src/HOL/Quotient.thy
author wenzelm
Thu Mar 15 19:02:34 2012 +0100 (2012-03-15)
changeset 46947 b8c7eb0c2f89
parent 46468 4db76d47b51a
child 46950 d0181abdbdac
permissions -rw-r--r--
declare minor keywords via theory header;
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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
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keywords "/"
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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 {* Function map and function relation *}
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notation map_fun (infixr "--->" 55)
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lemma map_fun_id:
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  "(id ---> id) = id"
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  by (simp add: fun_eq_iff)
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definition
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  fun_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" (infixr "===>" 55)
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where
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  "fun_rel R1 R2 = (\<lambda>f g. \<forall>x y. R1 x y \<longrightarrow> R2 (f x) (g y))"
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lemma fun_relI [intro]:
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  assumes "\<And>x y. R1 x y \<Longrightarrow> R2 (f x) (g y)"
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  shows "(R1 ===> R2) f g"
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  using assms by (simp add: fun_rel_def)
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lemma fun_relE:
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  assumes "(R1 ===> R2) f g" and "R1 x y"
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  obtains "R2 (f x) (g y)"
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  using assms by (simp add: fun_rel_def)
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lemma fun_rel_eq:
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  shows "((op =) ===> (op =)) = (op =)"
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  by (auto simp add: fun_eq_iff elim: fun_relE)
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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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  "Quotient 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 QuotientI:
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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 "Quotient R Abs Rep"
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  using assms unfolding Quotient_def by blast
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lemma Quotient_abs_rep:
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  assumes a: "Quotient R Abs Rep"
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  shows "Abs (Rep a) = a"
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  using a
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  unfolding Quotient_def
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  by simp
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lemma Quotient_rep_reflp:
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  assumes a: "Quotient R Abs Rep"
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  shows "R (Rep a) (Rep a)"
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  using a
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  unfolding Quotient_def
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  by blast
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lemma Quotient_rel:
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  assumes a: "Quotient 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 Quotient_def
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  by blast
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lemma Quotient_rel_rep:
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  assumes a: "Quotient 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 Quotient_def
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  by metis
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lemma Quotient_rep_abs:
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  assumes a: "Quotient R Abs Rep"
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  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_rel_abs:
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  assumes a: "Quotient R Abs Rep"
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  shows "R r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient_def
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  by blast
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lemma Quotient_symp:
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  assumes a: "Quotient R Abs Rep"
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  shows "symp R"
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  using a unfolding Quotient_def using sympI by metis
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lemma Quotient_transp:
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  assumes a: "Quotient R Abs Rep"
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  shows "transp R"
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  using a unfolding Quotient_def using transpI by metis
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lemma identity_quotient:
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  shows "Quotient (op =) id id"
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  unfolding Quotient_def id_def
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  by blast
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lemma fun_quotient:
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  assumes q1: "Quotient R1 abs1 rep1"
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  and     q2: "Quotient R2 abs2 rep2"
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  shows "Quotient (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: Quotient_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 Quotient_rel_abs [of R1 abs1 rep1] Quotient_rel_rep [of R2 abs2 rep2],
