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(* Title: Interval
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Author: Christoph Traut, TU Muenchen
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Fabian Immler, TU Muenchen
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*)
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section \<open>Interval Type\<close>
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theory Interval
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imports
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Complex_Main
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Lattice_Algebras
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Set_Algebras
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begin
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text \<open>A type of non-empty, closed intervals.\<close>
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typedef (overloaded) 'a interval =
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"{(a::'a::preorder, b). a \<le> b}"
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morphisms bounds_of_interval Interval
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by auto
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setup_lifting type_definition_interval
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lift_definition lower::"('a::preorder) interval \<Rightarrow> 'a" is fst .
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lift_definition upper::"('a::preorder) interval \<Rightarrow> 'a" is snd .
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lemma interval_eq_iff: "a = b \<longleftrightarrow> lower a = lower b \<and> upper a = upper b"
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by transfer auto
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lemma interval_eqI: "lower a = lower b \<Longrightarrow> upper a = upper b \<Longrightarrow> a = b"
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by (auto simp: interval_eq_iff)
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lemma lower_le_upper[simp]: "lower i \<le> upper i"
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by transfer auto
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lift_definition set_of :: "'a::preorder interval \<Rightarrow> 'a set" is "\<lambda>x. {fst x .. snd x}" .
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lemma set_of_eq: "set_of x = {lower x .. upper x}"
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by transfer simp
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context notes [[typedef_overloaded]] begin
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lift_definition(code_dt) Interval'::"'a::preorder \<Rightarrow> 'a::preorder \<Rightarrow> 'a interval option"
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is "\<lambda>a b. if a \<le> b then Some (a, b) else None"
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by auto
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end
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instantiation "interval" :: ("{preorder,equal}") equal
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begin
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definition "equal_class.equal a b \<equiv> (lower a = lower b) \<and> (upper a = upper b)"
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instance proof qed (simp add: equal_interval_def interval_eq_iff)
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end
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instantiation interval :: ("preorder") ord begin
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definition less_eq_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> bool"
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where "less_eq_interval a b \<longleftrightarrow> lower b \<le> lower a \<and> upper a \<le> upper b"
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definition less_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> bool"
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where "less_interval x y = (x \<le> y \<and> \<not> y \<le> x)"
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instance proof qed
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end
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instantiation interval :: ("lattice") semilattice_sup
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begin
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lift_definition sup_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
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is "\<lambda>(a, b) (c, d). (inf a c, sup b d)"
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by (auto simp: le_infI1 le_supI1)
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lemma lower_sup[simp]: "lower (sup A B) = inf (lower A) (lower B)"
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by transfer auto
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lemma upper_sup[simp]: "upper (sup A B) = sup (upper A) (upper B)"
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by transfer auto
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instance proof qed (auto simp: less_eq_interval_def less_interval_def interval_eq_iff)
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end
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lemma set_of_interval_union: "set_of A \<union> set_of B \<subseteq> set_of (sup A B)" for A::"'a::lattice interval"
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by (auto simp: set_of_eq)
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lemma interval_union_commute: "sup A B = sup B A" for A::"'a::lattice interval"
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by (auto simp add: interval_eq_iff inf.commute sup.commute)
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lemma interval_union_mono1: "set_of a \<subseteq> set_of (sup a A)" for A :: "'a::lattice interval"
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using set_of_interval_union by blast
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lemma interval_union_mono2: "set_of A \<subseteq> set_of (sup a A)" for A :: "'a::lattice interval"
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using set_of_interval_union by blast
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lift_definition interval_of :: "'a::preorder \<Rightarrow> 'a interval" is "\<lambda>x. (x, x)"
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by auto
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lemma lower_interval_of[simp]: "lower (interval_of a) = a"
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by transfer auto
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lemma upper_interval_of[simp]: "upper (interval_of a) = a"
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by transfer auto
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definition width :: "'a::{preorder,minus} interval \<Rightarrow> 'a"
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where "width i = upper i - lower i"
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instantiation "interval" :: ("ordered_ab_semigroup_add") ab_semigroup_add
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begin
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lift_definition plus_interval::"'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
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is "\<lambda>(a, b). \<lambda>(c, d). (a + c, b + d)"
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by (auto intro!: add_mono)
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lemma lower_plus[simp]: "lower (plus A B) = plus (lower A) (lower B)"
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by transfer auto
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lemma upper_plus[simp]: "upper (plus A B) = plus (upper A) (upper B)"
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by transfer auto
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instance proof qed (auto simp: interval_eq_iff less_eq_interval_def ac_simps)
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end
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instance "interval" :: ("{ordered_ab_semigroup_add, lattice}") ordered_ab_semigroup_add
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proof qed (auto simp: less_eq_interval_def intro!: add_mono)
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instantiation "interval" :: ("{preorder,zero}") zero
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begin
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lift_definition zero_interval::"'a interval" is "(0, 0)" by auto
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lemma lower_zero[simp]: "lower 0 = 0"
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by transfer auto
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lemma upper_zero[simp]: "upper 0 = 0"
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by transfer auto
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instance proof qed
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end
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instance "interval" :: ("{ordered_comm_monoid_add}") comm_monoid_add
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proof qed (auto simp: interval_eq_iff)
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instance "interval" :: ("{ordered_comm_monoid_add,lattice}") ordered_comm_monoid_add ..
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instantiation "interval" :: ("{ordered_ab_group_add}") uminus
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begin
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lift_definition uminus_interval::"'a interval \<Rightarrow> 'a interval" is "\<lambda>(a, b). (-b, -a)" by auto
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lemma lower_uminus[simp]: "lower (- A) = - upper A"
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by transfer auto
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lemma upper_uminus[simp]: "upper (- A) = - lower A"
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by transfer auto
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instance ..
