(*  Title:      HOL/Quotient.thy

     Author:     Cezary Kaliszyk and Christian Urban

 *)

 header {* Definition of Quotient Types *}

 theory Quotient

 imports Plain Hilbert_Choice Equiv_Relations

 uses

   ("Tools/Quotient/quotient_info.ML")

   ("Tools/Quotient/quotient_typ.ML")

   ("Tools/Quotient/quotient_def.ML")

   ("Tools/Quotient/quotient_term.ML")

   ("Tools/Quotient/quotient_tacs.ML")

 begin

 text {*

   Basic definition for equivalence relations

   that are represented by predicates.

 *}

 text {* Composition of Relations *}

 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"

 lemma eq_comp_r:

   shows "((op =) OOO R) = R"

   by (auto simp add: fun_eq_iff)

 subsection {* Respects predicate *}

 definition

   Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"

 where

   "Respects R x = R x x"

 lemma in_respects:

   shows "x \<in> Respects R \<longleftrightarrow> R x x"

   unfolding mem_def Respects_def

   by simp

 subsection {* Function map and function relation *}

 notation map_fun (infixr "--->" 55)

 lemma map_fun_id:

   "(id ---> id) = id"

   by (simp add: fun_eq_iff)

 definition

   fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)

 where

   "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"

 lemma fun_relI [intro]:

   assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"

   shows "(R1 ===> R2) f g"

   using assms by (simp add: fun_rel_def)

 lemma fun_relE:

   assumes "(R1 ===> R2) f g" and "R1 x y"

   obtains "R2 (f x) (g y)"

   using assms by (simp add: fun_rel_def)

 lemma fun_rel_eq:

   shows "((op =) ===> (op =)) = (op =)"

   by (auto simp add: fun_eq_iff elim: fun_relE)

 subsection {* Quotient Predicate *}

 definition

   "Quotient 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)"

 lemma QuotientI:

   assumes "\<And>a. Abs (Rep a) = a"

     and "\<And>a. R (Rep a) (Rep a)"

     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"

   shows "Quotient R Abs Rep"

   using assms unfolding Quotient_def by blast

 lemma Quotient_abs_rep:

   assumes a: "Quotient R Abs Rep"

   shows "Abs (Rep a) = a"

   using a

   unfolding Quotient_def

   by simp

 lemma Quotient_rep_reflp:

   assumes a: "Quotient R Abs Rep"

   shows "R (Rep a) (Rep a)"

   using a

   unfolding Quotient_def

   by blast

 lemma Quotient_rel:

   assumes a: "Quotient R Abs Rep"

   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}

   using a

   unfolding Quotient_def

   by blast

 lemma Quotient_rel_rep:

   assumes a: "Quotient R Abs Rep"

   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"

   using a

   unfolding Quotient_def

   by metis

 lemma Quotient_rep_abs:

   assumes a: "Quotient R Abs Rep"

   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"

   using a unfolding Quotient_def

   by blast

 lemma Quotient_rel_abs:

   assumes a: "Quotient R Abs Rep"

   shows "R r s \<Longrightarrow> Abs r = Abs s"

   using a unfolding Quotient_def

   by blast

 lemma Quotient_symp:

   assumes a: "Quotient R Abs Rep"

   shows "symp R"

   using a unfolding Quotient_def using sympI by metis

 lemma Quotient_transp:

   assumes a: "Quotient R Abs Rep"

   shows "transp R"

   using a unfolding Quotient_def using transpI by metis

 lemma identity_quotient:

   shows "Quotient (op =) id id"

   unfolding Quotient_def id_def

   by blast

 lemma fun_quotient:

   assumes q1: "Quotient R1 abs1 rep1"

   and     q2: "Quotient R2 abs2 rep2"

   shows "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"

 proof -

   have "\<And>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"

     using q1 q2 by (simp add: Quotient_def fun_eq_iff)

   moreover

   have "\<And>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"

     by (rule fun_relI)

