Names of files.
(* Title: Quotient.thy+ −
Author: Cezary Kaliszyk and Christian Urban+ −
*)+ −
+ −
theory Quotient+ −
imports Plain ATP_Linkup+ −
uses+ −
("quotient_info.ML")+ −
("quotient_typ.ML")+ −
("quotient_def.ML")+ −
("quotient_term.ML")+ −
("quotient_tacs.ML")+ −
begin+ −
+ −
+ −
text {*+ −
Basic definition for equivalence relations+ −
that are represented by predicates.+ −
*}+ −
+ −
definition+ −
"equivp E \<equiv> \<forall>x y. E x y = (E x = E y)"+ −
+ −
definition+ −
"reflp E \<equiv> \<forall>x. E x x"+ −
+ −
definition+ −
"symp E \<equiv> \<forall>x y. E x y \<longrightarrow> E y x"+ −
+ −
definition+ −
"transp E \<equiv> \<forall>x y z. E x y \<and> E y z \<longrightarrow> E x z"+ −
+ −
lemma equivp_reflp_symp_transp:+ −
shows "equivp E = (reflp E \<and> symp E \<and> transp E)"+ −
unfolding equivp_def reflp_def symp_def transp_def expand_fun_eq+ −
by blast+ −
+ −
lemma equivp_reflp:+ −
shows "equivp E \<Longrightarrow> E x x"+ −
by (simp only: equivp_reflp_symp_transp reflp_def)+ −
+ −
lemma equivp_symp:+ −
shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y x"+ −
by (metis equivp_reflp_symp_transp symp_def)+ −
+ −
lemma equivp_transp:+ −
shows "equivp E \<Longrightarrow> E x y \<Longrightarrow> E y z \<Longrightarrow> E x z"+ −
by (metis equivp_reflp_symp_transp transp_def)+ −
+ −
lemma equivpI:+ −
assumes "reflp R" "symp R" "transp R"+ −
shows "equivp R"+ −
using assms by (simp add: equivp_reflp_symp_transp)+ −
+ −
lemma identity_equivp:+ −
shows "equivp (op =)"+ −
unfolding equivp_def+ −
by auto+ −
+ −
text {* Partial equivalences: not yet used anywhere *}+ −
+ −
definition+ −
"part_equivp E \<equiv> (\<exists>x. E x x) \<and> (\<forall>x y. E x y = (E x x \<and> E y y \<and> (E x = E y)))"+ −
+ −
lemma equivp_implies_part_equivp:+ −
assumes a: "equivp E"+ −
shows "part_equivp E"+ −
using a+ −
unfolding equivp_def part_equivp_def+ −
by auto+ −
+ −
text {* Composition of Relations *}+ −
+ −
abbreviation+ −
rel_conj (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: expand_fun_eq)+ −
+ −
section {* Respects predicate *}+ −
+ −
definition+ −
Respects+ −
where+ −
"Respects R x \<equiv> R x x"+ −
+ −
lemma in_respects:+ −
shows "(x \<in> Respects R) = R x x"+ −
unfolding mem_def Respects_def+ −
by simp+ −
+ −
section {* Function map and function relation *}+ −
+ −
definition+ −
fun_map (infixr "--->" 55)+ −
where+ −
[simp]: "fun_map f g h x = g (h (f x))"+ −
+ −
definition+ −
fun_rel (infixr "===>" 55)+ −
where+ −
[simp]: "fun_rel E1 E2 f g = (\<forall>x y. E1 x y \<longrightarrow> E2 (f x) (g y))"+ −
+ −
+ −
lemma fun_map_id:+ −
shows "(id ---> id) = id"+ −
by (simp add: expand_fun_eq id_def)+ −
+ −
lemma fun_rel_eq:+ −
shows "((op =) ===> (op =)) = (op =)"+ −
by (simp add: expand_fun_eq)+ −
+ −
lemma fun_rel_id:+ −
assumes a: "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"+ −
shows "(R1 ===> R2) f g"+ −
using a by simp+ −
+ −
lemma fun_rel_id_asm:+ −
assumes a: "\<And>x y. R1 x y \<Longrightarrow> (A \<longrightarrow> R2 (f x) (g y))"+ −
shows "A \<longrightarrow> (R1 ===> R2) f g"+ −
using a by auto+ −
+ −
+ −
section {* Quotient Predicate *}+ −
+ −
definition+ −
"Quotient E Abs Rep \<equiv>+ −
(\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. E (Rep a) (Rep a)) \<and>+ −
