src/HOL/BNF/Basic_BNFs.thy
author traytel
Sat Jul 13 13:03:21 2013 +0200 (2013-07-13)
changeset 52635 4f84b730c489
parent 52545 d2ad6eae514f
child 52660 7f7311d04727
permissions -rw-r--r--
got rid of in_bd BNF property (derivable from set_bd+map_cong+map_comp+map_id)
blanchet@49509
     1
(*  Title:      HOL/BNF/Basic_BNFs.thy
blanchet@48975
     2
    Author:     Dmitriy Traytel, TU Muenchen
blanchet@48975
     3
    Author:     Andrei Popescu, TU Muenchen
blanchet@48975
     4
    Author:     Jasmin Blanchette, TU Muenchen
blanchet@48975
     5
    Copyright   2012
blanchet@48975
     6
blanchet@49309
     7
Registration of basic types as bounded natural functors.
blanchet@48975
     8
*)
blanchet@48975
     9
blanchet@49309
    10
header {* Registration of Basic Types as Bounded Natural Functors *}
blanchet@48975
    11
blanchet@48975
    12
theory Basic_BNFs
blanchet@49310
    13
imports BNF_Def
blanchet@48975
    14
begin
blanchet@48975
    15
blanchet@49312
    16
lemma wpull_id: "wpull UNIV B1 B2 id id id id"
blanchet@49312
    17
unfolding wpull_def by simp
blanchet@49312
    18
blanchet@48975
    19
lemmas natLeq_card_order = natLeq_Card_order[unfolded Field_natLeq]
blanchet@48975
    20
blanchet@48975
    21
lemma ctwo_card_order: "card_order ctwo"
blanchet@48975
    22
using Card_order_ctwo by (unfold ctwo_def Field_card_of)
blanchet@48975
    23
blanchet@48975
    24
lemma natLeq_cinfinite: "cinfinite natLeq"
blanchet@48975
    25
unfolding cinfinite_def Field_natLeq by (rule nat_infinite)
blanchet@48975
    26
traytel@51446
    27
lemma wpull_Gr_def: "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow> Gr B1 f1 O (Gr B2 f2)\<inverse> \<subseteq> (Gr A p1)\<inverse> O Gr A p2"
traytel@51446
    28
  unfolding wpull_def Gr_def relcomp_unfold converse_unfold by auto
traytel@51446
    29
traytel@51893
    30
lemma wpull_Grp_def: "wpull A B1 B2 f1 f2 p1 p2 \<longleftrightarrow> Grp B1 f1 OO (Grp B2 f2)\<inverse>\<inverse> \<le> (Grp A p1)\<inverse>\<inverse> OO Grp A p2"
traytel@51893
    31
  unfolding wpull_def Grp_def by auto
traytel@51893
    32
blanchet@51836
    33
bnf ID: "id :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" ["\<lambda>x. {x}"] "\<lambda>_:: 'a. natLeq" ["id :: 'a \<Rightarrow> 'a"]
traytel@51893
    34
  "id :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
traytel@51893
    35
apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)
blanchet@48975
    36
apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
blanchet@49453
    37
apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]
blanchet@48975
    38
done
blanchet@48975
    39
blanchet@51836
    40
bnf DEADID: "id :: 'a \<Rightarrow> 'a" [] "\<lambda>_:: 'a. natLeq +c |UNIV :: 'a set|" ["SOME x :: 'a. True"]
traytel@51893
    41
  "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
traytel@52635
    42
by (auto simp add: wpull_Grp_def Grp_def
traytel@51446
    43
  card_order_csum natLeq_card_order card_of_card_order_on
traytel@51446
    44
  cinfinite_csum natLeq_cinfinite)
blanchet@48975
    45
blanchet@49451
    46
definition setl :: "'a + 'b \<Rightarrow> 'a set" where
blanchet@49451
    47
"setl x = (case x of Inl z => {z} | _ => {})"
blanchet@48975
    48
blanchet@49451
    49
definition setr :: "'a + 'b \<Rightarrow> 'b set" where
blanchet@49451
    50
"setr x = (case x of Inr z => {z} | _ => {})"
blanchet@48975
    51
blanchet@49451
    52
lemmas sum_set_defs = setl_def[abs_def] setr_def[abs_def]
blanchet@48975
    53
blanchet@49507
    54
definition sum_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'c \<Rightarrow> 'b + 'd \<Rightarrow> bool" where
blanchet@49507
    55
"sum_rel \<phi> \<psi> x y =
blanchet@49463
    56