       (insert q1 q2 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],

         simp (no_asm) add: Quotient_def, simp)

   moreover

   have "\<And>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>

         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"

     apply(auto simp add: fun_rel_def fun_eq_iff)

     using q1 q2 unfolding Quotient_def

     apply(metis)

     using q1 q2 unfolding Quotient_def

     apply(metis)

     using q1 q2 unfolding Quotient_def

     apply(metis)

     using q1 q2 unfolding Quotient_def

     apply(metis)

     done

   ultimately

   show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"

     unfolding Quotient_def by blast

 qed

 lemma abs_o_rep:

   assumes a: "Quotient R Abs Rep"

   shows "Abs o Rep = id"

   unfolding fun_eq_iff

   by (simp add: Quotient_abs_rep[OF a])

 lemma equals_rsp:

   assumes q: "Quotient R Abs Rep"

   and     a: "R xa xb" "R ya yb"

   shows "R xa ya = R xb yb"

   using a Quotient_symp[OF q] Quotient_transp[OF q]

   by (blast elim: sympE transpE)

 lemma lambda_prs:

   assumes q1: "Quotient R1 Abs1 Rep1"

   and     q2: "Quotient R2 Abs2 Rep2"

   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"

   unfolding fun_eq_iff

   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

   by simp

 lemma lambda_prs1:

   assumes q1: "Quotient R1 Abs1 Rep1"

   and     q2: "Quotient R2 Abs2 Rep2"

   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"

   unfolding fun_eq_iff

   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

   by simp

 lemma rep_abs_rsp:

   assumes q: "Quotient R Abs Rep"

   and     a: "R x1 x2"

   shows "R x1 (Rep (Abs x2))"

   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]

   by metis

 lemma rep_abs_rsp_left:

   assumes q: "Quotient R Abs Rep"

   and     a: "R x1 x2"

   shows "R (Rep (Abs x1)) x2"

   using a Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_rep_reflp[OF q]

   by metis

 text{*

   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 *}

 lemma apply_rsp:

   fixes f g::"'a \<Rightarrow> 'c"

   assumes q: "Quotient R1 Abs1 Rep1"

   and     a: "(R1 ===> R2) f g" "R1 x y"

   shows "R2 (f x) (g y)"

   using a by (auto elim: fun_relE)

 lemma apply_rsp':

   assumes a: "(R1 ===> R2) f g" "R1 x y"

   shows "R2 (f x) (g y)"

   using a by (auto elim: fun_relE)

 subsection {* lemmas for regularisation of ball and bex *}

 lemma ball_reg_eqv:

   fixes P :: "'a \<Rightarrow> bool"

   assumes a: "equivp R"

   shows "Ball (Respects R) P = (All P)"

   using a

   unfolding equivp_def

   by (auto simp add: in_respects)

 lemma bex_reg_eqv:

   fixes P :: "'a \<Rightarrow> bool"

   assumes a: "equivp R"

   shows "Bex (Respects R) P = (Ex P)"

   using a

   unfolding equivp_def

   by (auto simp add: in_respects)

 lemma ball_reg_right:

   assumes a: "\<And>x. R x \<Longrightarrow> P x \<longrightarrow> Q x"

   shows "All P \<longrightarrow> Ball R Q"

   using a by (metis Collect_def Collect_mem_eq)

 lemma bex_reg_left:

   assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"

   shows "Bex R Q \<longrightarrow> Ex P"

   using a by (metis Collect_def Collect_mem_eq)

 lemma ball_reg_left:

   assumes a: "equivp R"

   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"

   using a by (metis equivp_reflp in_respects)

 lemma bex_reg_right:

   assumes a: "equivp R"

   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"

   using a by (metis equivp_reflp in_respects)

 lemma ball_reg_eqv_range:

   fixes P::"'a \<Rightarrow> bool"

   and x::"'a"

   assumes a: "equivp R2"

   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"

   apply(rule iffI)

   apply(rule allI)

   apply(drule_tac x="\<lambda>y. f x" in bspec)

   apply(simp add: in_respects fun_rel_def)

   apply(rule impI)

   using a equivp_reflp_symp_transp[of "R2"]

   apply (auto elim: equivpE reflpE)

   done

 lemma bex_reg_eqv_range:

   assumes a: "equivp R2"

