# Theory HOL-Library.Float

```(*  Title:      HOL/Library/Float.thy
Author:     Johannes Hölzl, Fabian Immler
Copyright   2012  TU München
*)

section ‹Floating-Point Numbers›

theory Float
imports Log_Nat 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

setup_lifting type_definition_float

declare real_of_float [code_unfold]

lemmas float_of_inject[simp]

declare [[coercion "real_of_float :: float ⇒ real"]]

lemma real_of_float_eq: "f1 = f2 ⟷ real_of_float f1 = real_of_float f2" for f1 f2 :: float
unfolding real_of_float_inject ..

declare real_of_float_inverse[simp] float_of_inverse [simp]
declare real_of_float [simp]

subsection ‹Real operations preserving the representation as floating point number›

lemma floatI: "m * 2 powr e = x ⟹ x ∈ float" for m e :: int
by (auto simp: float_def)

lemma zero_float[simp]: "0 ∈ float"
by (auto simp: float_def)

lemma one_float[simp]: "1 ∈ float"
by (intro floatI[of 1 0]) simp

lemma numeral_float[simp]: "numeral i ∈ float"
by (intro floatI[of "numeral i" 0]) simp

lemma neg_numeral_float[simp]: "- numeral i ∈ float"
by (intro floatI[of "- numeral i" 0]) simp

lemma real_of_int_float[simp]: "real_of_int x ∈ float" for x :: int
by (intro floatI[of x 0]) simp

lemma real_of_nat_float[simp]: "real x ∈ float" for x :: nat
by (intro floatI[of x 0]) simp

lemma two_powr_int_float[simp]: "2 powr (real_of_int i) ∈ float" for i :: int
by (intro floatI[of 1 i]) simp

lemma two_powr_nat_float[simp]: "2 powr (real i) ∈ float" for i :: nat
by (intro floatI[of 1 i]) simp

lemma two_powr_minus_int_float[simp]: "2 powr - (real_of_int i) ∈ float" for i :: int
by (intro floatI[of 1 "-i"]) simp

lemma two_powr_minus_nat_float[simp]: "2 powr - (real i) ∈ float" for i :: nat
by (intro floatI[of 1 "-i"]) simp

lemma two_powr_numeral_float[simp]: "2 powr numeral i ∈ float"
by (intro floatI[of 1 "numeral i"]) simp

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

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

lemma plus_float[simp]: "r ∈ float ⟹ p ∈ float ⟹ r + p ∈ float"
unfolding float_def
proof (safe, simp)
have *: "∃(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
if "e1 ≤ e2" for e1 m1 e2 m2 :: int
proof -
from that have "m1 * 2 powr e1 + m2 * 2 powr e2 = (m1 + m2 * 2 ^ nat (e2 - e1)) * 2 powr e1"
by (simp add: powr_diff field_simps flip: powr_realpow)
then show ?thesis
by blast
qed
fix e1 m1 e2 m2 :: int
consider "e2 ≤ e1" | "e1 ≤ e2" by (rule linorder_le_cases)
then show "∃(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
proof cases
case 1
from *[OF this, of m2 m1] show ?thesis
by (simp add: ac_simps)
next
case 2
then show ?thesis by (rule *)
qed
qed

lemma uminus_float[simp]: "x ∈ float ⟹ -x ∈ float"
by (simp add: float_def) (metis mult_minus_left of_int_minus)

lemma times_float[simp]: "x ∈ float ⟹ y ∈ float ⟹ x * y ∈ float"
apply (clarsimp simp: float_def)
by (metis (no_types, opaque_lifting) of_int_add powr_add mult.assoc mult.left_commute of_int_mult)

lemma minus_float[simp]: "x ∈ float ⟹ y ∈ float ⟹ x - y ∈ float"
using plus_float [of x "- y"] by simp

lemma abs_float[simp]: "x ∈ float ⟹ ¦x¦ ∈ float"
by (cases x rule: linorder_cases[of 0]) auto

lemma sgn_of_float[simp]: "x ∈ float ⟹ sgn x ∈ float"
by (cases x rule: linorder_cases[of 0]) (auto intro!: uminus_float)

lemma div_power_2_float[simp]: "x ∈ float ⟹ x / 2^d ∈ float"
by (simp add: float_def) (metis of_int_diff of_int_of_nat_eq powr_diff powr_realpow zero_less_numeral times_divide_eq_right)

lemma div_power_2_int_float[simp]: "x ∈ float ⟹ x / (2::int)^d ∈ float"
by simp

lemma div_numeral_Bit0_float[simp]:
assumes "x / numeral n ∈ float"
shows "x / (numeral (Num.Bit0 n)) ∈ float"
proof -
have "(x / numeral n) / 2^1 ∈ float"
by (intro assms 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 / numeral n ∈ float"
shows "x / (- numeral (Num.Bit0 n)) ∈ float"
using assms by force

lemma power_float[simp]:
assumes "a ∈ float"
shows "a ^ b ∈ float"
proof -
from assms obtain m e :: int where "a = m * 2 powr e"
by (auto simp: float_def)
then show ?thesis
by (auto intro!: floatI[where m="m^b" and e = "e*b"]
simp: power_mult_distrib powr_realpow[symmetric] powr_powr)
qed

lift_definition Float :: "int ⇒ int ⇒ float" is "λ(m::int) (e::int). m * 2 powr e"
by simp
declare Float.rep_eq[simp]

code_datatype Float

lemma compute_real_of_float[code]:
"real_of_float (Float m e) = (if e ≥ 0 then m * 2 ^ nat e else m / 2 ^ (nat (-e)))"
by (simp add: powr_int)

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 ⇒ float ⇒ float" is "(+)" by simp
declare plus_float.rep_eq[simp]

lift_definition times_float :: "float ⇒ float ⇒ float" is "(*)" by simp
declare times_float.rep_eq[simp]

lift_definition minus_float :: "float ⇒ float ⇒ float" is "(-)" by simp
declare minus_float.rep_eq[simp]

lift_definition uminus_float :: "float ⇒ float" is "uminus" by simp
declare uminus_float.rep_eq[simp]

lift_definition abs_float :: "float ⇒ float" is abs by simp
declare abs_float.rep_eq[simp]

lift_definition sgn_float :: "float ⇒ float" is sgn by simp
declare sgn_float.rep_eq[simp]

lift_definition equal_float :: "float ⇒ float ⇒ bool" is "(=) :: real ⇒ real ⇒ bool" .