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end
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instantiation "interval" :: ("{ordered_ab_group_add}") minus
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begin
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definition minus_interval::"'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
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where "minus_interval a b = a + - b"
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lemma lower_minus[simp]: "lower (minus A B) = minus (lower A) (upper B)"
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by (auto simp: minus_interval_def)
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lemma upper_minus[simp]: "upper (minus A B) = minus (upper A) (lower B)"
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by (auto simp: minus_interval_def)
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instance ..
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end
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instantiation "interval" :: (linordered_semiring) times
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begin
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lift_definition times_interval :: "'a interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval"
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is "\<lambda>(a1, a2). \<lambda>(b1, b2).
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(let x1 = a1 * b1; x2 = a1 * b2; x3 = a2 * b1; x4 = a2 * b2
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in (min x1 (min x2 (min x3 x4)), max x1 (max x2 (max x3 x4))))"
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by (auto simp: Let_def intro!: min.coboundedI1 max.coboundedI1)
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lemma lower_times:
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"lower (times A B) = Min {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
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by transfer (auto simp: Let_def)
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lemma upper_times:
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"upper (times A B) = Max {lower A * lower B, lower A * upper B, upper A * lower B, upper A * upper B}"
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by transfer (auto simp: Let_def)
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instance ..
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end
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lemma interval_eq_set_of_iff: "X = Y \<longleftrightarrow> set_of X = set_of Y" for X Y::"'a::order interval"
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by (auto simp: set_of_eq interval_eq_iff)
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subsection \<open>Membership\<close>
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abbreviation (in preorder) in_interval ("(_/ \<in>\<^sub>i _)" [51, 51] 50)
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where "in_interval x X \<equiv> x \<in> set_of X"
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lemma in_interval_to_interval[intro!]: "a \<in>\<^sub>i interval_of a"
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by (auto simp: set_of_eq)
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lemma plus_in_intervalI:
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fixes x y :: "'a :: ordered_ab_semigroup_add"
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shows "x \<in>\<^sub>i X \<Longrightarrow> y \<in>\<^sub>i Y \<Longrightarrow> x + y \<in>\<^sub>i X + Y"
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by (simp add: add_mono_thms_linordered_semiring(1) set_of_eq)
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lemma connected_set_of[intro, simp]:
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"connected (set_of X)" for X::"'a::linear_continuum_topology interval"
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by (auto simp: set_of_eq )
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lemma ex_sum_in_interval_lemma: "\<exists>xa\<in>{la .. ua}. \<exists>xb\<in>{lb .. ub}. x = xa + xb"
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if "la \<le> ua" "lb \<le> ub" "la + lb \<le> x" "x \<le> ua + ub"
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"ua - la \<le> ub - lb"
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for la b c d::"'a::linordered_ab_group_add"
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proof -
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define wa where "wa = ua - la"
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define wb where "wb = ub - lb"
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define w where "w = wa + wb"
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define d where "d = x - la - lb"
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define da where "da = max 0 (min wa (d - wa))"
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define db where "db = d - da"
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from that have nonneg: "0 \<le> wa" "0 \<le> wb" "0 \<le> w" "0 \<le> d" "d \<le> w"
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by (auto simp add: wa_def wb_def w_def d_def add.commute le_diff_eq)
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have "0 \<le> db"
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by (auto simp: da_def nonneg db_def intro!: min.coboundedI2)
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have "x = (la + da) + (lb + db)"
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by (simp add: da_def db_def d_def)
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moreover
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have "x - la - ub \<le> da"
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using that
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unfolding da_def
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by (intro max.coboundedI2) (auto simp: wa_def d_def diff_le_eq diff_add_eq)
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then have "db \<le> wb"
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by (auto simp: db_def d_def wb_def algebra_simps)
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with \<open>0 \<le> db\<close> that nonneg have "lb + db \<in> {lb..ub}"
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by (auto simp: wb_def algebra_simps)
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moreover
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have "da \<le> wa"
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by (auto simp: da_def nonneg)
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then have "la + da \<in> {la..ua}"
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by (auto simp: da_def wa_def algebra_simps)
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ultimately show ?thesis
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by force
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qed
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lemma ex_sum_in_interval: "\<exists>xa\<ge>la. xa \<le> ua \<and> (\<exists>xb\<ge>lb. xb \<le> ub \<and> x = xa + xb)"
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if a: "la \<le> ua" and b: "lb \<le> ub" and x: "la + lb \<le> x" "x \<le> ua + ub"
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for la b c d::"'a::linordered_ab_group_add"
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proof -
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from linear consider "ua - la \<le> ub - lb" | "ub - lb \<le> ua - la"
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by blast
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then show ?thesis
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proof cases
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case 1
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from ex_sum_in_interval_lemma[OF that 1]
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show ?thesis by auto
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next
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case 2
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from x have "lb + la \<le> x" "x \<le> ub + ua" by (simp_all add: ac_simps)
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from ex_sum_in_interval_lemma[OF b a this 2]
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show ?thesis by auto
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qed
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qed
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lemma Icc_plus_Icc:
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"{a .. b} + {c .. d} = {a + c .. b + d}"
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if "a \<le> b" "c \<le> d"
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for a b c d::"'a::linordered_ab_group_add"
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using ex_sum_in_interval[OF that]
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by (auto intro: add_mono simp: atLeastAtMost_iff Bex_def set_plus_def)
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lemma set_of_plus:
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fixes A :: "'a::linordered_ab_group_add interval"
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shows "set_of (A + B) = set_of A + set_of B"
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using Icc_plus_Icc[of "lower A" "upper A" "lower B" "upper B"]
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by (auto simp: set_of_eq)
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lemma plus_in_intervalE:
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fixes xy :: "'a :: linordered_ab_group_add"
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assumes "xy \<in>\<^sub>i X + Y"
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obtains x y where "xy = x + y" "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y"
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using assms
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unfolding set_of_plus set_plus_def
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by auto
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lemma set_of_uminus: "set_of (-X) = {- x | x. x \<in> set_of X}"
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for X :: "'a :: ordered_ab_group_add interval"
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by (auto simp: set_of_eq simp: le_minus_iff minus_le_iff
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intro!: exI[where x="-x" for x])
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lemma uminus_in_intervalI:
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fixes x :: "'a :: ordered_ab_group_add"
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shows "x \<in>\<^sub>i X \<Longrightarrow> -x \<in>\<^sub>i -X"
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by (auto simp: set_of_uminus)
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lemma uminus_in_intervalD:
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fixes x :: "'a :: ordered_ab_group_add"
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shows "x \<in>\<^sub>i - X \<Longrightarrow> - x \<in>\<^sub>i X"
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by (auto simp: set_of_uminus)
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lemma minus_in_intervalI:
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fixes x y :: "'a :: ordered_ab_group_add"
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shows "x \<in>\<^sub>i X \<Longrightarrow> y \<in>\<^sub>i Y \<Longrightarrow> x - y \<in>\<^sub>i X - Y"
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by (metis diff_conv_add_uminus minus_interval_def plus_in_intervalI uminus_in_intervalI)
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lemma set_of_minus: "set_of (X - Y) = {x - y | x y . x \<in> set_of X \<and> y \<in> set_of Y}"
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for X Y :: "'a :: linordered_ab_group_add interval"
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unfolding minus_interval_def set_of_plus set_of_uminus set_plus_def
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by force
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lemma times_in_intervalI:
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fixes x y::"'a::linordered_ring"
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assumes "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y"
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shows "x * y \<in>\<^sub>i X * Y"
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proof -
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define X1 where "X1 \<equiv> lower X"
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define X2 where "X2 \<equiv> upper X"
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define Y1 where "Y1 \<equiv> lower Y"
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define Y2 where "Y2 \<equiv> upper Y"
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from assms have assms: "X1 \<le> x" "x \<le> X2" "Y1 \<le> y" "y \<le> Y2"
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by (auto simp: X1_def X2_def Y1_def Y2_def set_of_eq)
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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>
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(X1 * Y1 \<ge> x * y \<or> X1 * Y2 \<ge> x * y \<or> X2 * Y1 \<ge> x * y \<or> X2 * Y2 \<ge> x * y)"
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proof (cases x "0::'a" rule: linorder_cases)
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case x0: less
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show ?thesis
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proof (cases "y < 0")
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case y0: True
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from y0 x0 assms have "x * y \<le> X1 * y" by (intro mult_right_mono_neg, auto)
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also from x0 y0 assms have "X1 * y \<le> X1 * Y1" by (intro mult_left_mono_neg, auto)
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finally have 1: "x * y \<le> X1 * Y1".
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show ?thesis proof(cases "X2 \<le> 0")
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case True
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with assms have "X2 * Y2 \<le> X2 * y" by (auto intro: mult_left_mono_neg)
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also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono_neg)
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finally have "X2 * Y2 \<le> x * y".
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with 1 show ?thesis by auto
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next
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case False
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with assms have "X2 * Y1 \<le> X2 * y" by (auto intro: mult_left_mono)
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also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono_neg)
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finally have "X2 * Y1 \<le> x * y".
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with 1 show ?thesis by auto
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qed
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next
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|
342 |
case False
|
|
343 |
then have y0: "y \<ge> 0" by auto
|
|
344 |
from x0 y0 assms have "X1 * Y2 \<le> x * Y2" by (intro mult_right_mono, auto)
|
|
345 |
also from y0 x0 assms have "... \<le> x * y" by (intro mult_left_mono_neg, auto)
|
|
346 |
finally have 1: "X1 * Y2 \<le> x * y".
|
|
347 |
show ?thesis
|
|
348 |
proof(cases "X2 \<le> 0")
|
|
349 |
case X2: True
|
|
350 |
from assms y0 have "x * y \<le> X2 * y" by (intro mult_right_mono)
|
|
351 |
also from assms X2 have "... \<le> X2 * Y1" by (auto intro: mult_left_mono_neg)
|
|
352 |
finally have "x * y \<le> X2 * Y1".
|
|
353 |
with 1 show ?thesis by auto
|
|
354 |
next
|
|
355 |
case X2: False
|
|
356 |
from assms y0 have "x * y \<le> X2 * y" by (intro mult_right_mono)
|
|
357 |
also from assms X2 have "... \<le> X2 * Y2" by (auto intro: mult_left_mono)
|
|
358 |
finally have "x * y \<le> X2 * Y2".
|
|
359 |
with 1 show ?thesis by auto
|
|
360 |
qed
|
|
361 |
qed
|
|
362 |
next
|
|
363 |
case [simp]: equal
|
|
364 |
with assms show ?thesis by (cases "Y2 \<le> 0", auto intro:mult_sign_intros)
|
|
365 |
next
|
|
366 |
case x0: greater
|
|
367 |
show ?thesis
|
|
368 |
proof (cases "y < 0")
|
|
369 |
case y0: True
|
|
370 |
from x0 y0 assms have "X2 * Y1 \<le> X2 * y" by (intro mult_left_mono, auto)
|
|
371 |
also from y0 x0 assms have "X2 * y \<le> x * y" by (intro mult_right_mono_neg, auto)
|
|
372 |
finally have 1: "X2 * Y1 \<le> x * y".
|
|
373 |
show ?thesis
|
|
374 |
proof(cases "Y2 \<le> 0")
|
|
375 |
case Y2: True
|
|
376 |
from x0 assms have "x * y \<le> x * Y2" by (auto intro: mult_left_mono)
|
|
377 |
also from assms Y2 have "... \<le> X1 * Y2" by (auto intro: mult_right_mono_neg)
|
|
378 |
finally have "x * y \<le> X1 * Y2".