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        simp (no_asm) add: Quotient_def, simp)
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  moreover
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  have "\<And>r s. (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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    apply(auto simp add: fun_rel_def fun_eq_iff)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    using q1 q2 unfolding Quotient_def
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    apply(metis)
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    done
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  ultimately
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  show "Quotient (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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    unfolding Quotient_def by blast
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qed
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lemma abs_o_rep:
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  assumes a: "Quotient 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: Quotient_abs_rep[OF a])
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lemma equals_rsp:
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  assumes q: "Quotient 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 Quotient_symp[OF q] Quotient_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: "Quotient R1 Abs1 Rep1"
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  and     q2: "Quotient 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 Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
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  by simp
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lemma lambda_prs1:
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  assumes q1: "Quotient R1 Abs1 Rep1"
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  and     q2: "Quotient 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 Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
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  by simp
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lemma rep_abs_rsp:
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  assumes q: "Quotient 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 Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_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: "Quotient 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 Quotient_rel[OF q] Quotient_abs_rep[OF q] Quotient_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_rsp:
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  fixes f g::"'a \<Rightarrow> 'c"
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  assumes q: "Quotient 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_rsp':
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  assumes 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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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"
kaliszyk@35222
   327
  shows "All Q"
huffman@44921
   328
  using a b by fast
kaliszyk@35222
   329
kaliszyk@35222
   330
lemma ex_reg:
kaliszyk@35222
   331
  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   332
  and     b: "Ex P"
kaliszyk@35222
   333
  shows "Ex Q"
huffman@44921
   334
  using a b by fast
kaliszyk@35222
   335
kaliszyk@35222
   336
lemma ball_reg:
haftmann@44553
   337
  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222
   338
  and     b: "Ball R P"
kaliszyk@35222
   339
  shows "Ball R Q"
huffman@44921
   340
  using a b by fast
kaliszyk@35222
   341
kaliszyk@35222
   342
lemma bex_reg:
haftmann@44553
   343
  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
kaliszyk@35222
   344
  and     b: "Bex R P"
kaliszyk@35222
   345
  shows "Bex R Q"
huffman@44921
   346
  using a b by fast
kaliszyk@35222
   347
kaliszyk@35222
   348
kaliszyk@35222
   349
lemma ball_all_comm:
kaliszyk@35222
   350
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   351
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   352
  using assms by auto
kaliszyk@35222
   353
kaliszyk@35222
   354
lemma bex_ex_comm:
kaliszyk@35222
   355
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   356
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   357
  using assms by auto
kaliszyk@35222
   358
huffman@35294
   359
subsection {* Bounded abstraction *}
kaliszyk@35222
   360
kaliszyk@35222
   361
definition
haftmann@40466
   362
  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   363
where
kaliszyk@35222
   364
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   365
kaliszyk@35222
   366
lemma babs_rsp:
kaliszyk@35222
   367
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   368
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   369
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   370
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   371
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   372
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   373
  apply (simp add: in_respects fun_rel_def)
kaliszyk@35222
   374
  using Quotient_rel[OF q]
kaliszyk@35222
   375
  by metis
kaliszyk@35222
   376
kaliszyk@35222
   377
lemma babs_prs:
kaliszyk@35222
   378
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   379
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   380
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   381
  apply (rule ext)
haftmann@40466
   382
  apply (simp add:)
kaliszyk@35222
   383
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kaliszyk@35222
   384
  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
kaliszyk@35222
   385
  apply (simp add: in_respects Quotient_rel_rep[OF q1])
kaliszyk@35222
   386
  done
kaliszyk@35222
   387
kaliszyk@35222
   388
lemma babs_simp:
kaliszyk@35222
   389
  assumes q: "Quotient R1 Abs Rep"
kaliszyk@35222
   390
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   391
  apply(rule iffI)
kaliszyk@35222
   392
