src/HOL/Basic_BNFs.thy
author blanchet
Mon Jan 20 20:21:12 2014 +0100 (2014-01-20)
changeset 55083 0a689157e3ce
parent 55075 b3d0a02a756d
child 55084 8ee9aabb2bca
permissions -rw-r--r--
move BNF_LFP up the dependency chain
blanchet@55075
     1
(*  Title:      HOL/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
traytel@54581
    14
   (*FIXME: define relators here, reuse in Lifting_* once this theory is in HOL*)
blanchet@48975
    15
begin
blanchet@48975
    16
traytel@54421
    17
bnf ID: 'a
traytel@54421
    18
  map: "id :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
traytel@54421
    19
  sets: "\<lambda>x. {x}"
traytel@54421
    20
  bd: natLeq
traytel@54421
    21
  rel: "id :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
traytel@51893
    22
apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)
blanchet@48975
    23
apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
blanchet@49453
    24
apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]
blanchet@48975
    25
done
blanchet@48975
    26
traytel@54421
    27
bnf DEADID: 'a
traytel@54421
    28
  map: "id :: 'a \<Rightarrow> 'a"
traytel@54421
    29
  bd: "natLeq +c |UNIV :: 'a set|"
traytel@54421
    30
  rel: "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
traytel@54841
    31
by (auto simp add: Grp_def
traytel@51446
    32
  card_order_csum natLeq_card_order card_of_card_order_on
traytel@51446
    33
  cinfinite_csum natLeq_cinfinite)
blanchet@48975
    34
blanchet@49451
    35
definition setl :: "'a + 'b \<Rightarrow> 'a set" where
blanchet@49451
    36
"setl x = (case x of Inl z => {z} | _ => {})"
blanchet@48975
    37
blanchet@49451
    38
definition setr :: "'a + 'b \<Rightarrow> 'b set" where
blanchet@49451
    39
"setr x = (case x of Inr z => {z} | _ => {})"
blanchet@48975
    40
blanchet@49451
    41
lemmas sum_set_defs = setl_def[abs_def] setr_def[abs_def]
blanchet@48975
    42
blanchet@55083
    43
definition
blanchet@55083
    44
   sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
blanchet@55083
    45
where
blanchet@55083
    46
   "sum_rel R1 R2 x y =
blanchet@55083
    47
     (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
blanchet@55083
    48
     | (Inr x, Inr y) \<Rightarrow> R2 x y
blanchet@55083
    49
     | _ \<Rightarrow> False)"
blanchet@55083
    50
blanchet@55083
    51
lemma sum_rel_simps[simp]:
blanchet@55083
    52
  "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
blanchet@55083
    53
  "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
blanchet@55083
    54
  "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
blanchet@55083
    55
  "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
blanchet@55083
    56
  unfolding sum_rel_def by simp_all
blanchet@55083
    57
traytel@54421
    58
bnf "'a + 'b"
traytel@54421
    59
  map: sum_map
traytel@54421
    60
  sets: setl setr
traytel@54421
    61
  bd: natLeq
traytel@54421
    62
  wits: Inl Inr
traytel@54421
    63
  rel: sum_rel
blanchet@48975
    64
proof -
blanchet@48975
    65
  show "sum_map id id = id" by (rule sum_map.id)
blanchet@48975
    66
next
blanchet@54486
    67
  fix f1 :: "'o \<Rightarrow> 's" and f2 :: "'p \<Rightarrow> 't" and g1 :: "'s \<Rightarrow> 'q" and g2 :: "'t \<Rightarrow> 'r"
blanchet@48975
    68
  show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"
blanchet@48975
    69
    by (rule sum_map.comp[symmetric])
blanchet@48975
    70
next
blanchet@54486
    71
  fix x and f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r" and g1 g2
blanchet@49451
    72
  assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and
blanchet@49451
    73
         a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"
blanchet@48975
    74
  thus "sum_map f1 f2 x = sum_map g1 g2 x"
blanchet@48975
    75
  proof (cases x)
blanchet@49451
    76
    case Inl thus ?thesis using a1 by (clarsimp simp: setl_def)
blanchet@48975
    77
  next
blanchet@49451
    78
