src/HOL/Quotient.thy
author Cezary Kaliszyk <kaliszyk@in.tum.de>
Tue Aug 23 03:34:17 2011 +0900 (2011-08-23)
changeset 44413 80d460bc6fa8
parent 44242 a5cb6aa77ffc
child 44553 4d39b032a021
permissions -rw-r--r--
Quotient Package: some infrastructure for lifting inside sets
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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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uses
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  ("Tools/Quotient/quotient_info.ML")
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  ("Tools/Quotient/quotient_typ.ML")
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  ("Tools/Quotient/quotient_def.ML")
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  ("Tools/Quotient/quotient_term.ML")
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  ("Tools/Quotient/quotient_tacs.ML")
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begin
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text {*
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  Basic definition for equivalence relations
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  that are represented by predicates.
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*}
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text {* Composition of Relations *}
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abbreviation
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  rel_conj :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool" (infixr "OOO" 75)
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where
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  "r1 OOO r2 \<equiv> r1 OO r2 OO r1"
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lemma eq_comp_r:
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  shows "((op =) OOO R) = R"
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  by (auto simp add: fun_eq_iff)
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subsection {* 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 mem_def Respects_def
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  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 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. R x \<Longrightarrow> P x \<longrightarrow> Q x"
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  shows "All P \<longrightarrow> Ball R Q"
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  using a by (metis Collect_def Collect_mem_eq)
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lemma bex_reg_left:
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  assumes a: "\<And>x. R x \<Longrightarrow> Q x \<longrightarrow> P x"
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  shows "Bex R Q \<longrightarrow> Ex P"
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  using a by (metis Collect_def Collect_mem_eq)
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lemma ball_reg_left:
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  assumes a: "equivp R"
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  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
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  using a by (metis equivp_reflp in_respects)
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lemma bex_reg_right:
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  assumes a: "equivp R"
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  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
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  using a by (metis equivp_reflp in_respects)
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lemma ball_reg_eqv_range:
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  fixes P::"'a \<Rightarrow> bool"
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  and x::"'a"
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  assumes a: "equivp R2"
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  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
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  apply(rule iffI)
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  apply(rule allI)
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  apply(drule_tac x="\<lambda>y. f x" in bspec)
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  apply(simp add: in_respects fun_rel_def)
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  apply(rule impI)
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  using a equivp_reflp_symp_transp[of "R2"]
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  apply (auto elim: equivpE reflpE)
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  done
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lemma bex_reg_eqv_range:
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  assumes a: "equivp R2"
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  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
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  apply(auto)
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  apply(rule_tac x="\<lambda>y. f x" in bexI)
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  apply(simp)
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  apply(simp add: Respects_def in_respects fun_rel_def)
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  apply(rule impI)
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  using a equivp_reflp_symp_transp[of "R2"]
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  apply (auto elim: equivpE reflpE)
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  done
