# Theory Quotient

```(*  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_defn and "/" and
"quotient_definition" :: thy_goal_defn and
"copy_bnf" :: thy_defn and
"lift_bnf" :: thy_goal_defn
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 "(rep1 ---> abs2) ((abs1 ---> rep2) a) = a" for a
using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
moreover
have "(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)" for a
by (rule rel_funI)
(use q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2]
in ‹simp (no_asm) add: Quotient3_def, simp›)
moreover
have "(R1 ===> R2) r s = ((R1 ===> R2) r r ∧ (R1 ===> R2) s s ∧
(rep1 ---> abs2) r  = (rep1 ---> abs2) s)" for r 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"
by (auto simp add: rel_fun_def fun_eq_iff)
(use q1 q2 in ‹unfold Quotient3_def, metis›)
moreover have "((R1 ===> R2) r r ∧ (R1 ===> R2) s s ∧
(rep1 ---> abs2) r  = (rep1 ---> abs2) s) ⟹ (R1 ===> R2) r s"
by (auto simp add: rel_fun_def fun_eq_iff)
(use q1 q2 in ‹unfold Quotient3_def, 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)))"
proof (intro allI iffI)
fix f
assume "∀f ∈ Respects (R1 ===> R2). P (f x)"
moreover have "(λy. f x) ∈ Respects (R1 ===> R2)"
using a equivp_reflp_symp_transp[of "R2"]
by(auto simp add: in_respects rel_fun_def elim: equivpE reflpE)
ultimately show "P (f x)"
by auto
qed auto

lemma bex_reg_eqv_range:
assumes a: "equivp R2"
shows   "(Bex (Respects (R1 ===> R2)) (λf. P (f x)) = Ex (λf. P (f x)))"
proof -
{ fix f
assume "P (f x)"
have "(λy. f x) ∈ Respects (R1 ===> R2)"
using a equivp_reflp_symp_transp[of "R2"]
by (auto simp add: Respects_def in_respects rel_fun_def elim: equivpE reflpE) }
then show ?thesis
by auto
qed

(* 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)"
proof
fix x y
assume "R1 x y"
then have "x ∈ Respects R1 ∧ y ∈ Respects R1"
unfolding in_respects rel_fun_def using Quotient3_rel[OF q]by metis
then show "R2 (Babs (Respects R1) f x) (Babs (Respects R1) g y)"
using ‹R1 x y› a by (simp add: Babs_def rel_fun_def)
qed

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"
proof -
{ fix x
have "Rep1 x ∈ Respects R1"
by (simp add: in_respects Quotient3_rel_rep[OF q1])
then have "Abs2 (Babs (Respects R1) ((Abs1 ---> Rep2) f) (Rep1 x)) = f x"
by (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
}
then show ?thesis
by force
qed

lemma babs_simp:
assumes q: "Quotient3 R1 Abs Rep"
shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding rel_fun_def by (metis Babs_def in_respects  Quotient3_rel[OF q])

text ‹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)

text ‹Two 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"
(is "?lhs = ?rhs")
using assms
apply (auto simp add: Bex1_rel_def Respects_def)
by (metis (full_types) Quotient3_def)+

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 iffI) 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
proof (intro iffI conjI; clarify)
show "(R1 OOO R2') r s"
if r: "R1 r a" "R2' a b" "R1 b r" and eq: "(Abs2 ∘ Abs1) r = (Abs2 ∘ Abs1) s"
