src/HOL/Quotient.thy
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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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section \<open>Definition of Quotient Types\<close>
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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_defn and "/" and
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  "quotient_definition" :: thy_goal_defn
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begin
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text \<open>
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  Basic definition for equivalence relations
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  that are represented by predicates.
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\<close>
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text \<open>Composition of Relations\<close>
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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 "((=) OOO R) = R"
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  by (auto simp add: fun_eq_iff)
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context includes lifting_syntax
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begin
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subsection \<open>Quotient Predicate\<close>
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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" \<comment> \<open>orientation does not loop on rewriting\<close>
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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 \<circ> 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 (=) 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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diff changeset
   144
  have "(rep1 ---> abs2) ((abs1 ---> rep2) a) = a" for a
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   145
    using q1 q2 by (simp add: Quotient3_def fun_eq_iff)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   146
  moreover
69990
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   147
  have "(R1 ===> R2) ((abs1 ---> rep2) a) ((abs1 ---> rep2) a)" for a
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   148
    by (rule rel_funI)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   149
      (insert q1 q2 Quotient3_rel_abs [of R1 abs1 rep1] Quotient3_rel_rep [of R2 abs2 rep2],
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   150
        simp (no_asm) add: Quotient3_def, simp)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   151
  moreover
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   152
  have "(R1 ===> R2) r s = ((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
69990
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   153
        (rep1 ---> abs2) r  = (rep1 ---> abs2) s)" for r s
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   154
  proof -
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   155
    have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) r r" unfolding rel_fun_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   156
      using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   157
      by (metis (full_types) part_equivp_def)
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   158
    moreover have "(R1 ===> R2) r s \<Longrightarrow> (R1 ===> R2) s s" unfolding rel_fun_def
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   159
      using Quotient3_part_equivp[OF q1] Quotient3_part_equivp[OF q2] 
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   160
      by (metis (full_types) part_equivp_def)
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   161
    moreover have "(R1 ===> R2) r s \<Longrightarrow> (rep1 ---> abs2) r  = (rep1 ---> abs2) s"
69990
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   162
      by (auto simp add: rel_fun_def fun_eq_iff)
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   163
        (use q1 q2 in \<open>unfold Quotient3_def, metis\<close>)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   164
    moreover have "((R1 ===> R2) r r \<and> (R1 ===> R2) s s \<and>
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   165
        (rep1 ---> abs2) r  = (rep1 ---> abs2) s) \<Longrightarrow> (R1 ===> R2) r s"
69990
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   166
      by (auto simp add: rel_fun_def fun_eq_iff)
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   167
        (use q1 q2 in \<open>unfold Quotient3_def, metis map_fun_apply\<close>)
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   168
    ultimately show ?thesis by blast
69990
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   169
  qed
eb072ce80f82 tuned proofs;
wenzelm
parents: 69913
diff changeset
   170
  ultimately show ?thesis by (intro Quotient3I) (assumption+)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   171
qed
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   172
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   173
lemma lambda_prs:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   174
  assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   175
  and     q2: "Quotient3 R2 Abs2 Rep2"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   176
  shows "(Rep1 ---> Abs2) (\<lambda>x. Rep2 (f (Abs1 x))) = (\<lambda>x. f x)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   177
  unfolding fun_eq_iff
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   178
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
40814
fa64f6278568 moved generic definitions about (partial) equivalence relations from Quotient to Equiv_Relations;
haftmann
parents: 40615
diff changeset
   179
  by simp
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   180
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   181
lemma lambda_prs1:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   182
  assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   183
  and     q2: "Quotient3 R2 Abs2 Rep2"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   184
  shows "(Rep1 ---> Abs2) (\<lambda>x. (Abs1 ---> Rep2) f x) = (\<lambda>x. f x)"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   185
  unfolding fun_eq_iff
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   186
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
40814
fa64f6278568 moved generic definitions about (partial) equivalence relations from Quotient to Equiv_Relations;
haftmann
parents: 40615
diff changeset
   187
  by simp
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   188
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   189
text\<open>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   190
  In the following theorem R1 can be instantiated with anything,
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   191
  but we know some of the types of the Rep and Abs functions;
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   192
  so by solving Quotient assumptions we can get a unique R1 that
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 60758
diff changeset
   193
  will be provable; which is why we need to use \<open>apply_rsp\<close> and
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   194
  not the primed version\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   195
