src/HOL/Quotient.thy
author bulwahn
Thu Apr 12 11:01:15 2012 +0200 (2012-04-12)
changeset 47436 d8fad618a67a
parent 47362 b1f099bdfbba
parent 47435 e1b761c216ac
child 47488 be6dd389639d
permissions -rw-r--r--
merged
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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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  An aside: contravariant functorial structure of sets.
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*}
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enriched_type vimage
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  by (simp_all add: fun_eq_iff vimage_compose)
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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 {* Respects predicate *}
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definition
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  Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set"
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where
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  "Respects R = {x. R x x}"
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lemma in_respects:
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  shows "x \<in> Respects R \<longleftrightarrow> R x x"
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  unfolding Respects_def by simp
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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 vimage_id:
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  "vimage id = id"
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  unfolding vimage_def fun_eq_iff by auto
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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"
kaliszyk@35222
   319
  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
kaliszyk@35222
   320
  apply(auto)
kaliszyk@35222
   321
  apply(rule_tac x="\<lambda>y. f x" in bexI)
kaliszyk@35222
   322
  apply(simp)
haftmann@40466
   323
  apply(simp add: Respects_def in_respects fun_rel_def)
kaliszyk@35222
   324
  apply(rule impI)
kaliszyk@35222
   325
  using a equivp_reflp_symp_transp[of "R2"]
haftmann@40814
   326
  apply (auto elim: equivpE reflpE)
kaliszyk@35222
   327
  done
kaliszyk@35222
   328
kaliszyk@35222
   329
(* Next four lemmas are unused *)
kaliszyk@35222
   330
lemma all_reg:
kaliszyk@35222
   331
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   332
  and     b: "All P"
kaliszyk@35222
   333
  shows "All Q"
huffman@44921
   334
  using a b by fast
kaliszyk@35222
   335
kaliszyk@35222
   336
lemma ex_reg:
kaliszyk@35222
   337
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   338
  and     b: "Ex P"
kaliszyk@35222
   339
  shows "Ex Q"
huffman@44921
   340
  using a b by fast
kaliszyk@35222
   341
kaliszyk@35222
   342
lemma ball_reg:
haftmann@44553
   343
  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222
   344
  and     b: "Ball R P"
kaliszyk@35222
   345
  shows "Ball R Q"
huffman@44921
   346
  using a b by fast
kaliszyk@35222
   347
kaliszyk@35222
   348
lemma bex_reg:
haftmann@44553
   349
  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222
   350
  and     b: "Bex R P"
kaliszyk@35222
   351
  shows "Bex R Q"
huffman@44921
   352
  using a b by fast
kaliszyk@35222
   353
kaliszyk@35222
   354
kaliszyk@35222
   355
lemma ball_all_comm:
kaliszyk@35222
   356
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   357
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   358
  using assms by auto
kaliszyk@35222
   359
kaliszyk@35222
   360
lemma bex_ex_comm:
kaliszyk@35222
   361
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   362
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   363
  using assms by auto
kaliszyk@35222
   364
huffman@35294
   365
subsection {* Bounded abstraction *}
kaliszyk@35222
   366
kaliszyk@35222
   367
definition
haftmann@40466
   368
  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   369
where
kaliszyk@35222
   370
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   371
kaliszyk@35222
   372
lemma babs_rsp:
kuncar@47308
   373
  assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222
   374
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   375
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   376
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   377
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   378
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   379
  apply (simp add: in_respects fun_rel_def)
kuncar@47308
   380
  using Quotient3_rel[OF q]
kaliszyk@35222
   381
  by metis
kaliszyk@35222
   382
kaliszyk@35222
   383
lemma babs_prs:
kuncar@47308
   384
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   385
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222
   386
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   387
  apply (rule ext)
haftmann@40466
   388
  apply (simp add:)
kaliszyk@35222
   389
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kuncar@47308
   390
  apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
