wenzelm@41959: (*  Title:      HOL/Quotient.thy
kaliszyk@35222:     Author:     Cezary Kaliszyk and Christian Urban
kaliszyk@35222: *)
kaliszyk@35222: 
huffman@35294: header {* Definition of Quotient Types *}
huffman@35294: 
kaliszyk@35222: theory Quotient
haftmann@51112: imports Hilbert_Choice Equiv_Relations Lifting
wenzelm@46950: keywords
kuncar@47308:   "print_quotmapsQ3" "print_quotientsQ3" "print_quotconsts" :: diag and
wenzelm@46950:   "quotient_type" :: thy_goal and "/" and
kuncar@47091:   "quotient_definition" :: thy_goal
kaliszyk@35222: begin
kaliszyk@35222: 
kaliszyk@35222: text {*
kaliszyk@35222:   Basic definition for equivalence relations
kaliszyk@35222:   that are represented by predicates.
kaliszyk@35222: *}
kaliszyk@35222: 
kaliszyk@35222: text {* Composition of Relations *}
kaliszyk@35222: 
kaliszyk@35222: abbreviation
haftmann@40818:   rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
kaliszyk@35222: where
kaliszyk@35222:   "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
kaliszyk@35222: 
kaliszyk@35222: lemma eq_comp_r:
kaliszyk@35222:   shows "((op =) OOO R) = R"
nipkow@39302:   by (auto simp add: fun_eq_iff)
kaliszyk@35222: 
kuncar@53011: context
kuncar@53011: begin
kuncar@53011: interpretation lifting_syntax .
kuncar@53011: 
huffman@35294: subsection {* Quotient Predicate *}
kaliszyk@35222: 
kaliszyk@35222: definition
kuncar@47308:   "Quotient3 R Abs Rep \<longleftrightarrow>
haftmann@40814:      (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
haftmann@40818:      (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
haftmann@40818: 
kuncar@47308: lemma Quotient3I:
haftmann@40818:   assumes "\<And>a. Abs (Rep a) = a"
haftmann@40818:     and "\<And>a. R (Rep a) (Rep a)"
haftmann@40818:     and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
kuncar@47308:   shows "Quotient3 R Abs Rep"
kuncar@47308:   using assms unfolding Quotient3_def by blast
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_abs_rep:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222:   shows "Abs (Rep a) = a"
kaliszyk@35222:   using a
kuncar@47308:   unfolding Quotient3_def
kaliszyk@35222:   by simp
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_rep_reflp:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
haftmann@40814:   shows "R (Rep a) (Rep a)"
kaliszyk@35222:   using a
kuncar@47308:   unfolding Quotient3_def
kaliszyk@35222:   by blast
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_rel:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
haftmann@40818:   shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
kaliszyk@35222:   using a
kuncar@47308:   unfolding Quotient3_def
kaliszyk@35222:   by blast
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_refl1: 
kuncar@47308:   assumes a: "Quotient3 R Abs Rep" 
kuncar@47096:   shows "R r s \<Longrightarrow> R r r"
kuncar@47308:   using a unfolding Quotient3_def 
kuncar@47096:   by fast
kuncar@47096: 
kuncar@47308: lemma Quotient3_refl2: 
kuncar@47308:   assumes a: "Quotient3 R Abs Rep" 
kuncar@47096:   shows "R r s \<Longrightarrow> R s s"
kuncar@47308:   using a unfolding Quotient3_def 
kuncar@47096:   by fast
kuncar@47096: 
kuncar@47308: lemma Quotient3_rel_rep:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
haftmann@40818:   shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
kaliszyk@35222:   using a
kuncar@47308:   unfolding Quotient3_def
kaliszyk@35222:   by metis
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_rep_abs:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222:   shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
kuncar@47308:   using a unfolding Quotient3_def
kuncar@47308:   by blast
kuncar@47308: 
kuncar@47308: lemma Quotient3_rel_abs:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kuncar@47308:   shows "R r s \<Longrightarrow> Abs r = Abs s"
kuncar@47308:   using a unfolding Quotient3_def
kaliszyk@35222:   by blast
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_symp:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
haftmann@40814:   shows "symp R"
kuncar@47308:   using a unfolding Quotient3_def using sympI by metis
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_transp:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
haftmann@40814:   shows "transp R"
kuncar@47308:   using a unfolding Quotient3_def using transpI by (metis (full_types))
kaliszyk@35222: 
kuncar@47308: lemma Quotient3_part_equivp:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kuncar@47308:   shows "part_equivp R"
kuncar@47308: by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp a part_equivpI)
kuncar@47308: 
kuncar@47308: lemma identity_quotient3:
kuncar@47308:   shows "Quotient3 (op =) id id"
kuncar@47308:   unfolding Quotient3_def id_def