lift_definition less_eq_float :: "float ⇒ float ⇒ bool" is "(≤)" .
declare less_eq_float.rep_eq[simp]

lift_definition less_float :: "float ⇒ float ⇒ bool" is "(<)" .
declare less_float.rep_eq[simp]

instance
by standard (transfer; fastforce simp add: field_simps intro: mult_left_mono mult_right_mono)+

end

lemma real_of_float [simp]: "real_of_float (of_nat n) = of_nat n"
by (induct n) simp_all

lemma real_of_float_of_int_eq [simp]: "real_of_float (of_int z) = of_int z"
by (cases z rule: int_diff_cases) (simp_all add: of_rat_diff)

lemma Float_0_eq_0[simp]: "Float 0 e = 0"
by transfer simp

lemma real_of_float_power[simp]: "real_of_float (f^n) = real_of_float f^n" for f :: float
by (induct n) simp_all

lemma real_of_float_min: "real_of_float (min x y) = min (real_of_float x) (real_of_float y)"
and real_of_float_max: "real_of_float (max x y) = max (real_of_float x) (real_of_float y)"
for x y :: float
by (simp_all add: min_def max_def)

instance float :: unbounded_dense_linorder
proof
fix a b :: float
show "∃c. a < c"
by (metis Float.real_of_float less_float.rep_eq reals_Archimedean2)
show "∃c. c < a"
show "∃c. a < c ∧ c < b" if "a < b"
apply (rule exI[of _ "(a + b) * Float 1 (- 1)"])
using that
apply transfer
apply (simp add: powr_minus)
done
qed

instantiation float :: lattice_ab_group_add
begin

definition inf_float :: "float ⇒ float ⇒ float"
where "inf_float a b = min a b"

definition sup_float :: "float ⇒ float ⇒ float"
where "sup_float a b = max a b"

instance
by standard (transfer; simp add: inf_float_def sup_float_def real_of_float_min real_of_float_max)+

end

lemma float_numeral[simp]: "real_of_float (numeral x :: float) = numeral x"
proof (induct x)
case One
then show ?case by simp
qed (metis of_int_numeral real_of_float_of_int_eq)+

lemma transfer_numeral [transfer_rule]:
"rel_fun (=) pcr_float (numeral :: _ ⇒ real) (numeral :: _ ⇒ float)"
by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)

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

lemma transfer_neg_numeral [transfer_rule]:
"rel_fun (=) pcr_float (- numeral :: _ ⇒ real) (- numeral :: _ ⇒ float)"
by (simp add: rel_fun_def float.pcr_cr_eq cr_float_def)

lemma float_of_numeral: "numeral k = float_of (numeral k)"
and float_of_neg_numeral: "- numeral k = float_of (- numeral k)"
unfolding real_of_float_eq by simp_all

subsection ‹Quickcheck›

instantiation float :: exhaustive
begin

definition exhaustive_float where
"exhaustive_float f d =
Quickcheck_Exhaustive.exhaustive (λx. Quickcheck_Exhaustive.exhaustive (λy. f (Float x y)) d) d"

instance ..

end

context
includes term_syntax
begin

definition [code_unfold]:
"valtermify_float x y = Code_Evaluation.valtermify Float {⋅} x {⋅} y"

end

instantiation float :: full_exhaustive
begin

definition
"full_exhaustive_float f d =
Quickcheck_Exhaustive.full_exhaustive
(λx. Quickcheck_Exhaustive.full_exhaustive (λy. f (valtermify_float x y)) d) d"

instance ..

end

instantiation float :: random
begin

definition "Quickcheck_Random.random i =
scomp (Quickcheck_Random.random (2 ^ nat_of_natural i))
(λman. scomp (Quickcheck_Random.random i) (λexp. Pair (valtermify_float man exp)))"

instance ..

end

subsection ‹Represent floats as unique mantissa and exponent›

lemma int_induct_abs[case_names less]:
fixes j :: int
assumes H: "⋀n. (⋀i. ¦i¦ < ¦n¦ ⟹ P i) ⟹ P n"
shows "P j"
proof (induct "nat ¦j¦" 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 ∨ (∃k i. n = k * r ^ i ∧ ¬ r dvd k)"
proof (induct n rule: int_induct_abs)
case (less n)
have "∃k i. n = k * r ^ Suc i ∧ ¬ r dvd k" if "n ≠ 0" "n = m * r" for m
proof -
from that have "¦m ¦ < ¦n¦"
using ‹1 < r› by (simp add: abs_mult)
from less[OF this] that show ?thesis by auto
qed
then show ?case
by (metis dvd_def monoid_mult_class.mult.right_neutral mult.commute power_0)
qed

lemma mult_powr_eq_mult_powr_iff_asym:
fixes m1 m2 e1 e2 :: int
assumes m1: "¬ 2 dvd m1"
and "e1 ≤ e2"
shows "m1 * 2 powr e1 = m2 * 2 powr e2 ⟷ m1 = m2 ∧ e1 = e2"
(is "?lhs ⟷ ?rhs")
proof
show ?rhs if eq: ?lhs
proof -
have "m1 ≠ 0"
using m1 unfolding dvd_def by auto
from ‹e1 ≤ e2› eq have "m1 = m2 * 2 powr nat (e2 - e1)"
by (simp add: powr_diff field_simps)
also have "… = m2 * 2^nat (e2 - e1)"
by (simp add: powr_realpow)
finally have m1_eq: "m1 = m2 * 2^nat (e2 - e1)"
by linarith
with m1 have "m1 = m2"
by (cases "nat (e2 - e1)") (auto simp add: dvd_def)
then show ?thesis
using eq ‹m1 ≠ 0› by (simp add: powr_inj)
qed
show ?lhs if ?rhs
using that by simp
qed

lemma mult_powr_eq_mult_powr_iff:
"¬ 2 dvd m1 ⟹ ¬ 2 dvd m2 ⟹ m1 * 2 powr e1 = m2 * 2 powr e2 ⟷ m1 = m2 ∧ e1 = e2"
for m1 m2 e1 e2 :: int
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 ∈ float"
obtains (zero) "x = 0"
| (powr) m e :: int where "x = m * 2 powr e" "¬ 2 dvd m" "x ≠ 0"
proof -
have "∃(m::int) (e::int). x = m * 2 powr e ∧ ¬ (2::int) dvd m" if "x ≠ 0"
proof -
from x obtain m e :: int where x: "x = m * 2 powr e"
by (auto simp: float_def)
with ‹x ≠ 0› int_cancel_factors[of 2 m] obtain k i where "m = k * 2 ^ i" "¬ 2 dvd k"
by auto
with ‹¬ 2 dvd k› x show ?thesis
apply (rule_tac exI[of _ "k"])
apply (rule_tac exI[of _ "e + int i"])
done
qed
with that show thesis by blast