kuncar@47308
   391
  apply (simp add: in_respects Quotient3_rel_rep[OF q1])
kaliszyk@35222
   392
  done
kaliszyk@35222
   393
kaliszyk@35222
   394
lemma babs_simp:
kuncar@47308
   395
  assumes q: "Quotient3 R1 Abs Rep"
kaliszyk@35222
   396
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   397
  apply(rule iffI)
kaliszyk@35222
   398
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   399
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   400
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   401
  apply(metis Babs_def)
kaliszyk@35222
   402
  apply (simp add: in_respects)
kuncar@47308
   403
  using Quotient3_rel[OF q]
kaliszyk@35222
   404
  by metis
kaliszyk@35222
   405
kaliszyk@35222
   406
(* If a user proves that a particular functional relation
kaliszyk@35222
   407
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   408
lemma babs_reg_eqv:
kaliszyk@35222
   409
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   410
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   411
kaliszyk@35222
   412
kaliszyk@35222
   413
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   414
lemma ball_rsp:
kaliszyk@35222
   415
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   416
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   417
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   418
kaliszyk@35222
   419
lemma bex_rsp:
kaliszyk@35222
   420
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   421
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   422
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   423
kaliszyk@35222
   424
lemma bex1_rsp:
kaliszyk@35222
   425
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   426
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   427
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   428
kaliszyk@35222
   429
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   430
lemma all_prs:
kuncar@47308
   431
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   432
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kuncar@47308
   433
  using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   434
  by metis
kaliszyk@35222
   435
kaliszyk@35222
   436
lemma ex_prs:
kuncar@47308
   437
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   438
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kuncar@47308
   439
  using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   440
  by metis
kaliszyk@35222
   441
huffman@35294
   442
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   443
kaliszyk@35222
   444
definition
kaliszyk@35222
   445
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   446
where
kaliszyk@35222
   447
  "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
   448
kaliszyk@35222
   449
lemma bex1_rel_aux:
kaliszyk@35222
   450
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   451
  unfolding Bex1_rel_def
kaliszyk@35222
   452
  apply (erule conjE)+
kaliszyk@35222
   453
  apply (erule bexE)
kaliszyk@35222
   454
  apply rule
kaliszyk@35222
   455
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   456
  apply metis
kaliszyk@35222
   457
  apply metis
kaliszyk@35222
   458
  apply rule+
kaliszyk@35222
   459
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   460
  prefer 2
kaliszyk@35222
   461
  apply (metis)
kaliszyk@35222
   462
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   463
  prefer 2
kaliszyk@35222
   464
  apply (metis)
kaliszyk@35222
   465
  apply (metis in_respects)
kaliszyk@35222
   466
  done
kaliszyk@35222
   467
kaliszyk@35222
   468
lemma bex1_rel_aux2:
kaliszyk@35222
   469
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   470
  unfolding Bex1_rel_def
kaliszyk@35222
   471
  apply (erule conjE)+
kaliszyk@35222
   472
  apply (erule bexE)
kaliszyk@35222
   473
  apply rule
kaliszyk@35222
   474
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   475
  apply metis
kaliszyk@35222
   476
  apply metis
kaliszyk@35222
   477
  apply rule+
kaliszyk@35222
   478
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   479
  prefer 2
kaliszyk@35222
   480
  apply (metis)
kaliszyk@35222
   481
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   482
  prefer 2
kaliszyk@35222
   483
  apply (metis)
kaliszyk@35222
   484
  apply (metis in_respects)
kaliszyk@35222
   485
  done
kaliszyk@35222
   486
kaliszyk@35222
   487
lemma bex1_rel_rsp:
kuncar@47308
   488
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   489
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   490
  apply (simp add: fun_rel_def)
kaliszyk@35222
   491
  apply clarify
kaliszyk@35222
   492
  apply rule
kaliszyk@35222
   493
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   494
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   495
  apply assumption
kaliszyk@35222
   496
  done
kaliszyk@35222
   497
kaliszyk@35222
   498
kaliszyk@35222
   499
lemma ex1_prs:
kuncar@47308
   500
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   501
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   502
apply (simp add:)
kaliszyk@35222
   503
