = 0 \<longleftrightarrow> p = (0, 0)" by auto

   then show ?E ?M

     by (auto simp add: mantissa_def exponent_def zero_float_def)

 qed

 lemma

   shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)

     and mantissa_not_dvd: "f \<noteq> (float_of 0) \<Longrightarrow> \<not> 2 dvd mantissa f" (is "_ \<Longrightarrow> ?D")

 proof cases

   assume [simp]: "f \<noteq> (float_of 0)"

   have "f = mantissa f * 2 powr exponent f \<and> \<not> 2 dvd mantissa f"

   proof (cases f rule: float_normed_cases)

     case (powr m e)

     then have "\<exists>p::int \<times> int. (f = 0 \<and> fst p = 0 \<and> snd p = 0)

      \<or> (f \<noteq> 0 \<and> real f = real (fst p) * 2 powr real (snd p) \<and> \<not> 2 dvd fst p)"

       by auto

     then show ?thesis

       unfolding exponent_def mantissa_def

       by (rule someI2_ex) (simp add: zero_float_def)

   qed (simp add: zero_float_def)

   then show ?E ?D by auto

 qed simp

 lemma mantissa_noteq_0: "f \<noteq> float_of 0 \<Longrightarrow> mantissa f \<noteq> 0"

   using mantissa_not_dvd[of f] by auto

 lemma

   fixes m e :: int

   defines "f \<equiv> float_of (m * 2 powr e)"

   assumes dvd: "\<not> 2 dvd m"

   shows mantissa_float: "mantissa f = m" (is "?M")

     and exponent_float: "m \<noteq> 0 \<Longrightarrow> exponent f = e" (is "_ \<Longrightarrow> ?E")

 proof cases

   assume "m = 0" with dvd show "mantissa f = m" by auto

 next

   assume "m \<noteq> 0"

   then have f_not_0: "f \<noteq> float_of 0" by (simp add: f_def)

   from mantissa_exponent[of f]

   have "m * 2 powr e = mantissa f * 2 powr exponent f"

     by (auto simp add: f_def)

   then show "?M" "?E"

     using mantissa_not_dvd[OF f_not_0] dvd

     by (auto simp: mult_powr_eq_mult_powr_iff)

 qed

 subsection {* Compute arithmetic operations *}

 lemma Float_mantissa_exponent: "Float (mantissa f) (exponent f) = f"

   unfolding real_of_float_eq mantissa_exponent[of f] by simp

 lemma Float_cases[case_names Float, cases type: float]:

   fixes f :: float

   obtains (Float) m e :: int where "f = Float m e"

   using Float_mantissa_exponent[symmetric]

   by (atomize_elim) auto

 lemma denormalize_shift:

   assumes f_def: "f \<equiv> Float m e" and not_0: "f \<noteq> float_of 0"

   obtains i where "m = mantissa f * 2 ^ i" "e = exponent f - i"

 proof

   from mantissa_exponent[of f] f_def

   have "m * 2 powr e = mantissa f * 2 powr exponent f"

     by simp

   then have eq: "m = mantissa f * 2 powr (exponent f - e)"

     by (simp add: powr_divide2[symmetric] field_simps)

   moreover

   have "e \<le> exponent f"

   proof (rule ccontr)

     assume "\<not> e \<le> exponent f"

     then have pos: "exponent f < e" by simp

     then have "2 powr (exponent f - e) = 2 powr - real (e - exponent f)"

       by simp

     also have "\<dots> = 1 / 2^nat (e - exponent f)"

       using pos by (simp add: powr_realpow[symmetric] powr_divide2[symmetric])

     finally have "m * 2^nat (e - exponent f) = real (mantissa f)"

       using eq by simp

     then have "mantissa f = m * 2^nat (e - exponent f)"

       unfolding real_of_int_inject by simp

     with `exponent f < e` have "2 dvd mantissa f"

       apply (intro dvdI[where k="m * 2^(nat (e-exponent f)) div 2"])

       apply (cases "nat (e - exponent f)")

       apply auto

       done

     then show False using mantissa_not_dvd[OF not_0] by simp

   qed

   ultimately have "real m = mantissa f * 2^nat (exponent f - e)"

     by (simp add: powr_realpow[symmetric])

   with `e \<le> exponent f`

   show "m = mantissa f * 2 ^ nat (exponent f - e)" "e = exponent f - nat (exponent f - e)"

     unfolding real_of_int_inject by auto

 qed

 lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"

   by transfer simp

 hide_fact (open) compute_float_zero

 lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"

   by transfer simp

 hide_fact (open) compute_float_one

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

   [simp]: "normfloat x = x"

 lemma compute_normfloat[code]: "normfloat (Float m e) =

   (if m mod 2 = 0 \<and> m \<noteq> 0 then normfloat (Float (m div 2) (e + 1))

                            else if m = 0 then 0 else Float m e)"

   unfolding normfloat_def

   by transfer (auto simp add: powr_add zmod_eq_0_iff)

 hide_fact (open) compute_normfloat

 lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"

   by transfer simp

 hide_fact (open) compute_float_numeral

 lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"

   by transfer simp

 hide_fact (open) compute_float_neg_numeral

 lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"

   by transfer simp

 hide_fact (open) compute_float_uminus

 lemma compute_float_times[code]: "Float m1 e1 * Float m2 e2 = Float (m1 * m2) (e1 + e2)"

   by transfer (simp add: field_simps powr_add)

 hide_fact (open) compute_float_times

 lemma compute_float_plus[code]: "Float m1 e1 + Float m2 e2 =

   (if m1 = 0 then Float m2 e2 else if m2 = 0 then Float m1 e1 else

   if e1 \<le> e2 then Float (m1 + m2 * 2^nat (e2 - e1)) e1

               else Float (m2 + m1 * 2^nat (e1 - e2)) e2)"

   by transfer (simp add: field_simps powr_realpow[symmetric] powr_divide2[symmetric])

 hide_fact (open) compute_float_plus

 lemma compute_float_minus[code]: fixes f g::float shows "f - g = f + (-g)"

   by simp

 hide_fact (open) compute_float_minus

 lemma compute_float_sgn[code]: "sgn (Float m1 e1) = (if 0 < m1 then 1 else if m1 < 0 then -1 else 0)"

   by transfer (simp add: sgn_times)

 hide_fact (open) compute_float_sgn

 lift_definition is_float_pos :: "float \<Rightarrow> bool" is "op < 0 :: real \<Rightarrow> bool" .

 lemma compute_is_float_pos[code]: "is_float_pos (Float m e) \<longleftrightarrow> 0 < m"

   by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])

 hide_fact (open) compute_is_float_pos

 lemma compute_float_less[code]: "a < b \<longleftrightarrow> is_float_pos (b - a)"

   by transfer (simp add: field_simps)

 hide_fact (open) compute_float_less

 lift_definition is_float_nonneg :: "float \<Rightarrow> bool" is "op \<le> 0 :: real \<Rightarrow> bool" .

 lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) \<longleftrightarrow> 0 \<le> m"

   by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])

 hide_fact (open) compute_is_float_nonneg

 lemma compute_float_le[code]: "a \<le> b \<longleftrightarrow> is_float_nonneg (b - a)"

   by transfer (simp add: field_simps)

 hide_fact (open) compute_float_le

 lift_definition is_float_zero :: "float \<Rightarrow> bool"  is "op = 0 :: real \<Rightarrow> bool" .

 lemma compute_is_float_zero[code]: "is_float_zero (Float m e) \<longleftrightarrow> 0 = m"

   by transfer (auto simp add: is_float_zero_def)

 hide_fact (open) compute_is_float_zero

 lemma compute_float_abs[code]: "abs (Float m e) = Float (abs m) e"

   by transfer (simp add: abs_mult)

 hide_fact (open) compute_float_abs

 lemma compute_float_eq[code]: "equal_class.equal f g = is_float_zero (f - g)"

   by transfer simp

 hide_fact (open) compute_float_eq

 subsection {* Rounding Real numbers *}

 definition round_down :: "int \<Rightarrow> real \<Rightarrow> real" where

   "round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"

 definition round_up :: "int \<Rightarrow> real \<Rightarrow> real" where

   "round_up prec x = ceiling (x * 2 powr prec) * 2 powr -prec"

 lemma round_down_float[simp]: "round_down prec x \<in> float"

   unfolding round_down_def

   by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)

 lemma round_up_float[simp]: "round_up prec x \<in> float"

   unfolding round_up_def

   by (auto intro!: times_float simp: real_of_int_minus[symmetric] simp del: real_of_int_minus)

 lemma round_up: "x \<le> round_up prec x"

   by (simp add: powr_minus_divide le_divide_eq round_up_def)

 lemma round_down: "round_down prec x \<le> x"

   by (simp add: powr_minus_divide divide_le_eq round_down_def)

 lemma round_up_0[simp]: "round_up p 0 = 0"

   unfolding round_up_def by simp

 lemma round_down_0[simp]: "round_down p 0 = 0"

   unfolding round_down_def by simp

 lemma round_up_diff_round_down:

   "round_up prec x - round_down prec x \<le> 2 powr -prec"

 proof -

   have "round_up prec x - round_down prec x =

     (ceiling (x * 2 powr prec) - floor (x * 2 powr prec)) * 2 powr -prec"

     by (simp add: round_up_def round_down_def field_simps)

   also have "\<dots> \<le> 1 * 2 powr -prec"

     by (rule mult_mono)

        (auto simp del: real_of_int_diff

              simp: real_of_int_diff[symmetric] real_of_int_le_one_cancel_iff ceiling_diff_floor_le_1)

   finally show ?thesis by simp

 qed

 lemma round_down_shift: "round_down p (x * 2 powr k) = 2 powr k * round_down (p + k) x"

   unfolding round_down_def

   by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])

     (simp add: powr_add[symmetric])

 lemma round_up_shift: "round_up p (x * 2 powr k) = 2 powr k * round_up (p + k) x"

   unfolding round_up_def

   by (simp add: powr_add powr_mult field_simps powr_divide2[symmetric])

     (simp add: powr_add[symmetric])

 subsection {* Rounding Floats *}

 lift_definition float_up :: "int \<Rightarrow> float \<Rightarrow> float" is round_up by simp

 declare float_up.rep_eq[simp]

 lemma round_up_correct:

   shows "round_up e f - f \<in> {0..2 powr -e}"

 unfolding atLeastAtMost_iff

 proof

   have "round_up e f - f \<le> round_up e f - round_down e f" using round_down by simp

   also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp

   finally show "round_up e f - f \<le> 2 powr real (- e)"

     by simp

 qed (simp add: algebra_simps round_up)

 lemma float_up_correct:

   shows "real (float_up e f) - real f \<in> {0..2 powr -e}"

   by transfer (rule round_up_correct)

 lift_definition float_down :: "int \<Rightarrow> float \<Rightarrow> float" is round_down by simp

 declare float_down.rep_eq[simp]

 lemma round_down_correct:

   shows "f - (round_down e f) \<in> {0..2 powr -e}"

 unfolding atLeastAtMost_iff

 proof

   have "f - round_down e f \<le> round_up e f - round_down e f" using round_up by simp

   also have "\<dots> \<le> 2 powr -e" using round_up_diff_round_down by simp

   finally show "f - round_down e f \<le> 2 powr real (- e)"

     by simp

 qed (simp add: algebra_simps round_down)

 lemma float_down_correct:

   shows "real f - real (float_down e f) \<in> {0..2 powr -e}"

   by transfer (rule round_down_correct)

 lemma compute_float_down[code]:

   "float_down p (Float m e) =

     (if p + e < 0 then Float (m div 2^nat (-(p + e))) (-p) else Float m e)"

 proof cases

   assume "p + e < 0"

   hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"

     using powr_realpow[of 2 "nat (-(p + e))"] by simp

   also have "... = 1 / 2 powr p / 2 powr e"

     unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)

   finally show ?thesis

     using `p + e < 0`

     by transfer (simp add: ac_simps round_down_def floor_divide_eq_div[symmetric])

 next

   assume "\<not> p + e < 0"

   then have r: "real e + real p = real (nat (e + p))" by simp

   have r: "\<lfloor>(m * 2 powr e) * 2 powr real p\<rfloor> = (m * 2 powr e) * 2 powr real p"

     by (auto intro: exI[where x="m*2^nat (e+p)"]

              simp add: ac_simps powr_add[symmetric] r powr_realpow)

   with `\<not> p + e < 0` show ?thesis

     by transfer

        (auto simp add: round_down_def field_simps powr_add powr_minus inverse_eq_divide)

 qed

 hide_fact (open) compute_float_down

 lemma abs_round_down_le: "\<bar>f - (round_down e f)\<bar> \<le> 2 powr -e"

   using round_down_correct[of f e] by simp

 lemma abs_round_up_le: "\<bar>f - (round_up e f)\<bar> \<le> 2 powr -e"

   using round_up_correct[of e f] by simp

 lemma round_down_nonneg: "0 \<le> s \<Longrightarrow> 0 \<le> round_down p s"

   by (auto simp: round_down_def)

 lemma ceil_divide_floor_conv:

 assumes "b \<noteq> 0"

 shows "\<lceil>real a / real b\<rceil> = (if b dvd a then a div b else \<lfloor>real a / real b\<rfloor> + 1)"

 proof cases

   assume "\<not> b dvd a"

   hence "a mod b \<noteq> 0" by auto

   hence ne: "real (a mod b) / real b \<noteq> 0" using `b \<noteq> 0` by auto

   have "\<lceil>real a / real b\<rceil> = \<lfloor>real a / real b\<rfloor> + 1"

   apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])

   proof -

     have "real \<lfloor>real a / real b\<rfloor> \<le> real a / real b" by simp

     moreover have "real \<lfloor>real a / real b\<rfloor> \<noteq> real a / real b"

     apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b \<noteq> 0` by auto

     ultimately show "real \<lfloor>real a / real b\<rfloor> < real a / real b" by arith

   qed

   thus ?thesis using `\<not> b dvd a` by simp

 qed (simp add: ceiling_def real_of_int_minus[symmetric] divide_minus_left[symmetric]

   floor_divide_eq_div dvd_neg_div del: divide_minus_left real_of_int_minus)

 lemma compute_float_up[code]:

