src/HOL/Quotient.thy
author huffman
Thu Apr 19 10:49:47 2012 +0200 (2012-04-19)
changeset 47579 28f6f4ad69bf
parent 47544 e455cdaac479
child 47626 f7b1034cb9ce
permissions -rw-r--r--
tuned lemmas (v)image_id;
removed duplicate of vimage_id
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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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header {* Definition of Quotient Types *}
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theory Quotient
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imports Plain Hilbert_Choice Equiv_Relations 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 and "/" and
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  "quotient_definition" :: thy_goal
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uses
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  ("Tools/Quotient/quotient_info.ML")
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  ("Tools/Quotient/quotient_type.ML")
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  ("Tools/Quotient/quotient_def.ML")
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  ("Tools/Quotient/quotient_term.ML")
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  ("Tools/Quotient/quotient_tacs.ML")
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begin
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text {*
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  Basic definition for equivalence relations
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  that are represented by predicates.
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*}
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text {* Composition of Relations *}
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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 "((op =) OOO R) = R"
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  by (auto simp add: fun_eq_iff)
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subsection {* set map (vimage) and set relation *}
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definition "set_rel R xs ys \<equiv> \<forall>x y. R x y \<longrightarrow> x \<in> xs \<longleftrightarrow> y \<in> ys"
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lemma set_rel_eq:
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  "set_rel op = = op ="
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  by (subst fun_eq_iff, subst fun_eq_iff) (simp add: set_eq_iff set_rel_def)
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lemma set_rel_equivp:
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  assumes e: "equivp R"
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  shows "set_rel R xs ys \<longleftrightarrow> xs = ys \<and> (\<forall>x y. x \<in> xs \<longrightarrow> R x y \<longrightarrow> y \<in> xs)"
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  unfolding set_rel_def
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  using equivp_reflp[OF e]
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  by auto (metis, metis equivp_symp[OF e])
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subsection {* Quotient Predicate *}
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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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lemma Quotient3_abs_rep:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
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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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  assumes a: "Quotient3 R Abs Rep" 
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  shows "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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  assumes a: "Quotient3 R Abs Rep" 
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "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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  assumes a: "Quotient3 R Abs Rep"
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  shows "part_equivp R"
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by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp a part_equivpI)
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lemma identity_quotient3:
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  shows "Quotient3 (op =) 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 fun_relI)
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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 fun_rel_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 fun_rel_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: fun_rel_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: fun_rel_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 abs_o_rep:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "Abs o Rep = id"
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  unfolding fun_eq_iff
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  by (simp add: Quotient3_abs_rep[OF a])
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lemma equals_rsp:
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  assumes q: "Quotient3 R Abs Rep"
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  and     a: "R xa xb" "R ya yb"
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  shows "R xa ya = R xb yb"
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  using a Quotient3_symp[OF q] Quotient3_transp[OF q]
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  by (blast elim: sympE transpE)
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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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lemma rep_abs_rsp:
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  assumes q: "Quotient3 R Abs Rep"
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  and     a: "R x1 x2"
