(* Title: Interval
Author: Christoph Traut, TU Muenchen
Fabian Immler, TU Muenchen
*)
section \<open>Interval Type\<close>
theory Interval
imports
Complex_Main
Lattice_Algebras
Set_Algebras
begin
text \<open>A type of non-empty, closed intervals.\<close>
typedef (overloaded) 'a interval =
"{(a::'a::preorder, b). a \<le> b}"
morphisms bounds_of_interval Interval
by auto
setup_lifting type_definition_interval
lift_definition lower::"('a::preorder) interval \<Rightarrow> 'a" is fst .
lift_definition upper::"('a::preorder) interval \<Rightarrow> 'a" is snd .
lemma interval_eq_iff: "a = b \<longleftrightarrow> lower a = lower b \<and> upper a = upper b"
by transfer auto
lemma interval_eqI: "lower a = lower b \<Longrightarrow> upper a = upper b \<Longrightarrow> a = b"
by (auto simp: interval_eq_iff)
lemma lower_le_upper[simp]: "lower i \<le> upper i"
by transfer auto
lift_definition set_of :: "'a::preorder interval \<Rightarrow> 'a set" is "\<lambda>x. {fst x .. snd x}" .
lemma set_of_eq: "set_of x = {lower x .. upper x}"
by transfer simp
context notes [[typedef_overloaded]] begin
lift_definition(code_dt) Interval'::"'a::preorder \<Rightarrow> 'a::preorder \<Rightarrow> 'a interval option"
is "\<lambda>a b. if a \<le> b then Some (a, b) else None"
by auto
lemma Interval'_split:
"P (Interval' a b) \<longleftrightarrow>
(\<forall>ivl. a \<le> b \<longrightarrow> lower ivl = a \<longrightarrow> upper ivl = b \<longrightarrow> P (Some ivl)) \<and> (\<not>a\<le>b \<longrightarrow> P None)"
by transfer auto
lemma Interval'_split_asm:
"P (Interval' a b) \<longleftrightarrow>
\<not>((\<exists>ivl. a \<le> b \<and> lower ivl = a \<and> upper ivl = b \<and> \<not>P (Some ivl)) \<or> (\<not>a\<le>b \<and> \<not>P None))"
unfolding Interval'_split
by auto
lemmas Interval'_splits = Interval'_split Interval'_split_asm
lemma Interval'_eq_Some: "Interval' a b = Some i \<Longrightarrow> lower i = a \<and> upper i = b"
by (simp split: Interval'_splits)
end
instantiation "interval" :: ("{preorder,equal}") equal
begin
definition "equal_class.equal a b \<equiv> (lower a = lower b) \<and> (upper a = upper b)"
instance proof qed (simp add: equal_interval_def interval_eq_iff)
end
instantiation interval :: ("preorder") ord begin
definition less_eq_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> bool"
where "less_eq_interval a b \<longleftrightarrow> lower b \<le> lower a \<and> upper a \<le> upper b"
definition less_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> bool"
where "less_interval x y = (x \<le> y \<and> \<not> y \<le> x)"
instance proof qed
end
instantiation interval :: ("lattice") semilattice_sup
begin
lift_definition sup_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
is "\<lambda>(a, b) (c, d). (inf a c, sup b d)"
by (auto simp: le_infI1 le_supI1)
lemma lower_sup[simp]: "lower (sup A B) = inf (lower A) (lower B)"
by transfer auto
lemma upper_sup[simp]: "upper (sup A B) = sup (upper A) (upper B)"
by transfer auto
instance proof qed (auto simp: less_eq_interval_def less_interval_def interval_eq_iff)
end
lemma set_of_interval_union: "set_of A \<union> set_of B \<subseteq> set_of (sup A B)" for A::"'a::lattice interval"
by (auto simp: set_of_eq)
lemma interval_union_commute: "sup A B = sup B A" for A::"'a::lattice interval"
by (auto simp add: interval_eq_iff inf.commute sup.commute)
lemma interval_union_mono1: "set_of a \<subseteq> set_of (sup a A)" for A :: "'a::lattice interval"
using set_of_interval_union by blast
lemma interval_union_mono2: "set_of A \<subseteq> set_of (sup a A)" for A :: "'a::lattice interval"
using set_of_interval_union by blast
lift_definition interval_of :: "'a::preorder \<Rightarrow> 'a interval" is "\<lambda>x. (x, x)"
by auto
lemma lower_interval_of[simp]: "lower (interval_of a) = a"
by transfer auto
lemma upper_interval_of[simp]: "upper (interval_of a) = a"
by transfer auto
definition width :: "'a::{preorder,minus} interval \<Rightarrow> 'a"
where "width i = upper i - lower i"
instantiation "interval" :: ("ordered_ab_semigroup_add") ab_semigroup_add
begin
lift_definition plus_interval::"'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
is "\<lambda>(a, b). \<lambda>(c, d). (a + c, b + d)"
by (auto intro!: add_mono)
lemma lower_plus[simp]: "lower (plus A B) = plus (lower A) (lower B)"
by transfer auto
lemma upper_plus[simp]: "upper (plus A B) = plus (upper A) (upper B)"
by transfer auto
instance proof qed (auto simp: interval_eq_iff less_eq_interval_def ac_simps)
end
instance "interval" :: ("{ordered_ab_semigroup_add, lattice}") ordered_ab_semigroup_add
proof qed (auto simp: less_eq_interval_def intro!: add_mono)
instantiation "interval" :: ("{preorder,zero}") zero
begin
lift_definition zero_interval::"'a interval" is "(0, 0)" by auto
lemma lower_zero[simp]: "lower 0 = 0"
by transfer auto
lemma upper_zero[simp]: "upper 0 = 0"
by transfer auto
instance proof qed
end
instance "interval" :: ("{ordered_comm_monoid_add}") comm_monoid_add
proof qed (auto simp: interval_eq_iff)
instance "interval" :: ("{ordered_comm_monoid_add,lattice}") ordered_comm_monoid_add ..