kaliszyk@35222:   by blast
kaliszyk@35222: 
kuncar@47308: lemma fun_quotient3:
kuncar@47308:   assumes q1: "Quotient3 R1 abs1 rep1"
kuncar@47308:   and     q2: "Quotient3 R2 abs2 rep2"
kuncar@47308:   shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
kaliszyk@35222: proof -
kuncar@47308:   have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
kuncar@47308:     using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
kaliszyk@35222:   moreover
kuncar@47308:   have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
haftmann@40466:     by (rule fun_relI)
kuncar@47308:       (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
kuncar@47308:         simp (no_asm) add: Quotient3_def, simp)
kuncar@47308:   
kaliszyk@35222:   moreover
kuncar@47308:   {
kuncar@47308:   fix r s
kuncar@47308:   have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
kaliszyk@35222:         (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
kuncar@47308:   proof -
kuncar@47308:     
kuncar@47308:     have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding fun_rel_def
kuncar@47308:       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
kuncar@47308:       by (metis (full_types) part_equivp_def)
kuncar@47308:     moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding fun_rel_def
kuncar@47308:       using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
kuncar@47308:       by (metis (full_types) part_equivp_def)
kuncar@47308:     moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
kuncar@47308:       apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
kuncar@47308:     moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
kuncar@47308:         (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
kuncar@47308:       apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def 
kuncar@47308:     by (metis map_fun_apply)
kuncar@47308:   
kuncar@47308:     ultimately show ?thesis by blast
kuncar@47308:  qed
kuncar@47308:  }
kuncar@47308:  ultimately show ?thesis by (intro Quotient3I) (assumption+)
kaliszyk@35222: qed
kaliszyk@35222: 
kaliszyk@35222: lemma abs_o_rep:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222:   shows "Abs o Rep = id"
nipkow@39302:   unfolding fun_eq_iff
kuncar@47308:   by (simp add: Quotient3_abs_rep[OF a])
kaliszyk@35222: 
kaliszyk@35222: lemma equals_rsp:
kuncar@47308:   assumes q: "Quotient3 R Abs Rep"
kaliszyk@35222:   and     a: "R xa xb" "R ya yb"
kaliszyk@35222:   shows "R xa ya = R xb yb"
kuncar@47308:   using a Quotient3_symp[OF q] Quotient3_transp[OF q]
haftmann@40814:   by (blast elim: sympE transpE)
kaliszyk@35222: 
kaliszyk@35222: lemma lambda_prs:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
nipkow@39302:   unfolding fun_eq_iff
kuncar@47308:   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
haftmann@40814:   by simp
kaliszyk@35222: 
kaliszyk@35222: lemma lambda_prs1:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
nipkow@39302:   unfolding fun_eq_iff
kuncar@47308:   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
haftmann@40814:   by simp
kaliszyk@35222: 
kaliszyk@35222: lemma rep_abs_rsp:
kuncar@47308:   assumes q: "Quotient3 R Abs Rep"
kaliszyk@35222:   and     a: "R x1 x2"
kaliszyk@35222:   shows "R x1 (Rep (Abs x2))"
kuncar@47308:   using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
kaliszyk@35222:   by metis
kaliszyk@35222: 
kaliszyk@35222: lemma rep_abs_rsp_left:
kuncar@47308:   assumes q: "Quotient3 R Abs Rep"
kaliszyk@35222:   and     a: "R x1 x2"
kaliszyk@35222:   shows "R (Rep (Abs x1)) x2"
kuncar@47308:   using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
kaliszyk@35222:   by metis
kaliszyk@35222: 
kaliszyk@35222: text{*
kaliszyk@35222:   In the following theorem R1 can be instantiated with anything,
kaliszyk@35222:   but we know some of the types of the Rep and Abs functions;
kaliszyk@35222:   so by solving Quotient assumptions we can get a unique R1 that
kaliszyk@35236:   will be provable; which is why we need to use @{text apply_rsp} and
kaliszyk@35222:   not the primed version *}
kaliszyk@35222: 
kuncar@47308: lemma apply_rspQ3:
kaliszyk@35222:   fixes f g::"'a \<Rightarrow> 'c"
kuncar@47308:   assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222:   and     a: "(R1 ===> R2) f g" "R1 x y"
kaliszyk@35222:   shows "R2 (f x) (g y)"
haftmann@40466:   using a by (auto elim: fun_relE)
kaliszyk@35222: 
kuncar@47308: lemma apply_rspQ3'':
kuncar@47308:   assumes "Quotient3 R Abs Rep"
kuncar@47096:   and "(R ===> S) f f"
kuncar@47096:   shows "S (f (Rep x)) (f (Rep x))"
kuncar@47096: proof -
kuncar@47308:   from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
kuncar@47096:   then show ?thesis using assms(2) by (auto intro: apply_rsp')
kuncar@47096: qed