    case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)
blanchet@48975
    79
  qed
blanchet@48975
    80
next
blanchet@54486
    81
  fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
blanchet@49451
    82
  show "setl o sum_map f1 f2 = image f1 o setl"
blanchet@49451
    83
    by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)
blanchet@48975
    84
next
blanchet@54486
    85
  fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
blanchet@49451
    86
  show "setr o sum_map f1 f2 = image f2 o setr"
blanchet@49451
    87
    by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)
blanchet@48975
    88
next
blanchet@48975
    89
  show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975
    90
next
blanchet@48975
    91
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975
    92
next
blanchet@54486
    93
  fix x :: "'o + 'p"
blanchet@49451
    94
  show "|setl x| \<le>o natLeq"
blanchet@48975
    95
    apply (rule ordLess_imp_ordLeq)
blanchet@48975
    96
    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451
    97
    by (simp add: setl_def split: sum.split)
blanchet@48975
    98
next
blanchet@54486
    99
  fix x :: "'o + 'p"
blanchet@49451
   100
  show "|setr x| \<le>o natLeq"
blanchet@48975
   101
    apply (rule ordLess_imp_ordLeq)
blanchet@48975
   102
    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451
   103
    by (simp add: setr_def split: sum.split)
blanchet@48975
   104
next
traytel@54841
   105
  fix R1 R2 S1 S2
traytel@54841
   106
  show "sum_rel R1 R2 OO sum_rel S1 S2 \<le> sum_rel (R1 OO S1) (R2 OO S2)"
traytel@54841
   107
    by (auto simp: sum_rel_def split: sum.splits)
blanchet@49453
   108
next
blanchet@49453
   109
  fix R S
traytel@51893
   110
  show "sum_rel R S =
traytel@51893
   111
        (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map fst fst))\<inverse>\<inverse> OO
traytel@51893
   112
        Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map snd snd)"
traytel@51893
   113
  unfolding setl_def setr_def sum_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453
   114
  by (fastforce split: sum.splits)
blanchet@48975
   115
qed (auto simp: sum_set_defs)
blanchet@48975
   116
blanchet@48975
   117
definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where
blanchet@48975
   118
"fsts x = {fst x}"
blanchet@48975
   119
blanchet@48975
   120
definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where
blanchet@48975
   121
"snds x = {snd x}"
blanchet@48975
   122
blanchet@48975
   123
lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]
blanchet@48975
   124
blanchet@55083
   125
definition
blanchet@55083
   126
  prod_rel :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a \<times> 'c \<Rightarrow> 'b \<times> 'd \<Rightarrow> bool"
blanchet@55083
   127
where
blanchet@55083
   128
  "prod_rel R1 R2 = (\<lambda>(a, b) (c, d). R1 a c \<and> R2 b d)"
blanchet@55083
   129
blanchet@55083
   130
lemma prod_rel_apply [simp]:
blanchet@55083
   131
  "prod_rel R1 R2 (a, b) (c, d) \<longleftrightarrow> R1 a c \<and> R2 b d"
blanchet@55083
   132
  by (simp add: prod_rel_def)
blanchet@55083
   133
traytel@54421
   134
bnf "'a \<times> 'b"
traytel@54421
   135
  map: map_pair
traytel@54421
   136
  sets: fsts snds
traytel@54421
   137
  bd: natLeq
traytel@54421
   138
  rel: prod_rel
blanchet@48975
   139
proof (unfold prod_set_defs)
blanchet@48975
   140
  show "map_pair id id = id" by (rule map_pair.id)
blanchet@48975
   141
next
blanchet@48975
   142
  fix f1 f2 g1 g2
blanchet@48975
   143
  show "map_pair (g1 o f1) (g2 o f2) = map_pair g1 g2 o map_pair f1 f2"
blanchet@48975
   144
    by (rule map_pair.comp[symmetric])
blanchet@48975
   145
next
blanchet@48975
   146
  fix x f1 f2 g1 g2
blanchet@48975
   147
  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
   148
  thus "map_pair f1 f2 x = map_pair g1 g2 x" by (cases x) simp
blanchet@48975
   149
next
blanchet@48975
   150
  fix f1 f2
blanchet@48975
   151