|
|
379 |
with 1 show ?thesis by auto
|
|
380 |
next
|
|
381 |
case Y2: False
|
|
382 |
from x0 assms have "x * y \<le> x * Y2" by (auto intro: mult_left_mono)
|
|
383 |
also from assms Y2 have "... \<le> X2 * Y2" by (auto intro: mult_right_mono)
|
|
384 |
finally have "x * y \<le> X2 * Y2".
|
|
385 |
with 1 show ?thesis by auto
|
|
386 |
qed
|
|
387 |
next
|
|
388 |
case y0: False
|
|
389 |
from x0 y0 assms have "x * y \<le> X2 * y" by (intro mult_right_mono, auto)
|
|
390 |
also from y0 x0 assms have "... \<le> X2 * Y2" by (intro mult_left_mono, auto)
|
|
391 |
finally have 1: "x * y \<le> X2 * Y2".
|
|
392 |
show ?thesis
|
|
393 |
proof(cases "X1 \<le> 0")
|
|
394 |
case True
|
|
395 |
with assms have "X1 * Y2 \<le> X1 * y" by (auto intro: mult_left_mono_neg)
|
|
396 |
also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono)
|
|
397 |
finally have "X1 * Y2 \<le> x * y".
|
|
398 |
with 1 show ?thesis by auto
|
|
399 |
next
|
|
400 |
case False
|
|
401 |
with assms have "X1 * Y1 \<le> X1 * y" by (auto intro: mult_left_mono)
|
|
402 |
also from assms y0 have "... \<le> x * y" by (auto intro: mult_right_mono)
|
|
403 |
finally have "X1 * Y1 \<le> x * y".
|
|
404 |
with 1 show ?thesis by auto
|
|
405 |
qed
|
|
406 |
qed
|
|
407 |
qed
|
|
408 |
hence min:"min (X1 * Y1) (min (X1 * Y2) (min (X2 * Y1) (X2 * Y2))) \<le> x * y"
|
|
409 |
and max:"x * y \<le> max (X1 * Y1) (max (X1 * Y2) (max (X2 * Y1) (X2 * Y2)))"
|
|
410 |
by (auto simp:min_le_iff_disj le_max_iff_disj)
|
|
411 |
show ?thesis using min max
|
|
412 |
by (auto simp: Let_def X1_def X2_def Y1_def Y2_def set_of_eq lower_times upper_times)
|
|
413 |
qed
|
|
414 |
|
|
415 |
lemma times_in_intervalE:
|
|
416 |
fixes xy :: "'a :: {linordered_semiring, real_normed_algebra, linear_continuum_topology}"
|
|
417 |
\<comment> \<open>TODO: linear continuum topology is pretty strong\<close>
|
|
418 |
assumes "xy \<in>\<^sub>i X * Y"
|
|
419 |
obtains x y where "xy = x * y" "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y"
|
|
420 |
proof -
|
|
421 |
let ?mult = "\<lambda>(x, y). x * y"
|
|
422 |
let ?XY = "set_of X \<times> set_of Y"
|
|
423 |
have cont: "continuous_on ?XY ?mult"
|
|
424 |
by (auto intro!: tendsto_eq_intros simp: continuous_on_def split_beta')
|
|
425 |
have conn: "connected (?mult ` ?XY)"
|
|
426 |
by (rule connected_continuous_image[OF cont]) auto
|
|
427 |
have "lower (X * Y) \<in> ?mult ` ?XY" "upper (X * Y) \<in> ?mult ` ?XY"
|
|
428 |
by (auto simp: set_of_eq lower_times upper_times min_def max_def split: if_splits)
|
|
429 |
from connectedD_interval[OF conn this, of xy] assms
|
|
430 |
obtain x y where "xy = x * y" "x \<in>\<^sub>i X" "y \<in>\<^sub>i Y" by (auto simp: set_of_eq)
|
|
431 |
then show ?thesis ..
|
|
432 |
qed
|
|
433 |
|
|
434 |
lemma set_of_times: "set_of (X * Y) = {x * y | x y. x \<in> set_of X \<and> y \<in> set_of Y}"
|
|
435 |
for X Y::"'a :: {linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
|
|
436 |
by (auto intro!: times_in_intervalI elim!: times_in_intervalE)
|
|
437 |
|
|
438 |
instance "interval" :: (linordered_idom) cancel_semigroup_add
|
|
439 |
proof qed (auto simp: interval_eq_iff)
|
|
440 |
|
|
441 |
lemma interval_mul_commute: "A * B = B * A" for A B:: "'a::linordered_idom interval"
|
|
442 |
by (simp add: interval_eq_iff lower_times upper_times ac_simps)
|
|
443 |
|
|
444 |
lemma interval_times_zero_right[simp]: "A * 0 = 0" for A :: "'a::linordered_ring interval"
|
|
445 |
by (simp add: interval_eq_iff lower_times upper_times ac_simps)
|
|
446 |
|
|
447 |
lemma interval_times_zero_left[simp]:
|
|
448 |
"0 * A = 0" for A :: "'a::linordered_ring interval"
|
|
449 |
by (simp add: interval_eq_iff lower_times upper_times ac_simps)
|
|
450 |
|
|
451 |
instantiation "interval" :: ("{preorder,one}") one
|
|
452 |
begin
|
|
453 |
|
|
454 |
lift_definition one_interval::"'a interval" is "(1, 1)" by auto
|
|
455 |
lemma lower_one[simp]: "lower 1 = 1"
|
|
456 |
by transfer auto
|
|
457 |
lemma upper_one[simp]: "upper 1 = 1"
|
|
458 |
by transfer auto
|
|
459 |
instance proof qed
|
|
460 |
end
|
|
461 |
|
|
462 |
instance interval :: ("{one, preorder, linordered_semiring}") power
|
|
463 |
proof qed
|
|
464 |
|
|
465 |
lemma set_of_one[simp]: "set_of (1::'a::{one, order} interval) = {1}"
|
|
466 |
by (auto simp: set_of_eq)
|
|
467 |
|
|
468 |
