src/HOL/Basic_BNFs.thy
author blanchet
Mon Jan 20 18:24:56 2014 +0100 (2014-01-20)
changeset 55075 b3d0a02a756d
parent 55062 6d3fad6f01c9
child 55083 0a689157e3ce
permissions -rw-r--r--
dissolved BNF session
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*)
traytel@54581
    15
  Lifting_Sum
traytel@54581
    16
  Lifting_Product
blanchet@48975
    17
begin
blanchet@48975
    18
traytel@54421
    19
bnf ID: 'a
traytel@54421
    20
  map: "id :: ('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b"
traytel@54421
    21
  sets: "\<lambda>x. {x}"
traytel@54421
    22
  bd: natLeq
traytel@54421
    23
  rel: "id :: ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
traytel@51893
    24
apply (auto simp: Grp_def fun_eq_iff relcompp.simps natLeq_card_order natLeq_cinfinite)
blanchet@48975
    25
apply (rule ordLess_imp_ordLeq[OF finite_ordLess_infinite[OF _ natLeq_Well_order]])
blanchet@49453
    26
apply (auto simp add: Field_card_of Field_natLeq card_of_well_order_on)[3]
blanchet@48975
    27
done
blanchet@48975
    28
traytel@54421
    29
bnf DEADID: 'a
traytel@54421
    30
  map: "id :: 'a \<Rightarrow> 'a"
traytel@54421
    31
  bd: "natLeq +c |UNIV :: 'a set|"
traytel@54421
    32
  rel: "op = :: 'a \<Rightarrow> 'a \<Rightarrow> bool"
traytel@54841
    33
by (auto simp add: Grp_def
traytel@51446
    34
  card_order_csum natLeq_card_order card_of_card_order_on
traytel@51446
    35
  cinfinite_csum natLeq_cinfinite)
blanchet@48975
    36
blanchet@49451
    37
definition setl :: "'a + 'b \<Rightarrow> 'a set" where
blanchet@49451
    38
"setl x = (case x of Inl z => {z} | _ => {})"
blanchet@48975
    39
blanchet@49451
    40
definition setr :: "'a + 'b \<Rightarrow> 'b set" where
blanchet@49451
    41
"setr x = (case x of Inr z => {z} | _ => {})"
blanchet@48975
    42
blanchet@49451
    43
lemmas sum_set_defs = setl_def[abs_def] setr_def[abs_def]
blanchet@48975
    44
traytel@54421
    45
bnf "'a + 'b"
traytel@54421
    46
  map: sum_map
traytel@54421
    47
  sets: setl setr
traytel@54421
    48
  bd: natLeq
traytel@54421
    49
  wits: Inl Inr
traytel@54421
    50
  rel: sum_rel
blanchet@48975
    51
proof -
blanchet@48975
    52
  show "sum_map id id = id" by (rule sum_map.id)
blanchet@48975
    53
next
blanchet@54486
    54
  fix f1 :: "'o \<Rightarrow> 's" and f2 :: "'p \<Rightarrow> 't" and g1 :: "'s \<Rightarrow> 'q" and g2 :: "'t \<Rightarrow> 'r"
blanchet@48975
    55
  show "sum_map (g1 o f1) (g2 o f2) = sum_map g1 g2 o sum_map f1 f2"
blanchet@48975
    56
    by (rule sum_map.comp[symmetric])
blanchet@48975
    57
next
blanchet@54486
    58
  fix x and f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r" and g1 g2
blanchet@49451
    59
  assume a1: "\<And>z. z \<in> setl x \<Longrightarrow> f1 z = g1 z" and
blanchet@49451
    60
         a2: "\<And>z. z \<in> setr x \<Longrightarrow> f2 z = g2 z"
blanchet@48975
    61
  thus "sum_map f1 f2 x = sum_map g1 g2 x"
blanchet@48975
    62
  proof (cases x)
blanchet@49451
    63
    case Inl thus ?thesis using a1 by (clarsimp simp: setl_def)
blanchet@48975
    64
  next
blanchet@49451
    65
    case Inr thus ?thesis using a2 by (clarsimp simp: setr_def)
blanchet@48975
    66
  qed
blanchet@48975
    67
next
blanchet@54486
    68
  fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
blanchet@49451
    69
  show "setl o sum_map f1 f2 = image f1 o setl"
blanchet@49451
    70
    by (rule ext, unfold o_apply) (simp add: setl_def split: sum.split)
blanchet@48975
    71
next
blanchet@54486
    72
  fix f1 :: "'o \<Rightarrow> 'q" and f2 :: "'p \<Rightarrow> 'r"
blanchet@49451