(\<forall>r s. E r s = (E r r \<and> E s s \<and> (Abs r = Abs s)))"+ −
+ −
lemma Quotient_abs_rep:+ −
assumes a: "Quotient E Abs Rep"+ −
shows "Abs (Rep a) = a"+ −
using a+ −
unfolding Quotient_def+ −
by simp+ −
+ −
lemma Quotient_rep_reflp:+ −
assumes a: "Quotient E Abs Rep"+ −
shows "E (Rep a) (Rep a)"+ −
using a+ −
unfolding Quotient_def+ −
by blast+ −
+ −
lemma Quotient_rel:+ −
assumes a: "Quotient E Abs Rep"+ −
shows " E r s = (E r r \<and> E s s \<and> (Abs r = Abs s))"+ −
using a+ −
unfolding Quotient_def+ −
by blast+ −
+ −
lemma Quotient_rel_rep:+ −
assumes a: "Quotient R Abs Rep"+ −
shows "R (Rep a) (Rep b) = (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 E Abs Rep"+ −
shows "E r s \<Longrightarrow> Abs r = Abs s"+ −
using a unfolding Quotient_def+ −
by blast+ −
+ −
lemma Quotient_symp:+ −
assumes a: "Quotient E Abs Rep"+ −
shows "symp E"+ −
using a unfolding Quotient_def symp_def+ −
by metis+ −
+ −
lemma Quotient_transp:+ −
assumes a: "Quotient E Abs Rep"+ −
shows "transp E"+ −
using a unfolding Quotient_def transp_def+ −
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 "\<forall>a. (rep1 ---> abs2) ((abs1 ---> rep2) a) = a"+ −
using q1 q2+ −
unfolding Quotient_def+ −
unfolding expand_fun_eq+ −
by simp+ −
moreover+ −
have "\<forall>a. (R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"+ −
using q1 q2+ −
unfolding Quotient_def+ −
by (simp (no_asm)) (metis)+ −
moreover+ −
have "\<forall>r s. (R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>+ −
(rep1 ---> abs2) r = (rep1 ---> abs2) s)"+ −
unfolding expand_fun_eq+ −
apply(auto)+ −
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 expand_fun_eq+ −
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]+ −
unfolding symp_def transp_def+ −
by blast+ −
+ −
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 expand_fun_eq+ −
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 expand_fun_eq+ −
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 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 simp+ −
+ −
lemma apply_rsp':+ −
assumes a: "(R1 ===> R2) f g" "R1 x y"+ −
shows "R2 (f x) (g y)"+ −
using a by simp+ −
+ −
section {* 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 COMBC_def 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 COMBC_def 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)+ −
apply(rule impI)+ −
using a equivp_reflp_symp_transp[of "R2"]+ −
apply(simp add: reflp_def)+ −
apply(simp)+ −
apply(simp)+ −
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)+ −
apply(rule impI)+ −
using a equivp_reflp_symp_transp[of "R2"]+ −
apply(simp add: reflp_def)+ −
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 COMBC_def 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 COMBC_def 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+ −
+ −
section {* Bounded abstraction *}+ −
+ −
definition+ −
Babs :: "('a \<Rightarrow> bool) \<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)+ −
apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")+ −
using a apply (simp add: Babs_def)+ −
apply (simp add: in_respects)+ −
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)+ −
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)+ −
