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theory Defs
imports Main
begin
datatype 'a mtree = Leaf | Node (left: "'a mtree") (minimum: 'a) (element: 'a) (right: "'a mtree")
consts set_mtree2 :: "'a mtree \<Rightarrow> 'a set"
consts mbst :: "'a::{linorder,zero} mtree \<Rightarrow> bool"
consts min_val :: "'a::{linorder,zero} mtree \<Rightarrow> 'a"
consts mins :: "'a::{linorder,zero} \<Rightarrow> 'a mtree \<Rightarrow> 'a mtree"
consts misin :: "'a::linorder \<Rightarrow> 'a mtree \<Rightarrow> bool"
consts mtree_in_range :: "'a::linorder mtree \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a list"
end
theory Submission
imports Defs
begin
fun set_mtree2 :: "'a mtree \<Rightarrow> 'a set" where
"set_mtree2 _ = undefined"
fun mbst :: "'a::{linorder,zero} mtree \<Rightarrow> bool" where
"mbst _ = undefined"
fun min_val :: "'a::{linorder,zero} mtree \<Rightarrow> 'a" where
"min_val _ = undefined"
lemma mbst_min: "mbst (Node l m a r) \<Longrightarrow> min_val (Node l m a r) = m"
sorry
fun mins :: "'a::{linorder,zero} \<Rightarrow> 'a mtree \<Rightarrow> 'a mtree" where
"mins _ = undefined"
lemma mbst_mins: "mbst t \<Longrightarrow> mbst (mins x t)"
sorry
fun misin :: "'a::linorder \<Rightarrow> 'a mtree \<Rightarrow> bool" where
"misin _ = undefined"
lemma misin_set: "mbst t \<Longrightarrow> misin x t \<longleftrightarrow> x\<in>set_mtree2 t"
sorry
fun mtree_in_range :: "'a::linorder mtree \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a list" where
"mtree_in_range _ = undefined"
lemma mbst_range: "mbst t \<Longrightarrow> set (mtree_in_range t u v) = {x\<in>set_mtree2 t. u\<le>x \<and> x\<le>v}"
sorry
end
theory Check
imports Submission
begin
lemma mbst_min: "mbst (Node l m a r) \<Longrightarrow> min_val (Node l m a r) = m"
by (rule Submission.mbst_min)
lemma mbst_mins: "mbst t \<Longrightarrow> mbst (mins x t)"
by (rule Submission.mbst_mins)
lemma misin_set: "mbst t \<Longrightarrow> misin x t \<longleftrightarrow> x\<in>set_mtree2 t"
by (rule Submission.misin_set)
lemma mbst_range: "mbst t \<Longrightarrow> set (mtree_in_range t u v) = {x\<in>set_mtree2 t. u\<le>x \<and> x\<le>v}"
by (rule Submission.mbst_range)
end