    73
  show "setr o sum_map f1 f2 = image f2 o setr"
blanchet@49451
    74
    by (rule ext, unfold o_apply) (simp add: setr_def split: sum.split)
blanchet@48975
    75
next
blanchet@48975
    76
  show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975
    77
next
blanchet@48975
    78
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975
    79
next
blanchet@54486
    80
  fix x :: "'o + 'p"
blanchet@49451
    81
  show "|setl x| \<le>o natLeq"
blanchet@48975
    82
    apply (rule ordLess_imp_ordLeq)
blanchet@48975
    83
    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451
    84
    by (simp add: setl_def split: sum.split)
blanchet@48975
    85
next
blanchet@54486
    86
  fix x :: "'o + 'p"
blanchet@49451
    87
  show "|setr x| \<le>o natLeq"
blanchet@48975
    88
    apply (rule ordLess_imp_ordLeq)
blanchet@48975
    89
    apply (rule finite_iff_ordLess_natLeq[THEN iffD1])
blanchet@49451
    90
    by (simp add: setr_def split: sum.split)
blanchet@48975
    91
next
traytel@54841
    92
  fix R1 R2 S1 S2
traytel@54841
    93
  show "sum_rel R1 R2 OO sum_rel S1 S2 \<le> sum_rel (R1 OO S1) (R2 OO S2)"
traytel@54841
    94
    by (auto simp: sum_rel_def split: sum.splits)
blanchet@49453
    95
next
blanchet@49453
    96
  fix R S
traytel@51893
    97
  show "sum_rel R S =
traytel@51893
    98
        (Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map fst fst))\<inverse>\<inverse> OO
traytel@51893
    99
        Grp {x. setl x \<subseteq> Collect (split R) \<and> setr x \<subseteq> Collect (split S)} (sum_map snd snd)"
traytel@51893
   100
  unfolding setl_def setr_def sum_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453
   101
  by (fastforce split: sum.splits)
blanchet@48975
   102
qed (auto simp: sum_set_defs)
blanchet@48975
   103
blanchet@48975
   104
definition fsts :: "'a \<times> 'b \<Rightarrow> 'a set" where
blanchet@48975
   105
"fsts x = {fst x}"
blanchet@48975
   106
blanchet@48975
   107
definition snds :: "'a \<times> 'b \<Rightarrow> 'b set" where
blanchet@48975
   108
"snds x = {snd x}"
blanchet@48975
   109
blanchet@48975
   110
lemmas prod_set_defs = fsts_def[abs_def] snds_def[abs_def]
blanchet@48975
   111
traytel@54421
   112
bnf "'a \<times> 'b"
traytel@54421
   113
  map: map_pair
traytel@54421
   114
  sets: fsts snds
traytel@54421
   115
  bd: natLeq
traytel@54421
   116
  rel: prod_rel
blanchet@48975
   117
proof (unfold prod_set_defs)
blanchet@48975
   118
  show "map_pair id id = id" by (rule map_pair.id)
blanchet@48975
   119
next
blanchet@48975
   120
  fix f1 f2 g1 g2
blanchet@48975
   121
  show "map_pair (g1 o f1) (g2 o f2) = map_pair g1 g2 o map_pair f1 f2"
blanchet@48975
   122
    by (rule map_pair.comp[symmetric])
blanchet@48975
   123
next
blanchet@48975
   124
  fix x f1 f2 g1 g2
blanchet@48975
   125
  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
   126
  thus "map_pair f1 f2 x = map_pair g1 g2 x" by (cases x) simp
blanchet@48975
   127
next
blanchet@48975
   128
  fix f1 f2
blanchet@48975
   129
  show "(\<lambda>x. {fst x}) o map_pair f1 f2 = image f1 o (\<lambda>x. {fst x})"
blanchet@48975
   130
    by (rule ext, unfold o_apply) simp
blanchet@48975
   131
next
blanchet@48975
   132
  fix f1 f2
blanchet@48975
   133
  show "(\<lambda>x. {snd x}) o map_pair f1 f2 = image f2 o (\<lambda>x. {snd x})"
blanchet@48975
   134
    by (rule ext, unfold o_apply) simp
blanchet@48975
   135
next
traytel@52635
   136
  show "card_order natLeq" by (rule natLeq_card_order)
blanchet@48975
   137
next
traytel@52635
   138
  show "cinfinite natLeq" by (rule natLeq_cinfinite)