qed

lemma float_normed_cases:
fixes f :: float
obtains (zero) "f = 0"
| (powr) m e :: int where "real_of_float f = m * 2 powr e" "¬ 2 dvd m" "f ≠ 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 ⇒ int"
where "mantissa f =
fst (SOME p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0) ∨
(f ≠ 0 ∧ real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) ∧ ¬ 2 dvd fst p))"

definition exponent :: "float ⇒ int"
where "exponent f =
snd (SOME p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0) ∨
(f ≠ 0 ∧ real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) ∧ ¬ 2 dvd fst p))"

lemma exponent_0[simp]: "exponent 0 = 0" (is ?E)
and mantissa_0[simp]: "mantissa 0 = 0" (is ?M)
proof -
have "⋀p::int × int. fst p = 0 ∧ snd p = 0 ⟷ p = (0, 0)"
by auto
then show ?E ?M
by (auto simp add: mantissa_def exponent_def zero_float_def)
qed

lemma mantissa_exponent: "real_of_float f = mantissa f * 2 powr exponent f" (is ?E)
and mantissa_not_dvd: "f ≠ 0 ⟹ ¬ 2 dvd mantissa f" (is "_ ⟹ ?D")
proof cases
assume [simp]: "f ≠ 0"
have "f = mantissa f * 2 powr exponent f ∧ ¬ 2 dvd mantissa f"
proof (cases f rule: float_normed_cases)
case zero
then show ?thesis by simp
next
case (powr m e)
then have "∃p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0) ∨
(f ≠ 0 ∧ real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) ∧ ¬ 2 dvd fst p)"
by auto
then show ?thesis
unfolding exponent_def mantissa_def
by (rule someI2_ex) simp
qed
then show ?E ?D by auto
qed simp

lemma mantissa_noteq_0: "f ≠ 0 ⟹ mantissa f ≠ 0"
using mantissa_not_dvd[of f] by auto

lemma mantissa_eq_zero_iff: "mantissa x = 0 ⟷ x = 0"
(is "?lhs ⟷ ?rhs")
proof
show ?rhs if ?lhs
proof -
from that have z: "0 = real_of_float x"
using mantissa_exponent by simp
show ?thesis
by (simp add: zero_float_def z)
qed
show ?lhs if ?rhs
using that by simp
qed

lemma mantissa_pos_iff: "0 < mantissa x ⟷ 0 < x"
by (auto simp: mantissa_exponent algebra_split_simps)

lemma mantissa_nonneg_iff: "0 ≤ mantissa x ⟷ 0 ≤ x"
by (auto simp: mantissa_exponent algebra_split_simps)

lemma mantissa_neg_iff: "0 > mantissa x ⟷ 0 > x"
by (auto simp: mantissa_exponent algebra_split_simps)

lemma
fixes m e :: int
defines "f ≡ float_of (m * 2 powr e)"
assumes dvd: "¬ 2 dvd m"
shows mantissa_float: "mantissa f = m" (is "?M")
and exponent_float: "m ≠ 0 ⟹ exponent f = e" (is "_ ⟹ ?E")
proof cases
assume "m = 0"
with dvd show "mantissa f = m" by auto
next
assume "m ≠ 0"
then have f_not_0: "f ≠ 0" by (simp add: f_def zero_float_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 [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 = Float m e"
and not_0: "f ≠ 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_diff field_simps)
moreover
have "e ≤ exponent f"
proof (rule ccontr)
assume "¬ e ≤ exponent f"
then have pos: "exponent f < e" by simp
then have "2 powr (exponent f - e) = 2 powr - real_of_int (e - exponent f)"
by simp
also have "… = 1 / 2^nat (e - exponent f)"
using pos by (simp flip: powr_realpow add: powr_diff)
finally have "m * 2^nat (e - exponent f) = real_of_int (mantissa f)"
using eq by simp
then have "mantissa f = m * 2^nat (e - exponent f)"
by linarith
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_of_int m = mantissa f * 2^nat (exponent f - e)"
by (simp flip: powr_realpow)
with ‹e ≤ exponent f›
show "m = mantissa f * 2 ^ nat (exponent f - e)"
by linarith
show "e = exponent f - nat (exponent f - e)"
using ‹e ≤ exponent f› by auto
qed

context
begin

qualified lemma compute_float_zero[code_unfold, code]: "0 = Float 0 0"
by transfer simp

qualified lemma compute_float_one[code_unfold, code]: "1 = Float 1 0"
by transfer simp

lift_definition normfloat :: "float ⇒ float" is "λx. x" .
lemma normloat_id[simp]: "normfloat x = x" by transfer rule

qualified lemma compute_normfloat[code]:
"normfloat (Float m e) =
(if m mod 2 = 0 ∧ m ≠ 0 then normfloat (Float (m div 2) (e + 1))
else if m = 0 then 0 else Float m e)"
by transfer (auto simp add: powr_add zmod_eq_0_iff)

qualified lemma compute_float_numeral[code_abbrev]: "Float (numeral k) 0 = numeral k"
by transfer simp

qualified lemma compute_float_neg_numeral[code_abbrev]: "Float (- numeral k) 0 = - numeral k"
by transfer simp

qualified lemma compute_float_uminus[code]: "- Float m1 e1 = Float (- m1) e1"
by transfer simp

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

qualified 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 ≤ 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_diff)

qualified lemma compute_float_minus[code]: "f - g = f + (-g)" for f g :: float
by simp

qualified 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_mult)

lift_definition is_float_pos :: "float ⇒ bool" is "(<) 0 :: real ⇒ bool" .

qualified lemma compute_is_float_pos[code]: "is_float_pos (Float m e) ⟷ 0 < m"
by transfer (auto simp add: zero_less_mult_iff not_le[symmetric, of _ 0])

lift_definition is_float_nonneg :: "float ⇒ bool" is "(≤) 0 :: real ⇒ bool" .

qualified lemma compute_is_float_nonneg[code]: "is_float_nonneg (Float m e) ⟷ 0 ≤ m"
by transfer (auto simp add: zero_le_mult_iff not_less[symmetric, of _ 0])

lift_definition is_float_zero :: "float ⇒ bool"  is "(=) 0 :: real ⇒ bool" .

