src/HOL/Library/Quotient.thy
 changeset 10483 eb93ace45a6e parent 10477 c21bee84cefe child 10491 e4a408728012
```     1.1 --- a/src/HOL/Library/Quotient.thy	Fri Nov 17 18:48:00 2000 +0100
1.2 +++ b/src/HOL/Library/Quotient.thy	Fri Nov 17 18:48:50 2000 +0100
1.3 @@ -1,11 +1,11 @@
1.4  (*  Title:      HOL/Library/Quotient.thy
1.5      ID:         \$Id\$
1.6 -    Author:     Gertrud Bauer and Markus Wenzel, TU Muenchen
1.7 +    Author:     Markus Wenzel, TU Muenchen
1.8  *)
1.9
1.11    \title{Quotient types}
1.12 -  \author{Gertrud Bauer and Markus Wenzel}
1.13 +  \author{Markus Wenzel}
1.14  *}
1.15
1.16  theory Quotient = Main:
1.17 @@ -160,7 +160,7 @@
1.18    qed
1.19  qed
1.20
1.21 -theorem pick_inverse: "\<lfloor>pick A\<rfloor> = A"
1.22 +theorem pick_inverse [intro]: "\<lfloor>pick A\<rfloor> = A"
1.23  proof (cases A)
1.24    fix a assume a: "A = \<lfloor>a\<rfloor>"
1.25    hence "pick A \<sim> a" by (simp only: pick_equiv)
1.26 @@ -170,145 +170,45 @@
1.27
1.28  text {*
1.29   \medskip The following rules support canonical function definitions
1.30 - on quotient types.
1.31 + on quotient types (with up to two arguments).  Note that the
1.32 + stripped-down version without additional conditions is sufficient
1.33 + most of the time.
1.34  *}
1.35
1.36 -theorem quot_cond_function1:
1.37 -  "(!!X. f X == g (pick X)) ==>
1.38 -    (!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x = g x') ==>
1.39 -    (!!x x'. x \<sim> x' ==> P x = P x') ==>
1.40 -  P a ==> f \<lfloor>a\<rfloor> = g a"
1.41 -proof -
1.42 -  assume cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x = g x'"
1.43 -  assume cong_P: "!!x x'. x \<sim> x' ==> P x = P x'"
1.44 -  assume P: "P a"
1.45 -  assume "!!X. f X == g (pick X)"
1.46 -  hence "f \<lfloor>a\<rfloor> = g (pick \<lfloor>a\<rfloor>)" by (simp only:)
1.47 -  also have "\<dots> = g a"
1.48 -  proof (rule cong_g)
1.49 -    show "pick \<lfloor>a\<rfloor> \<sim> a" ..
1.50 -    hence "P (pick \<lfloor>a\<rfloor>) = P a" by (rule cong_P)
1.51 -    also note P
1.52 -    finally show "P (pick \<lfloor>a\<rfloor>)" .
1.53 -  qed
1.54 -  finally show ?thesis .
1.55 -qed
1.56 -
1.57 -theorem quot_function1:
1.58 -  "(!!X. f X == g (pick X)) ==>
1.59 -    (!!x x'. x \<sim> x' ==> g x = g x') ==>
1.60 -    f \<lfloor>a\<rfloor> = g a"
1.61 +theorem quot_cond_function:
1.62 +  "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
1.63 +    (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> P x y ==> P x' y'
1.64 +      ==> g x y = g x' y') ==>
1.65 +    (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> P x y = P x' y') ==>
1.66 +    P a b ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
1.67 +  (is "PROP ?eq ==> PROP ?cong_g ==> PROP ?cong_P ==> _ ==> _")
1.68  proof -
1.69 -  case antecedent from this refl TrueI
1.70 -  show ?thesis by (rule quot_cond_function1)
1.71 -qed
1.72 -
1.73 -theorem quot_cond_operation1:
1.74 -  "(!!X. f X == \<lfloor>g (pick X)\<rfloor>) ==>
1.75 -    (!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x') ==>
1.76 -    (!!x x'. x \<sim> x' ==> P x = P x') ==>
1.77 -  P a ==> f \<lfloor>a\<rfloor> = \<lfloor>g a\<rfloor>"
1.78 -proof -
1.79 -  assume defn: "!!X. f X == \<lfloor>g (pick X)\<rfloor>"
1.80 -  assume "!!x x'. x \<sim> x' ==> P x ==> P x' ==> g x \<sim> g x'"
1.81 -  hence cong_g: "!!x x'. x \<sim> x' ==> P x ==> P x' ==> \<lfloor>g x\<rfloor> = \<lfloor>g x'\<rfloor>" ..
1.82 -  assume "!!x x'. x \<sim> x' ==> P x = P x'" and "P a"
1.83 -  with defn cong_g show ?thesis by (rule quot_cond_function1)
1.84 -qed
1.85 -
1.86 -theorem quot_operation1:
1.87 -  "(!!X. f X == \<lfloor>g (pick X)\<rfloor>) ==>
1.88 -    (!!x x'. x \<sim> x' ==> g x \<sim> g x') ==>
1.89 -    f \<lfloor>a\<rfloor> = \<lfloor>g a\<rfloor>"
1.90 -proof -
1.91 -  case antecedent from this refl TrueI
1.92 -  show ?thesis by (rule quot_cond_operation1)
1.93 -qed
1.94 -
1.95 -theorem quot_cond_function2:
1.96 -  "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
1.97 -    (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
1.98 -      ==> g x y = g x' y') ==>
1.99 -    (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y') ==>
1.100 -    P a b ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
1.101 -proof -
1.102 -  assume cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
1.103 -    ==> g x y = g x' y'"
1.104 -  assume cong_P: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y'"
1.105 -  assume P: "P a b"
1.106 -  assume "!!X Y. f X Y == g (pick X) (pick Y)"
1.107 -  hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" by (simp only:)
1.108 +  assume cong_g: "PROP ?cong_g"
1.109 +    and cong_P: "PROP ?cong_P" and P: "P a b"
1.110 +  assume "PROP ?eq"
1.111 +  hence "f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)"
1.112 +    by (simp only:)
1.113    also have "\<dots> = g a b"
1.114    proof (rule cong_g)
1.115 -    show "pick \<lfloor>a\<rfloor> \<sim> a" ..