   "float_up p (Float m e) =

     (let P = 2^nat (-(p + e)); r = m mod P in

       if p + e < 0 then Float (m div P + (if r = 0 then 0 else 1)) (-p) else Float m e)"

 proof cases

   assume "p + e < 0"

   hence "real ((2::int) ^ nat (-(p + e))) = 2 powr (-(p + e))"

     using powr_realpow[of 2 "nat (-(p + e))"] by simp

   also have "... = 1 / 2 powr p / 2 powr e"

   unfolding powr_minus_divide real_of_int_minus by (simp add: powr_add)

   finally have twopow_rewrite:

     "real ((2::int) ^ nat (- (p + e))) = 1 / 2 powr real p / 2 powr real e" .

   with `p + e < 0` have powr_rewrite:

     "2 powr real e * 2 powr real p = 1 / real ((2::int) ^ nat (- (p + e)))"

     unfolding powr_divide2 by simp

   show ?thesis

   proof cases

     assume "2^nat (-(p + e)) dvd m"

     with `p + e < 0` twopow_rewrite show ?thesis

       by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div dvd_eq_mod_eq_0)

   next

     assume ndvd: "\<not> 2 ^ nat (- (p + e)) dvd m"

     have one_div: "real m * (1 / real ((2::int) ^ nat (- (p + e)))) =

       real m / real ((2::int) ^ nat (- (p + e)))"

       by (simp add: field_simps)

     have "real \<lceil>real m * (2 powr real e * 2 powr real p)\<rceil> =

       real \<lfloor>real m * (2 powr real e * 2 powr real p)\<rfloor> + 1"

       using ndvd unfolding powr_rewrite one_div

       by (subst ceil_divide_floor_conv) (auto simp: field_simps)

     thus ?thesis using `p + e < 0` twopow_rewrite

       by transfer (auto simp: ac_simps round_up_def floor_divide_eq_div[symmetric])

   qed

 next

   assume "\<not> p + e < 0"

   then have r1: "real e + real p = real (nat (e + p))" by simp

   have r: "\<lceil>(m * 2 powr e) * 2 powr real p\<rceil> = (m * 2 powr e) * 2 powr real p"

     by (auto simp add: ac_simps powr_add[symmetric] r1 powr_realpow

       intro: exI[where x="m*2^nat (e+p)"])

   then show ?thesis using `\<not> p + e < 0`

     by transfer

        (simp add: round_up_def floor_divide_eq_div field_simps powr_add powr_minus inverse_eq_divide)

 qed

 hide_fact (open) compute_float_up

 lemmas real_of_ints =

   real_of_int_zero

   real_of_one

   real_of_int_add

   real_of_int_minus

   real_of_int_diff

   real_of_int_mult

   real_of_int_power

   real_numeral

 lemmas real_of_nats =

   real_of_nat_zero

   real_of_nat_one

   real_of_nat_1

   real_of_nat_add

   real_of_nat_mult

   real_of_nat_power

 lemmas int_of_reals = real_of_ints[symmetric]

 lemmas nat_of_reals = real_of_nats[symmetric]

 lemma two_real_int: "(2::real) = real (2::int)" by simp

 lemma two_real_nat: "(2::real) = real (2::nat)" by simp

 lemma mult_cong: "a = c ==> b = d ==> a*b = c*d" by simp

 subsection {* Compute bitlen of integers *}

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

   "bitlen a = (if a > 0 then \<lfloor>log 2 a\<rfloor> + 1 else 0)"

 lemma bitlen_nonneg: "0 \<le> bitlen x"

 proof -

   {

     assume "0 > x"

     have "-1 = log 2 (inverse 2)" by (subst log_inverse) simp_all

     also have "... < log 2 (-x)" using `0 > x` by auto

     finally have "-1 < log 2 (-x)" .

   } thus "0 \<le> bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)

 qed

 lemma bitlen_bounds:

   assumes "x > 0"

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

 proof

   have "(2::real) ^ nat \<lfloor>log 2 (real x)\<rfloor> = 2 powr real (floor (log 2 (real x)))"

     using powr_realpow[symmetric, of 2 "nat \<lfloor>log 2 (real x)\<rfloor>"] `x > 0`

     using real_nat_eq_real[of "floor (log 2 (real x))"]

     by simp

   also have "... \<le> 2 powr log 2 (real x)"

     by simp

   also have "... = real x"

     using `0 < x` by simp

   finally have "2 ^ nat \<lfloor>log 2 (real x)\<rfloor> \<le> real x" by simp

   thus "2 ^ nat (bitlen x - 1) \<le> x" using `x > 0`

     by (simp add: bitlen_def)

 next

   have "x \<le> 2 powr (log 2 x)" using `x > 0` by simp

   also have "... < 2 ^ nat (\<lfloor>log 2 (real x)\<rfloor> + 1)"

     apply (simp add: powr_realpow[symmetric])

     using `x > 0` by simp

   finally show "x < 2 ^ nat (bitlen x)" using `x > 0`

     by (simp add: bitlen_def ac_simps int_of_reals del: real_of_ints)

 qed

 lemma bitlen_pow2[simp]:

   assumes "b > 0"

   shows "bitlen (b * 2 ^ c) = bitlen b + c"

 proof -

   from assms have "b * 2 ^ c > 0" by auto

   thus ?thesis

     using floor_add[of "log 2 b" c] assms

     by (auto simp add: log_mult log_nat_power bitlen_def)

 qed

 lemma bitlen_Float:

   fixes m e

   defines "f \<equiv> Float m e"

   shows "bitlen (\<bar>mantissa f\<bar>) + exponent f = (if m = 0 then 0 else bitlen \<bar>m\<bar> + e)"

 proof (cases "m = 0")

   case True

   then show ?thesis by (simp add: f_def bitlen_def Float_def)

 next

   case False

   hence "f \<noteq> float_of 0"

     unfolding real_of_float_eq by (simp add: f_def)

   hence "mantissa f \<noteq> 0"

     by (simp add: mantissa_noteq_0)

   moreover

   obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"

     by (rule f_def[THEN denormalize_shift, OF `f \<noteq> float_of 0`])

   ultimately show ?thesis by (simp add: abs_mult)

 qed

 lemma compute_bitlen[code]:

   shows "bitlen x = (if x > 0 then bitlen (x div 2) + 1 else 0)"

 proof -

   { assume "2 \<le> x"

     then have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 (x - x mod 2)\<rfloor>"

       by (simp add: log_mult zmod_zdiv_equality')

     also have "\<dots> = \<lfloor>log 2 (real x)\<rfloor>"

     proof cases

       assume "x mod 2 = 0" then show ?thesis by simp

     next

       def n \<equiv> "\<lfloor>log 2 (real x)\<rfloor>"

       then have "0 \<le> n"

         using `2 \<le> x` by simp

       assume "x mod 2 \<noteq> 0"

       with `2 \<le> x` have "x mod 2 = 1" "\<not> 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)

       with `2 \<le> x` have "x \<noteq> 2^nat n" by (cases "nat n") auto

       moreover

       { have "real (2^nat n :: int) = 2 powr (nat n)"

           by (simp add: powr_realpow)

         also have "\<dots> \<le> 2 powr (log 2 x)"

           using `2 \<le> x` by (simp add: n_def del: powr_log_cancel)

         finally have "2^nat n \<le> x" using `2 \<le> x` by simp }

       ultimately have "2^nat n \<le> x - 1" by simp

       then have "2^nat n \<le> real (x - 1)"

         unfolding real_of_int_le_iff[symmetric] by simp

       { have "n = \<lfloor>log 2 (2^nat n)\<rfloor>"

           using `0 \<le> n` by (simp add: log_nat_power)

         also have "\<dots> \<le> \<lfloor>log 2 (x - 1)\<rfloor>"

           using `2^nat n \<le> real (x - 1)` `0 \<le> n` `2 \<le> x` by (auto intro: floor_mono)

         finally have "n \<le> \<lfloor>log 2 (x - 1)\<rfloor>" . }

       moreover have "\<lfloor>log 2 (x - 1)\<rfloor> \<le> n"

         using `2 \<le> x` by (auto simp add: n_def intro!: floor_mono)

       ultimately show "\<lfloor>log 2 (x - x mod 2)\<rfloor> = \<lfloor>log 2 x\<rfloor>"

         unfolding n_def `x mod 2 = 1` by auto

     qed

     finally have "\<lfloor>log 2 (x div 2)\<rfloor> + 1 = \<lfloor>log 2 x\<rfloor>" . }

   moreover

   { assume "x < 2" "0 < x"

     then have "x = 1" by simp

     then have "\<lfloor>log 2 (real x)\<rfloor> = 0" by simp }

   ultimately show ?thesis

     unfolding bitlen_def

     by (auto simp: pos_imp_zdiv_pos_iff not_le)

 qed

 hide_fact (open) compute_bitlen

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

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

 proof -

   have "0 < Float m e" using assms by auto

   hence "0 < m" using powr_gt_zero[of 2 e]

     by (auto simp: zero_less_mult_iff)

   hence "m \<noteq> 0" by auto

   show ?thesis

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

     case True thus ?thesis using `0 < m`  by (simp add: bitlen_def)

   next

     have "(1::int) < 2" by simp

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

     have "inverse (2 ^ nat (- e)) = 2 powr e" using assms False powr_realpow[of 2 "nat (-e)"]

       by (auto simp: powr_minus field_simps inverse_eq_divide)

     hence "1 \<le> real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]

       by (auto simp: powr_minus)

     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 by auto

     thus ?thesis by auto

   qed

 qed

 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 `0 < m` by (auto simp: bitlen_def)

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

   also have "\<dots> = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)

   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

 subsection {* Approximation of positive rationals *}

 lemma zdiv_zmult_twopow_eq: fixes a b::int shows "a div b div (2 ^ n) = a div (b * 2 ^ n)"

 by (simp add: zdiv_zmult2_eq)

 lemma div_mult_twopow_eq: fixes a b::nat shows "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)"

   by (cases "b=0") (simp_all add: div_mult2_eq[symmetric] ac_simps)

 lemma real_div_nat_eq_floor_of_divide:

   fixes a b::nat

   shows "a div b = real (floor (a/b))"

 by (metis floor_divide_eq_div real_of_int_of_nat_eq zdiv_int)

 definition "rat_precision prec x y = int prec - (bitlen x - bitlen y)"

 lift_definition lapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"

   is "\<lambda>prec (x::nat) (y::nat). round_down (rat_precision prec x y) (x / y)" by simp

 lemma compute_lapprox_posrat[code]:

   fixes prec x y

   shows "lapprox_posrat prec x y =

    (let

        l = rat_precision prec x y;

        d = if 0 \<le> l then x * 2^nat l div y else x div 2^nat (- l) div y

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

     unfolding div_mult_twopow_eq normfloat_def

     by transfer

        (simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def

              del: two_powr_minus_int_float)

 hide_fact (open) compute_lapprox_posrat

 lift_definition rapprox_posrat :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> float"

   is "\<lambda>prec (x::nat) (y::nat). round_up (rat_precision prec x y) (x / y)" by simp

 (* TODO: optimize using zmod_zmult2_eq, pdivmod ? *)

 lemma compute_rapprox_posrat[code]:

   fixes prec x y

   defines "l \<equiv> rat_precision prec x y"

   shows "rapprox_posrat prec x y = (let

      l = l ;

      X = if 0 \<le> l then (x * 2^nat l, y) else (x, y * 2^nat(-l)) ;

      d = fst X div snd X ;

      m = fst X mod snd X

    in normfloat (Float (d + (if m = 0 \<or> y = 0 then 0 else 1)) (- l)))"

 proof (cases "y = 0")

   assume "y = 0" thus ?thesis unfolding normfloat_def by transfer simp

 next

   assume "y \<noteq> 0"

   show ?thesis

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

     assume "0 \<le> l"

     def x' \<equiv> "x * 2 ^ nat l"

     have "int x * 2 ^ nat l = x'" by (simp add: x'_def int_mult int_power)

     moreover have "real x * 2 powr real l = real x'"

       by (simp add: powr_realpow[symmetric] `0 \<le> l` x'_def)

     ultimately show ?thesis

       unfolding normfloat_def

       using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] `0 \<le> l` `y \<noteq> 0`

         l_def[symmetric, THEN meta_eq_to_obj_eq]

       by transfer

          (simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)

    next

     assume "\<not> 0 \<le> l"

     def y' \<equiv> "y * 2 ^ nat (- l)"

     from `y \<noteq> 0` have "y' \<noteq> 0" by (simp add: y'_def)

     have "int y * 2 ^ nat (- l) = y'" by (simp add: y'_def int_mult int_power)

     moreover have "real x * real (2::int) powr real l / real y = x / real y'"

       using `\<not> 0 \<le> l`

       by (simp add: powr_realpow[symmetric] powr_minus y'_def field_simps inverse_eq_divide)

     ultimately show ?thesis

       unfolding normfloat_def

       using ceil_divide_floor_conv[of y' x] `\<not> 0 \<le> l` `y' \<noteq> 0` `y \<noteq> 0`

         l_def[symmetric, THEN meta_eq_to_obj_eq]

       by transfer

          (simp add: round_up_def ceil_divide_floor_conv floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0)

   qed

 qed

 hide_fact (open) compute_rapprox_posrat

 lemma rat_precision_pos:

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

   shows "rat_precision n (int x) (int y) > 0"

 proof -

   { assume "0 < x" hence "log 2 x + 1 = log 2 (2 * x)" by (simp add: log_mult) }

   hence "bitlen (int x) < bitlen (int y)" using assms

     by (simp add: bitlen_def del: floor_add_one)

       (auto intro!: floor_mono simp add: floor_add_one[symmetric] simp del: floor_add floor_add_one)

   thus ?thesis

     using assms by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)

 qed

 lemma power_aux:

   assumes "x > 0"

   shows "(2::int) ^ nat (x - 1) \<le> 2 ^ nat x - 1"

 proof -

   def y \<equiv> "nat (x - 1)"

   moreover

   have "(2::int) ^ y \<le> (2 ^ (y + 1)) - 1" by simp

   ultimately show ?thesis using assms by simp

 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 -

   have powr1: "2 powr real (rat_precision n (int x) (int y)) =

     2 ^ nat (rat_precision n (int x) (int y))" using rat_precision_pos[of x y n] assms

     by (simp add: powr_realpow[symmetric])

   have "x * 2 powr real (rat_precision n (int x) (int y)) / y = (x / y) *

      2 powr real (rat_precision n (int x) (int y))" by simp

   also have "... < (1 / 2) * 2 powr real (rat_precision n (int x) (int y))"

     apply (rule mult_strict_right_mono) by (insert assms) auto

   also have "\<dots> = 2 powr real (rat_precision n (int x) (int y) - 1)"

     using powr_add [of 2 _ "- 1", simplified add_uminus_conv_diff] by (simp add: powr_minus)

   also have "\<dots> = 2 ^ nat (rat_precision n (int x) (int y) - 1)"

     using rat_precision_pos[of x y n] assms by (simp add: powr_realpow[symmetric])

   also have "\<dots> \<le> 2 ^ nat (rat_precision n (int x) (int y)) - 1"

     unfolding int_of_reals real_of_int_le_iff

     using rat_precision_pos[OF assms] by (rule power_aux)

   finally show ?thesis

     apply (transfer fixing: n x y)

     apply (simp add: round_up_def field_simps powr_minus inverse_eq_divide powr1)

     unfolding int_of_reals real_of_int_less_iff

     apply (simp add: ceiling_less_eq)

     done

 qed

 lift_definition lapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is

   "\<lambda>prec (x::int) (y::int). round_down (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp

 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 (nat x) (nat y)

       else - (rapprox_posrat prec (nat x) (nat (-y))))

       else (if 0 < y

         then - (rapprox_posrat prec (nat (-x)) (nat y))

         else lapprox_posrat prec (nat (-x)) (nat (-y))))"

   by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)

 hide_fact (open) compute_lapprox_rat

 lift_definition rapprox_rat :: "nat \<Rightarrow> int \<Rightarrow> int \<Rightarrow> float" is

   "\<lambda>prec (x::int) (y::int). round_up (rat_precision prec \<bar>x\<bar> \<bar>y\<bar>) (x / y)" by simp

 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 (nat x) (nat y)

       else - (lapprox_posrat prec (nat x) (nat (-y))))

       else (if 0 < y

         then - (lapprox_posrat prec (nat (-x)) (nat y))

         else rapprox_posrat prec (nat (-x)) (nat (-y))))"

   by transfer (auto simp: round_up_def round_down_def ceiling_def ac_simps)

 hide_fact (open) compute_rapprox_rat

 subsection {* Division *}

 definition "real_divl prec a b = round_down (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"

 definition "real_divr prec a b = round_up (int prec + \<lfloor> log 2 \<bar>b\<bar> \<rfloor> - \<lfloor> log 2 \<bar>a\<bar> \<rfloor>) (a / b)"

 lift_definition float_divl :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divl

   by (simp add: real_divl_def)

 lemma compute_float_divl[code]:

   "float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"

 proof cases

   let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"

   let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"

   assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"

   then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"

     by (simp add: abs_mult log_mult rat_precision_def bitlen_def)

   have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"

     by (simp add: field_simps powr_divide2[symmetric])

   show ?thesis

     using not_0

     by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_down_shift real_divl_def,

       simp add: field_simps)

 qed (transfer, auto simp: real_divl_def)

 hide_fact (open) compute_float_divl

 lift_definition float_divr :: "nat \<Rightarrow> float \<Rightarrow> float \<Rightarrow> float" is real_divr

   by (simp add: real_divr_def)

 lemma compute_float_divr[code]:

   "float_divr prec (Float m1 s1) (Float m2 s2) = rapprox_rat prec m1 m2 * Float 1 (s1 - s2)"

 proof cases

   let ?f1 = "real m1 * 2 powr real s1" and ?f2 = "real m2 * 2 powr real s2"

   let ?m = "real m1 / real m2" and ?s = "2 powr real (s1 - s2)"

   assume not_0: "m1 \<noteq> 0 \<and> m2 \<noteq> 0"

   then have eq2: "(int prec + \<lfloor>log 2 \<bar>?f2\<bar>\<rfloor> - \<lfloor>log 2 \<bar>?f1\<bar>\<rfloor>) = rat_precision prec \<bar>m1\<bar> \<bar>m2\<bar> + (s2 - s1)"

     by (simp add: abs_mult log_mult rat_precision_def bitlen_def)

   have eq1: "real m1 * 2 powr real s1 / (real m2 * 2 powr real s2) = ?m * ?s"

     by (simp add: field_simps powr_divide2[symmetric])

   show ?thesis

     using not_0

     by (transfer fixing: m1 s1 m2 s2 prec) (unfold eq1 eq2 round_up_shift real_divr_def,

       simp add: field_simps)

 qed (transfer, auto simp: real_divr_def)

 hide_fact (open) compute_float_divr

 subsection {* Lemmas needed by Approximate *}

 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"

 using two_powr_int_float[of 2] two_powr_int_float[of "-1"] two_powr_int_float[of "-2"] two_powr_int_float[of "-3"]

 using powr_realpow[of 2 2] powr_realpow[of 2 3]

 using powr_minus[of 2 1] powr_minus[of 2 2] powr_minus[of 2 3]

 by auto

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

 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

 lemma lapprox_rat:

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

   using round_down by (simp add: lapprox_rat_def)

 lemma mult_div_le: fixes a b:: int assumes "b > 0" shows "a \<ge> b * (a div b)"

 proof -

   from zmod_zdiv_equality'[of a b]

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

   also have "... \<ge> b * (a div b) + 0" apply (rule add_left_mono) apply (rule pos_mod_sign)

   using assms by simp

   finally show ?thesis by simp

 qed

 lemma lapprox_rat_nonneg:

   fixes n x y

   defines "p \<equiv> int n - ((bitlen \<bar>x\<bar>) - (bitlen \<bar>y\<bar>))"

   assumes "0 \<le> x" and "0 < y"

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

 using assms unfolding lapprox_rat_def p_def[symmetric] round_down_def real_of_int_minus[symmetric]

    powr_int[of 2, simplified]

   by auto

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

   using round_up by (simp add: rapprox_rat_def)

 lemma rapprox_rat_le1:

   fixes n x y

   assumes xy: "0 \<le> x" "0 < y" "x \<le> y"

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

 proof -

   have "bitlen \<bar>x\<bar> \<le> bitlen \<bar>y\<bar>"

     using xy unfolding bitlen_def by (auto intro!: floor_mono)

   then have "0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>" by (simp add: rat_precision_def)

   have "real \<lceil>real x / real y * 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>

       \<le> real \<lceil>2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)\<rceil>"

     using xy by (auto intro!: ceiling_mono simp: field_simps)

   also have "\<dots> = 2 powr real (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"

     using `0 \<le> rat_precision n \<bar>x\<bar> \<bar>y\<bar>`

     by (auto intro!: exI[of _ "2^nat (rat_precision n \<bar>x\<bar> \<bar>y\<bar>)"] simp: powr_int)

   finally show ?thesis

     by (simp add: rapprox_rat_def round_up_def)

        (simp add: powr_minus inverse_eq_divide)

 qed

 lemma rapprox_rat_nonneg_neg:

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

   unfolding rapprox_rat_def round_up_def

   by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)

 lemma rapprox_rat_neg:

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

   unfolding rapprox_rat_def round_up_def

   by (auto simp: field_simps mult_le_0_iff)

 lemma rapprox_rat_nonpos_pos:

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

   unfolding rapprox_rat_def round_up_def

   by (auto simp: field_simps mult_le_0_iff)

 lemma real_divl: "real_divl prec x y \<le> x / y"

   by (simp add: real_divl_def round_down)

 lemma real_divr: "x / y \<le> real_divr prec x y"

   using round_up by (simp add: real_divr_def)

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

   by transfer (rule real_divl)

 lemma real_divl_lower_bound:

   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real_divl prec x y"

   by (simp add: real_divl_def round_down_def zero_le_mult_iff zero_le_divide_iff)

 lemma float_divl_lower_bound:

   "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> real (float_divl prec x y)"

   by transfer (rule real_divl_lower_bound)

 lemma exponent_1: "exponent 1 = 0"

   using exponent_float[of 1 0] by (simp add: one_float_def)

 lemma mantissa_1: "mantissa 1 = 1"

   using mantissa_float[of 1 0] by (simp add: one_float_def)

 lemma bitlen_1: "bitlen 1 = 1"

   by (simp add: bitlen_def)

 lemma mantissa_eq_zero_iff: "mantissa x = 0 \<longleftrightarrow> x = 0"

 proof

   assume "mantissa x = 0" hence z: "0 = real x" using mantissa_exponent by simp

   show "x = 0" by (simp add: zero_float_def z)

 qed (simp add: zero_float_def)

 lemma float_upper_bound: "x \<le> 2 powr (bitlen \<bar>mantissa x\<bar> + exponent x)"

 proof (cases "x = 0", simp)

   assume "x \<noteq> 0" hence "mantissa x \<noteq> 0" using mantissa_eq_zero_iff by auto

   have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)

   also have "mantissa x \<le> \<bar>mantissa x\<bar>" by simp

   also have "... \<le> 2 powr (bitlen \<bar>mantissa x\<bar>)"

     using bitlen_bounds[of "\<bar>mantissa x\<bar>"] bitlen_nonneg `mantissa x \<noteq> 0`

     by (simp add: powr_int) (simp only: two_real_int int_of_reals real_of_int_abs[symmetric]

       real_of_int_le_iff less_imp_le)

   finally show ?thesis by (simp add: powr_add)

 qed

 lemma real_divl_pos_less1_bound:

   "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real_divl prec 1 x"

 proof (unfold real_divl_def)

   fix prec :: nat and x :: real assume x: "0 < x" "x < 1" and prec: "1 \<le> prec"

   def p \<equiv> "int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor>"

   show "1 \<le> round_down (int prec + \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - \<lfloor>log 2 \<bar>1\<bar>\<rfloor>) (1 / x) "

   proof cases

     assume nonneg: "0 \<le> p"

     hence "2 powr real (p) = floor (real ((2::int) ^ nat p)) * floor (1::real)"

       by (simp add: powr_int del: real_of_int_power) simp

     also have "floor (1::real) \<le> floor (1 / x)" using x prec by simp

     also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) \<le>

       floor (real ((2::int) ^ nat p) * (1 / x))"

       by (rule le_mult_floor) (auto simp: x prec less_imp_le)

     finally have "2 powr real p \<le> floor (2 powr nat p / x)" by (simp add: powr_realpow)

     thus ?thesis unfolding p_def[symmetric]

       using x prec nonneg by (simp add: powr_minus inverse_eq_divide round_down_def)

   next

     assume neg: "\<not> 0 \<le> p"

     have "x = 2 powr (log 2 x)"

       using x by simp

     also have "2 powr (log 2 x) \<le> 2 powr p"

     proof (rule powr_mono)

       have "log 2 x \<le> \<lceil>log 2 x\<rceil>"

         by simp

       also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + 1"

         using ceiling_diff_floor_le_1[of "log 2 x"] by simp

       also have "\<dots> \<le> \<lfloor>log 2 x\<rfloor> + prec"

         using prec by simp

       finally show "log 2 x \<le> real p"

         using x by (simp add: p_def)

     qed simp

     finally have x_le: "x \<le> 2 powr p" .

     from neg have "2 powr real p \<le> 2 powr 0"

       by (intro powr_mono) auto

     also have "\<dots> \<le> \<lfloor>2 powr 0\<rfloor>" by simp

     also have "\<dots> \<le> \<lfloor>2 powr real p / x\<rfloor>" unfolding real_of_int_le_iff

       using x x_le by (intro floor_mono) (simp add:  pos_le_divide_eq)

     finally show ?thesis

       using prec x unfolding p_def[symmetric]

       by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)

   qed

 qed

 lemma float_divl_pos_less1_bound:

   "0 < real x \<Longrightarrow> real x < 1 \<Longrightarrow> prec \<ge> 1 \<Longrightarrow> 1 \<le> real (float_divl prec 1 x)"

   by (transfer, rule real_divl_pos_less1_bound)

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

   by transfer (rule real_divr)

 lemma real_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 \<le> real_divr prec 1 x"

 proof -

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

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

   finally show ?thesis by auto

 qed

 lemma float_divr_pos_less1_lower_bound: "0 < x \<Longrightarrow> x < 1 \<Longrightarrow> 1 \<le> float_divr prec 1 x"

   by transfer (rule real_divr_pos_less1_lower_bound)

 lemma real_divr_nonpos_pos_upper_bound:

   "x \<le> 0 \<Longrightarrow> 0 < y \<Longrightarrow> real_divr prec x y \<le> 0"

   by (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def real_divr_def)

 lemma float_divr_nonpos_pos_upper_bound:

   "real x \<le> 0 \<Longrightarrow> 0 < real y \<Longrightarrow> real (float_divr prec x y) \<le> 0"

   by transfer (rule real_divr_nonpos_pos_upper_bound)

 lemma real_divr_nonneg_neg_upper_bound:

   "0 \<le> x \<Longrightarrow> y < 0 \<Longrightarrow> real_divr prec x y \<le> 0"

   by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def real_divr_def)

 lemma float_divr_nonneg_neg_upper_bound:

   "0 \<le> real x \<Longrightarrow> real y < 0 \<Longrightarrow> real (float_divr prec x y) \<le> 0"

   by transfer (rule real_divr_nonneg_neg_upper_bound)

 definition truncate_down::"nat \<Rightarrow> real \<Rightarrow> real" where

   "truncate_down prec x = round_down (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"

 lemma truncate_down: "truncate_down prec x \<le> x"

   using round_down by (simp add: truncate_down_def)

 lemma truncate_down_le: "x \<le> y \<Longrightarrow> truncate_down prec x \<le> y"

   by (rule order_trans[OF truncate_down])

 definition truncate_up::"nat \<Rightarrow> real \<Rightarrow> real" where

   "truncate_up prec x = round_up (prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) x"

 lemma truncate_up: "x \<le> truncate_up prec x"

   using round_up by (simp add: truncate_up_def)

 lemma truncate_up_le: "x \<le> y \<Longrightarrow> x \<le> truncate_up prec y"

   by (rule order_trans[OF _ truncate_up])

 lemma truncate_up_zero[simp]: "truncate_up prec 0 = 0"

   by (simp add: truncate_up_def)

 lift_definition float_round_up :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_up

   by (simp add: truncate_up_def)

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

   using truncate_up by transfer simp

 lift_definition float_round_down :: "nat \<Rightarrow> float \<Rightarrow> float" is truncate_down

   by (simp add: truncate_down_def)

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

   using truncate_down by transfer simp

 lemma floor_add2[simp]: "\<lfloor> real i + x \<rfloor> = i + \<lfloor> x \<rfloor>"

   using floor_add[of x i] by (simp del: floor_add add: ac_simps)

 lemma compute_float_round_down[code]:

   "float_round_down prec (Float m e) = (let d = bitlen (abs 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)"

   using Float.compute_float_down[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]

   by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_down_def

     cong del: if_weak_cong)

 hide_fact (open) compute_float_round_down

 lemma compute_float_round_up[code]:

   "float_round_up prec (Float m e) = (let d = (bitlen (abs 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)"

   using Float.compute_float_up[of "prec - bitlen \<bar>m\<bar> - e" m e, symmetric]

   unfolding Let_def

   by transfer (simp add: field_simps abs_mult log_mult bitlen_def truncate_up_def

     cong del: if_weak_cong)

 hide_fact (open) compute_float_round_up

 lemma round_up_mono: "x \<le> y \<Longrightarrow> round_up p x \<le> round_up p y"

   by (auto intro!: ceiling_mono simp: round_up_def)

 lemma truncate_up_nonneg_mono:

   assumes "0 \<le> x" "x \<le> y"

   shows "truncate_up prec x \<le> truncate_up prec y"

 proof -

   {

     assume "\<lfloor>log 2 x\<rfloor> = \<lfloor>log 2 y\<rfloor>"

     hence ?thesis

       using assms

       by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)

   } moreover {

     assume "0 < x"

     hence "log 2 x \<le> log 2 y" using assms by auto

     moreover

     assume "\<lfloor>log 2 x\<rfloor> \<noteq> \<lfloor>log 2 y\<rfloor>"

     ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"

       unfolding atomize_conj

       by (metis floor_less_cancel linorder_cases not_le)

     have "truncate_up prec x =

       real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> * 2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1)"

       using assms by (simp add: truncate_up_def round_up_def)

     also have "\<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> (2 ^ prec)"

     proof (unfold ceiling_le_eq)

       have "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> x * (2 powr real prec / (2 powr log 2 x))"

         using real_of_int_floor_add_one_ge[of "log 2 x"] assms

         by (auto simp add: algebra_simps powr_divide2 intro!: mult_left_mono)

       thus "x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> real ((2::int) ^ prec)"

         using `0 < x` by (simp add: powr_realpow)

     qed

     hence "real \<lceil>x * 2 powr real (int prec - \<lfloor>log 2 x\<rfloor> - 1)\<rceil> \<le> 2 powr int prec"

       by (auto simp: powr_realpow)

     also

     have "2 powr - real (int prec - \<lfloor>log 2 x\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"

       using logless flogless by (auto intro!: floor_mono)

     also have "2 powr real (int prec) \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>))"

       using assms `0 < x`

       by (auto simp: algebra_simps)

     finally have "truncate_up prec x \<le> 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>)) * 2 powr - real (int prec - \<lfloor>log 2 y\<rfloor>)"

       by simp

     also have "\<dots> = 2 powr (log 2 y + real (int prec - \<lfloor>log 2 y\<rfloor>) - real (int prec - \<lfloor>log 2 y\<rfloor>))"

       by (subst powr_add[symmetric]) simp

     also have "\<dots> = y"

       using `0 < x` assms

       by (simp add: powr_add)

     also have "\<dots> \<le> truncate_up prec y"

       by (rule truncate_up)

     finally have ?thesis .

   } moreover {

     assume "~ 0 < x"

     hence ?thesis

       using assms

       by (auto intro!: truncate_up_le)

   } ultimately show ?thesis

     by blast

 qed

 lemma truncate_up_nonpos: "x \<le> 0 \<Longrightarrow> truncate_up prec x \<le> 0"

   by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)

 lemma truncate_down_nonpos: "x \<le> 0 \<Longrightarrow> truncate_down prec x \<le> 0"

   by (auto simp: truncate_down_def round_down_def intro!: mult_nonpos_nonneg

     order_le_less_trans[of _ 0, OF mult_nonpos_nonneg])

 lemma truncate_up_switch_sign_mono:

   assumes "x \<le> 0" "0 \<le> y"

   shows "truncate_up prec x \<le> truncate_up prec y"

 proof -

   note truncate_up_nonpos[OF `x \<le> 0`]

   also note truncate_up_le[OF `0 \<le> y`]

   finally show ?thesis .

 qed

 lemma truncate_down_zeroprec_mono:

   assumes "0 < x" "x \<le> y"

   shows "truncate_down 0 x \<le> truncate_down 0 y"

 proof -

   have "x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1) = x * inverse (2 powr ((real \<lfloor>log 2 x\<rfloor> + 1)))"

     by (simp add: powr_divide2[symmetric] powr_add powr_minus inverse_eq_divide)

   also have "\<dots> = 2 powr (log 2 x - (real \<lfloor>log 2 x\<rfloor>) - 1)"

     using `0 < x`

     by (auto simp: inverse_eq_divide field_simps powr_add powr_divide2[symmetric])

   also have "\<dots> < 2 powr 0"

     using real_of_int_floor_add_one_gt

     unfolding neg_less_iff_less

     by (intro powr_less_mono) (auto simp: algebra_simps)

   finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> < 1"

     unfolding less_ceiling_eq real_of_int_minus real_of_one

     by simp

   moreover

   have "0 \<le> \<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor>"

     using `x > 0` by auto

   ultimately have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> \<in> {0 ..< 1}"

     by simp

   also have "\<dots> \<subseteq> {0}" by auto

   finally have "\<lfloor>x * 2 powr (- real \<lfloor>log 2 x\<rfloor> - 1)\<rfloor> = 0" by simp

   with assms show ?thesis

     by (auto simp: truncate_down_def round_down_def)

 qed

 lemma truncate_down_nonneg: "0 \<le> y \<Longrightarrow> 0 \<le> truncate_down prec y"

   by (auto simp: truncate_down_def round_down_def)

 lemma truncate_down_zero: "truncate_down prec 0 = 0"

   by (auto simp: truncate_down_def round_down_def)

 lemma truncate_down_switch_sign_mono:

   assumes "x \<le> 0" "0 \<le> y"

   assumes "x \<le> y"

   shows "truncate_down prec x \<le> truncate_down prec y"

 proof -

   note truncate_down_nonpos[OF `x \<le> 0`]

   also note truncate_down_nonneg[OF `0 \<le> y`]

   finally show ?thesis .

 qed

 lemma truncate_up_uminus_truncate_down:

   "truncate_up prec x = - truncate_down prec (- x)"

   "truncate_up prec (-x) = - truncate_down prec x"

   by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)

 lemma truncate_down_uminus_truncate_up:

   "truncate_down prec x = - truncate_up prec (- x)"

   "truncate_down prec (-x) = - truncate_up prec x"

   by (auto simp: truncate_up_def round_up_def truncate_down_def round_down_def ceiling_def)

 lemma truncate_down_nonneg_mono:

   assumes "0 \<le> x" "x \<le> y"

   shows "truncate_down prec x \<le> truncate_down prec y"

 proof -

   {

     assume "0 < x" "prec = 0"

     with assms have ?thesis

       by (simp add: truncate_down_zeroprec_mono)

   } moreover {

     assume "~ 0 < x"

     with assms have "x = 0" "0 \<le> y" by simp_all

     hence ?thesis

       by (auto simp add: truncate_down_zero intro!: truncate_down_nonneg)

   } moreover {

     assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> = \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"

     hence ?thesis

       using assms

       by (auto simp: truncate_down_def round_down_def intro!: floor_mono)

   } moreover {

     assume "0 < x"

     hence "log 2 x \<le> log 2 y" "0 < y" "0 \<le> y" using assms by auto

     moreover

     assume "\<lfloor>log 2 \<bar>x\<bar>\<rfloor> \<noteq> \<lfloor>log 2 \<bar>y\<bar>\<rfloor>"

     ultimately have logless: "log 2 x < log 2 y" and flogless: "\<lfloor>log 2 x\<rfloor> < \<lfloor>log 2 y\<rfloor>"

       unfolding atomize_conj abs_of_pos[OF `0 < x`] abs_of_pos[OF `0 < y`]

       by (metis floor_less_cancel linorder_cases not_le)

     assume "prec \<noteq> 0" hence [simp]: "prec \<ge> Suc 0" by auto

     have "2 powr (prec - 1) \<le> y * 2 powr real (prec - 1) / (2 powr log 2 y)"

       using `0 < y`

       by simp

     also have "\<dots> \<le> y * 2 powr real prec / (2 powr (real \<lfloor>log 2 y\<rfloor> + 1))"

       using `0 \<le> y` `0 \<le> x` assms(2)

       by (auto intro!: powr_mono divide_left_mono

         simp: real_of_nat_diff powr_add

         powr_divide2[symmetric])

     also have "\<dots> = y * 2 powr real prec / (2 powr real \<lfloor>log 2 y\<rfloor> * 2)"

       by (auto simp: powr_add)

     finally have "(2 ^ (prec - 1)) \<le> \<lfloor>y * 2 powr real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)\<rfloor>"

       using `0 \<le> y`

       by (auto simp: powr_divide2[symmetric] le_floor_eq powr_realpow)

     hence "(2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1) \<le> truncate_down prec y"

       by (auto simp: truncate_down_def round_down_def)

     moreover

     {

       have "x = 2 powr (log 2 \<bar>x\<bar>)" using `0 < x` by simp

       also have "\<dots> \<le> (2 ^ (prec )) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1)"

         using real_of_int_floor_add_one_ge[of "log 2 \<bar>x\<bar>"]

         by (auto simp: powr_realpow[symmetric] powr_add[symmetric] algebra_simps)

       also

       have "2 powr - real (int prec - \<lfloor>log 2 \<bar>x\<bar>\<rfloor> - 1) \<le> 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor>)"

         using logless flogless `x > 0` `y > 0`

         by (auto intro!: floor_mono)

       finally have "x \<le> (2 ^ (prec - 1)) * 2 powr - real (int prec - \<lfloor>log 2 \<bar>y\<bar>\<rfloor> - 1)"

         by (auto simp: powr_realpow[symmetric] powr_divide2[symmetric] assms real_of_nat_diff)

     } ultimately have ?thesis

       by (metis dual_order.trans truncate_down)

   } ultimately show ?thesis by blast

 qed

 lemma truncate_down_mono: "x \<le> y \<Longrightarrow> truncate_down p x \<le> truncate_down p y"

   apply (cases "0 \<le> x")

   apply (rule truncate_down_nonneg_mono, assumption+)

   apply (simp add: truncate_down_uminus_truncate_up)

   apply (cases "0 \<le> y")

   apply (auto intro: truncate_up_nonneg_mono truncate_up_switch_sign_mono)

   done

 lemma truncate_up_mono: "x \<le> y \<Longrightarrow> truncate_up p x \<le> truncate_up p y"

   by (simp add: truncate_up_uminus_truncate_down truncate_down_mono)

 lemma Float_le_zero_iff: "Float a b \<le> 0 \<longleftrightarrow> a \<le> 0"

  apply (auto simp: zero_float_def mult_le_0_iff)

  using powr_gt_zero[of 2 b] by simp

 lemma real_of_float_pprt[simp]: fixes a::float shows "real (pprt a) = pprt (real a)"

   unfolding pprt_def sup_float_def max_def sup_real_def by auto

 lemma real_of_float_nprt[simp]: fixes a::float shows "real (nprt a) = nprt (real a)"

   unfolding nprt_def inf_float_def min_def inf_real_def by auto

 lift_definition int_floor_fl :: "float \<Rightarrow> int" is floor .

 lemma compute_int_floor_fl[code]:

   "int_floor_fl (Float m e) = (if 0 \<le> e then m * 2 ^ nat e else m div (2 ^ (nat (-e))))"

   by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)

 hide_fact (open) compute_int_floor_fl

 lift_definition floor_fl :: "float \<Rightarrow> float" is "\<lambda>x. real (floor x)" by simp

 lemma compute_floor_fl[code]:

   "floor_fl (Float m e) = (if 0 \<le> e then Float m e else Float (m div (2 ^ (nat (-e)))) 0)"

   by transfer (simp add: powr_int int_of_reals floor_divide_eq_div del: real_of_ints)

 hide_fact (open) compute_floor_fl

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

 lemma int_floor_fl: "real (int_floor_fl x) \<le> real x" by transfer simp

 lemma floor_pos_exp: "exponent (floor_fl x) \<ge> 0"

 proof (cases "floor_fl x = float_of 0")

   case True

   then show ?thesis by (simp add: floor_fl_def)

 next

   case False

   have eq: "floor_fl x = Float \<lfloor>real x\<rfloor> 0" by transfer simp

   obtain i where "\<lfloor>real x\<rfloor> = mantissa (floor_fl x) * 2 ^ i" "0 = exponent (floor_fl x) - int i"

     by (rule denormalize_shift[OF eq[THEN eq_reflection] False])

   then show ?thesis by simp

 qed

 end