apply (subst Bex1_rel_def)
kaliszyk@35222
   504
apply (subst Bex_def)
kaliszyk@35222
   505
apply (subst Ex1_def)
kaliszyk@35222
   506
apply simp
kaliszyk@35222
   507
apply rule
kaliszyk@35222
   508
 apply (erule conjE)+
kaliszyk@35222
   509
 apply (erule_tac exE)
kaliszyk@35222
   510
 apply (erule conjE)
kaliszyk@35222
   511
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   512
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   513
  apply (simp)
kaliszyk@35222
   514
  apply rule+
kuncar@47308
   515
  using a unfolding Quotient3_def
kaliszyk@35222
   516
  apply metis
kaliszyk@35222
   517
 apply rule+
kaliszyk@35222
   518
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   519
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   520
   apply simp
kaliszyk@35222
   521
  apply (simp add: in_respects)
kaliszyk@35222
   522
 apply (simp add: in_respects)
kaliszyk@35222
   523
apply (erule_tac exE)
kaliszyk@35222
   524
 apply rule
kaliszyk@35222
   525
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   526
 apply (simp only: in_respects)
kaliszyk@35222
   527
  apply rule
kuncar@47308
   528
 apply (metis Quotient3_rel_rep[OF a])
kuncar@47308
   529
using a unfolding Quotient3_def apply (simp)
kaliszyk@35222
   530
apply rule+
kuncar@47308
   531
using a unfolding Quotient3_def in_respects
kaliszyk@35222
   532
apply metis
kaliszyk@35222
   533
done
kaliszyk@35222
   534
kaliszyk@38702
   535
lemma bex1_bexeq_reg:
kaliszyk@38702
   536
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   537
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   538
  apply clarify
kaliszyk@35222
   539
  apply auto
kaliszyk@35222
   540
  apply (rule bexI)
kaliszyk@35222
   541
  apply assumption
kaliszyk@35222
   542
  apply (simp add: in_respects)
kaliszyk@35222
   543
  apply (simp add: in_respects)
kaliszyk@35222
   544
  apply auto
kaliszyk@35222
   545
  done
kaliszyk@35222
   546
kaliszyk@38702
   547
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   548
  assumes a: "equivp R"
kaliszyk@38702
   549
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   550
  using equivp_reflp[OF a]
kaliszyk@38702
   551
  apply (intro impI)
kaliszyk@38702
   552
  apply (elim ex1E)
kaliszyk@38702
   553
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   554
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   555
  apply (subst in_respects)
kaliszyk@38702
   556
  apply (rule conjI)
kaliszyk@38702
   557
  apply assumption
kaliszyk@38702
   558
  apply assumption
kaliszyk@38702
   559
  apply clarify
kaliszyk@38702
   560
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   561
  apply simp
kaliszyk@38702
   562
  done
kaliszyk@38702
   563
huffman@35294
   564
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   565
kaliszyk@35222
   566
lemma quot_rel_rsp:
kuncar@47308
   567
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222
   568
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   569
  apply(rule fun_relI)+
kaliszyk@35222
   570
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   571
  apply(assumption)+
kaliszyk@35222
   572
  done
kaliszyk@35222
   573
kaliszyk@35222
   574
lemma o_prs:
kuncar@47308
   575
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   576
  and     q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308
   577
  and     q3: "Quotient3 R3 Abs3 Rep3"
kaliszyk@36215
   578
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   579
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kuncar@47308
   580
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
haftmann@40466
   581
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   582
kaliszyk@35222
   583
lemma o_rsp:
kaliszyk@36215
   584
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   585
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
huffman@44921
   586
  by (force elim: fun_relE)+
kaliszyk@35222
   587
kaliszyk@35222
   588
lemma cond_prs:
kuncar@47308
   589
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   590
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kuncar@47308
   591
  using a unfolding Quotient3_def by auto
kaliszyk@35222
   592
kaliszyk@35222
   593
lemma if_prs:
kuncar@47308
   594
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   595
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kuncar@47308
   596
  using Quotient3_abs_rep[OF q]
nipkow@39302
   597
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   598
kaliszyk@35222
   599
lemma if_rsp:
kuncar@47308
   600
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   601
  shows "(op = ===> R ===> R ===> R) If If"
huffman@44921
   602
  by force
kaliszyk@35222
   603
kaliszyk@35222
   604
lemma let_prs:
kuncar@47308
   605
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   606
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@37049
   607
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kuncar@47308
   608
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
nipkow@39302
   609
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   610
kaliszyk@35222
   611
lemma let_rsp:
kaliszyk@37049
   612
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
huffman@44921
   613
  by (force elim: fun_relE)
kaliszyk@35222
   614
kaliszyk@39669
   615
lemma id_rsp:
kaliszyk@39669
   616
  shows "(R ===> R) id id"
huffman@44921
   617
  by auto
kaliszyk@39669
   618
kaliszyk@39669
   619
lemma id_prs:
kuncar@47308
   620
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@39669
   621
  shows "(Rep ---> Abs) id = id"
kuncar@47308
   622
  by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
kaliszyk@39669
   623
kaliszyk@39669
   624
kaliszyk@35222
   625
locale quot_type =
kaliszyk@35222
   626
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   627
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   628
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   629
  assumes equivp: "part_equivp R"
kaliszyk@44204
   630
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   631
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   632
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   633
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   634
begin
kaliszyk@35222
   635
kaliszyk@35222
   636
definition
haftmann@40466
   637
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   638
where
kaliszyk@44204
   639
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   640
kaliszyk@35222
   641
definition
haftmann@40466
   642
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   643
where
kaliszyk@44204
   644
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   645
kaliszyk@44204
   646
lemma some_collect:
kaliszyk@37493
   647
  assumes "R r r"
kaliszyk@44204
   648
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   649
  apply simp
kaliszyk@44204
   650
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   651
kaliszyk@35222
   652
lemma Quotient:
kuncar@47308
   653
  shows "Quotient3 R abs rep"
kuncar@47308
   654
  unfolding Quotient3_def abs_def rep_def
kaliszyk@37493
   655
  proof (intro conjI allI)
kaliszyk@37493
   656
    fix a r s
kaliszyk@44204
   657
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   658
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   659
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   660
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   661
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204
   662
        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493
   663
    qed
kaliszyk@44204
   664
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   665
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   666
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   667
    proof -
haftmann@44242
   668
      assume "R r r" and "R s s"
haftmann@44242
   669
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   670
        by (metis abs_inverse)
haftmann@44242
   671
      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
   672
        by rule simp_all
haftmann@44242
   673
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   674
    qed
kaliszyk@44204
   675
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   676
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   677
    qed
haftmann@44242
   678
kaliszyk@35222
   679
end
kaliszyk@35222
   680
kuncar@47096
   681
subsection {* Quotient composition *}
kuncar@47096
   682
kuncar@47308
   683
lemma OOO_quotient3:
kuncar@47096
   684
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   685
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   686
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47096
   687
  fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   688
  fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
kuncar@47308
   689
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   690
  assumes R2: "Quotient3 R2 Abs2 Rep2"
kuncar@47096
   691
  assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
kuncar@47096
   692
  assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
kuncar@47308
   693
  shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47308
   694
apply (rule Quotient3I)
kuncar@47308
   695
   apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
kuncar@47096
   696
  apply simp
griff@47434
   697
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
kuncar@47308
   698
   apply (rule Quotient3_rep_reflp [OF R1])
griff@47434
   699
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
kuncar@47308
   700
   apply (rule Quotient3_rep_reflp [OF R1])
kuncar@47096
   701
  apply (rule Rep1)
kuncar@47308
   702
  apply (rule Quotient3_rep_reflp [OF R2])
kuncar@47096
   703
 apply safe
kuncar@47096
   704
    apply (rename_tac x y)
kuncar@47096
   705
    apply (drule Abs1)
kuncar@47308
   706
      apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   707
     apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   708
    apply (drule Quotient3_refl1 [OF R2], drule Rep1)
kuncar@47096
   709
    apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
griff@47434
   710
     apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
griff@47434
   711
     apply (erule relcomppI)