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   196
lemma apply_rspQ3:
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   197
  fixes f g::"'a \<Rightarrow> 'c"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   198
  assumes q: "Quotient3 R1 Abs1 Rep1"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   199
  and     a: "(R1 ===> R2) f g" "R1 x y"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   200
  shows "R2 (f x) (g y)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   201
  using a by (auto elim: rel_funE)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   202
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   203
lemma apply_rspQ3'':
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   204
  assumes "Quotient3 R Abs Rep"
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   205
  and "(R ===> S) f f"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   206
  shows "S (f (Rep x)) (f (Rep x))"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   207
proof -
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   208
  from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient3_rep_reflp)
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   209
  then show ?thesis using assms(2) by (auto intro: apply_rsp')
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   210
qed
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   211
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   212
subsection \<open>lemmas for regularisation of ball and bex\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   213
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   214
lemma ball_reg_eqv:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   215
  fixes P :: "'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   216
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   217
  shows "Ball (Respects R) P = (All P)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   218
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   219
  unfolding equivp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   220
  by (auto simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   221
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   222
lemma bex_reg_eqv:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   223
  fixes P :: "'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   224
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   225
  shows "Bex (Respects R) P = (Ex P)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   226
  using a
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   227
  unfolding equivp_def
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   228
  by (auto simp add: in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   229
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   230
lemma ball_reg_right:
44553
4d39b032a021 avoid intermixing set and predicates; dropped lemmas mem_rsp and mem_prs (now in Quotient_Set.thy)
haftmann
parents: 44413
diff changeset
   231
  assumes a: "\<And>x. x \<in> R \<Longrightarrow> P x \<longrightarrow> Q x"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   232
  shows "All P \<longrightarrow> Ball R Q"
44921
58eef4843641 tuned proofs
huffman
parents: 44553
diff changeset
   233
  using a by fast
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   234
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   235
lemma bex_reg_left:
44553
4d39b032a021 avoid intermixing set and predicates; dropped lemmas mem_rsp and mem_prs (now in Quotient_Set.thy)
haftmann
parents: 44413
diff changeset
   236
  assumes a: "\<And>x. x \<in> R \<Longrightarrow> Q x \<longrightarrow> P x"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   237
  shows "Bex R Q \<longrightarrow> Ex P"
44921
58eef4843641 tuned proofs
huffman
parents: 44553
diff changeset
   238
  using a by fast
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   239
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   240
lemma ball_reg_left:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   241
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   242
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ball (Respects R) Q \<longrightarrow> All P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   243
  using a by (metis equivp_reflp in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   244
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   245
lemma bex_reg_right:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   246
  assumes a: "equivp R"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   247
  shows "(\<And>x. (Q x \<longrightarrow> P x)) \<Longrightarrow> Ex Q \<longrightarrow> Bex (Respects R) P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   248
  using a by (metis equivp_reflp in_respects)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   249
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   250
lemma ball_reg_eqv_range:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   251
  fixes P::"'a \<Rightarrow> bool"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   252
  and x::"'a"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   253
  assumes a: "equivp R2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   254
  shows   "(Ball (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = All (\<lambda>f. P (f x)))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   255
  apply(rule iffI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   256
  apply(rule allI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   257
  apply(drule_tac x="\<lambda>y. f x" in bspec)
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   258
  apply(simp add: in_respects rel_fun_def)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   259
  apply(rule impI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   260
  using a equivp_reflp_symp_transp[of "R2"]
40814
fa64f6278568 moved generic definitions about (partial) equivalence relations from Quotient to Equiv_Relations;
haftmann
parents: 40615
diff changeset
   261
  apply (auto elim: equivpE reflpE)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   262
  done
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   263
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   264
lemma bex_reg_eqv_range:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   265
  assumes a: "equivp R2"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   266
  shows   "(Bex (Respects (R1 ===> R2)) (\<lambda>f. P (f x)) = Ex (\<lambda>f. P (f x)))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   267
  apply(auto)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   268
  apply(rule_tac x="\<lambda>y. f x" in bexI)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   269
  apply(simp)
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   270
  apply(simp add: Respects_def in_respects rel_fun_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: "\<forall>x :: 'a. (P x \<longrightarrow> 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: "\<forall>x :: 'a. (P x \<longrightarrow> 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: "\<forall>x :: 'a. (x \<in> R \<longrightarrow> P x \<longrightarrow> 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: "\<forall>x :: 'a. (x \<in> R \<longrightarrow> P x \<longrightarrow> 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 \<open>Bounded abstraction\<close>
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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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lemma babs_rsp:
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  assumes q: "Quotient3 R1 Abs1 Rep1"
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  and     a: "(R1 ===> R2) f g"
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  shows      "(R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)"
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  apply (auto simp add: Babs_def in_respects rel_fun_def)
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  apply (subgoal_tac "x \<in> Respects R1 \<and> y \<in> Respects R1")
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  using a apply (simp add: Babs_def rel_fun_def)
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  apply (simp add: in_respects rel_fun_def)
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  using Quotient3_rel[OF q]
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  by metis
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lemma babs_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) (Babs (Respects R1) ((Abs1 ---> Rep2) f))) = f"
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  apply (rule ext)
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  apply (simp add:)
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  apply (subgoal_tac "Rep1 x \<in> Respects R1")
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  apply (simp add: Babs_def Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2])
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  apply (simp add: in_respects Quotient3_rel_rep[OF q1])
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  done
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lemma babs_simp:
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  assumes q: "Quotient3 R1 Abs Rep"
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  shows "((R1 ===> R2) (Babs (Respects R1) f) (Babs (Respects R1) g)) = ((R1 ===> R2) f g)"
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  apply(rule iffI)
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  apply(simp_all only: babs_rsp[OF q])
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   346
  apply(auto simp add: Babs_def rel_fun_def)
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   347
  apply(metis Babs_def in_respects  Quotient3_rel[OF q])
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   348
  done
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(* If a user proves that a particular functional relation
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   is an equivalence this may be useful in regularising *)
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lemma babs_reg_eqv:
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   353
  shows "equivp R \<Longrightarrow> Babs (Respects R) P = P"
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
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   354
  by (simp add: fun_eq_iff Babs_def in_respects equivp_reflp)
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   355
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   356
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(* 3 lemmas needed for proving repabs_inj *)
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lemma ball_rsp:
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   359
  assumes a: "(R ===> (=)) f g"
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   360
  shows "Ball (Respects R) f = Ball (Respects R) g"
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   361
  using a by (auto simp add: Ball_def in_respects elim: rel_funE)
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   362
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   363
lemma bex_rsp:
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   364
  assumes a: "(R ===> (=)) f g"
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   365
  shows "(Bex (Respects R) f = Bex (Respects R) g)"
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blanchet
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   366
  using a by (auto simp add: Bex_def in_respects elim: rel_funE)
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   367
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lemma bex1_rsp:
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   369
  assumes a: "(R ===> (=)) f g"
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   370
  shows "Ex1 (\<lambda>x. x \<in> Respects R \<and> f x) = Ex1 (\<lambda>x. x \<in> Respects R \<and> g x)"
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   371
  using a by (auto elim: rel_funE simp add: Ex1_def in_respects) 
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(* 2 lemmas needed for cleaning of quantifiers *)
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lemma all_prs:
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  assumes a: "Quotient3 R absf repf"
35222
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   376
  shows "Ball (Respects R) ((absf ---> id) f) = All f"
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   377
  using a unfolding Quotient3_def Ball_def in_respects id_apply comp_def map_fun_def
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   378
  by metis
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   379
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lemma ex_prs:
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   381
  assumes a: "Quotient3 R absf repf"
35222
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   382
  shows "Bex (Respects R) ((absf ---> id) f) = Ex f"
47308
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parents: 47105
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   383
  using a unfolding Quotient3_def Bex_def in_respects id_apply comp_def map_fun_def
35222
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   384
  by metis
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   385
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   386
subsection \<open>\<open>Bex1_rel\<close> quantifier\<close>
35222
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   387
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   388
definition
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   389
  Bex1_rel :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> bool) \<Rightarrow> bool"
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   390
where
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   391
  "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)))"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   392
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parents:
diff changeset
   393
lemma bex1_rel_aux:
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parents:
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   394
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R x\<rbrakk> \<Longrightarrow> Bex1_rel R y"
4f1fba00f66d Initial version of HOL quotient package.
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parents:
diff changeset
   395
  unfolding Bex1_rel_def
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   396
  by (metis in_respects)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   397
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   398
lemma bex1_rel_aux2:
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   399
  "\<lbrakk>\<forall>xa ya. R xa ya \<longrightarrow> x xa = y ya; Bex1_rel R y\<rbrakk> \<Longrightarrow> Bex1_rel R x"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   400
  unfolding Bex1_rel_def
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   401
  by (metis in_respects)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   402
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   403
lemma bex1_rel_rsp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   404
  assumes a: "Quotient3 R absf repf"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   405
  shows "((R ===> (=)) ===> (=)) (Bex1_rel R) (Bex1_rel R)"
68615
3ed4ff0b7ac4 de-applying
paulson <lp15@cam.ac.uk>
parents: 67399
diff changeset
   406
  unfolding rel_fun_def by (metis bex1_rel_aux bex1_rel_aux2)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   407
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   408
lemma ex1_prs:
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   409
  assumes "Quotient3 R absf repf"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   410
  shows "((absf ---> id) ---> id) (Bex1_rel R) f = Ex1 f"
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   411
  apply (auto simp: Bex1_rel_def Respects_def)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   412
  apply (metis Quotient3_def assms)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   413
  apply (metis (full_types) Quotient3_def assms)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   414
  by (meson Quotient3_rel assms)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   415
38702
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   416
lemma bex1_bexeq_reg:
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   417
  shows "(\<exists>!x\<in>Respects R. P x) \<longrightarrow> (Bex1_rel R (\<lambda>x. P x))"
56073
29e308b56d23 enhanced simplifier solver for preconditions of rewrite rule, can now deal with conjunctions
nipkow
parents: 55945
diff changeset
   418
  by (auto simp add: Ex1_def Bex1_rel_def Bex_def Ball_def in_respects)
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   419
 
38702
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   420
lemma bex1_bexeq_reg_eqv:
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   421
  assumes a: "equivp R"
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   422
  shows "(\<exists>!x. P x) \<longrightarrow> Bex1_rel R P"
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   423
  using equivp_reflp[OF a]
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   424
  by (metis (full_types) Bex1_rel_def in_respects)
38702
72fd257f4343 Quotient Package / lemma for regularization of bex1_rel for equivalence relations
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 38317
diff changeset
   425
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   426
subsection \<open>Various respects and preserve lemmas\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   427
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   428
lemma quot_rel_rsp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   429
  assumes a: "Quotient3 R Abs Rep"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   430
  shows "(R ===> R ===> (=)) R R"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   431
  apply(rule rel_funI)+
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   432
  by (meson assms equals_rsp)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   433
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   434
lemma o_prs:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   435
  assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   436
  and     q2: "Quotient3 R2 Abs2 Rep2"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   437
  and     q3: "Quotient3 R3 Abs3 Rep3"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   438
  shows "((Abs2 ---> Rep3) ---> (Abs1 ---> Rep2) ---> (Rep1 ---> Abs3)) (\<circ>) = (\<circ>)"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   439
  and   "(id ---> (Abs1 ---> id) ---> Rep1 ---> id) (\<circ>) = (\<circ>)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   440
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2] Quotient3_abs_rep[OF q3]
40466
c6587375088e type annotations in specifications; fun_rel_def is no simp rule by default; slightly changed fun_map_def; more on predicates on relation functions; proper HOL equations in definitions
haftmann
parents: 40031
diff changeset
   441
  by (simp_all add: fun_eq_iff)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   442
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   443
lemma o_rsp:
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   444
  "((R2 ===> R3) ===> (R1 ===> R2) ===> (R1 ===> R3)) (\<circ>) (\<circ>)"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   445
  "((=) ===> (R1 ===> (=)) ===> R1 ===> (=)) (\<circ>) (\<circ>)"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   446
  by (force elim: rel_funE)+
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   447
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   448
lemma cond_prs:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   449
  assumes a: "Quotient3 R absf repf"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   450
  shows "absf (if a then repf b else repf c) = (if a then b else c)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   451
  using a unfolding Quotient3_def by auto
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   452
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   453
lemma if_prs:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   454
  assumes q: "Quotient3 R Abs Rep"
36123
7f877bbad5b2 add If respectfullness and preservation to Quotient package database
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36116
diff changeset
   455
  shows "(id ---> Rep ---> Rep ---> Abs) If = If"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   456
  using Quotient3_abs_rep[OF q]
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   457
  by (auto simp add: fun_eq_iff)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   458
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   459
lemma if_rsp:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   460
  assumes q: "Quotient3 R Abs Rep"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   461
  shows "((=) ===> R ===> R ===> R) If If"
44921
58eef4843641 tuned proofs
huffman
parents: 44553
diff changeset
   462
  by force
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   463
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   464
lemma let_prs:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   465
  assumes q1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   466
  and     q2: "Quotient3 R2 Abs2 Rep2"
37049
ca1c293e521e Let rsp and prs in fun_rel/fun_map format
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36276
diff changeset
   467
  shows "(Rep2 ---> (Abs2 ---> Rep1) ---> Abs1) Let = Let"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   468
  using Quotient3_abs_rep[OF q1] Quotient3_abs_rep[OF q2]
39302
d7728f65b353 renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents: 39198
diff changeset
   469
  by (auto simp add: fun_eq_iff)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   470
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   471
lemma let_rsp:
37049
ca1c293e521e Let rsp and prs in fun_rel/fun_map format
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 36276
diff changeset
   472
  shows "(R1 ===> (R1 ===> R2) ===> R2) Let Let"
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   473
  by (force elim: rel_funE)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   474
39669
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   475
lemma id_rsp:
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   476
  shows "(R ===> R) id id"
44921
58eef4843641 tuned proofs
huffman
parents: 44553
diff changeset
   477
  by auto
39669
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   478
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   479
lemma id_prs:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   480
  assumes a: "Quotient3 R Abs Rep"
39669
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   481
  shows "(Rep ---> Abs) id = id"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   482
  by (simp add: fun_eq_iff Quotient3_abs_rep [OF a])
39669
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   483
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 51112
diff changeset
   484
end
39669
9e3b035841e4 quotient package: respectfulness and preservation of identity.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 39302
diff changeset
   485
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   486
locale quot_type =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   487
  fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   488
  and   Abs :: "'a set \<Rightarrow> 'b"
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   489
  and   Rep :: "'b \<Rightarrow> 'a set"
37493
2377d246a631 Quotient package now uses Partial Equivalence instead place of equivalence
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 37049
diff changeset
   490
  assumes equivp: "part_equivp R"
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   491
  and     rep_prop: "\<And>y. \<exists>x. R x x \<and> Rep y = Collect (R x)"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   492
  and     rep_inverse: "\<And>x. Abs (Rep x) = x"
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   493
  and     abs_inverse: "\<And>c. (\<exists>x. ((R x x) \<and> (c = Collect (R x)))) \<Longrightarrow> (Rep (Abs c)) = c"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   494
  and     rep_inject: "\<And>x y. (Rep x = Rep y) = (x = y)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   495
begin
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   496
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   497
definition
40466
c6587375088e type annotations in specifications; fun_rel_def is no simp rule by default; slightly changed fun_map_def; more on predicates on relation functions; proper HOL equations in definitions
haftmann
parents: 40031
diff changeset
   498
  abs :: "'a \<Rightarrow> 'b"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   499
where
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   500
  "abs x = Abs (Collect (R x))"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   501
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   502
definition
40466
c6587375088e type annotations in specifications; fun_rel_def is no simp rule by default; slightly changed fun_map_def; more on predicates on relation functions; proper HOL equations in definitions
haftmann
parents: 40031
diff changeset
   503
  rep :: "'b \<Rightarrow> 'a"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   504
where
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   505
  "rep a = (SOME x. x \<in> Rep a)"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   506
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   507
lemma some_collect:
37493
2377d246a631 Quotient package now uses Partial Equivalence instead place of equivalence
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 37049
diff changeset
   508
  assumes "R r r"
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   509
  shows "R (SOME x. x \<in> Collect (R r)) = R r"
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   510
  apply simp
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   511
  by (metis assms exE_some equivp[simplified part_equivp_def])
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   512
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   513
lemma Quotient:
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   514
  shows "Quotient3 R abs rep"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   515
  unfolding Quotient3_def abs_def rep_def
37493
2377d246a631 Quotient package now uses Partial Equivalence instead place of equivalence
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 37049
diff changeset
   516
  proof (intro conjI allI)
2377d246a631 Quotient package now uses Partial Equivalence instead place of equivalence
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 37049
diff changeset
   517
    fix a r s
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   518
    show x: "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)" proof -
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   519
      obtain x where r: "R x x" and rep: "Rep a = Collect (R x)" using rep_prop[of a] by auto
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   520
      have "R (SOME x. x \<in> Rep a) x"  using r rep some_collect by metis
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   521
      then have "R x (SOME x. x \<in> Rep a)" using part_equivp_symp[OF equivp] by fast
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   522
      then show "R (SOME x. x \<in> Rep a) (SOME x. x \<in> Rep a)"
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   523
        using part_equivp_transp[OF equivp] by (metis \<open>R (SOME x. x \<in> Rep a) x\<close>)
37493
2377d246a631 Quotient package now uses Partial Equivalence instead place of equivalence
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 37049
diff changeset
   524
    qed
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   525
    have "Collect (R (SOME x. x \<in> Rep a)) = (Rep a)" by (metis some_collect rep_prop)
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   526
    then show "Abs (Collect (R (SOME x. x \<in> Rep a))) = a" using rep_inverse by auto
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   527
    have "R r r \<Longrightarrow> R s s \<Longrightarrow> Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s"
44242
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   528
    proof -
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   529
      assume "R r r" and "R s s"
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   530
      then have "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> Collect (R r) = Collect (R s)"
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   531
        by (metis abs_inverse)
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   532
      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))"
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   533
        by rule simp_all
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   534
      finally show "Abs (Collect (R r)) = Abs (Collect (R s)) \<longleftrightarrow> R r = R s" by simp
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   535
    qed
44204
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   536
    then show "R r s \<longleftrightarrow> R r r \<and> R s s \<and> (Abs (Collect (R r)) = Abs (Collect (R s)))"
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   537
      using equivp[simplified part_equivp_def] by metis
3cdc4176638c Quotient Package: make quotient_type work with separate set type
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 42814
diff changeset
   538
    qed
44242
a5cb6aa77ffc avoid Collect_def in proof
haftmann
parents: 44204
diff changeset
   539
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   540
end
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   541
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   542
subsection \<open>Quotient composition\<close>
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   543
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   544
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   545
lemma OOO_quotient3:
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   546
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   547
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   548
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   549
  fixes R2' :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   550
  fixes R2 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   551
  assumes R1: "Quotient3 R1 Abs1 Rep1"
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   552
  assumes R2: "Quotient3 R2 Abs2 Rep2"
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   553
  assumes Abs1: "\<And>x y. R2' x y \<Longrightarrow> R1 x x \<Longrightarrow> R1 y y \<Longrightarrow> R2 (Abs1 x) (Abs1 y)"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   554
  assumes Rep1: "\<And>x y. R2 x y \<Longrightarrow> R2' (Rep1 x) (Rep1 y)"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   555
  shows "Quotient3 (R1 OO R2' OO R1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
68616
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   556
proof -
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   557
  have *: "(R1 OOO R2') r r \<and> (R1 OOO R2') s s \<and> (Abs2 \<circ> Abs1) r = (Abs2 \<circ> Abs1) s 
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   558
           \<longleftrightarrow> (R1 OOO R2') r s" for r s
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   559
    apply safe
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   560
    subgoal for a b c d
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   561
      apply simp
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   562
      apply (rule_tac b="Rep1 (Abs1 r)" in relcomppI)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   563
      using Quotient3_refl1 R1 rep_abs_rsp apply fastforce
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   564
      apply (rule_tac b="Rep1 (Abs1 s)" in relcomppI)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   565
       apply (metis (full_types) Rep1 Abs1 Quotient3_rel R2  Quotient3_refl1 [OF R1] Quotient3_refl2 [OF R1] Quotient3_rel_abs [OF R1])
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   566
      by (metis Quotient3_rel R1 rep_abs_rsp_left)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   567
    subgoal for x y
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   568
      apply (drule Abs1)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   569
        apply (erule Quotient3_refl2 [OF R1])
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   570
       apply (erule Quotient3_refl1 [OF R1])
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   571
      apply (drule Quotient3_refl1 [OF R2], drule Rep1)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   572
      by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   573
    subgoal for x y
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   574
      apply (drule Abs1)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   575
        apply (erule Quotient3_refl2 [OF R1])
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   576
       apply (erule Quotient3_refl1 [OF R1])
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   577
      apply (drule Quotient3_refl2 [OF R2], drule Rep1)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   578
      by (metis (full_types) Quotient3_def R1 relcompp.relcompI)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   579
    subgoal for x y
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   580
      by simp (metis (full_types) Abs1 Quotient3_rel R1 R2)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   581
    done
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   582
  show ?thesis
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   583
    apply (rule Quotient3I)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   584
    using * apply (simp_all add: o_def Quotient3_abs_rep [OF R2] Quotient3_abs_rep [OF R1])
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   585
    apply (metis Quotient3_rep_reflp R1 R2 Rep1 relcompp.relcompI)
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   586
    done
cedf3480fdad de-applying (mostly Quotient)
paulson <lp15@cam.ac.uk>
parents: 68615
diff changeset
   587
qed
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   588
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   589
lemma OOO_eq_quotient3:
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   590
  fixes R1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   591
  fixes Abs1 :: "'a \<Rightarrow> 'b" and Rep1 :: "'b \<Rightarrow> 'a"
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   592
  fixes Abs2 :: "'b \<Rightarrow> 'c" and Rep2 :: "'c \<Rightarrow> 'b"
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   593
  assumes R1: "Quotient3 R1 Abs1 Rep1"
67399
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   594
  assumes R2: "Quotient3 (=) Abs2 Rep2"
eab6ce8368fa ran isabelle update_op on all sources
nipkow
parents: 67091
diff changeset
   595
  shows "Quotient3 (R1 OOO (=)) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2)"
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   596
using assms
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   597
by (rule OOO_quotient3) auto
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   598
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   599
subsection \<open>Quotient3 to Quotient\<close>
47362
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   600
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   601
lemma Quotient3_to_Quotient:
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   602
assumes "Quotient3 R Abs Rep"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   603
and "T \<equiv> \<lambda>x y. R x x \<and> Abs x = y"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   604
shows "Quotient R Abs Rep T"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   605
using assms unfolding Quotient3_def by (intro QuotientI) blast+
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   606
47362
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   607
lemma Quotient3_to_Quotient_equivp:
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   608
assumes q: "Quotient3 R Abs Rep"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   609
and T_def: "T \<equiv> \<lambda>x y. Abs x = y"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   610
and eR: "equivp R"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   611
shows "Quotient R Abs Rep T"
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   612
proof (intro QuotientI)
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   613
  fix a
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   614
  show "Abs (Rep a) = a" using q by(rule Quotient3_abs_rep)
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   615
next
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   616
  fix a
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   617
  show "R (Rep a) (Rep a)" using q by(rule Quotient3_rep_reflp)
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   618
next
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   619
  fix r s
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   620
  show "R r s = (R r r \<and> R s s \<and> Abs r = Abs s)" using q by(rule Quotient3_rel[symmetric])
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   621
next
b1f099bdfbba connect the Quotient package to the Lifting package
kuncar
parents: 47361
diff changeset
   622
  show "T = (\<lambda>x y. R x x \<and> Abs x = y)" using T_def equivp_reflp[OF eR] by simp
47096
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   623
qed
3ea48c19673e generation of a code certificate from a respectfulness theorem for constants lifted by the quotient_definition command & setup_lifting command: setups Quotient infrastructure from a typedef theorem
kuncar
parents: 47094
diff changeset
   624
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   625
subsection \<open>ML setup\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   626
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   627
text \<open>Auxiliary data for the quotient package\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   628
57960
ee1ba4848896 updated to named_theorems;
wenzelm
parents: 56073
diff changeset
   629
named_theorems quot_equiv "equivalence relation theorems"
59028
df7476e79558 named_theorems: multiple args;
wenzelm
parents: 58889
diff changeset
   630
  and quot_respect "respectfulness theorems"
df7476e79558 named_theorems: multiple args;
wenzelm
parents: 58889
diff changeset
   631
  and quot_preserve "preservation theorems"
df7476e79558 named_theorems: multiple args;
wenzelm
parents: 58889
diff changeset
   632
  and id_simps "identity simp rules for maps"
df7476e79558 named_theorems: multiple args;
wenzelm
parents: 58889
diff changeset
   633
  and quot_thm "quotient theorems"
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 68616
diff changeset
   634
ML_file \<open>Tools/Quotient/quotient_info.ML\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   635
55945
e96383acecf9 renamed 'fun_rel' to 'rel_fun'
blanchet
parents: 54867
diff changeset
   636
declare [[mapQ3 "fun" = (rel_fun, fun_quotient3)]]
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   637
47308
9caab698dbe4 new package Lifting - initial commit
kuncar
parents: 47105
diff changeset
   638
lemmas [quot_thm] = fun_quotient3
44553
4d39b032a021 avoid intermixing set and predicates; dropped lemmas mem_rsp and mem_prs (now in Quotient_Set.thy)
haftmann
parents: 44413
diff changeset
   639
lemmas [quot_respect] = quot_rel_rsp if_rsp o_rsp let_rsp id_rsp
4d39b032a021 avoid intermixing set and predicates; dropped lemmas mem_rsp and mem_prs (now in Quotient_Set.thy)
haftmann
parents: 44413
diff changeset
   640
lemmas [quot_preserve] = if_prs o_prs let_prs id_prs
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   641
lemmas [quot_equiv] = identity_equivp
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   642
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   643
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   644
text \<open>Lemmas about simplifying id's.\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   645
lemmas [id_simps] =
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   646
  id_def[symmetric]
40602
91e583511113 map_fun combinator in theory Fun
haftmann
parents: 40466
diff changeset
   647
  map_fun_id
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   648
  id_apply
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   649
  id_o
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   650
  o_id
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   651
  eq_comp_r
44413
80d460bc6fa8 Quotient Package: some infrastructure for lifting inside sets
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents: 44242
diff changeset
   652
  vimage_id
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   653
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   654
text \<open>Translation functions for the lifting process.\<close>
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 68616
diff changeset
   655
ML_file \<open>Tools/Quotient/quotient_term.ML\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   656
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   657
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   658
text \<open>Definitions of the quotient types.\<close>
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 68616
diff changeset
   659
ML_file \<open>Tools/Quotient/quotient_type.ML\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   660
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   661
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   662
text \<open>Definitions for quotient constants.\<close>
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 68616
diff changeset
   663
ML_file \<open>Tools/Quotient/quotient_def.ML\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   664
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   665
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   666
text \<open>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   667
  An auxiliary constant for recording some information
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   668
  about the lifted theorem in a tactic.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   669
\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   670
definition
40466
c6587375088e type annotations in specifications; fun_rel_def is no simp rule by default; slightly changed fun_map_def; more on predicates on relation functions; proper HOL equations in definitions
haftmann
parents: 40031
diff changeset
   671
  Quot_True :: "'a \<Rightarrow> bool"
c6587375088e type annotations in specifications; fun_rel_def is no simp rule by default; slightly changed fun_map_def; more on predicates on relation functions; proper HOL equations in definitions
haftmann
parents: 40031
diff changeset
   672
where
c6587375088e type annotations in specifications; fun_rel_def is no simp rule by default; slightly changed fun_map_def; more on predicates on relation functions; proper HOL equations in definitions
haftmann
parents: 40031
diff changeset
   673
  "Quot_True x \<longleftrightarrow> True"
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   674
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   675
lemma
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   676
  shows QT_all: "Quot_True (All P) \<Longrightarrow> Quot_True P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   677
  and   QT_ex:  "Quot_True (Ex P) \<Longrightarrow> Quot_True P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   678