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: expand_fun_eq 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 (simp add: Ball_def in_respects)+ −
+ −
lemma bex_rsp:+ −
assumes a: "(R ===> (op =)) f g"+ −
shows "(Bex (Respects R) f = Bex (Respects R) g)"+ −
using a by (simp add: Bex_def in_respects)+ −
+ −
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 (simp add: Ex1_def in_respects) auto+ −
+ −
(* 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 fun_map_def id_apply+ −
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 fun_map_def id_apply+ −
by metis+ −
+ −
section {* 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+ −
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+ −
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: "(\<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+ −
+ −
section {* Various respects and preserve lemmas *}+ −
+ −
lemma quot_rel_rsp:+ −
assumes a: "Quotient R Abs Rep"+ −
shows "(R ===> R ===> op =) R R"+ −
apply(rule fun_rel_id)++ −
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 "(Rep1 ---> Abs3) (((Abs2 ---> Rep3) f) o ((Abs1 ---> Rep2) g)) = f o g"+ −
using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]+ −
unfolding o_def expand_fun_eq by simp+ −
+ −
lemma o_rsp:+ −
assumes q1: "Quotient R1 Abs1 Rep1"+ −
and q2: "Quotient R2 Abs2 Rep2"+ −
and q3: "Quotient R3 Abs3 Rep3"+ −
and a1: "(R2 ===> R3) f1 f2"+ −
and a2: "(R1 ===> R2) g1 g2"+ −
shows "(R1 ===> R3) (f1 o g1) (f2 o g2)"+ −
using a1 a2 unfolding o_def expand_fun_eq+ −
by (auto)+ −
+ −
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 "Abs (If a (Rep b) (Rep c)) = If a b c"+ −
using Quotient_abs_rep[OF q] by auto+ −
+ −
(* q not used *)+ −
lemma if_rsp:+ −
assumes q: "Quotient R Abs Rep"+ −
and a: "a1 = a2" "R b1 b2" "R c1 c2"+ −
shows "R (If a1 b1 c1) (If a2 b2 c2)"+ −
using a by auto+ −
+ −
lemma let_prs:+ −
assumes q1: "Quotient R1 Abs1 Rep1"+ −
and q2: "Quotient R2 Abs2 Rep2"+ −
shows "Abs2 (Let (Rep1 x) ((Abs1 ---> Rep2) f)) = Let x f"+ −
using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] by auto+ −
+ −
lemma let_rsp:+ −
assumes q1: "Quotient R1 Abs1 Rep1"+ −
and a1: "(R1 ===> R2) f g"+ −
and a2: "R1 x y"+ −
shows "R2 ((Let x f)::'c) ((Let y g)::'c)"+ −
using apply_rsp[OF q1 a1] a2 by auto+ −
+ −
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: "equivp R"+ −
and rep_prop: "\<And>y. \<exists>x. Rep y = R x"+ −
and rep_inverse: "\<And>x. Abs (Rep x) = x"+ −
and abs_inverse: "\<And>x. (Rep (Abs (R x))) = (R x)"+ −
and rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"+ −
begin+ −
+ −
definition+ −
abs::"'a \<Rightarrow> 'b"+ −
where+ −
"abs x \<equiv> Abs (R x)"+ −
+ −
definition+ −
rep::"'b \<Rightarrow> 'a"+ −
where+ −
"rep a = Eps (Rep a)"+ −
+ −
lemma homeier_lem9:+ −
shows "R (Eps (R x)) = R x"+ −
proof -+ −
have a: "R x x" using equivp by (simp add: equivp_reflp_symp_transp reflp_def)+ −
then have "R x (Eps (R x))" by (rule someI)+ −
then show "R (Eps (R x)) = R x"+ −
using equivp unfolding equivp_def by simp+ −
qed+ −
+ −
theorem homeier_thm10:+ −
shows "abs (rep a) = a"+ −
unfolding abs_def rep_def+ −
proof -+ −
from rep_prop+ −
obtain x where eq: "Rep a = R x" by auto+ −
have "Abs (R (Eps (Rep a))) = Abs (R (Eps (R x)))" using eq by simp+ −
