# Theory Quotient

theory Quotient
imports Lifting
```(*  Title:      HOL/Quotient.thy
Author:     Cezary Kaliszyk and Christian Urban
*)

section ‹Definition of Quotient Types›

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 ‹
Basic definition for equivalence relations
that are represented by predicates.
›

text ‹Composition of Relations›

abbreviation
rel_conj :: "('a ⇒ 'b ⇒ bool) ⇒ ('b ⇒ 'a ⇒ bool) ⇒ 'a ⇒ 'b ⇒ bool" (infixr "OOO" 75)
where
"r1 OOO r2 ≡ r1 OO r2 OO r1"

lemma eq_comp_r:
shows "((=) OOO R) = R"

context includes lifting_syntax
begin

subsection ‹Quotient Predicate›

definition
"Quotient3 R Abs Rep ⟷
(∀a. Abs (Rep a) = a) ∧ (∀a. R (Rep a) (Rep a)) ∧
(∀r s. R r s ⟷ R r r ∧ R s s ∧ Abs r = Abs s)"

lemma Quotient3I:
assumes "⋀a. Abs (Rep a) = a"
and "⋀a. R (Rep a) (Rep a)"
and "⋀r s. R r s ⟷ R r r ∧ R s s ∧ Abs r = Abs s"
shows "Quotient3 R Abs Rep"
using assms unfolding Quotient3_def by blast

context
fixes R Abs Rep
assumes a: "Quotient3 R Abs Rep"
begin

lemma Quotient3_abs_rep:
"Abs (Rep a) = a"
using a
unfolding Quotient3_def
by simp

lemma Quotient3_rep_reflp:
"R (Rep a) (Rep a)"
using a
unfolding Quotient3_def
by blast

lemma Quotient3_rel:
"R r r ∧ R s s ∧ Abs r = Abs s ⟷ R r s" ― ‹orientation does not loop on rewriting›
using a
unfolding Quotient3_def
by blast

lemma Quotient3_refl1:
"R r s ⟹ R r r"
using a unfolding Quotient3_def
by fast

lemma Quotient3_refl2:
"R r s ⟹ R s s"
using a unfolding Quotient3_def
by fast

lemma Quotient3_rel_rep:
"R (Rep a) (Rep b) ⟷ a = b"
using a
unfolding Quotient3_def
by metis

lemma Quotient3_rep_abs:
"R r r ⟹ R (Rep (Abs r)) r"
using a unfolding Quotient3_def
by blast

lemma Quotient3_rel_abs:
"R r s ⟹ Abs r = Abs s"
using a unfolding Quotient3_def
by blast

lemma Quotient3_symp:
"symp R"
using a unfolding Quotient3_def using sympI by metis

lemma Quotient3_transp:
"transp R"
using a unfolding Quotient3_def using transpI by (metis (full_types))

lemma Quotient3_part_equivp:
"part_equivp R"
by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp part_equivpI)

lemma abs_o_rep:
"Abs ∘ Rep = id"
unfolding fun_eq_iff

lemma equals_rsp:
assumes b: "R xa xb" "R ya yb"
shows "R xa ya = R xb yb"
using b Quotient3_symp Quotient3_transp
by (blast elim: sympE transpE)

lemma rep_abs_rsp:
assumes b: "R x1 x2"
shows "R x1 (Rep (Abs x2))"
using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
by metis

lemma rep_abs_rsp_left:
assumes b: "R x1 x2"
shows "R (Rep (Abs x1)) x2"
using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
by metis

end

lemma identity_quotient3:
"Quotient3 (=) 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 "⋀a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
moreover
have "⋀a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
by (rule rel_funI)
(insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],

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

have "(R1 ===> R2) r s ⟹ (R1 ===> R2) r r" unfolding rel_fun_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 ⟹ (R1 ===> R2) s s" unfolding rel_fun_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 ⟹ (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
apply(auto simp add: rel_fun_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
moreover have "((R1 ===> R2) r r ∧ (R1 ===> R2) s s ∧
