(* Title: Quotient.thy Author: Cezary Kaliszyk and Christian Urban*)theory Quotientimports Plain ATP_Linkupuses ("quotient_info.ML") ("quotient_typ.ML") ("quotient_def.ML") ("quotient_term.ML") ("quotient_tacs.ML")begintext {* 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 blastlemma 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 autotext {* 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 autotext {* 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 Respectswhere "Respects R x \<equiv> R x x"lemma in_respects: shows "(x \<in> Respects R) = R x x" unfolding mem_def Respects_def by simpsection {* 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 simplemma 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 autosection {* 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 simplemma Quotient_rep_reflp: assumes a: "Quotient E Abs Rep" shows "E (Rep a) (Rep a)" using a unfolding Quotient_def by blastlemma 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 blastlemma Quotient_rel_rep: assumes a: "Quotient R Abs Rep" shows "R (Rep a) (Rep b) = (a = b)" using a unfolding Quotient_def by metislemma 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 blastlemma Quotient_rel_abs: assumes a: "Quotient E Abs Rep" shows "E r s \<Longrightarrow> Abs r = Abs s" using a unfolding Quotient_def by blastlemma Quotient_symp: assumes a: "Quotient E Abs Rep" shows "symp E" using a unfolding Quotient_def symp_def by metislemma Quotient_transp: assumes a: "Quotient E Abs Rep" shows "transp E" using a unfolding Quotient_def transp_def by metislemma identity_quotient: shows "Quotient (op =) id id" unfolding Quotient_def id_def by blastlemma 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 blastqedlemma 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 blastlemma 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 simplemma 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 simplemma 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 metislemma 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 metistext{* 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 simplemma apply_rsp': assumes a: "(R1 ===> R2) f g" "R1 x y" shows "R2 (f x) (g y)" using a by simpsection {* 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) donelemma 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 metislemma 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 autolemma 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 autosection {* 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 metislemma 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]) donelemma 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 metislemma 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 metissection {* 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) donelemma 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) donelemma 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 donelemma ex1_prs: assumes a: "Quotient R absf repf" shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"apply simpapply (subst Bex1_rel_def)apply (subst Bex_def)apply (subst Ex1_def)apply simpapply 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_respectsapply metisdonelemma 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 donesection {* 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)+ donelemma 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 simplemma 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 autolemma 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 autolemma 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 autolemma 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 autolocale 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)"begindefinition 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 simpqedtheorem homeier_thm10: shows "abs (rep a) = a" unfolding abs_def rep_defproof - 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 simpqedlemma 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 simpqedtheorem 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]) doneendsection {* ML setup *}text {* Auxiliary data for the quotient package *}use "quotient_info.ML"declare [[map "fun" = (fun_map, fun_rel)]]lemmas [quot_thm] = fun_quotientlemmas [quot_respect] = quot_rel_rsplemmas [quot_equiv] = identity_equivptext {* Lemmas about simplifying id's. *}lemmas [id_simps] = id_def[symmetric] fun_map_id id_apply id_o o_id eq_comp_rtext {* 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