blanchet@48975
   139
next
blanchet@48975
   140
  fix x
traytel@52635
   141
  show "|{fst x}| \<le>o natLeq"
traytel@52635
   142
    by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
blanchet@48975
   143
next
traytel@52635
   144
  fix x
traytel@52635
   145
  show "|{snd x}| \<le>o natLeq"
traytel@52635
   146
    by (metis ordLess_imp_ordLeq finite_iff_ordLess_natLeq finite.emptyI finite_insert)
blanchet@48975
   147
next
traytel@54841
   148
  fix R1 R2 S1 S2
traytel@54841
   149
  show "prod_rel R1 R2 OO prod_rel S1 S2 \<le> prod_rel (R1 OO S1) (R2 OO S2)" by auto
blanchet@49453
   150
next
blanchet@49453
   151
  fix R S
traytel@51893
   152
  show "prod_rel R S =
traytel@51893
   153
        (Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair fst fst))\<inverse>\<inverse> OO
traytel@51893
   154
        Grp {x. {fst x} \<subseteq> Collect (split R) \<and> {snd x} \<subseteq> Collect (split S)} (map_pair snd snd)"
traytel@51893
   155
  unfolding prod_set_defs prod_rel_def Grp_def relcompp.simps conversep.simps fun_eq_iff
blanchet@49453
   156
  by auto
traytel@54189
   157
qed
blanchet@48975
   158
traytel@54421
   159
bnf "'a \<Rightarrow> 'b"
traytel@54421
   160
  map: "op \<circ>"
traytel@54421
   161
  sets: range
traytel@54421
   162
  bd: "natLeq +c |UNIV :: 'a set|"
traytel@54421
   163
  rel: "fun_rel op ="
blanchet@48975
   164
proof
blanchet@48975
   165
  fix f show "id \<circ> f = id f" by simp
blanchet@48975
   166
next
blanchet@48975
   167
  fix f g show "op \<circ> (g \<circ> f) = op \<circ> g \<circ> op \<circ> f"
blanchet@48975
   168
  unfolding comp_def[abs_def] ..
blanchet@48975
   169
next
blanchet@48975
   170
  fix x f g
blanchet@48975
   171
  assume "\<And>z. z \<in> range x \<Longrightarrow> f z = g z"
blanchet@48975
   172
  thus "f \<circ> x = g \<circ> x" by auto
blanchet@48975
   173
next
blanchet@48975
   174
  fix f show "range \<circ> op \<circ> f = op ` f \<circ> range"
blanchet@48975
   175
  unfolding image_def comp_def[abs_def] by auto
blanchet@48975
   176
next
blanchet@48975
   177
  show "card_order (natLeq +c |UNIV| )" (is "_ (_ +c ?U)")
blanchet@48975
   178
  apply (rule card_order_csum)
blanchet@48975
   179
  apply (rule natLeq_card_order)
blanchet@48975
   180
  by (rule card_of_card_order_on)
blanchet@48975
   181
(*  *)
blanchet@48975
   182
  show "cinfinite (natLeq +c ?U)"
blanchet@48975
   183
    apply (rule cinfinite_csum)
blanchet@48975
   184
    apply (rule disjI1)
blanchet@48975
   185
    by (rule natLeq_cinfinite)
blanchet@48975
   186
next
blanchet@48975
   187
  fix f :: "'d => 'a"
blanchet@48975
   188
  have "|range f| \<le>o | (UNIV::'d set) |" (is "_ \<le>o ?U") by (rule card_of_image)
blanchet@54486
   189
  also have "?U \<le>o natLeq +c ?U" by (rule ordLeq_csum2) (rule card_of_Card_order)
blanchet@48975
   190
  finally show "|range f| \<le>o natLeq +c ?U" .
blanchet@48975
   191
next
traytel@54841
   192
  fix R S
traytel@54841
   193
  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
   194
next
blanchet@49463
   195
  fix R
traytel@51893
   196
  show "fun_rel op = R =
traytel@51893
   197
        (Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> fst))\<inverse>\<inverse> OO
traytel@51893
   198
         Grp {x. range x \<subseteq> Collect (split R)} (op \<circ> snd)"
traytel@51893
   199
  unfolding fun_rel_def Grp_def fun_eq_iff relcompp.simps conversep.simps  subset_iff image_iff
blanchet@54486
   200
  by auto (force, metis (no_types) pair_collapse)
traytel@54189
   201
qed
traytel@54191
   202
blanchet@48975
   203
end