(rep1 ---> abs2) r  = (rep1 ---> abs2) s) ⟹ (R1 ===> R2) r s"
apply(auto simp add: rel_fun_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 lambda_prs:
assumes q1: "Quotient3 R1 Abs1 Rep1"
and     q2: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> Abs2) (λx. Rep2 (f (Abs1 x))) = (λx. 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) (λx. (Abs1 ---> Rep2) f x) = (λx. f x)"
unfolding fun_eq_iff
using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
by simp

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 ‹apply_rsp› and
not the primed version›

lemma apply_rspQ3:
fixes f g::"'a ⇒ '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: rel_funE)

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 ⇒ bool"
assumes a: "equivp R"
shows "Ball (Respects R) P = (All P)"
using a
unfolding equivp_def

lemma bex_reg_eqv:
fixes P :: "'a ⇒ bool"
assumes a: "equivp R"
shows "Bex (Respects R) P = (Ex P)"
using a
unfolding equivp_def

lemma ball_reg_right:
assumes a: "⋀x. x ∈ R ⟹ P x ⟶ Q x"
shows "All P ⟶ Ball R Q"
using a by fast

lemma bex_reg_left:
assumes a: "⋀x. x ∈ R ⟹ Q x ⟶ P x"
shows "Bex R Q ⟶ Ex P"
using a by fast

lemma ball_reg_left:
assumes a: "equivp R"
shows "(⋀x. (Q x ⟶ P x)) ⟹ Ball (Respects R) Q ⟶ All P"
using a by (metis equivp_reflp in_respects)

lemma bex_reg_right:
assumes a: "equivp R"
shows "(⋀x. (Q x ⟶ P x)) ⟹ Ex Q ⟶ Bex (Respects R) P"
using a by (metis equivp_reflp in_respects)

lemma ball_reg_eqv_range:
fixes P::"'a ⇒ bool"
and x::"'a"
assumes a: "equivp R2"
shows   "(Ball (Respects (R1 ===> R2)) (λf. P (f x)) = All (λf. P (f x)))"
apply(rule iffI)
apply(rule allI)
apply(drule_tac x="λy. f x" in bspec)
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)) (λf. P (f x)) = Ex (λf. P (f x)))"
apply(auto)
apply(rule_tac x="λy. f x" in bexI)
apply(simp)
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 ∈ 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 ∈ R ⟶ P x ⟶ Q x)"
and     b: "Bex R P"
shows "Bex R Q"
using a b by fast

lemma ball_all_comm:
assumes "⋀y. (∀x∈P. A x y) ⟶ (∀x. B x y)"
shows "(∀x∈P. ∀y. A x y) ⟶ (∀x. ∀y. B x y)"
using assms by auto

lemma bex_ex_comm:
assumes "(∃y. ∃x. A x y) ⟶ (∃y. ∃x∈P. B x y)"
shows "(∃x. ∃y. A x y) ⟶ (∃x∈P. ∃y. B x y)"
using assms by auto

subsection ‹Bounded abstraction›

definition
Babs :: "'a set ⇒ ('a ⇒ 'b) ⇒ 'a ⇒ 'b"
where
"x ∈ p ⟹ 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 rel_fun_def)
apply (subgoal_tac "x ∈ Respects R1 ∧ y ∈ Respects R1")
using a apply (simp add: Babs_def rel_fun_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 (subgoal_tac "Rep1 x ∈ 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(metis Babs_def in_respects  Quotient3_rel[OF q])
done

(* 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 ⟹ 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 ===> (=)) f g"
shows "Ball (Respects R) f = Ball (Respects R) g"
using a by (auto simp add: Ball_def in_respects elim: rel_funE)

lemma bex_rsp:
assumes a: "(R ===> (=)) f g"
shows "(Bex (Respects R) f = Bex (Respects R) g)"
using a by (auto simp add: Bex_def in_respects elim: rel_funE)

lemma bex1_rsp:
assumes a: "(R ===> (=)) f g"
shows "Ex1 (λx. x ∈ Respects R ∧ f x) = Ex1 (λx. x ∈ Respects R ∧ g x)"
using a by (auto elim: rel_funE 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 ‹‹Bex1_rel› quantifier›

definition