instance "interval" ::
|
|
469 |
("{linordered_idom,linordered_ring, real_normed_algebra, linear_continuum_topology}") monoid_mult
|
|
470 |
apply standard
|
|
471 |
unfolding interval_eq_set_of_iff set_of_times
|
|
472 |
subgoal
|
|
473 |
by (auto simp: interval_eq_set_of_iff set_of_times; metis mult.assoc)
|
|
474 |
by auto
|
|
475 |
|
|
476 |
lemma one_times_ivl_left[simp]: "1 * A = A" for A :: "'a::linordered_idom interval"
|
|
477 |
by (simp add: interval_eq_iff lower_times upper_times ac_simps min_def max_def)
|
|
478 |
|
|
479 |
lemma one_times_ivl_right[simp]: "A * 1 = A" for A :: "'a::linordered_idom interval"
|
|
480 |
by (metis interval_mul_commute one_times_ivl_left)
|
|
481 |
|
|
482 |
lemma set_of_power_mono: "a^n \<in> set_of (A^n)" if "a \<in> set_of A"
|
|
483 |
for a :: "'a::linordered_idom"
|
|
484 |
using that
|
|
485 |
by (induction n) (auto intro!: times_in_intervalI)
|
|
486 |
|
|
487 |
lemma set_of_add_cong:
|
|
488 |
"set_of (A + B) = set_of (A' + B')"
|
|
489 |
if "set_of A = set_of A'" "set_of B = set_of B'"
|
|
490 |
for A :: "'a::linordered_ab_group_add interval"
|
|
491 |
unfolding set_of_plus that ..
|
|
492 |
|
|
493 |
lemma set_of_add_inc_left:
|
|
494 |
"set_of (A + B) \<subseteq> set_of (A' + B)"
|
|
495 |
if "set_of A \<subseteq> set_of A'"
|
|
496 |
for A :: "'a::linordered_ab_group_add interval"
|
|
497 |
unfolding set_of_plus using that by (auto simp: set_plus_def)
|
|
498 |
|
|
499 |
lemma set_of_add_inc_right:
|
|
500 |
"set_of (A + B) \<subseteq> set_of (A + B')"
|
|
501 |
if "set_of B \<subseteq> set_of B'"
|
|
502 |
for A :: "'a::linordered_ab_group_add interval"
|
|
503 |
using set_of_add_inc_left[OF that]
|
|
504 |
by (simp add: add.commute)
|
|
505 |
|
|
506 |
lemma set_of_add_inc:
|
|
507 |
"set_of (A + B) \<subseteq> set_of (A' + B')"
|
|
508 |
if "set_of A \<subseteq> set_of A'" "set_of B \<subseteq> set_of B'"
|
|
509 |
for A :: "'a::linordered_ab_group_add interval"
|
|
510 |
using set_of_add_inc_left[OF that(1)] set_of_add_inc_right[OF that(2)]
|
|
511 |
by auto
|
|
512 |
|
|
513 |
lemma set_of_neg_inc:
|
|
514 |
"set_of (-A) \<subseteq> set_of (-A')"
|
|
515 |
if "set_of A \<subseteq> set_of A'"
|
|
516 |
for A :: "'a::ordered_ab_group_add interval"
|
|
517 |
using that
|
|
518 |
unfolding set_of_uminus
|
|
519 |
by auto
|
|
520 |
|
|
521 |
lemma set_of_sub_inc_left:
|
|
522 |
"set_of (A - B) \<subseteq> set_of (A' - B)"
|
|
523 |
if "set_of A \<subseteq> set_of A'"
|
|
524 |
for A :: "'a::linordered_ab_group_add interval"
|
|
525 |
using that
|
|
526 |
unfolding set_of_minus
|
|
527 |
by auto
|
|
528 |
|
|
529 |
lemma set_of_sub_inc_right:
|
|
530 |
"set_of (A - B) \<subseteq> set_of (A - B')"
|
|
531 |
if "set_of B \<subseteq> set_of B'"
|
|
532 |
for A :: "'a::linordered_ab_group_add interval"
|
|
533 |
using that
|
|
534 |
unfolding set_of_minus
|
|
535 |
by auto
|
|
536 |
|
|
537 |
lemma set_of_sub_inc:
|
|
538 |
"set_of (A - B) \<subseteq> set_of (A' - B')"
|
|
539 |
if "set_of A \<subseteq> set_of A'" "set_of B \<subseteq> set_of B'"
|
|
540 |
for A :: "'a::linordered_idom interval"
|
|
541 |
using set_of_sub_inc_left[OF that(1)] set_of_sub_inc_right[OF that(2)]
|
|
542 |
by auto
|
|
543 |
|
|
544 |
lemma set_of_mul_inc_right:
|
|
545 |
"set_of (A * B) \<subseteq> set_of (A * B')"
|
|
546 |
if "set_of B \<subseteq> set_of B'"
|
|
547 |
for A :: "'a::linordered_ring interval"
|
|
548 |
using that
|
|
549 |
apply transfer
|
|
550 |
apply (clarsimp simp add: Let_def)
|
|
551 |
apply (intro conjI)
|
|
552 |
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
553 |
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
554 |
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
555 |
apply (metis linear min.coboundedI1 min.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
556 |
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
557 |
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
558 |
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
559 |
apply (metis linear max.coboundedI1 max.coboundedI2 mult_left_mono mult_left_mono_neg order_trans)
|
|
560 |
done
|
|
561 |
|
|
562 |
lemma set_of_distrib_left:
|
|
563 |
"set_of (B * (A1 + A2)) \<subseteq> set_of (B * A1 + B * A2)"
|
|
564 |
for A1 :: "'a::linordered_ring interval"
|
|
565 |
apply transfer
|
|
566 |
