Theory Float

theory Float
imports Lattice_Algebras
(*  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 ≡ 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 => real"]]

lemma real_of_float_eq:
fixes f1 f2 :: float shows "f1 = f2 <-> 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 ∈ float ==> 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 ==> x ∈ float"
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]: "neg_numeral i ∈ float" by (intro floatI[of "neg_numeral i" 0]) simp
lemma real_of_int_float[simp]: "real (x :: int) ∈ float" by (intro floatI[of x 0]) simp
lemma real_of_nat_float[simp]: "real (x :: nat) ∈ float" by (intro floatI[of x 0]) simp
lemma two_powr_int_float[simp]: "2 powr (real (i::int)) ∈ float" by (intro floatI[of 1 i]) simp
lemma two_powr_nat_float[simp]: "2 powr (real (i::nat)) ∈ float" by (intro floatI[of 1 i]) simp
lemma two_powr_minus_int_float[simp]: "2 powr - (real (i::int)) ∈ float" by (intro floatI[of 1 "-i"]) simp
lemma two_powr_minus_nat_float[simp]: "2 powr - (real (i::nat)) ∈ float" 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 neg_numeral i ∈ float" by (intro floatI[of 1 "neg_numeral i"]) simp
lemma two_pow_float[simp]: "2 ^ n ∈ float" by (intro floatI[of 1 "n"]) (simp add: powr_realpow)
lemma real_of_float_float[simp]: "real (f::float) ∈ float" by (cases f) simp

lemma plus_float[simp]: "r ∈ float ==> p ∈ float ==> r + p ∈ float"
unfolding float_def
proof (safe, simp)
fix e1 m1 e2 m2 :: int
{ fix e1 m1 e2 m2 :: int assume "e1 ≤ 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 "∃(m::int) (e::int). m1 * 2 powr e1 + m2 * 2 powr e2 = m * 2 powr e"
by blast }
note * = this
show "∃(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 ≤ e1" from *[OF this, of m2 m1] show ?thesis by (simp add: ac_simps)
qed (rule *)
qed

lemma uminus_float[simp]: "x ∈ float ==> -x ∈ float"
apply (auto simp: float_def)
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 ∈ float ==> y ∈ float ==> x * y ∈ float"
apply (auto simp: float_def)
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 ∈ float ==> y ∈ float ==> x - y ∈ float"
unfolding ab_diff_minus by (intro uminus_float plus_float)

lemma abs_float[simp]: "x ∈ float ==> abs 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"
apply (auto simp add: float_def)
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 ∈ float ==> x / (2::int)^d ∈ float"
apply (auto simp add: float_def)
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 ∈ float" shows "x / (numeral (Num.Bit0 n)) ∈ float"
proof -
have "(x / numeral n) / 2^1 ∈ 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 ∈ float" shows "x / (neg_numeral (Num.Bit0 n)) ∈ float"
proof -
have "- (x / numeral (Num.Bit0 n)) ∈ float" using x by simp
also have "- (x / numeral (Num.Bit0 n)) = x / neg_numeral (Num.Bit0 n)"
unfolding neg_numeral_def by (simp del: minus_numeral)
finally show ?thesis .
qed

lift_definition Float :: "int => int => float" is "λ(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 ≥ 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 => float => float" is "op +" by simp
declare plus_float.rep_eq[simp]
lift_definition times_float :: "float => float => float" is "op *" by simp
declare times_float.rep_eq[simp]
lift_definition minus_float :: "float => float => float" is "op -" 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 "op = :: real => real => bool" ..

lift_definition less_eq_float :: "float => float => bool" is "op ≤" ..
declare less_eq_float.rep_eq[simp]
lift_definition less_float :: "float => float => 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 "∃c. a < c"
apply (intro exI[of _ "a + 1"])
apply transfer
apply simp
done
show "∃c. c < a"
apply (intro exI[of _ "a - 1"])
apply transfer
apply simp
done
assume "a < b"
then show "∃c. a < c ∧ c < b"
apply (intro exI[of _ "(a + b) * Float 1 -1"])
apply transfer
apply (simp add: powr_neg_numeral)
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 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]:
"fun_rel (op =) pcr_float (numeral :: _ => real) (numeral :: _ => float)"
unfolding fun_rel_def float.pcr_cr_eq cr_float_def by simp

