src/HOL/Quotient.thy
author wenzelm
Thu Mar 14 16:55:06 2019 +0100 (5 weeks ago)
changeset 69913 ca515cf61651
parent 69605 a96320074298
child 69990 eb072ce80f82
permissions -rw-r--r--
more specific keyword kinds;
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(*  Title:      HOL/Quotient.thy
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    Author:     Cezary Kaliszyk and Christian Urban
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*)
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section \<open>Definition of Quotient Types\<close>
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theory Quotient
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imports Lifting
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keywords
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  "print_quotmapsQ3" "print_quotientsQ3" "print_quotconsts" :: diag and
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  "quotient_type" :: thy_goal_defn and "/" and
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  "quotient_definition" :: thy_goal_defn
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begin
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text \<open>
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  Basic definition for equivalence relations
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  that are represented by predicates.
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\<close>
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text \<open>Composition of Relations\<close>
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abbreviation
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  rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
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where
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  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
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lemma eq_comp_r:
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  shows "((=) OOO R) = R"
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  by (auto simp add: fun_eq_iff)
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context includes lifting_syntax
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begin
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subsection \<open>Quotient Predicate\<close>
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definition
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  "Quotient3 R Abs Rep \<longleftrightarrow>
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     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
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     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
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lemma Quotient3I:
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  assumes "\<And>a. Abs (Rep a) = a"
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    and "\<And>a. R (Rep a) (Rep a)"
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    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
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  shows "Quotient3 R Abs Rep"
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  using assms unfolding Quotient3_def by blast
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context
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  fixes R Abs Rep
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  assumes a: "Quotient3 R Abs Rep"
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begin
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lemma Quotient3_abs_rep:
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  "Abs (Rep a) = a"
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  using a
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  unfolding Quotient3_def
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  by simp
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lemma Quotient3_rep_reflp:
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  "R (Rep a) (Rep a)"
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  using a
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  unfolding Quotient3_def
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  by blast
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lemma Quotient3_rel:
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  "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" \<comment> \<open>orientation does not loop on rewriting\<close>
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  using a
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  unfolding Quotient3_def
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  by blast
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lemma Quotient3_refl1: 
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  "R r s \<Longrightarrow> R r r"
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  using a unfolding Quotient3_def 
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  by fast
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lemma Quotient3_refl2: 
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  "R r s \<Longrightarrow> R s s"
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  using a unfolding Quotient3_def 
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  by fast
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lemma Quotient3_rel_rep:
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  "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
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  using a
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  unfolding Quotient3_def
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  by metis
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lemma Quotient3_rep_abs:
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  "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient3_def
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  by blast
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lemma Quotient3_rel_abs:
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  "R r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient3_def
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  by blast
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lemma Quotient3_symp:
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  "symp R"
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  using a unfolding Quotient3_def using sympI by metis
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lemma Quotient3_transp:
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  "transp R"
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  using a unfolding Quotient3_def using transpI by (metis (full_types))