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  shows "R x1 (Rep (Abs x2))"
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  using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
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  by metis
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lemma rep_abs_rsp_left:
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  assumes q: "Quotient3 R Abs Rep"
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  and     a: "R x1 x2"
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  shows "R (Rep (Abs x1)) x2"
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  using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
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  by metis
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text{*
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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 @{text apply_rsp} and
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  not the primed version *}
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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: fun_relE)
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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 {* lemmas for regularisation of ball and bex *}
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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 fun_rel_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 fun_rel_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: "!x :: 'a. (P x --> 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:
kaliszyk@35222
   315
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   316
  and     b: "Ex P"
kaliszyk@35222
   317
  shows "Ex Q"
huffman@44921
   318
  using a b by fast
kaliszyk@35222
   319
kaliszyk@35222
   320
lemma ball_reg:
haftmann@44553
   321
  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222
   322
  and     b: "Ball R P"
kaliszyk@35222
   323
  shows "Ball R Q"
huffman@44921
   324
  using a b by fast
kaliszyk@35222
   325
kaliszyk@35222
   326
lemma bex_reg:
haftmann@44553
   327
  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222
   328
  and     b: "Bex R P"
kaliszyk@35222
   329
  shows "Bex R Q"
huffman@44921
   330
  using a b by fast
kaliszyk@35222
   331
kaliszyk@35222
   332
kaliszyk@35222
   333
lemma ball_all_comm:
kaliszyk@35222
   334
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   335
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   336
  using assms by auto
kaliszyk@35222
   337
kaliszyk@35222
   338
lemma bex_ex_comm:
kaliszyk@35222
   339
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   340
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   341
  using assms by auto
kaliszyk@35222
   342
huffman@35294
   343
subsection {* Bounded abstraction *}
kaliszyk@35222
   344
kaliszyk@35222
   345
definition
haftmann@40466
   346
  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   347
where
kaliszyk@35222
   348
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   349
kaliszyk@35222
   350
lemma babs_rsp:
kuncar@47308
   351
  assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222
   352
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   353
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   354
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   355
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   356
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   357
  apply (simp add: in_respects fun_rel_def)
kuncar@47308
   358
  using Quotient3_rel[OF q]
kaliszyk@35222
   359
  by metis
kaliszyk@35222
   360
kaliszyk@35222
   361
lemma babs_prs:
kuncar@47308
   362
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   363
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222
   364
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   365
  apply (rule ext)
haftmann@40466
   366
  apply (simp add:)
kaliszyk@35222
   367
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kuncar@47308
   368
  apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
kuncar@47308
   369
  apply (simp add: in_respects Quotient3_rel_rep[OF q1])
kaliszyk@35222
   370
  done
kaliszyk@35222
   371
kaliszyk@35222
   372
lemma babs_simp:
kuncar@47308
   373
  assumes q: "Quotient3 R1 Abs Rep"
kaliszyk@35222
   374
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   375
  apply(rule iffI)
kaliszyk@35222
   376
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   377
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   378
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   379
  apply(metis Babs_def)
kaliszyk@35222
   380
  apply (simp add: in_respects)
kuncar@47308
   381
  using Quotient3_rel[OF q]
kaliszyk@35222
   382
  by metis
kaliszyk@35222
   383
kaliszyk@35222
   384
(* If a user proves that a particular functional relation
kaliszyk@35222
   385
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   386
lemma babs_reg_eqv:
kaliszyk@35222
   387
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   388
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   389
kaliszyk@35222
   390
kaliszyk@35222
   391
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   392
lemma ball_rsp:
kaliszyk@35222
   393
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   394
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   395
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   396
kaliszyk@35222
   397
lemma bex_rsp:
kaliszyk@35222
   398
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   399
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   400
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   401
kaliszyk@35222
   402
lemma bex1_rsp:
kaliszyk@35222
   403
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   404
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   405
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   406
kaliszyk@35222
