src/HOL/Quotient.thy

 (*  Title:      HOL/Quotient.thy
     Author:     Cezary Kaliszyk and Christian Urban
 *)
 
-section {* Definition of Quotient Types *}
+section \<open>Definition of Quotient Types\<close>
 
 theory Quotient
 imports Lifting
 keywords
   "print_quotmapsQ3" "print_quotientsQ3" "print_quotconsts" :: diag and
   "quotient_type" :: thy_goal and "/" and
   "quotient_definition" :: thy_goal
 begin
 
-text {*
+text \<open>
   Basic definition for equivalence relations
   that are represented by predicates.
-*}
-
-text {* Composition of Relations *}
+\<close>
+
+text \<open>Composition of Relations\<close>
 
 abbreviation
   rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
 where
   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"

 
 context
 begin
 interpretation lifting_syntax .
 
-subsection {* Quotient Predicate *}
+subsection \<open>Quotient Predicate\<close>
 
 definition
   "Quotient3 R Abs Rep \<longleftrightarrow>
      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"

   using a
   unfolding Quotient3_def
   by blast
 
 lemma Quotient3_rel:
-  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
+  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- \<open>orientation does not loop on rewriting\<close>
   using a
   unfolding Quotient3_def
   by blast
 
 lemma Quotient3_refl1: 

   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
   unfolding fun_eq_iff
   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
   by simp
 
-text{*
+text\<open>
   In the following theorem R1 can be instantiated with anything,
   but we know some of the types of the Rep and Abs functions;
   so by solving Quotient assumptions we can get a unique R1 that
   will be provable; which is why we need to use @{text apply_rsp} and
-  not the primed version *}
+  not the primed version\<close>
 
 lemma apply_rspQ3:
   fixes f g::"'a \<Rightarrow> 'c"
   assumes q: "Quotient3 R1 Abs1 Rep1"
   and     a: "(R1 ===> R2) f g" "R1 x y"

 proof -
   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
   then show ?thesis using assms(2) by (auto intro: apply_rsp')
 qed
 
-subsection {* lemmas for regularisation of ball and bex *}
+subsection \<open>lemmas for regularisation of ball and bex\<close>
 
 lemma ball_reg_eqv:
   fixes P :: "'a \<Rightarrow> bool"
   assumes a: "equivp R"
   shows "Ball (Respects R) P = (All P)"

 lemma bex_ex_comm:
   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
   using assms by auto
 
-subsection {* Bounded abstraction *}
+subsection \<open>Bounded abstraction\<close>
 
 definition
   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
 where
   "x \<in> p \<Longrightarrow> Babs p m x = m x"

   assumes a: "Quotient3 R absf repf"
   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
   using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
   by metis
 
-subsection {* @{text Bex1_rel} quantifier *}
+subsection \<open>@{text Bex1_rel} quantifier\<close>
 
 definition
   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
 where
   "Bex1_rel R P \<longleftrightarrow> (\<exists>x \<in> Respects R. P x) \<and> (\<forall>x \<in> Respects R. \<forall>y \<in> Respects R. ((P x \<and> P y) \<longrightarrow> (R x y)))"

   apply clarify
   apply (erule_tac x="xa" in allE)
   apply simp
   done
 
-subsection {* Various respects and preserve lemmas *}
+subsection \<open>Various respects and preserve lemmas\<close>
 
 lemma quot_rel_rsp:
   assumes a: "Quotient3 R Abs Rep"
   shows "(R ===> R ===> op =) R R"
   apply(rule rel_funI)+

     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
-        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
+        using part_equivp_transp[OF equivp] by (metis \<open>R (SOME x. x \<in> Rep a) x\<close>)
     qed
     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
     proof -

       using equivp[simplified part_equivp_def] by metis
     qed
 
 end
 
-subsection {* Quotient composition *}
+subsection \<open>Quotient composition\<close>
 
 lemma OOO_quotient3:
   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"

   assumes R2: "Quotient3 op= Abs2 Rep2"
   shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
 using assms
 by (rule OOO_quotient3) auto
 
-subsection {* Quotient3 to Quotient *}
+subsection \<open>Quotient3 to Quotient\<close>
 
 lemma Quotient3_to_Quotient:
 assumes "Quotient3 R Abs Rep"
 and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
 shows "Quotient R Abs Rep T"

   show "R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
 next
   show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
 qed
 