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lemma Quotient3_part_equivp:
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  "part_equivp R"
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  by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp part_equivpI)
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lemma abs_o_rep:
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  "Abs \<circ> Rep = id"
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  unfolding fun_eq_iff
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  by (simp add: Quotient3_abs_rep)
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lemma equals_rsp:
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  assumes b: "R xa xb" "R ya yb"
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  shows "R xa ya = R xb yb"
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  using b Quotient3_symp Quotient3_transp
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  by (blast elim: sympE transpE)
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lemma rep_abs_rsp:
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  assumes b: "R x1 x2"
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  shows "R x1 (Rep (Abs x2))"
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  using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
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  by metis
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lemma rep_abs_rsp_left:
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  assumes b: "R x1 x2"
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  shows "R (Rep (Abs x1)) x2"
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  using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
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  by metis
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end
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lemma identity_quotient3:
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  "Quotient3 (=) id id"
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  unfolding Quotient3_def id_def
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  by blast
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lemma fun_quotient3:
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  assumes q1: "Quotient3 R1 abs1 rep1"
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  and     q2: "Quotient3 R2 abs2 rep2"
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  shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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proof -
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  have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
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    using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
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  moreover
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  have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
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    by (rule rel_funI)
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      (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
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        simp (no_asm) add: Quotient3_def, simp)
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  moreover
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  {
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  fix r s
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  have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
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        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
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  proof -
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    have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding rel_fun_def
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      using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
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      by (metis (full_types) part_equivp_def)
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    moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding rel_fun_def
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      using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
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      by (metis (full_types) part_equivp_def)
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    moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
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      apply(auto simp add: rel_fun_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
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    moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
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        (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
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      apply(auto simp add: rel_fun_def fun_eq_iff) using q1 q2 unfolding Quotient3_def 
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    by (metis map_fun_apply)
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    ultimately show ?thesis by blast
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 qed
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 }
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 ultimately show ?thesis by (intro Quotient3I) (assumption+)
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qed
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lemma lambda_prs:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
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  unfolding fun_eq_iff
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  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
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  by simp
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lemma lambda_prs1:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
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  unfolding fun_eq_iff
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  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
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  by simp
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text\<open>
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  In the following theorem R1 can be instantiated with anything,
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  but we know some of the types of the Rep and Abs functions;
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  so by solving Quotient assumptions we can get a unique R1 that
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  will be provable; which is why we need to use \<open>apply_rsp\<close> and