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(* Next four lemmas are unused *)
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lemma all_reg:
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  assumes a: "!x :: 'a. (P x --> Q x)"
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  and     b: "All P"
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  shows "All Q"
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  using a b by (metis)
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lemma ex_reg:
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  assumes a: "!x :: 'a. (P x --> Q x)"
kaliszyk@35222
   325
  and     b: "Ex P"
kaliszyk@35222
   326
  shows "Ex Q"
kaliszyk@35222
   327
  using a b by metis
kaliszyk@35222
   328
kaliszyk@35222
   329
lemma ball_reg:
kaliszyk@35222
   330
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   331
  and     b: "Ball R P"
kaliszyk@35222
   332
  shows "Ball R Q"
blanchet@39956
   333
  using a b by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   334
kaliszyk@35222
   335
lemma bex_reg:
kaliszyk@35222
   336
  assumes a: "!x :: 'a. (R x --> P x --> Q x)"
kaliszyk@35222
   337
  and     b: "Bex R P"
kaliszyk@35222
   338
  shows "Bex R Q"
blanchet@39956
   339
  using a b by (metis Collect_def Collect_mem_eq)
kaliszyk@35222
   340
kaliszyk@35222
   341
kaliszyk@35222
   342
lemma ball_all_comm:
kaliszyk@35222
   343
  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
kaliszyk@35222
   344
  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
kaliszyk@35222
   345
  using assms by auto
kaliszyk@35222
   346
kaliszyk@35222
   347
lemma bex_ex_comm:
kaliszyk@35222
   348
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   349
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   350
  using assms by auto
kaliszyk@35222
   351
huffman@35294
   352
subsection {* Bounded abstraction *}
kaliszyk@35222
   353
kaliszyk@35222
   354
definition
haftmann@40466
   355
  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   356
where
kaliszyk@35222
   357
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   358
kaliszyk@35222
   359
lemma babs_rsp:
kaliszyk@35222
   360
  assumes q: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   361
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   362
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   363
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   364
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   365
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   366
  apply (simp add: in_respects fun_rel_def)
kaliszyk@35222
   367
  using Quotient_rel[OF q]
kaliszyk@35222
   368
  by metis
kaliszyk@35222
   369
kaliszyk@35222
   370
lemma babs_prs:
kaliszyk@35222
   371
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   372
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   373
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   374
  apply (rule ext)
haftmann@40466
   375
  apply (simp add:)
kaliszyk@35222
   376
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kaliszyk@35222
   377
  apply (simp add: Babs_def Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])
kaliszyk@35222
   378
  apply (simp add: in_respects Quotient_rel_rep[OF q1])
kaliszyk@35222
   379
  done
kaliszyk@35222
   380
kaliszyk@35222
   381
lemma babs_simp:
kaliszyk@35222
   382
  assumes q: "Quotient R1 Abs Rep"
kaliszyk@35222
   383
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   384
  apply(rule iffI)
kaliszyk@35222
   385
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   386
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   387
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   388
  apply(metis Babs_def)
kaliszyk@35222
   389
  apply (simp add: in_respects)
kaliszyk@35222
   390
  using Quotient_rel[OF q]
kaliszyk@35222
   391
  by metis
kaliszyk@35222
   392
kaliszyk@35222
   393
(* If a user proves that a particular functional relation
kaliszyk@35222
   394
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   395
lemma babs_reg_eqv:
kaliszyk@35222
   396
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   397
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   398
kaliszyk@35222
   399
kaliszyk@35222
   400
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   401
lemma ball_rsp:
kaliszyk@35222
   402
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   403
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   404
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   405
kaliszyk@35222
   406
lemma bex_rsp:
kaliszyk@35222
   407
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   408
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   409
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   410
kaliszyk@35222
   411
lemma bex1_rsp:
kaliszyk@35222
   412
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   413
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   414
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   415
kaliszyk@35222
   416
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   417
lemma all_prs:
kaliszyk@35222
   418
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   419
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
haftmann@40602
   420
  using a unfolding Quotient_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   421
  by metis
kaliszyk@35222
   422
kaliszyk@35222
   423
lemma ex_prs:
kaliszyk@35222
   424
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   425
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
haftmann@40602
   426
  using a unfolding Quotient_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   427
