--- a/src/HOL/Library/Quotient.thy Fri Nov 17 18:48:00 2000 +0100
+++ b/src/HOL/Library/Quotient.thy Fri Nov 17 18:48:50 2000 +0100
@@ -1,11 +1,11 @@
(* Title: HOL/Library/Quotient.thy
ID: $Id$
- Author: Gertrud Bauer and Markus Wenzel, TU Muenchen
+ Author: Markus Wenzel, TU Muenchen
*)
header {*
\title{Quotient types}
- \author{Gertrud Bauer and Markus Wenzel}
+ \author{Markus Wenzel}
*}
theory Quotient = Main:
@@ -160,7 +160,7 @@
qed
qed
-theorem pick_inverse: "\<lfloor>pick A\<rfloor> = A"
+theorem pick_inverse [intro]: "\<lfloor>pick A\<rfloor> = A"
proof (cases A)
fix a assume a: "A = \<lfloor>a\<rfloor>"
hence "pick A \<sim> a" by (simp only: pick_equiv)
@@ -170,145 +170,45 @@
text {*
\medskip The following rules support canonical function definitions
- on quotient types.
+ on quotient types (with up to two arguments). Note that the
+ stripped-down version without additional conditions is sufficient
+ most of the time.
*}
-theorem quot_cond_function1:
- "(!!X. f X == g (pick X)) ==>
- (!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x = g x') ==>
- (!!x x'. x \<sim> x' ==> P x = P x') ==>
- P a ==> f \<lfloor>a\<rfloor> = g a"
-proof -
- assume cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x = g x'"
- assume cong_P: "!!x x'. x \<sim> x' ==> P x = P x'"
- assume P: "P a"
- assume "!!X. f X == g (pick X)"
- hence "f \<lfloor>a\<rfloor> = g (pick \<lfloor>a\<rfloor>)" by (simp only:)
- also have "\<dots> = 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 note P
- finally show "P (pick \<lfloor>a\<rfloor>)" .
- qed
- finally show ?thesis .
-qed
-
-theorem quot_function1:
- "(!!X. f X == g (pick X)) ==>
- (!!x x'. x \<sim> x' ==> g x = g x') ==>
- f \<lfloor>a\<rfloor> = g a"
+theorem quot_cond_function:
+ "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
+ (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> P x y ==> P x' y'
+ ==> g x y = g x' y') ==>
+ (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> P x y = P x' y') ==>
+ P a b ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
+ (is "PROP ?eq ==> PROP ?cong_g ==> PROP ?cong_P ==> _ ==> _")
proof -
- case antecedent from this refl TrueI
- show ?thesis by (rule quot_cond_function1)
-qed
-
-theorem quot_cond_operation1:
- "(!!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 defn: "!!X. f X == \<lfloor>g (pick X)\<rfloor>"
- assume "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x'"
- hence cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> \<lfloor>g x\<rfloor> = \<lfloor>g x'\<rfloor>" ..
- assume "!!x x'. x \<sim> x' ==> P x = P x'" and "P a"
- with defn cong_g show ?thesis by (rule quot_cond_function1)
-qed
-
-theorem quot_operation1:
- "(!!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 -
- case antecedent from this refl TrueI
- show ?thesis by (rule quot_cond_operation1)
-qed
-
-theorem quot_cond_function2:
- "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
- (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
- ==> g x y = 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> = g a b"
-proof -
- assume cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
- ==> g x y = 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 == g (pick X) (pick Y)"
- hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)
+ assume cong_g: "PROP ?cong_g"
+ and cong_P: "PROP ?cong_P" and P: "P a b"
+ assume "PROP ?eq"
+ hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)"
+ by (simp only:)
also have "\<dots> = 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" .
+ show "\<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> = \<lfloor>a\<rfloor>" ..
+ moreover
+ show "\<lfloor>pick \<lfloor>b\<rfloor>\<rfloor> = \<lfloor>b\<rfloor>" ..
+ ultimately
+ have "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) = P a b"
+ by (rule cong_P)
+ also show \<dots> .
finally show "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" .
qed
finally show ?thesis .
qed
-theorem quot_function2:
+theorem quot_function:
"(!!X Y. f X Y == g (pick X) (pick Y)) ==>
- (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y = g x' y') ==>
+ (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>
f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
proof -
case antecedent from this refl TrueI
- show ?thesis by (rule quot_cond_function2)
-qed
-
-theorem quot_cond_operation2:
- "(!!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 defn: "!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>"
- assume "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
- ==> g x y \<sim> g x' y'"
- hence cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
- ==> \<lfloor>g x y\<rfloor> = \<lfloor>g x' y'\<rfloor>" ..
- assume "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y'" and "P a b"
- with defn cong_g show ?thesis by (rule quot_cond_function2)
-qed
-
-theorem quot_operation2:
- "(!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>) ==>
- (!!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 -
- case antecedent from this refl TrueI
- show ?thesis by (rule quot_cond_operation2)
+ show ?thesis by (rule quot_cond_function)
qed
-text {*
- \medskip HOL's collection of overloaded standard operations is lifted
- to quotient types in the canonical manner.
-*}
-
-instance quot :: (zero) zero ..
-instance quot :: (plus) plus ..
-instance quot :: (minus) minus ..
-instance quot :: (times) times ..
-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>"
- add_quot_def: "X + Y == \<lfloor>pick X + pick Y\<rfloor>"
- diff_quot_def: "X - Y == \<lfloor>pick X - pick Y\<rfloor>"
- minus_quot_def: "- X == \<lfloor>- pick X\<rfloor>"
- abs_quot_def: "abs X == \<lfloor>abs (pick X)\<rfloor>"
- mult_quot_def: "X * Y == \<lfloor>pick X * pick Y\<rfloor>"
- inverse_quot_def: "inverse X == \<lfloor>inverse (pick X)\<rfloor>"
- 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