 (case x of Inl a1 \<Rightarrow> (case y of Inl a2 \<Rightarrow> \<phi> a1 a2 | Inr _ \<Rightarrow> False)
blanchet@49463
    57
          | Inr b1 \<Rightarrow> (case y of Inl _ \<Rightarrow> False | Inr b2 \<Rightarrow> \<psi> b1 b2))"
blanchet@49453
    58
blanchet@51836
    59
bnf sum_map [setl, setr] "\<lambda>_::'a + 'b. natLeq" [Inl, Inr] sum_rel
blanchet@48975
    60
proof -
blanchet@48975
    61
  show "sum_map id id = id" by (rule sum_map.id)
blanchet@48975
    62
next
blanchet@48975
    63
  fix f1 f2 g1 g2
blanchet@48975
    64
  show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"
blanchet@48975
    65
    by (rule sum_map.comp[symmetric])
blanchet@48975
    66
next
blanchet@48975
    67
  fix x f1 f2 g1 g2
blanchet@49451
    68
  assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and
blanchet@49451
    69
         a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"
blanchet@48975
    70
  thus "sum_map f1 f2 x = sum_map g1 g2 x"
blanchet@48975
    71
  proof (cases x)
blanchet@49451
    72
    case Inl thus ?thesis using a1 by (clarsimp simp: setl_def)
blanchet@48975
    73
  next
blanchet@49451
    74
    case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)
blanchet@48975
    75
  qed
blanchet@48975
    76
next
blanchet@48975
    77
  fix f1 f2
blanchet@49451
    78
  show "setl o sum_map f1 f2 = image f1 o setl"
blanchet@49451
    79
    by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)
blanchet@48975
    80
next
blanchet@48975
    81
  fix f1 f2
blanchet@49451
    82
  show "setr o sum_map f1 f2 = image f2 o setr"
blanchet@49451
    83
    by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)
blanchet@48975
    84
next
blanchet@48975
    85
  show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975
    86
next
blanchet@48975
    87
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975
    88
next
blanchet@48975
    89
  fix x
blanchet@49451
    90
  show "|setl x| \<le>o natLeq"
blanchet@48975
    91
    apply (rule ordLess_imp_ordLeq)
blanchet@48975
    92
    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451
    93
    by (simp add: setl_def split: sum.split)
blanchet@48975
    94
next
blanchet@48975
    95
  fix x
blanchet@49451
    96
  show "|setr x| \<le>o natLeq"
blanchet@48975
    97
    apply (rule ordLess_imp_ordLeq)
blanchet@48975
    98
    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451
    99
    by (simp add: setr_def split: sum.split)
blanchet@48975
   100
next
blanchet@48975
   101
  fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22
blanchet@48975
   102
  assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"
blanchet@48975
   103
  hence
blanchet@48975
   104
    pull1: "\<And>b1 b2. \<lbrakk>b1 \<in> B11; b2 \<in> B21; f11 b1 = f21 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A1. p11 a = b1 \<and> p21 a = b2"
blanchet@48975
   105
    and pull2: "\<And>b1 b2. \<lbrakk>b1 \<in> B12; b2 \<in> B22; f12 b1 = f22 b2\<rbrakk> \<Longrightarrow> \<exists>a \<in> A2. p12 a = b1 \<and> p22 a = b2"
blanchet@48975
   106
    unfolding wpull_def by blast+
blanchet@49451
   107
  show "wpull {x. setl x \<subseteq> A1 \<and> setr x \<subseteq> A2}
blanchet@49451
   108
  {x. setl x \<subseteq> B11 \<and> setr x \<subseteq> B12} {x. setl x \<subseteq> B21 \<and> setr x \<subseteq> B22}
blanchet@48975
   109
  (sum_map f11 f12) (sum_map f21 f22) (sum_map p11 p12) (sum_map p21 p22)"
blanchet@48975
   110
    (is "wpull ?in ?in1 ?in2 ?mapf1 ?mapf2 ?mapp1 ?mapp2")
blanchet@48975
   111
  proof (unfold wpull_def)
blanchet@48975
   112
    { fix B1 B2
blanchet@48975
   113
      assume *: "B1 \<in> ?in1" "B2 \<in> ?in2" "?mapf1 B1 = ?mapf2 B2"
blanchet@48975
   114
      have "\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2"
blanchet@48975
   115
      proof (cases B1)
blanchet@48975
   116
        case (Inl b1)
blanchet@48975
   117
        { fix b2 assume "B2 = Inr b2"
blanchet@48975
   118
          with Inl *(3) have False by simp
blanchet@48975
   119
        } then obtain b2 where Inl': "B2 = Inl b2" by (cases B2) (simp, blast)
blanchet@48975
   120
        with Inl * have "b1 \<in> B11" "b2 \<in> B21" "f11 b1 = f21 b2"
blanchet@49451
   121
        by (simp add: setl_def)+
blanchet@48975
   122
        with pull1 obtain a where "a \<in> A1" "p11 a = b1" "p21 a = b2" by blast+
blanchet@48975
   123
        with Inl Inl' have "Inl a \<in> ?in" "?mapp1 (Inl a) = B1 \<and> ?mapp2 (Inl a) = B2"
blanchet@48975
   124
        by (simp add: sum_set_defs)+
blanchet@48975
   125
        thus ?thesis by blast
blanchet@48975
   126
      next
blanchet@48975
   127
        case (Inr b1)
blanchet@48975
   128
        { fix b2 assume "B2 = Inl b2"
blanchet@48975
   129
          with Inr *(3) have False by simp
blanchet@48975
   130
        } then obtain b2 where Inr': "B2 = Inr b2" by (cases B2) (simp, blast)
blanchet@48975
   131
        with Inr * have "b1 \<in> B12" "b2 \<in> B22" "f12 b1 = f22 b2"
blanchet@48975
   132
        by (simp add: sum_set_defs)+
blanchet@48975
   133
        with pull2 obtain a where "a \<in> A2" "p12 a = b1" "p22 a = b2" by blast+
blanchet@48975
   134
        with Inr Inr' have "Inr a \<in> ?in" "?mapp1 (Inr a) = B1 \<and> ?mapp2 (Inr a) = B2"
blanchet@48975
   135
        by (simp add: sum_set_defs)+
blanchet@48975
   136
        thus ?thesis by blast
blanchet@48975
   137
      qed
blanchet@48975
   138
    }
blanchet@48975
   139
    thus "\<forall>B1 B2. B1 \<in> ?in1 \<and> B2 \<in> ?in2 \<and> ?mapf1 B1 = ?mapf2 B2 \<longrightarrow>
blanchet@48975
   140
      (\<exists>A \<in> ?in. ?mapp1 A = B1 \<and> ?mapp2 A = B2)" by fastforce
blanchet@48975
   141
  qed
blanchet@49453
   142
next
blanchet@49453
   143
  fix R S
traytel@51893
   144
  show "sum_rel R S =
traytel@51893
   145
        (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map fst fst))\<inverse>\<inverse> OO
traytel@51893
   146
        Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map snd snd)"
traytel@51893
   147
  unfolding setl_def setr_def sum_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453
   148
  by (fastforce split: sum.splits)
blanchet@48975
   149
qed (auto simp: sum_set_defs)
blanchet@48975
   150
blanchet@48975
   151
definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where
blanchet@48975
   152
"fsts x = {fst x}"
blanchet@48975
   153
blanchet@48975
   154
definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where
blanchet@48975
   155
"snds x = {snd x}"
blanchet@48975
   156
blanchet@48975
   157
lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]
blanchet@48975
   158
blanchet@49507
   159
definition prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool" where
blanchet@49507
   160
"prod_rel \<phi> \<psi> p1 p2 = (case p1 of (a1, b1) \<Rightarrow> case p2 of (a2, b2) \<Rightarrow> \<phi> a1 a2 \<and> \<psi> b1 b2)"
blanchet@49453
   161
traytel@52635
   162
bnf map_pair [fsts, snds] "\<lambda>_::'a \<times> 'b. natLeq" [Pair] prod_rel
blanchet@48975
   163
proof (unfold prod_set_defs)
blanchet@48975
   164
  show "map_pair id id = id" by (rule map_pair.id)
blanchet@48975
   165
next
blanchet@48975
   166
  fix f1 f2 g1 g2
blanchet@48975
   167
  show "map_pair (g1 o f1) (g2 o f2) = map_pair g1 g2 o map_pair f1 f2"
blanchet@48975
   168
    by (rule map_pair.comp[symmetric])
blanchet@48975
   169
next
blanchet@48975
   170
  fix x f1 f2 g1 g2
blanchet@48975
   171
  assume "\<And>z. z \<in> {fst x} \<Longrightarrow> f1 z = g1 z" "\<And>z. z \<in> {snd x} \<Longrightarrow> f2 z = g2 z"
blanchet@48975
   172
  thus "map_pair f1 f2 x = map_pair g1 g2 x" by (cases x) simp
blanchet@48975
   173
next
blanchet@48975
   174
  fix f1 f2
blanchet@48975
   175
  show "(\<lambda>x. {fst x}) o map_pair f1 f2 = image f1 o (\<lambda>x. {fst x})"
blanchet@48975
   176
    by (rule ext, unfold o_apply) simp
blanchet@48975
   177
next
blanchet@48975
   178
  fix f1 f2
blanchet@48975
   179
  show "(\<lambda>x. {snd x}) o map_pair f1 f2 = image f2 o (\<lambda>x. {snd x})"
blanchet@48975
   180
    by (rule ext, unfold o_apply) simp
blanchet@48975
   181
next
traytel@52635
   182
  show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975
   183
next
traytel@52635
   184
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975
   185
next
blanchet@48975
   186
  fix x
traytel@52635
   187
  show "|{fst x}| \<le>o natLeq"
traytel@52635
   188
    by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
blanchet@48975
   189
next
traytel@52635
   190
  fix x
traytel@52635
   191
  show "|{snd x}| \<le>o natLeq"
traytel@52635
   192
    by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
blanchet@48975
   193
next
blanchet@48975
   194
  fix A1 A2 B11 B12 B21 B22 f11 f12 f21 f22 p11 p12 p21 p22
blanchet@48975
   195
  assume "wpull A1 B11 B21 f11 f21 p11 p21" "wpull A2 B12 B22 f12 f22 p12 p22"
blanchet@48975
   196
  thus "wpull {x. {fst x} \<subseteq> A1 \<and> {snd x} \<subseteq> A2}
blanchet@48975
   197
    {x. {fst x} \<subseteq> B11 \<and> {snd x} \<subseteq> B12} {x. {fst x} \<subseteq> B21 \<and> {snd x} \<subseteq> B22}
blanchet@48975
   198
   (map_pair f11 f12) (map_pair f21 f22) (map_pair p11 p12) (map_pair p21 p22)"
blanchet@48975
   199
    unfolding wpull_def by simp fast
blanchet@49453
   200
next
blanchet@49453
   201
  fix R S
traytel@51893
   202
  show "prod_rel R S =
traytel@51893
   203
        (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair fst fst))\<inverse>\<inverse> OO
traytel@51893
   204
        Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair snd snd)"
traytel@51893
   205
  unfolding prod_set_defs prod_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453
   206
  by auto
blanchet@48975
   207
qed simp+
blanchet@48975
   208
blanchet@48975
   209
(* Categorical version of pullback: *)
blanchet@48975
   210
lemma wpull_cat:
blanchet@48975
   211
assumes p: "wpull A B1 B2 f1 f2 p1 p2"
blanchet@48975
   212
and c: "f1 o q1 = f2 o q2"
blanchet@48975
   213
and r: "range q1 \<subseteq> B1" "range q2 \<subseteq> B2"
blanchet@48975
   214
obtains h where "range h \<subseteq> A \<and> q1 = p1 o h \<and> q2 = p2 o h"
blanchet@48975
   215
proof-
blanchet@48975
   216
  have *: "\<forall>d. \<exists>a \<in> A. p1 a = q1 d & p2 a = q2 d"
blanchet@48975
   217
  proof safe
blanchet@48975
   218
    fix d
blanchet@48975
   219
    have "f1 (q1 d) = f2 (q2 d)" using c unfolding comp_def[abs_def] by (rule fun_cong)
blanchet@48975
   220
    moreover
blanchet@48975
   221
    have "q1 d : B1" "q2 d : B2" using r unfolding image_def by auto
blanchet@48975
   222
    ultimately show "\<exists>a \<in> A. p1 a = q1 d \<and> p2 a = q2 d"
blanchet@48975
   223
      using p unfolding wpull_def by auto
blanchet@48975
   224
  qed
blanchet@48975
   225
  then obtain h where "!! d. h d \<in> A & p1 (h d) = q1 d & p2 (h d) = q2 d" by metis
blanchet@48975
   226
  thus ?thesis using that by fastforce
blanchet@48975
   227
qed
blanchet@48975
   228
blanchet@48975
   229
lemma card_of_bounded_range:
blanchet@48975
   230
  "|{f :: 'd \<Rightarrow> 'a. range f \<subseteq> B}| \<le>o |Func (UNIV :: 'd set) B|" (is "|?LHS| \<le>o |?RHS|")
blanchet@48975
   231
proof -
traytel@52545
   232
  let ?f = "\<lambda>f. %x. if f x \<in> B then f x else undefined"
blanchet@48975
   233
  have "inj_on ?f ?LHS" unfolding inj_on_def
blanchet@48975
   234
  proof (unfold fun_eq_iff, safe)
blanchet@48975
   235
    fix g :: "'d \<Rightarrow> 'a" and f :: "'d \<Rightarrow> 'a" and x
blanchet@48975
   236
    assume "range f \<subseteq> B" "range g \<subseteq> B" and eq: "\<forall>x. ?f f x = ?f g x"
blanchet@48975
   237
    hence "f x \<in> B" "g x \<in> B" by auto
blanchet@48975
   238
    with eq have "Some (f x) = Some (g x)" by metis
blanchet@48975
   239
    thus "f x = g x" by simp
blanchet@48975
   240
  qed
blanchet@48975
   241
  moreover have "?f ` ?LHS \<subseteq> ?RHS" unfolding Func_def by fastforce
blanchet@48975
   242
  ultimately show ?thesis using card_of_ordLeq by fast
blanchet@48975
   243
qed
blanchet@48975
   244
blanchet@51836
   245
bnf "op \<circ>" [range] "\<lambda>_:: 'a \<Rightarrow> 'b. natLeq +c |UNIV :: 'a set|" ["%c x::'b::type. c::'a::type"]
traytel@51446
   246
  "fun_rel op ="
blanchet@48975
   247
proof
blanchet@48975
   248
  fix f show "id \<circ> f = id f" by simp
blanchet@48975
   249
next
blanchet@48975
   250
  fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
blanchet@48975
   251
  unfolding comp_def[abs_def] ..
blanchet@48975
   252
next
blanchet@48975
   253
  fix x f g
blanchet@48975
   254
  assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
blanchet@48975
   255
  thus "f \<circ> x = g \<circ> x" by auto
blanchet@48975
   256
next
blanchet@48975
   257
  fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
blanchet@48975
   258
  unfolding image_def comp_def[abs_def] by auto
blanchet@48975
   259
next
blanchet@48975
   260
  show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
blanchet@48975
   261
  apply (rule card_order_csum)
blanchet@48975
   262
  apply (rule natLeq_card_order)
blanchet@48975
   263
  by (rule card_of_card_order_on)
blanchet@48975
   264
(*  *)
blanchet@48975
   265
  show "cinfinite (natLeq +c ?U)"
blanchet@48975
   266
    apply (rule cinfinite_csum)
blanchet@48975
   267
    apply (rule disjI1)
blanchet@48975
   268
    by (rule natLeq_cinfinite)
blanchet@48975
   269
next
blanchet@48975
   270
  fix f :: "'d => 'a"
blanchet@48975
   271
  have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
blanchet@48975
   272
  also have "?U \<le>o natLeq +c ?U"  by (rule ordLeq_csum2) (rule card_of_Card_order)
blanchet@48975
   273
  finally show "|range f| \<le>o natLeq +c ?U" .
blanchet@48975
   274
next
blanchet@48975
   275
  fix A B1 B2 f1 f2 p1 p2 assume p: "wpull A B1 B2 f1 f2 p1 p2"
blanchet@48975
   276
  show "wpull {h. range h \<subseteq> A} {g1. range g1 \<subseteq> B1} {g2. range g2 \<subseteq> B2}
blanchet@48975
   277
    (op \<circ> f1) (op \<circ> f2) (op \<circ> p1) (op \<circ> p2)"
blanchet@48975
   278
  unfolding wpull_def
blanchet@48975
   279
  proof safe
blanchet@48975
   280
    fix g1 g2 assume r: "range g1 \<subseteq> B1" "range g2 \<subseteq> B2"
blanchet@48975
   281
    and c: "f1 \<circ> g1 = f2 \<circ> g2"
blanchet@48975
   282
    show "\<exists>h \<in> {h. range h \<subseteq> A}. p1 \<circ> h = g1 \<and> p2 \<circ> h = g2"
blanchet@48975
   283
    using wpull_cat[OF p c r] by simp metis
blanchet@48975
   284
  qed
blanchet@49453
   285
next
blanchet@49463
   286
  fix R
traytel@51893
   287
  show "fun_rel op = R =
traytel@51893
   288
        (Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO
traytel@51893
   289
         Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"
traytel@51893
   290
  unfolding fun_rel_def Grp_def fun_eq_iff relcompp.simps conversep.simps  subset_iff image_iff
traytel@51893
   291
  by auto (force, metis pair_collapse)
blanchet@48975
   292
qed auto
blanchet@48975
   293
blanchet@48975
   294
end