   407
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   408
lemma all_prs:
kuncar@47308
   409
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   410
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kuncar@47308
   411
  using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   412
  by metis
kaliszyk@35222
   413
kaliszyk@35222
   414
lemma ex_prs:
kuncar@47308
   415
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   416
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kuncar@47308
   417
  using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   418
  by metis
kaliszyk@35222
   419
huffman@35294
   420
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   421
kaliszyk@35222
   422
definition
kaliszyk@35222
   423
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   424
where
kaliszyk@35222
   425
  "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
   426
kaliszyk@35222
   427
lemma bex1_rel_aux:
kaliszyk@35222
   428
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   429
  unfolding Bex1_rel_def
kaliszyk@35222
   430
  apply (erule conjE)+
kaliszyk@35222
   431
  apply (erule bexE)
kaliszyk@35222
   432
  apply rule
kaliszyk@35222
   433
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   434
  apply metis
kaliszyk@35222
   435
  apply metis
kaliszyk@35222
   436
  apply rule+
kaliszyk@35222
   437
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   438
  prefer 2
kaliszyk@35222
   439
  apply (metis)
kaliszyk@35222
   440
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   441
  prefer 2
kaliszyk@35222
   442
  apply (metis)
kaliszyk@35222
   443
  apply (metis in_respects)
kaliszyk@35222
   444
  done
kaliszyk@35222
   445
kaliszyk@35222
   446
lemma bex1_rel_aux2:
kaliszyk@35222
   447
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   448
  unfolding Bex1_rel_def
kaliszyk@35222
   449
  apply (erule conjE)+
kaliszyk@35222
   450
  apply (erule bexE)
kaliszyk@35222
   451
  apply rule
kaliszyk@35222
   452
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   453
  apply metis
kaliszyk@35222
   454
  apply metis
kaliszyk@35222
   455
  apply rule+
kaliszyk@35222
   456
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   457
  prefer 2
kaliszyk@35222
   458
  apply (metis)
kaliszyk@35222
   459
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   460
  prefer 2
kaliszyk@35222
   461
  apply (metis)
kaliszyk@35222
   462
  apply (metis in_respects)
kaliszyk@35222
   463
  done
kaliszyk@35222
   464
kaliszyk@35222
   465
lemma bex1_rel_rsp:
kuncar@47308
   466
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   467
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   468
  apply (simp add: fun_rel_def)
kaliszyk@35222
   469
  apply clarify
kaliszyk@35222
   470
  apply rule
kaliszyk@35222
   471
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   472
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   473
  apply assumption
kaliszyk@35222
   474
  done
kaliszyk@35222
   475
kaliszyk@35222
   476
kaliszyk@35222
   477
lemma ex1_prs:
kuncar@47308
   478
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   479
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   480
apply (simp add:)
kaliszyk@35222
   481
apply (subst Bex1_rel_def)
kaliszyk@35222
   482
apply (subst Bex_def)
kaliszyk@35222
   483
apply (subst Ex1_def)
kaliszyk@35222
   484
apply simp
kaliszyk@35222
   485
apply rule
kaliszyk@35222
   486
 apply (erule conjE)+
kaliszyk@35222
   487
 apply (erule_tac exE)
kaliszyk@35222
   488
 apply (erule conjE)
kaliszyk@35222
   489
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   490
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   491
  apply (simp)
kaliszyk@35222
   492
  apply rule+
kuncar@47308
   493
  using a unfolding Quotient3_def
kaliszyk@35222
   494
  apply metis
kaliszyk@35222
   495
 apply rule+
kaliszyk@35222
   496
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   497
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   498
   apply simp
kaliszyk@35222
   499
  apply (simp add: in_respects)
kaliszyk@35222
   500
 apply (simp add: in_respects)
kaliszyk@35222
   501
apply (erule_tac exE)
kaliszyk@35222
   502
 apply rule
kaliszyk@35222
   503
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   504
 apply (simp only: in_respects)
kaliszyk@35222
   505
  apply rule
kuncar@47308
   506
 apply (metis Quotient3_rel_rep[OF a])
kuncar@47308
   507
using a unfolding Quotient3_def apply (simp)
kaliszyk@35222
   508
apply rule+
kuncar@47308
   509
using a unfolding Quotient3_def in_respects
kaliszyk@35222
   510
apply metis
kaliszyk@35222
   511
done
kaliszyk@35222
   512
kaliszyk@38702
   513
lemma bex1_bexeq_reg:
kaliszyk@38702
   514
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   515
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   516
  apply clarify
kaliszyk@35222
   517
  apply auto
kaliszyk@35222
   518
  apply (rule bexI)
kaliszyk@35222
   519
  apply assumption
kaliszyk@35222
   520
  apply (simp add: in_respects)
kaliszyk@35222
   521
  apply (simp add: in_respects)
kaliszyk@35222
   522
  apply auto
kaliszyk@35222
   523
  done
kaliszyk@35222
   524
kaliszyk@38702
   525
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   526
  assumes a: "equivp R"
kaliszyk@38702
   527
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   528
  using equivp_reflp[OF a]
kaliszyk@38702
   529
  apply (intro impI)
kaliszyk@38702
   530
  apply (elim ex1E)
kaliszyk@38702
   531
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   532
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   533
  apply (subst in_respects)
kaliszyk@38702
   534
  apply (rule conjI)
kaliszyk@38702
   535
  apply assumption
kaliszyk@38702
   536
  apply assumption
kaliszyk@38702
   537
  apply clarify
kaliszyk@38702
   538
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   539
  apply simp
kaliszyk@38702
