src/HOL/Quotient.thy
 author webertj Fri, 19 Oct 2012 15:12:52 +0200 changeset 49962 a8cc904a6820 parent 48891 c0eafbd55de3 child 51112 da97167e03f7 permissions -rw-r--r--
Renamed {left,right}_distrib to distrib_{right,left}.
```
(*  Title:      HOL/Quotient.thy
Author:     Cezary Kaliszyk and Christian Urban
*)

header {* Definition of Quotient Types *}

theory Quotient
imports Plain Hilbert_Choice Equiv_Relations Lifting
keywords
"print_quotmapsQ3" "print_quotientsQ3" "print_quotconsts" :: diag and
"quotient_type" :: thy_goal and "/" and
"quotient_definition" :: thy_goal
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 {* Quotient Predicate *}

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)"

lemma Quotient3I:
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 "Quotient3 R Abs Rep"
using assms unfolding Quotient3_def by blast

lemma Quotient3_abs_rep:
assumes a: "Quotient3 R Abs Rep"
shows "Abs (Rep a) = a"
using a
unfolding Quotient3_def
by simp

lemma Quotient3_rep_reflp:
assumes a: "Quotient3 R Abs Rep"
shows "R (Rep a) (Rep a)"
using a
unfolding Quotient3_def
by blast

lemma Quotient3_rel:
assumes a: "Quotient3 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 Quotient3_def
by blast

lemma Quotient3_refl1:
assumes a: "Quotient3 R Abs Rep"
shows "R r s \<Longrightarrow> R r r"
using a unfolding Quotient3_def
by fast

lemma Quotient3_refl2:
assumes a: "Quotient3 R Abs Rep"
shows "R r s \<Longrightarrow> R s s"
using a unfolding Quotient3_def
by fast

lemma Quotient3_rel_rep:
assumes a: "Quotient3 R Abs Rep"
shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
using a
unfolding Quotient3_def
by metis

lemma Quotient3_rep_abs:
assumes a: "Quotient3 R Abs Rep"
shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
using a unfolding Quotient3_def
by blast

lemma Quotient3_rel_abs:
assumes a: "Quotient3 R Abs Rep"
shows "R r s \<Longrightarrow> Abs r = Abs s"
using a unfolding Quotient3_def
by blast

lemma Quotient3_symp:
assumes a: "Quotient3 R Abs Rep"
shows "symp R"
using a unfolding Quotient3_def using sympI by metis

lemma Quotient3_transp:
assumes a: "Quotient3 R Abs Rep"
shows "transp R"
using a unfolding Quotient3_def using transpI by (metis (full_types))

lemma Quotient3_part_equivp:
assumes a: "Quotient3 R Abs Rep"
shows "part_equivp R"
by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp a part_equivpI)

lemma identity_quotient3:
shows "Quotient3 (op =) id id"
unfolding Quotient3_def id_def
by blast

lemma fun_quotient3:
assumes q1: "Quotient3 R1 abs1 rep1"
and     q2: "Quotient3 R2 abs2 rep2"
shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
proof -
have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
moreover
have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
by (rule fun_relI)
(insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
simp (no_asm) add: Quotient3_def, simp)

moreover
{
fix r s
have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
(rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
proof -

have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding fun_rel_def
using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
by (metis (full_types) part_equivp_def)
moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding fun_rel_def
using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2]
by (metis (full_types) part_equivp_def)
moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
(rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def
by (metis map_fun_apply)

ultimately show ?thesis by blast
qed
}
ultimately show ?thesis by (intro Quotient3I) (assumption+)
qed

lemma abs_o_rep:
assumes a: "Quotient3 R Abs Rep"
shows "Abs o Rep = id"
unfolding fun_eq_iff
by (simp add: Quotient3_abs_rep[OF a])

lemma equals_rsp:
assumes q: "Quotient3 R Abs Rep"
and     a: "R xa xb" "R ya yb"
shows "R xa ya = R xb yb"
using a Quotient3_symp[OF q] Quotient3_transp[OF q]
by (blast elim: sympE transpE)

lemma lambda_prs:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
unfolding fun_eq_iff
using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
by simp

lemma lambda_prs1:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 R2 Abs2 Rep2"
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

lemma rep_abs_rsp:
assumes q: "Quotient3 R Abs Rep"
and     a: "R x1 x2"
shows "R x1 (Rep (Abs x2))"
using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
by metis

lemma rep_abs_rsp_left:
assumes q: "Quotient3 R Abs Rep"
and     a: "R x1 x2"
shows "R (Rep (Abs x1)) x2"
using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_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_rspQ3:
fixes f g::"'a \<Rightarrow> 'c"
assumes q: "Quotient3 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_rspQ3'':