1.116 -    moreover show "pick \<lfloor>b\<rfloor> \<sim> b" ..
1.117 -    ultimately have "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) = P a b" by (rule cong_P)
1.118 -    also show "P a b" .
1.119 +    show "\<lfloor>pick \<lfloor>a\<rfloor>\<rfloor> = \<lfloor>a\<rfloor>" ..
1.120 +    moreover
1.121 +    show "\<lfloor>pick \<lfloor>b\<rfloor>\<rfloor> = \<lfloor>b\<rfloor>" ..
1.122 +    ultimately
1.123 +    have "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>) = P a b"
1.124 +      by (rule cong_P)
1.125 +    also show \<dots> .
1.126      finally show "P (pick \<lfloor>a\<rfloor>) (pick \<lfloor>b\<rfloor>)" .
1.127    qed
1.128    finally show ?thesis .
1.129  qed
1.130
1.131 -theorem quot_function2:
1.132 +theorem quot_function:
1.133    "(!!X Y. f X Y == g (pick X) (pick Y)) ==>
1.134 -    (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y = g x' y') ==>
1.135 +    (!!x x' y y'. \<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor> ==> \<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor> ==> g x y = g x' y') ==>
1.136      f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = g a b"
1.137  proof -
1.138    case antecedent from this refl TrueI
1.139 -  show ?thesis by (rule quot_cond_function2)
1.140 -qed
1.141 -
1.142 -theorem quot_cond_operation2:
1.143 -  "(!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>) ==>
1.144 -    (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
1.145 -      ==> g x y \<sim> g x' y') ==>
1.146 -    (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y') ==>
1.147 -    P a b ==> f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = \<lfloor>g a b\<rfloor>"
1.148 -proof -
1.149 -  assume defn: "!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>"
1.150 -  assume "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
1.151 -    ==> g x y \<sim> g x' y'"
1.152 -  hence cong_g: "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y ==> P x' y'
1.153 -    ==> \<lfloor>g x y\<rfloor> = \<lfloor>g x' y'\<rfloor>" ..
1.154 -  assume "!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> P x y = P x' y'" and "P a b"
1.155 -  with defn cong_g show ?thesis by (rule quot_cond_function2)
1.156 -qed
1.157 -
1.158 -theorem quot_operation2:
1.159 -  "(!!X Y. f X Y == \<lfloor>g (pick X) (pick Y)\<rfloor>) ==>
1.160 -    (!!x x' y y'. x \<sim> x' ==> y \<sim> y' ==> g x y \<sim> g x' y') ==>
1.161 -    f \<lfloor>a\<rfloor> \<lfloor>b\<rfloor> = \<lfloor>g a b\<rfloor>"
1.162 -proof -
1.163 -  case antecedent from this refl TrueI
1.164 -  show ?thesis by (rule quot_cond_operation2)
1.165 +  show ?thesis by (rule quot_cond_function)
1.166  qed
1.167
1.168 -text {*
1.169 - \medskip HOL's collection of overloaded standard operations is lifted
1.170 - to quotient types in the canonical manner.
1.171 -*}
1.172 -
1.173 -instance quot :: (zero) zero ..
1.174 -instance quot :: (plus) plus ..
1.175 -instance quot :: (minus) minus ..
1.176 -instance quot :: (times) times ..
1.177 -instance quot :: (inverse) inverse ..
1.178 -instance quot :: (power) power ..
1.179 -instance quot :: (number) number ..
1.180 -instance quot :: (ord) ord ..
1.181 -
1.183 -  zero_quot_def: "0 == \<lfloor>0\<rfloor>"
1.184 -  add_quot_def: "X + Y == \<lfloor>pick X + pick Y\<rfloor>"
1.185 -  diff_quot_def: "X - Y == \<lfloor>pick X - pick Y\<rfloor>"
1.186 -  minus_quot_def: "- X == \<lfloor>- pick X\<rfloor>"
1.187 -  abs_quot_def: "abs X == \<lfloor>abs (pick X)\<rfloor>"
1.188 -  mult_quot_def: "X * Y == \<lfloor>pick X * pick Y\<rfloor>"
1.189 -  inverse_quot_def: "inverse X == \<lfloor>inverse (pick X)\<rfloor>"
1.190 -  divide_quot_def: "X / Y == \<lfloor>pick X / pick Y\<rfloor>"
1.191 -  power_quot_def: "X^n == \<lfloor>(pick X)^n\<rfloor>"
1.192 -  number_of_quot_def: "number_of b == \<lfloor>number_of b\<rfloor>"
1.193 -  le_quot_def: "X \<le> Y == pick X \<le> pick Y"
1.194 -  less_quot_def: "X < Y == pick X < pick Y"
1.195 -
1.196  end
```