instantiation "interval" :: ("{ordered_ab_group_add}") uminus
begin
lift_definition uminus_interval::"'a interval \<Rightarrow> 'a interval" is "\<lambda>(a, b). (-b, -a)" by auto
lemma lower_uminus[simp]: "lower (- A) = - upper A"
by transfer auto
lemma upper_uminus[simp]: "upper (- A) = - lower A"
by transfer auto
instance ..
end
instantiation "interval" :: ("{ordered_ab_group_add}") minus
begin
definition minus_interval::"'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
where "minus_interval a b = a + - b"
lemma lower_minus[simp]: "lower (minus A B) = minus (lower A) (upper B)"
by (auto simp: minus_interval_def)
lemma upper_minus[simp]: "upper (minus A B) = minus (upper A) (lower B)"
by (auto simp: minus_interval_def)
instance ..
end
instantiation "interval" :: (linordered_semiring) times
begin
lift_definition times_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
is "\<lambda>(a1, a2). \<lambda>(b1, b2).
(let x1 = a1 * b1; x2 = a1 * b2; x3 = a2 * b1; x4 = a2 * b2
in (min x1 (min x2 (min x3 x4)), max x1 (max x2 (max x3 x4))))"
by (auto simp: Let_def intro!: min.coboundedI1 max.coboundedI1)
lemma lower_times:
"lower (times A B) = Min {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
by transfer (auto simp: Let_def)
lemma upper_times:
"upper (times A B) = Max {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
by transfer (auto simp: Let_def)
instance ..
end
lemma interval_eq_set_of_iff: "X = Y \<longleftrightarrow> set_of X = set_of Y" for X Y::"'a::order interval"
by (auto simp: set_of_eq interval_eq_iff)
subsection \<open>Membership\<close>
abbreviation (in preorder) in_interval ("(_/ \<in>\<^sub>i _)" [51, 51] 50)
where "in_interval x X \<equiv> x \<in> set_of X"
lemma in_interval_to_interval[intro!]: "a \<in>\<^sub>i interval_of a"
by (auto simp: set_of_eq)
lemma plus_in_intervalI:
fixes x y :: "'a :: ordered_ab_semigroup_add"
shows "x \<in>\<^sub>i X \<Longrightarrow> y \<in>\<^sub>i Y \<Longrightarrow> x + y \<in>\<^sub>i X + Y"
by (simp add: add_mono_thms_linordered_semiring(1) set_of_eq)
lemma connected_set_of[intro, simp]:
"connected (set_of X)" for X::"'a::linear_continuum_topology interval"
by (auto simp: set_of_eq )
lemma ex_sum_in_interval_lemma: "\<exists>xa\<in>{la .. ua}. \<exists>xb\<in>{lb .. ub}. x = xa + xb"
if "la \<le> ua" "lb \<le> ub" "la + lb \<le> x" "x \<le> ua + ub"
"ua - la \<le> ub - lb"
for la b c d::"'a::linordered_ab_group_add"
proof -
define wa where "wa = ua - la"
define wb where "wb = ub - lb"
define w where "w = wa + wb"
define d where "d = x - la - lb"
define da where "da = max 0 (min wa (d - wa))"
define db where "db = d - da"
from that have nonneg: "0 \<le> wa" "0 \<le> wb" "0 \<le> w" "0 \<le> d" "d \<le> w"
by (auto simp add: wa_def wb_def w_def d_def add.commute le_diff_eq)
have "0 \<le> db"
by (auto simp: da_def nonneg db_def intro!: min.coboundedI2)
have "x = (la + da) + (lb + db)"
by (simp add: da_def db_def d_def)
moreover
have "x - la - ub \<le> da"
using that
unfolding da_def
by (intro max.coboundedI2) (auto simp: wa_def d_def diff_le_eq diff_add_eq)
then have "db \<le> wb"
by (auto simp: db_def d_def wb_def algebra_simps)
with \<open>0 \<le> db\<close> that nonneg have "lb + db \<in> {lb..ub}"
by (auto simp: wb_def algebra_simps)
moreover
have "da \<le> wa"
by (auto simp: da_def nonneg)
then have "la + da \<in> {la..ua}"