  show "(\<lambda>x. {fst x}) o map_pair f1 f2 = image f1 o (\<lambda>x. {fst x})"
blanchet@48975
   152
    by (rule ext, unfold o_apply) simp
blanchet@48975
   153
next
blanchet@48975
   154
  fix f1 f2
blanchet@48975
   155
  show "(\<lambda>x. {snd x}) o map_pair f1 f2 = image f2 o (\<lambda>x. {snd x})"
blanchet@48975
   156
    by (rule ext, unfold o_apply) simp
blanchet@48975
   157
next
traytel@52635
   158
  show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975
   159
next
traytel@52635
   160
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975
   161
next
blanchet@48975
   162
  fix x
traytel@52635
   163
  show "|{fst x}| \<le>o natLeq"
traytel@52635
   164
    by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
blanchet@48975
   165
next
traytel@52635
   166
  fix x
traytel@52635
   167
  show "|{snd x}| \<le>o natLeq"
traytel@52635
   168
    by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
blanchet@48975
   169
next
traytel@54841
   170
  fix R1 R2 S1 S2
traytel@54841
   171
  show "prod_rel R1 R2 OO prod_rel S1 S2 \<le> prod_rel (R1 OO S1) (R2 OO S2)" by auto
blanchet@49453
   172
next
blanchet@49453
   173
  fix R S
traytel@51893
   174
  show "prod_rel R S =
traytel@51893
   175
        (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair fst fst))\<inverse>\<inverse> OO
traytel@51893
   176
        Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair snd snd)"
traytel@51893
   177
  unfolding prod_set_defs prod_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453
   178
  by auto
traytel@54189
   179
qed
blanchet@48975
   180
traytel@54421
   181
bnf "'a \<Rightarrow> 'b"
traytel@54421
   182
  map: "op \<circ>"
traytel@54421
   183
  sets: range
traytel@54421
   184
  bd: "natLeq +c |UNIV :: 'a set|"
traytel@54421
   185
  rel: "fun_rel op ="
blanchet@48975
   186
proof
blanchet@48975
   187
  fix f show "id \<circ> f = id f" by simp
blanchet@48975
   188
next
blanchet@48975
   189
  fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
blanchet@48975
   190
  unfolding comp_def[abs_def] ..
blanchet@48975
   191
next
blanchet@48975
   192
  fix x f g
blanchet@48975
   193
  assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
blanchet@48975
   194
  thus "f \<circ> x = g \<circ> x" by auto
blanchet@48975
   195
next
blanchet@48975
   196
  fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
blanchet@48975
   197
  unfolding image_def comp_def[abs_def] by auto
blanchet@48975
   198
next
blanchet@48975
   199
  show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
blanchet@48975
   200
  apply (rule card_order_csum)
blanchet@48975
   201
  apply (rule natLeq_card_order)
blanchet@48975
   202
  by (rule card_of_card_order_on)
blanchet@48975
   203
(*  *)
blanchet@48975
   204
  show "cinfinite (natLeq +c ?U)"
blanchet@48975
   205
    apply (rule cinfinite_csum)
blanchet@48975
   206
    apply (rule disjI1)
blanchet@48975
   207
    by (rule natLeq_cinfinite)
blanchet@48975
   208
next
blanchet@48975
   209
  fix f :: "'d => 'a"
blanchet@48975
   210
  have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
blanchet@54486
   211
  also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
blanchet@48975
   212
  finally show "|range f| \<le>o natLeq +c ?U" .
blanchet@48975
   213
next
traytel@54841
   214
  fix R S
traytel@54841
   215
  show "fun_rel op = R OO fun_rel op = S \<le> fun_rel op = (R OO S)" by (auto simp: fun_rel_def)
blanchet@49453
   216
next
blanchet@49463
   217
  fix R
traytel@51893
   218
  show "fun_rel op = R =
traytel@51893
   219
        (Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO
traytel@51893
   220
         Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"
traytel@51893
   221
  unfolding fun_rel_def Grp_def fun_eq_iff relcompp.simps conversep.simps  subset_iff image_iff
blanchet@54486
   222
  by auto (force, metis (no_types) pair_collapse)
traytel@54189
   223
qed
traytel@54191
   224
blanchet@48975
   225
end