   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"

   apply(auto)

   apply(rule_tac x="\<lambda>y. f x" in bexI)

   apply(simp)

   apply(simp add: Respects_def in_respects fun_rel_def)

   apply(rule impI)

   using a equivp_reflp_symp_transp[of "R2"]

   apply (auto elim: equivpE reflpE)

   done

 (* Next four lemmas are unused *)

 lemma all_reg:

   assumes a: "!x :: 'a. (P x --> Q x)"

   and     b: "All P"

   shows "All Q"

   using a b by (metis)

 lemma ex_reg:

   assumes a: "!x :: 'a. (P x --> Q x)"

   and     b: "Ex P"

   shows "Ex Q"

   using a b by metis

 lemma ball_reg:

   assumes a: "!x :: 'a. (R x --> P x --> Q x)"

   and     b: "Ball R P"

   shows "Ball R Q"

   using a b by (metis Collect_def Collect_mem_eq)

 lemma bex_reg:

   assumes a: "!x :: 'a. (R x --> P x --> Q x)"

   and     b: "Bex R P"

   shows "Bex R Q"

   using a b by (metis Collect_def Collect_mem_eq)

 lemma ball_all_comm:

   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"

   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"

   using assms by auto

 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 *}

 definition

   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"

 where

   "x \<in> p \<Longrightarrow> Babs p m x = m x"

 lemma babs_rsp:

   assumes q: "Quotient R1 Abs1 Rep1"

   and     a: "(R1 ===> R2) f g"

   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"

   apply (auto simp add: Babs_def in_respects fun_rel_def)

   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

   using a apply (simp add: Babs_def fun_rel_def)

   apply (simp add: in_respects fun_rel_def)

   using Quotient_rel[OF q]

   by metis

 lemma babs_prs:

   assumes q1: "Quotient R1 Abs1 Rep1"

   and     q2: "Quotient R2 Abs2 Rep2"

   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"

   apply (rule ext)

   apply (simp add:)

   apply (subgoal_tac "Rep1 x \<in> Respects R1")

   apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

   apply (simp add: in_respects Quotient_rel_rep[OF q1])

   done

 lemma babs_simp:

   assumes q: "Quotient R1 Abs Rep"

   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"

   apply(rule iffI)

   apply(simp_all only: babs_rsp[OF q])

   apply(auto simp add: Babs_def fun_rel_def)

   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")

   apply(metis Babs_def)

   apply (simp add: in_respects)

   using Quotient_rel[OF q]

   by metis

 (* If a user proves that a particular functional relation

    is an equivalence this may be useful in regularising *)

 lemma babs_reg_eqv:

   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"

   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)

 (* 3 lemmas needed for proving repabs_inj *)

 lemma ball_rsp:

   assumes a: "(R ===> (op =)) f g"

   shows "Ball (Respects R) f = Ball (Respects R) g"

   using a by (auto simp add: Ball_def in_respects elim: fun_relE)

 lemma bex_rsp:

   assumes a: "(R ===> (op =)) f g"

   shows "(Bex (Respects R) f = Bex (Respects R) g)"

   using a by (auto simp add: Bex_def in_respects elim: fun_relE)

 lemma bex1_rsp:

   assumes a: "(R ===> (op =)) f g"

   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"

   using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 

 (* 2 lemmas needed for cleaning of quantifiers *)

 lemma all_prs:

   assumes a: "Quotient R absf repf"

   shows "Ball (Respects R) ((absf ---> id) f) = All f"

   using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def

   by metis

 lemma ex_prs:

   assumes a: "Quotient R absf repf"

   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"

   using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def

   by metis

 subsection {* @{text Bex1_rel} quantifier *}

 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)))"

 lemma bex1_rel_aux:

   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"

   unfolding Bex1_rel_def

   apply (erule conjE)+

   apply (erule bexE)

   apply rule

   apply (rule_tac x="xa" in bexI)

   apply metis

   apply metis

   apply rule+

   apply (erule_tac x="xaa" in ballE)

   prefer 2

   apply (metis)

   apply (erule_tac x="ya" in ballE)

   prefer 2

   apply (metis)

   apply (metis in_respects)

   done

 lemma bex1_rel_aux2:

   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"

   unfolding Bex1_rel_def

   apply (erule conjE)+

   apply (erule bexE)

   apply rule

   apply (rule_tac x="xa" in bexI)

   apply metis

   apply metis

   apply rule+

   apply (erule_tac x="xaa" in ballE)

   prefer 2

   apply (metis)

   apply (erule_tac x="ya" in ballE)

   prefer 2

   apply (metis)

   apply (metis in_respects)

   done

 lemma bex1_rel_rsp:

   assumes a: "Quotient R absf repf"

   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"

   apply (simp add: fun_rel_def)

   apply clarify

   apply rule

   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)

   apply (erule bex1_rel_aux2)

   apply assumption

   done

 lemma ex1_prs:

   assumes a: "Quotient R absf repf"