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  not the primed version\<close>
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lemma apply_rspQ3:
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  fixes f g::"'a \<Rightarrow> 'c"
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  assumes q: "Quotient3 R1 Abs1 Rep1"
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  and     a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: rel_funE)
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lemma apply_rspQ3'':
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  assumes "Quotient3 R Abs Rep"
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  and "(R ===> S) f f"
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  shows "S (f (Rep x)) (f (Rep x))"
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proof -
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  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
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  then show ?thesis using assms(2) by (auto intro: apply_rsp')
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qed
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subsection \<open>lemmas for regularisation of ball and bex\<close>
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lemma ball_reg_eqv:
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  fixes P :: "'a \<Rightarrow> bool"
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  assumes a: "equivp R"
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  shows "Ball (Respects R) P = (All P)"
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  using a
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  unfolding equivp_def
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  by (auto simp add: in_respects)
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lemma bex_reg_eqv:
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  fixes P :: "'a \<Rightarrow> bool"
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  assumes a: "equivp R"
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  shows "Bex (Respects R) P = (Ex P)"
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  using a
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  unfolding equivp_def
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  by (auto simp add: in_respects)
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lemma ball_reg_right:
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  assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
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  shows "All P \<longrightarrow> Ball R Q"
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  using a by fast
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lemma bex_reg_left:
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  assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
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  shows "Bex R Q \<longrightarrow> Ex P"
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  using a by fast
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lemma ball_reg_left:
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  assumes a: "equivp R"
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  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
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  using a by (metis equivp_reflp in_respects)
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lemma bex_reg_right:
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  assumes a: "equivp R"
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  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
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  using a by (metis equivp_reflp in_respects)
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lemma ball_reg_eqv_range:
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  fixes P::"'a \<Rightarrow> bool"
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  and x::"'a"
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  assumes a: "equivp R2"
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  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
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  apply(rule iffI)
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  apply(rule allI)
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  apply(drule_tac x="\<lambda>y. f x" in bspec)
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  apply(simp add: in_respects rel_fun_def)
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  apply(rule impI)
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  using a equivp_reflp_symp_transp[of "R2"]
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  apply (auto elim: equivpE reflpE)
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  done
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lemma bex_reg_eqv_range:
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  assumes a: "equivp R2"
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  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
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  apply(auto)
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  apply(rule_tac x="\<lambda>y. f x" in bexI)
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  apply(simp)
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  apply(simp add: Respects_def in_respects rel_fun_def)
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  apply(rule impI)
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  using a equivp_reflp_symp_transp[of "R2"]
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  apply (auto elim: equivpE reflpE)
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  done
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(* Next four lemmas are unused *)
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lemma all_reg:
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  assumes a: "\<forall>x :: 'a. (P x \<longrightarrow> Q x)"
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  and     b: "All P"
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  shows "All Q"
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  using a b by fast
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lemma ex_reg:
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  assumes a: "\<forall>x :: 'a. (P x \<longrightarrow> Q x)"
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  and     b: "Ex P"
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  shows "Ex Q"
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  using a b by fast
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lemma ball_reg:
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  assumes a: "\<forall>x :: 'a. (x \<in> R \<longrightarrow> P x \<longrightarrow> Q x)"
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  and     b: "Ball R P"
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  shows "Ball R Q"
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  using a b by fast
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lemma bex_reg:
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  assumes a: "\<forall>x :: 'a. (x \<in> R \<longrightarrow> P x \<longrightarrow> Q x)"
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  and     b: "Bex R P"
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  shows "Bex R Q"
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  using a b by fast
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lemma ball_all_comm:
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  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
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  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
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  using assms by auto
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lemma bex_ex_comm:
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  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
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  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
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  using assms by auto
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subsection \<open>Bounded abstraction\<close>
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definition
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  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
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where
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  "x \<in> p \<Longrightarrow> Babs p m x = m x"
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lemma babs_rsp:
kuncar@47308
   325
  assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222
   326
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   327
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
blanchet@55945
   328
  apply (auto simp add: Babs_def in_respects rel_fun_def)
kaliszyk@35222
   329
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
blanchet@55945
   330
  using a apply (simp add: Babs_def rel_fun_def)
blanchet@55945
   331
  apply (simp add: in_respects rel_fun_def)
kuncar@47308
   332
  using Quotient3_rel[OF q]
kaliszyk@35222
   333
  by metis
kaliszyk@35222
   334
kaliszyk@35222
   335
lemma babs_prs:
kuncar@47308
   336
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   337
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222
   338
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   339
  apply (rule ext)
haftmann@40466
   340
  apply (simp add:)
kaliszyk@35222
   341
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kuncar@47308
   342
  apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
kuncar@47308
   343
  apply (simp add: in_respects Quotient3_rel_rep[OF q1])
kaliszyk@35222
   344
  done
kaliszyk@35222
   345
kaliszyk@35222
   346
lemma babs_simp:
kuncar@47308
   347
  assumes q: "Quotient3 R1 Abs Rep"
kaliszyk@35222
   348
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   349
  apply(rule iffI)
kaliszyk@35222
   350
  apply(simp_all only: babs_rsp[OF q])
blanchet@55945
   351
  apply(auto simp add: Babs_def rel_fun_def)
lp15@68615
   352
  apply(metis Babs_def in_respects  Quotient3_rel[OF q])
lp15@68615
   353
  done
kaliszyk@35222
   354
kaliszyk@35222
   355
(* If a user proves that a particular functional relation
kaliszyk@35222
   356
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   357
lemma babs_reg_eqv:
kaliszyk@35222
   358
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   359
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   360
kaliszyk@35222
   361
kaliszyk@35222
   362
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   363
lemma ball_rsp:
nipkow@67399
   364
  assumes a: "(R ===> (=)) f g"
kaliszyk@35222
   365
  shows "Ball (Respects R) f = Ball (Respects R) g"
blanchet@55945
   366
  using a by (auto simp add: Ball_def in_respects elim: rel_funE)
kaliszyk@35222
   367
kaliszyk@35222
   368
lemma bex_rsp:
nipkow@67399
   369
  assumes a: "(R ===> (=)) f g"
kaliszyk@35222
   370
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
blanchet@55945
   371
  using a by (auto simp add: Bex_def in_respects elim: rel_funE)
kaliszyk@35222
   372
kaliszyk@35222
   373
lemma bex1_rsp:
nipkow@67399
   374
  assumes a: "(R ===> (=)) f g"
kaliszyk@35222
   375
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
blanchet@55945
   376
  using a by (auto elim: rel_funE simp add: Ex1_def in_respects) 
kaliszyk@35222
   377
kaliszyk@35222
   378
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   379
lemma all_prs:
kuncar@47308
   380
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   381
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kuncar@47308
   382
  using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   383
  by metis
kaliszyk@35222
   384
kaliszyk@35222
   385
lemma ex_prs:
kuncar@47308
   386
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   387
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kuncar@47308
   388
  using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   389
  by metis
kaliszyk@35222
   390
wenzelm@61799
   391
subsection \<open>\<open>Bex1_rel\<close> quantifier\<close>
kaliszyk@35222
   392
kaliszyk@35222
   393
definition
kaliszyk@35222
   394
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   395
where
kaliszyk@35222
   396
  "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
   397
kaliszyk@35222
   398
lemma bex1_rel_aux:
kaliszyk@35222
   399
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   400
  unfolding Bex1_rel_def
lp15@68615
   401
  by (metis in_respects)
kaliszyk@35222
   402
kaliszyk@35222
   403
lemma bex1_rel_aux2:
kaliszyk@35222
   404
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   405
  unfolding Bex1_rel_def
lp15@68615
   406
  by (metis in_respects)
kaliszyk@35222
   407
kaliszyk@35222
   408
lemma bex1_rel_rsp:
kuncar@47308
   409
  assumes a: "Quotient3 R absf repf"
nipkow@67399
   410
  shows "((R ===> (=)) ===> (=)) (Bex1_rel R) (Bex1_rel R)"
lp15@68615
   411
  unfolding rel_fun_def by (metis bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   412
kaliszyk@35222
   413
lemma ex1_prs:
lp15@68616
   414
  assumes "Quotient3 R absf repf"
kaliszyk@35222
   415
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
lp15@68616
   416
  apply (auto simp: Bex1_rel_def Respects_def)
lp15@68616
   417
  apply (metis Quotient3_def assms)
lp15@68616
   418