   540
  done
kaliszyk@38702
   541
huffman@35294
   542
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   543
kaliszyk@35222
   544
lemma quot_rel_rsp:
kuncar@47308
   545
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222
   546
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   547
  apply(rule fun_relI)+
kaliszyk@35222
   548
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   549
  apply(assumption)+
kaliszyk@35222
   550
  done
kaliszyk@35222
   551
kaliszyk@35222
   552
lemma o_prs:
kuncar@47308
   553
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   554
  and     q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308
   555
  and     q3: "Quotient3 R3 Abs3 Rep3"
kaliszyk@36215
   556
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   557
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kuncar@47308
   558
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
haftmann@40466
   559
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   560
kaliszyk@35222
   561
lemma o_rsp:
kaliszyk@36215
   562
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   563
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
huffman@44921
   564
  by (force elim: fun_relE)+
kaliszyk@35222
   565
kaliszyk@35222
   566
lemma cond_prs:
kuncar@47308
   567
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   568
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kuncar@47308
   569
  using a unfolding Quotient3_def by auto
kaliszyk@35222
   570
kaliszyk@35222
   571
lemma if_prs:
kuncar@47308
   572
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   573
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kuncar@47308
   574
  using Quotient3_abs_rep[OF q]
nipkow@39302
   575
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   576
kaliszyk@35222
   577
lemma if_rsp:
kuncar@47308
   578
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   579
  shows "(op = ===> R ===> R ===> R) If If"
huffman@44921
   580
  by force
kaliszyk@35222
   581
kaliszyk@35222
   582
lemma let_prs:
kuncar@47308
   583
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   584
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@37049
   585
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kuncar@47308
   586
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
nipkow@39302
   587
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   588
kaliszyk@35222
   589
lemma let_rsp:
kaliszyk@37049
   590
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
huffman@44921
   591
  by (force elim: fun_relE)
kaliszyk@35222
   592
kaliszyk@39669
   593
lemma id_rsp:
kaliszyk@39669
   594
  shows "(R ===> R) id id"
huffman@44921
   595
  by auto
kaliszyk@39669
   596
kaliszyk@39669
   597
lemma id_prs:
kuncar@47308
   598
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@39669
   599
  shows "(Rep ---> Abs) id = id"
kuncar@47308
   600
  by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
kaliszyk@39669
   601
kaliszyk@39669
   602
kaliszyk@35222
   603
locale quot_type =
kaliszyk@35222
   604
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   605
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   606
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   607
  assumes equivp: "part_equivp R"
kaliszyk@44204
   608
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   609
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   610
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   611
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   612
begin
kaliszyk@35222
   613
kaliszyk@35222
   614
definition
haftmann@40466
   615
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   616
where
kaliszyk@44204
   617
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   618
kaliszyk@35222
   619
definition
haftmann@40466
   620
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   621
where
kaliszyk@44204
   622
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   623
kaliszyk@44204
   624
lemma some_collect:
kaliszyk@37493
   625
  assumes "R r r"
kaliszyk@44204
   626
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   627
  apply simp
kaliszyk@44204
   628
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   629
kaliszyk@35222
   630
lemma Quotient:
kuncar@47308
   631
  shows "Quotient3 R abs rep"
kuncar@47308
   632
  unfolding Quotient3_def abs_def rep_def
kaliszyk@37493
   633
  proof (intro conjI allI)
kaliszyk@37493
   634
    fix a r s
kaliszyk@44204
   635
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   636
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   637
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   638
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   639
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204
   640
        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493
   641
    qed
kaliszyk@44204
   642
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   643
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   644
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   645
    proof -
haftmann@44242
   646
      assume "R r r" and "R s s"
haftmann@44242
   647
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   648
        by (metis abs_inverse)
haftmann@44242
   649
      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
   650
        by rule simp_all
haftmann@44242
   651
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   652
    qed
kaliszyk@44204
   653
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   654
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   655
    qed
haftmann@44242
   656
kaliszyk@35222
   657
end
kaliszyk@35222
   658
kuncar@47096
   659
subsection {* Quotient composition *}
kuncar@47096
   660
kuncar@47308
   661
lemma OOO_quotient3:
kuncar@47096
   662
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   663
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   664
