(* 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