kuncar@47096: 
huffman@35294: subsection {* lemmas for regularisation of ball and bex *}
kaliszyk@35222: 
kaliszyk@35222: lemma ball_reg_eqv:
kaliszyk@35222:   fixes P :: "'a \<Rightarrow> bool"
kaliszyk@35222:   assumes a: "equivp R"
kaliszyk@35222:   shows "Ball (Respects R) P = (All P)"
kaliszyk@35222:   using a
kaliszyk@35222:   unfolding equivp_def
kaliszyk@35222:   by (auto simp add: in_respects)
kaliszyk@35222: 
kaliszyk@35222: lemma bex_reg_eqv:
kaliszyk@35222:   fixes P :: "'a \<Rightarrow> bool"
kaliszyk@35222:   assumes a: "equivp R"
kaliszyk@35222:   shows "Bex (Respects R) P = (Ex P)"
kaliszyk@35222:   using a
kaliszyk@35222:   unfolding equivp_def
kaliszyk@35222:   by (auto simp add: in_respects)
kaliszyk@35222: 
kaliszyk@35222: lemma ball_reg_right:
haftmann@44553:   assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
kaliszyk@35222:   shows "All P \<longrightarrow> Ball R Q"
huffman@44921:   using a by fast
kaliszyk@35222: 
kaliszyk@35222: lemma bex_reg_left:
haftmann@44553:   assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
kaliszyk@35222:   shows "Bex R Q \<longrightarrow> Ex P"
huffman@44921:   using a by fast
kaliszyk@35222: 
kaliszyk@35222: lemma ball_reg_left:
kaliszyk@35222:   assumes a: "equivp R"
kaliszyk@35222:   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
kaliszyk@35222:   using a by (metis equivp_reflp in_respects)
kaliszyk@35222: 
kaliszyk@35222: lemma bex_reg_right:
kaliszyk@35222:   assumes a: "equivp R"
kaliszyk@35222:   shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
kaliszyk@35222:   using a by (metis equivp_reflp in_respects)
kaliszyk@35222: 
kaliszyk@35222: lemma ball_reg_eqv_range:
kaliszyk@35222:   fixes P::"'a \<Rightarrow> bool"
kaliszyk@35222:   and x::"'a"
kaliszyk@35222:   assumes a: "equivp R2"
kaliszyk@35222:   shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
kaliszyk@35222:   apply(rule iffI)
kaliszyk@35222:   apply(rule allI)
kaliszyk@35222:   apply(drule_tac x="\<lambda>y. f x" in bspec)
haftmann@40466:   apply(simp add: in_respects fun_rel_def)
kaliszyk@35222:   apply(rule impI)
kaliszyk@35222:   using a equivp_reflp_symp_transp[of "R2"]
haftmann@40814:   apply (auto elim: equivpE reflpE)
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: lemma bex_reg_eqv_range:
kaliszyk@35222:   assumes a: "equivp R2"
kaliszyk@35222:   shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
kaliszyk@35222:   apply(auto)
kaliszyk@35222:   apply(rule_tac x="\<lambda>y. f x" in bexI)
kaliszyk@35222:   apply(simp)
haftmann@40466:   apply(simp add: Respects_def in_respects fun_rel_def)
kaliszyk@35222:   apply(rule impI)
kaliszyk@35222:   using a equivp_reflp_symp_transp[of "R2"]
haftmann@40814:   apply (auto elim: equivpE reflpE)
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: (* Next four lemmas are unused *)
kaliszyk@35222: lemma all_reg:
kaliszyk@35222:   assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222:   and     b: "All P"
kaliszyk@35222:   shows "All Q"
huffman@44921:   using a b by fast
kaliszyk@35222: 
kaliszyk@35222: lemma ex_reg:
kaliszyk@35222:   assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222:   and     b: "Ex P"
kaliszyk@35222:   shows "Ex Q"
huffman@44921:   using a b by fast
kaliszyk@35222: 
kaliszyk@35222: lemma ball_reg:
haftmann@44553:   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222:   and     b: "Ball R P"
kaliszyk@35222:   shows "Ball R Q"
huffman@44921:   using a b by fast
kaliszyk@35222: 
kaliszyk@35222: lemma bex_reg:
haftmann@44553:   assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222:   and     b: "Bex R P"
kaliszyk@35222:   shows "Bex R Q"
huffman@44921:   using a b by fast
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: lemma ball_all_comm:
kaliszyk@35222:   assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222:   shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222:   using assms by auto
kaliszyk@35222: 
kaliszyk@35222: lemma bex_ex_comm:
kaliszyk@35222:   assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222:   shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222:   using assms by auto
kaliszyk@35222: 
huffman@35294: subsection {* Bounded abstraction *}
kaliszyk@35222: 
kaliszyk@35222: definition
haftmann@40466:   Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222: where
kaliszyk@35222:   "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222: 
kaliszyk@35222: lemma babs_rsp:
kuncar@47308:   assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222:   and     a: "(R1 ===> R2) f g"
kaliszyk@35222:   shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466:   apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222:   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466:   using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466:   apply (simp add: in_respects fun_rel_def)