qualified lemma compute_is_float_zero[code]: "is_float_zero (Float m e) ⟷ 0 = m"
by transfer (auto simp add: is_float_zero_def)

qualified lemma compute_float_abs[code]: "¦Float m e¦ = Float ¦m¦ e"
by transfer (simp add: abs_mult)

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

end

subsection ‹Lemmas for types \<^typ>‹real›, \<^typ>‹nat›, \<^typ>‹int››

lemmas real_of_ints =
of_int_minus
of_int_diff
of_int_mult
of_int_power
of_int_numeral of_int_neg_numeral

lemmas int_of_reals = real_of_ints[symmetric]

subsection ‹Rounding Real Numbers›

definition round_down :: "int ⇒ real ⇒ real"
where "round_down prec x = ⌊x * 2 powr prec⌋ * 2 powr -prec"

definition round_up :: "int ⇒ real ⇒ real"
where "round_up prec x = ⌈x * 2 powr prec⌉ * 2 powr -prec"

lemma round_down_float[simp]: "round_down prec x ∈ float"
unfolding round_down_def
by (auto intro!: times_float simp flip: of_int_minus)

lemma round_up_float[simp]: "round_up prec x ∈ float"
unfolding round_up_def
by (auto intro!: times_float simp flip: of_int_minus)

lemma round_up: "x ≤ round_up prec x"
by (simp add: powr_minus_divide le_divide_eq round_up_def ceiling_correct)

lemma round_down: "round_down prec x ≤ 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 ≤ 2 powr -prec"
proof -
have "round_up prec x - round_down prec x = (⌈x * 2 powr prec⌉ - ⌊x * 2 powr prec⌋) * 2 powr -prec"
by (simp add: round_up_def round_down_def field_simps)
also have "… ≤ 1 * 2 powr -prec"
by (rule mult_mono)
(auto simp flip: of_int_diff simp: 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_diff)

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_diff)

lemma round_up_uminus_eq: "round_up p (-x) = - round_down p x"
and round_down_uminus_eq: "round_down p (-x) = - round_up p x"
by (auto simp: round_up_def round_down_def ceiling_def)

lemma round_up_mono: "x ≤ y ⟹ round_up p x ≤ round_up p y"
by (auto intro!: ceiling_mono simp: round_up_def)

lemma round_up_le1:
assumes "x ≤ 1" "prec ≥ 0"
shows "round_up prec x ≤ 1"
proof -
have "real_of_int ⌈x * 2 powr prec⌉ ≤ real_of_int ⌈2 powr real_of_int prec⌉"
using assms by (auto intro!: ceiling_mono)
also have "… = 2 powr prec" using assms by (auto simp: powr_int intro!: exI[where x="2^nat prec"])
finally show ?thesis
by (simp add: round_up_def) (simp add: powr_minus inverse_eq_divide)
qed

lemma round_up_less1:
assumes "x < 1 / 2" "p > 0"
shows "round_up p x < 1"
proof -
have "x * 2 powr p < 1 / 2 * 2 powr p"
using assms by simp
also have "… ≤ 2 powr p - 1" using ‹p > 0›
by (auto simp: powr_diff powr_int field_simps self_le_power)
finally show ?thesis using ‹p > 0›
by (simp add: round_up_def field_simps powr_minus powr_int ceiling_less_iff)
qed

lemma round_down_ge1:
assumes x: "x ≥ 1"
assumes prec: "p ≥ - log 2 x"
shows "1 ≤ round_down p x"
proof cases
assume nonneg: "0 ≤ p"
have "2 powr p = real_of_int ⌊2 powr real_of_int p⌋"
using nonneg by (auto simp: powr_int)
also have "… ≤ real_of_int ⌊x * 2 powr p⌋"
using assms by (auto intro!: floor_mono)
finally show ?thesis
by (simp add: round_down_def) (simp add: powr_minus inverse_eq_divide)
next
assume neg: "¬ 0 ≤ p"
have "x = 2 powr (log 2 x)"
using x by simp
also have "2 powr (log 2 x) ≥ 2 powr - p"
using prec by auto
finally have x_le: "x ≥ 2 powr -p" .

from neg have "2 powr real_of_int p ≤ 2 powr 0"
by (intro powr_mono) auto
also have "… ≤ ⌊2 powr 0::real⌋" by simp
also have "… ≤ ⌊x * 2 powr (real_of_int p)⌋"
unfolding of_int_le_iff
using x x_le by (intro floor_mono) (simp add: powr_minus_divide field_simps)
finally show ?thesis
using prec x
by (simp add: round_down_def powr_minus_divide pos_le_divide_eq)
qed

lemma round_up_le0: "x ≤ 0 ⟹ round_up p x ≤ 0"
unfolding round_up_def
by (auto simp: field_simps mult_le_0_iff zero_le_mult_iff)

subsection ‹Rounding Floats›

definition div_twopow :: "int ⇒ nat ⇒ int"
where [simp]: "div_twopow x n = x div (2 ^ n)"

definition mod_twopow :: "int ⇒ nat ⇒ int"
where [simp]: "mod_twopow x n = x mod (2 ^ n)"

lemma compute_div_twopow[code]:
"div_twopow x n = (if x = 0 ∨ x = -1 ∨ n = 0 then x else div_twopow (x div 2) (n - 1))"
by (cases n) (auto simp: zdiv_zmult2_eq div_eq_minus1)

lemma compute_mod_twopow[code]:
"mod_twopow x n = (if n = 0 then 0 else x mod 2 + 2 * mod_twopow (x div 2) (n - 1))"
by (cases n) (auto simp: zmod_zmult2_eq)

lift_definition float_up :: "int ⇒ float ⇒ float" is round_up by simp
declare float_up.rep_eq[simp]

lemma round_up_correct: "round_up e f - f ∈ {0..2 powr -e}"
unfolding atLeastAtMost_iff
proof
have "round_up e f - f ≤ round_up e f - round_down e f"
using round_down by simp
also have "… ≤ 2 powr -e"
using round_up_diff_round_down by simp
finally show "round_up e f - f ≤ 2 powr - (real_of_int e)"
by simp
qed (simp add: algebra_simps round_up)

lemma float_up_correct: "real_of_float (float_up e f) - real_of_float f ∈ {0..2 powr -e}"
by transfer (rule round_up_correct)

lift_definition float_down :: "int ⇒ float ⇒ float" is round_down by simp
declare float_down.rep_eq[simp]

lemma round_down_correct: "f - (round_down e f) ∈ {0..2 powr -e}"
unfolding atLeastAtMost_iff
proof
have "f - round_down e f ≤ round_up e f - round_down e f"
using round_up by simp
also have "… ≤ 2 powr -e"
using round_up_diff_round_down by simp
finally show "f - round_down e f ≤ 2 powr - (real_of_int e)"
by simp
qed (simp add: algebra_simps round_down)

lemma float_down_correct: "real_of_float f - real_of_float (float_down e f) ∈ {0..2 powr -e}"
by transfer (rule round_down_correct)

context
begin

qualified lemma compute_float_down[code]:
"float_down p (Float m e) =
(if p + e < 0 then Float (div_twopow m (nat (-(p + e)))) (-p) else Float m e)"
proof (cases "p + e < 0")
case True
then have "real_of_int ((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 of_int_minus by (simp add: powr_add)
finally show ?thesis
using ‹p + e < 0›
apply transfer
apply (simp add: round_down_def field_simps flip: floor_divide_of_int_eq powr_add)
apply (metis (no_types, opaque_lifting) Float.rep_eq
floor_divide_of_int_eq int_of_reals(1) linorder_not_le
done
next
case False
then have r: "real_of_int e + real_of_int p = real (nat (e + p))"
by simp
have r: "⌊(m * 2 powr e) * 2 powr real_of_int p⌋ = (m * 2 powr e) * 2 powr real_of_int p"
by (auto intro: exI[where x="m*2^nat (e+p)"]
with ‹¬ p + e < 0› show ?thesis
by transfer (auto simp add: round_down_def field_simps powr_add powr_minus)
qed

lemma abs_round_down_le: "¦f - (round_down e f)¦ ≤ 2 powr -e"
using round_down_correct[of f e] by simp

lemma abs_round_up_le: "¦f - (round_up e f)¦ ≤ 2 powr -e"
using round_up_correct[of e f] by simp

lemma round_down_nonneg: "0 ≤ s ⟹ 0 ≤ round_down p s"
by (auto simp: round_down_def)

lemma ceil_divide_floor_conv:
assumes "b ≠ 0"
shows "⌈real_of_int a / real_of_int b⌉ =
(if b dvd a then a div b else ⌊real_of_int a / real_of_int b⌋ + 1)"
proof (cases "b dvd a")
case True
then show ?thesis
by (simp add: ceiling_def floor_divide_of_int_eq dvd_neg_div
flip: of_int_minus divide_minus_left)
next
case False
then have "a mod b ≠ 0"
by auto
then have ne: "real_of_int (a mod b) / real_of_int b ≠ 0"
using ‹b ≠ 0› by auto
have "⌈real_of_int a / real_of_int b⌉ = ⌊real_of_int a / real_of_int b⌋ + 1"
apply (rule ceiling_eq)
apply (auto simp flip: floor_divide_of_int_eq)
proof -
have "real_of_int ⌊real_of_int a / real_of_int b⌋ ≤ real_of_int a / real_of_int b"
by simp
moreover have "real_of_int ⌊real_of_int a / real_of_int b⌋ ≠ real_of_int a / real_of_int b"
by (smt (verit) floor_divide_of_int_eq ne real_of_int_div_aux)
ultimately show "real_of_int ⌊real_of_int a / real_of_int b⌋ < real_of_int a / real_of_int b" by arith
qed
then show ?thesis
using ‹¬ b dvd a› by simp
qed

qualified lemma compute_float_up[code]: "float_up p x = - float_down p (-x)"
by transfer (simp add: round_down_uminus_eq)

end

lemma bitlen_Float:
fixes m e
defines [THEN meta_eq_to_obj_eq]: "f ≡ Float m e"
shows "bitlen ¦mantissa f¦ + exponent f = (if m = 0 then 0 else bitlen ¦m¦ + e)"
proof (cases "m = 0")
case True
then show ?thesis by (simp add: f_def bitlen_alt_def)
next
case False
then have "f ≠ 0"
unfolding real_of_float_eq by (simp add: f_def)
then have "mantissa f ≠ 0"
by (simp add: mantissa_eq_zero_iff)
moreover
obtain i where "m = mantissa f * 2 ^ i" "e = exponent f - int i"
by (rule f_def[THEN denormalize_shift, OF ‹f ≠ 0›])
ultimately show ?thesis by (simp add: abs_mult)
qed

lemma float_gt1_scale:
assumes "1 ≤ Float m e"
shows "0 ≤ e + (bitlen m - 1)"
proof -
have "0 < Float m e" using assms by auto
then have "0 < m" using powr_gt_zero[of 2 e]
by (auto simp: zero_less_mult_iff)
then have "m ≠ 0" by auto
show ?thesis
proof (cases "0 ≤ e")
case True
then show ?thesis
using ‹0 < m› by (simp add: bitlen_alt_def)
next
case False
have "(1::int) < 2" by simp
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)
then have "1 ≤ real_of_int m * inverse ?S"
using assms False powr_realpow[of 2 "nat (-e)"]
by (auto simp: powr_minus)
then have "1 * ?S ≤ real_of_int m * inverse ?S * ?S"
by (rule mult_right_mono) auto
then have "?S ≤ real_of_int m"
unfolding mult.assoc by auto
then have "?S ≤ m"
unfolding 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)
then have "-e < bitlen m"
using False by auto
then show ?thesis
by auto
qed
qed

subsection ‹Truncating Real Numbers›

definition truncate_down::"nat ⇒ real ⇒ real"
where "truncate_down prec x = round_down (prec - ⌊log 2 ¦x¦⌋) x"

lemma truncate_down: "truncate_down prec x ≤ x"
using round_down by (simp add: truncate_down_def)

lemma truncate_down_le: "x ≤ y ⟹ truncate_down prec x ≤ y"
by (rule order_trans[OF truncate_down])

lemma truncate_down_zero[simp]: "truncate_down prec 0 = 0"
by (simp add: truncate_down_def)

lemma truncate_down_float[simp]: "truncate_down p x ∈ float"
by (auto simp: truncate_down_def)

definition truncate_up::"nat ⇒ real ⇒ real"
where "truncate_up prec x = round_up (prec - ⌊log 2 ¦x¦⌋) x"

lemma truncate_up: "x ≤ truncate_up prec x"
using round_up by (simp add: truncate_up_def)

lemma truncate_up_le: "x ≤ y ⟹ x ≤ 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)

lemma truncate_up_uminus_eq: "truncate_up prec (-x) = - truncate_down prec x"
and truncate_down_uminus_eq: "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_up_float[simp]: "truncate_up p x ∈ float"
by (auto simp: truncate_up_def)

lemma mult_powr_eq: "0 < b ⟹ b ≠ 1 ⟹ 0 < x ⟹ x * b powr y = b powr (y + log b x)"

lemma truncate_down_pos:
assumes "x > 0"
shows "truncate_down p x > 0"
proof -
have "0 ≤ log 2 x - real_of_int ⌊log 2 x⌋"
by (simp add: algebra_simps)
with assms
show ?thesis
apply (auto simp: truncate_down_def round_down_def mult_powr_eq
intro!: ge_one_powr_ge_zero mult_pos_pos)
by linarith
qed

lemma truncate_down_nonneg: "0 ≤ y ⟹ 0 ≤ truncate_down prec y"
by (auto simp: truncate_down_def round_down_def)

lemma truncate_down_ge1: "1 ≤ x ⟹ 1 ≤ truncate_down p x"
apply (auto simp: truncate_down_def algebra_simps intro!: round_down_ge1)
apply linarith
done

lemma truncate_up_nonpos: "x ≤ 0 ⟹ truncate_up prec x ≤ 0"