  apply (metis (full_types) Quotient3_def assms)
lp15@68616
   419
  by (meson Quotient3_rel assms)
kaliszyk@35222
   420
kaliszyk@38702
   421
lemma bex1_bexeq_reg:
kaliszyk@38702
   422
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
nipkow@56073
   423
  by (auto simp add: Ex1_def Bex1_rel_def Bex_def Ball_def in_respects)
lp15@68616
   424
 
kaliszyk@38702
   425
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   426
  assumes a: "equivp R"
kaliszyk@38702
   427
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   428
  using equivp_reflp[OF a]
lp15@68616
   429
  by (metis (full_types) Bex1_rel_def in_respects)
kaliszyk@38702
   430
wenzelm@60758
   431
subsection \<open>Various respects and preserve lemmas\<close>
kaliszyk@35222
   432
kaliszyk@35222
   433
lemma quot_rel_rsp:
kuncar@47308
   434
  assumes a: "Quotient3 R Abs Rep"
nipkow@67399
   435
  shows "(R ===> R ===> (=)) R R"
blanchet@55945
   436
  apply(rule rel_funI)+
lp15@68616
   437
  by (meson assms equals_rsp)
kaliszyk@35222
   438
kaliszyk@35222
   439
lemma o_prs:
kuncar@47308
   440
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   441
  and     q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308
   442
  and     q3: "Quotient3 R3 Abs3 Rep3"
nipkow@67399
   443
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) (\<circ>) = (\<circ>)"
nipkow@67399
   444
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) (\<circ>) = (\<circ>)"
kuncar@47308
   445
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
haftmann@40466
   446
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   447
kaliszyk@35222
   448
lemma o_rsp:
nipkow@67399
   449
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) (\<circ>) (\<circ>)"
nipkow@67399
   450
  "((=) ===> (R1 ===> (=)) ===> R1 ===> (=)) (\<circ>) (\<circ>)"
blanchet@55945
   451
  by (force elim: rel_funE)+
kaliszyk@35222
   452
kaliszyk@35222
   453
lemma cond_prs:
kuncar@47308
   454
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   455
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kuncar@47308
   456
  using a unfolding Quotient3_def by auto
kaliszyk@35222
   457
kaliszyk@35222
   458
lemma if_prs:
kuncar@47308
   459
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   460
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kuncar@47308
   461
  using Quotient3_abs_rep[OF q]
nipkow@39302
   462
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   463
kaliszyk@35222
   464
lemma if_rsp:
kuncar@47308
   465
  assumes q: "Quotient3 R Abs Rep"
nipkow@67399
   466
  shows "((=) ===> R ===> R ===> R) If If"
huffman@44921
   467
  by force
kaliszyk@35222
   468
kaliszyk@35222
   469
lemma let_prs:
kuncar@47308
   470
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   471
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@37049
   472
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kuncar@47308
   473
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
nipkow@39302
   474
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   475
kaliszyk@35222
   476
lemma let_rsp:
kaliszyk@37049
   477
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
blanchet@55945
   478
  by (force elim: rel_funE)
kaliszyk@35222
   479
kaliszyk@39669
   480
lemma id_rsp:
kaliszyk@39669
   481
  shows "(R ===> R) id id"
huffman@44921
   482
  by auto
kaliszyk@39669
   483
kaliszyk@39669
   484
lemma id_prs:
kuncar@47308
   485
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@39669
   486
  shows "(Rep ---> Abs) id = id"
kuncar@47308
   487
  by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
kaliszyk@39669
   488
kuncar@53011
   489
end
kaliszyk@39669
   490
kaliszyk@35222
   491
locale quot_type =
kaliszyk@35222
   492
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   493
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   494
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   495
  assumes equivp: "part_equivp R"
kaliszyk@44204
   496
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   497
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   498
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   499
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   500
begin
kaliszyk@35222
   501
kaliszyk@35222
   502
definition
haftmann@40466
   503
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   504
where
kaliszyk@44204
   505
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   506
kaliszyk@35222
   507
definition
haftmann@40466
   508
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   509
where
kaliszyk@44204
   510
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   511
kaliszyk@44204
   512
lemma some_collect:
kaliszyk@37493
   513
  assumes "R r r"
kaliszyk@44204
   514
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   515
  apply simp
kaliszyk@44204
   516
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   517
kaliszyk@35222
   518
lemma Quotient:
kuncar@47308
   519
  shows "Quotient3 R abs rep"
kuncar@47308
   520
  unfolding Quotient3_def abs_def rep_def
kaliszyk@37493
   521
  proof (intro conjI allI)
kaliszyk@37493
   522
    fix a r s
kaliszyk@44204
   523
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   524
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   525
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   526
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   527
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
wenzelm@60758
   528
        using part_equivp_transp[OF equivp] by (metis \<open>R (SOME x. x \<in> Rep a) x\<close>)
kaliszyk@37493
   529
    qed
kaliszyk@44204
   530
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   531
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   532
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   533
    proof -
haftmann@44242
   534
      assume "R r r" and "R s s"
haftmann@44242
   535
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   536
        by (metis abs_inverse)
haftmann@44242
   537
      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
   538
        by rule simp_all
haftmann@44242
   539
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   540
    qed
kaliszyk@44204
   541
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   542
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   543
    qed
haftmann@44242
   544
kaliszyk@35222
   545
end
kaliszyk@35222
   546
wenzelm@60758
   547
subsection \<open>Quotient composition\<close>
kuncar@47096
   548
lp15@68616
   549
kuncar@47308
   550
lemma OOO_quotient3:
kuncar@47096
   551
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   552
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   553
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47096
   554
  fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   555
  fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
kuncar@47308
   556
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   557
  assumes R2: "Quotient3 R2 Abs2 Rep2"
kuncar@47096
   558
  assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
kuncar@47096
   559
  assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
kuncar@47308
   560
  shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
lp15@68616
   561
proof -
lp15@68616
   562
  have *: "(R1 OOO R2') r r \<and> (R1 OOO R2') s s \<and> (Abs2 \<circ> Abs1) r = (Abs2 \<circ> Abs1) s 
lp15@68616
   563
           \<longleftrightarrow> (R1 OOO R2') r s" for r s
lp15@68616
   564
    apply safe
lp15@68616
   565
    subgoal for a b c d
lp15@68616
   566
      apply simp
lp15@68616
   567
      apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
lp15@68616
   568
      using Quotient3_refl1 R1 rep_abs_rsp apply fastforce
lp15@68616
   569
      apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI)
lp15@68616
   570
       apply (metis (full_types) Rep1 Abs1 Quotient3_rel R2  Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1] Quotient3_rel_abs [OF R1])
lp15@68616
   571
      by (metis Quotient3_rel R1 rep_abs_rsp_left)
lp15@68616
   572
    subgoal for x y
lp15@68616
   573
      apply (drule Abs1)
lp15@68616
   574
        apply (erule Quotient3_refl2 [OF R1])
lp15@68616
   575
       apply (erule Quotient3_refl1 [OF R1])
lp15@68616
   576
      apply (drule Quotient3_refl1 [OF R2], drule Rep1)
lp15@68616
   577
      by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
lp15@68616
   578
    subgoal for x y
lp15@68616
   579
      apply (drule Abs1)
lp15@68616
   580
        apply (erule Quotient3_refl2 [OF R1])
lp15@68616
   581
       apply (erule Quotient3_refl1 [OF R1])
lp15@68616
   582
      apply (drule Quotient3_refl2 [OF R2], drule Rep1)
lp15@68616
   583
      by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
lp15@68616
   584
    subgoal for x y
lp15@68616
   585
      by simp (metis (full_types) Abs1 Quotient3_rel R1 R2)
lp15@68616
   586
    done
lp15@68616
   587
  show ?thesis
lp15@68616
   588
    apply (rule Quotient3I)
lp15@68616
   589
    using * apply (simp_all add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
lp15@68616
   590
    apply (metis Quotient3_rep_reflp R1 R2 Rep1 relcompp.relcompI)
lp15@68616
   591
    done
lp15@68616
   592
qed
kuncar@47096
   593
kuncar@47308
   594
lemma OOO_eq_quotient3:
kuncar@47096
   595
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   596
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   597
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47308
   598
  assumes R1: "Quotient3 R1 Abs1 Rep1"
nipkow@67399
   599
  assumes R2: "Quotient3 (=) Abs2 Rep2"
nipkow@67399
   600
  shows "Quotient3 (R1 OOO (=)) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47096
   601
using assms
kuncar@47308
   602
by (rule OOO_quotient3) auto
kuncar@47096
   603
wenzelm@60758
   604
subsection \<open>Quotient3 to Quotient\<close>
kuncar@47362
   605
kuncar@47362
   606
lemma Quotient3_to_Quotient:
kuncar@47362
   607
assumes "Quotient3 R Abs Rep"
kuncar@47362
   608
and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
kuncar@47362
   609
shows "Quotient R Abs Rep T"
kuncar@47362
   610
using assms unfolding Quotient3_def by (intro QuotientI) blast+
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   611
kuncar@47362
   612
lemma Quotient3_to_Quotient_equivp:
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   613
assumes q: "Quotient3 R Abs Rep"
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   614
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
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   615
and eR: "equivp R"
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   616
shows "Quotient R Abs Rep T"
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   617
proof (intro QuotientI)
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   618
  fix a
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   619
  show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
kuncar@47362
   620
next
kuncar@47362
   621
  fix a
kuncar@47362
   622
  show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
kuncar@47362
   623
next
kuncar@47362
   624
  fix r s
kuncar@47362
   625
  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
   626
next
kuncar@47362
   627
  show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
kuncar@47096
   628
qed
kuncar@47096
   629
wenzelm@60758
   630
subsection \<open>ML setup\<close>
kaliszyk@35222
   631
wenzelm@60758
   632
text \<open>Auxiliary data for the quotient package\<close>
kaliszyk@35222
   633
wenzelm@57960
   634
named_theorems quot_equiv "equivalence relation theorems"
wenzelm@59028
   635
  and quot_respect "respectfulness theorems"
wenzelm@59028
   636
  and quot_preserve "preservation theorems"
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   637
  and id_simps "identity simp rules for maps"
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   638
  and quot_thm "quotient theorems"
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   639
ML_file \<open>Tools/Quotient/quotient_info.ML\<close>
kaliszyk@35222
   640
blanchet@55945
   641
declare [[mapQ3 "fun" = (rel_fun, fun_quotient3)]]
kaliszyk@35222
   642
kuncar@47308
   643
lemmas [quot_thm] = fun_quotient3
haftmann@44553
   644
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553
   645
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222
   646
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   647
kaliszyk@35222
   648
wenzelm@60758
   649
text \<open>Lemmas about simplifying id's.\<close>
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   650
lemmas [id_simps] =
kaliszyk@35222
   651
  id_def[symmetric]
haftmann@40602
   652
  map_fun_id
kaliszyk@35222
   653
  id_apply
kaliszyk@35222
   654
  id_o
kaliszyk@35222
   655
  o_id
kaliszyk@35222
   656
  eq_comp_r
kaliszyk@44413
   657
  vimage_id
kaliszyk@35222
   658
wenzelm@60758
   659
text \<open>Translation functions for the lifting process.\<close>
wenzelm@69605
   660
ML_file \<open>Tools/Quotient/quotient_term.ML\<close>
kaliszyk@35222
   661
kaliszyk@35222
   662
wenzelm@60758
   663
text \<open>Definitions of the quotient types.\<close>
wenzelm@69605
   664
ML_file \<open>Tools/Quotient/quotient_type.ML\<close>
kaliszyk@35222
   665
kaliszyk@35222
   666
wenzelm@60758
   667
text \<open>Definitions for quotient constants.\<close>
wenzelm@69605
   668
ML_file \<open>Tools/Quotient/quotient_def.ML\<close>
kaliszyk@35222
   669
kaliszyk@35222
   670
wenzelm@60758
   671
text \<open>
kaliszyk@35222
   672
  An auxiliary constant for recording some information
kaliszyk@35222
   673
  about the lifted theorem in a tactic.