kuncar@47308:   using Quotient3_rel[OF q]
kaliszyk@35222:   by metis
kaliszyk@35222: 
kaliszyk@35222: lemma babs_prs:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222:   shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222:   apply (rule ext)
haftmann@40466:   apply (simp add:)
kaliszyk@35222:   apply (subgoal_tac "Rep1 x \<in> Respects R1")
kuncar@47308:   apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
kuncar@47308:   apply (simp add: in_respects Quotient3_rel_rep[OF q1])
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: lemma babs_simp:
kuncar@47308:   assumes q: "Quotient3 R1 Abs Rep"
kaliszyk@35222:   shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222:   apply(rule iffI)
kaliszyk@35222:   apply(simp_all only: babs_rsp[OF q])
haftmann@40466:   apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222:   apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222:   apply(metis Babs_def)
kaliszyk@35222:   apply (simp add: in_respects)
kuncar@47308:   using Quotient3_rel[OF q]
kaliszyk@35222:   by metis
kaliszyk@35222: 
kaliszyk@35222: (* If a user proves that a particular functional relation
kaliszyk@35222:    is an equivalence this may be useful in regularising *)
kaliszyk@35222: lemma babs_reg_eqv:
kaliszyk@35222:   shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302:   by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: (* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222: lemma ball_rsp:
kaliszyk@35222:   assumes a: "(R ===> (op =)) f g"
kaliszyk@35222:   shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466:   using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222: 
kaliszyk@35222: lemma bex_rsp:
kaliszyk@35222:   assumes a: "(R ===> (op =)) f g"
kaliszyk@35222:   shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466:   using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222: 
kaliszyk@35222: lemma bex1_rsp:
kaliszyk@35222:   assumes a: "(R ===> (op =)) f g"
kaliszyk@35222:   shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466:   using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222: 
kaliszyk@35222: (* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222: lemma all_prs:
kuncar@47308:   assumes a: "Quotient3 R absf repf"
kaliszyk@35222:   shows "Ball (Respects R) ((absf ---> id) f) = All f"
kuncar@47308:   using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222:   by metis
kaliszyk@35222: 
kaliszyk@35222: lemma ex_prs:
kuncar@47308:   assumes a: "Quotient3 R absf repf"
kaliszyk@35222:   shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kuncar@47308:   using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222:   by metis
kaliszyk@35222: 
huffman@35294: subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222: 
kaliszyk@35222: definition
kaliszyk@35222:   Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222: where
kaliszyk@35222:   "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)))"
kaliszyk@35222: 
kaliszyk@35222: lemma bex1_rel_aux:
kaliszyk@35222:   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222:   unfolding Bex1_rel_def
kaliszyk@35222:   apply (erule conjE)+
kaliszyk@35222:   apply (erule bexE)
kaliszyk@35222:   apply rule
kaliszyk@35222:   apply (rule_tac x="xa" in bexI)
kaliszyk@35222:   apply metis
kaliszyk@35222:   apply metis
kaliszyk@35222:   apply rule+
kaliszyk@35222:   apply (erule_tac x="xaa" in ballE)
kaliszyk@35222:   prefer 2
kaliszyk@35222:   apply (metis)
kaliszyk@35222:   apply (erule_tac x="ya" in ballE)
kaliszyk@35222:   prefer 2
kaliszyk@35222:   apply (metis)
kaliszyk@35222:   apply (metis in_respects)
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: lemma bex1_rel_aux2:
kaliszyk@35222:   "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222:   unfolding Bex1_rel_def
kaliszyk@35222:   apply (erule conjE)+
kaliszyk@35222:   apply (erule bexE)
kaliszyk@35222:   apply rule
kaliszyk@35222:   apply (rule_tac x="xa" in bexI)
kaliszyk@35222:   apply metis
kaliszyk@35222:   apply metis
kaliszyk@35222:   apply rule+
kaliszyk@35222:   apply (erule_tac x="xaa" in ballE)
kaliszyk@35222:   prefer 2
kaliszyk@35222:   apply (metis)
kaliszyk@35222:   apply (erule_tac x="ya" in ballE)
kaliszyk@35222:   prefer 2
kaliszyk@35222:   apply (metis)
kaliszyk@35222:   apply (metis in_respects)
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: lemma bex1_rel_rsp:
kuncar@47308:   assumes a: "Quotient3 R absf repf"
kaliszyk@35222:   shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466:   apply (simp add: fun_rel_def)
kaliszyk@35222:   apply clarify
kaliszyk@35222:   apply rule