and s: "R1 s c" "R2' c d" "R1 d s" for a b c d
proof -
have "R1 r (Rep1 (Abs1 r))"
using r Quotient3_refl1 R1 rep_abs_rsp by fastforce
moreover have "R2' (Rep1 (Abs1 r)) (Rep1 (Abs1 s))"
using that
apply simp
apply (metis (full_types) Rep1 Abs1 Quotient3_rel R2 Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1] Quotient3_rel_abs [OF R1])
done
moreover have "R1 (Rep1 (Abs1 s)) s"
by (metis s Quotient3_rel R1 rep_abs_rsp_left)
ultimately show ?thesis
by (metis relcomppI)
qed
next
fix x y
assume xy: "R1 r x" "R2' x y" "R1 y s"
then have "R2 (Abs1 x) (Abs1 y)"
by (iprover dest: Abs1 elim: Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1])
then have "R2' (Rep1 (Abs1 x)) (Rep1 (Abs1 x))" "R2' (Rep1 (Abs1 y)) (Rep1 (Abs1 y))"
by (simp_all add: Quotient3_refl1 [OF R2] Quotient3_refl2 [OF R2] Rep1)
with ‹R1 r x› ‹R1 y s› show "(R1 OOO R2') r r" "(R1 OOO R2') s s"
by (metis (full_types) Quotient3_def R1 relcompp.relcompI)+
show "(Abs2 ∘ Abs1) r = (Abs2 ∘ Abs1) s"
using xy by simp (metis (full_types) Abs1 Quotient3_rel R1 R2)
qed
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)

section ‹Lifting of BNFs›

lemma sum_insert_Inl_unit: "x ∈ A ⟹ (⋀y. x = Inr y ⟹ Inr y ∈ B) ⟹ x ∈ insert (Inl ()) B"
by (cases x) (simp_all)

lemma lift_sum_unit_vimage_commute:
"insert (Inl ()) (Inr ` f -` A) = map_sum id f -` insert (Inl ()) (Inr ` A)"
by (auto simp: map_sum_def split: sum.splits)

lemma insert_Inl_int_map_sum_unit: "insert (Inl ()) A ∩ range (map_sum id f) ≠ {}"
by (auto simp: map_sum_def split: sum.splits)

lemma image_map_sum_unit_subset:
"A ⊆ insert (Inl ()) (Inr ` B) ⟹ map_sum id f ` A ⊆ insert (Inl ()) (Inr ` f ` B)"
by auto

lemma subset_lift_sum_unitD: "A ⊆ insert (Inl ()) (Inr ` B) ⟹ Inr x ∈ A ⟹ x ∈ B"
unfolding insert_def by auto

lemma UNIV_sum_unit_conv: "insert (Inl ()) (range Inr) = UNIV"
unfolding UNIV_sum UNIV_unit image_insert image_empty Un_insert_left sup_bot.left_neutral..

lemma subset_vimage_image_subset: "A ⊆ f -` B ⟹ f ` A ⊆ B"
by auto

lemma relcompp_mem_Grp_neq_bot:
"A ∩ range f ≠ {} ⟹ (λx y. x ∈ A ∧ y ∈ A) OO (Grp UNIV f)¯¯ ≠ bot"
unfolding Grp_def relcompp_apply fun_eq_iff by blast

lemma comp_projr_Inr: "projr ∘ Inr = id"
by auto

lemma in_rel_sum_in_image_projr:
"B ⊆ {(x,y). rel_sum ((=) :: unit ⇒ unit ⇒ bool) A x y} ⟹
Inr ` C = fst ` B ⟹ snd ` B = Inr ` D ⟹ map_prod projr projr ` B ⊆ {(x,y). A x y}"
by (force simp: projr_def image_iff dest!: spec[of _ "Inl ()"]  split: sum.splits)

lemma subset_rel_sumI: "B ⊆ {(x,y). A x y} ⟹ rel_sum ((=) :: unit => unit => bool) A
(if x ∈ B then Inr (fst x) else Inl ())
(if x ∈ B then Inr (snd x) else Inl ())"
by auto

lemma relcompp_eq_Grp_neq_bot: "(=) OO (Grp UNIV f)¯¯ ≠ bot"
unfolding Grp_def relcompp_apply fun_eq_iff by blast

lemma rel_fun_rel_OO1: "(rel_fun Q (rel_fun R (=))) A B ⟹ conversep Q OO A OO R ≤ B"
by (auto simp: rel_fun_def)

lemma rel_fun_rel_OO2: "(rel_fun Q (rel_fun R (=))) A B ⟹ Q OO B OO conversep R ≤ A"
by (auto simp: rel_fun_def)