wenzelm@60758
   674
\<close>
kaliszyk@35222
   675
definition
haftmann@40466
   676
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   677
where
haftmann@40466
   678
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   679
kaliszyk@35222
   680
lemma
kaliszyk@35222
   681
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   682
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   683
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   684
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   685
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   686
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   687
kaliszyk@35222
   688
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   689
  by (simp add: Quot_True_def)
kaliszyk@35222
   690
wenzelm@63343
   691
context includes lifting_syntax
kuncar@53011
   692
begin
kaliszyk@35222
   693
wenzelm@60758
   694
text \<open>Tactics for proving the lifted theorems\<close>
wenzelm@69605
   695
ML_file \<open>Tools/Quotient/quotient_tacs.ML\<close>
kaliszyk@35222
   696
kuncar@53011
   697
end
kuncar@53011
   698
wenzelm@60758
   699
subsection \<open>Methods / Interface\<close>
kaliszyk@35222
   700
kaliszyk@35222
   701
method_setup lifting =
wenzelm@60758
   702
  \<open>Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@60758
   703
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms))\<close>
wenzelm@60758
   704
  \<open>lift theorems to quotient types\<close>
kaliszyk@35222
   705
kaliszyk@35222
   706
method_setup lifting_setup =
wenzelm@60758
   707
  \<open>Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@60758
   708
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm))\<close>
wenzelm@60758
   709
  \<open>set up the three goals for the quotient lifting procedure\<close>
kaliszyk@35222
   710
urbanc@37593
   711
method_setup descending =
wenzelm@60758
   712
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt []))\<close>
wenzelm@60758
   713
  \<open>decend theorems to the raw level\<close>
urbanc@37593
   714
urbanc@37593
   715
method_setup descending_setup =
wenzelm@60758
   716
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt []))\<close>
wenzelm@60758
   717
  \<open>set up the three goals for the decending theorems\<close>
urbanc@37593
   718
urbanc@45782
   719
method_setup partiality_descending =
wenzelm@60758
   720
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt []))\<close>
wenzelm@60758
   721
  \<open>decend theorems to the raw level\<close>
urbanc@45782
   722
urbanc@45782
   723
method_setup partiality_descending_setup =
wenzelm@60758
   724
  \<open>Scan.succeed (fn ctxt => 
wenzelm@60758
   725
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt []))\<close>
wenzelm@60758
   726
  \<open>set up the three goals for the decending theorems\<close>
urbanc@45782
   727
kaliszyk@35222
   728
method_setup regularize =
wenzelm@60758
   729
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt))\<close>
wenzelm@60758
   730
  \<open>prove the regularization goals from the quotient lifting procedure\<close>
kaliszyk@35222
   731
kaliszyk@35222
   732
method_setup injection =
wenzelm@60758
   733
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt))\<close>
wenzelm@60758
   734
  \<open>prove the rep/abs injection goals from the quotient lifting procedure\<close>
kaliszyk@35222
   735
kaliszyk@35222
   736
method_setup cleaning =
wenzelm@60758
   737
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt))\<close>
wenzelm@60758
   738
  \<open>prove the cleaning goals from the quotient lifting procedure\<close>
kaliszyk@35222
   739
kaliszyk@35222
   740
attribute_setup quot_lifted =
wenzelm@60758
   741
  \<open>Scan.succeed Quotient_Tacs.lifted_attrib\<close>
wenzelm@60758
   742
  \<open>lift theorems to quotient types\<close>
kaliszyk@35222
   743
kaliszyk@35222
   744
no_notation
kuncar@53011
   745
  rel_conj (infixr "OOO" 75)
kaliszyk@35222
   746
kaliszyk@35222
   747
end
haftmann@47488
   748