--- a/src/HOL/ex/Classpackage.thy Fri Jul 21 14:47:22 2006 +0200
+++ b/src/HOL/ex/Classpackage.thy Fri Jul 21 14:47:44 2006 +0200
@@ -13,26 +13,26 @@
assumes assoc: "x \<^loc>\<otimes> y \<^loc>\<otimes> z = x \<^loc>\<otimes> (y \<^loc>\<otimes> z)"
instance nat :: semigroup
- "m \<otimes> n == m + n"
+ "m \<otimes> n \<equiv> m + n"
proof
fix m n q :: nat
from semigroup_nat_def show "m \<otimes> n \<otimes> q = m \<otimes> (n \<otimes> q)" by simp
qed
instance int :: semigroup
- "k \<otimes> l == k + l"
+ "k \<otimes> l \<equiv> k + l"
proof
fix k l j :: int
from semigroup_int_def show "k \<otimes> l \<otimes> j = k \<otimes> (l \<otimes> j)" by simp
qed
instance (type) list :: semigroup
- "xs \<otimes> ys == xs @ ys"
+ "xs \<otimes> ys \<equiv> xs @ ys"
proof
fix xs ys zs :: "'a list"
show "xs \<otimes> ys \<otimes> zs = xs \<otimes> (ys \<otimes> zs)"
proof -
- from semigroup_list_def have "\<And>xs ys::'a list. xs \<otimes> ys == xs @ ys".
+ from semigroup_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
thus ?thesis by simp
qed
qed
@@ -41,28 +41,25 @@
fixes one :: 'a ("\<^loc>\<one>")
assumes neutl: "\<^loc>\<one> \<^loc>\<otimes> x = x"
-instance nat :: monoidl
- "\<one> == 0"
+instance monoidl_num_def: nat :: monoidl and int :: monoidl
+ "\<one> \<equiv> 0"
+ "\<one> \<equiv> 0"
proof
fix n :: nat
- from semigroup_nat_def monoidl_nat_def show "\<one> \<otimes> n = n" by simp
-qed
-
-instance int :: monoidl
- "\<one> == 0"
-proof
+ from monoidl_num_def show "\<one> \<otimes> n = n" by simp
+next
fix k :: int
- from semigroup_int_def monoidl_int_def show "\<one> \<otimes> k = k" by simp
+ from monoidl_num_def show "\<one> \<otimes> k = k" by simp
qed
instance (type) list :: monoidl
- "\<one> == []"
+ "\<one> \<equiv> []"
proof
fix xs :: "'a list"
show "\<one> \<otimes> xs = xs"
proof -
- from semigroup_list_def have "\<And>xs ys::'a list. xs \<otimes> ys == xs @ ys" .
- moreover from monoidl_list_def have "\<one> == []::'a list" by simp
+ from mult_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
+ moreover from monoidl_list_def have "\<one> \<equiv> []\<Colon>'a list" by simp
ultimately show ?thesis by simp
qed
qed
@@ -70,13 +67,13 @@
class monoid = monoidl +
assumes neutr: "x \<^loc>\<otimes> \<^loc>\<one> = x"
-instance (type) list :: monoid
+instance monoid_list_def: (type) list :: monoid
proof
fix xs :: "'a list"
show "xs \<otimes> \<one> = xs"
proof -
- from semigroup_list_def have "\<And>xs ys::'a list. xs \<otimes> ys == xs @ ys" .
- moreover from monoidl_list_def have "\<one> == []::'a list" by simp
+ from mult_list_def have "\<And>xs ys\<Colon>'a list. xs \<otimes> ys \<equiv> xs @ ys" .