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   393
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   394
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   395
  apply(metis Babs_def)
kaliszyk@35222
   396
  apply (simp add: in_respects)
kaliszyk@35222
   397
  using Quotient_rel[OF q]
kaliszyk@35222
   398
  by metis
kaliszyk@35222
   399
kaliszyk@35222
   400
(* If a user proves that a particular functional relation
kaliszyk@35222
   401
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   402
lemma babs_reg_eqv:
kaliszyk@35222
   403
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   404
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   405
kaliszyk@35222
   406
kaliszyk@35222
   407
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   408
lemma ball_rsp:
kaliszyk@35222
   409
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   410
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   411
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   412
kaliszyk@35222
   413
lemma bex_rsp:
kaliszyk@35222
   414
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   415
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   416
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   417
kaliszyk@35222
   418
lemma bex1_rsp:
kaliszyk@35222
   419
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   420
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   421
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   422
kaliszyk@35222
   423
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   424
lemma all_prs:
kaliszyk@35222
   425
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   426
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
haftmann@40602
   427
  using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   428
  by metis
kaliszyk@35222
   429
kaliszyk@35222
   430
lemma ex_prs:
kaliszyk@35222
   431
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   432
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
haftmann@40602
   433
  using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   434
  by metis
kaliszyk@35222
   435
huffman@35294
   436
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   437
kaliszyk@35222
   438
definition
kaliszyk@35222
   439
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   440
where
kaliszyk@35222
   441
  "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
   442
kaliszyk@35222
   443
lemma bex1_rel_aux:
kaliszyk@35222
   444
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   445
  unfolding Bex1_rel_def
kaliszyk@35222
   446
  apply (erule conjE)+
kaliszyk@35222
   447
  apply (erule bexE)
kaliszyk@35222
   448
  apply rule
kaliszyk@35222
   449
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   450
  apply metis
kaliszyk@35222
   451
  apply metis
kaliszyk@35222
   452
  apply rule+
kaliszyk@35222
   453
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   454
  prefer 2
kaliszyk@35222
   455
  apply (metis)
kaliszyk@35222
   456
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   457
  prefer 2
kaliszyk@35222
   458
  apply (metis)
kaliszyk@35222
   459
  apply (metis in_respects)
kaliszyk@35222
   460
  done
kaliszyk@35222
   461
kaliszyk@35222
   462
lemma bex1_rel_aux2:
kaliszyk@35222
   463
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   464
  unfolding Bex1_rel_def
kaliszyk@35222
   465
  apply (erule conjE)+
kaliszyk@35222
   466
  apply (erule bexE)
kaliszyk@35222
   467
  apply rule
kaliszyk@35222
   468
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   469
  apply metis
kaliszyk@35222
   470
  apply metis
kaliszyk@35222
   471
  apply rule+
kaliszyk@35222
   472
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   473
  prefer 2
kaliszyk@35222
   474
  apply (metis)
kaliszyk@35222
   475
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   476
  prefer 2
kaliszyk@35222
   477
  apply (metis)
kaliszyk@35222
   478
  apply (metis in_respects)
kaliszyk@35222
   479
  done
kaliszyk@35222
   480
kaliszyk@35222
   481
lemma bex1_rel_rsp:
kaliszyk@35222
   482
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   483
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   484
  apply (simp add: fun_rel_def)
kaliszyk@35222
   485
  apply clarify
kaliszyk@35222
   486
  apply rule
kaliszyk@35222
   487
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   488
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   489
  apply assumption
kaliszyk@35222
   490
  done
kaliszyk@35222
   491
kaliszyk@35222
   492
kaliszyk@35222
   493
lemma ex1_prs:
kaliszyk@35222
   494
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   495
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   496
apply (simp add:)
kaliszyk@35222
   497
apply (subst Bex1_rel_def)
kaliszyk@35222
   498
apply (subst Bex_def)
kaliszyk@35222
   499
apply (subst Ex1_def)
kaliszyk@35222
   500
apply simp
kaliszyk@35222
   501
apply rule
kaliszyk@35222
   502
 apply (erule conjE)+
kaliszyk@35222
   503
 apply (erule_tac exE)
kaliszyk@35222
   504
 apply (erule conjE)
kaliszyk@35222
   505
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   506
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   507
  apply (simp)
kaliszyk@35222
   508
  apply rule+
kaliszyk@35222
   509
  using a unfolding Quotient_def
kaliszyk@35222
   510
  apply metis
kaliszyk@35222
   511
 apply rule+
kaliszyk@35222
   512
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   513
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   514
   apply simp
kaliszyk@35222
   515
  apply (simp add: in_respects)
kaliszyk@35222
   516
 apply (simp add: in_respects)
kaliszyk@35222
   517
apply (erule_tac exE)
kaliszyk@35222
   518
 apply rule
kaliszyk@35222
   519
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   520
 apply (simp only: in_respects)
kaliszyk@35222
   521
  apply rule
kaliszyk@35222
   522
 apply (metis Quotient_rel_rep[OF a])
kaliszyk@35222
   523
using a unfolding Quotient_def apply (simp)
kaliszyk@35222
   524
apply rule+
kaliszyk@35222
   525
using a unfolding Quotient_def in_respects
kaliszyk@35222
   526
apply metis
kaliszyk@35222
   527
done
kaliszyk@35222