by (auto simp: truncate_up_def round_up_def intro!: mult_nonpos_nonneg)

lemma truncate_up_le1:
assumes "x ≤ 1"
shows "truncate_up p x ≤ 1"
proof -
consider "x ≤ 0" | "x > 0"
by arith
then show ?thesis
proof cases
case 1
with truncate_up_nonpos[OF this, of p] show ?thesis
by simp
next
case 2
then have le: "⌊log 2 ¦x¦⌋ ≤ 0"
using assms by (auto simp: log_less_iff)
from assms have "0 ≤ int p" by simp
from add_mono[OF this le]
show ?thesis
using assms by (simp add: truncate_up_def round_up_le1 add_mono)
qed
qed

lemma truncate_down_shift_int:
"truncate_down p (x * 2 powr real_of_int k) = truncate_down p x * 2 powr k"
by (cases "x = 0")
(simp_all add: algebra_simps abs_mult log_mult truncate_down_def
round_down_shift[of _ _ k, simplified])

lemma truncate_down_shift_nat: "truncate_down p (x * 2 powr real k) = truncate_down p x * 2 powr k"
by (metis of_int_of_nat_eq truncate_down_shift_int)

lemma truncate_up_shift_int: "truncate_up p (x * 2 powr real_of_int k) = truncate_up p x * 2 powr k"
by (cases "x = 0")
(simp_all add: algebra_simps abs_mult log_mult truncate_up_def
round_up_shift[of _ _ k, simplified])

lemma truncate_up_shift_nat: "truncate_up p (x * 2 powr real k) = truncate_up p x * 2 powr k"
by (metis of_int_of_nat_eq truncate_up_shift_int)

subsection ‹Truncating Floats›

lift_definition float_round_up :: "nat ⇒ float ⇒ float" is truncate_up
by (simp add: truncate_up_def)

lemma float_round_up: "real_of_float x ≤ real_of_float (float_round_up prec x)"
using truncate_up by transfer simp

lemma float_round_up_zero[simp]: "float_round_up prec 0 = 0"
by transfer simp

lift_definition float_round_down :: "nat ⇒ float ⇒ float" is truncate_down
by (simp add: truncate_down_def)

lemma float_round_down: "real_of_float (float_round_down prec x) ≤ real_of_float x"
using truncate_down by transfer simp

lemma float_round_down_zero[simp]: "float_round_down prec 0 = 0"
by transfer simp

lemmas float_round_up_le = order_trans[OF _ float_round_up]
and float_round_down_le = order_trans[OF float_round_down]

lemma minus_float_round_up_eq: "- float_round_up prec x = float_round_down prec (- x)"
and minus_float_round_down_eq: "- float_round_down prec x = float_round_up prec (- x)"
by (transfer; simp add: truncate_down_uminus_eq truncate_up_uminus_eq)+

context
begin

qualified lemma compute_float_round_down[code]:
"float_round_down prec (Float m e) =
(let d = bitlen ¦m¦ - int prec - 1 in
if 0 < d then Float (div_twopow m (nat d)) (e + d)
else Float m e)"
using Float.compute_float_down[of "Suc prec - bitlen ¦m¦ - e" m e, symmetric]
by transfer
(simp add: field_simps abs_mult log_mult bitlen_alt_def truncate_down_def
cong del: if_weak_cong)

qualified lemma compute_float_round_up[code]:
"float_round_up prec x = - float_round_down prec (-x)"
by transfer (simp add: truncate_down_uminus_eq)

end

lemma truncate_up_nonneg_mono:
assumes "0 ≤ x" "x ≤ y"
shows "truncate_up prec x ≤ truncate_up prec y"
proof -
consider "⌊log 2 x⌋ = ⌊log 2 y⌋" | "⌊log 2 x⌋ ≠ ⌊log 2 y⌋" "0 < x" | "x ≤ 0"
by arith
then show ?thesis
proof cases
case 1
then show ?thesis
using assms
by (auto simp: truncate_up_def round_up_def intro!: ceiling_mono)
next
case 2
from assms ‹0 < x› have "log 2 x ≤ log 2 y"
by auto
with ‹⌊log 2 x⌋ ≠ ⌊log 2 y⌋›
have logless: "log 2 x < log 2 y"
by linarith
have flogless: "⌊log 2 x⌋ < ⌊log 2 y⌋"
using ‹⌊log 2 x⌋ ≠ ⌊log 2 y⌋› ‹log 2 x ≤ log 2 y› by linarith
have "truncate_up prec x =
real_of_int ⌈x * 2 powr real_of_int (int prec - ⌊log 2 x⌋ )⌉ * 2 powr - real_of_int (int prec - ⌊log 2 x⌋)"
using assms by (simp add: truncate_up_def round_up_def)
also have "⌈x * 2 powr real_of_int (int prec - ⌊log 2 x⌋)⌉ ≤ (2 ^ (Suc prec))"
proof (simp only: ceiling_le_iff)
have "x * 2 powr real_of_int (int prec - ⌊log 2 x⌋) ≤
x * (2 powr real (Suc prec) / (2 powr log 2 x))"
using real_of_int_floor_add_one_ge[of "log 2 x"] assms
by (auto simp: algebra_simps simp flip: powr_diff intro!: mult_left_mono)
then show "x * 2 powr real_of_int (int prec - ⌊log 2 x⌋) ≤ real_of_int ((2::int) ^ (Suc prec))"
using ‹0 < x› by (simp add: powr_realpow powr_add)
qed
then have "real_of_int ⌈x * 2 powr real_of_int (int prec - ⌊log 2 x⌋)⌉ ≤ 2 powr int (Suc prec)"
by (auto simp: powr_realpow powr_add)
(metis power_Suc of_int_le_numeral_power_cancel_iff)
also
have "2 powr - real_of_int (int prec - ⌊log 2 x⌋) ≤ 2 powr - real_of_int (int prec - ⌊log 2 y⌋ + 1)"
using logless flogless by (auto intro!: floor_mono)
also have "2 powr real_of_int (int (Suc prec)) ≤
2 powr (log 2 y + real_of_int (int prec - ⌊log 2 y⌋ + 1))"
using assms ‹0 < x›
by (auto simp: algebra_simps)
finally have "truncate_up prec x ≤
2 powr (log 2 y + real_of_int (int prec - ⌊log 2 y⌋ + 1)) * 2 powr - real_of_int (int prec - ⌊log 2 y⌋ + 1)"
by simp
also have "… = 2 powr (log 2 y + real_of_int (int prec - ⌊log 2 y⌋) - real_of_int (int prec - ⌊log 2 y⌋))"
by (subst powr_add[symmetric]) simp
also have "… = y"
using ‹0 < x› assms
also have "… ≤ truncate_up prec y"
by (rule truncate_up)
finally show ?thesis .
next
case 3
then show ?thesis
using assms
by (auto intro!: truncate_up_le)
qed
qed

lemma truncate_up_switch_sign_mono:
assumes "x ≤ 0" "0 ≤ y"
shows "truncate_up prec x ≤ truncate_up prec y"
proof -
note truncate_up_nonpos[OF ‹x ≤ 0›]
also note truncate_up_le[OF ‹0 ≤ y›]
finally show ?thesis .
qed

lemma truncate_down_switch_sign_mono:
assumes "x ≤ 0"
and "0 ≤ y"
and "x ≤ y"
shows "truncate_down prec x ≤ truncate_down prec y"
proof -
note truncate_down_le[OF ‹x ≤ 0›]
also note truncate_down_nonneg[OF ‹0 ≤ y›]
finally show ?thesis .