kuncar@47308
   712
     apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   713
    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   714
    apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   715
    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   716
    apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   717
    apply (erule Quotient3_rel_abs [OF R1])
kuncar@47096
   718
   apply (rename_tac x y)
kuncar@47096
   719
   apply (drule Abs1)
kuncar@47308
   720
     apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   721
    apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   722
   apply (drule Quotient3_refl2 [OF R2], drule Rep1)
kuncar@47096
   723
   apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
griff@47434
   724
    apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
griff@47434
   725
    apply (erule relcomppI)
kuncar@47308
   726
    apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   727
   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   728
   apply (rule conjI, erule Quotient3_refl2 [OF R1])
kuncar@47308
   729
   apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   730
   apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   731
   apply (erule Quotient3_rel_abs [OF R1, THEN sym])
kuncar@47096
   732
  apply simp
kuncar@47308
   733
  apply (rule Quotient3_rel_abs [OF R2])
kuncar@47308
   734
  apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
kuncar@47308
   735
  apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
kuncar@47096
   736
  apply (erule Abs1)
kuncar@47308
   737
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   738
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   739
 apply (rename_tac a b c d)
kuncar@47096
   740
 apply simp
griff@47434
   741
 apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
kuncar@47308
   742
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   743
  apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   744
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
griff@47434
   745
 apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
kuncar@47308
   746
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   747
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
kuncar@47308
   748
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47096
   749
 apply (rule Rep1)
kuncar@47096
   750
 apply (drule Abs1)
kuncar@47308
   751
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   752
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   753
 apply (drule Abs1)
kuncar@47308
   754
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   755
 apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   756
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   757
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   758
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   759
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47096
   760
 apply simp
kuncar@47308
   761
 apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
kuncar@47096
   762
 apply simp
kuncar@47096
   763
done
kuncar@47096
   764
kuncar@47308
   765
lemma OOO_eq_quotient3:
kuncar@47096
   766
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   767
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   768
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47308
   769
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   770
  assumes R2: "Quotient3 op= Abs2 Rep2"
kuncar@47308
   771
  shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47096
   772
using assms
kuncar@47308
   773
by (rule OOO_quotient3) auto
kuncar@47096
   774
kuncar@47362
   775
subsection {* Quotient3 to Quotient *}
kuncar@47362
   776
kuncar@47362
   777
lemma Quotient3_to_Quotient:
kuncar@47362
   778
assumes "Quotient3 R Abs Rep"
kuncar@47362
   779
and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
kuncar@47362
   780
shows "Quotient R Abs Rep T"
kuncar@47362
   781
using assms unfolding Quotient3_def by (intro QuotientI) blast+
kuncar@47096
   782
kuncar@47362
   783
lemma Quotient3_to_Quotient_equivp:
kuncar@47362
   784
assumes q: "Quotient3 R Abs Rep"
kuncar@47362
   785
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
kuncar@47362
   786
and eR: "equivp R"
kuncar@47362
   787
shows "Quotient R Abs Rep T"
kuncar@47362
   788
proof (intro QuotientI)
kuncar@47362
   789
  fix a
kuncar@47362
   790
  show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
kuncar@47362
   791
next
kuncar@47362
   792
  fix a
kuncar@47362
   793
  show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
kuncar@47362
   794
next
kuncar@47362
   795
  fix r s
kuncar@47362
   796
  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
   797
next
kuncar@47362
   798
  show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
kuncar@47096
   799
qed
kuncar@47096
   800
huffman@35294
   801
subsection {* ML setup *}
kaliszyk@35222
   802
kaliszyk@35222
   803
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   804
wenzelm@37986
   805
use "Tools/Quotient/quotient_info.ML"
wenzelm@41452
   806
