--- a/src/HOL/Library/Abstract_Rat.thy Thu Jul 02 13:48:39 2009 +0200
+++ b/src/HOL/Library/Abstract_Rat.thy Sat Jul 04 15:19:29 2009 +0200
@@ -5,7 +5,7 @@
header {* Abstract rational numbers *}
theory Abstract_Rat
-imports GCD Complex_Main
+imports GCD Main
begin
types Num = "int \<times> int"
@@ -404,14 +404,16 @@
qed
lemma Nadd_commute:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "x +\<^sub>N y = y +\<^sub>N x"
proof-
have n: "isnormNum (x +\<^sub>N y)" "isnormNum (y +\<^sub>N x)" by simp_all
- have "(INum (x +\<^sub>N y)::rat) = INum (y +\<^sub>N x)" by simp
+ have "(INum (x +\<^sub>N y)::'a) = INum (y +\<^sub>N x)" by simp
with isnormNum_unique[OF n] show ?thesis by simp
qed
lemma [simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "(0, b) +\<^sub>N y = normNum y"
and "(a, 0) +\<^sub>N y = normNum y"
and "x +\<^sub>N (0, b) = normNum x"
@@ -423,17 +425,19 @@
done
lemma normNum_nilpotent_aux[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
assumes nx: "isnormNum x"
shows "normNum x = x"
proof-
let ?a = "normNum x"
have n: "isnormNum ?a" by simp
- have th:"INum ?a = (INum x :: 'a::{ring_char_0, division_by_zero, field})" by simp
+ have th:"INum ?a = (INum x ::'a)" by simp
with isnormNum_unique[OF n nx]
show ?thesis by simp
qed
lemma normNum_nilpotent[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "normNum (normNum x) = normNum x"
by simp
@@ -441,31 +445,35 @@
by (simp_all add: normNum_def)
lemma normNum_Nadd:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "normNum (x +\<^sub>N y) = x +\<^sub>N y" by simp
lemma Nadd_normNum1[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "normNum x +\<^sub>N y = x +\<^sub>N y"
proof-
have n: "isnormNum (normNum x +\<^sub>N y)" "isnormNum (x +\<^sub>N y)" by simp_all
- have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: real)" by simp
+ have "INum (normNum x +\<^sub>N y) = INum x + (INum y :: 'a)" by simp
also have "\<dots> = INum (x +\<^sub>N y)" by simp
finally show ?thesis using isnormNum_unique[OF n] by simp
qed
lemma Nadd_normNum2[simp]:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "x +\<^sub>N normNum y = x +\<^sub>N y"
proof-
have n: "isnormNum (x +\<^sub>N normNum y)" "isnormNum (x +\<^sub>N y)" by simp_all
- have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: real)" by simp
+ have "INum (x +\<^sub>N normNum y) = INum x + (INum y :: 'a)" by simp
also have "\<dots> = INum (x +\<^sub>N y)" by simp
finally show ?thesis using isnormNum_unique[OF n] by simp
qed
lemma Nadd_assoc:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
shows "x +\<^sub>N y +\<^sub>N z = x +\<^sub>N (y +\<^sub>N z)"
proof-
have n: "isnormNum (x +\<^sub>N y +\<^sub>N z)" "isnormNum (x +\<^sub>N (y +\<^sub>N z))" by simp_all
- have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: real)" by simp
+ have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
with isnormNum_unique[OF n] show ?thesis by simp
qed
@@ -473,22 +481,24 @@
by (simp add: Nmul_def split_def Let_def int_gcd_commute mult_commute)
lemma Nmul_assoc:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
assumes nx: "isnormNum x" and ny:"isnormNum y" and nz:"isnormNum z"
shows "x *\<^sub>N y *\<^sub>N z = x *\<^sub>N (y *\<^sub>N z)"
proof-
from nx ny nz have n: "isnormNum (x *\<^sub>N y *\<^sub>N z)" "isnormNum (x *\<^sub>N (y *\<^sub>N z))"
by simp_all
- have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: real)" by simp
+ have "INum (x +\<^sub>N y +\<^sub>N z) = (INum (x +\<^sub>N (y +\<^sub>N z)) :: 'a)" by simp
with isnormNum_unique[OF n] show ?thesis by simp
qed
lemma Nsub0:
+ assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
assumes x: "isnormNum x" and y:"isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
proof-
{ fix h :: 'a
- from isnormNum_unique[where 'a = real, OF Nsub_normN[OF y], where y="0\<^sub>N"]
- have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: real)) " by simp
- also have "\<dots> = (INum x = (INum y :: real))" by simp
+ from isnormNum_unique[where 'a = 'a, OF Nsub_normN[OF y], where y="0\<^sub>N"]
+ have "(x -\<^sub>N y = 0\<^sub>N) = (INum (x -\<^sub>N y) = (INum 0\<^sub>N :: 'a)) " by simp
+ also have "\<dots> = (INum x = (INum y :: 'a))" by simp
also have "\<dots> = (x = y)" using x y by simp
finally show ?thesis . }
qed