also have "\<dots> = Abs (R x)" using homeier_lem9 by simp+ −
also have "\<dots> = Abs (Rep a)" using eq by simp+ −
also have "\<dots> = a" using rep_inverse by simp+ −
finally+ −
show "Abs (R (Eps (Rep a))) = a" by simp+ −
qed+ −
+ −
lemma homeier_lem7:+ −
shows "(R x = R y) = (Abs (R x) = Abs (R y))" (is "?LHS = ?RHS")+ −
proof -+ −
have "?RHS = (Rep (Abs (R x)) = Rep (Abs (R y)))" by (simp add: rep_inject)+ −
also have "\<dots> = ?LHS" by (simp add: abs_inverse)+ −
finally show "?LHS = ?RHS" by simp+ −
qed+ −
+ −
theorem homeier_thm11:+ −
shows "R r r' = (abs r = abs r')"+ −
unfolding abs_def+ −
by (simp only: equivp[simplified equivp_def] homeier_lem7)+ −
+ −
lemma rep_refl:+ −
shows "R (rep a) (rep a)"+ −
unfolding rep_def+ −
by (simp add: equivp[simplified equivp_def])+ −
+ −
+ −
lemma rep_abs_rsp:+ −
shows "R f (rep (abs g)) = R f g"+ −
and "R (rep (abs g)) f = R g f"+ −
by (simp_all add: homeier_thm10 homeier_thm11)+ −
+ −
lemma Quotient:+ −
shows "Quotient R abs rep"+ −
unfolding Quotient_def+ −
apply(simp add: homeier_thm10)+ −
apply(simp add: rep_refl)+ −
apply(subst homeier_thm11[symmetric])+ −
apply(simp add: equivp[simplified equivp_def])+ −
done+ −
+ −
end+ −
+ −
section {* ML setup *}+ −
+ −
text {* Auxiliary data for the quotient package *}+ −
+ −
use "quotient_info.ML"+ −
+ −
declare [[map "fun" = (fun_map, fun_rel)]]+ −
+ −
lemmas [quot_thm] = fun_quotient+ −
lemmas [quot_respect] = quot_rel_rsp+ −
lemmas [quot_equiv] = identity_equivp+ −
+ −
+ −
text {* Lemmas about simplifying id's. *}+ −
lemmas [id_simps] =+ −
id_def[symmetric]+ −
fun_map_id+ −
id_apply+ −
id_o+ −
o_id+ −
eq_comp_r+ −
+ −
text {* Translation functions for the lifting process. *}+ −
use "quotient_term.ML"+ −
+ −
+ −
text {* Definitions of the quotient types. *}+ −
use "quotient_typ.ML"+ −
+ −
+ −
text {* Definitions for quotient constants. *}+ −
use "quotient_def.ML"+ −
+ −
+ −
text {*+ −
An auxiliary constant for recording some information+ −
about the lifted theorem in a tactic.+ −
*}+ −
definition+ −
"Quot_True x \<equiv> 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 "quotient_tacs.ML"+ −
+ −
section {* Methods / Interface *}+ −
+ −
method_setup lifting =+ −
{* Attrib.thms >> (fn thms => fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt thms))) *}+ −
{* lifts theorems to quotient types *}+ −
+ −
method_setup lifting_setup =+ −
{* Attrib.thm >> (fn thms => fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.procedure_tac ctxt thms))) *}+ −
{* sets up the three goals for the quotient lifting procedure *}+ −
+ −
method_setup regularize =+ −
{* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}+ −
{* proves the regularization goals from the quotient lifting procedure *}+ −
+ −
method_setup injection =+ −
{* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}+ −
{* proves 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))) *}+ −
{* proves the cleaning goals from the quotient lifting procedure *}+ −
+ −
attribute_setup quot_lifted =+ −
{* Scan.succeed Quotient_Tacs.lifted_attrib *}+ −
{* lifts theorems to quotient types *}+ −
+ −
no_notation+ −
rel_conj (infixr "OOO" 75) and+ −
fun_map (infixr "--->" 55) and+ −
fun_rel (infixr "===>" 55)+ −
+ −
end+ −
+ −