+ moreover from monoid_list_def have "\<one> \<equiv> []\<Colon>'a list" by simp
ultimately show ?thesis by simp
qed
qed
@@ -84,22 +81,19 @@
class monoid_comm = monoid +
assumes comm: "x \<^loc>\<otimes> y = y \<^loc>\<otimes> x"
-instance nat :: monoid_comm
+instance monoid_comm_num_def: nat :: monoid_comm and int :: monoid_comm
proof
fix n :: nat
- from semigroup_nat_def monoidl_nat_def show "n \<otimes> \<one> = n" by simp
+ from monoid_comm_num_def show "n \<otimes> \<one> = n" by simp
next
fix n m :: nat
- from semigroup_nat_def monoidl_nat_def show "n \<otimes> m = m \<otimes> n" by simp
-qed
-
-instance int :: monoid_comm
-proof
+ from monoid_comm_num_def show "n \<otimes> m = m \<otimes> n" by simp
+next
fix k :: int
- from semigroup_int_def monoidl_int_def show "k \<otimes> \<one> = k" by simp
+ from monoid_comm_num_def show "k \<otimes> \<one> = k" by simp
next
fix k l :: int
- from semigroup_int_def monoidl_int_def show "k \<otimes> l = l \<otimes> k" by simp
+ from monoid_comm_num_def show "k \<otimes> l = l \<otimes> k" by simp
qed
definition (in monoid)
@@ -150,7 +144,7 @@
abbreviation (in monoid)
abbrev_npow :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infix "\<^loc>\<up>" 75)
- "x \<^loc>\<up> n == npow n x"
+ "x \<^loc>\<up> n \<equiv> npow n x"
lemma (in monoid) npow_def:
"x \<^loc>\<up> 0 = \<^loc>\<one>"
@@ -179,11 +173,11 @@
class group_comm = group + monoid_comm
-instance int :: group_comm
- "\<div> k == - (k::int)"
+instance group_comm_int_def: int :: group_comm
+ "\<div> k \<equiv> - (k\<Colon>int)"
proof
fix k :: int
- from semigroup_int_def monoidl_int_def group_comm_int_def show "\<div> k \<otimes> k = \<one>" by simp
+ from group_comm_int_def show "\<div> k \<otimes> k = \<one>" by simp
qed
lemma (in group) cancel:
@@ -225,12 +219,12 @@
instance group < monoid
proof
- fix x :: "'a::group"
+ fix x :: "'a\<Colon>group"
from group.neutr show "x \<otimes> \<one> = x" .
qed
lemma (in group) all_inv [intro]:
- "(x::'a) \<in> monoid.units (op \<^loc>\<otimes>) \<^loc>\<one>"
+ "(x\<Colon>'a) \<in> monoid.units (op \<^loc>\<otimes>) \<^loc>\<one>"
unfolding units_def
proof -
fix x :: "'a"
@@ -293,31 +287,32 @@
"x \<^loc>\<up> k \<equiv> pow k x"
lemma (in group) int_pow_zero:
- "x \<^loc>\<up> (0::int) = \<^loc>\<one>"
+ "x \<^loc>\<up> (0\<Colon>int) = \<^loc>\<one>"
using npow_def pow_def by simp
lemma (in group) int_pow_one:
- "\<^loc>\<one> \<^loc>\<up> (k::int) = \<^loc>\<one>"
+ "\<^loc>\<one> \<^loc>\<up> (k\<Colon>int) = \<^loc>\<one>"
using pow_def nat_pow_one inv_one by simp
instance group_prod_def: (group, group) * :: group
- mult_prod_def: "x \<otimes> y == let (x1, x2) = x; (y1, y2) = y in
+ mult_prod_def: "x \<otimes> y \<equiv> let (x1, x2) = x; (y1, y2) = y in
(x1 \<otimes> y1, x2 \<otimes> y2)"
- mult_one_def: "\<one> == (\<one>, \<one>)"
- mult_inv_def: "\<div> x == let (x1, x2) = x in (\<div> x1, \<div> x2)"
+ mult_one_def: "\<one> \<equiv> (\<one>, \<one>)"
+ mult_inv_def: "\<div> x \<equiv> let (x1, x2) = x in (\<div> x1, \<div> x2)"
by default (simp_all add: split_paired_all group_prod_def assoc neutl invl)
instance group_comm_prod_def: (group_comm, group_comm) * :: group_comm
by default (simp_all add: split_paired_all group_prod_def assoc neutl invl comm)
definition
- "x = ((2::nat) \<otimes> \<one> \<otimes> 3, (2::int) \<otimes> \<one> \<otimes> \<div> 3, [1::nat, 2] \<otimes> \<one> \<otimes> [1, 2, 3])"
- "y = (2 :: int, \<div> 2 :: int) \<otimes> \<one> \<otimes> (3, \<div> 3)"
+ "x = ((2\<Colon>nat) \<otimes> \<one> \<otimes> 3, (2\<Colon>int) \<otimes> \<one> \<otimes> \<div> 3, [1\<Colon>nat, 2] \<otimes> \<one> \<otimes> [1, 2, 3])"
+ "y = (2 \<Colon> int, \<div> 2 \<Colon> int) \<otimes> \<one> \<otimes> (3, \<div> 3)"
-code_generate "op \<otimes>" "\<one>" "inv"
+code_generate "op \<otimes>" \<one> inv
code_generate (ml, haskell) x
code_generate (ml, haskell) y
+code_serialize ml (_)
code_serialize ml (-)
end