   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"

 apply (simp add:)

 apply (subst Bex1_rel_def)

 apply (subst Bex_def)

 apply (subst Ex1_def)

 apply simp

 apply rule

  apply (erule conjE)+

  apply (erule_tac exE)

  apply (erule conjE)

  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")

   apply (rule_tac x="absf x" in exI)

   apply (simp)

   apply rule+

   using a unfolding Quotient_def

   apply metis

  apply rule+

  apply (erule_tac x="x" in ballE)

   apply (erule_tac x="y" in ballE)

    apply simp

   apply (simp add: in_respects)

  apply (simp add: in_respects)

 apply (erule_tac exE)

  apply rule

  apply (rule_tac x="repf x" in exI)

  apply (simp only: in_respects)

   apply rule

  apply (metis Quotient_rel_rep[OF a])

 using a unfolding Quotient_def apply (simp)

 apply rule+

 using a unfolding Quotient_def in_respects

 apply metis

 done

 lemma bex1_bexeq_reg:

   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"

   apply (simp add: Ex1_def Bex1_rel_def in_respects)

   apply clarify

   apply auto

   apply (rule bexI)

   apply assumption

   apply (simp add: in_respects)

   apply (simp add: in_respects)

   apply auto

   done

 lemma bex1_bexeq_reg_eqv:

   assumes a: "equivp R"

   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"

   using equivp_reflp[OF a]

   apply (intro impI)

   apply (elim ex1E)

   apply (rule mp[OF bex1_bexeq_reg])

   apply (rule_tac a="x" in ex1I)

   apply (subst in_respects)

   apply (rule conjI)

   apply assumption

   apply assumption

   apply clarify

   apply (erule_tac x="xa" in allE)

   apply simp

   done

 subsection {* Various respects and preserve lemmas *}

 lemma quot_rel_rsp:

   assumes a: "Quotient R Abs Rep"

   shows "(R ===> R ===> op =) R R"

   apply(rule fun_relI)+

   apply(rule equals_rsp[OF a])

   apply(assumption)+

   done

 lemma o_prs:

   assumes q1: "Quotient R1 Abs1 Rep1"

   and     q2: "Quotient R2 Abs2 Rep2"

   and     q3: "Quotient R3 Abs3 Rep3"

   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"

   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"

   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]

   by (simp_all add: fun_eq_iff)

 lemma o_rsp:

   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"

   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"

   by (auto intro!: fun_relI elim: fun_relE)

 lemma cond_prs:

   assumes a: "Quotient R absf repf"

   shows "absf (if a then repf b else repf c) = (if a then b else c)"

   using a unfolding Quotient_def by auto

 lemma if_prs:

   assumes q: "Quotient R Abs Rep"

   shows "(id ---> Rep ---> Rep ---> Abs) If = If"

   using Quotient_abs_rep[OF q]

   by (auto simp add: fun_eq_iff)

 lemma if_rsp:

   assumes q: "Quotient R Abs Rep"

   shows "(op = ===> R ===> R ===> R) If If"

   by (auto intro!: fun_relI)

 lemma let_prs:

   assumes q1: "Quotient R1 Abs1 Rep1"

   and     q2: "Quotient R2 Abs2 Rep2"

   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"

   using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]

   by (auto simp add: fun_eq_iff)

 lemma let_rsp:

   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"

   by (auto intro!: fun_relI elim: fun_relE)

 lemma mem_rsp:

   shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"

   by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)

 lemma mem_prs:

   assumes a1: "Quotient R1 Abs1 Rep1"

   and     a2: "Quotient R2 Abs2 Rep2"

   shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"

   by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])

 lemma id_rsp:

   shows "(R ===> R) id id"

   by (auto intro: fun_relI)

 lemma id_prs:

   assumes a: "Quotient R Abs Rep"

   shows "(Rep ---> Abs) id = id"

   by (simp add: fun_eq_iff Quotient_abs_rep [OF a])

 locale quot_type =

   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"

   and   Abs :: "('a \<Rightarrow> bool) \<Rightarrow> 'b"

   and   Rep :: "'b \<Rightarrow> ('a \<Rightarrow> bool)"

   assumes equivp: "part_equivp R"

   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = R x"

   and     rep_inverse: "\<And>x. Abs (Rep x) = x"

   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = R x))) \<Longrightarrow> (Rep (Abs c)) = c"