kaliszyk@35222:   apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222:   apply (erule bex1_rel_aux2)
kaliszyk@35222:   apply assumption
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: lemma ex1_prs:
kuncar@47308:   assumes a: "Quotient3 R absf repf"
kaliszyk@35222:   shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466: apply (simp add:)
kaliszyk@35222: apply (subst Bex1_rel_def)
kaliszyk@35222: apply (subst Bex_def)
kaliszyk@35222: apply (subst Ex1_def)
kaliszyk@35222: apply simp
kaliszyk@35222: apply rule
kaliszyk@35222:  apply (erule conjE)+
kaliszyk@35222:  apply (erule_tac exE)
kaliszyk@35222:  apply (erule conjE)
kaliszyk@35222:  apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222:   apply (rule_tac x="absf x" in exI)
kaliszyk@35222:   apply (simp)
kaliszyk@35222:   apply rule+
kuncar@47308:   using a unfolding Quotient3_def
kaliszyk@35222:   apply metis
kaliszyk@35222:  apply rule+
kaliszyk@35222:  apply (erule_tac x="x" in ballE)
kaliszyk@35222:   apply (erule_tac x="y" in ballE)
kaliszyk@35222:    apply simp
kaliszyk@35222:   apply (simp add: in_respects)
kaliszyk@35222:  apply (simp add: in_respects)
kaliszyk@35222: apply (erule_tac exE)
kaliszyk@35222:  apply rule
kaliszyk@35222:  apply (rule_tac x="repf x" in exI)
kaliszyk@35222:  apply (simp only: in_respects)
kaliszyk@35222:   apply rule
kuncar@47308:  apply (metis Quotient3_rel_rep[OF a])
kuncar@47308: using a unfolding Quotient3_def apply (simp)
kaliszyk@35222: apply rule+
kuncar@47308: using a unfolding Quotient3_def in_respects
kaliszyk@35222: apply metis
kaliszyk@35222: done
kaliszyk@35222: 
kaliszyk@38702: lemma bex1_bexeq_reg:
kaliszyk@38702:   shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222:   apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222:   apply clarify
kaliszyk@35222:   apply auto
kaliszyk@35222:   apply (rule bexI)
kaliszyk@35222:   apply assumption
kaliszyk@35222:   apply (simp add: in_respects)
kaliszyk@35222:   apply (simp add: in_respects)
kaliszyk@35222:   apply auto
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@38702: lemma bex1_bexeq_reg_eqv:
kaliszyk@38702:   assumes a: "equivp R"
kaliszyk@38702:   shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702:   using equivp_reflp[OF a]
kaliszyk@38702:   apply (intro impI)
kaliszyk@38702:   apply (elim ex1E)
kaliszyk@38702:   apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702:   apply (rule_tac a="x" in ex1I)
kaliszyk@38702:   apply (subst in_respects)
kaliszyk@38702:   apply (rule conjI)
kaliszyk@38702:   apply assumption
kaliszyk@38702:   apply assumption
kaliszyk@38702:   apply clarify
kaliszyk@38702:   apply (erule_tac x="xa" in allE)
kaliszyk@38702:   apply simp
kaliszyk@38702:   done
kaliszyk@38702: 
huffman@35294: subsection {* Various respects and preserve lemmas *}
kaliszyk@35222: 
kaliszyk@35222: lemma quot_rel_rsp:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222:   shows "(R ===> R ===> op =) R R"
urbanc@38317:   apply(rule fun_relI)+
kaliszyk@35222:   apply(rule equals_rsp[OF a])
kaliszyk@35222:   apply(assumption)+
kaliszyk@35222:   done
kaliszyk@35222: 
kaliszyk@35222: lemma o_prs:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   and     q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308:   and     q3: "Quotient3 R3 Abs3 Rep3"
kaliszyk@36215:   shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215:   and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kuncar@47308:   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
haftmann@40466:   by (simp_all add: fun_eq_iff)
kaliszyk@35222: 
kaliszyk@35222: lemma o_rsp:
kaliszyk@36215:   "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215:   "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
huffman@44921:   by (force elim: fun_relE)+
kaliszyk@35222: 
kaliszyk@35222: lemma cond_prs:
kuncar@47308:   assumes a: "Quotient3 R absf repf"
kaliszyk@35222:   shows "absf (if a then repf b else repf c) = (if a then b else c)"
kuncar@47308:   using a unfolding Quotient3_def by auto
kaliszyk@35222: 
kaliszyk@35222: lemma if_prs:
kuncar@47308:   assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123:   shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kuncar@47308:   using Quotient3_abs_rep[OF q]
nipkow@39302:   by (auto simp add: fun_eq_iff)
kaliszyk@35222: 
kaliszyk@35222: lemma if_rsp:
kuncar@47308:   assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123:   shows "(op = ===> R ===> R ===> R) If If"
huffman@44921:   by force
kaliszyk@35222: 
kaliszyk@35222: lemma let_prs:
kuncar@47308:   assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@37049:   shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kuncar@47308:   using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
nipkow@39302:   by (auto simp add: fun_eq_iff)