lemma rel_sum_eq2_nonempty: "rel_sum (=) A OO rel_sum (=) B ≠ bot"
by (auto simp: fun_eq_iff relcompp_apply intro!: exI[of _ "Inl _"])

lemma rel_sum_eq3_nonempty: "rel_sum (=) A OO (rel_sum (=) B OO rel_sum (=) C) ≠ bot"
by (auto simp: fun_eq_iff relcompp_apply intro!: exI[of _ "Inl _"])

lemma hypsubst: "A = B ⟹ x ∈ B ⟹ (x ∈ A ⟹ P) ⟹ P" by simp

lemma Quotient_crel_quotient: "Quotient R Abs Rep T ⟹ equivp R ⟹ T ≡ (λx y. Abs x = y)"
by (drule Quotient_cr_rel) (auto simp: fun_eq_iff equivp_reflp intro!: eq_reflection)

lemma Quotient_crel_typedef: "Quotient (eq_onp P) Abs Rep T ⟹ T ≡ (λx y. x = Rep y)"
unfolding Quotient_def
by (auto 0 4 simp: fun_eq_iff eq_onp_def intro: sym intro!: eq_reflection)

lemma Quotient_crel_typecopy: "Quotient (=) Abs Rep T ⟹ T ≡ (λx y. x = Rep y)"
by (subst (asm) eq_onp_True[symmetric]) (rule Quotient_crel_typedef)

assumes equiv: "equivp R" "equivp R'" and le: "S OO T OO U ≤ R OO STU OO R'"
shows "R OO S OO T OO U OO R' ≤ R OO STU OO R'"
proof -
have trans: "R OO R ≤ R" "R' OO R' ≤ R'"
using equiv unfolding equivp_reflp_symp_transp transp_relcompp by blast+
have "R OO S OO T OO U OO R' = R OO (S OO T OO U) OO R'"
unfolding relcompp_assoc ..
also have "… ≤ R OO (R OO STU OO R') OO R'"
by (intro le relcompp_mono order_refl)
also have "… ≤ (R OO R) OO STU OO (R' OO R')"
unfolding relcompp_assoc ..
also have "… ≤ R OO STU OO R'"
by (intro trans relcompp_mono order_refl)
finally show ?thesis .
qed

lemma Grp_conversep_eq_onp: "((BNF_Def.Grp UNIV f)¯¯ OO BNF_Def.Grp UNIV f) = eq_onp (λx. x ∈ range f)"
by (auto simp: fun_eq_iff Grp_def eq_onp_def image_iff)

lemma Grp_conversep_nonempty: "(BNF_Def.Grp UNIV f)¯¯ OO BNF_Def.Grp UNIV f ≠ bot"
by (auto simp: fun_eq_iff Grp_def)

lemma relcomppI2: "r a b ⟹ s b c ⟹ t c d ⟹ (r OO s OO t) a d"
by (auto)

lemma rel_conj_eq_onp: "equivp R ⟹ rel_conj R (eq_onp P) ≤ R"
by (auto simp: eq_onp_def transp_def equivp_def)

lemma Quotient_Quotient3: "Quotient R Abs Rep T ⟹ Quotient3 R Abs Rep"
unfolding Quotient_def Quotient3_def by blast

lemma Quotient_reflp_imp_equivp: "Quotient R Abs Rep T ⟹ reflp R ⟹ equivp R"
using Quotient_symp Quotient_transp equivpI by blast

lemma Quotient_eq_onp_typedef:
"Quotient (eq_onp P) Abs Rep cr ⟹ type_definition Rep Abs {x. P x}"
unfolding Quotient_def eq_onp_def
by unfold_locales auto

lemma Quotient_eq_onp_type_copy:
"Quotient (=) Abs Rep cr ⟹ type_definition Rep Abs UNIV"
unfolding Quotient_def eq_onp_def
by unfold_locales auto

ML_file ‹Tools/BNF/bnf_lift.ML›

hide_fact
sum_insert_Inl_unit lift_sum_unit_vimage_commute insert_Inl_int_map_sum_unit
image_map_sum_unit_subset subset_lift_sum_unitD UNIV_sum_unit_conv subset_vimage_image_subset
relcompp_mem_Grp_neq_bot comp_projr_Inr in_rel_sum_in_image_projr subset_rel_sumI
relcompp_eq_Grp_neq_bot rel_fun_rel_OO1 rel_fun_rel_OO2 rel_sum_eq2_nonempty rel_sum_eq3_nonempty
hypsubst equivp_add_relconj Grp_conversep_eq_onp Grp_conversep_nonempty relcomppI2 rel_conj_eq_onp
Quotient_reflp_imp_equivp Quotient_Quotient3

end
```