   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"

 begin

 definition

   abs :: "'a \<Rightarrow> 'b"

 where

   "abs x = Abs (R x)"

 definition

   rep :: "'b \<Rightarrow> 'a"

 where

   "rep a = Eps (Rep a)"

 lemma homeier5:

   assumes a: "R r r"

   shows "Rep (Abs (R r)) = R r"

   apply (subst abs_inverse)

   using a by auto

 theorem homeier6:

   assumes a: "R r r"

   and b: "R s s"

   shows "Abs (R r) = Abs (R s) \<longleftrightarrow> R r = R s"

   by (metis a b homeier5)

 theorem homeier8:

   assumes "R r r"

   shows "R (Eps (R r)) = R r"

   using assms equivp[simplified part_equivp_def]

   apply clarify

   by (metis assms exE_some)

 lemma Quotient:

   shows "Quotient R abs rep"

   unfolding Quotient_def abs_def rep_def

   proof (intro conjI allI)

     fix a r s

     show "Abs (R (Eps (Rep a))) = a"

       using [[metis_new_skolemizer = false]]

       by (metis equivp exE_some part_equivp_def rep_inverse rep_prop)

     show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (R r) = Abs (R s))"

       by (metis homeier6 equivp[simplified part_equivp_def])

     show "R (Eps (Rep a)) (Eps (Rep a))" proof -

       obtain x where r: "R x x" and rep: "Rep a = R x" using rep_prop[of a] by auto

       have "R (Eps (R x)) x" using homeier8 r by simp

       then have "R x (Eps (R x))" using part_equivp_symp[OF equivp] by fast

       then have "R (Eps (R x)) (Eps (R x))" using homeier8[OF r] by simp

       then show "R (Eps (Rep a)) (Eps (Rep a))" using rep by simp

     qed

   qed

 end

 subsection {* ML setup *}

 text {* Auxiliary data for the quotient package *}

 use "Tools/Quotient/quotient_info.ML"

 setup Quotient_Info.setup

 declare [[map "fun" = (map_fun, fun_rel)]]

 lemmas [quot_thm] = fun_quotient

 lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp

 lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs

 lemmas [quot_equiv] = identity_equivp

 text {* Lemmas about simplifying id's. *}

 lemmas [id_simps] =

   id_def[symmetric]

   map_fun_id

   id_apply

   id_o

   o_id

   eq_comp_r

 text {* Translation functions for the lifting process. *}

 use "Tools/Quotient/quotient_term.ML"

 text {* Definitions of the quotient types. *}

 use "Tools/Quotient/quotient_typ.ML"

 text {* Definitions for quotient constants. *}

 use "Tools/Quotient/quotient_def.ML"

 text {*

   An auxiliary constant for recording some information

   about the lifted theorem in a tactic.

 *}

 definition

   Quot_True :: "'a \<Rightarrow> bool"

 where

   "Quot_True x \<longleftrightarrow> True"

 lemma

   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"

   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"

   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"

   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"

   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"

   by (simp_all add: Quot_True_def ext)

 lemma QT_imp: "Quot_True a \<equiv> Quot_True b"

   by (simp add: Quot_True_def)

 text {* Tactics for proving the lifted theorems *}

 use "Tools/Quotient/quotient_tacs.ML"

 subsection {* Methods / Interface *}

 method_setup lifting =

   {* Attrib.thms >> (fn thms => fn ctxt => 

        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}

   {* lift theorems to quotient types *}

 method_setup lifting_setup =

   {* Attrib.thm >> (fn thm => fn ctxt => 

        SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}

   {* set up the three goals for the quotient lifting procedure *}

 method_setup descending =

   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}

   {* decend theorems to the raw level *}

 method_setup descending_setup =

   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}

   {* set up the three goals for the decending theorems *}

 method_setup regularize =

   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}

   {* prove the regularization goals from the quotient lifting procedure *}

 method_setup injection =

   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}

   {* prove the rep/abs injection goals from the quotient lifting procedure *}

 method_setup cleaning =

   {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}

   {* prove the cleaning goals from the quotient lifting procedure *}

 attribute_setup quot_lifted =

   {* Scan.succeed Quotient_Tacs.lifted_attrib *}

   {* lift theorems to quotient types *}

 no_notation

   rel_conj (infixr "OOO" 75) and

   map_fun (infixr "--->" 55) and

   fun_rel (infixr "===>" 55)

 end