kaliszyk@35222: 
kaliszyk@35222: lemma let_rsp:
kaliszyk@37049:   shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
huffman@44921:   by (force elim: fun_relE)
kaliszyk@35222: 
kaliszyk@39669: lemma id_rsp:
kaliszyk@39669:   shows "(R ===> R) id id"
huffman@44921:   by auto
kaliszyk@39669: 
kaliszyk@39669: lemma id_prs:
kuncar@47308:   assumes a: "Quotient3 R Abs Rep"
kaliszyk@39669:   shows "(Rep ---> Abs) id = id"
kuncar@47308:   by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
kaliszyk@39669: 
kuncar@53011: end
kaliszyk@39669: 
kaliszyk@35222: locale quot_type =
kaliszyk@35222:   fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204:   and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204:   and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493:   assumes equivp: "part_equivp R"
kaliszyk@44204:   and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222:   and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204:   and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222:   and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222: begin
kaliszyk@35222: 
kaliszyk@35222: definition
haftmann@40466:   abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222: where
kaliszyk@44204:   "abs x = Abs (Collect (R x))"
kaliszyk@35222: 
kaliszyk@35222: definition
haftmann@40466:   rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222: where
kaliszyk@44204:   "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222: 
kaliszyk@44204: lemma some_collect:
kaliszyk@37493:   assumes "R r r"
kaliszyk@44204:   shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204:   apply simp
kaliszyk@44204:   by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222: 
kaliszyk@35222: lemma Quotient:
kuncar@47308:   shows "Quotient3 R abs rep"
kuncar@47308:   unfolding Quotient3_def abs_def rep_def
kaliszyk@37493:   proof (intro conjI allI)
kaliszyk@37493:     fix a r s
kaliszyk@44204:     show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204:       obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204:       have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204:       then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204:       then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204:         using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493:     qed
kaliszyk@44204:     have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204:     then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204:     have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242:     proof -
haftmann@44242:       assume "R r r" and "R s s"
haftmann@44242:       then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242:         by (metis abs_inverse)
haftmann@44242:       also have "Collect (R r) = Collect (R s) \<longleftrightarrow> (\<lambda>A x. x \<in> A) (Collect (R r)) = (\<lambda>A x. x \<in> A) (Collect (R s))"
haftmann@44242:         by rule simp_all
haftmann@44242:       finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242:     qed
kaliszyk@44204:     then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204:       using equivp[simplified part_equivp_def] by metis
kaliszyk@44204:     qed
haftmann@44242: 
kaliszyk@35222: end
kaliszyk@35222: 
kuncar@47096: subsection {* Quotient composition *}
kuncar@47096: 
kuncar@47308: lemma OOO_quotient3:
kuncar@47096:   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096:   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096:   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47096:   fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096:   fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
kuncar@47308:   assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes R2: "Quotient3 R2 Abs2 Rep2"
kuncar@47096:   assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
kuncar@47096:   assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
kuncar@47308:   shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47308: apply (rule Quotient3I)
kuncar@47308:    apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
kuncar@47096:   apply simp
griff@47434:   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
kuncar@47308:    apply (rule Quotient3_rep_reflp [OF R1])
griff@47434:   apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
kuncar@47308:    apply (rule Quotient3_rep_reflp [OF R1])
kuncar@47096:   apply (rule Rep1)
kuncar@47308:   apply (rule Quotient3_rep_reflp [OF R2])
kuncar@47096:  apply safe
kuncar@47096:     apply (rename_tac x y)
kuncar@47096:     apply (drule Abs1)
kuncar@47308:       apply (erule Quotient3_refl2 [OF R1])
kuncar@47308:      apply (erule Quotient3_refl1 [OF R1])
kuncar@47308:     apply (drule Quotient3_refl1 [OF R2], drule Rep1)