-subsection {* ML setup *}
-
-text {* Auxiliary data for the quotient package *}
+subsection \<open>ML setup\<close>
+
+text \<open>Auxiliary data for the quotient package\<close>
 
 named_theorems quot_equiv "equivalence relation theorems"
   and quot_respect "respectfulness theorems"
   and quot_preserve "preservation theorems"
   and id_simps "identity simp rules for maps"

 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
 lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
 lemmas [quot_equiv] = identity_equivp
 
 
-text {* Lemmas about simplifying id's. *}
+text \<open>Lemmas about simplifying id's.\<close>
 lemmas [id_simps] =
   id_def[symmetric]
   map_fun_id
   id_apply
   id_o
   o_id
   eq_comp_r
   vimage_id
 
-text {* Translation functions for the lifting process. *}
+text \<open>Translation functions for the lifting process.\<close>
 ML_file "Tools/Quotient/quotient_term.ML"
 
 
-text {* Definitions of the quotient types. *}
+text \<open>Definitions of the quotient types.\<close>
 ML_file "Tools/Quotient/quotient_type.ML"
 
 
-text {* Definitions for quotient constants. *}
+text \<open>Definitions for quotient constants.\<close>
 ML_file "Tools/Quotient/quotient_def.ML"
 
 
-text {*
+text \<open>
   An auxiliary constant for recording some information
   about the lifted theorem in a tactic.
-*}
+\<close>
 definition
   Quot_True :: "'a \<Rightarrow> bool"
 where
   "Quot_True x \<longleftrightarrow> True"
 

 
 context 
 begin
 interpretation lifting_syntax .
 
-text {* Tactics for proving the lifted theorems *}
+text \<open>Tactics for proving the lifted theorems\<close>
 ML_file "Tools/Quotient/quotient_tacs.ML"
 
 end
 
-subsection {* Methods / Interface *}
+subsection \<open>Methods / Interface\<close>
 
 method_setup lifting =
-  {* Attrib.thms >> (fn thms => fn ctxt => 
-       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
-  {* lift theorems to quotient types *}
+  \<open>Attrib.thms >> (fn thms => fn ctxt => 
+       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms))\<close>
+  \<open>lift theorems to quotient types\<close>
 
 method_setup lifting_setup =
-  {* Attrib.thm >> (fn thm => fn ctxt => 
-       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
-  {* set up the three goals for the quotient lifting procedure *}
+  \<open>Attrib.thm >> (fn thm => fn ctxt => 
+       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm))\<close>
+  \<open>set up the three goals for the quotient lifting procedure\<close>
 
 method_setup descending =
-  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
-  {* decend theorems to the raw level *}
+  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt []))\<close>
+  \<open>decend theorems to the raw level\<close>
 
 method_setup descending_setup =
-  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
-  {* set up the three goals for the decending theorems *}
+  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt []))\<close>
+  \<open>set up the three goals for the decending theorems\<close>
 
 method_setup partiality_descending =
-  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
-  {* decend theorems to the raw level *}
+  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt []))\<close>
+  \<open>decend theorems to the raw level\<close>
 
 method_setup partiality_descending_setup =
-  {* Scan.succeed (fn ctxt => 
-       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
-  {* set up the three goals for the decending theorems *}
+  \<open>Scan.succeed (fn ctxt => 
+       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt []))\<close>
+  \<open>set up the three goals for the decending theorems\<close>
 
 method_setup regularize =
-  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
-  {* prove the regularization goals from the quotient lifting procedure *}
+  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt))\<close>
+  \<open>prove the regularization goals from the quotient lifting procedure\<close>
 
 method_setup injection =
-  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
-  {* prove the rep/abs injection goals from the quotient lifting procedure *}
+  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt))\<close>
+  \<open>prove the rep/abs injection goals from the quotient lifting procedure\<close>
 
 method_setup cleaning =
-  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
-  {* prove the cleaning goals from the quotient lifting procedure *}
+  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt))\<close>
+  \<open>prove the cleaning goals from the quotient lifting procedure\<close>
 
 attribute_setup quot_lifted =
-  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
-  {* lift theorems to quotient types *}
+  \<open>Scan.succeed Quotient_Tacs.lifted_attrib\<close>
+  \<open>lift theorems to quotient types\<close>
 
 no_notation
   rel_conj (infixr "OOO" 75)
 
 end