  and   QT_ex1: "Quot_True (Ex1 P) \<Longrightarrow> Quot_True P"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   679
  and   QT_lam: "Quot_True (\<lambda>x. P x) \<Longrightarrow> (\<And>x. Quot_True (P x))"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   680
  and   QT_ext: "(\<And>x. Quot_True (a x) \<Longrightarrow> f x = g x) \<Longrightarrow> (Quot_True a \<Longrightarrow> f = g)"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   681
  by (simp_all add: Quot_True_def ext)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   682
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   683
lemma QT_imp: "Quot_True a \<equiv> Quot_True b"
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   684
  by (simp add: Quot_True_def)
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   685
63343
fb5d8a50c641 bundle lifting_syntax;
wenzelm
parents: 61799
diff changeset
   686
context includes lifting_syntax
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 51112
diff changeset
   687
begin
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   688
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   689
text \<open>Tactics for proving the lifted theorems\<close>
69605
a96320074298 isabelle update -u path_cartouches;
wenzelm
parents: 68616
diff changeset
   690
ML_file \<open>Tools/Quotient/quotient_tacs.ML\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   691
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 51112
diff changeset
   692
end
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 51112
diff changeset
   693
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   694
subsection \<open>Methods / Interface\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   695
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   696
method_setup lifting =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   697
  \<open>Attrib.thms >> (fn thms => fn ctxt => 
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   698
       SIMPLE_METHOD' (Quotient_Tacs.lift_tac ctxt [] thms))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   699
  \<open>lift theorems to quotient types\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   700
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   701
method_setup lifting_setup =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   702
  \<open>Attrib.thm >> (fn thm => fn ctxt => 
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   703
       SIMPLE_METHOD' (Quotient_Tacs.lift_procedure_tac ctxt [] thm))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   704
  \<open>set up the three goals for the quotient lifting procedure\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   705
37593
2505feaf2d70 separated the lifting and descending procedures in the quotient package
Christian Urban <urbanc@in.tum.de>
parents: 37564
diff changeset
   706
method_setup descending =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   707
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_tac ctxt []))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   708
  \<open>decend theorems to the raw level\<close>
37593
2505feaf2d70 separated the lifting and descending procedures in the quotient package
Christian Urban <urbanc@in.tum.de>
parents: 37564
diff changeset
   709
2505feaf2d70 separated the lifting and descending procedures in the quotient package
Christian Urban <urbanc@in.tum.de>
parents: 37564
diff changeset
   710
method_setup descending_setup =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   711
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.descend_procedure_tac ctxt []))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   712
  \<open>set up the three goals for the decending theorems\<close>
37593
2505feaf2d70 separated the lifting and descending procedures in the quotient package
Christian Urban <urbanc@in.tum.de>
parents: 37564
diff changeset
   713
45782
f82020ca3248 added a specific tactic and method that deal with partial equivalence relations
Christian Urban <urbanc@in.tum.de>
parents: 45680
diff changeset
   714
method_setup partiality_descending =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   715
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_tac ctxt []))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   716
  \<open>decend theorems to the raw level\<close>
45782
f82020ca3248 added a specific tactic and method that deal with partial equivalence relations
Christian Urban <urbanc@in.tum.de>
parents: 45680
diff changeset
   717
f82020ca3248 added a specific tactic and method that deal with partial equivalence relations
Christian Urban <urbanc@in.tum.de>
parents: 45680
diff changeset
   718
method_setup partiality_descending_setup =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   719
  \<open>Scan.succeed (fn ctxt => 
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   720
       SIMPLE_METHOD' (Quotient_Tacs.partiality_descend_procedure_tac ctxt []))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   721
  \<open>set up the three goals for the decending theorems\<close>
45782
f82020ca3248 added a specific tactic and method that deal with partial equivalence relations
Christian Urban <urbanc@in.tum.de>
parents: 45680
diff changeset
   722
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   723
method_setup regularize =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   724
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.regularize_tac ctxt))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   725
  \<open>prove the regularization goals from the quotient lifting procedure\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   726
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   727
method_setup injection =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   728
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.all_injection_tac ctxt))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   729
  \<open>prove the rep/abs injection goals from the quotient lifting procedure\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   730
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   731
method_setup cleaning =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   732
  \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD' (Quotient_Tacs.clean_tac ctxt))\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   733
  \<open>prove the cleaning goals from the quotient lifting procedure\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   734
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   735
attribute_setup quot_lifted =
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   736
  \<open>Scan.succeed Quotient_Tacs.lifted_attrib\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 59028
diff changeset
   737
  \<open>lift theorems to quotient types\<close>
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   738
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   739
no_notation
53011
aeee0a4be6cf introduce locale with syntax for fun_rel and map_fun and make thus ===> and ---> local
kuncar
parents: 51112
diff changeset
   740
  rel_conj (infixr "OOO" 75)
35222
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   741
4f1fba00f66d Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff changeset
   742
end
47488
be6dd389639d centralized enriched_type declaration, thanks to in-situ available Isar commands
haftmann
parents: 47436
diff changeset
   743