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(*
File: Interval.thy
Author: Bohua Zhan
*)
section \<open>Intervals\<close>
theory Interval
imports "Auto2_HOL.Auto2_Main"
begin
text \<open>Basic definition of intervals.\<close>
subsection \<open>Definition of interval\<close>
datatype 'a interval = Interval (low: 'a) (high: 'a)
setup \<open>add_simple_datatype "interval"\<close>
instantiation interval :: (linorder) linorder begin
definition int_less: "(a < b) = (low a < low b | (low a = low b \<and> high a < high b))"
definition int_less_eq: "(a \<le> b) = (low a < low b | (low a = low b \<and> high a \<le> high b))"
instance proof
fix x y z :: "'a interval"
show a: "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
using int_less int_less_eq by force
show b: "x \<le> x"
by (simp add: int_less_eq)
show c: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
by (smt int_less_eq dual_order.trans less_trans)
show d: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
using int_less_eq a interval.expand int_less by fastforce
show e: "x \<le> y \<or> y \<le> x"
by (meson int_less_eq leI not_less_iff_gr_or_eq)
qed end
definition is_interval :: "('a::linorder) interval \<Rightarrow> bool" where [rewrite]:
"is_interval it \<longleftrightarrow> (low it \<le> high it)"
subsection \<open>Definition of interval with an index\<close>
datatype 'a idx_interval = IdxInterval (int: "'a interval") (idx: nat)
setup \<open>add_simple_datatype "idx_interval"\<close>
instantiation idx_interval :: (linorder) linorder begin
definition iint_less: "(a < b) = (int a < int b | (int a = int b \<and> idx a < idx b))"
definition iint_less_eq: "(a \<le> b) = (int a < int b | (int a = int b \<and> idx a \<le> idx b))"
instance proof
fix x y z :: "'a idx_interval"
show a: "(x < y) = (x \<le> y \<and> \<not> y \<le> x)"
using iint_less iint_less_eq by force
show b: "x \<le> x"
by (simp add: iint_less_eq)
show c: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
by (smt iint_less_eq dual_order.trans less_trans)
show d: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
using a idx_interval.expand iint_less iint_less_eq by auto
show e: "x \<le> y \<or> y \<le> x"
by (meson iint_less_eq leI not_less_iff_gr_or_eq)
qed end
lemma interval_less_to_le_low [forward]:
"(a::('a::linorder idx_interval)) < b \<Longrightarrow> low (int a) \<le> low (int b)"
by (metis eq_iff iint_less int_less less_imp_le)
subsection \<open>Overlapping intervals\<close>
definition is_overlap :: "('a::linorder) interval \<Rightarrow> 'a interval \<Rightarrow> bool" where [rewrite]:
"is_overlap x y \<longleftrightarrow> (high x \<ge> low y \<and> high y \<ge> low x)"
definition has_overlap :: "('a::linorder) idx_interval set \<Rightarrow> 'a interval \<Rightarrow> bool" where [rewrite]:
"has_overlap xs y \<longleftrightarrow> (\<exists>x\<in>xs. is_overlap (int x) y)"
end
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