kuncar@47096:     apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
griff@47434:      apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
griff@47434:      apply (erule relcomppI)
kuncar@47308:      apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308:     apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308:     apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308:     apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308:     apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308:     apply (erule Quotient3_rel_abs [OF R1])
kuncar@47096:    apply (rename_tac x y)
kuncar@47096:    apply (drule Abs1)
kuncar@47308:      apply (erule Quotient3_refl2 [OF R1])
kuncar@47308:     apply (erule Quotient3_refl1 [OF R1])
kuncar@47308:    apply (drule Quotient3_refl2 [OF R2], drule Rep1)
kuncar@47096:    apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
griff@47434:     apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
griff@47434:     apply (erule relcomppI)
kuncar@47308:     apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308:    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308:    apply (rule conjI, erule Quotient3_refl2 [OF R1])
kuncar@47308:    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308:    apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308:    apply (erule Quotient3_rel_abs [OF R1, THEN sym])
kuncar@47096:   apply simp
kuncar@47308:   apply (rule Quotient3_rel_abs [OF R2])
kuncar@47308:   apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
kuncar@47308:   apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
kuncar@47096:   apply (erule Abs1)
kuncar@47308:    apply (erule Quotient3_refl2 [OF R1])
kuncar@47308:   apply (erule Quotient3_refl1 [OF R1])
kuncar@47096:  apply (rename_tac a b c d)
kuncar@47096:  apply simp
griff@47434:  apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
kuncar@47308:   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308:   apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308:   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
griff@47434:  apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
kuncar@47308:   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308:   apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
kuncar@47308:   apply (erule Quotient3_refl2 [OF R1])
kuncar@47096:  apply (rule Rep1)
kuncar@47096:  apply (drule Abs1)
kuncar@47308:    apply (erule Quotient3_refl2 [OF R1])
kuncar@47308:   apply (erule Quotient3_refl1 [OF R1])
kuncar@47096:  apply (drule Abs1)
kuncar@47308:   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308:  apply (erule Quotient3_refl1 [OF R1])
kuncar@47308:  apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308:  apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308:  apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308:  apply (drule Quotient3_rel_abs [OF R1])
kuncar@47096:  apply simp
kuncar@47308:  apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
kuncar@47096:  apply simp
kuncar@47096: done
kuncar@47096: 
kuncar@47308: lemma OOO_eq_quotient3:
kuncar@47096:   fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096:   fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096:   fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47308:   assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308:   assumes R2: "Quotient3 op= Abs2 Rep2"
kuncar@47308:   shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47096: using assms
kuncar@47308: by (rule OOO_quotient3) auto
kuncar@47096: 
kuncar@47362: subsection {* Quotient3 to Quotient *}
kuncar@47362: 
kuncar@47362: lemma Quotient3_to_Quotient:
kuncar@47362: assumes "Quotient3 R Abs Rep"
kuncar@47362: and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
kuncar@47362: shows "Quotient R Abs Rep T"
kuncar@47362: using assms unfolding Quotient3_def by (intro QuotientI) blast+
kuncar@47096: 
kuncar@47362: lemma Quotient3_to_Quotient_equivp:
kuncar@47362: assumes q: "Quotient3 R Abs Rep"
kuncar@47362: and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
kuncar@47362: and eR: "equivp R"
kuncar@47362: shows "Quotient R Abs Rep T"
kuncar@47362: proof (intro QuotientI)
kuncar@47362:   fix a
kuncar@47362:   show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
kuncar@47362: next
kuncar@47362:   fix a
kuncar@47362:   show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
kuncar@47362: next
kuncar@47362:   fix r s
kuncar@47362:   show "R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
kuncar@47362: next
kuncar@47362:   show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
kuncar@47096: qed
kuncar@47096: 
huffman@35294: subsection {* ML setup *}
kaliszyk@35222: 
kaliszyk@35222: text {* Auxiliary data for the quotient package *}