lemma float_neg_numeral[simp]: "real (neg_numeral x :: float) = neg_numeral x"
by (simp add: minus_numeral[symmetric] del: minus_numeral)

lemma transfer_neg_numeral [transfer_rule]:
"fun_rel (op =) pcr_float (neg_numeral :: _ => real) (neg_numeral :: _ => float)"
unfolding fun_rel_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]: "neg_numeral k = float_of (neg_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: "!!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)
{ fix m assume n: "n ≠ 0" "n = m * r"
then have "¦m ¦ < ¦n¦"
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 "∃k i. n = k * r ^ Suc i ∧ ¬ 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: "¬ 2 dvd m1" and "e1 ≤ e2"
shows "m1 * 2 powr e1 = m2 * 2 powr e2 <-> m1 = m2 ∧ e1 = e2"
proof
have "m1 ≠ 0" using m1 unfolding dvd_def by auto
assume eq: "m1 * 2 powr e1 = m2 * 2 powr e2"
with `e1 ≤ e2` have "m1 = m2 * 2 powr nat (e2 - e1)"
by (simp add: powr_divide2[symmetric] field_simps)
also have "… = 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 ∧ e1 = e2"
using eq `m1 ≠ 0` by (simp add: powr_inj)
qed simp

lemma mult_powr_eq_mult_powr_iff:
fixes m1 m2 e1 e2 :: int
shows "¬ 2 dvd m1 ==> ¬ 2 dvd m2 ==> m1 * 2 powr e1 = m2 * 2 powr e2 <-> m1 = m2 ∧ 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 ∈ float"
obtains (zero) "x = 0"
| (powr) m e :: int where "x = m * 2 powr e" "¬ 2 dvd m" "x ≠ 0"
proof atomize_elim
{ assume "x ≠ 0"
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 have "∃(m::int) (e::int). x = m * 2 powr e ∧ ¬ (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 ∨ (∃(m::int) (e::int). x = m * 2 powr e ∧ ¬ (2::int) dvd m ∧ x ≠ 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" "¬ 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 f = real (fst p) * 2 powr real (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 f = real (fst p) * 2 powr real (snd p) ∧ ¬ 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 "!!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
shows mantissa_exponent: "real f = mantissa f * 2 powr exponent f" (is ?E)
and mantissa_not_dvd: "f ≠ (float_of 0) ==> ¬ 2 dvd mantissa f" (is "_ ==> ?D")
proof cases
assume [simp]: "f ≠ (float_of 0)"
have "f = mantissa f * 2 powr exponent f ∧ ¬ 2 dvd mantissa f"
proof (cases f rule: float_normed_cases)
case (powr m e)
then have "∃p::int × int. (f = 0 ∧ fst p = 0 ∧ snd p = 0)
∨ (f ≠ 0 ∧ real f = real (fst p) * 2 powr real (snd p) ∧ ¬ 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 ≠ float_of 0 ==> mantissa f ≠ 0"
using mantissa_not_dvd[of f] by auto

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 ≠ 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 ≡ Float m e" and not_0: "f ≠ 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 ≤ 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 (e - exponent f)"
by simp
also have "… = 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 ≤ 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 => float" where
[simp]: "normfloat x = x"

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)"

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 (neg_numeral k) 0 = neg_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 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_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 => bool" is "op < 0 :: real => bool" ..

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])
hide_fact (open) compute_is_float_pos

lemma compute_float_less[code]: "a < b <-> is_float_pos (b - a)"
by transfer (simp add: field_simps)
hide_fact (open) compute_float_less

lift_definition is_float_nonneg :: "float => bool" is "op ≤ 0 :: real => bool" ..

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])
hide_fact (open) compute_is_float_nonneg

lemma compute_float_le[code]: "a ≤ b <-> is_float_nonneg (b - a)"
by transfer (simp add: field_simps)
hide_fact (open) compute_float_le

lift_definition is_float_zero :: "float => bool" is "op = 0 :: real => bool" by simp

lemma compute_is_float_zero[code]: "is_float_zero (Float m e) <-> 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 => real => real" where
"round_down prec x = floor (x * 2 powr prec) * 2 powr -prec"

definition round_up :: "int => real => real" where
"round_up prec x = ceiling (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: real_of_int_minus[symmetric] simp del: real_of_int_minus)

lemma round_up_float[simp]: "round_up prec x ∈ 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 ≤ round_up prec x"
by (simp add: powr_minus_divide le_divide_eq round_up_def)