qed

lemma truncate_down_nonneg_mono:
assumes "0 ≤ x" "x ≤ y"
shows "truncate_down prec x ≤ truncate_down prec y"
proof -
consider "x ≤ 0" | "⌊log 2 ¦x¦⌋ = ⌊log 2 ¦y¦⌋" |
"0 < x" "⌊log 2 ¦x¦⌋ ≠ ⌊log 2 ¦y¦⌋"
by arith
then show ?thesis
proof cases
case 1
with assms have "x = 0" "0 ≤ y" by simp_all
then show ?thesis
by (auto intro!: truncate_down_nonneg)
next
case 2
then show ?thesis
using assms
by (auto simp: truncate_down_def round_down_def intro!: floor_mono)
next
case 3
from ‹0 < x› have "log 2 x ≤ log 2 y" "0 < y" "0 ≤ y"
using assms by auto
with ‹⌊log 2 ¦x¦⌋ ≠ ⌊log 2 ¦y¦⌋›
have logless: "log 2 x < log 2 y" and flogless: "⌊log 2 x⌋ < ⌊log 2 y⌋"
unfolding atomize_conj abs_of_pos[OF ‹0 < x›] abs_of_pos[OF ‹0 < y›]
by (metis floor_less_cancel linorder_cases not_le)
have "2 powr prec ≤ y * 2 powr real prec / (2 powr log 2 y)"
using ‹0 < y› by simp
also have "… ≤ y * 2 powr real (Suc prec) / (2 powr (real_of_int ⌊log 2 y⌋ + 1))"
using ‹0 ≤ y› ‹0 ≤ x› assms(2)
by (auto intro!: powr_mono divide_left_mono
simp: of_nat_diff powr_add powr_diff)
also have "… = y * 2 powr real (Suc prec) / (2 powr real_of_int ⌊log 2 y⌋ * 2)"
by (auto simp: powr_add)
finally have "(2 ^ prec) ≤ ⌊y * 2 powr real_of_int (int (Suc prec) - ⌊log 2 ¦y¦⌋ - 1)⌋"
using ‹0 ≤ y›
by (auto simp: powr_diff le_floor_iff powr_realpow powr_add)
then have "(2 ^ (prec)) * 2 powr - real_of_int (int prec - ⌊log 2 ¦y¦⌋) ≤ truncate_down prec y"
by (auto simp: truncate_down_def round_down_def)
moreover have "x ≤ (2 ^ prec) * 2 powr - real_of_int (int prec - ⌊log 2 ¦y¦⌋)"
proof -
have "x = 2 powr (log 2 ¦x¦)" using ‹0 < x› by simp
also have "… ≤ (2 ^ (Suc prec )) * 2 powr - real_of_int (int prec - ⌊log 2 ¦x¦⌋)"
using real_of_int_floor_add_one_ge[of "log 2 ¦x¦"] ‹0 < x›
by (auto simp flip: powr_realpow powr_add simp: algebra_simps powr_mult_base le_powr_iff)
also
have "2 powr - real_of_int (int prec - ⌊log 2 ¦x¦⌋) ≤ 2 powr - real_of_int (int prec - ⌊log 2 ¦y¦⌋ + 1)"
using logless flogless ‹x > 0› ‹y > 0›
by (auto intro!: floor_mono)
finally show ?thesis
by (auto simp flip: powr_realpow simp: powr_diff assms of_nat_diff)
qed
ultimately show ?thesis
by (metis dual_order.trans truncate_down)
qed
qed

lemma truncate_down_eq_truncate_up: "truncate_down p x = - truncate_up p (-x)"
and truncate_up_eq_truncate_down: "truncate_up p x = - truncate_down p (-x)"
by (auto simp: truncate_up_uminus_eq truncate_down_uminus_eq)

lemma truncate_down_mono: "x ≤ y ⟹ truncate_down p x ≤ truncate_down p y"
by (smt (verit) truncate_down_nonneg_mono truncate_up_nonneg_mono truncate_up_uminus_eq)

lemma truncate_up_mono: "x ≤ y ⟹ truncate_up p x ≤ truncate_up p y"
by (simp add: truncate_up_eq_truncate_down truncate_down_mono)

lemma truncate_up_nonneg: "0 ≤ truncate_up p x" if "0 ≤ x"
by (simp add: that truncate_up_le)

lemma truncate_up_pos: "0 < truncate_up p x" if "0 < x"
by (meson less_le_trans that truncate_up)

lemma truncate_up_less_zero_iff[simp]: "truncate_up p x < 0 ⟷ x < 0"
by (smt (verit) truncate_down_pos truncate_down_uminus_eq truncate_up_nonneg)

lemma truncate_up_nonneg_iff[simp]: "truncate_up p x ≥ 0 ⟷ x ≥ 0"
using truncate_up_less_zero_iff[of p x] truncate_up_nonneg[of x]
by linarith

lemma truncate_down_less_zero_iff[simp]: "truncate_down p x < 0 ⟷ x < 0"
by (metis le_less_trans not_less_iff_gr_or_eq truncate_down truncate_down_pos truncate_down_zero)

lemma truncate_down_nonneg_iff[simp]: "truncate_down p x ≥ 0 ⟷ x ≥ 0"
using truncate_down_less_zero_iff[of p x] truncate_down_nonneg[of x p]
by linarith

lemma truncate_down_eq_zero_iff[simp]: "truncate_down prec x = 0 ⟷ x = 0"
by (metis not_less_iff_gr_or_eq truncate_down_less_zero_iff truncate_down_pos truncate_down_zero)

lemma truncate_up_eq_zero_iff[simp]: "truncate_up prec x = 0 ⟷ x = 0"
by (metis not_less_iff_gr_or_eq truncate_up_less_zero_iff truncate_up_pos truncate_up_zero)

subsection ‹Approximation of positive rationals›

lemma div_mult_twopow_eq: "a div ((2::nat) ^ n) div b = a div (b * 2 ^ n)" for a b :: nat
by (cases "b = 0") (simp_all add: div_mult2_eq[symmetric] ac_simps)

lemma real_div_nat_eq_floor_of_divide: "a div b = real_of_int ⌊a / b⌋" for a b :: nat
by (simp add: floor_divide_of_nat_eq [of a b])

definition "rat_precision prec x y =
(let d = bitlen x - bitlen y
in int prec - d + (if Float (abs x) 0 < Float (abs y) d then 1 else 0))"

lemma floor_log_divide_eq:
assumes "i > 0" "j > 0" "p > 1"
shows "⌊log p (i / j)⌋ = floor (log p i) - floor (log p j) -
(if i ≥ j * p powr (floor (log p i) - floor (log p j)) then 0 else 1)"
proof -
let ?l = "log p"
let ?fl = "λx. floor (?l x)"
have "⌊?l (i / j)⌋ = ⌊?l i - ?l j⌋" using assms
by (auto simp: log_divide)
also have "… = floor (real_of_int (?fl i - ?fl j) + (?l i - ?fl i - (?l j - ?fl j)))"
(is "_ = floor (_ + ?r)")
by (simp add: algebra_simps)
also note ‹p > 1›
note powr = powr_le_cancel_iff[symmetric, OF ‹1 < p›, THEN iffD2]
note powr_strict = powr_less_cancel_iff[symmetric, OF ‹1 < p›, THEN iffD2]
have "floor ?r = (if i ≥ j * p powr (?fl i - ?fl j) then 0 else -1)" (is "_ = ?if")
using assms
by (linarith |
auto
intro!: floor_eq2
intro: powr_strict powr
simp: powr_diff powr_add field_split_simps algebra_simps)+
finally
show ?thesis by simp
qed

lemma truncate_down_rat_precision:
"truncate_down prec (real x / real y) = round_down (rat_precision prec x y) (real x / real y)"
and truncate_up_rat_precision:
"truncate_up prec (real x / real y) = round_up (rat_precision prec x y) (real x / real y)"
unfolding truncate_down_def truncate_up_def rat_precision_def
