--- a/NEWS Fri Aug 19 19:01:00 2011 -0700
+++ b/NEWS Sat Aug 20 09:30:23 2011 +0200
@@ -70,7 +70,8 @@
generalized theorems INF_cong and SUP_cong. New type classes for complete
boolean algebras and complete linear orders. Lemmas Inf_less_iff,
less_Sup_iff, INF_less_iff, less_SUP_iff now reside in class complete_linorder.
-Changed proposition of lemmas Inf_fun_def, Sup_fun_def, Inf_apply, Sup_apply.
+Changed proposition of lemmas Inf_bool_def, Sup_bool_def, Inf_fun_def, Sup_fun_def,
+Inf_apply, Sup_apply.
Redundant lemmas Inf_singleton, Sup_singleton, Inf_binary, Sup_binary,
INF_eq, SUP_eq, INF_UNIV_range, SUP_UNIV_range, Int_eq_Inter,
INTER_eq_Inter_image, Inter_def, INT_eq, Un_eq_Union, UNION_eq_Union_image,
--- a/src/HOL/Complete_Lattice.thy Fri Aug 19 19:01:00 2011 -0700
+++ b/src/HOL/Complete_Lattice.thy Sat Aug 20 09:30:23 2011 +0200
@@ -414,8 +414,7 @@
apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
done
-subclass distrib_lattice proof -- {* Question: is it sufficient to include @{class distrib_lattice}
- and prove @{text inf_Sup} and @{text sup_Inf} from that? *}
+subclass distrib_lattice proof
fix a b c
from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_insert)
@@ -556,13 +555,13 @@
begin
definition
- "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
+ [simp]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
definition
- "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
+ [simp]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
instance proof
-qed (auto simp add: Inf_bool_def Sup_bool_def)
+qed (auto intro: bool_induct)
end
@@ -572,7 +571,7 @@
fix A :: "'a set"
fix P :: "'a \<Rightarrow> bool"
show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
- by (auto simp add: Ball_def INF_def Inf_bool_def)
+ by (auto simp add: INF_def)
qed
lemma SUP_bool_eq [simp]:
@@ -581,11 +580,11 @@
fix A :: "'a set"
fix P :: "'a \<Rightarrow> bool"
show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
- by (auto simp add: Bex_def SUP_def Sup_bool_def)
+ by (auto simp add: SUP_def)
qed
instance bool :: complete_boolean_algebra proof
-qed (auto simp add: Inf_bool_def Sup_bool_def)
+qed (auto intro: bool_induct)
instantiation "fun" :: (type, complete_lattice) complete_lattice
begin
@@ -638,7 +637,7 @@
have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
by auto
then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
- by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
+ by (simp add: Inf_fun_def) (simp add: mem_def)
qed
lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
@@ -821,7 +820,7 @@
have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
by auto
then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
- by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
+ by (simp add: Sup_fun_def) (simp add: mem_def)
qed
lemma Union_iff [simp, no_atp]:
--- a/src/HOL/Library/More_Set.thy Fri Aug 19 19:01:00 2011 -0700
+++ b/src/HOL/Library/More_Set.thy Sat Aug 20 09:30:23 2011 +0200
@@ -50,7 +50,7 @@
lemma remove_set_compl:
"remove x (- set xs) = - set (List.insert x xs)"
- by (auto simp del: mem_def simp add: remove_def List.insert_def)
+ by (auto simp add: remove_def List.insert_def)
lemma image_set:
"image f (set xs) = set (map f xs)"
--- a/src/HOL/Main.thy Fri Aug 19 19:01:00 2011 -0700
+++ b/src/HOL/Main.thy Sat Aug 20 09:30:23 2011 +0200
@@ -11,4 +11,17 @@
text {* See further \cite{Nipkow-et-al:2002:tutorial} *}
+text {* Compatibility layer -- to be dropped *}
+
+lemma Inf_bool_def:
+ "Inf A \<longleftrightarrow> (\<forall>x\<in>A. x)"
+ by (auto intro: bool_induct)
+
+lemma Sup_bool_def:
+ "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)"
+ by auto
+
+declare Complete_Lattice.Inf_bool_def [simp del]
+declare Complete_Lattice.Sup_bool_def [simp del]
+
end
--- a/src/HOL/Nat.thy Fri Aug 19 19:01:00 2011 -0700
+++ b/src/HOL/Nat.thy Sat Aug 20 09:30:23 2011 +0200
@@ -22,10 +22,7 @@
typedecl ind
-axiomatization
- Zero_Rep :: ind and
- Suc_Rep :: "ind => ind"
-where
+axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
-- {* the axiom of infinity in 2 parts *}
Suc_Rep_inject: "Suc_Rep x = Suc_Rep y ==> x = y" and
Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
@@ -34,10 +31,9 @@
text {* Type definition *}
-inductive Nat :: "ind \<Rightarrow> bool"
-where
- Zero_RepI: "Nat Zero_Rep"
- | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
+inductive Nat :: "ind \<Rightarrow> bool" where
+ Zero_RepI: "Nat Zero_Rep"
+| Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
typedef (open Nat) nat = "{n. Nat n}"
using Nat.Zero_RepI by auto
@@ -142,10 +138,9 @@
instantiation nat :: "{minus, comm_monoid_add}"
begin
-primrec plus_nat
-where
+primrec plus_nat where
add_0: "0 + n = (n\<Colon>nat)"
- | add_Suc: "Suc m + n = Suc (m + n)"
+| add_Suc: "Suc m + n = Suc (m + n)"
lemma add_0_right [simp]: "m + 0 = (m::nat)"
by (induct m) simp_all
@@ -158,8 +153,7 @@
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
by simp
-primrec minus_nat
-where
+primrec minus_nat where
diff_0 [code]: "m - 0 = (m\<Colon>nat)"
| diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
@@ -188,10 +182,9 @@
definition
One_nat_def [simp]: "1 = Suc 0"
-primrec times_nat
-where
+primrec times_nat where
mult_0: "0 * n = (0\<Colon>nat)"
- | mult_Suc: "Suc m * n = n + (m * n)"
+| mult_Suc: "Suc m * n = n + (m * n)"
lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
by (induct m) simp_all
@@ -364,7 +357,7 @@
primrec less_eq_nat where
"(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
- | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
+| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
declare less_eq_nat.simps [simp del]
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
@@ -1200,8 +1193,8 @@
begin
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
- "funpow 0 f = id"
- | "funpow (Suc n) f = f o funpow n f"
+ "funpow 0 f = id"
+| "funpow (Suc n) f = f o funpow n f"
end
@@ -1267,7 +1260,7 @@
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
"of_nat_aux inc 0 i = i"
- | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
+| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
lemma of_nat_code:
"of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"