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 =
(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 "… ≤ 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 => float => float" is round_up by simp
declare float_up.rep_eq[simp]

lemma float_up_correct:
shows "real (float_up e f) - real 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 "real (float_up e f) - real f ≤ 2 powr real (- e)"
by simp
qed (simp add: algebra_simps round_up)

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

lemma float_down_correct:
shows "real f - real (float_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 "real f - real (float_down e f) ≤ 2 powr real (- e)"
by simp
qed (simp add: algebra_simps round_down)

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 "¬ p + e < 0"
then have r: "real e + real p = real (nat (e + p))" by simp
have r: "⌊(m * 2 powr e) * 2 powr real p⌋ = (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 `¬ 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 ceil_divide_floor_conv:
assumes "b ≠ 0"
shows "⌈real a / real b⌉ = (if b dvd a then a div b else ⌊real a / real b⌋ + 1)"
proof cases
assume "¬ b dvd a"
hence "a mod b ≠ 0" by auto
hence ne: "real (a mod b) / real b ≠ 0" using `b ≠ 0` by auto
have "⌈real a / real b⌉ = ⌊real a / real b⌋ + 1"
apply (rule ceiling_eq) apply (auto simp: floor_divide_eq_div[symmetric])
proof -
have "real ⌊real a / real b⌋ ≤ real a / real b" by simp
moreover have "real ⌊real a / real b⌋ ≠ real a / real b"
apply (subst (2) real_of_int_div_aux) unfolding floor_divide_eq_div using ne `b ≠ 0` by auto
ultimately show "real ⌊real a / real b⌋ < real a / real b" by arith
qed
thus ?thesis using `¬ 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: "¬ 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 ⌈real m * (2 powr real e * 2 powr real p)⌉ =
real ⌊real m * (2 powr real e * 2 powr real p)⌋ + 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 "¬ p + e < 0"
then have r1: "real e + real p = real (nat (e + p))" by simp
have r: "⌈(m * 2 powr e) * 2 powr real p⌉ = (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 `¬ 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 => int" where
"bitlen a = (if a > 0 then ⌊log 2 a⌋ + 1 else 0)"

lemma bitlen_nonneg: "0 ≤ 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 ≤ bitlen x" unfolding bitlen_def by (auto intro!: add_nonneg_nonneg)
qed