apply (clarsimp simp: Let_def distrib_left distrib_right)
|
|
567 |
apply (intro conjI)
|
|
568 |
apply (metis add_mono min.cobounded1 min.left_commute)
|
|
569 |
apply (metis add_mono min.cobounded1 min.left_commute)
|
|
570 |
apply (metis add_mono min.cobounded1 min.left_commute)
|
|
571 |
apply (metis add_mono min.assoc min.cobounded2)
|
|
572 |
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
|
|
573 |
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
|
|
574 |
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
|
|
575 |
apply (meson add_mono order.trans max.cobounded1 max.cobounded2)
|
|
576 |
done
|
|
577 |
|
|
578 |
lemma set_of_distrib_right:
|
|
579 |
"set_of ((A1 + A2) * B) \<subseteq> set_of (A1 * B + A2 * B)"
|
|
580 |
for A1 A2 B :: "'a::{linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
|
|
581 |
unfolding set_of_times set_of_plus set_plus_def
|
|
582 |
apply clarsimp
|
|
583 |
subgoal for b a1 a2
|
|
584 |
apply (rule exI[where x="a1 * b"])
|
|
585 |
apply (rule conjI)
|
|
586 |
subgoal by force
|
|
587 |
subgoal
|
|
588 |
apply (rule exI[where x="a2 * b"])
|
|
589 |
apply (rule conjI)
|
|
590 |
subgoal by force
|
|
591 |
subgoal by (simp add: algebra_simps)
|
|
592 |
done
|
|
593 |
done
|
|
594 |
done
|
|
595 |
|
|
596 |
lemma set_of_mul_inc_left:
|
|
597 |
"set_of (A * B) \<subseteq> set_of (A' * B)"
|
|
598 |
if "set_of A \<subseteq> set_of A'"
|
|
599 |
for A :: "'a::{linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
|
|
600 |
using that
|
|
601 |
unfolding set_of_times
|
|
602 |
by auto
|
|
603 |
|
|
604 |
lemma set_of_mul_inc:
|
|
605 |
"set_of (A * B) \<subseteq> set_of (A' * B')"
|
|
606 |
if "set_of A \<subseteq> set_of A'" "set_of B \<subseteq> set_of B'"
|
|
607 |
for A :: "'a::{linordered_ring, real_normed_algebra, linear_continuum_topology} interval"
|
|
608 |
using that unfolding set_of_times by auto
|
|
609 |
|
|
610 |
lemma set_of_pow_inc:
|
|
611 |
"set_of (A^n) \<subseteq> set_of (A'^n)"
|
|
612 |
if "set_of A \<subseteq> set_of A'"
|
|
613 |
for A :: "'a::{linordered_idom, real_normed_algebra, linear_continuum_topology} interval"
|
|
614 |
using that
|
|
615 |
by (induction n, simp_all add: set_of_mul_inc)
|
|
616 |
|
|
617 |
lemma set_of_distrib_right_left:
|
|
618 |
"set_of ((A1 + A2) * (B1 + B2)) \<subseteq> set_of (A1 * B1 + A1 * B2 + A2 * B1 + A2 * B2)"
|
|
619 |
for A1 :: "'a::{linordered_idom, real_normed_algebra, linear_continuum_topology} interval"
|
|
620 |
proof-
|
|
621 |
have "set_of ((A1 + A2) * (B1 + B2)) \<subseteq> set_of (A1 * (B1 + B2) + A2 * (B1 + B2))"
|
|
622 |
by (rule set_of_distrib_right)
|
|
623 |
also have "... \<subseteq> set_of ((A1 * B1 + A1 * B2) + A2 * (B1 + B2))"
|
|
624 |
by (rule set_of_add_inc_left[OF set_of_distrib_left])
|
|
625 |
also have "... \<subseteq> set_of ((A1 * B1 + A1 * B2) + (A2 * B1 + A2 * B2))"
|
|
626 |
by (rule set_of_add_inc_right[OF set_of_distrib_left])
|
|
627 |
finally show ?thesis
|
|
628 |
by (simp add: add.assoc)
|
|
629 |
qed
|
|
630 |
|
|
631 |
lemma mult_bounds_enclose_zero1:
|
|
632 |
"min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \<le> 0"
|
|
633 |
"0 \<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))"
|
|
634 |
if "la \<le> 0" "0 \<le> ua"
|
|
635 |
for la lb ua ub:: "'a::linordered_idom"
|
|
636 |
subgoal by (metis (no_types, hide_lams) that eq_iff min_le_iff_disj mult_zero_left mult_zero_right
|
|
637 |
zero_le_mult_iff)
|
|
638 |
subgoal by (metis that le_max_iff_disj mult_zero_right order_refl zero_le_mult_iff)
|
|
639 |
done
|
|
640 |
|
|
641 |
lemma mult_bounds_enclose_zero2:
|
|
642 |
"min (la * lb) (min (la * ub) (min (lb * ua) (ua * ub))) \<le> 0"
|
|
643 |
"0 \<le> max (la * lb) (max (la * ub) (max (lb * ua) (ua * ub)))"
|
|
644 |
if "lb \<le> 0" "0 \<le> ub"
|
|
645 |
for la lb ua ub:: "'a::linordered_idom"
|
|
646 |
using mult_bounds_enclose_zero1[OF that, of la ua]
|
|
647 |
by (simp_all add: ac_simps)
|
|
648 |
|
|
649 |
lemma set_of_mul_contains_zero:
|
|
650 |
"0 \<in> set_of (A * B)"
|
|
651 |
if "0 \<in> set_of A \<or> 0 \<in> set_of B"
|
|
652 |
for A :: "'a::linordered_idom interval"
|
|
653 |
using that
|
|
654 |
by (auto simp: set_of_eq lower_times upper_times algebra_simps mult_le_0_iff
|
|
655 |
mult_bounds_enclose_zero1 mult_bounds_enclose_zero2)
|
|
656 |
|
|
657 |
instance "interval" :: (linordered_semiring) mult_zero