kaliszyk@35222: 
wenzelm@48891: ML_file "Tools/Quotient/quotient_info.ML"
wenzelm@41452: setup Quotient_Info.setup
kaliszyk@35222: 
kuncar@47308: declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
kaliszyk@35222: 
kuncar@47308: lemmas [quot_thm] = fun_quotient3
haftmann@44553: lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553: lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222: lemmas [quot_equiv] = identity_equivp
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: text {* Lemmas about simplifying id's. *}
kaliszyk@35222: lemmas [id_simps] =
kaliszyk@35222:   id_def[symmetric]
haftmann@40602:   map_fun_id
kaliszyk@35222:   id_apply
kaliszyk@35222:   id_o
kaliszyk@35222:   o_id
kaliszyk@35222:   eq_comp_r
kaliszyk@44413:   vimage_id
kaliszyk@35222: 
kaliszyk@35222: text {* Translation functions for the lifting process. *}
wenzelm@48891: ML_file "Tools/Quotient/quotient_term.ML"
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: text {* Definitions of the quotient types. *}
wenzelm@48891: ML_file "Tools/Quotient/quotient_type.ML"
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: text {* Definitions for quotient constants. *}
wenzelm@48891: ML_file "Tools/Quotient/quotient_def.ML"
kaliszyk@35222: 
kaliszyk@35222: 
kaliszyk@35222: text {*
kaliszyk@35222:   An auxiliary constant for recording some information
kaliszyk@35222:   about the lifted theorem in a tactic.
kaliszyk@35222: *}
kaliszyk@35222: definition
haftmann@40466:   Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466: where
haftmann@40466:   "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222: 
kaliszyk@35222: lemma
kaliszyk@35222:   shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222:   and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222:   and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222:   and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222:   and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222:   by (simp_all add: Quot_True_def ext)
kaliszyk@35222: 
kaliszyk@35222: lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222:   by (simp add: Quot_True_def)
kaliszyk@35222: 
kuncar@53011: context 
kuncar@53011: begin
kuncar@53011: interpretation lifting_syntax .
kaliszyk@35222: 
kaliszyk@35222: text {* Tactics for proving the lifted theorems *}
wenzelm@48891: ML_file "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222: 
kuncar@53011: end
kuncar@53011: 
huffman@35294: subsection {* Methods / Interface *}
kaliszyk@35222: 
kaliszyk@35222: method_setup lifting =
urbanc@37593:   {* Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@46468:        SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
wenzelm@42814:   {* lift theorems to quotient types *}
kaliszyk@35222: 
kaliszyk@35222: method_setup lifting_setup =
urbanc@37593:   {* Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@46468:        SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
wenzelm@42814:   {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222: 
urbanc@37593: method_setup descending =
wenzelm@46468:   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
wenzelm@42814:   {* decend theorems to the raw level *}
urbanc@37593: 
urbanc@37593: method_setup descending_setup =
wenzelm@46468:   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
wenzelm@42814:   {* set up the three goals for the decending theorems *}
urbanc@37593: 
urbanc@45782: method_setup partiality_descending =
wenzelm@46468:   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
urbanc@45782:   {* decend theorems to the raw level *}
urbanc@45782: 
urbanc@45782: method_setup partiality_descending_setup =
urbanc@45782:   {* Scan.succeed (fn ctxt => 
wenzelm@46468:        SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
urbanc@45782:   {* set up the three goals for the decending theorems *}
urbanc@45782: 
kaliszyk@35222: method_setup regularize =
wenzelm@46468:   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
wenzelm@42814:   {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222: 
kaliszyk@35222: method_setup injection =
wenzelm@46468:   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
wenzelm@42814:   {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222: 
kaliszyk@35222: method_setup cleaning =
wenzelm@46468:   {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
wenzelm@42814:   {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222: 
kaliszyk@35222: attribute_setup quot_lifted =
kaliszyk@35222:   {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814:   {* lift theorems to quotient types *}
kaliszyk@35222: 
kaliszyk@35222: no_notation
kuncar@53011:   rel_conj (infixr "OOO" 75)
kaliszyk@35222: 
kaliszyk@35222: end
haftmann@47488: