src/HOL/Quotient.thy
author haftmann
Thu Dec 26 22:47:49 2013 +0100 (2013-12-26)
changeset 54867 c21a2465cac1
parent 54555 e8c5e95d338b
child 55945 e96383acecf9
permissions -rw-r--r--
prefer ephemeral interpretation over interpretation in proof contexts;
prefer context begin ... end blocks for often-occuring assumptions;
slightly more complete interpretations into abstract algebraic structures for gcd/lcm
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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 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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context
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  fixes R Abs Rep
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  assumes a: "Quotient3 R Abs Rep"
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begin
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lemma Quotient3_abs_rep:
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  "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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  "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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  "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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  "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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  "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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  "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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  "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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  "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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  "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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  "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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  "part_equivp R"
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  by (metis Quotient3_rep_reflp Quotient3_symp Quotient3_transp part_equivpI)
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lemma abs_o_rep:
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  "Abs o Rep = id"
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  unfolding fun_eq_iff
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  by (simp add: Quotient3_abs_rep)
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lemma equals_rsp:
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  assumes b: "R xa xb" "R ya yb"
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  shows "R xa ya = R xb yb"
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  using b Quotient3_symp Quotient3_transp
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  by (blast elim: sympE transpE)
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lemma rep_abs_rsp:
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  assumes b: "R x1 x2"
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  shows "R x1 (Rep (Abs x2))"
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  using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
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  by metis
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lemma rep_abs_rsp_left:
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  assumes b: "R x1 x2"
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  shows "R (Rep (Abs x1)) x2"
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  using b Quotient3_rel Quotient3_abs_rep Quotient3_rep_reflp
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  by metis
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end
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lemma identity_quotient3:
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  "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 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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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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  using assms by auto
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lemma bex_ex_comm:
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  assumes "(\<exists>y. \<exists>x. A x y) \<longrightarrow> (\<exists>y. \<exists>x\<in>P. B x y)"
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  shows "(\<exists>x. \<exists>y. A x y) \<longrightarrow> (\<exists>x\<in>P. \<exists>y. B x y)"
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  using assms by auto
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subsection {* Bounded abstraction *}
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definition
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  Babs :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
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where
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  "x \<in> p \<Longrightarrow> Babs p m x = m x"
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   325
lemma babs_rsp:
kuncar@47308
   326
  assumes q: "Quotient3 R1 Abs1 Rep1"
kaliszyk@35222
   327
  and     a: "(R1 ===> R2) f g"
kaliszyk@35222
   328
  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
haftmann@40466
   329
  apply (auto simp add: Babs_def in_respects fun_rel_def)
kaliszyk@35222
   330
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
haftmann@40466
   331
  using a apply (simp add: Babs_def fun_rel_def)
haftmann@40466
   332
  apply (simp add: in_respects fun_rel_def)
kuncar@47308
   333
  using Quotient3_rel[OF q]
kaliszyk@35222
   334
  by metis
kaliszyk@35222
   335
kaliszyk@35222
   336
lemma babs_prs:
kuncar@47308
   337
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   338
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@35222
   339
  shows "((Rep1 ---> Abs2) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
kaliszyk@35222
   340
  apply (rule ext)
haftmann@40466
   341
  apply (simp add:)
kaliszyk@35222
   342
  apply (subgoal_tac "Rep1 x \<in> Respects R1")
kuncar@47308
   343
  apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
kuncar@47308
   344
  apply (simp add: in_respects Quotient3_rel_rep[OF q1])
kaliszyk@35222
   345
  done
kaliszyk@35222
   346
kaliszyk@35222
   347
lemma babs_simp:
kuncar@47308
   348
  assumes q: "Quotient3 R1 Abs Rep"
kaliszyk@35222
   349
  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
kaliszyk@35222
   350
  apply(rule iffI)
kaliszyk@35222
   351
  apply(simp_all only: babs_rsp[OF q])
haftmann@40466
   352
  apply(auto simp add: Babs_def fun_rel_def)
kaliszyk@35222
   353
  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
kaliszyk@35222
   354
  apply(metis Babs_def)
kaliszyk@35222
   355
  apply (simp add: in_respects)
kuncar@47308
   356
  using Quotient3_rel[OF q]
kaliszyk@35222
   357
  by metis
kaliszyk@35222
   358
kaliszyk@35222
   359
(* If a user proves that a particular functional relation
kaliszyk@35222
   360
   is an equivalence this may be useful in regularising *)
kaliszyk@35222
   361
lemma babs_reg_eqv:
kaliszyk@35222
   362
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
nipkow@39302
   363
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
kaliszyk@35222
   364
kaliszyk@35222
   365
kaliszyk@35222
   366
(* 3 lemmas needed for proving repabs_inj *)
kaliszyk@35222
   367
lemma ball_rsp:
kaliszyk@35222
   368
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   369
  shows "Ball (Respects R) f = Ball (Respects R) g"
haftmann@40466
   370
  using a by (auto simp add: Ball_def in_respects elim: fun_relE)
kaliszyk@35222
   371
kaliszyk@35222
   372
lemma bex_rsp:
kaliszyk@35222
   373
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   374
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
haftmann@40466
   375
  using a by (auto simp add: Bex_def in_respects elim: fun_relE)
kaliszyk@35222
   376
kaliszyk@35222
   377
lemma bex1_rsp:
kaliszyk@35222
   378
  assumes a: "(R ===> (op =)) f g"
kaliszyk@35222
   379
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
haftmann@40466
   380
  using a by (auto elim: fun_relE simp add: Ex1_def in_respects) 
kaliszyk@35222
   381
kaliszyk@35222
   382
(* 2 lemmas needed for cleaning of quantifiers *)
kaliszyk@35222
   383
lemma all_prs:
kuncar@47308
   384
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   385
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
kuncar@47308
   386
  using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   387
  by metis
kaliszyk@35222
   388
kaliszyk@35222
   389
lemma ex_prs:
kuncar@47308
   390
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   391
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
kuncar@47308
   392
  using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
kaliszyk@35222
   393
  by metis
kaliszyk@35222
   394
huffman@35294
   395
subsection {* @{text Bex1_rel} quantifier *}
kaliszyk@35222
   396
kaliszyk@35222
   397
definition
kaliszyk@35222
   398
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
kaliszyk@35222
   399
where
kaliszyk@35222
   400
  "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
   401
kaliszyk@35222
   402
lemma bex1_rel_aux:
kaliszyk@35222
   403
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
kaliszyk@35222
   404
  unfolding Bex1_rel_def
kaliszyk@35222
   405
  apply (erule conjE)+
kaliszyk@35222
   406
  apply (erule bexE)
kaliszyk@35222
   407
  apply rule
kaliszyk@35222
   408
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   409
  apply metis
kaliszyk@35222
   410
  apply metis
kaliszyk@35222
   411
  apply rule+
kaliszyk@35222
   412
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   413
  prefer 2
kaliszyk@35222
   414
  apply (metis)
kaliszyk@35222
   415
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   416
  prefer 2
kaliszyk@35222
   417
  apply (metis)
kaliszyk@35222
   418
  apply (metis in_respects)
kaliszyk@35222
   419
  done
kaliszyk@35222
   420
kaliszyk@35222
   421
lemma bex1_rel_aux2:
kaliszyk@35222
   422
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
kaliszyk@35222
   423
  unfolding Bex1_rel_def
kaliszyk@35222
   424
  apply (erule conjE)+
kaliszyk@35222
   425
  apply (erule bexE)
kaliszyk@35222
   426
  apply rule
kaliszyk@35222
   427
  apply (rule_tac x="xa" in bexI)
kaliszyk@35222
   428
  apply metis
kaliszyk@35222
   429
  apply metis
kaliszyk@35222
   430
  apply rule+
kaliszyk@35222
   431
  apply (erule_tac x="xaa" in ballE)
kaliszyk@35222
   432
  prefer 2
kaliszyk@35222
   433
  apply (metis)
kaliszyk@35222
   434
  apply (erule_tac x="ya" in ballE)
kaliszyk@35222
   435
  prefer 2
kaliszyk@35222
   436
  apply (metis)
kaliszyk@35222
   437
  apply (metis in_respects)
kaliszyk@35222
   438
  done
kaliszyk@35222
   439
kaliszyk@35222
   440
lemma bex1_rel_rsp:
kuncar@47308
   441
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   442
  shows "((R ===> op =) ===> op =) (Bex1_rel R) (Bex1_rel R)"
haftmann@40466
   443
  apply (simp add: fun_rel_def)
kaliszyk@35222
   444
  apply clarify
kaliszyk@35222
   445
  apply rule
kaliszyk@35222
   446
  apply (simp_all add: bex1_rel_aux bex1_rel_aux2)
kaliszyk@35222
   447
  apply (erule bex1_rel_aux2)
kaliszyk@35222
   448
  apply assumption
kaliszyk@35222
   449
  done
kaliszyk@35222
   450
kaliszyk@35222
   451
kaliszyk@35222
   452
lemma ex1_prs:
kuncar@47308
   453
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   454
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
haftmann@40466
   455
apply (simp add:)
kaliszyk@35222
   456
apply (subst Bex1_rel_def)
kaliszyk@35222
   457
apply (subst Bex_def)
kaliszyk@35222
   458
apply (subst Ex1_def)
kaliszyk@35222
   459
apply simp
kaliszyk@35222
   460
apply rule
kaliszyk@35222
   461
 apply (erule conjE)+
kaliszyk@35222
   462
 apply (erule_tac exE)
kaliszyk@35222
   463
 apply (erule conjE)
kaliszyk@35222
   464
 apply (subgoal_tac "\<forall>y. R y y \<longrightarrow> f (absf y) \<longrightarrow> R x y")
kaliszyk@35222
   465
  apply (rule_tac x="absf x" in exI)
kaliszyk@35222
   466
  apply (simp)
kaliszyk@35222
   467
  apply rule+
kuncar@47308
   468
  using a unfolding Quotient3_def
kaliszyk@35222
   469
  apply metis
kaliszyk@35222
   470
 apply rule+
kaliszyk@35222
   471
 apply (erule_tac x="x" in ballE)
kaliszyk@35222
   472
  apply (erule_tac x="y" in ballE)
kaliszyk@35222
   473
   apply simp
kaliszyk@35222
   474
  apply (simp add: in_respects)
kaliszyk@35222
   475
 apply (simp add: in_respects)
kaliszyk@35222
   476
apply (erule_tac exE)
kaliszyk@35222
   477
 apply rule
kaliszyk@35222
   478
 apply (rule_tac x="repf x" in exI)
kaliszyk@35222
   479
 apply (simp only: in_respects)
kaliszyk@35222
   480
  apply rule
kuncar@47308
   481
 apply (metis Quotient3_rel_rep[OF a])
kuncar@47308
   482
using a unfolding Quotient3_def apply (simp)
kaliszyk@35222
   483
apply rule+
kuncar@47308
   484
using a unfolding Quotient3_def in_respects
kaliszyk@35222
   485
apply metis
kaliszyk@35222
   486
done
kaliszyk@35222
   487
kaliszyk@38702
   488
lemma bex1_bexeq_reg:
kaliszyk@38702
   489
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
kaliszyk@35222
   490
  apply (simp add: Ex1_def Bex1_rel_def in_respects)
kaliszyk@35222
   491
  apply clarify
kaliszyk@35222
   492
  apply auto
kaliszyk@35222
   493
  apply (rule bexI)
kaliszyk@35222
   494
  apply assumption
kaliszyk@35222
   495
  apply (simp add: in_respects)
kaliszyk@35222
   496
  apply (simp add: in_respects)
kaliszyk@35222
   497
  apply auto
kaliszyk@35222
   498
  done
kaliszyk@35222
   499
kaliszyk@38702
   500
lemma bex1_bexeq_reg_eqv:
kaliszyk@38702
   501
  assumes a: "equivp R"
kaliszyk@38702
   502
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
kaliszyk@38702
   503
  using equivp_reflp[OF a]
kaliszyk@38702
   504
  apply (intro impI)
kaliszyk@38702
   505
  apply (elim ex1E)
kaliszyk@38702
   506
  apply (rule mp[OF bex1_bexeq_reg])
kaliszyk@38702
   507
  apply (rule_tac a="x" in ex1I)
kaliszyk@38702
   508
  apply (subst in_respects)
kaliszyk@38702
   509
  apply (rule conjI)
kaliszyk@38702
   510
  apply assumption
kaliszyk@38702
   511
  apply assumption
kaliszyk@38702
   512
  apply clarify
kaliszyk@38702
   513
  apply (erule_tac x="xa" in allE)
kaliszyk@38702
   514
  apply simp
kaliszyk@38702
   515
  done
kaliszyk@38702
   516
huffman@35294
   517
subsection {* Various respects and preserve lemmas *}
kaliszyk@35222
   518
kaliszyk@35222
   519
lemma quot_rel_rsp:
kuncar@47308
   520
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@35222
   521
  shows "(R ===> R ===> op =) R R"
urbanc@38317
   522
  apply(rule fun_relI)+
kaliszyk@35222
   523
  apply(rule equals_rsp[OF a])
kaliszyk@35222
   524
  apply(assumption)+
kaliszyk@35222
   525
  done
kaliszyk@35222
   526
kaliszyk@35222
   527
lemma o_prs:
kuncar@47308
   528
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   529
  and     q2: "Quotient3 R2 Abs2 Rep2"
kuncar@47308
   530
  and     q3: "Quotient3 R3 Abs3 Rep3"
kaliszyk@36215
   531
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) op \<circ> = op \<circ>"
kaliszyk@36215
   532
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) op \<circ> = op \<circ>"
kuncar@47308
   533
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
haftmann@40466
   534
  by (simp_all add: fun_eq_iff)
kaliszyk@35222
   535
kaliszyk@35222
   536
lemma o_rsp:
kaliszyk@36215
   537
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) op \<circ> op \<circ>"
kaliszyk@36215
   538
  "(op = ===> (R1 ===> op =) ===> R1 ===> op =) op \<circ> op \<circ>"
huffman@44921
   539
  by (force elim: fun_relE)+
kaliszyk@35222
   540
kaliszyk@35222
   541
lemma cond_prs:
kuncar@47308
   542
  assumes a: "Quotient3 R absf repf"
kaliszyk@35222
   543
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
kuncar@47308
   544
  using a unfolding Quotient3_def by auto
kaliszyk@35222
   545
kaliszyk@35222
   546
lemma if_prs:
kuncar@47308
   547
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   548
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
kuncar@47308
   549
  using Quotient3_abs_rep[OF q]
nipkow@39302
   550
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   551
kaliszyk@35222
   552
lemma if_rsp:
kuncar@47308
   553
  assumes q: "Quotient3 R Abs Rep"
kaliszyk@36123
   554
  shows "(op = ===> R ===> R ===> R) If If"
huffman@44921
   555
  by force
kaliszyk@35222
   556
kaliszyk@35222
   557
lemma let_prs:
kuncar@47308
   558
  assumes q1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   559
  and     q2: "Quotient3 R2 Abs2 Rep2"
kaliszyk@37049
   560
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
kuncar@47308
   561
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
nipkow@39302
   562
  by (auto simp add: fun_eq_iff)
kaliszyk@35222
   563
kaliszyk@35222
   564
lemma let_rsp:
kaliszyk@37049
   565
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
huffman@44921
   566
  by (force elim: fun_relE)
kaliszyk@35222
   567
kaliszyk@39669
   568
lemma id_rsp:
kaliszyk@39669
   569
  shows "(R ===> R) id id"
huffman@44921
   570
  by auto
kaliszyk@39669
   571
kaliszyk@39669
   572
lemma id_prs:
kuncar@47308
   573
  assumes a: "Quotient3 R Abs Rep"
kaliszyk@39669
   574
  shows "(Rep ---> Abs) id = id"
kuncar@47308
   575
  by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
kaliszyk@39669
   576
kuncar@53011
   577
end
kaliszyk@39669
   578
kaliszyk@35222
   579
locale quot_type =
kaliszyk@35222
   580
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kaliszyk@44204
   581
  and   Abs :: "'a set \<Rightarrow> 'b"
kaliszyk@44204
   582
  and   Rep :: "'b \<Rightarrow> 'a set"
kaliszyk@37493
   583
  assumes equivp: "part_equivp R"
kaliszyk@44204
   584
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
kaliszyk@35222
   585
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
kaliszyk@44204
   586
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
kaliszyk@35222
   587
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
kaliszyk@35222
   588
begin
kaliszyk@35222
   589
kaliszyk@35222
   590
definition
haftmann@40466
   591
  abs :: "'a \<Rightarrow> 'b"
kaliszyk@35222
   592
where
kaliszyk@44204
   593
  "abs x = Abs (Collect (R x))"
kaliszyk@35222
   594
kaliszyk@35222
   595
definition
haftmann@40466
   596
  rep :: "'b \<Rightarrow> 'a"
kaliszyk@35222
   597
where
kaliszyk@44204
   598
  "rep a = (SOME x. x \<in> Rep a)"
kaliszyk@35222
   599
kaliszyk@44204
   600
lemma some_collect:
kaliszyk@37493
   601
  assumes "R r r"
kaliszyk@44204
   602
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
kaliszyk@44204
   603
  apply simp
kaliszyk@44204
   604
  by (metis assms exE_some equivp[simplified part_equivp_def])
kaliszyk@35222
   605
kaliszyk@35222
   606
lemma Quotient:
kuncar@47308
   607
  shows "Quotient3 R abs rep"
kuncar@47308
   608
  unfolding Quotient3_def abs_def rep_def
kaliszyk@37493
   609
  proof (intro conjI allI)
kaliszyk@37493
   610
    fix a r s
kaliszyk@44204
   611
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
kaliszyk@44204
   612
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
kaliszyk@44204
   613
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
kaliszyk@44204
   614
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
kaliszyk@44204
   615
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
kaliszyk@44204
   616
        using part_equivp_transp[OF equivp] by (metis `R (SOME x. x \<in> Rep a) x`)
kaliszyk@37493
   617
    qed
kaliszyk@44204
   618
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
kaliszyk@44204
   619
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
kaliszyk@44204
   620
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
haftmann@44242
   621
    proof -
haftmann@44242
   622
      assume "R r r" and "R s s"
haftmann@44242
   623
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
haftmann@44242
   624
        by (metis abs_inverse)
haftmann@44242
   625
      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
   626
        by rule simp_all
haftmann@44242
   627
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
haftmann@44242
   628
    qed
kaliszyk@44204
   629
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
kaliszyk@44204
   630
      using equivp[simplified part_equivp_def] by metis
kaliszyk@44204
   631
    qed
haftmann@44242
   632
kaliszyk@35222
   633
end
kaliszyk@35222
   634
kuncar@47096
   635
subsection {* Quotient composition *}
kuncar@47096
   636
kuncar@47308
   637
lemma OOO_quotient3:
kuncar@47096
   638
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   639
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   640
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47096
   641
  fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   642
  fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
kuncar@47308
   643
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   644
  assumes R2: "Quotient3 R2 Abs2 Rep2"
kuncar@47096
   645
  assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
kuncar@47096
   646
  assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
kuncar@47308
   647
  shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47308
   648
apply (rule Quotient3I)
kuncar@47308
   649
   apply (simp add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
kuncar@47096
   650
  apply simp
griff@47434
   651
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI)
kuncar@47308
   652
   apply (rule Quotient3_rep_reflp [OF R1])
griff@47434
   653
  apply (rule_tac b="Rep1 (Rep2 a)" in relcomppI [rotated])
kuncar@47308
   654
   apply (rule Quotient3_rep_reflp [OF R1])
kuncar@47096
   655
  apply (rule Rep1)
kuncar@47308
   656
  apply (rule Quotient3_rep_reflp [OF R2])
kuncar@47096
   657
 apply safe
kuncar@47096
   658
    apply (rename_tac x y)
kuncar@47096
   659
    apply (drule Abs1)
kuncar@47308
   660
      apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   661
     apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   662
    apply (drule Quotient3_refl1 [OF R2], drule Rep1)
kuncar@47096
   663
    apply (subgoal_tac "R1 r (Rep1 (Abs1 x))")
griff@47434
   664
     apply (rule_tac b="Rep1 (Abs1 x)" in relcomppI, assumption)
griff@47434
   665
     apply (erule relcomppI)
kuncar@47308
   666
     apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   667
    apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   668
    apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   669
    apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   670
    apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   671
    apply (erule Quotient3_rel_abs [OF R1])
kuncar@47096
   672
   apply (rename_tac x y)
kuncar@47096
   673
   apply (drule Abs1)
kuncar@47308
   674
     apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   675
    apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   676
   apply (drule Quotient3_refl2 [OF R2], drule Rep1)
kuncar@47096
   677
   apply (subgoal_tac "R1 s (Rep1 (Abs1 y))")
griff@47434
   678
    apply (rule_tac b="Rep1 (Abs1 y)" in relcomppI, assumption)
griff@47434
   679
    apply (erule relcomppI)
kuncar@47308
   680
    apply (erule Quotient3_symp [OF R1, THEN sympD])
kuncar@47308
   681
   apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   682
   apply (rule conjI, erule Quotient3_refl2 [OF R1])
kuncar@47308
   683
   apply (rule conjI, rule Quotient3_rep_reflp [OF R1])
kuncar@47308
   684
   apply (subst Quotient3_abs_rep [OF R1])
kuncar@47308
   685
   apply (erule Quotient3_rel_abs [OF R1, THEN sym])
kuncar@47096
   686
  apply simp
kuncar@47308
   687
  apply (rule Quotient3_rel_abs [OF R2])
kuncar@47308
   688
  apply (rule Quotient3_rel_abs [OF R1, THEN ssubst], assumption)
kuncar@47308
   689
  apply (rule Quotient3_rel_abs [OF R1, THEN subst], assumption)
kuncar@47096
   690
  apply (erule Abs1)
kuncar@47308
   691
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   692
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   693
 apply (rename_tac a b c d)
kuncar@47096
   694
 apply simp
griff@47434
   695
 apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
kuncar@47308
   696
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   697
  apply (rule conjI, erule Quotient3_refl1 [OF R1])
kuncar@47308
   698
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
griff@47434
   699
 apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI [rotated])
kuncar@47308
   700
  apply (rule Quotient3_rel[symmetric, OF R1, THEN iffD2])
kuncar@47308
   701
  apply (simp add: Quotient3_abs_rep [OF R1] Quotient3_rep_reflp [OF R1])
kuncar@47308
   702
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47096
   703
 apply (rule Rep1)
kuncar@47096
   704
 apply (drule Abs1)
kuncar@47308
   705
   apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   706
  apply (erule Quotient3_refl1 [OF R1])
kuncar@47096
   707
 apply (drule Abs1)
kuncar@47308
   708
  apply (erule Quotient3_refl2 [OF R1])
kuncar@47308
   709
 apply (erule Quotient3_refl1 [OF R1])
kuncar@47308
   710
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   711
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   712
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47308
   713
 apply (drule Quotient3_rel_abs [OF R1])
kuncar@47096
   714
 apply simp
kuncar@47308
   715
 apply (rule Quotient3_rel[symmetric, OF R2, THEN iffD2])
kuncar@47096
   716
 apply simp
kuncar@47096
   717
done
kuncar@47096
   718
kuncar@47308
   719
lemma OOO_eq_quotient3:
kuncar@47096
   720
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
kuncar@47096
   721
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
kuncar@47096
   722
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
kuncar@47308
   723
  assumes R1: "Quotient3 R1 Abs1 Rep1"
kuncar@47308
   724
  assumes R2: "Quotient3 op= Abs2 Rep2"
kuncar@47308
   725
  shows "Quotient3 (R1 OOO op=) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
kuncar@47096
   726
using assms
kuncar@47308
   727
by (rule OOO_quotient3) auto
kuncar@47096
   728
kuncar@47362
   729
subsection {* Quotient3 to Quotient *}
kuncar@47362
   730
kuncar@47362
   731
lemma Quotient3_to_Quotient:
kuncar@47362
   732
assumes "Quotient3 R Abs Rep"
kuncar@47362
   733
and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
kuncar@47362
   734
shows "Quotient R Abs Rep T"
kuncar@47362
   735
using assms unfolding Quotient3_def by (intro QuotientI) blast+
kuncar@47096
   736
kuncar@47362
   737
lemma Quotient3_to_Quotient_equivp:
kuncar@47362
   738
assumes q: "Quotient3 R Abs Rep"
kuncar@47362
   739
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
kuncar@47362
   740
and eR: "equivp R"
kuncar@47362
   741
shows "Quotient R Abs Rep T"
kuncar@47362
   742
proof (intro QuotientI)
kuncar@47362
   743
  fix a
kuncar@47362
   744
  show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
kuncar@47362
   745
next
kuncar@47362
   746
  fix a
kuncar@47362
   747
  show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
kuncar@47362
   748
next
kuncar@47362
   749
  fix r s
kuncar@47362
   750
  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
   751
next
kuncar@47362
   752
  show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
kuncar@47096
   753
qed
kuncar@47096
   754
huffman@35294
   755
subsection {* ML setup *}
kaliszyk@35222
   756
kaliszyk@35222
   757
text {* Auxiliary data for the quotient package *}
kaliszyk@35222
   758
wenzelm@48891
   759
ML_file "Tools/Quotient/quotient_info.ML"
wenzelm@41452
   760
setup Quotient_Info.setup
kaliszyk@35222
   761
kuncar@47308
   762
declare [[mapQ3 "fun" = (fun_rel, fun_quotient3)]]
kaliszyk@35222
   763
kuncar@47308
   764
lemmas [quot_thm] = fun_quotient3
haftmann@44553
   765
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
haftmann@44553
   766
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
kaliszyk@35222
   767
lemmas [quot_equiv] = identity_equivp
kaliszyk@35222
   768
kaliszyk@35222
   769
kaliszyk@35222
   770
text {* Lemmas about simplifying id's. *}
kaliszyk@35222
   771
lemmas [id_simps] =
kaliszyk@35222
   772
  id_def[symmetric]
haftmann@40602
   773
  map_fun_id
kaliszyk@35222
   774
  id_apply
kaliszyk@35222
   775
  id_o
kaliszyk@35222
   776
  o_id
kaliszyk@35222
   777
  eq_comp_r
kaliszyk@44413
   778
  vimage_id
kaliszyk@35222
   779
kaliszyk@35222
   780
text {* Translation functions for the lifting process. *}
wenzelm@48891
   781
ML_file "Tools/Quotient/quotient_term.ML"
kaliszyk@35222
   782
kaliszyk@35222
   783
kaliszyk@35222
   784
text {* Definitions of the quotient types. *}
wenzelm@48891
   785
ML_file "Tools/Quotient/quotient_type.ML"
kaliszyk@35222
   786
kaliszyk@35222
   787
kaliszyk@35222
   788
text {* Definitions for quotient constants. *}
wenzelm@48891
   789
ML_file "Tools/Quotient/quotient_def.ML"
kaliszyk@35222
   790
kaliszyk@35222
   791
kaliszyk@35222
   792
text {*
kaliszyk@35222
   793
  An auxiliary constant for recording some information
kaliszyk@35222
   794
  about the lifted theorem in a tactic.