assumes "Quotient3 R Abs Rep"
and "(R ===> S) f f"
shows "S (f (Rep x)) (f (Rep x))"
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 *}

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. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
shows "All P \<longrightarrow> Ball R Q"
using a by fast

lemma bex_reg_left:
assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
shows "Bex R Q \<longrightarrow> Ex P"
using a by fast

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 fast

lemma ex_reg:
assumes a: "!x :: 'a. (P x --> Q x)"
and     b: "Ex P"
shows "Ex Q"
using a b by fast

lemma ball_reg:
assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
and     b: "Ball R P"
shows "Ball R Q"
using a b by fast

lemma bex_reg:
assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
and     b: "Bex R P"
shows "Bex R Q"
using a b by fast

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: "Quotient3 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 Quotient3_rel[OF q]
by metis

lemma babs_prs:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 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 Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
apply (simp add: in_respects Quotient3_rel_rep[OF q1])
done

lemma babs_simp:
assumes q: "Quotient3 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 Quotient3_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: "Quotient3 R absf repf"
shows "Ball (Respects R) ((absf ---> id) f) = All f"
using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
by metis

lemma ex_prs:
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 *}

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: "Quotient3 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: "Quotient3 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 Quotient3_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 Quotient3_rel_rep[OF a])
using a unfolding Quotient3_def apply (simp)
apply rule+
using a unfolding Quotient3_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: "Quotient3 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: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 R2 Abs2 Rep2"
and     q3: "Quotient3 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 Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_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 (force elim: fun_relE)+

lemma cond_prs:
assumes a: "Quotient3 R absf repf"
shows "absf (if a then repf b else repf c) = (if a then b else c)"
using a unfolding Quotient3_def by auto

lemma if_prs:
assumes q: "Quotient3 R Abs Rep"
shows "(id ---> Rep ---> Rep ---> Abs) If = If"
using Quotient3_abs_rep[OF q]
by (auto simp add: fun_eq_iff)

lemma if_rsp:
assumes q: "Quotient3 R Abs Rep"
shows "(op = ===> R ===> R ===> R) If If"
by force

lemma let_prs:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 R2 Abs2 Rep2"
shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
by (auto simp add: fun_eq_iff)

lemma let_rsp:
shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
by (force elim: fun_relE)

lemma id_rsp:
shows "(R ===> R) id id"
by auto

lemma id_prs:
assumes a: "Quotient3 R Abs Rep"
shows "(Rep ---> Abs) id = id"
by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])

locale quot_type =
fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
and   Abs :: "'a set \<Rightarrow> 'b"
and   Rep :: "'b \<Rightarrow> 'a set"
assumes equivp: "part_equivp R"
and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
and     rep_inverse: "\<And>x. Abs (Rep x) = x"
and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (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 (Collect (R x))"

definition
rep :: "'b \<Rightarrow> 'a"
where
"rep a = (SOME x. x \<in> Rep a)"

lemma some_collect:
assumes "R r r"
shows "R (SOME x. x \<in> Collect (R r)) = R r"
apply simp
by (metis assms exE_some equivp[simplified part_equivp_def])

lemma Quotient:
shows "Quotient3 R abs rep"
unfolding Quotient3_def abs_def rep_def
proof (intro conjI allI)
fix a r s
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`)
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 -
assume "R r r" and "R s s"
then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
by (metis abs_inverse)
also have "Collect (R r) = Collect (R s) \<longleftrightarrow> (\<lambda>A x. x \<in> A) (Collect (R r)) = (\<lambda>A x. x \<in> A) (Collect (R s))"
by rule simp_all
finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
qed
then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
using equivp[simplified part_equivp_def] by metis
qed

end

subsection {* Quotient composition *}

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"
fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
assumes R1: "Quotient3 R1 Abs1 Rep1"
assumes R2: "Quotient3 R2 Abs2 Rep2"
assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
apply (rule Quotient3I)
apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
apply simp
apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
apply (rule Quotient3_rep_reflp [OF R1])
apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
apply (rule Quotient3_rep_reflp [OF R1])
apply (rule Rep1)
apply (rule Quotient3_rep_reflp [OF R2])
apply safe
apply (rename_tac x y)
apply (drule Abs1)
apply (erule Quotient3_refl2 [OF R1])
apply (erule Quotient3_refl1 [OF R1])
apply (drule Quotient3_refl1 [OF R2], drule Rep1)
apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
apply (erule relcomppI)
apply (erule Quotient3_symp [OF R1, THEN sympD])
apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
apply (rule conjI, erule Quotient3_refl1 [OF R1])
apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
apply (subst Quotient3_abs_rep [OF R1])
apply (erule Quotient3_rel_abs [OF R1])
apply (rename_tac x y)
apply (drule Abs1)
apply (erule Quotient3_refl2 [OF R1])
apply (erule Quotient3_refl1 [OF R1])
apply (drule Quotient3_refl2 [OF R2], drule Rep1)
apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
apply (erule relcomppI)
apply (erule Quotient3_symp [OF R1, THEN sympD])
apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
apply (rule conjI, erule Quotient3_refl2 [OF R1])
apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
apply (subst Quotient3_abs_rep [OF R1])
apply (erule Quotient3_rel_abs [OF R1, THEN sym])
apply simp
apply (rule Quotient3_rel_abs [OF R2])
apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
apply (erule Abs1)
apply (erule Quotient3_refl2 [OF R1])
apply (erule Quotient3_refl1 [OF R1])
apply (rename_tac a b c d)
apply simp
apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
apply (rule conjI, erule Quotient3_refl1 [OF R1])
apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
apply (erule Quotient3_refl2 [OF R1])
apply (rule Rep1)
apply (drule Abs1)
apply (erule Quotient3_refl2 [OF R1])
apply (erule Quotient3_refl1 [OF R1])
apply (drule Abs1)
apply (erule Quotient3_refl2 [OF R1])
apply (erule Quotient3_refl1 [OF R1])
apply (drule Quotient3_rel_abs [OF R1])
apply (drule Quotient3_rel_abs [OF R1])
apply (drule Quotient3_rel_abs [OF R1])
apply (drule Quotient3_rel_abs [OF R1])
apply simp
apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
apply simp
done

lemma OOO_eq_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 R1: "Quotient3 R1 Abs1 Rep1"
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 *}

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"
using assms unfolding Quotient3_def by (intro QuotientI) blast+

lemma Quotient3_to_Quotient_equivp:
assumes q: "Quotient3 R Abs Rep"
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
and eR: "equivp R"
shows "Quotient R Abs Rep T"
proof (intro QuotientI)
fix a
show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
next
fix a
show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
next
fix r s
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 *}

ML_file "Tools/Quotient/quotient_info.ML"
setup Quotient_Info.setup

declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]

lemmas [quot_thm] = fun_quotient3
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. *}
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. *}
ML_file "Tools/Quotient/quotient_term.ML"

text {* Definitions of the quotient types. *}
ML_file "Tools/Quotient/quotient_type.ML"

text {* Definitions for quotient constants. *}
ML_file "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 *}
ML_file "Tools/Quotient/quotient_tacs.ML"

subsection {* Methods / Interface *}

method_setup lifting =
{* Attrib.thms >> (fn thms => fn ctxt =>
SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
{* lift theorems to quotient types *}

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 *}

method_setup descending =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
{* decend theorems to the raw level *}

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 *}

method_setup partiality_descending =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
{* decend theorems to the raw level *}

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 *}

method_setup regularize =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
{* prove the regularization goals from the quotient lifting procedure *}

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 *}

method_setup cleaning =
{* Scan.succeed (fn ctxt => SIMPLE_METHOD' (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

```