--- a/src/HOL/Library/Quotient.thy Sun Nov 12 14:49:37 2000 +0100
+++ b/src/HOL/Library/Quotient.thy Sun Nov 12 14:50:26 2000 +0100
@@ -73,7 +73,7 @@
relation.
*}
-theorem equivalence_class_eq [iff?]: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
+theorem equivalence_class_iff [iff?]: "(\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>) = (a \<sim> b)"
proof
assume eq: "\<lfloor>a\<rfloor> = \<lfloor>b\<rfloor>"
show "a \<sim> b"
@@ -136,19 +136,59 @@
on quotient types.
*}
+theorem quot_cond_definition1:
+ "(!!X. f X == \<lfloor>g (pick X)\<rfloor>) ==>
+ (!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x') ==>
+ (!!x x'. x \<sim> x' ==> P x = P x') ==>
+ P a ==> f \<lfloor>a\<rfloor> = \<lfloor>g a\<rfloor>"
+proof -
+ assume cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x'"
+ assume cong_P: "!!x x'. x \<sim> x' ==> P x = P x'"
+ assume P: "P a"
+ assume "!!X. f X == \<lfloor>g (pick X)\<rfloor>"
+ hence "f \<lfloor>a\<rfloor> = \<lfloor>g (pick \<lfloor>a\<rfloor>)\<rfloor>" by (simp only:)
+ also have "\<dots> = \<lfloor>g a\<rfloor>"
+ proof
+ show "g (pick \<lfloor>a\<rfloor>) \<sim> g a"
+ proof (rule cong_g)
+ show "pick \<lfloor>a\<rfloor> \<sim> a" ..
+ hence "P (pick \<lfloor>a\<rfloor>) = P a" by (rule cong_P)
+ also show "P a" .
+ finally show "P (pick \<lfloor>a\<rfloor>)" .
+ qed
+ qed
+ finally show ?thesis .
+qed
+
theorem quot_definition1:
"(!!X. f X == \<lfloor>g (pick X)\<rfloor>) ==>
(!!x x'. x \<sim> x' ==> g x \<sim> g x') ==>
f \<lfloor>a\<rfloor> = \<lfloor>g a\<rfloor>"
proof -
- assume cong: "!!x x'. x \<sim> x' ==> g x \<sim> g x'"
- assume "!!X. f X == \<lfloor>g (pick X)\<rfloor>"
- hence "f \<lfloor>a\<rfloor> = \<lfloor>g (pick \<lfloor>a\<rfloor>)\<rfloor>" by (simp only:)
- also have "\<dots> = \<lfloor>g a\<rfloor>"
+ case antecedent from this refl TrueI
+ show ?thesis by (rule quot_cond_definition1)
+qed
+
+theorem quot_cond_definition2:
+ "(!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>) ==>
+ (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y' ==> g x y \<sim> g x' y') ==>
+ (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y') ==>
+ P a b ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = \<lfloor>g a b\<rfloor>"
+proof -
+ assume cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y' ==> g x y \<sim> g x' y'"
+ assume cong_P: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y'"
+ assume P: "P a b"
+ assume "!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>"
+ hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = \<lfloor>g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)\<rfloor>" by (simp only:)
+ also have "\<dots> = \<lfloor>g a b\<rfloor>"
proof
- show "g (pick \<lfloor>a\<rfloor>) \<sim> g a"
- proof (rule cong)
+ show "g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) \<sim> g a b"
+ proof (rule cong_g)
show "pick \<lfloor>a\<rfloor> \<sim> a" ..
+ moreover show "pick \<lfloor>b\<rfloor> \<sim> b" ..
+ ultimately have "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) = P a b" by (rule cong_P)
+ also show "P a b" .
+ finally show "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" .
qed
qed
finally show ?thesis .
@@ -159,21 +199,10 @@
(!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y \<sim> g x' y') ==>
f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = \<lfloor>g a b\<rfloor>"
proof -
- assume cong: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y \<sim> g x' y'"
- assume "!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>"
- hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = \<lfloor>g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)\<rfloor>" by (simp only:)
- also have "\<dots> = \<lfloor>g a b\<rfloor>"
- proof
- show "g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) \<sim> g a b"
- proof (rule cong)
- show "pick \<lfloor>a\<rfloor> \<sim> a" ..
- show "pick \<lfloor>b\<rfloor> \<sim> b" ..
- qed
- qed
- finally show ?thesis .
+ case antecedent from this refl TrueI
+ show ?thesis by (rule quot_cond_definition2)
qed
-
text {*
\medskip HOL's collection of overloaded standard operations is lifted
to quotient types in the canonical manner.
@@ -186,6 +215,7 @@
instance quot :: (inverse) inverse ..
instance quot :: (power) power ..
instance quot :: (number) number ..
+instance quot :: (ord) ord ..
defs (overloaded)
zero_quot_def: "0 == \<lfloor>0\<rfloor>"
@@ -198,5 +228,7 @@
divide_quot_def: "X / Y == \<lfloor>pick X / pick Y\<rfloor>"
power_quot_def: "X^n == \<lfloor>(pick X)^n\<rfloor>"
number_of_quot_def: "number_of b == \<lfloor>number_of b\<rfloor>"
+ le_quot_def: "X \<le> Y == pick X \<le> pick Y"
+ less_quot_def: "X < Y == pick X < pick Y"
end