kaliszyk@35222
   795
*}
kaliszyk@35222
   796
definition
haftmann@40466
   797
  Quot_True :: "'a \<Rightarrow> bool"
haftmann@40466
   798
where
haftmann@40466
   799
  "Quot_True x \<longleftrightarrow> True"
kaliszyk@35222
   800
kaliszyk@35222
   801
lemma
kaliszyk@35222
   802
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   803
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   804
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
kaliszyk@35222
   805
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
kaliszyk@35222
   806
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
kaliszyk@35222
   807
  by (simp_all add: Quot_True_def ext)
kaliszyk@35222
   808
kaliszyk@35222
   809
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
kaliszyk@35222
   810
  by (simp add: Quot_True_def)
kaliszyk@35222
   811
kuncar@53011
   812
context 
kuncar@53011
   813
begin
kuncar@53011
   814
interpretation lifting_syntax .
kaliszyk@35222
   815
kaliszyk@35222
   816
text {* Tactics for proving the lifted theorems *}
wenzelm@48891
   817
ML_file "Tools/Quotient/quotient_tacs.ML"
kaliszyk@35222
   818
kuncar@53011
   819
end
kuncar@53011
   820
huffman@35294
   821
subsection {* Methods / Interface *}
kaliszyk@35222
   822
kaliszyk@35222
   823
method_setup lifting =
urbanc@37593
   824
  {* Attrib.thms >> (fn thms => fn ctxt => 
wenzelm@46468
   825
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms)) *}
wenzelm@42814
   826
  {* lift theorems to quotient types *}
kaliszyk@35222
   827
kaliszyk@35222
   828
method_setup lifting_setup =
urbanc@37593
   829
  {* Attrib.thm >> (fn thm => fn ctxt => 
wenzelm@46468
   830
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm)) *}
wenzelm@42814
   831
  {* set up the three goals for the quotient lifting procedure *}
kaliszyk@35222
   832
urbanc@37593
   833
method_setup descending =
wenzelm@46468
   834
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt [])) *}
wenzelm@42814
   835
  {* decend theorems to the raw level *}
urbanc@37593
   836
urbanc@37593
   837
method_setup descending_setup =
wenzelm@46468
   838
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt [])) *}
wenzelm@42814
   839
  {* set up the three goals for the decending theorems *}
urbanc@37593
   840
urbanc@45782
   841
method_setup partiality_descending =
wenzelm@46468
   842
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt [])) *}
urbanc@45782
   843
  {* decend theorems to the raw level *}
urbanc@45782
   844
urbanc@45782
   845
method_setup partiality_descending_setup =
urbanc@45782
   846
  {* Scan.succeed (fn ctxt => 
wenzelm@46468
   847
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt [])) *}
urbanc@45782
   848
  {* set up the three goals for the decending theorems *}
urbanc@45782
   849
kaliszyk@35222
   850
method_setup regularize =
wenzelm@46468
   851
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt)) *}
wenzelm@42814
   852
  {* prove the regularization goals from the quotient lifting procedure *}
kaliszyk@35222
   853
kaliszyk@35222
   854
method_setup injection =
wenzelm@46468
   855
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt)) *}
wenzelm@42814
   856
  {* prove the rep/abs injection goals from the quotient lifting procedure *}
kaliszyk@35222
   857
kaliszyk@35222
   858
method_setup cleaning =
wenzelm@46468
   859
  {* Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt)) *}
wenzelm@42814
   860
  {* prove the cleaning goals from the quotient lifting procedure *}
kaliszyk@35222
   861
kaliszyk@35222
   862
attribute_setup quot_lifted =
kaliszyk@35222
   863
  {* Scan.succeed Quotient_Tacs.lifted_attrib *}
wenzelm@42814
   864
  {* lift theorems to quotient types *}
kaliszyk@35222
   865
kaliszyk@35222
   866
no_notation
kuncar@53011
   867
  rel_conj (infixr "OOO" 75)
kaliszyk@35222
   868
kaliszyk@35222
   869
end
haftmann@47488
   870