by (auto simp: da_def wa_def algebra_simps)
ultimately show ?thesis
by force
qed
lemma ex_sum_in_interval: "\<exists>xa\<ge>la. xa \<le> ua \<and> (\<exists>xb\<ge>lb. xb \<le> ub \<and> x = xa + xb)"
if a: "la \<le> ua" and b: "lb \<le> ub" and x: "la + lb \<le> x" "x \<le> ua + ub"
for la b c d::"'a::linordered_ab_group_add"
proof -
from linear consider "ua - la \<le> ub - lb" | "ub - lb \<le> ua - la"
by blast
then show ?thesis
proof cases
case 1
from ex_sum_in_interval_lemma[OF that 1]
show ?thesis by auto
next
case 2
from x have "lb + la \<le> x" "x \<le> ub + ua" by (simp_all add: ac_simps)
from ex_sum_in_interval_lemma[OF b a this 2]
show ?thesis by auto
qed
qed
lemma Icc_plus_Icc:
"{a .. b} + {c .. d} = {a + c .. b + d}"
if "a \<le> b" "c \<le> d"
for a b c d::"'a::linordered_ab_group_add"
using ex_sum_in_interval[OF that]
by (auto intro: add_mono simp: atLeastAtMost_iff Bex_def set_plus_def)
lemma set_of_plus:
fixes A :: "'a::linordered_ab_group_add interval"
shows "set_of (A + B) = set_of A + set_of B"
using Icc_plus_Icc[of "lower A" "upper A" "lower B" "upper B"]
by (auto simp: set_of_eq)
lemma plus_in_intervalE:
fixes xy :: "'a :: linordered_ab_group_add"
assumes "xy \<in>\<^sub>i X + Y"
obtains x y where "xy = x + y" "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y"
using assms
unfolding set_of_plus set_plus_def
by auto
lemma set_of_uminus: "set_of (-X) = {- x | x. x \<in> set_of X}"
for X :: "'a :: ordered_ab_group_add interval"
by (auto simp: set_of_eq simp: le_minus_iff minus_le_iff
intro!: exI[where x="-x" for x])
lemma uminus_in_intervalI:
fixes x :: "'a :: ordered_ab_group_add"
shows "x \<in>\<^sub>i X \<Longrightarrow> -x \<in>\<^sub>i -X"
by (auto simp: set_of_uminus)
lemma uminus_in_intervalD:
fixes x :: "'a :: ordered_ab_group_add"
shows "x \<in>\<^sub>i - X \<Longrightarrow> - x \<in>\<^sub>i X"
by (auto simp: set_of_uminus)
lemma minus_in_intervalI:
fixes x y :: "'a :: ordered_ab_group_add"
shows "x \<in>\<^sub>i X \<Longrightarrow> y \<in>\<^sub>i Y \<Longrightarrow> x - y \<in>\<^sub>i X - Y"
by (metis diff_conv_add_uminus minus_interval_def plus_in_intervalI uminus_in_intervalI)
lemma set_of_minus: "set_of (X - Y) = {x - y | x y . x \<in> set_of X \<and> y \<in> set_of Y}"
for X Y :: "'a :: linordered_ab_group_add interval"
unfolding minus_interval_def set_of_plus set_of_uminus set_plus_def
by force
lemma times_in_intervalI:
fixes x y::"'a::linordered_ring"
assumes "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y"
shows "x * y \<in>\<^sub>i X * Y"
proof -
define X1 where "X1 \<equiv> lower X"
define X2 where "X2 \<equiv> upper X"
define Y1 where "Y1 \<equiv> lower Y"
define Y2 where "Y2 \<equiv> upper Y"
from assms have assms: "X1 \<le> x" "x \<le> X2" "Y1 \<le> y" "y \<le> Y2"
by (auto simp: X1_def X2_def Y1_def Y2_def set_of_eq)
have "(X1 * Y1 \<le> x * y \<or> X1 * Y2 \<le> x * y \<or> X2 * Y1 \<le> x * y \<or> X2 * Y2 \<le> x * y) \<and>
(X1 * Y1 \<ge> x * y \<or> X1 * Y2 \<ge> x * y \<or> X2 * Y1 \<ge> x * y \<or> X2 * Y2 \<ge> x * y)"
proof (cases x "0::'a" rule: linorder_cases)
case x0: less
show ?thesis
proof (cases "y < 0")
case y0: True
from y0 x0 assms have "x * y \<le> X1 * y" by (intro mult_right_mono_neg, auto)
also from x0 y0 assms have "X1 * y \<le> X1 * Y1" by (intro mult_left_mono_neg, auto)
finally have 1: "x * y \<le> X1 * Y1".