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47096
   665
  fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   666
  fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
kuncar@47308
   667
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   668
  assumes R2: "Quotient3 R2 Abs2 Rep2"
kuncar@47096
   669
  assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
kuncar@47096
   670
  assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
kuncar@47308
   671
  shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47308
   672
apply (rule Quotient3I)
kuncar@47308
   673
   apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
kuncar@47096
   674
  apply simp
griff@47434
   675
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
kuncar@47308
   676
   apply (rule Quotient3_rep_reflp [OF R1])
griff@47434
   677
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
kuncar@47308
   678
   apply (rule Quotient3_rep_reflp [OF R1])
kuncar@47096
   679
  apply (rule Rep1)
kuncar@47308
   680
  apply (rule Quotient3_rep_reflp [OF R2])
kuncar@47096
   681
 apply safe
kuncar@47096
   682
    apply (rename_tac x y)
kuncar@47096
   683
    apply (drule Abs1)
kuncar@47308
   684
      apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   685
     apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   686
    apply (drule Quotient3_refl1 [OF R2], drule Rep1)
kuncar@47096
   687
    apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
griff@47434
   688
     apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
griff@47434
   689
     apply (erule relcomppI)
kuncar@47308
   690
     apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   691
    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   692
    apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   693
    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   694
    apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   695
    apply (erule Quotient3_rel_abs [OF R1])
kuncar@47096
   696
   apply (rename_tac x y)
kuncar@47096
   697
   apply (drule Abs1)
kuncar@47308
   698
     apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   699
    apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   700
   apply (drule Quotient3_refl2 [OF R2], drule Rep1)
kuncar@47096
   701
   apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
griff@47434
   702
    apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
griff@47434
   703
    apply (erule relcomppI)
kuncar@47308
   704
    apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   705
   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   706
   apply (rule conjI, erule Quotient3_refl2 [OF R1])
kuncar@47308
   707
   apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   708
   apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   709
   apply (erule Quotient3_rel_abs [OF R1, THEN sym])
kuncar@47096
   710
  apply simp
kuncar@47308
   711
  apply (rule Quotient3_rel_abs [OF R2])
kuncar@47308
   712
  apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
kuncar@47308
   713
  apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
kuncar@47096
   714
  apply (erule Abs1)
kuncar@47308
   715
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   716
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   717
 apply (rename_tac a b c d)
kuncar@47096
   718
 apply simp
griff@47434
   719
 apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
kuncar@47308
   720
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   721
  apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   722
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
griff@47434
   723
 apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
kuncar@47308
   724
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   725
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
kuncar@47308
   726
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47096
   727
 apply (rule Rep1)
kuncar@47096
   728
 apply (drule Abs1)
kuncar@47308
   729
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   730
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   731
 apply (drule Abs1)
kuncar@47308
   732
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   733
 apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   734
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   735
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   736
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   737
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47096
   738
 apply simp
kuncar@47308
   739
 apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
kuncar@47096
   740
 apply simp
kuncar@47096
   741
done
kuncar@47096
   742
kuncar@47308
   743
lemma OOO_eq_quotient3:
kuncar@47096
   744
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   745
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   746
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47308
   747
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   748
  assumes R2: "Quotient3 op= Abs2 Rep2"
kuncar@47308
   749
  shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47096
   750
using assms
kuncar@47308
   751
by (rule OOO_quotient3) auto
kuncar@47096
   752
kuncar@47362
   753
subsection {* Quotient3 to Quotient *}
kuncar@47362
   754
kuncar@47362
   755
lemma Quotient3_to_Quotient:
kuncar@47362
   756
assumes "Quotient3 R Abs Rep"
kuncar@47362
   757
and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
kuncar@47362
   758
shows "Quotient R Abs Rep T"
kuncar@47362
   759
using assms unfolding Quotient3_def by (intro QuotientI) blast+
kuncar@47096
   760
kuncar@47362
   761
lemma Quotient3_to_Quotient_equivp:
kuncar@47362
   762
assumes q: "Quotient3 R Abs Rep"
kuncar@47362
   763
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
kuncar@47362
   764
and eR: "equivp R"
kuncar@47362
   765
shows "Quotient R Abs Rep T"