  by metis
kaliszyk@35222
   428
huffman@35294
   429
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   430
kaliszyk@35222
   431
definition
kaliszyk@35222
   432
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   433
where
kaliszyk@35222
   434
  "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
   435
kaliszyk@35222
   436
lemma bex1_rel_aux:
kaliszyk@35222
   437
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   438
  unfolding Bex1_rel_def
kaliszyk@35222
   439
  apply (erule conjE)+
kaliszyk@35222
   440
  apply (erule bexE)
kaliszyk@35222
   441
  apply rule
kaliszyk@35222
   442
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   443
  apply metis
kaliszyk@35222
   444
  apply metis
kaliszyk@35222
   445
  apply rule+
kaliszyk@35222
   446
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   447
  prefer 2
kaliszyk@35222
   448
  apply (metis)
kaliszyk@35222
   449
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   450
  prefer 2
kaliszyk@35222
   451
  apply (metis)
kaliszyk@35222
   452
  apply (metis in_respects)
kaliszyk@35222
   453
  done
kaliszyk@35222
   454
kaliszyk@35222
   455
lemma bex1_rel_aux2:
kaliszyk@35222
   456
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   457
  unfolding Bex1_rel_def
kaliszyk@35222
   458
  apply (erule conjE)+
kaliszyk@35222
   459
  apply (erule bexE)
kaliszyk@35222
   460
  apply rule
kaliszyk@35222
   461
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   462
  apply metis
kaliszyk@35222
   463
  apply metis
kaliszyk@35222
   464
  apply rule+
kaliszyk@35222
   465
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   466
  prefer 2
kaliszyk@35222
   467
  apply (metis)
kaliszyk@35222
   468
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   469
  prefer 2
kaliszyk@35222
   470
  apply (metis)
kaliszyk@35222
   471
  apply (metis in_respects)
kaliszyk@35222
   472
  done
kaliszyk@35222
   473
kaliszyk@35222
   474
lemma bex1_rel_rsp:
kaliszyk@35222
   475
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   476
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   477
  apply (simp add: fun_rel_def)
kaliszyk@35222
   478
  apply clarify
kaliszyk@35222
   479
  apply rule
kaliszyk@35222
   480
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   481
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   482
  apply assumption
kaliszyk@35222
   483
  done
kaliszyk@35222
   484
kaliszyk@35222
   485
kaliszyk@35222
   486
lemma ex1_prs:
kaliszyk@35222
   487
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   488
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   489
apply (simp add:)
kaliszyk@35222
   490
apply (subst Bex1_rel_def)
kaliszyk@35222
   491
apply (subst Bex_def)
kaliszyk@35222
   492
apply (subst Ex1_def)
kaliszyk@35222
   493
apply simp
kaliszyk@35222
   494
apply rule
kaliszyk@35222
   495
 apply (erule conjE)+
kaliszyk@35222
   496
 apply (erule_tac exE)
kaliszyk@35222
   497
 apply (erule conjE)
kaliszyk@35222
   498
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   499
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   500
  apply (simp)
kaliszyk@35222
   501
  apply rule+
kaliszyk@35222
   502
  using a unfolding Quotient_def
kaliszyk@35222
   503
  apply metis
kaliszyk@35222
   504
 apply rule+
kaliszyk@35222
   505
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   506
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   507
   apply simp
kaliszyk@35222
   508
  apply (simp add: in_respects)
kaliszyk@35222
   509
 apply (simp add: in_respects)
kaliszyk@35222
   510
apply (erule_tac exE)
kaliszyk@35222
   511
 apply rule
kaliszyk@35222
   512
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   513
 apply (simp only: in_respects)
kaliszyk@35222
   514
  apply rule
kaliszyk@35222
   515
 apply (metis Quotient_rel_rep[OF a])
kaliszyk@35222
   516
using a unfolding Quotient_def apply (simp)
kaliszyk@35222
   517
apply rule+
kaliszyk@35222
   518
using a unfolding Quotient_def in_respects
kaliszyk@35222
   519
apply metis
kaliszyk@35222
   520
done
kaliszyk@35222
   521
kaliszyk@38702
   522
lemma bex1_bexeq_reg:
kaliszyk@38702
   523
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   524
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   525
  apply clarify
kaliszyk@35222
   526
  apply auto
kaliszyk@35222
   527
  apply (rule bexI)
kaliszyk@35222
   528
  apply assumption
kaliszyk@35222
   529
  apply (simp add: in_respects)
kaliszyk@35222
   530
  apply (simp add: in_respects)
kaliszyk@35222
   531
  apply auto
kaliszyk@35222
   532
  done
kaliszyk@35222
   533
kaliszyk@38702
   534
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   535
  assumes a: "equivp R"
kaliszyk@38702
   536
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   537
  using equivp_reflp[OF a]
kaliszyk@38702
   538
  apply (intro impI)
kaliszyk@38702
   539
  apply (elim ex1E)
kaliszyk@38702
   540
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   541
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   542
  apply (subst in_respects)
kaliszyk@38702
   543
  apply (rule conjI)
kaliszyk@38702
   544
  apply assumption
kaliszyk@38702
   545
  apply assumption
kaliszyk@38702
   546
  apply clarify
kaliszyk@38702
   547
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   548
  apply simp
kaliszyk@38702
   549
  done
kaliszyk@38702
   550
huffman@35294
   551
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   552
kaliszyk@35222
   553
lemma quot_rel_rsp:
kaliszyk@35222
   554
  assumes a: "Quotient R Abs Rep"
kaliszyk@35222
   555