show ?thesis proof(cases "X2 \<le> 0")
case True
with assms have "X2 * Y2 \<le> X2 * y" by (auto intro: mult_left_mono_neg)
also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono_neg)
finally have "X2 * Y2 \<le> x * y".
with 1 show ?thesis by auto
next
case False
with assms have "X2 * Y1 \<le> X2 * y" by (auto intro: mult_left_mono)
also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono_neg)
finally have "X2 * Y1 \<le> x * y".
with 1 show ?thesis by auto
qed
next
case False
then have y0: "y \<ge> 0" by auto
from x0 y0 assms have "X1 * Y2 \<le> x * Y2" by (intro mult_right_mono, auto)
also from y0 x0 assms have "... \<le> x * y" by (intro mult_left_mono_neg, auto)
finally have 1: "X1 * Y2 \<le> x * y".
show ?thesis
proof(cases "X2 \<le> 0")
case X2: True
from assms y0 have "x * y \<le> X2 * y" by (intro mult_right_mono)
also from assms X2 have "... \<le> X2 * Y1" by (auto intro: mult_left_mono_neg)
finally have "x * y \<le> X2 * Y1".
with 1 show ?thesis by auto
next
case X2: False
from assms y0 have "x * y \<le> X2 * y" by (intro mult_right_mono)
also from assms X2 have "... \<le> X2 * Y2" by (auto intro: mult_left_mono)
finally have "x * y \<le> X2 * Y2".
with 1 show ?thesis by auto
qed
qed
next
case [simp]: equal
with assms show ?thesis by (cases "Y2 \<le> 0", auto intro:mult_sign_intros)
next
case x0: greater
show ?thesis
proof (cases "y < 0")
case y0: True
from x0 y0 assms have "X2 * Y1 \<le> X2 * y" by (intro mult_left_mono, auto)
also from y0 x0 assms have "X2 * y \<le> x * y" by (intro mult_right_mono_neg, auto)
finally have 1: "X2 * Y1 \<le> x * y".
show ?thesis
proof(cases "Y2 \<le> 0")
case Y2: True
from x0 assms have "x * y \<le> x * Y2" by (auto intro: mult_left_mono)
also from assms Y2 have "... \<le> X1 * Y2" by (auto intro: mult_right_mono_neg)
finally have "x * y \<le> X1 * Y2".
with 1 show ?thesis by auto
next
case Y2: False
from x0 assms have "x * y \<le> x * Y2" by (auto intro: mult_left_mono)
also from assms Y2 have "... \<le> X2 * Y2" by (auto intro: mult_right_mono)
finally have "x * y \<le> X2 * Y2".
with 1 show ?thesis by auto
qed
next
case y0: False
from x0 y0 assms have "x * y \<le> X2 * y" by (intro mult_right_mono, auto)
also from y0 x0 assms have "... \<le> X2 * Y2" by (intro mult_left_mono, auto)
finally have 1: "x * y \<le> X2 * Y2".
show ?thesis
proof(cases "X1 \<le> 0")
case True
with assms have "X1 * Y2 \<le> X1 * y" by (auto intro: mult_left_mono_neg)
also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono)
finally have "X1 * Y2 \<le> x * y".
with 1 show ?thesis by auto
next
case False
with assms have "X1 * Y1 \<le> X1 * y" by (auto intro: mult_left_mono)
also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono)
finally have "X1 * Y1 \<le> x * y".
with 1 show ?thesis by auto
qed
qed
qed
hence min:"min (X1 * Y1) (min (X1 * Y2) (min (X2 * Y1) (X2 * Y2))) \<le> x * y"
and max:"x * y \<le> max (X1 * Y1) (max (X1 * Y2) (max (X2 * Y1) (X2 * Y2)))"
by (auto simp:min_le_iff_disj le_max_iff_disj)
show ?thesis using min max
by (auto simp: Let_def X1_def X2_def Y1_def Y2_def set_of_eq lower_times upper_times)
qed
lemma times_in_intervalE:
fixes xy :: "'a :: {linordered_semiring, real_normed_algebra, linear_continuum_topology}"
\<comment> \<open>TODO: linear continuum topology is pretty strong\<close>
assumes "xy \<in>\<^sub>i X * Y"
obtains x y where "xy = x * y" "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y"
proof -
let ?mult = "\<lambda>(x, y). x * y"
let ?XY = "set_of X \<times> set_of Y"
have cont: "continuous_on ?XY ?mult"
by (auto intro!: tendsto_eq_intros simp: continuous_on_def split_beta')
have conn: "connected (?mult ` ?XY)"
by (rule connected_continuous_image[OF cont]) auto
have "lower (X * Y) \<in> ?mult ` ?XY" "upper (X * Y) \<in> ?mult ` ?XY"
by (auto simp: set_of_eq lower_times upper_times min_def max_def split: if_splits)
from connectedD_interval[OF conn this, of xy] assms
obtain x y where "xy = x * y" "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y" by (auto simp: set_of_eq)
then show ?thesis ..
qed
lemma set_of_times: "set_of (X * Y) = {x * y | x y. x \<in> set_of X \<and> y \<in> set_of Y}"
for X Y::"'a :: {linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
by (auto intro!: times_in_intervalI elim!: times_in_intervalE)
instance "interval" :: (linordered_idom) cancel_semigroup_add
proof qed (auto simp: interval_eq_iff)
lemma interval_mul_commute: "A * B = B * A" for A B:: "'a::linordered_idom interval"
by (simp add: interval_eq_iff lower_times upper_times ac_simps)
lemma interval_times_zero_right[simp]: "A * 0 = 0" for A :: "'a::linordered_ring interval"
by (simp add: interval_eq_iff lower_times upper_times ac_simps)
lemma interval_times_zero_left[simp]:
"0 * A = 0" for A :: "'a::linordered_ring interval"
by (simp add: interval_eq_iff lower_times upper_times ac_simps)
instantiation "interval" :: ("{preorder,one}") one
begin
lift_definition one_interval::"'a interval" is "(1, 1)" by auto
lemma lower_one[simp]: "lower 1 = 1"
by transfer auto
lemma upper_one[simp]: "upper 1 = 1"
by transfer auto
instance proof qed
end
instance interval :: ("{one, preorder, linordered_semiring}") power
proof qed
lemma set_of_one[simp]: "set_of (1::'a::{one, order} interval) = {1}"
by (auto simp: set_of_eq)
instance "interval" ::
("{linordered_idom,linordered_ring, real_normed_algebra, linear_continuum_topology}") monoid_mult
apply standard
unfolding interval_eq_set_of_iff set_of_times
subgoal
by (auto simp: interval_eq_set_of_iff set_of_times; metis mult.assoc)
by auto
lemma one_times_ivl_left[simp]: "1 * A = A" for A :: "'a::linordered_idom interval"
by (simp add: interval_eq_iff lower_times upper_times ac_simps min_def max_def)
lemma one_times_ivl_right[simp]: "A * 1 = A" for A :: "'a::linordered_idom interval"
by (metis interval_mul_commute one_times_ivl_left)
lemma set_of_power_mono: "a^n \<in> set_of (A^n)" if "a \<in> set_of A"
for a :: "'a::linordered_idom"
using that
by (induction n) (auto intro!: times_in_intervalI)
lemma set_of_add_cong:
"set_of (A + B) = set_of (A' + B')"
if "set_of A = set_of A'" "set_of B = set_of B'"
for A :: "'a::linordered_ab_group_add interval"
unfolding set_of_plus that ..
lemma set_of_add_inc_left:
"set_of (A + B) \<subseteq> set_of (A' + B)"
if "set_of A \<subseteq> set_of A'"
for A :: "'a::linordered_ab_group_add interval"
unfolding set_of_plus using that by (auto simp: set_plus_def)
lemma set_of_add_inc_right:
"set_of (A + B) \<subseteq> set_of (A + B')"
if "set_of B \<subseteq> set_of B'"
for A :: "'a::linordered_ab_group_add interval"
using set_of_add_inc_left[OF that]
by (simp add: add.commute)
lemma set_of_add_inc:
"set_of (A + B) \<subseteq> set_of (A' + B')"
if "set_of A \<subseteq> set_of A'" "set_of B \<subseteq> set_of B'"
for A :: "'a::linordered_ab_group_add interval"
using set_of_add_inc_left[OF that(1)] set_of_add_inc_right[OF that(2)]
by auto
lemma set_of_neg_inc:
"set_of (-A) \<subseteq> set_of (-A')"
if "set_of A \<subseteq> set_of A'"
for A :: "'a::ordered_ab_group_add interval"
using that
unfolding set_of_uminus
by auto
lemma set_of_sub_inc_left:
"set_of (A - B) \<subseteq> set_of (A' - B)"
if "set_of A \<subseteq> set_of A'"
for A :: "'a::linordered_ab_group_add interval"
using that
unfolding set_of_minus
by auto
lemma set_of_sub_inc_right:
"set_of (A - B) \<subseteq> set_of (A - B')"
if "set_of B \<subseteq> set_of B'"
for A :: "'a::linordered_ab_group_add interval"
using that
unfolding set_of_minus
by auto
lemma set_of_sub_inc:
"set_of (A - B) \<subseteq> set_of (A' - B')"
if "set_of A \<subseteq> set_of A'" "set_of B \<subseteq> set_of B'"
for A :: "'a::linordered_idom interval"
using set_of_sub_inc_left[OF that(1)] set_of_sub_inc_right[OF that(2)]
by auto
lemma set_of_mul_inc_right:
"set_of (A * B) \<subseteq> set_of (A * B')"
if "set_of B \<subseteq> set_of B'"
for A :: "'a::linordered_ring interval"
using that
apply transfer
apply (clarsimp simp add: Let_def)
apply (intro conjI)
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
done
lemma set_of_distrib_left:
"set_of (B * (A1 + A2)) \<subseteq> set_of (B * A1 + B * A2)"
for A1 :: "'a::linordered_ring interval"
apply transfer
apply (clarsimp simp: Let_def distrib_left distrib_right)
apply (intro conjI)
apply (metis add_mono min.cobounded1 min.left_commute)
apply (metis add_mono min.cobounded1 min.left_commute)
apply (metis add_mono min.cobounded1 min.left_commute)
apply (metis add_mono min.assoc min.cobounded2)
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
done
lemma set_of_distrib_right:
"set_of ((A1 + A2) * B) \<subseteq> set_of (A1 * B + A2 * B)"
for A1 A2 B :: "'a::{linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
unfolding set_of_times set_of_plus set_plus_def
apply clarsimp
subgoal for b a1 a2
apply (rule exI[where x="a1 * b"])
apply (rule conjI)
subgoal by force
subgoal
apply (rule exI[where x="a2 * b"])
apply (rule conjI)
subgoal by force
subgoal by (simp add: algebra_simps)
done
done
done
lemma set_of_mul_inc_left:
"set_of (A * B) \<subseteq> set_of (A' * B)"
if "set_of A \<subseteq> set_of A'"
for A :: "'a::{linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
using that
unfolding set_of_times
by auto
lemma set_of_mul_inc:
"set_of (A * B) \<subseteq> set_of (A' * B')"
if "set_of A \<subseteq> set_of A'" "set_of B \<subseteq> set_of B'"
for A :: "'a::{linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
using that unfolding set_of_times by auto
lemma set_of_pow_inc:
"set_of (A^n) \<subseteq> set_of (A'^n)"
if "set_of A \<subseteq> set_of A'"
for A :: "'a::{linordered_idom, real_normed_algebra, linear_continuum_topology} interval"
using that
by (induction n, simp_all add: set_of_mul_inc)
lemma set_of_distrib_right_left:
"set_of ((A1 + A2) * (B1 + B2)) \<subseteq> set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)"
for A1 :: "'a::{linordered_idom, real_normed_algebra, linear_continuum_topology} interval"
proof-
have "set_of ((A1 + A2) * (B1 + B2)) \<subseteq> set_of (A1 * (B1 + B2) + A2 * (B1 + B2))"
by (rule set_of_distrib_right)
also have "... \<subseteq> set_of ((A1 * B1 + A1 * B2) + A2 * (B1 + B2))"
by (rule set_of_add_inc_left[OF set_of_distrib_left])
also have "... \<subseteq> set_of ((A1 * B1 + A1 * B2) + (A2 * B1 + A2 * B2))"
by (rule set_of_add_inc_right[OF set_of_distrib_left])
finally show ?thesis
by (simp add: add.assoc)
qed
lemma mult_bounds_enclose_zero1:
"min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \<le> 0"
"0 \<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))"
if "la \<le> 0" "0 \<le> ua"
for la lb ua ub:: "'a::linordered_idom"
subgoal by (metis (no_types, hide_lams) that eq_iff min_le_iff_disj mult_zero_left mult_zero_right
zero_le_mult_iff)
subgoal by (metis that le_max_iff_disj mult_zero_right order_refl zero_le_mult_iff)
done
lemma mult_bounds_enclose_zero2:
"min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \<le> 0"