Bex1_rel :: "('a ⇒ 'a ⇒ bool) ⇒ ('a ⇒ bool) ⇒ bool"
where
"Bex1_rel R P ⟷ (∃x ∈ Respects R. P x) ∧ (∀x ∈ Respects R. ∀y ∈ Respects R. ((P x ∧ P y) ⟶ (R x y)))"

lemma bex1_rel_aux:
"⟦∀xa ya. R xa ya ⟶ x xa = y ya; Bex1_rel R x⟧ ⟹ Bex1_rel R y"
unfolding Bex1_rel_def
by (metis in_respects)

lemma bex1_rel_aux2:
"⟦∀xa ya. R xa ya ⟶ x xa = y ya; Bex1_rel R y⟧ ⟹ Bex1_rel R x"
unfolding Bex1_rel_def
by (metis in_respects)

lemma bex1_rel_rsp:
assumes a: "Quotient3 R absf repf"
shows "((R ===> (=)) ===> (=)) (Bex1_rel R) (Bex1_rel R)"
unfolding rel_fun_def by (metis bex1_rel_aux bex1_rel_aux2)

lemma ex1_prs:
assumes "Quotient3 R absf repf"
shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
apply (auto simp: Bex1_rel_def Respects_def)
apply (metis Quotient3_def assms)
apply (metis (full_types) Quotient3_def assms)
by (meson Quotient3_rel assms)

lemma bex1_bexeq_reg:
shows "(∃!x∈Respects R. P x) ⟶ (Bex1_rel R (λx. P x))"
by (auto simp add: Ex1_def Bex1_rel_def Bex_def Ball_def in_respects)

lemma bex1_bexeq_reg_eqv:
assumes a: "equivp R"
shows "(∃!x. P x) ⟶ Bex1_rel R P"
using equivp_reflp[OF a]
by (metis (full_types) Bex1_rel_def in_respects)

subsection ‹Various respects and preserve lemmas›

lemma quot_rel_rsp:
assumes a: "Quotient3 R Abs Rep"
shows "(R ===> R ===> (=)) R R"
apply(rule rel_funI)+
by (meson assms equals_rsp)

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)) (∘) = (∘)"
and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) (∘) = (∘)"
using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]

lemma o_rsp:
"((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) (∘) (∘)"
"((=) ===> (R1 ===> (=)) ===> R1 ===> (=)) (∘) (∘)"
by (force elim: rel_funE)+

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]

lemma if_rsp:
assumes q: "Quotient3 R Abs Rep"
shows "((=) ===> 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]

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

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])

end

locale quot_type =
fixes R :: "'a ⇒ 'a ⇒ bool"
and   Abs :: "'a set ⇒ 'b"
and   Rep :: "'b ⇒ 'a set"
assumes equivp: "part_equivp R"
and     rep_prop: "⋀y. ∃x. R x x ∧ Rep y = Collect (R x)"
and     rep_inverse: "⋀x. Abs (Rep x) = x"
and     abs_inverse: "⋀c. (∃x. ((R x x) ∧ (c = Collect (R x)))) ⟹ (Rep (Abs c)) = c"
and     rep_inject: "⋀x y. (Rep x = Rep y) = (x = y)"
begin

definition
abs :: "'a ⇒ 'b"
where
"abs x = Abs (Collect (R x))"

definition
rep :: "'b ⇒ 'a"
where
"rep a = (SOME x. x ∈ Rep a)"

lemma some_collect:
assumes "R r r"
shows "R (SOME x. x ∈ 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 ∈ Rep a) (SOME x. x ∈ 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 ∈ Rep a) x"  using r rep some_collect by metis
then have "R x (SOME x. x ∈ Rep a)" using part_equivp_symp[OF equivp] by fast
then show "R (SOME x. x ∈ Rep a) (SOME x. x ∈ Rep a)"
using part_equivp_transp[OF equivp] by (metis ‹R (SOME x. x ∈ Rep a) x›)
qed
have "Collect (R (SOME x. x ∈ Rep a)) = (Rep a)" by (metis some_collect rep_prop)
then show "Abs (Collect (R (SOME x. x ∈ Rep a))) = a" using rep_inverse by auto
have "R r r ⟹ R s s ⟹ Abs (Collect (R r)) = Abs (Collect (R s)) ⟷ R r = R s"
proof -
assume "R r r" and "R s s"
then have "Abs (Collect (R r)) = Abs (Collect (R s)) ⟷ Collect (R r) = Collect (R s)"
by (metis abs_inverse)
