--- a/src/HOL/Library/Numeral_Type.thy Mon Jan 14 18:35:03 2019 +0000
+++ b/src/HOL/Library/Numeral_Type.thy Wed Jan 16 21:27:33 2019 +0000
@@ -45,6 +45,8 @@
unfolding type_definition.card [OF type_definition_bit1]
by simp
+subsection \<open>@{typ num1}\<close>
+
instance num1 :: finite
proof
show "finite (UNIV::num1 set)"
@@ -52,6 +54,52 @@
using finite by (rule finite_imageI)
qed
+instantiation num1 :: CARD_1
+begin
+
+instance
+proof
+ show "CARD(num1) = 1" by auto
+qed
+
+end
+
+lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
+ by (induct x, induct y) simp
+
+instantiation num1 :: "{comm_ring,comm_monoid_mult,numeral}"
+begin
+
+instance
+ by standard (simp_all add: num1_eq_iff)
+
+end
+
+lemma num1_eqI:
+ fixes a::num1 shows "a = b"
+by(simp add: num1_eq_iff)
+
+lemma num1_eq1 [simp]:
+ fixes a::num1 shows "a = 1"
+ by (rule num1_eqI)
+
+lemma forall_1[simp]: "(\<forall>i::num1. P i) \<longleftrightarrow> P 1"
+ by (metis (full_types) num1_eq_iff)
+
+lemma ex_1[simp]: "(\<exists>x::num1. P x) \<longleftrightarrow> P 1"
+ by auto (metis (full_types) num1_eq_iff)
+
+instantiation num1 :: linorder begin
+definition "a < b \<longleftrightarrow> Rep_num1 a < Rep_num1 b"
+definition "a \<le> b \<longleftrightarrow> Rep_num1 a \<le> Rep_num1 b"
+instance
+ by intro_classes (auto simp: less_eq_num1_def less_num1_def intro: num1_eqI)
+end
+
+instance num1 :: wellorder
+ by intro_classes (auto simp: less_eq_num1_def less_num1_def)
+
+
instance bit0 :: (finite) card2
proof
show "finite (UNIV::'a bit0 set)"
@@ -185,17 +233,6 @@
\<^typ>\<open>num1\<close>, since 0 and 1 are not distinct.
\<close>
-instantiation num1 :: "{comm_ring,comm_monoid_mult,numeral}"
-begin
-
-lemma num1_eq_iff: "(x::num1) = (y::num1) \<longleftrightarrow> True"
- by (induct x, induct y) simp
-
-instance
- by standard (simp_all add: num1_eq_iff)
-
-end
-
instantiation
bit0 and bit1 :: (finite) "{zero,one,plus,times,uminus,minus}"
begin
@@ -350,8 +387,7 @@
definition "enum_class.enum_ex P = P (1 :: num1)"
instance
by intro_classes
- (auto simp add: enum_num1_def enum_all_num1_def enum_ex_num1_def num1_eq_iff Ball_def,
- (metis (full_types) num1_eq_iff)+)
+ (auto simp add: enum_num1_def enum_all_num1_def enum_ex_num1_def num1_eq_iff Ball_def)
end
instantiation num0 and num1 :: card_UNIV begin