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   556
  apply(rule fun_relI)+
kaliszyk@35222
   557
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   558
  apply(assumption)+
kaliszyk@35222
   559
  done
kaliszyk@35222
   560
kaliszyk@35222
   561
lemma o_prs:
kaliszyk@35222
   562
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   563
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@35222
   564
  and     q3: "Quotient R3 Abs3 Rep3"
kaliszyk@36215
   565
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   566
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kaliszyk@35222
   567
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2] Quotient_abs_rep[OF q3]
haftmann@40466
   568
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   569
kaliszyk@35222
   570
lemma o_rsp:
kaliszyk@36215
   571
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   572
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
haftmann@40466
   573
  by (auto intro!: fun_relI elim: fun_relE)
kaliszyk@35222
   574
kaliszyk@35222
   575
lemma cond_prs:
kaliszyk@35222
   576
  assumes a: "Quotient R absf repf"
kaliszyk@35222
   577
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kaliszyk@35222
   578
  using a unfolding Quotient_def by auto
kaliszyk@35222
   579
kaliszyk@35222
   580
lemma if_prs:
kaliszyk@35222
   581
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   582
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kaliszyk@36123
   583
  using Quotient_abs_rep[OF q]
nipkow@39302
   584
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   585
kaliszyk@35222
   586
lemma if_rsp:
kaliszyk@35222
   587
  assumes q: "Quotient R Abs Rep"
kaliszyk@36123
   588
  shows "(op = ===> R ===> R ===> R) If If"
haftmann@40466
   589
  by (auto intro!: fun_relI)
kaliszyk@35222
   590
kaliszyk@35222
   591
lemma let_prs:
kaliszyk@35222
   592
  assumes q1: "Quotient R1 Abs1 Rep1"
kaliszyk@35222
   593
  and     q2: "Quotient R2 Abs2 Rep2"
kaliszyk@37049
   594
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kaliszyk@37049
   595
  using Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2]
nipkow@39302
   596
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   597
kaliszyk@35222
   598
lemma let_rsp:
kaliszyk@37049
   599
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
haftmann@40466
   600
  by (auto intro!: fun_relI elim: fun_relE)
kaliszyk@35222
   601
kaliszyk@38861
   602
lemma mem_rsp:
kaliszyk@38861
   603
  shows "(R1 ===> (R1 ===> R2) ===> R2) op \<in> op \<in>"
haftmann@40466
   604
  by (auto intro!: fun_relI elim: fun_relE simp add: mem_def)
kaliszyk@38861
   605
kaliszyk@38861
   606
lemma mem_prs:
kaliszyk@38861
   607
  assumes a1: "Quotient R1 Abs1 Rep1"
kaliszyk@38861
   608
  and     a2: "Quotient R2 Abs2 Rep2"
kaliszyk@38861
   609
  shows "(Rep1 ---> (Abs1 ---> Rep2) ---> Abs2) op \<in> = op \<in>"
nipkow@39302
   610
  by (simp add: fun_eq_iff mem_def Quotient_abs_rep[OF a1] Quotient_abs_rep[OF a2])
kaliszyk@38861
   611
kaliszyk@39669
   612
lemma id_rsp:
kaliszyk@39669
   613
  shows "(R ===> R) id id"
haftmann@40466
   614
  by (auto intro: fun_relI)
kaliszyk@39669
   615
kaliszyk@39669
   616
lemma id_prs:
kaliszyk@39669
   617
  assumes a: "Quotient R Abs Rep"
kaliszyk@39669
   618
  shows "(Rep ---> Abs) id = id"
haftmann@40466
   619
  by (simp add: fun_eq_iff Quotient_abs_rep [OF a])
kaliszyk@39669
   620
kaliszyk@39669
   621
kaliszyk@35222
   622
locale quot_type =
kaliszyk@35222
   623
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   624
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   625
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   626
  assumes equivp: "part_equivp R"
kaliszyk@44204
   627
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   628
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   629
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   630
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   631
begin
kaliszyk@35222
   632
kaliszyk@35222
   633
definition
haftmann@40466
   634
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   635
where
kaliszyk@44204
   636
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   637
kaliszyk@35222
   638
definition
haftmann@40466
   639
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   640
where
kaliszyk@44204
   641
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   642
kaliszyk@44204
   643
lemma some_collect:
kaliszyk@37493
   644
  assumes "R r r"
kaliszyk@44204
   645
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   646
  apply simp
kaliszyk@44204
   647
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   648
kaliszyk@35222
   649
lemma Quotient:
kaliszyk@35222
   650
  shows "Quotient R abs rep"
kaliszyk@37493
   651
  unfolding Quotient_def abs_def rep_def
kaliszyk@37493
   652
  proof (intro conjI allI)
kaliszyk@37493
   653
    fix a r s
kaliszyk@44204
   654
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   655
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   656
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   657
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   658
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204
   659
        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493
   660
    qed
kaliszyk@44204
   661
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   662
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   663
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   664
    proof -
haftmann@44242
   665
      assume "R r r" and "R s s"
haftmann@44242
   666
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   667