|
|
658 |
apply standard
|
|
659 |
subgoal by transfer auto
|
|
660 |
subgoal by transfer auto
|
|
661 |
done
|
|
662 |
|
|
663 |
lift_definition min_interval::"'a::linorder interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval" is
|
|
664 |
"\<lambda>(l1, u1). \<lambda>(l2, u2). (min l1 l2, min u1 u2)"
|
|
665 |
by (auto simp: min_def)
|
|
666 |
lemma lower_min_interval[simp]: "lower (min_interval x y) = min (lower x) (lower y)"
|
|
667 |
by transfer auto
|
|
668 |
lemma upper_min_interval[simp]: "upper (min_interval x y) = min (upper x) (upper y)"
|
|
669 |
by transfer auto
|
|
670 |
|
|
671 |
lemma min_intervalI:
|
|
672 |
"a \<in>\<^sub>i A \<Longrightarrow> b \<in>\<^sub>i B \<Longrightarrow> min a b \<in>\<^sub>i min_interval A B"
|
|
673 |
by (auto simp: set_of_eq min_def)
|
|
674 |
|
|
675 |
lift_definition max_interval::"'a::linorder interval \<Rightarrow> 'a interval \<Rightarrow> 'a interval" is
|
|
676 |
"\<lambda>(l1, u1). \<lambda>(l2, u2). (max l1 l2, max u1 u2)"
|
|
677 |
by (auto simp: max_def)
|
|
678 |
lemma lower_max_interval[simp]: "lower (max_interval x y) = max (lower x) (lower y)"
|
|
679 |
by transfer auto
|
|
680 |
lemma upper_max_interval[simp]: "upper (max_interval x y) = max (upper x) (upper y)"
|
|
681 |
by transfer auto
|
|
682 |
|
|
683 |
lemma max_intervalI:
|
|
684 |
"a \<in>\<^sub>i A \<Longrightarrow> b \<in>\<^sub>i B \<Longrightarrow> max a b \<in>\<^sub>i max_interval A B"
|
|
685 |
by (auto simp: set_of_eq max_def)
|
|
686 |
|
|
687 |
lift_definition abs_interval::"'a::linordered_idom interval \<Rightarrow> 'a interval" is
|
|
688 |
"(\<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>))"
|
|
689 |
by auto
|
|
690 |
|
|
691 |
lemma lower_abs_interval[simp]:
|
|
692 |
"lower (abs_interval x) = (if lower x < 0 \<and> 0 < upper x then 0 else min \<bar>lower x\<bar> \<bar>upper x\<bar>)"
|
|
693 |
by transfer auto
|
|
694 |
lemma upper_abs_interval[simp]: "upper (abs_interval x) = max \<bar>lower x\<bar> \<bar>upper x\<bar>"
|
|
695 |
by transfer auto
|
|
696 |
|
|
697 |
lemma in_abs_intervalI1:
|
|
698 |
"lx < 0 \<Longrightarrow> 0 < ux \<Longrightarrow> 0 \<le> xa \<Longrightarrow> xa \<le> max (- lx) (ux) \<Longrightarrow> xa \<in> abs ` {lx..ux}"
|
|
699 |
for xa::"'a::linordered_idom"
|
|
700 |
by (metis abs_minus_cancel abs_of_nonneg atLeastAtMost_iff image_eqI le_less le_max_iff_disj
|
|
701 |
le_minus_iff neg_le_0_iff_le order_trans)
|
|
702 |
|
|
703 |
lemma in_abs_intervalI2:
|
|
704 |
"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>
|
|
705 |
xa \<in> abs ` {lx..ux}"
|
|
706 |
for xa::"'a::linordered_idom"
|
|
707 |
by (force intro: image_eqI[where x="-xa"] image_eqI[where x="xa"])
|
|
708 |
|
|
709 |
lemma set_of_abs_interval: "set_of (abs_interval x) = abs ` set_of x"
|
|
710 |
by (auto simp: set_of_eq not_less intro: in_abs_intervalI1 in_abs_intervalI2 cong del: image_cong_simp)
|
|
711 |
|
|
712 |
fun split_domain :: "('a::preorder interval \<Rightarrow> 'a interval list) \<Rightarrow> 'a interval list \<Rightarrow> 'a interval list list"
|
|
713 |
where "split_domain split [] = [[]]"
|
|
714 |
| "split_domain split (I#Is) = (
|
|
715 |
let S = split I;
|
|
716 |
D = split_domain split Is
|
|
717 |
in concat (map (\<lambda>d. map (\<lambda>s. s # d) S) D)
|
|
718 |
)"
|
|
719 |
|
|
720 |
context notes [[typedef_overloaded]] begin
|
|
721 |
lift_definition(code_dt) split_interval::"'a::linorder interval \<Rightarrow> 'a \<Rightarrow> ('a interval \<times> 'a interval)"
|
|
722 |
is "\<lambda>(l, u) x. ((min l x, max l x), (min u x, max u x))"
|
|
723 |
by (auto simp: min_def)
|
|
724 |
end
|
|
725 |
|
|
726 |
lemma split_domain_nonempty:
|
|
727 |
assumes "\<And>I. split I \<noteq> []"
|
|
728 |
shows "split_domain split I \<noteq> []"
|
|
729 |
using last_in_set assms
|
|
730 |
by (induction I, auto)
|
|
731 |
|
|
732 |
|
|
733 |
lemma split_intervalD: "split_interval X x = (A, B) \<Longrightarrow> set_of X \<subseteq> set_of A \<union> set_of B"
|
|
734 |
unfolding set_of_eq
|
|
735 |
by transfer (auto simp: min_def max_def split: if_splits)
|
|
736 |
|
|
737 |
instantiation interval :: ("{topological_space, preorder}") topological_space
|
|
738 |
begin
|
|
739 |
|
|
740 |
definition open_interval_def[code del]: "open (X::'a interval set) =
|
|
741 |
(\<forall>x\<in>X.