   528
kaliszyk@38702
   529
lemma bex1_bexeq_reg:
kaliszyk@38702
   530
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   531
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   532
  apply clarify
kaliszyk@35222
   533
  apply auto
kaliszyk@35222
   534
  apply (rule bexI)
kaliszyk@35222
   535
  apply assumption
kaliszyk@35222
   536
  apply (simp add: in_respects)
kaliszyk@35222
   537
  apply (simp add: in_respects)
kaliszyk@35222
   538
  apply auto
kaliszyk@35222
   539
  done
kaliszyk@35222
   540
kaliszyk@38702
   541
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   542
  assumes a: "equivp R"
kaliszyk@38702
   543
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   544
  using equivp_reflp[OF a]
kaliszyk@38702
   545
  apply (intro impI)
kaliszyk@38702
   546
  apply (elim ex1E)
kaliszyk@38702
   547
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   548
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   549
  apply (subst in_respects)
kaliszyk@38702
   550
  apply (rule conjI)
kaliszyk@38702
   551
  apply assumption
kaliszyk@38702
   552
  apply assumption
kaliszyk@38702
   553
  apply clarify
kaliszyk@38702
   554
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   555
  apply simp
kaliszyk@38702
   556
  done
kaliszyk@38702
   557
huffman@35294
   558
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   559
kaliszyk@35222
   560
lemma quot_rel_rsp:
kaliszyk@35222
   561
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   562
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   563
  apply(rule fun_relI)+
kaliszyk@35222
   564
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   565
  apply(assumption)+
kaliszyk@35222
   566
  done
kaliszyk@35222
   567
kaliszyk@35222
   568
lemma o_prs:
kaliszyk@35222
   569
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   570
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   571
  and     q3: "Quotient R3 Abs3 Rep3"
kaliszyk@36215
   572
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   573
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kaliszyk@35222
   574
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
haftmann@40466
   575
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   576
kaliszyk@35222
   577
lemma o_rsp:
kaliszyk@36215
   578
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   579
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
huffman@44921
   580
  by (force elim: fun_relE)+
kaliszyk@35222
   581
kaliszyk@35222
   582
lemma cond_prs:
kaliszyk@35222
   583
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   584
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kaliszyk@35222
   585
  using a unfolding Quotient_def by auto
kaliszyk@35222
   586
kaliszyk@35222
   587
lemma if_prs:
kaliszyk@35222
   588
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   589
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kaliszyk@36123
   590
  using Quotient_abs_rep[OF q]
nipkow@39302
   591
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   592
kaliszyk@35222
   593
lemma if_rsp:
kaliszyk@35222
   594
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   595
  shows "(op = ===> R ===> R ===> R) If If"
huffman@44921
   596
  by force
kaliszyk@35222
   597
kaliszyk@35222
   598
lemma let_prs:
kaliszyk@35222
   599
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   600
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@37049
   601
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kaliszyk@37049
   602
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
nipkow@39302
   603
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   604
kaliszyk@35222
   605
lemma let_rsp:
kaliszyk@37049
   606
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
huffman@44921
   607
  by (force elim: fun_relE)
kaliszyk@35222
   608
kaliszyk@39669
   609
lemma id_rsp:
kaliszyk@39669
   610
  shows "(R ===> R) id id"
huffman@44921
   611
  by auto
kaliszyk@39669
   612
kaliszyk@39669
   613
lemma id_prs:
kaliszyk@39669
   614
  assumes a: "Quotient R Abs Rep"
kaliszyk@39669
   615
  shows "(Rep ---> Abs) id = id"
haftmann@40466
   616
  by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
kaliszyk@39669
   617
kaliszyk@39669
   618
kaliszyk@35222
   619
locale quot_type =
kaliszyk@35222
   620
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   621
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   622
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   623
  assumes equivp: "part_equivp R"
kaliszyk@44204
   624
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   625
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   626
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   627
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   628
begin
kaliszyk@35222
   629
kaliszyk@35222
   630
definition
haftmann@40466
   631
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   632
where
kaliszyk@44204
   633
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   634
kaliszyk@35222
   635
definition
haftmann@40466
   636
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   637
where
kaliszyk@44204
   638
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   639
kaliszyk@44204
   640
lemma some_collect:
kaliszyk@37493
   641
  assumes "R r r"
kaliszyk@44204
   642
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   643
  apply simp
kaliszyk@44204
   644
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   645
kaliszyk@35222
   646
lemma Quotient:
kaliszyk@35222
   647
  shows "Quotient R abs rep"
kaliszyk@37493
   648
  unfolding Quotient_def abs_def rep_def
kaliszyk@37493
   649
  proof (intro conjI allI)
kaliszyk@37493
   650
    fix a r s
kaliszyk@44204
   651
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   652
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   653
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   654
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   655
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204
   656
        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493
   657
    qed
kaliszyk@44204
   658