"0 \<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))"
if "lb \<le> 0" "0 \<le> ub"
for la lb ua ub:: "'a::linordered_idom"
using mult_bounds_enclose_zero1[OF that, of la ua]
by (simp_all add: ac_simps)
lemma set_of_mul_contains_zero:
"0 \<in> set_of (A * B)"
if "0 \<in> set_of A \<or> 0 \<in> set_of B"
for A :: "'a::linordered_idom interval"
using that
by (auto simp: set_of_eq lower_times upper_times algebra_simps mult_le_0_iff
mult_bounds_enclose_zero1 mult_bounds_enclose_zero2)
instance "interval" :: (linordered_semiring) mult_zero
apply standard
subgoal by transfer auto
subgoal by transfer auto
done
lift_definition min_interval::"'a::linorder interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval" is
"\<lambda>(l1, u1). \<lambda>(l2, u2). (min l1 l2, min u1 u2)"
by (auto simp: min_def)
lemma lower_min_interval[simp]: "lower (min_interval x y) = min (lower x) (lower y)"
by transfer auto
lemma upper_min_interval[simp]: "upper (min_interval x y) = min (upper x) (upper y)"
by transfer auto
lemma min_intervalI:
"a \<in>\<^sub>i A \<Longrightarrow> b \<in>\<^sub>i B \<Longrightarrow> min a b \<in>\<^sub>i min_interval A B"
by (auto simp: set_of_eq min_def)
lift_definition max_interval::"'a::linorder interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval" is
"\<lambda>(l1, u1). \<lambda>(l2, u2). (max l1 l2, max u1 u2)"
by (auto simp: max_def)
lemma lower_max_interval[simp]: "lower (max_interval x y) = max (lower x) (lower y)"
by transfer auto
lemma upper_max_interval[simp]: "upper (max_interval x y) = max (upper x) (upper y)"
by transfer auto
lemma max_intervalI:
"a \<in>\<^sub>i A \<Longrightarrow> b \<in>\<^sub>i B \<Longrightarrow> max a b \<in>\<^sub>i max_interval A B"
by (auto simp: set_of_eq max_def)
lift_definition abs_interval::"'a::linordered_idom interval \<Rightarrow> 'a interval" is
"(\<lambda>(l,u). (if l < 0 \<and> 0 < u then 0 else min \<bar>l\<bar> \<bar>u\<bar>, max \<bar>l\<bar> \<bar>u\<bar>))"
by auto
lemma lower_abs_interval[simp]:
"lower (abs_interval x) = (if lower x < 0 \<and> 0 < upper x then 0 else min \<bar>lower x\<bar> \<bar>upper x\<bar>)"
by transfer auto
lemma upper_abs_interval[simp]: "upper (abs_interval x) = max \<bar>lower x\<bar> \<bar>upper x\<bar>"
by transfer auto
lemma in_abs_intervalI1:
"lx < 0 \<Longrightarrow> 0 < ux \<Longrightarrow> 0 \<le> xa \<Longrightarrow> xa \<le> max (- lx) (ux) \<Longrightarrow> xa \<in> abs ` {lx..ux}"
for xa::"'a::linordered_idom"
by (metis abs_minus_cancel abs_of_nonneg atLeastAtMost_iff image_eqI le_less le_max_iff_disj
le_minus_iff neg_le_0_iff_le order_trans)
lemma in_abs_intervalI2:
"min (\<bar>lx\<bar>) \<bar>ux\<bar> \<le> xa \<Longrightarrow> xa \<le> max \<bar>lx\<bar> \<bar>ux\<bar> \<Longrightarrow> lx \<le> ux \<Longrightarrow> 0 \<le> lx \<or> ux \<le> 0 \<Longrightarrow>
xa \<in> abs ` {lx..ux}"
for xa::"'a::linordered_idom"
by (force intro: image_eqI[where x="-xa"] image_eqI[where x="xa"])
lemma set_of_abs_interval: "set_of (abs_interval x) = abs ` set_of x"
by (auto simp: set_of_eq not_less intro: in_abs_intervalI1 in_abs_intervalI2 cong del: image_cong_simp)
fun split_domain :: "('a::preorder interval \<Rightarrow> 'a interval list) \<Rightarrow> 'a interval list \<Rightarrow> 'a interval list list"
where "split_domain split [] = [[]]"
| "split_domain split (I#Is) = (
let S = split I;
D = split_domain split Is
in concat (map (\<lambda>d. map (\<lambda>s. s # d) S) D)
)"
context notes [[typedef_overloaded]] begin
lift_definition(code_dt) split_interval::"'a::linorder interval \<Rightarrow> 'a \<Rightarrow> ('a interval \<times> 'a interval)"
is "\<lambda>(l, u) x. ((min l x, max l x), (min u x, max u x))"
by (auto simp: min_def)
end
lemma split_domain_nonempty:
assumes "\<And>I. split I \<noteq> []"
shows "split_domain split I \<noteq> []"
using last_in_set assms
by (induction I, auto)
lemma lower_split_interval1: "lower (fst (split_interval X m)) = min (lower X) m"
and lower_split_interval2: "lower (snd (split_interval X m)) = min (upper X) m"
and upper_split_interval1: "upper (fst (split_interval X m)) = max (lower X) m"
and upper_split_interval2: "upper (snd (split_interval X m)) = max (upper X) m"
subgoal by transfer auto
subgoal by transfer (auto simp: min.commute)
subgoal by transfer (auto simp: )
subgoal by transfer (auto simp: )
done
lemma split_intervalD: "split_interval X x = (A, B) \<Longrightarrow> set_of X \<subseteq> set_of A \<union> set_of B"
unfolding set_of_eq
by transfer (auto simp: min_def max_def split: if_splits)
instantiation interval :: ("{topological_space, preorder}") topological_space
begin
definition open_interval_def[code del]: "open (X::'a interval set) =
(\<forall>x\<in>X.
