--- a/src/HOL/Data_Structures/Brother12_Set.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/Brother12_Set.thy Fri Mar 23 11:39:41 2018 +0100
@@ -150,13 +150,9 @@
subsubsection "Proofs for isin"
-lemma
- "t \<in> T h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
-by(induction h arbitrary: t) (fastforce simp: elems_simps1 split: if_splits)+
-
-lemma isin_set: "t \<in> T h \<Longrightarrow>
- sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
-by(induction h arbitrary: t) (auto simp: elems_simps2 split: if_splits)
+lemma isin_set:
+ "t \<in> T h \<Longrightarrow> sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set(inorder t))"
+by(induction h arbitrary: t) (fastforce simp: isin_simps split: if_splits)+
subsubsection "Proofs for insertion"
--- a/src/HOL/Data_Structures/Isin2.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/Isin2.thy Fri Mar 23 11:39:41 2018 +0100
@@ -17,10 +17,7 @@
EQ \<Rightarrow> True |
GT \<Rightarrow> isin r x)"
-lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
-by (induction t) (auto simp: elems_simps1)
-
-lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems(inorder t))"
-by (induction t) (auto simp: elems_simps2)
+lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set(inorder t))"
+by (induction t) (auto simp: isin_simps)
end
--- a/src/HOL/Data_Structures/List_Ins_Del.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/List_Ins_Del.thy Fri Mar 23 11:39:41 2018 +0100
@@ -8,38 +8,29 @@
subsection \<open>Elements in a list\<close>
-fun elems :: "'a list \<Rightarrow> 'a set" where
-"elems [] = {}" |
-"elems (x#xs) = Set.insert x (elems xs)"
-
-lemma elems_app: "elems (xs @ ys) = (elems xs \<union> elems ys)"
-by (induction xs) auto
-
-lemma elems_eq_set: "elems xs = set xs"
-by (induction xs) auto
-
lemma sorted_Cons_iff:
- "sorted(x # xs) = ((\<forall>y \<in> elems xs. x < y) \<and> sorted xs)"
-by(simp add: elems_eq_set sorted_wrt_Cons)
+ "sorted(x # xs) = ((\<forall>y \<in> set xs. x < y) \<and> sorted xs)"
+by(simp add: sorted_wrt_Cons)
lemma sorted_snoc_iff:
- "sorted(xs @ [x]) = (sorted xs \<and> (\<forall>y \<in> elems xs. y < x))"
-by(simp add: elems_eq_set sorted_wrt_append)
-
+ "sorted(xs @ [x]) = (sorted xs \<and> (\<forall>y \<in> set xs. y < x))"
+by(simp add: sorted_wrt_append)
+(*
text\<open>The above two rules introduce quantifiers. It turns out
that in practice this is not a problem because of the simplicity of
-the "isin" functions that implement @{const elems}. Nevertheless
+the "isin" functions that implement @{const set}. Nevertheless
it is possible to avoid the quantifiers with the help of some rewrite rules:\<close>
-lemma sorted_ConsD: "sorted (y # xs) \<Longrightarrow> x \<le> y \<Longrightarrow> x \<notin> elems xs"
+lemma sorted_ConsD: "sorted (y # xs) \<Longrightarrow> x \<le> y \<Longrightarrow> x \<notin> set xs"
by (auto simp: sorted_Cons_iff)
-lemma sorted_snocD: "sorted (xs @ [y]) \<Longrightarrow> y \<le> x \<Longrightarrow> x \<notin> elems xs"
+lemma sorted_snocD: "sorted (xs @ [y]) \<Longrightarrow> y \<le> x \<Longrightarrow> x \<notin> set xs"
by (auto simp: sorted_snoc_iff)
-lemmas elems_simps = sorted_lems elems_app
-lemmas elems_simps1 = elems_simps sorted_Cons_iff sorted_snoc_iff
-lemmas elems_simps2 = elems_simps sorted_ConsD sorted_snocD
+lemmas isin_simps2 = sorted_lems sorted_ConsD sorted_snocD
+*)
+
+lemmas isin_simps = sorted_lems sorted_Cons_iff sorted_snoc_iff
subsection \<open>Inserting into an ordered list without duplicates:\<close>
@@ -49,7 +40,7 @@
"ins_list x (a#xs) =
(if x < a then x#a#xs else if x=a then a#xs else a # ins_list x xs)"
-lemma set_ins_list: "elems (ins_list x xs) = insert x (elems xs)"
+lemma set_ins_list: "set (ins_list x xs) = insert x (set xs)"
by(induction xs) auto
lemma distinct_if_sorted: "sorted xs \<Longrightarrow> distinct xs"
@@ -93,16 +84,12 @@
"del_list x [] = []" |
"del_list x (a#xs) = (if x=a then xs else a # del_list x xs)"
-lemma del_list_idem: "x \<notin> elems xs \<Longrightarrow> del_list x xs = xs"
+lemma del_list_idem: "x \<notin> set xs \<Longrightarrow> del_list x xs = xs"
by (induct xs) simp_all
-lemma elems_del_list_eq:
- "distinct xs \<Longrightarrow> elems (del_list x xs) = elems xs - {x}"
-apply(induct xs)
- apply simp
-apply (simp add: elems_eq_set)
-apply blast
-done
+lemma set_del_list_eq:
+ "distinct xs \<Longrightarrow> set (del_list x xs) = set xs - {x}"
+by(induct xs) auto
lemma sorted_del_list: "sorted xs \<Longrightarrow> sorted(del_list x xs)"
apply(induction xs rule: induct_list012)
@@ -112,7 +99,7 @@
lemma del_list_sorted: "sorted (xs @ a # ys) \<Longrightarrow>
del_list x (xs @ a # ys) = (if x < a then del_list x xs @ a # ys else xs @ del_list x (a # ys))"
by(induction xs)
- (fastforce simp: sorted_lems sorted_Cons_iff elems_eq_set intro!: del_list_idem)+
+ (fastforce simp: sorted_lems sorted_Cons_iff intro!: del_list_idem)+
text\<open>In principle, @{thm del_list_sorted} suffices, but the following
corollaries speed up proofs.\<close>
--- a/src/HOL/Data_Structures/Set_by_Ordered.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/Set_by_Ordered.thy Fri Mar 23 11:39:41 2018 +0100
@@ -30,7 +30,7 @@
fixes inv :: "'t \<Rightarrow> bool"
assumes empty: "inorder empty = []"
assumes isin: "inv t \<and> sorted(inorder t) \<Longrightarrow>
- isin t x = (x \<in> elems (inorder t))"
+ isin t x = (x \<in> set (inorder t))"
assumes insert: "inv t \<and> sorted(inorder t) \<Longrightarrow>
inorder(insert x t) = ins_list x (inorder t)"
assumes delete: "inv t \<and> sorted(inorder t) \<Longrightarrow>
@@ -41,7 +41,7 @@
begin
sublocale Set
- empty insert delete isin "elems o inorder" "\<lambda>t. inv t \<and> sorted(inorder t)"
+ empty insert delete isin "set o inorder" "\<lambda>t. inv t \<and> sorted(inorder t)"
proof(standard, goal_cases)
case 1 show ?case by (auto simp: empty)
next
@@ -50,7 +50,7 @@
case 3 thus ?case by(simp add: insert set_ins_list)
next
case (4 s x) thus ?case
- using delete[OF 4, of x] by (auto simp: distinct_if_sorted elems_del_list_eq)
+ using delete[OF 4, of x] by (auto simp: distinct_if_sorted set_del_list_eq)
next
case 5 thus ?case by(simp add: empty inv_empty)
next
--- a/src/HOL/Data_Structures/Tree234_Set.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/Tree234_Set.thy Fri Mar 23 11:39:41 2018 +0100
@@ -202,11 +202,8 @@
subsubsection \<open>Functional correctness of isin:\<close>
-lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps1 ball_Un)
-
-lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps2 split!: if_splits)
+lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"
+by (induction t) (auto simp: isin_simps ball_Un)
subsubsection \<open>Functional correctness of insert:\<close>
--- a/src/HOL/Data_Structures/Tree23_Set.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/Tree23_Set.thy Fri Mar 23 11:39:41 2018 +0100
@@ -142,11 +142,8 @@
subsubsection "Proofs for isin"
-lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps1 ball_Un)
-
-lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps2)
+lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"
+by (induction t) (auto simp: isin_simps ball_Un)
subsubsection "Proofs for insert"
--- a/src/HOL/Data_Structures/Tree_Set.thy Fri Mar 23 10:52:00 2018 +0100
+++ b/src/HOL/Data_Structures/Tree_Set.thy Fri Mar 23 11:39:41 2018 +0100
@@ -42,12 +42,8 @@
subsection "Functional Correctness Proofs"
-lemma "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps1)
-
-lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> elems (inorder t))"
-by (induction t) (auto simp: elems_simps2)
-
+lemma isin_set: "sorted(inorder t) \<Longrightarrow> isin t x = (x \<in> set (inorder t))"
+by (induction t) (auto simp: isin_simps)
lemma inorder_insert:
"sorted(inorder t) \<Longrightarrow> inorder(insert x t) = ins_list x (inorder t)"