        by (metis abs_inverse)
haftmann@44242
   668
      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
   669
        by rule simp_all
haftmann@44242
   670
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   671
    qed
kaliszyk@44204
   672
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   673
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   674
    qed
haftmann@44242
   675
kaliszyk@35222
   676
end
kaliszyk@35222
   677
huffman@35294
   678
subsection {* ML setup *}
kaliszyk@35222
   679
kaliszyk@35222
   680
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   681
wenzelm@37986
   682
use "Tools/Quotient/quotient_info.ML"
wenzelm@41452
   683
setup Quotient_Info.setup
kaliszyk@35222
   684
haftmann@40602
   685
declare [[map "fun" = (map_fun, fun_rel)]]
kaliszyk@44413
   686
declare [[map set = (vimage, set_rel)]]
kaliszyk@35222
   687
kaliszyk@35222
   688
lemmas [quot_thm] = fun_quotient
kaliszyk@39669
   689
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp mem_rsp id_rsp
kaliszyk@39669
   690
lemmas [quot_preserve] = if_prs o_prs let_prs mem_prs id_prs
kaliszyk@35222
   691
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   692
kaliszyk@35222
   693
kaliszyk@35222
   694
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   695
lemmas [id_simps] =
kaliszyk@35222
   696
  id_def[symmetric]
haftmann@40602
   697
  map_fun_id
kaliszyk@35222
   698
  id_apply
kaliszyk@35222
   699
  id_o
kaliszyk@35222
   700
  o_id
kaliszyk@35222
   701
  eq_comp_r
kaliszyk@44413
   702
  set_rel_eq
kaliszyk@44413
   703
  vimage_id
kaliszyk@35222
   704
kaliszyk@35222
   705
text {* Translation functions for the lifting process. *}
wenzelm@37986
   706
use "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   707
kaliszyk@35222
   708
kaliszyk@35222
   709
text {* Definitions of the quotient types. *}
wenzelm@37986
   710
use "Tools/Quotient/quotient_typ.ML"
kaliszyk@35222
   711
kaliszyk@35222
   712
kaliszyk@35222
   713
text {* Definitions for quotient constants. *}
wenzelm@37986
   714
use "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   715
kaliszyk@35222
   716
kaliszyk@35222
   717
text {*
kaliszyk@35222
   718
  An auxiliary constant for recording some information
kaliszyk@35222
   719
  about the lifted theorem in a tactic.
kaliszyk@35222
   720
*}
kaliszyk@35222
   721
definition
haftmann@40466
   722
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   723
where
haftmann@40466
   724
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   725
kaliszyk@35222
   726
lemma
kaliszyk@35222
   727
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   728
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   729
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   730
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   731
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   732
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   733
kaliszyk@35222
   734
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   735
  by (simp add: Quot_True_def)
kaliszyk@35222
   736
kaliszyk@35222
   737
kaliszyk@35222
   738
text {* Tactics for proving the lifted theorems *}
wenzelm@37986
   739
use "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   740
huffman@35294
   741
subsection {* Methods / Interface *}
kaliszyk@35222
   742
kaliszyk@35222
   743
method_setup lifting =
urbanc@37593
   744
  {* Attrib.thms >> (fn thms => fn ctxt => 
urbanc@38859
   745
       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_tac ctxt [] thms))) *}
wenzelm@42814
   746
  {* lift theorems to quotient types *}
kaliszyk@35222
   747
kaliszyk@35222
   748
method_setup lifting_setup =
urbanc@37593
   749
  {* Attrib.thm >> (fn thm => fn ctxt => 
urbanc@38859
   750
       SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.lift_procedure_tac ctxt [] thm))) *}
wenzelm@42814
   751
  {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   752
urbanc@37593
   753
method_setup descending =
urbanc@38859
   754
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_tac ctxt []))) *}
wenzelm@42814
   755
  {* decend theorems to the raw level *}
urbanc@37593
   756
urbanc@37593
   757
method_setup descending_setup =
urbanc@38859
   758
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.descend_procedure_tac ctxt []))) *}
wenzelm@42814
   759
  {* set up the three goals for the decending theorems *}
urbanc@37593
   760
kaliszyk@35222
   761
method_setup regularize =
kaliszyk@35222
   762
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.regularize_tac ctxt))) *}
wenzelm@42814
   763
  {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   764
kaliszyk@35222
   765
method_setup injection =
kaliszyk@35222
   766
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.all_injection_tac ctxt))) *}
wenzelm@42814
   767
  {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   768
kaliszyk@35222
   769
method_setup cleaning =
kaliszyk@35222
   770
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD (HEADGOAL (Quotient_Tacs.clean_tac ctxt))) *}
wenzelm@42814
   771
  {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   772
kaliszyk@35222
   773
attribute_setup quot_lifted =
kaliszyk@35222
   774
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814
   775
  {* lift theorems to quotient types *}
kaliszyk@35222
   776
kaliszyk@35222
   777
no_notation
kaliszyk@35222
   778
  rel_conj (infixr "OOO" 75) and
haftmann@40602
   779
  map_fun (infixr "--->" 55) and
kaliszyk@35222
   780
  fun_rel (infixr "===>" 55)
kaliszyk@35222
   781
kaliszyk@35222
   782
end