src/HOL/Quotient.thy
author kuncar
Tue Aug 13 15:59:22 2013 +0200 (2013-08-13)
changeset 53011 aeee0a4be6cf
parent 51112 da97167e03f7
child 54555 e8c5e95d338b
permissions -rw-r--r--
introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
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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 Hilbert_Choice Equiv_Relations Lifting
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keywords
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  "print_quotmapsQ3" "print_quotientsQ3" "print_quotconsts" :: diag and
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  "quotient_type" :: thy_goal and "/" and
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  "quotient_definition" :: thy_goal
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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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context
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begin
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interpretation lifting_syntax .
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subsection {* Quotient Predicate *}
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definition
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  "Quotient3 R Abs Rep \<longleftrightarrow>
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     (\<forall>a. Abs (Rep a) = a) \<and> (\<forall>a. R (Rep a) (Rep a)) \<and>
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     (\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s)"
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lemma Quotient3I:
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  assumes "\<And>a. Abs (Rep a) = a"
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    and "\<And>a. R (Rep a) (Rep a)"
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    and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s"
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  shows "Quotient3 R Abs Rep"
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  using assms unfolding Quotient3_def by blast
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lemma Quotient3_abs_rep:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "Abs (Rep a) = a"
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  using a
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  unfolding Quotient3_def
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  by simp
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lemma Quotient3_rep_reflp:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "R (Rep a) (Rep a)"
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  using a
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  unfolding Quotient3_def
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  by blast
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lemma Quotient3_rel:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s" -- {* orientation does not loop on rewriting *}
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  using a
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  unfolding Quotient3_def
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  by blast
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lemma Quotient3_refl1: 
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  assumes a: "Quotient3 R Abs Rep" 
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  shows "R r s \<Longrightarrow> R r r"
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  using a unfolding Quotient3_def 
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  by fast
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lemma Quotient3_refl2: 
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  assumes a: "Quotient3 R Abs Rep" 
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  shows "R r s \<Longrightarrow> R s s"
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  using a unfolding Quotient3_def 
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  by fast
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lemma Quotient3_rel_rep:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "R (Rep a) (Rep b) \<longleftrightarrow> a = b"
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  using a
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  unfolding Quotient3_def
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  by metis
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lemma Quotient3_rep_abs:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "R r r \<Longrightarrow> R (Rep (Abs r)) r"
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  using a unfolding Quotient3_def
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  by blast
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lemma Quotient3_rel_abs:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "R r s \<Longrightarrow> Abs r = Abs s"
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  using a unfolding Quotient3_def
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  by blast
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lemma Quotient3_symp:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "symp R"
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  using a unfolding Quotient3_def using sympI by metis
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lemma Quotient3_transp:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "transp R"
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  using a unfolding Quotient3_def using transpI by (metis (full_types))
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lemma Quotient3_part_equivp:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "part_equivp R"
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by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp a part_equivpI)
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lemma identity_quotient3:
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  shows "Quotient3 (op =) id id"
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  unfolding Quotient3_def id_def
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  by blast
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lemma fun_quotient3:
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  assumes q1: "Quotient3 R1 abs1 rep1"
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  and     q2: "Quotient3 R2 abs2 rep2"
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  shows "Quotient3 (R1 ===> R2) (rep1 ---> abs2) (abs1 ---> rep2)"
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proof -
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  have "\<And>a.(rep1 ---> abs2) ((abs1 ---> rep2) a) = a"
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    using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
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  moreover
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  have "\<And>a.(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)"
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    by (rule fun_relI)
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      (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
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        simp (no_asm) add: Quotient3_def, simp)
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  moreover
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  {
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  fix r s
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  have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
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        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)"
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  proof -
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    have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding fun_rel_def
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      using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
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      by (metis (full_types) part_equivp_def)
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    moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding fun_rel_def
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      using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
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      by (metis (full_types) part_equivp_def)
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    moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
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      apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def by metis
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    moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
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        (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
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      apply(auto simp add: fun_rel_def fun_eq_iff) using q1 q2 unfolding Quotient3_def 
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    by (metis map_fun_apply)
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    ultimately show ?thesis by blast
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 qed
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 }
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 ultimately show ?thesis by (intro Quotient3I) (assumption+)
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qed
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lemma abs_o_rep:
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  assumes a: "Quotient3 R Abs Rep"
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  shows "Abs o Rep = id"
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  unfolding fun_eq_iff
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  by (simp add: Quotient3_abs_rep[OF a])
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lemma equals_rsp:
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  assumes q: "Quotient3 R Abs Rep"
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  and     a: "R xa xb" "R ya yb"
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  shows "R xa ya = R xb yb"
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  using a Quotient3_symp[OF q] Quotient3_transp[OF q]
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  by (blast elim: sympE transpE)
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lemma lambda_prs:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
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  unfolding fun_eq_iff
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  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
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  by simp
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lemma lambda_prs1:
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  assumes q1: "Quotient3 R1 Abs1 Rep1"
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  and     q2: "Quotient3 R2 Abs2 Rep2"
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  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
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  unfolding fun_eq_iff
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  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
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  by simp
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lemma rep_abs_rsp:
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  assumes q: "Quotient3 R Abs Rep"
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  and     a: "R x1 x2"
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  shows "R x1 (Rep (Abs x2))"
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  using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
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  by metis
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lemma rep_abs_rsp_left:
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  assumes q: "Quotient3 R Abs Rep"
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  and     a: "R x1 x2"
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  shows "R (Rep (Abs x1)) x2"
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  using a Quotient3_rel[OF q] Quotient3_abs_rep[OF q] Quotient3_rep_reflp[OF q]
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  by metis
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text{*
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  In the following theorem R1 can be instantiated with anything,
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  but we know some of the types of the Rep and Abs functions;
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  so by solving Quotient assumptions we can get a unique R1 that
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  will be provable; which is why we need to use @{text apply_rsp} and
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  not the primed version *}
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lemma apply_rspQ3:
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  fixes f g::"'a \<Rightarrow> 'c"
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  assumes q: "Quotient3 R1 Abs1 Rep1"
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  and     a: "(R1 ===> R2) f g" "R1 x y"
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  shows "R2 (f x) (g y)"
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  using a by (auto elim: fun_relE)
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lemma apply_rspQ3'':
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  assumes "Quotient3 R Abs Rep"
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  and "(R ===> S) f f"
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  shows "S (f (Rep x)) (f (Rep x))"
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proof -
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  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
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  then show ?thesis using assms(2) by (auto intro: apply_rsp')
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qed
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subsection {* lemmas for regularisation of ball and bex *}
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lemma ball_reg_eqv:
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  fixes P :: "'a \<Rightarrow> bool"
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  assumes a: "equivp R"
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  shows "Ball (Respects R) P = (All P)"
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  using a
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  unfolding equivp_def
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  by (auto simp add: in_respects)
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lemma bex_reg_eqv:
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  fixes P :: "'a \<Rightarrow> bool"
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  assumes a: "equivp R"
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  shows "Bex (Respects R) P = (Ex P)"
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  using a
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  unfolding equivp_def
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  by (auto simp add: in_respects)
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lemma ball_reg_right:
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  assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
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  shows "All P \<longrightarrow> Ball R Q"
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  using a by fast
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lemma bex_reg_left:
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  assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
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  shows "Bex R Q \<longrightarrow> Ex P"
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  using a by fast
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lemma ball_reg_left:
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  assumes a: "equivp R"
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  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
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  using a by (metis equivp_reflp in_respects)
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lemma bex_reg_right:
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  assumes a: "equivp R"
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  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
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  using a by (metis equivp_reflp in_respects)
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lemma ball_reg_eqv_range:
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  fixes P::"'a \<Rightarrow> bool"
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  and x::"'a"
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  assumes a: "equivp R2"
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  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
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  apply(rule iffI)
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  apply(rule allI)
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  apply(drule_tac x="\<lambda>y. f x" in bspec)
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  apply(simp add: in_respects fun_rel_def)
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  apply(rule impI)
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  using a equivp_reflp_symp_transp[of "R2"]
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  apply (auto elim: equivpE reflpE)
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  done
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lemma bex_reg_eqv_range:
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  assumes a: "equivp R2"
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  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
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  apply(auto)
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  apply(rule_tac x="\<lambda>y. f x" in bexI)
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  apply(simp)
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  apply(simp add: Respects_def in_respects fun_rel_def)
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  apply(rule impI)
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  using a equivp_reflp_symp_transp[of "R2"]
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  apply (auto elim: equivpE reflpE)
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  done
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(* Next four lemmas are unused *)
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lemma all_reg:
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  assumes a: "!x :: 'a. (P x --> Q x)"
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  and     b: "All P"
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  shows "All Q"
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  using a b by fast
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lemma ex_reg:
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  assumes a: "!x :: 'a. (P x --> Q x)"
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  and     b: "Ex P"
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  shows "Ex Q"
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  using a b by fast
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lemma ball_reg:
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  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
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  and     b: "Ball R P"
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  shows "Ball R Q"
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  using a b by fast
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lemma bex_reg:
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  assumes a: "!x :: 'a. (x \<in> R --> P x --> Q x)"
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  and     b: "Bex R P"
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  shows "Bex R Q"
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  using a b by fast
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lemma ball_all_comm:
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  assumes "\<And>y. (\<forall>x\<in>P. A x y) \<longrightarrow> (\<forall>x. B x y)"
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  shows "(\<forall>x\<in>P. \<forall>y. A x y) \<longrightarrow> (\<forall>x. \<forall>y. B x y)"
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   319
  using assms by auto
kaliszyk@35222
   320
kaliszyk@35222
   321
lemma bex_ex_comm:
kaliszyk@35222
   322
  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
kaliszyk@35222
   323
  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
kaliszyk@35222
   324
  using assms by auto
kaliszyk@35222
   325
huffman@35294
   326
subsection {* Bounded abstraction *}
kaliszyk@35222
   327
kaliszyk@35222
   328
definition
haftmann@40466
   329
  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
kaliszyk@35222
   330
where
kaliszyk@35222
   331
  "x \<in> p \<Longrightarrow> Babs p m x = m x"
kaliszyk@35222
   332
kaliszyk@35222
   333
lemma babs_rsp:
kuncar@47308
   334
  assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222
   335
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   336
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   337
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   338
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   339
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   340
  apply (simp add: in_respects fun_rel_def)
kuncar@47308
   341
  using Quotient3_rel[OF q]
kaliszyk@35222
   342
  by metis
kaliszyk@35222
   343
kaliszyk@35222
   344
lemma babs_prs:
kuncar@47308
   345
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   346
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222
   347
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   348
  apply (rule ext)
haftmann@40466
   349
  apply (simp add:)
kaliszyk@35222
   350
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kuncar@47308
   351
  apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
kuncar@47308
   352
  apply (simp add: in_respects Quotient3_rel_rep[OF q1])
kaliszyk@35222
   353
  done
kaliszyk@35222
   354
kaliszyk@35222
   355
lemma babs_simp:
kuncar@47308
   356
  assumes q: "Quotient3 R1 Abs Rep"
kaliszyk@35222
   357
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   358
  apply(rule iffI)
kaliszyk@35222
   359
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   360
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   361
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   362
  apply(metis Babs_def)
kaliszyk@35222
   363
  apply (simp add: in_respects)
kuncar@47308
   364
  using Quotient3_rel[OF q]
kaliszyk@35222
   365
  by metis
kaliszyk@35222
   366
kaliszyk@35222
   367
(* If a user proves that a particular functional relation
kaliszyk@35222
   368
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   369
lemma babs_reg_eqv:
kaliszyk@35222
   370
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   371
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   372
kaliszyk@35222
   373
kaliszyk@35222
   374
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   375
lemma ball_rsp:
kaliszyk@35222
   376
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   377
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   378
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   379
kaliszyk@35222
   380
lemma bex_rsp:
kaliszyk@35222
   381
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   382
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   383
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   384
kaliszyk@35222
   385
lemma bex1_rsp:
kaliszyk@35222
   386
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   387
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   388
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   389
kaliszyk@35222
   390
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   391
lemma all_prs:
kuncar@47308
   392
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   393
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kuncar@47308
   394
  using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   395
  by metis
kaliszyk@35222
   396
kaliszyk@35222
   397
lemma ex_prs:
kuncar@47308
   398
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   399
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kuncar@47308
   400
  using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   401
  by metis
kaliszyk@35222
   402
huffman@35294
   403
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   404
kaliszyk@35222
   405
definition
kaliszyk@35222
   406
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   407
where
kaliszyk@35222
   408
  "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
   409
kaliszyk@35222
   410
lemma bex1_rel_aux:
kaliszyk@35222
   411
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   412
  unfolding Bex1_rel_def
kaliszyk@35222
   413
  apply (erule conjE)+
kaliszyk@35222
   414
  apply (erule bexE)
kaliszyk@35222
   415
  apply rule
kaliszyk@35222
   416
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   417
  apply metis
kaliszyk@35222
   418
  apply metis
kaliszyk@35222
   419
  apply rule+
kaliszyk@35222
   420
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   421
  prefer 2
kaliszyk@35222
   422
  apply (metis)
kaliszyk@35222
   423
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   424
  prefer 2
kaliszyk@35222
   425
  apply (metis)
kaliszyk@35222
   426
  apply (metis in_respects)
kaliszyk@35222
   427
  done
kaliszyk@35222
   428
kaliszyk@35222
   429
lemma bex1_rel_aux2:
kaliszyk@35222
   430
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   431
  unfolding Bex1_rel_def
kaliszyk@35222
   432
  apply (erule conjE)+
kaliszyk@35222
   433
  apply (erule bexE)
kaliszyk@35222
   434
  apply rule
kaliszyk@35222
   435
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   436
  apply metis
kaliszyk@35222
   437
  apply metis
kaliszyk@35222
   438
  apply rule+
kaliszyk@35222
   439
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   440
  prefer 2
kaliszyk@35222
   441
  apply (metis)
kaliszyk@35222
   442
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   443
  prefer 2
kaliszyk@35222
   444
  apply (metis)
kaliszyk@35222
   445
  apply (metis in_respects)
kaliszyk@35222
   446
  done
kaliszyk@35222
   447
kaliszyk@35222
   448
lemma bex1_rel_rsp:
kuncar@47308
   449
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   450
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   451
  apply (simp add: fun_rel_def)
kaliszyk@35222
   452
  apply clarify
kaliszyk@35222
   453
  apply rule
kaliszyk@35222
   454
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   455
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   456
  apply assumption
kaliszyk@35222
   457
  done
kaliszyk@35222
   458
kaliszyk@35222
   459
kaliszyk@35222
   460
lemma ex1_prs:
kuncar@47308
   461
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   462
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   463
apply (simp add:)
kaliszyk@35222
   464
apply (subst Bex1_rel_def)
kaliszyk@35222
   465
apply (subst Bex_def)
kaliszyk@35222
   466
apply (subst Ex1_def)
kaliszyk@35222
   467
apply simp
kaliszyk@35222
   468
apply rule
kaliszyk@35222
   469
 apply (erule conjE)+
kaliszyk@35222
   470
 apply (erule_tac exE)
kaliszyk@35222
   471
 apply (erule conjE)
kaliszyk@35222
   472
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   473
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   474
  apply (simp)
kaliszyk@35222
   475
  apply rule+
kuncar@47308
   476
  using a unfolding Quotient3_def
kaliszyk@35222
   477
  apply metis
kaliszyk@35222
   478
 apply rule+
kaliszyk@35222
   479
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   480
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   481
   apply simp
kaliszyk@35222
   482
  apply (simp add: in_respects)
kaliszyk@35222
   483
 apply (simp add: in_respects)
kaliszyk@35222
   484
apply (erule_tac exE)
kaliszyk@35222
   485
 apply rule
kaliszyk@35222
   486
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   487
 apply (simp only: in_respects)
kaliszyk@35222
   488
  apply rule
kuncar@47308
   489
 apply (metis Quotient3_rel_rep[OF a])
kuncar@47308
   490
using a unfolding Quotient3_def apply (simp)
kaliszyk@35222
   491
apply rule+
kuncar@47308
   492
using a unfolding Quotient3_def in_respects
kaliszyk@35222
   493
apply metis
kaliszyk@35222
   494
done
kaliszyk@35222
   495
kaliszyk@38702
   496
lemma bex1_bexeq_reg:
kaliszyk@38702
   497
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   498
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   499
  apply clarify
kaliszyk@35222
   500
  apply auto
kaliszyk@35222
   501
  apply (rule bexI)
kaliszyk@35222
   502
  apply assumption
kaliszyk@35222
   503
  apply (simp add: in_respects)
kaliszyk@35222
   504
  apply (simp add: in_respects)
kaliszyk@35222
   505
  apply auto
kaliszyk@35222
   506
  done
kaliszyk@35222
   507
kaliszyk@38702
   508
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   509
  assumes a: "equivp R"
kaliszyk@38702
   510
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   511
  using equivp_reflp[OF a]
kaliszyk@38702
   512
  apply (intro impI)
kaliszyk@38702
   513
  apply (elim ex1E)
kaliszyk@38702
   514
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   515
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   516
  apply (subst in_respects)
kaliszyk@38702
   517
  apply (rule conjI)
kaliszyk@38702
   518
  apply assumption
kaliszyk@38702
   519
  apply assumption
kaliszyk@38702
   520
  apply clarify
kaliszyk@38702
   521
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   522
  apply simp
kaliszyk@38702
   523
  done
kaliszyk@38702
   524
huffman@35294
   525
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   526
kaliszyk@35222
   527
lemma quot_rel_rsp:
kuncar@47308
   528
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222
   529
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   530
  apply(rule fun_relI)+
kaliszyk@35222
   531
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   532
  apply(assumption)+
kaliszyk@35222
   533
  done
kaliszyk@35222
   534
kaliszyk@35222
   535
lemma o_prs:
kuncar@47308
   536
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   537
  and     q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308
   538
  and     q3: "Quotient3 R3 Abs3 Rep3"
kaliszyk@36215
   539
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   540
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kuncar@47308
   541
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
haftmann@40466
   542
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   543
kaliszyk@35222
   544
lemma o_rsp:
kaliszyk@36215
   545
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   546
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
huffman@44921
   547
  by (force elim: fun_relE)+
kaliszyk@35222
   548
kaliszyk@35222
   549
lemma cond_prs:
kuncar@47308
   550
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   551
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kuncar@47308
   552
  using a unfolding Quotient3_def by auto
kaliszyk@35222
   553
kaliszyk@35222
   554
lemma if_prs:
kuncar@47308
   555
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   556
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kuncar@47308
   557
  using Quotient3_abs_rep[OF q]
nipkow@39302
   558
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   559
kaliszyk@35222
   560
lemma if_rsp:
kuncar@47308
   561
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   562
  shows "(op = ===> R ===> R ===> R) If If"
huffman@44921
   563
  by force
kaliszyk@35222
   564
kaliszyk@35222
   565
lemma let_prs:
kuncar@47308
   566
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   567
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@37049
   568
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kuncar@47308
   569
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
nipkow@39302
   570
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   571
kaliszyk@35222
   572
lemma let_rsp:
kaliszyk@37049
   573
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
huffman@44921
   574
  by (force elim: fun_relE)
kaliszyk@35222
   575
kaliszyk@39669
   576
lemma id_rsp:
kaliszyk@39669
   577
  shows "(R ===> R) id id"
huffman@44921
   578
  by auto
kaliszyk@39669
   579
kaliszyk@39669
   580
lemma id_prs:
kuncar@47308
   581
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@39669
   582
  shows "(Rep ---> Abs) id = id"
kuncar@47308
   583
  by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
kaliszyk@39669
   584
kuncar@53011
   585
end
kaliszyk@39669
   586
kaliszyk@35222
   587
locale quot_type =
kaliszyk@35222
   588
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   589
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   590
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   591
  assumes equivp: "part_equivp R"
kaliszyk@44204
   592
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   593
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   594
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   595
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   596
begin
kaliszyk@35222
   597
kaliszyk@35222
   598
definition
haftmann@40466
   599
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   600
where
kaliszyk@44204
   601
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   602
kaliszyk@35222
   603
definition
haftmann@40466
   604
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   605
where
kaliszyk@44204
   606
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   607
kaliszyk@44204
   608
lemma some_collect:
kaliszyk@37493
   609
  assumes "R r r"
kaliszyk@44204
   610
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   611
  apply simp
kaliszyk@44204
   612
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   613
kaliszyk@35222
   614
lemma Quotient:
kuncar@47308
   615
  shows "Quotient3 R abs rep"
kuncar@47308
   616
  unfolding Quotient3_def abs_def rep_def
kaliszyk@37493
   617
  proof (intro conjI allI)
kaliszyk@37493
   618
    fix a r s
kaliszyk@44204
   619
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   620
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   621
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   622
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   623
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204
   624
        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493
   625
    qed
kaliszyk@44204
   626
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   627
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   628
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   629
    proof -
haftmann@44242
   630
      assume "R r r" and "R s s"
haftmann@44242
   631
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   632
        by (metis abs_inverse)
haftmann@44242
   633
      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
   634
        by rule simp_all
haftmann@44242
   635
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   636
    qed
kaliszyk@44204
   637
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   638
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   639
    qed
haftmann@44242
   640
kaliszyk@35222
   641
end
kaliszyk@35222
   642
kuncar@47096
   643
subsection {* Quotient composition *}
kuncar@47096
   644
kuncar@47308
   645
lemma OOO_quotient3:
kuncar@47096
   646
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   647
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   648
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47096
   649
  fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   650
  fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
kuncar@47308
   651
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   652
  assumes R2: "Quotient3 R2 Abs2 Rep2"
kuncar@47096
   653
  assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
kuncar@47096
   654
  assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
kuncar@47308
   655
  shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47308
   656
apply (rule Quotient3I)
kuncar@47308
   657
   apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
kuncar@47096
   658
  apply simp
griff@47434
   659
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
kuncar@47308
   660
   apply (rule Quotient3_rep_reflp [OF R1])
griff@47434
   661
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
kuncar@47308
   662
   apply (rule Quotient3_rep_reflp [OF R1])
kuncar@47096
   663
  apply (rule Rep1)
kuncar@47308
   664
  apply (rule Quotient3_rep_reflp [OF R2])
kuncar@47096
   665
 apply safe
kuncar@47096
   666
    apply (rename_tac x y)
kuncar@47096
   667
    apply (drule Abs1)
kuncar@47308
   668
      apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   669
     apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   670
    apply (drule Quotient3_refl1 [OF R2], drule Rep1)
kuncar@47096
   671
    apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
griff@47434
   672
     apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
griff@47434
   673
     apply (erule relcomppI)
kuncar@47308
   674
     apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   675
    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   676
    apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   677
    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   678
    apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   679
    apply (erule Quotient3_rel_abs [OF R1])
kuncar@47096
   680
   apply (rename_tac x y)
kuncar@47096
   681
   apply (drule Abs1)
kuncar@47308
   682
     apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   683
    apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   684
   apply (drule Quotient3_refl2 [OF R2], drule Rep1)
kuncar@47096
   685
   apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
griff@47434
   686
    apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
griff@47434
   687
    apply (erule relcomppI)
kuncar@47308
   688
    apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   689
   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   690
   apply (rule conjI, erule Quotient3_refl2 [OF R1])
kuncar@47308
   691
   apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   692
   apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   693
   apply (erule Quotient3_rel_abs [OF R1, THEN sym])
kuncar@47096
   694
  apply simp
kuncar@47308
   695
  apply (rule Quotient3_rel_abs [OF R2])
kuncar@47308
   696
  apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
kuncar@47308
   697
  apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
kuncar@47096
   698
  apply (erule Abs1)
kuncar@47308
   699
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   700
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   701
 apply (rename_tac a b c d)
kuncar@47096
   702
 apply simp
griff@47434
   703
 apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
kuncar@47308
   704
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   705
  apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   706
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
griff@47434
   707
 apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
kuncar@47308
   708
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   709
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
kuncar@47308
   710
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47096
   711
 apply (rule Rep1)
kuncar@47096
   712
 apply (drule Abs1)
kuncar@47308
   713
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   714
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   715
 apply (drule Abs1)
kuncar@47308
   716
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   717
 apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   718
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   719
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   720
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   721
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47096
   722
 apply simp
kuncar@47308
   723
 apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
kuncar@47096
   724
 apply simp
kuncar@47096
   725
done
kuncar@47096
   726
kuncar@47308
   727
lemma OOO_eq_quotient3:
kuncar@47096
   728
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   729
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   730
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47308
   731
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   732
  assumes R2: "Quotient3 op= Abs2 Rep2"
kuncar@47308
   733
  shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47096
   734
using assms
kuncar@47308
   735
by (rule OOO_quotient3) auto
kuncar@47096
   736
kuncar@47362
   737
subsection {* Quotient3 to Quotient *}
kuncar@47362
   738
kuncar@47362
   739
lemma Quotient3_to_Quotient:
kuncar@47362
   740
assumes "Quotient3 R Abs Rep"
kuncar@47362
   741
and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
kuncar@47362
   742
shows "Quotient R Abs Rep T"
kuncar@47362
   743
using assms unfolding Quotient3_def by (intro QuotientI) blast+
kuncar@47096
   744
kuncar@47362
   745
lemma Quotient3_to_Quotient_equivp:
kuncar@47362
   746
assumes q: "Quotient3 R Abs Rep"
kuncar@47362
   747
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
kuncar@47362
   748
and eR: "equivp R"
kuncar@47362
   749
shows "Quotient R Abs Rep T"
kuncar@47362
   750
proof (intro QuotientI)
kuncar@47362
   751
  fix a
kuncar@47362
   752
  show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
kuncar@47362
   753
next
kuncar@47362
   754
  fix a
kuncar@47362
   755
  show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
kuncar@47362
   756
next
kuncar@47362
   757
  fix r s
kuncar@47362
   758
  show "R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
kuncar@47362
   759
next
kuncar@47362
   760
  show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
kuncar@47096
   761
qed
kuncar@47096
   762
huffman@35294
   763
subsection {* ML setup *}
kaliszyk@35222
   764
kaliszyk@35222
   765
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   766
wenzelm@48891
   767
ML_file "Tools/Quotient/quotient_info.ML"
wenzelm@41452
   768
setup Quotient_Info.setup
kaliszyk@35222
   769
kuncar@47308
   770
declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
kaliszyk@35222
   771
kuncar@47308
   772
lemmas [quot_thm] = fun_quotient3
haftmann@44553
   773
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553
   774
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222
   775
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   776
kaliszyk@35222
   777
kaliszyk@35222
   778
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   779
lemmas [id_simps] =
kaliszyk@35222
   780
  id_def[symmetric]
haftmann@40602
   781
  map_fun_id
kaliszyk@35222
   782
  id_apply
kaliszyk@35222
   783
  id_o
kaliszyk@35222
   784
  o_id
kaliszyk@35222
   785
  eq_comp_r
kaliszyk@44413
   786
  vimage_id
kaliszyk@35222
   787
kaliszyk@35222
   788
text {* Translation functions for the lifting process. *}
wenzelm@48891
   789
ML_file "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   790
kaliszyk@35222
   791
kaliszyk@35222
   792
text {* Definitions of the quotient types. *}
wenzelm@48891
   793
ML_file "Tools/Quotient/quotient_type.ML"
kaliszyk@35222
   794
kaliszyk@35222
   795
kaliszyk@35222
   796
text {* Definitions for quotient constants. *}
wenzelm@48891
   797
ML_file "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   798
kaliszyk@35222
   799
kaliszyk@35222
   800
text {*
kaliszyk@35222
   801
  An auxiliary constant for recording some information
kaliszyk@35222
   802
  about the lifted theorem in a tactic.
kaliszyk@35222
   803
*}
kaliszyk@35222
   804
definition
haftmann@40466
   805
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   806
where
haftmann@40466
   807
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   808
kaliszyk@35222
   809
lemma
kaliszyk@35222
   810
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   811
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   812
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   813
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   814
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   815
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   816
kaliszyk@35222
   817
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   818
  by (simp add: Quot_True_def)
kaliszyk@35222
   819
kuncar@53011
   820
context 
kuncar@53011
   821
begin
kuncar@53011
   822
interpretation lifting_syntax .
kaliszyk@35222
   823
kaliszyk@35222
   824
text {* Tactics for proving the lifted theorems *}
wenzelm@48891
   825
ML_file "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   826
kuncar@53011
   827
end
kuncar@53011
   828
huffman@35294
   829
subsection {* Methods / Interface *}
kaliszyk@35222
   830
kaliszyk@35222
   831
method_setup lifting =
urbanc@37593
   832
  {* Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@46468
   833
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
wenzelm@42814
   834
  {* lift theorems to quotient types *}
kaliszyk@35222
   835
kaliszyk@35222
   836
method_setup lifting_setup =
urbanc@37593
   837
  {* Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@46468
   838
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
wenzelm@42814
   839
  {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   840
urbanc@37593
   841
method_setup descending =
wenzelm@46468
   842
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
wenzelm@42814
   843
  {* decend theorems to the raw level *}
urbanc@37593
   844
urbanc@37593
   845
method_setup descending_setup =
wenzelm@46468
   846
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
wenzelm@42814
   847
  {* set up the three goals for the decending theorems *}
urbanc@37593
   848
urbanc@45782
   849
method_setup partiality_descending =
wenzelm@46468
   850
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
urbanc@45782
   851
  {* decend theorems to the raw level *}
urbanc@45782
   852
urbanc@45782
   853
method_setup partiality_descending_setup =
urbanc@45782
   854
  {* Scan.succeed (fn ctxt => 
wenzelm@46468
   855
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
urbanc@45782
   856
  {* set up the three goals for the decending theorems *}
urbanc@45782
   857
kaliszyk@35222
   858
method_setup regularize =
wenzelm@46468
   859
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
wenzelm@42814
   860
  {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   861
kaliszyk@35222
   862
method_setup injection =
wenzelm@46468
   863
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
wenzelm@42814
   864
  {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   865
kaliszyk@35222
   866
method_setup cleaning =
wenzelm@46468
   867
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
wenzelm@42814
   868
  {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   869
kaliszyk@35222
   870
attribute_setup quot_lifted =
kaliszyk@35222
   871
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814
   872
  {* lift theorems to quotient types *}
kaliszyk@35222
   873
kaliszyk@35222
   874
no_notation
kuncar@53011
   875
  rel_conj (infixr "OOO" 75)
kaliszyk@35222
   876
kaliszyk@35222
   877
end
haftmann@47488
   878