also have "Collect (R r) = Collect (R s) ⟷ (λA x. x ∈ A) (Collect (R r)) = (λA x. x ∈ A) (Collect (R s))"
by rule simp_all
finally show "Abs (Collect (R r)) = Abs (Collect (R s)) ⟷ R r = R s" by simp
qed
then show "R r s ⟷ R r r ∧ R s s ∧ (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 ⇒ 'a ⇒ bool"
fixes Abs1 :: "'a ⇒ 'b" and Rep1 :: "'b ⇒ 'a"
fixes Abs2 :: "'b ⇒ 'c" and Rep2 :: "'c ⇒ 'b"
fixes R2' :: "'a ⇒ 'a ⇒ bool"
fixes R2 :: "'b ⇒ 'b ⇒ bool"
assumes R1: "Quotient3 R1 Abs1 Rep1"
assumes R2: "Quotient3 R2 Abs2 Rep2"
assumes Abs1: "⋀x y. R2' x y ⟹ R1 x x ⟹ R1 y y ⟹ R2 (Abs1 x) (Abs1 y)"
assumes Rep1: "⋀x y. R2 x y ⟹ R2' (Rep1 x) (Rep1 y)"
shows "Quotient3 (R1 OO R2' OO R1) (Abs2 ∘ Abs1) (Rep1 ∘ Rep2)"
proof -
have *: "(R1 OOO R2') r r ∧ (R1 OOO R2') s s ∧ (Abs2 ∘ Abs1) r = (Abs2 ∘ Abs1) s
⟷ (R1 OOO R2') r s" for r s
apply safe
subgoal for a b c d
apply simp
apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
using Quotient3_refl1 R1 rep_abs_rsp apply fastforce
apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI)
apply (metis (full_types) Rep1 Abs1 Quotient3_rel R2  Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1] Quotient3_rel_abs [OF R1])
by (metis Quotient3_rel R1 rep_abs_rsp_left)
subgoal for 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)
by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
subgoal for 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)
by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
subgoal for x y
by simp (metis (full_types) Abs1 Quotient3_rel R1 R2)
done
show ?thesis
apply (rule Quotient3I)
using * apply (simp_all add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
apply (metis Quotient3_rep_reflp R1 R2 Rep1 relcompp.relcompI)
done
qed

lemma OOO_eq_quotient3:
fixes R1 :: "'a ⇒ 'a ⇒ bool"
fixes Abs1 :: "'a ⇒ 'b" and Rep1 :: "'b ⇒ 'a"
fixes Abs2 :: "'b ⇒ 'c" and Rep2 :: "'c ⇒ 'b"
assumes R1: "Quotient3 R1 Abs1 Rep1"
assumes R2: "Quotient3 (=) Abs2 Rep2"
shows "Quotient3 (R1 OOO (=)) (Abs2 ∘ Abs1) (Rep1 ∘ Rep2)"
using assms
by (rule OOO_quotient3) auto

subsection ‹Quotient3 to Quotient›

lemma Quotient3_to_Quotient:
assumes "Quotient3 R Abs Rep"
and "T ≡ λx y. R x x ∧ 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 ≡ λx 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 ∧ R s s ∧ Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
next
show "T = (λx y. R x x ∧ Abs x = y)" using T_def equivp_reflp[OF eR] by simp
qed

subsection ‹ML setup›

text ‹Auxiliary data for the quotient package›

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"
and quot_thm "quotient theorems"
ML_file "Tools/Quotient/quotient_info.ML"

declare [[mapQ3 "fun" = (rel_fun, 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

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 ⇒ bool"
where
"Quot_True x ⟷ True"

lemma
shows QT_all: "Quot_True (All P) ⟹ Quot_True P"
and   QT_ex:  "Quot_True (Ex P) ⟹ Quot_True P"
and   QT_ex1: "Quot_True (Ex1 P) ⟹ Quot_True P"
and   QT_lam: "Quot_True (λx. P x) ⟹ (⋀x. Quot_True (P x))"
and   QT_ext: "(⋀x. Quot_True (a x) ⟹ f x = g x) ⟹ (Quot_True a ⟹ f = g)"

lemma QT_imp: "Quot_True a ≡ Quot_True b"

context includes lifting_syntax
begin

text ‹Tactics for proving the lifted theorems›
ML_file "Tools/Quotient/quotient_tacs.ML"

end

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)

end

```