lemma bitlen_bounds:
assumes "x > 0"
shows "2 ^ nat (bitlen x - 1) ≤ x ∧ x < 2 ^ nat (bitlen x)"
proof
have "(2::real) ^ nat ⌊log 2 (real x)⌋ = 2 powr real (floor (log 2 (real x)))"
using powr_realpow[symmetric, of 2 "nat ⌊log 2 (real x)⌋"] `x > 0`
using real_nat_eq_real[of "floor (log 2 (real x))"]
by simp
also have "... ≤ 2 powr log 2 (real x)"
by simp
also have "... = real x"
using `0 < x` by simp
finally have "2 ^ nat ⌊log 2 (real x)⌋ ≤ real x" by simp
thus "2 ^ nat (bitlen x - 1) ≤ x" using `x > 0`
by (simp add: bitlen_def)
next
have "x ≤ 2 powr (log 2 x)" using `x > 0` by simp
also have "... < 2 ^ nat (⌊log 2 (real x)⌋ + 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 intro: mult_pos_pos)
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 ≡ 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_def Float_def)
next
case False
hence "f ≠ float_of 0"
unfolding real_of_float_eq by (simp add: f_def)
hence "mantissa f ≠ 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 ≠ 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 ≤ x"
then have "⌊log 2 (x div 2)⌋ + 1 = ⌊log 2 (x - x mod 2)⌋"
by (simp add: log_mult zmod_zdiv_equality')
also have "… = ⌊log 2 (real x)⌋"
proof cases
assume "x mod 2 = 0" then show ?thesis by simp
next
def n "⌊log 2 (real x)⌋"
then have "0 ≤ n"
using `2 ≤ x` by simp
assume "x mod 2 ≠ 0"
with `2 ≤ x` have "x mod 2 = 1" "¬ 2 dvd x" by (auto simp add: dvd_eq_mod_eq_0)
with `2 ≤ x` have "x ≠ 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 "… ≤ 2 powr (log 2 x)"
using `2 ≤ x` by (simp add: n_def del: powr_log_cancel)
finally have "2^nat n ≤ x" using `2 ≤ x` by simp }
ultimately have "2^nat n ≤ x - 1" by simp
then have "2^nat n ≤ real (x - 1)"
unfolding real_of_int_le_iff[symmetric] by simp
{ have "n = ⌊log 2 (2^nat n)⌋"
using `0 ≤ n` by (simp add: log_nat_power)
also have "… ≤ ⌊log 2 (x - 1)⌋"
using `2^nat n ≤ real (x - 1)` `0 ≤ n` `2 ≤ x` by (auto intro: floor_mono)
finally have "n ≤ ⌊log 2 (x - 1)⌋" . }
moreover have "⌊log 2 (x - 1)⌋ ≤ n"
using `2 ≤ x` by (auto simp add: n_def intro!: floor_mono)
ultimately show "⌊log 2 (x - x mod 2)⌋ = ⌊log 2 x⌋"
unfolding n_def `x mod 2 = 1` by auto
qed
finally have "⌊log 2 (x div 2)⌋ + 1 = ⌊log 2 x⌋" . }
moreover
{ assume "x < 2" "0 < x"
then have "x = 1" by simp
then have "⌊log 2 (real x)⌋ = 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 ≤ Float m e"
shows "0 ≤ 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 ≠ 0" by auto
show ?thesis
proof (cases "0 ≤ 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 ≤ real m * inverse ?S" using assms False powr_realpow[of 2 "nat (-e)"]
by (auto simp: powr_minus)
hence "1 * ?S ≤ real m * inverse ?S * ?S" by (rule mult_right_mono, auto)
hence "?S ≤ real m" unfolding mult_assoc by auto
hence "?S ≤ 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 ≤ 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 ≤ m" using bitlen_bounds[OF `0 <m`] ..
hence "1 * ?B ≤ real m" unfolding real_of_int_le_iff[symmetric] by auto
thus "1 ≤ real m / ?B" by auto

have "m ≠ 0" using assms by auto
have "0 ≤ 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 "… = 2^nat(bitlen m - 1 + 1)" using `0 < m` by (auto simp: bitlen_def)
also have "… = ?B * 2" unfolding nat_add_distrib[OF `0 ≤ 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 => nat => nat => float"
is "λprec (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 ≤ 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 => nat => nat => float"
is "λprec (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 ≡ rat_precision prec x y"
shows "rapprox_posrat prec x y = (let
l = l ;
X = if 0 ≤ 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 ∨ 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 ≠ 0"
show ?thesis
proof (cases "0 ≤ l")
assume "0 ≤ l"
def x' == "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 ≤ l` x'_def)
ultimately show ?thesis
unfolding normfloat_def
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]
by transfer
(simp add: floor_divide_eq_div[symmetric] dvd_eq_mod_eq_0 round_up_def)
next
assume "¬ 0 ≤ l"
def 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 int_mult int_power)
moreover have "real x * real (2::int) powr real l / real y = x / real y'"
using `¬ 0 ≤ 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] `¬ 0 ≤ l` `y' ≠ 0` `y ≠ 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 ≤ 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) ≤ 2 ^ nat x - 1"
proof -
def y "nat (x - 1)" moreover
have "(2::int) ^ y ≤ (2 ^ (y + 1)) - 1" by simp
ultimately show ?thesis using assms by simp
qed

lemma rapprox_posrat_less1:
assumes "0 ≤ 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 "… = 2 powr real (rat_precision n (int x) (int y) - 1)"
by (simp add: powr_add diff_def powr_neg_numeral)
also have "… = 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 "… ≤ 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 => int => int => float" is
"λprec (x::int) (y::int). round_down (rat_precision prec ¦x¦ ¦y¦) (x / y)" by simp

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 (auto simp: round_up_def round_down_def ceiling_def ac_simps)
hide_fact (open) compute_lapprox_rat

lift_definition rapprox_rat :: "nat => int => int => float" is
"λprec (x::int) (y::int). round_up (rat_precision prec ¦x¦ ¦y¦) (x / y)" by simp

lemma compute_rapprox_rat[code]:
"rapprox_rat prec x y =
(if y = 0 then 0
else if 0 ≤ 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 *}

lift_definition float_divl :: "nat => float => float => float" is
"λ(prec::nat) a b. round_down (prec + ⌊ log 2 ¦b¦ ⌋ - ⌊ log 2 ¦a¦ ⌋) (a / b)" by simp