setup Quotient_Info.setup
kaliszyk@35222
   807
kuncar@47308
   808
declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
kaliszyk@35222
   809
kuncar@47308
   810
lemmas [quot_thm] = fun_quotient3
haftmann@44553
   811
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553
   812
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222
   813
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   814
kaliszyk@35222
   815
kaliszyk@35222
   816
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   817
lemmas [id_simps] =
kaliszyk@35222
   818
  id_def[symmetric]
haftmann@40602
   819
  map_fun_id
kaliszyk@35222
   820
  id_apply
kaliszyk@35222
   821
  id_o
kaliszyk@35222
   822
  o_id
kaliszyk@35222
   823
  eq_comp_r
kaliszyk@44413
   824
  set_rel_eq
kaliszyk@44413
   825
  vimage_id
kaliszyk@35222
   826
kaliszyk@35222
   827
text {* Translation functions for the lifting process. *}
wenzelm@37986
   828
use "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   829
kaliszyk@35222
   830
kaliszyk@35222
   831
text {* Definitions of the quotient types. *}
wenzelm@45680
   832
use "Tools/Quotient/quotient_type.ML"
kaliszyk@35222
   833
kaliszyk@35222
   834
kaliszyk@35222
   835
text {* Definitions for quotient constants. *}
wenzelm@37986
   836
use "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   837
kaliszyk@35222
   838
kaliszyk@35222
   839
text {*
kaliszyk@35222
   840
  An auxiliary constant for recording some information
kaliszyk@35222
   841
  about the lifted theorem in a tactic.
kaliszyk@35222
   842
*}
kaliszyk@35222
   843
definition
haftmann@40466
   844
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   845
where
haftmann@40466
   846
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   847
kaliszyk@35222
   848
lemma
kaliszyk@35222
   849
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   850
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   851
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   852
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   853
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   854
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   855
kaliszyk@35222
   856
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   857
  by (simp add: Quot_True_def)
kaliszyk@35222
   858
kaliszyk@35222
   859
kaliszyk@35222
   860
text {* Tactics for proving the lifted theorems *}
wenzelm@37986
   861
use "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   862
huffman@35294
   863
subsection {* Methods / Interface *}
kaliszyk@35222
   864
kaliszyk@35222
   865
method_setup lifting =
urbanc@37593
   866
  {* Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@46468
   867
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
wenzelm@42814
   868
  {* lift theorems to quotient types *}
kaliszyk@35222
   869
kaliszyk@35222
   870
method_setup lifting_setup =
urbanc@37593
   871
  {* Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@46468
   872
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
wenzelm@42814
   873
  {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   874
urbanc@37593
   875
method_setup descending =
wenzelm@46468
   876
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
wenzelm@42814
   877
  {* decend theorems to the raw level *}
urbanc@37593
   878
urbanc@37593
   879
method_setup descending_setup =
wenzelm@46468
   880
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
wenzelm@42814
   881
  {* set up the three goals for the decending theorems *}
urbanc@37593
   882
urbanc@45782
   883
method_setup partiality_descending =
wenzelm@46468
   884
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
urbanc@45782
   885
  {* decend theorems to the raw level *}
urbanc@45782
   886
urbanc@45782
   887
method_setup partiality_descending_setup =
urbanc@45782
   888
  {* Scan.succeed (fn ctxt => 
wenzelm@46468
   889
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
urbanc@45782
   890
  {* set up the three goals for the decending theorems *}
urbanc@45782
   891
kaliszyk@35222
   892
method_setup regularize =
wenzelm@46468
   893
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
wenzelm@42814
   894
  {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   895
kaliszyk@35222
   896
method_setup injection =
wenzelm@46468
   897
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
wenzelm@42814
   898
  {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   899
kaliszyk@35222
   900
method_setup cleaning =
wenzelm@46468
   901
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
wenzelm@42814
   902
  {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   903
kaliszyk@35222
   904
attribute_setup quot_lifted =
kaliszyk@35222
   905
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814
   906
  {* lift theorems to quotient types *}
kaliszyk@35222
   907
kaliszyk@35222
   908
no_notation
kaliszyk@35222
   909
  rel_conj (infixr "OOO" 75) and
haftmann@40602
   910
  map_fun (infixr "--->" 55) and
kaliszyk@35222
   911
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   912
kaliszyk@35222
   913
end