by (cases x; cases y) (auto simp: floor_log_divide_eq algebra_simps bitlen_alt_def)

lift_definition lapprox_posrat :: "nat ⇒ nat ⇒ nat ⇒ float"
is "λprec (x::nat) (y::nat). truncate_down prec (x / y)"
by simp

context
begin

qualified lemma compute_lapprox_posrat[code]:
"lapprox_posrat prec x y =
(let
l = rat_precision prec x y;
d = if 0 ≤ 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
by transfer
(simp add: round_down_def powr_int real_div_nat_eq_floor_of_divide field_simps Let_def
truncate_down_rat_precision del: two_powr_minus_int_float)

end

lift_definition rapprox_posrat :: "nat ⇒ nat ⇒ nat ⇒ float"
is "λprec (x::nat) (y::nat). truncate_up prec (x / y)"
by simp

context
begin

qualified lemma compute_rapprox_posrat[code]:
fixes prec x y
defines "l ≡ rat_precision prec x y"
shows "rapprox_posrat prec x y =
(let
l = l;
(r, s) = if 0 ≤ l then (x * 2^nat l, y) else (x, y * 2^nat(-l));
d = r div s;
m = r mod s
in normfloat (Float (d + (if m = 0 ∨ y = 0 then 0 else 1)) (- l)))"
proof (cases "y = 0")
assume "y = 0"
then show ?thesis by transfer simp
next
assume "y ≠ 0"
show ?thesis
proof (cases "0 ≤ l")
case True
define x' where "x' = x * 2 ^ nat l"
have "int x * 2 ^ nat l = x'"
by (simp add: x'_def)
moreover have "real x * 2 powr l = real x'"
by (simp flip: powr_realpow add: ‹0 ≤ l› x'_def)
ultimately show ?thesis
using ceil_divide_floor_conv[of y x'] powr_realpow[of 2 "nat l"] ‹0 ≤ l› ‹y ≠ 0›
l_def[symmetric, THEN meta_eq_to_obj_eq]
apply transfer
apply (auto simp add: round_up_def truncate_up_rat_precision)
apply (metis dvd_triv_left of_nat_dvd_iff)
apply (metis floor_divide_of_int_eq of_int_of_nat_eq)
done
next
case False
define y' where "y' = y * 2 ^ nat (- l)"
from ‹y ≠ 0› have "y' ≠ 0" by (simp add: y'_def)
have "int y * 2 ^ nat (- l) = y'"
by (simp add: y'_def)
moreover have "real x * real_of_int (2::int) powr real_of_int l / real y = x / real y'"
using ‹¬ 0 ≤ l› by (simp flip: powr_realpow add: powr_minus y'_def field_simps)
ultimately show ?thesis
using ceil_divide_floor_conv[of y' x] ‹¬ 0 ≤ l› ‹y' ≠ 0› ‹y ≠ 0›
l_def[symmetric, THEN meta_eq_to_obj_eq]
apply transfer
apply (auto simp add: round_up_def ceil_divide_floor_conv truncate_up_rat_precision)
apply (metis dvd_triv_left of_nat_dvd_iff)
apply (metis floor_divide_of_int_eq of_int_of_nat_eq)
done
qed
qed

end

lemma rat_precision_pos:
assumes "0 ≤ x"
and "0 < y"
and "2 * x < y"
shows "rat_precision n (int x) (int y) > 0"
proof -
have "0 < x ⟹ log 2 x + 1 = log 2 (2 * x)"
by (simp add: log_mult)
then have "bitlen (int x) < bitlen (int y)"
using assms
by (simp add: bitlen_alt_def)
then show ?thesis
using assms
by (auto intro!: pos_add_strict simp add: field_simps rat_precision_def)
qed

lemma rapprox_posrat_less1:
"0 ≤ x ⟹ 0 < y ⟹ 2 * x < y ⟹ real_of_float (rapprox_posrat n x y) < 1"
by transfer (simp add: rat_precision_pos round_up_less1 truncate_up_rat_precision)

lift_definition lapprox_rat :: "nat ⇒ int ⇒ int ⇒ float" is
"λprec (x::int) (y::int). truncate_down prec (x / y)"
by simp

context
begin

qualified lemma compute_lapprox_rat[code]:
"lapprox_rat prec x y =
(if y = 0 then 0
else if 0 ≤ 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 (simp add: truncate_up_uminus_eq)

lift_definition rapprox_rat :: "nat ⇒ int ⇒ int ⇒ float" is
"λprec (x::int) (y::int). truncate_up prec (x / y)"
by simp

lemma "rapprox_rat = rapprox_posrat"
by transfer auto

lemma "lapprox_rat = lapprox_posrat"
by transfer auto

qualified lemma compute_rapprox_rat[code]:
"rapprox_rat prec x y = - lapprox_rat prec (-x) y"
by transfer (simp add: truncate_down_uminus_eq)

qualified lemma compute_truncate_down[code]:
"truncate_down p (Ratreal r) = (let (a, b) = quotient_of r in lapprox_rat p a b)"
by transfer (auto split: prod.split simp: of_rat_divide dest!: quotient_of_div)

qualified lemma compute_truncate_up[code]:
"truncate_up p (Ratreal r) = (let (a, b) = quotient_of r in rapprox_rat p a b)"
by transfer (auto split: prod.split simp:  of_rat_divide dest!: quotient_of_div)

end

subsection ‹Division›

definition "real_divl prec a b = truncate_down prec (a / b)"

definition "real_divr prec a b = truncate_up prec (a / b)"

lift_definition float_divl :: "nat ⇒ float ⇒ float ⇒ float" is real_divl
by (simp add: real_divl_def)

context
begin

qualified lemma compute_float_divl[code]:
"float_divl prec (Float m1 s1) (Float m2 s2) = lapprox_rat prec m1 m2 * Float 1 (s1 - s2)"
apply transfer
unfolding real_divl_def of_int_1 mult_1 truncate_down_shift_int[symmetric]
apply (simp add: powr_diff powr_minus)
done

lift_definition float_divr :: "nat ⇒ float ⇒ float ⇒ float" is real_divr
by (simp add: real_divr_def)

qualified lemma compute_float_divr[code]:
"float_divr prec x y = - float_divl prec (-x) y"
by transfer (simp add: real_divr_def real_divl_def truncate_down_uminus_eq)

end

definition "plus_down prec x y = truncate_down prec (x + y)"

definition "plus_up prec x y = truncate_up prec (x + y)"

lemma float_plus_down_float[intro, simp]: "x ∈ float ⟹ y ∈ float ⟹ plus_down p x y ∈ float"
by (simp add: plus_down_def)

lemma float_plus_up_float[intro, simp]: "x ∈ float ⟹ y ∈ float ⟹ plus_up p x y ∈ float"
by (simp add: plus_up_def)

lift_definition float_plus_down :: "nat ⇒ float ⇒ float ⇒ float" is plus_down ..

lift_definition float_plus_up :: "nat ⇒ float ⇒ float ⇒ float" is plus_up ..

lemma plus_down: "plus_down prec x y ≤ x + y"
and plus_up: "x + y ≤ plus_up prec x y"
by (auto simp: plus_down_def truncate_down plus_up_def truncate_up)

lemma float_plus_down: "real_of_float (float_plus_down prec x y) ≤ x + y"
and float_plus_up: "x + y ≤ ```