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 ≠ 0 ∧ m2 ≠ 0"
then have eq2: "(int prec + ⌊log 2 ¦?f2¦⌋ - ⌊log 2 ¦?f1¦⌋) = rat_precision prec ¦m1¦ ¦m2¦ + (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, simp add: field_simps)
qed (transfer, auto)
hide_fact (open) compute_float_divl

lift_definition float_divr :: "nat => float => float => float" is
"λ(prec::nat) a b. round_up (prec + ⌊ log 2 ¦b¦ ⌋ - ⌊ log 2 ¦a¦ ⌋) (a / b)" by simp

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 ≠ 0 ∧ m2 ≠ 0"
then have eq2: "(int prec + ⌊log 2 ¦?f2¦⌋ - ⌊log 2 ¦?f1¦⌋) = rat_precision prec ¦m1¦ ¦m2¦ + (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, simp add: field_simps)
qed (transfer, auto)
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 ≠ 0 ==> a ≠ -1 ==> abs((a::int) div 2) < abs a"
by arith

lemma lapprox_rat:
shows "real (lapprox_rat prec x y) ≤ 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 ≥ 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 "... ≥ 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 == int n - ((bitlen ¦x¦) - (bitlen ¦y¦))"
assumes "0 ≤ x" "0 < y"
shows "0 ≤ 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 simp add: inverse_eq_divide intro!: mult_nonneg_nonneg divide_nonneg_pos mult_pos_pos)

lemma rapprox_rat: "real x / real y ≤ 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 ≤ x" "0 < y" "x ≤ y"
shows "real (rapprox_rat n x y) ≤ 1"
proof -
have "bitlen ¦x¦ ≤ bitlen ¦y¦"
using xy unfolding bitlen_def by (auto intro!: floor_mono)
then have "0 ≤ rat_precision n ¦x¦ ¦y¦" by (simp add: rat_precision_def)
have "real ⌈real x / real y * 2 powr real (rat_precision n ¦x¦ ¦y¦)⌉
≤ real ⌈2 powr real (rat_precision n ¦x¦ ¦y¦)⌉"

using xy by (auto intro!: ceiling_mono simp: field_simps)
also have "… = 2 powr real (rat_precision n ¦x¦ ¦y¦)"
using `0 ≤ rat_precision n ¦x¦ ¦y¦`
by (auto intro!: exI[of _ "2^nat (rat_precision n ¦x¦ ¦y¦)"] 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 ≤ x ==> y < 0 ==> real (rapprox_rat n x y) ≤ 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 ==> 0 < y ==> real (rapprox_rat n x y) ≤ 0"
unfolding rapprox_rat_def round_up_def
by (auto simp: field_simps mult_le_0_iff)

lemma rapprox_rat_nonpos_pos:
"x ≤ 0 ==> 0 < y ==> real (rapprox_rat n x y) ≤ 0"
unfolding rapprox_rat_def round_up_def
by (auto simp: field_simps mult_le_0_iff)

lemma float_divl: "real (float_divl prec x y) ≤ real x / real y"
by transfer (simp add: round_down)

lemma float_divl_lower_bound:
"0 ≤ x ==> 0 < y ==> 0 ≤ real (float_divl prec x y)"
by transfer (simp add: round_down_def zero_le_mult_iff zero_le_divide_iff)

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 <-> 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 ≤ 2 powr (bitlen ¦mantissa x¦ + exponent x)"
proof (cases "x = 0", simp)
assume "x ≠ 0" hence "mantissa x ≠ 0" using mantissa_eq_zero_iff by auto
have "x = mantissa x * 2 powr (exponent x)" by (rule mantissa_exponent)
also have "mantissa x ≤ ¦mantissa x¦" by simp
also have "... ≤ 2 powr (bitlen ¦mantissa x¦)"
using bitlen_bounds[of "¦mantissa x¦"] bitlen_nonneg `mantissa x ≠ 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 float_divl_pos_less1_bound:
"0 < real x ==> real x < 1 ==> prec ≥ 1 ==> 1 ≤ real (float_divl prec 1 x)"
proof transfer
fix prec :: nat and x :: real assume x: "0 < x" "x < 1" "x ∈ float" and prec: "1 ≤ prec"
def p "int prec + ⌊log 2 ¦x¦⌋"
show "1 ≤ round_down (int prec + ⌊log 2 ¦x¦⌋ - ⌊log 2 ¦1¦⌋) (1 / x) "
proof cases
assume nonneg: "0 ≤ 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) ≤ floor (1 / x)" using x prec by simp
also have "floor (real ((2::int) ^ nat p)) * floor (1 / x) ≤
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 ≤ 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: "¬ 0 ≤ p"

have "x = 2 powr (log 2 x)"
using x by simp
also have "2 powr (log 2 x) ≤ 2 powr p"
proof (rule powr_mono)
have "log 2 x ≤ ⌈log 2 x⌉"
by simp
also have "… ≤ ⌊log 2 x⌋ + 1"
using ceiling_diff_floor_le_1[of "log 2 x"] by simp
also have "… ≤ ⌊log 2 x⌋ + prec"
using prec by simp
finally show "log 2 x ≤ real p"
using x by (simp add: p_def)
qed simp
finally have x_le: "x ≤ 2 powr p" .

from neg have "2 powr real p ≤ 2 powr 0"
by (intro powr_mono) auto
also have "… ≤ ⌊2 powr 0⌋" by simp
also have "… ≤ ⌊2 powr real p / x⌋" unfolding real_of_int_le_iff
using x x_le by (intro floor_mono) (simp add: pos_le_divide_eq mult_pos_pos)
finally show ?thesis
using prec x unfolding p_def[symmetric]
by (simp add: round_down_def powr_minus_divide pos_le_divide_eq mult_pos_pos)
qed
qed

lemma float_divr: "real x / real y ≤ real (float_divr prec x y)"
using round_up by transfer simp

lemma float_divr_pos_less1_lower_bound: assumes "0 < x" and "x < 1" shows "1 ≤ float_divr prec 1 x"
proof -
have "1 ≤ 1 / real x" using `0 < x` and `x < 1` by auto
also have "… ≤ real (float_divr prec 1 x)" using float_divr[where x=1 and y=x] by auto
finally show ?thesis by auto
qed

lemma float_divr_nonpos_pos_upper_bound:
"real x ≤ 0 ==> 0 < real y ==> real (float_divr prec x y) ≤ 0"
by transfer (auto simp: field_simps mult_le_0_iff divide_le_0_iff round_up_def)

lemma float_divr_nonneg_neg_upper_bound:
"0 ≤ real x ==> real y < 0 ==> real (float_divr prec x y) ≤ 0"
by transfer (auto simp: field_simps mult_le_0_iff zero_le_mult_iff divide_le_0_iff round_up_def)

lift_definition float_round_up :: "nat => float => float" is
"λ(prec::nat) x. round_up (prec - ⌊log 2 ¦x¦⌋ - 1) x" by simp

lemma float_round_up: "real x ≤ real (float_round_up prec x)"
using round_up by transfer simp

lift_definition float_round_down :: "nat => float => float" is
"λ(prec::nat) x. round_down (prec - ⌊log 2 ¦x¦⌋ - 1) x" by simp

lemma float_round_down: "real (float_round_down prec x) ≤ real x"
using round_down by transfer simp

lemma floor_add2[simp]: "⌊ real i + x ⌋ = i + ⌊ x ⌋"
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 ¦m¦ - e" m e, symmetric]
by transfer (simp add: field_simps abs_mult log_mult bitlen_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 ¦m¦ - e" m e, symmetric]
unfolding Let_def
by transfer (simp add: field_simps abs_mult log_mult bitlen_def cong del: if_weak_cong)
hide_fact (open) compute_float_round_up

lemma Float_le_zero_iff: "Float a b ≤ 0 <-> a ≤ 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 => int" is floor by simp

lemma compute_int_floor_fl[code]:
"int_floor_fl (Float m e) = (if 0 ≤ 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 => float" is "λx. real (floor x)" by simp

lemma compute_floor_fl[code]:
"floor_fl (Float m e) = (if 0 ≤ 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) ≤ real x" by transfer simp

lemma int_floor_fl: "real (int_floor_fl x) ≤ real x" by transfer simp

lemma floor_pos_exp: "exponent (floor_fl x) ≥ 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 ⌊real x⌋ 0" by transfer simp
obtain i where "⌊real x⌋ = 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