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   659
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   660
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   661
    proof -
haftmann@44242
   662
      assume "R r r" and "R s s"
haftmann@44242
   663
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   664
        by (metis abs_inverse)
haftmann@44242
   665
      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
   666
        by rule simp_all
haftmann@44242
   667
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   668
    qed
kaliszyk@44204
   669
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   670
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   671
    qed
haftmann@44242
   672
kaliszyk@35222
   673
end
kaliszyk@35222
   674
huffman@35294
   675
subsection {* ML setup *}
kaliszyk@35222
   676
kaliszyk@35222
   677
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   678
wenzelm@37986
   679
use "Tools/Quotient/quotient_info.ML"
wenzelm@41452
   680
setup Quotient_Info.setup
kaliszyk@35222
   681
kuncar@45802
   682
declare [[map "fun" = fun_rel]]
kuncar@45802
   683
declare [[map set = set_rel]]
kaliszyk@35222
   684
kaliszyk@35222
   685
lemmas [quot_thm] = fun_quotient
haftmann@44553
   686
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553
   687
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222
   688
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   689
kaliszyk@35222
   690
kaliszyk@35222
   691
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   692
lemmas [id_simps] =
kaliszyk@35222
   693
  id_def[symmetric]
haftmann@40602
   694
  map_fun_id
kaliszyk@35222
   695
  id_apply
kaliszyk@35222
   696
  id_o
kaliszyk@35222
   697
  o_id
kaliszyk@35222
   698
  eq_comp_r
kaliszyk@44413
   699
  set_rel_eq
kaliszyk@44413
   700
  vimage_id
kaliszyk@35222
   701
kaliszyk@35222
   702
text {* Translation functions for the lifting process. *}
wenzelm@37986
   703
use "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   704
kaliszyk@35222
   705
kaliszyk@35222
   706
text {* Definitions of the quotient types. *}
wenzelm@45680
   707
use "Tools/Quotient/quotient_type.ML"
kaliszyk@35222
   708
kaliszyk@35222
   709
kaliszyk@35222
   710
text {* Definitions for quotient constants. *}
wenzelm@37986
   711
use "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   712
kaliszyk@35222
   713
kaliszyk@35222
   714
text {*
kaliszyk@35222
   715
  An auxiliary constant for recording some information
kaliszyk@35222
   716
  about the lifted theorem in a tactic.
kaliszyk@35222
   717
*}
kaliszyk@35222
   718
definition
haftmann@40466
   719
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   720
where
haftmann@40466
   721
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   722
kaliszyk@35222
   723
lemma
kaliszyk@35222
   724
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   725
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   726
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   727
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   728
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   729
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   730
kaliszyk@35222
   731
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   732
  by (simp add: Quot_True_def)
kaliszyk@35222
   733
kaliszyk@35222
   734
kaliszyk@35222
   735
text {* Tactics for proving the lifted theorems *}
wenzelm@37986
   736
use "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   737
huffman@35294
   738
subsection {* Methods / Interface *}
kaliszyk@35222
   739
kaliszyk@35222
   740
method_setup lifting =
urbanc@37593
   741
  {* Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@46468
   742
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
wenzelm@42814
   743
  {* lift theorems to quotient types *}
kaliszyk@35222
   744
kaliszyk@35222
   745
method_setup lifting_setup =
urbanc@37593
   746
  {* Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@46468
   747
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
wenzelm@42814
   748
  {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   749
urbanc@37593
   750
method_setup descending =
wenzelm@46468
   751
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
wenzelm@42814
   752
  {* decend theorems to the raw level *}
urbanc@37593
   753
urbanc@37593
   754
method_setup descending_setup =
wenzelm@46468
   755
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
wenzelm@42814
   756
  {* set up the three goals for the decending theorems *}
urbanc@37593
   757
urbanc@45782
   758
method_setup partiality_descending =
wenzelm@46468
   759
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
urbanc@45782
   760
  {* decend theorems to the raw level *}
urbanc@45782
   761
urbanc@45782
   762
method_setup partiality_descending_setup =
urbanc@45782
   763
  {* Scan.succeed (fn ctxt => 
wenzelm@46468
   764
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
urbanc@45782
   765
  {* set up the three goals for the decending theorems *}
urbanc@45782
   766
kaliszyk@35222
   767
method_setup regularize =
wenzelm@46468
   768
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
wenzelm@42814
   769
  {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   770
kaliszyk@35222
   771
method_setup injection =
wenzelm@46468
   772
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
wenzelm@42814
   773
  {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   774
kaliszyk@35222
   775
method_setup cleaning =
wenzelm@46468
   776
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
wenzelm@42814
   777
  {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   778
kaliszyk@35222
   779
attribute_setup quot_lifted =
kaliszyk@35222
   780
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814
   781
  {* lift theorems to quotient types *}
kaliszyk@35222
   782
kaliszyk@35222
   783
no_notation
kaliszyk@35222
   784
  rel_conj (infixr "OOO" 75) and
haftmann@40602
   785
  map_fun (infixr "--->" 55) and
kaliszyk@35222
   786
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   787
kaliszyk@35222
   788
end