|
|
742 |
\<exists>A B.
|
|
743 |
open A \<and>
|
|
744 |
open B \<and>
|
|
745 |
lower x \<in> A \<and> upper x \<in> B \<and> Interval ` (A \<times> B) \<subseteq> X)"
|
|
746 |
|
|
747 |
instance
|
|
748 |
proof
|
|
749 |
show "open (UNIV :: ('a interval) set)"
|
|
750 |
unfolding open_interval_def by auto
|
|
751 |
next
|
|
752 |
fix S T :: "('a interval) set"
|
|
753 |
assume "open S" "open T"
|
|
754 |
show "open (S \<inter> T)"
|
|
755 |
unfolding open_interval_def
|
|
756 |
proof (safe)
|
|
757 |
fix x assume "x \<in> S" "x \<in> T"
|
|
758 |
from \<open>x \<in> S\<close> \<open>open S\<close> obtain Sl Su where S:
|
|
759 |
"open Sl" "open Su" "lower x \<in> Sl" "upper x \<in> Su" "Interval ` (Sl \<times> Su) \<subseteq> S"
|
|
760 |
by (auto simp: open_interval_def)
|
|
761 |
from \<open>x \<in> T\<close> \<open>open T\<close> obtain Tl Tu where T:
|
|
762 |
"open Tl" "open Tu" "lower x \<in> Tl" "upper x \<in> Tu" "Interval ` (Tl \<times> Tu) \<subseteq> T"
|
|
763 |
by (auto simp: open_interval_def)
|
|
764 |
|
|
765 |
let ?L = "Sl \<inter> Tl" and ?U = "Su \<inter> Tu"
|
|
766 |
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"
|
|
767 |
using S T by (auto simp add: open_Int)
|
|
768 |
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"
|
|
769 |
by fast
|
|
770 |
qed
|
|
771 |
qed (unfold open_interval_def, fast)
|
|
772 |
|
|
773 |
end
|
|
774 |
|
|
775 |
|
|
776 |
subsection \<open>Quickcheck\<close>
|
|
777 |
|
|
778 |
lift_definition Ivl::"'a \<Rightarrow> 'a::preorder \<Rightarrow> 'a interval" is "\<lambda>a b. (min a b, b)"
|
|
779 |
by (auto simp: min_def)
|
|
780 |
|
|
781 |
instantiation interval :: ("{exhaustive,preorder}") exhaustive
|
|
782 |
begin
|
|
783 |
|
|
784 |
definition exhaustive_interval::"('a interval \<Rightarrow> (bool \<times> term list) option)
|
|
785 |
\<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option"
|
|
786 |
where
|
|
787 |
"exhaustive_interval f d =
|
|
788 |
Quickcheck_Exhaustive.exhaustive (\<lambda>x. Quickcheck_Exhaustive.exhaustive (\<lambda>y. f (Ivl x y)) d) d"
|
|
789 |
|
|
790 |
instance ..
|
|
791 |
|
|
792 |
end
|
|
793 |
|
|
794 |
definition (in term_syntax) [code_unfold]:
|
|
795 |
"valtermify_interval x y = Code_Evaluation.valtermify (Ivl::'a::{preorder,typerep}\<Rightarrow>_) {\<cdot>} x {\<cdot>} y"
|
|
796 |
|
|
797 |
instantiation interval :: ("{full_exhaustive,preorder,typerep}") full_exhaustive
|
|
798 |
begin
|
|
799 |
|
|
800 |
definition full_exhaustive_interval::
|
|
801 |
"('a interval \<times> (unit \<Rightarrow> term) \<Rightarrow> (bool \<times> term list) option)
|
|
802 |
\<Rightarrow> natural \<Rightarrow> (bool \<times> term list) option" where
|
|
803 |
"full_exhaustive_interval f d =
|
|
804 |
Quickcheck_Exhaustive.full_exhaustive
|
|
805 |
(\<lambda>x. Quickcheck_Exhaustive.full_exhaustive (\<lambda>y. f (valtermify_interval x y)) d) d"
|
|
806 |
|
|
807 |
instance ..
|
|
808 |
|
|
809 |
end
|
|
810 |
|
|
811 |
instantiation interval :: ("{random,preorder,typerep}") random
|
|
812 |
begin
|
|
813 |
|
|
814 |
definition random_interval ::
|
|
815 |
"natural
|
|
816 |
\<Rightarrow> natural \<times> natural
|
|
817 |
\<Rightarrow> ('a interval \<times> (unit \<Rightarrow> term)) \<times> natural \<times> natural" where
|
|
818 |
"random_interval i =
|
|
819 |
scomp (Quickcheck_Random.random i)
|
|
820 |
(\<lambda>man. scomp (Quickcheck_Random.random i) (\<lambda>exp. Pair (valtermify_interval man exp)))"
|
|
821 |
|
|
822 |
instance ..
|
|
823 |
|
|
824 |
end
|
|
825 |
|
|
826 |
lifting_update interval.lifting
|
|
827 |
lifting_forget interval.lifting
|
|
828 |
|
|
829 |
end
|