\<exists>A B.
open A \<and>
open B \<and>
lower x \<in> A \<and> upper x \<in> B \<and> Interval ` (A \<times> B) \<subseteq> X)"
instance
proof
show "open (UNIV :: ('a interval) set)"
unfolding open_interval_def by auto
next
fix S T :: "('a interval) set"
assume "open S" "open T"
show "open (S \<inter> T)"
unfolding open_interval_def
proof (safe)
fix x assume "x \<in> S" "x \<in> T"
from \<open>x \<in> S\<close> \<open>open S\<close> obtain Sl Su where S:
"open Sl" "open Su" "lower x \<in> Sl" "upper x \<in> Su" "Interval ` (Sl \<times> Su) \<subseteq> S"
by (auto simp: open_interval_def)
from \<open>x \<in> T\<close> \<open>open T\<close> obtain Tl Tu where T:
"open Tl" "open Tu" "lower x \<in> Tl" "upper x \<in> Tu" "Interval ` (Tl \<times> Tu) \<subseteq> T"
by (auto simp: open_interval_def)
let ?L = "Sl \<inter> Tl" and ?U = "Su \<inter> Tu"
have "open ?L \<and> open ?U \<and> lower x \<in> ?L \<and> upper x \<in> ?U \<and> Interval ` (?L \<times> ?U) \<subseteq> S \<inter> T"
using S T by (auto simp add: open_Int)
then show "\<exists>A B. open A \<and> open B \<and> lower x \<in> A \<and> upper x \<in> B \<and> Interval ` (A \<times> B) \<subseteq> S \<inter> T"
by fast
qed
qed (unfold open_interval_def, fast)
end
subsection \<open>Quickcheck\<close>
lift_definition Ivl::"'a \<Rightarrow> 'a::preorder \<Rightarrow> 'a interval" is "\<lambda>a b. (min a b, b)"
by (auto simp: min_def)
instantiation interval :: ("{exhaustive,preorder}") exhaustive
begin
definition exhaustive_interval::"('a interval \<Rightarrow> (bool \<times> term list) option)
\<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
where
"exhaustive_interval f d =
Quickcheck_Exhaustive.exhaustive (\<lambda>x. Quickcheck_Exhaustive.exhaustive (\<lambda>y. f (Ivl x y)) d) d"
instance ..
end
definition (in term_syntax) [code_unfold]:
"valtermify_interval x y = Code_Evaluation.valtermify (Ivl::'a::{preorder,typerep}\<Rightarrow>_) {\<cdot>} x {\<cdot>} y"
instantiation interval :: ("{full_exhaustive,preorder,typerep}") full_exhaustive
begin
definition full_exhaustive_interval::
"('a interval \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option)
\<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option" where
"full_exhaustive_interval f d =
Quickcheck_Exhaustive.full_exhaustive
(\<lambda>x. Quickcheck_Exhaustive.full_exhaustive (\<lambda>y. f (valtermify_interval x y)) d) d"
instance ..
end
instantiation interval :: ("{random,preorder,typerep}") random
begin
definition random_interval ::
"natural
\<Rightarrow> natural \<times> natural
\<Rightarrow> ('a interval \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where
"random_interval i =
scomp (Quickcheck_Random.random i)
(\<lambda>man. scomp (Quickcheck_Random.random i) (\<lambda>exp. Pair (valtermify_interval man exp)))"
instance ..
end
lifting_update interval.lifting
lifting_forget interval.lifting
end