kuncar@47362
   766
proof (intro QuotientI)
kuncar@47362
   767
  fix a
kuncar@47362
   768
  show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
kuncar@47362
   769
next
kuncar@47362
   770
  fix a
kuncar@47362
   771
  show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
kuncar@47362
   772
next
kuncar@47362
   773
  fix r s
kuncar@47362
   774
  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
   775
next
kuncar@47362
   776
  show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
kuncar@47096
   777
qed
kuncar@47096
   778
huffman@35294
   779
subsection {* ML setup *}
kaliszyk@35222
   780
kaliszyk@35222
   781
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   782
wenzelm@37986
   783
use "Tools/Quotient/quotient_info.ML"
wenzelm@41452
   784
setup Quotient_Info.setup
kaliszyk@35222
   785
kuncar@47308
   786
declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
kaliszyk@35222
   787
kuncar@47308
   788
lemmas [quot_thm] = fun_quotient3
haftmann@44553
   789
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553
   790
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222
   791
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   792
kaliszyk@35222
   793
kaliszyk@35222
   794
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   795
lemmas [id_simps] =
kaliszyk@35222
   796
  id_def[symmetric]
haftmann@40602
   797
  map_fun_id
kaliszyk@35222
   798
  id_apply
kaliszyk@35222
   799
  id_o
kaliszyk@35222
   800
  o_id
kaliszyk@35222
   801
  eq_comp_r
kaliszyk@44413
   802
  set_rel_eq
kaliszyk@44413
   803
  vimage_id
kaliszyk@35222
   804
kaliszyk@35222
   805
text {* Translation functions for the lifting process. *}
wenzelm@37986
   806
use "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   807
kaliszyk@35222
   808
kaliszyk@35222
   809
text {* Definitions of the quotient types. *}
wenzelm@45680
   810
use "Tools/Quotient/quotient_type.ML"
kaliszyk@35222
   811
kaliszyk@35222
   812
kaliszyk@35222
   813
text {* Definitions for quotient constants. *}
wenzelm@37986
   814
use "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   815
kaliszyk@35222
   816
kaliszyk@35222
   817
text {*
kaliszyk@35222
   818
  An auxiliary constant for recording some information
kaliszyk@35222
   819
  about the lifted theorem in a tactic.
kaliszyk@35222
   820
*}
kaliszyk@35222
   821
definition
haftmann@40466
   822
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   823
where
haftmann@40466
   824
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   825
kaliszyk@35222
   826
lemma
kaliszyk@35222
   827
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   828
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   829
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   830
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   831
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   832
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   833
kaliszyk@35222
   834
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   835
  by (simp add: Quot_True_def)
kaliszyk@35222
   836
kaliszyk@35222
   837
kaliszyk@35222
   838
text {* Tactics for proving the lifted theorems *}
wenzelm@37986
   839
use "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   840
huffman@35294
   841
subsection {* Methods / Interface *}
kaliszyk@35222
   842
kaliszyk@35222
   843
method_setup lifting =
urbanc@37593
   844
  {* Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@46468
   845
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
wenzelm@42814
   846
  {* lift theorems to quotient types *}
kaliszyk@35222
   847
kaliszyk@35222
   848
method_setup lifting_setup =
urbanc@37593
   849
  {* Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@46468
   850
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
wenzelm@42814
   851
  {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   852
urbanc@37593
   853
method_setup descending =
wenzelm@46468
   854
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
wenzelm@42814
   855
  {* decend theorems to the raw level *}
urbanc@37593
   856
urbanc@37593
   857
method_setup descending_setup =
wenzelm@46468
   858
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
wenzelm@42814
   859
  {* set up the three goals for the decending theorems *}
urbanc@37593
   860
urbanc@45782
   861
method_setup partiality_descending =
wenzelm@46468
   862
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
urbanc@45782
   863
  {* decend theorems to the raw level *}
urbanc@45782
   864
urbanc@45782
   865
method_setup partiality_descending_setup =
urbanc@45782
   866
  {* Scan.succeed (fn ctxt => 
wenzelm@46468
   867
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
urbanc@45782
   868
  {* set up the three goals for the decending theorems *}
urbanc@45782
   869
kaliszyk@35222
   870
method_setup regularize =
wenzelm@46468
   871
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
wenzelm@42814
   872
  {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   873
kaliszyk@35222
   874
method_setup injection =
wenzelm@46468
   875
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
wenzelm@42814
   876
  {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   877
kaliszyk@35222
   878
method_setup cleaning =
wenzelm@46468
   879
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
wenzelm@42814
   880
  {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   881
kaliszyk@35222
   882
attribute_setup quot_lifted =
kaliszyk@35222
   883
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814
   884
  {* lift theorems to quotient types *}
kaliszyk@35222
   885
kaliszyk@35222
   886
no_notation
kaliszyk@35222
   887
  rel_conj (infixr "OOO" 75) and
haftmann@40602
   888
  map_fun (infixr "--->" 55) and
kaliszyk@35222
   889
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   890
kaliszyk@35222
   891
end
haftmann@47488
   892