# Theory HOL-Library.Interval_Float

```section ‹Approximate Operations on Intervals of Floating Point Numbers›
theory Interval_Float
imports
Interval
Float
begin

definition mid :: "float interval ⇒ float"
where "mid i = (lower i + upper i) * Float 1 (-1)"

lemma mid_in_interval: "mid i ∈⇩i i"
using lower_le_upper[of i]
by (auto simp: mid_def set_of_eq powr_minus)

lemma mid_le: "lower i ≤ mid i" "mid i ≤ upper i"
using mid_in_interval
by (auto simp: set_of_eq)

definition centered :: "float interval ⇒ float interval"
where "centered i = i - interval_of (mid i)"

definition "split_float_interval x = split_interval x ((lower x + upper x) * Float 1 (-1))"

lemma split_float_intervalD: "split_float_interval X = (A, B) ⟹ set_of X ⊆ set_of A ∪ set_of B"
by (auto dest!: split_intervalD simp: split_float_interval_def)

lemma split_float_interval_bounds:
shows
lower_split_float_interval1: "lower (fst (split_float_interval X)) = lower X"
and lower_split_float_interval2: "lower (snd (split_float_interval X)) = mid X"
and upper_split_float_interval1: "upper (fst (split_float_interval X)) = mid X"
and upper_split_float_interval2: "upper (snd (split_float_interval X)) = upper X"
using mid_le[of X]
by (auto simp: split_float_interval_def mid_def[symmetric] min_def max_def real_of_float_eq
lower_split_interval1 lower_split_interval2
upper_split_interval1 upper_split_interval2)

lemmas float_round_down_le[intro] = order_trans[OF float_round_down]
and float_round_up_ge[intro] = order_trans[OF _ float_round_up]

text ‹TODO: many of the lemmas should move to theories Float or Approximation
(the latter should be based on type @{type interval}.›

subsection "Intervals with Floating Point Bounds"

context includes interval.lifting begin

lift_definition round_interval :: "nat ⇒ float interval ⇒ float interval"
is "λp. λ(l, u). (float_round_down p l, float_round_up p u)"
by (auto simp: intro!: float_round_down_le float_round_up_le)

lemma lower_round_ivl[simp]: "lower (round_interval p x) = float_round_down p (lower x)"
by transfer auto
lemma upper_round_ivl[simp]: "upper (round_interval p x) = float_round_up p (upper x)"
by transfer auto

lemma round_ivl_correct: "set_of A ⊆ set_of (round_interval prec A)"
by (auto simp: set_of_eq float_round_down_le float_round_up_le)

lift_definition truncate_ivl :: "nat ⇒ real interval ⇒ real interval"
is "λp. λ(l, u). (truncate_down p l, truncate_up p u)"
by (auto intro!: truncate_down_le truncate_up_le)

lemma lower_truncate_ivl[simp]: "lower (truncate_ivl p x) = truncate_down p (lower x)"
by transfer auto
lemma upper_truncate_ivl[simp]: "upper (truncate_ivl p x) = truncate_up p (upper x)"
by transfer auto

lemma truncate_ivl_correct: "set_of A ⊆ set_of (truncate_ivl prec A)"
by (auto simp: set_of_eq intro!: truncate_down_le truncate_up_le)

lift_definition real_interval::"float interval ⇒ real interval"
is "λ(l, u). (real_of_float l, real_of_float u)"
by auto

lemma lower_real_interval[simp]: "lower (real_interval x) = lower x"
by transfer auto
lemma upper_real_interval[simp]: "upper (real_interval x) = upper x"
by transfer auto

definition "set_of' x = (case x of None ⇒ UNIV | Some i ⇒ set_of (real_interval i))"

lemma real_interval_min_interval[simp]:
"real_interval (min_interval a b) = min_interval (real_interval a) (real_interval b)"
by (auto simp: interval_eq_set_of_iff set_of_eq real_of_float_min)

lemma real_interval_max_interval[simp]:
"real_interval (max_interval a b) = max_interval (real_interval a) (real_interval b)"
by (auto simp: interval_eq_set_of_iff set_of_eq real_of_float_max)

lemma in_intervalI:
"x ∈⇩i X" if "lower X ≤ x" "x ≤ upper X"
using that by (auto simp: set_of_eq)

abbreviation in_real_interval ("(_/ ∈⇩r _)" [51, 51] 50) where
"x ∈⇩r X ≡ x ∈⇩i real_interval X"

lemma in_real_intervalI:
"x ∈⇩r X" if "lower X ≤ x" "x ≤ upper X" for x::real and X::"float interval"
using that
by (intro in_intervalI) auto

subsection ‹intros for ‹real_interval››

lemma in_round_intervalI: "x ∈⇩r A  ⟹ x ∈⇩r (round_interval prec A)"
by (auto simp: set_of_eq float_round_down_le float_round_up_le)

lemma zero_in_float_intervalI: "0 ∈⇩r 0"
by (auto simp: set_of_eq)

lemma plus_in_float_intervalI: "a + b ∈⇩r A + B" if "a ∈⇩r A" "b ∈⇩r B"
using that
by (auto simp: set_of_eq)

lemma minus_in_float_intervalI: "a - b ∈⇩r A - B" if "a ∈⇩r A" "b ∈⇩r B"
using that
by (auto simp: set_of_eq)

lemma uminus_in_float_intervalI: "-a ∈⇩r -A" if "a ∈⇩r A"
using that
by (auto simp: set_of_eq)

lemma real_interval_times: "real_interval (A * B) = real_interval A * real_interval B"
by (auto simp: interval_eq_iff lower_times upper_times min_def max_def)

lemma times_in_float_intervalI: "a * b ∈⇩r A * B" if "a ∈⇩r A" "b ∈⇩r B"
using times_in_intervalI[OF that]
by (auto simp: real_interval_times)

lemma real_interval_abs: "real_interval (abs_interval A) = abs_interval (real_interval A)"
by (auto simp: interval_eq_iff min_def max_def)

lemma abs_in_float_intervalI: "abs a ∈⇩r abs_interval A" if "a ∈⇩r A"
by (auto simp: set_of_abs_interval real_interval_abs intro!: imageI that)

lemma interval_of[intro,simp]: "x ∈⇩r interval_of x"
by (auto simp: set_of_eq)

lemma split_float_interval_realD: "split_float_interval X = (A, B) ⟹ x ∈⇩r X ⟹ x ∈⇩r A ∨ x ∈⇩r B"
by (auto simp: set_of_eq prod_eq_iff split_float_interval_bounds)

subsection ‹bounds for lists›

lemma lower_Interval: "lower (Interval x) = fst x"
and upper_Interval: "upper (Interval x) = snd x"
if "fst x ≤ snd x"
using that
by (auto simp: lower_def upper_def Interval_inverse split_beta')

definition all_in_i :: "'a::preorder list ⇒ 'a interval list ⇒ bool"
(infix "(all'_in⇩i)" 50)
where "x all_in⇩i I = (length x = length I ∧ (∀i < length I. x ! i ∈⇩i I ! i))"

definition all_in :: "real list ⇒ float interval list ⇒ bool"
(infix "(all'_in)" 50)
where "x all_in I = (length x = length I ∧ (∀i < length I. x ! i ∈⇩r I ! i))"

definition all_subset :: "'a::order interval list ⇒ 'a interval list ⇒ bool"
(infix "(all'_subset)" 50)
where "I all_subset J = (length I = length J ∧ (∀i < length I. set_of (I!i) ⊆ set_of (J!i)))"

lemmas [simp] = all_in_def all_subset_def

lemma all_subsetD:
assumes "I all_subset J"
assumes "x all_in I"
shows "x all_in J"
using assms
by (auto simp: set_of_eq; fastforce)

lemma round_interval_mono: "set_of (round_interval prec X) ⊆ set_of (round_interval prec Y)"
if "set_of X ⊆ set_of Y"
using that
by transfer
(auto simp: float_round_down.rep_eq float_round_up.rep_eq truncate_down_mono truncate_up_mono)

lemma Ivl_simps[simp]: "lower (Ivl a b) = min a b" "upper (Ivl a b) = b"
subgoal by transfer simp
subgoal by transfer simp
done

lemma set_of_subset_iff: "set_of X ⊆ set_of Y ⟷ lower Y ≤ lower X ∧ upper X ≤ upper Y"
for X Y::"'a::linorder interval"
by (auto simp: set_of_eq subset_iff)

lemma set_of_subset_iff':
"set_of a ⊆ set_of (b :: 'a :: linorder interval) ⟷ a ≤ b"
unfolding less_eq_interval_def set_of_subset_iff ..

lemma bounds_of_interval_eq_lower_upper:
"bounds_of_interval ivl = (lower ivl, upper ivl)" if "lower ivl ≤ upper ivl"
using that
by (auto simp: lower.rep_eq upper.rep_eq)

lemma real_interval_Ivl: "real_interval (Ivl a b) = Ivl a b"
by transfer (auto simp: min_def)

lemma set_of_mul_contains_real_zero:
"0 ∈⇩r (A * B)" if "0 ∈⇩r A ∨ 0 ∈⇩r B"
using that set_of_mul_contains_zero[of A B]
by (auto simp: set_of_eq)

fun subdivide_interval :: "nat ⇒ float interval ⇒ float interval list"
where "subdivide_interval 0 I = [I]"
| "subdivide_interval (Suc n) I = (
let m = mid I
in (subdivide_interval n (Ivl (lower I) m)) @ (subdivide_interval n (Ivl m (upper I)))
)"

lemma subdivide_interval_length:
shows "length (subdivide_interval n I) = 2^n"
by(induction n arbitrary: I, simp_all add: Let_def)

lemma lower_le_mid: "lower x ≤ mid x" "real_of_float (lower x) ≤ mid x"
and mid_le_upper: "mid x ≤ upper x" "real_of_float (mid x) ≤ upper x"
unfolding mid_def
subgoal by transfer (auto simp: powr_neg_one)
subgoal by transfer (auto simp: powr_neg_one)
subgoal by transfer (auto simp: powr_neg_one)
subgoal by transfer (auto simp: powr_neg_one)
done

lemma subdivide_interval_correct:
"list_ex (λi. x ∈⇩r i) (subdivide_interval n I)" if "x ∈⇩r I" for x::real
using that
proof(induction n arbitrary: x I)
case 0
then show ?case by simp
next
case (Suc n)
from ‹x ∈⇩r I› consider "x ∈⇩r Ivl (lower I) (mid I)" | "x ∈⇩r Ivl (mid I) (upper I)"
by (cases "x ≤ real_of_float (mid I)")
(auto simp: set_of_eq min_def lower_le_mid mid_le_upper)
from this[case_names lower upper] show ?case
by cases (use Suc.IH in ‹auto simp: Let_def›)
qed

fun interval_list_union :: "'a::lattice interval list ⇒ 'a interval"
where "interval_list_union [] = undefined"
| "interval_list_union [I] = I"
| "interval_list_union (I#Is) = sup I (interval_list_union Is)"

lemma interval_list_union_correct:
assumes "S ≠ []"
assumes "i < length S"
shows "set_of (S!i) ⊆ set_of (interval_list_union S)"
using assms
proof(induction S arbitrary: i)
case (Cons a S i)
thus ?case
proof(cases S)
fix b S'
assume "S = b # S'"
hence "S ≠ []"
by simp
show ?thesis
proof(cases i)
case 0
show ?thesis
apply(cases S)
using interval_union_mono1
next
case (Suc i_prev)
hence "i_prev < length S"
using Cons(3) by simp

from Cons(1)[OF ‹S ≠ []› this] Cons(1)
have "set_of ((a # S) ! i) ⊆ set_of (interval_list_union S)"
by (simp add: ‹i = Suc i_prev›)
also have "... ⊆ set_of (interval_list_union (a # S))"
using ‹S ≠ []›
apply(cases S)
using interval_union_mono2
by auto
finally show ?thesis .
qed
qed simp
qed simp

lemma split_domain_correct:
fixes x :: "real list"
assumes "x all_in I"
assumes split_correct: "⋀x a I. x ∈⇩r I ⟹ list_ex (λi::float interval. x ∈⇩r i) (split I)"
shows "list_ex (λs. x all_in s) (split_domain split I)"
using assms(1)
proof(induction I arbitrary: x)
case (Cons I Is x)
have "x ≠ []"
using Cons(2) by auto
obtain x' xs where x_decomp: "x = x' # xs"
using ‹x ≠ []› list.exhaust by auto
hence "x' ∈⇩r I" "xs all_in Is"
using Cons(2)
by auto
show ?case
using Cons(1)[OF ‹xs all_in Is›]
split_correct[OF ‹x' ∈⇩r I›]
apply (auto simp add: list_ex_iff set_of_eq)
by (smt (verit, ccfv_SIG) One_nat_def Suc_pred ‹x ≠ []› le_simps(3) length_greater_0_conv length_tl linorder_not_less list.sel(3) neq0_conv nth_Cons' x_decomp)
qed simp

lift_definition(code_dt) inverse_float_interval::"nat ⇒ float interval ⇒ float interval option" is
"λprec (l, u). if (0 < l ∨ u < 0) then Some (float_divl prec 1 u, float_divr prec 1 l) else None"
by (auto intro!: order_trans[OF float_divl] order_trans[OF _ float_divr]
simp: divide_simps)

lemma inverse_float_interval_eq_Some_conv:
defines "one ≡ (1::float)"
shows
"inverse_float_interval p X = Some R ⟷
(lower X > 0 ∨ upper X < 0) ∧
lower R = float_divl p one (upper X) ∧
upper R = float_divr p one (lower X)"
by clarsimp (transfer fixing: one, force simp: one_def split: if_splits)

lemma inverse_float_interval:
"inverse ` set_of (real_interval X) ⊆ set_of (real_interval Y)"
if "inverse_float_interval p X = Some Y"
using that
apply (clarsimp simp: set_of_eq inverse_float_interval_eq_Some_conv)
by (intro order_trans[OF float_divl] order_trans[OF _ float_divr] conjI)
(auto simp: divide_simps)

lemma inverse_float_intervalI:
"x ∈⇩r X ⟹ inverse x ∈ set_of' (inverse_float_interval p X)"
using inverse_float_interval[of p X]
by (auto simp: set_of'_def split: option.splits)

lemma inverse_float_interval_eqI: "inverse_float_interval p X = Some IVL ⟹ x ∈⇩r X ⟹ inverse x ∈⇩r IVL"
using inverse_float_intervalI[of x X p]
by (auto simp: set_of'_def)

lemma real_interval_abs_interval[simp]:
"real_interval (abs_interval x) = abs_interval (real_interval x)"
by (auto simp: interval_eq_set_of_iff set_of_eq real_of_float_max real_of_float_min)

lift_definition floor_float_interval::"float interval ⇒ float interval" is
"λ(l, u). (floor_fl l, floor_fl u)"
by (auto intro!: floor_mono simp: floor_fl.rep_eq)

lemma lower_floor_float_interval[simp]: "lower (floor_float_interval x) = floor_fl (lower x)"
by transfer auto
lemma upper_floor_float_interval[simp]: "upper (floor_float_interval x) = floor_fl (upper x)"
by transfer auto

lemma floor_float_intervalI: "⌊x⌋ ∈⇩r floor_float_interval X" if "x ∈⇩r X"
using that by (auto simp: set_of_eq floor_fl_def floor_mono)

end

subsection ‹constants for code generation›

definition lowerF::"float interval ⇒ float" where "lowerF = lower"
definition upperF::"float interval ⇒ float" where "upperF = upper"

end```