src/HOLCF/Cfun.thy
author huffman
Sun Mar 07 16:12:01 2010 -0800 (2010-03-07)
changeset 35641 a17bc4cec23a
parent 35547 991a6af75978
child 35794 8cd7134275cc
permissions -rw-r--r--
add simp rule LAM_strict
huffman@15600
     1
(*  Title:      HOLCF/Cfun.thy
huffman@15576
     2
    Author:     Franz Regensburger
huffman@15576
     3
huffman@15576
     4
Definition of the type ->  of continuous functions.
huffman@15576
     5
*)
huffman@15576
     6
huffman@15576
     7
header {* The type of continuous functions *}
huffman@15576
     8
huffman@15577
     9
theory Cfun
huffman@29533
    10
imports Pcpodef Ffun Product_Cpo
huffman@15577
    11
begin
huffman@15576
    12
huffman@15576
    13
defaultsort cpo
huffman@15576
    14
huffman@15589
    15
subsection {* Definition of continuous function type *}
huffman@15589
    16
huffman@16699
    17
lemma Ex_cont: "\<exists>f. cont f"
huffman@16699
    18
by (rule exI, rule cont_const)
huffman@16699
    19
huffman@16699
    20
lemma adm_cont: "adm cont"
huffman@16699
    21
by (rule admI, rule cont_lub_fun)
huffman@16699
    22
huffman@35525
    23
cpodef (CFun)  ('a, 'b) cfun (infixr "->" 0) = "{f::'a => 'b. cont f}"
wenzelm@29063
    24
by (simp_all add: Ex_cont adm_cont)
huffman@15576
    25
wenzelm@35427
    26
type_notation (xsymbols)
huffman@35525
    27
  cfun  ("(_ \<rightarrow>/ _)" [1, 0] 0)
huffman@17816
    28
wenzelm@25131
    29
notation
wenzelm@25131
    30
  Rep_CFun  ("(_$/_)" [999,1000] 999)
huffman@15576
    31
wenzelm@25131
    32
notation (xsymbols)
wenzelm@25131
    33
  Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
huffman@15576
    34
wenzelm@25131
    35
notation (HTML output)
wenzelm@25131
    36
  Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
huffman@17816
    37
huffman@17832
    38
subsection {* Syntax for continuous lambda abstraction *}
huffman@17832
    39
huffman@18078
    40
syntax "_cabs" :: "'a"
huffman@18078
    41
huffman@18078
    42
parse_translation {*
wenzelm@35115
    43
(* rewrite (_cabs x t) => (Abs_CFun (%x. t)) *)
wenzelm@35115
    44
  [mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_CFun})];
huffman@18078
    45
*}
huffman@17816
    46
huffman@17832
    47
text {* To avoid eta-contraction of body: *}
huffman@18087
    48
typed_print_translation {*
huffman@18078
    49
  let
huffman@18087
    50
    fun cabs_tr' _ _ [Abs abs] = let
huffman@18087
    51
          val (x,t) = atomic_abs_tr' abs
wenzelm@35115
    52
        in Syntax.const @{syntax_const "_cabs"} $ x $ t end
huffman@18087
    53
huffman@18087
    54
      | cabs_tr' _ T [t] = let
huffman@18087
    55
          val xT = domain_type (domain_type T);
huffman@18087
    56
          val abs' = ("x",xT,(incr_boundvars 1 t)$Bound 0);
huffman@18087
    57
          val (x,t') = atomic_abs_tr' abs';
wenzelm@35115
    58
        in Syntax.const @{syntax_const "_cabs"} $ x $ t' end;
huffman@18087
    59
wenzelm@25131
    60
  in [(@{const_syntax Abs_CFun}, cabs_tr')] end;
huffman@17816
    61
*}
huffman@17816
    62
huffman@18087
    63
text {* Syntax for nested abstractions *}
huffman@17832
    64
huffman@17832
    65
syntax
huffman@18078
    66
  "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic"  ("(3LAM _./ _)" [1000, 10] 10)
huffman@17832
    67
huffman@17832
    68
syntax (xsymbols)
huffman@25927
    69
  "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic" ("(3\<Lambda> _./ _)" [1000, 10] 10)
huffman@17832
    70
huffman@17816
    71
parse_ast_translation {*
wenzelm@35115
    72
(* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
wenzelm@35115
    73
(* cf. Syntax.lambda_ast_tr from src/Pure/Syntax/syn_trans.ML *)
huffman@18078
    74
  let
huffman@18078
    75
    fun Lambda_ast_tr [pats, body] =
wenzelm@35115
    76
          Syntax.fold_ast_p @{syntax_const "_cabs"}
wenzelm@35115
    77
            (Syntax.unfold_ast @{syntax_const "_cargs"} pats, body)
huffman@18078
    78
      | Lambda_ast_tr asts = raise Syntax.AST ("Lambda_ast_tr", asts);
wenzelm@35115
    79
  in [(@{syntax_const "_Lambda"}, Lambda_ast_tr)] end;
huffman@17816
    80
*}
huffman@17816
    81
huffman@17816
    82
print_ast_translation {*
wenzelm@35115
    83
(* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
wenzelm@35115
    84
(* cf. Syntax.abs_ast_tr' from src/Pure/Syntax/syn_trans.ML *)
huffman@18078
    85
  let
huffman@18078
    86
    fun cabs_ast_tr' asts =
wenzelm@35115
    87
      (case Syntax.unfold_ast_p @{syntax_const "_cabs"}
wenzelm@35115
    88
          (Syntax.Appl (Syntax.Constant @{syntax_const "_cabs"} :: asts)) of
huffman@18078
    89
        ([], _) => raise Syntax.AST ("cabs_ast_tr'", asts)
huffman@18078
    90
      | (xs, body) => Syntax.Appl
wenzelm@35115
    91
          [Syntax.Constant @{syntax_const "_Lambda"},
wenzelm@35115
    92
           Syntax.fold_ast @{syntax_const "_cargs"} xs, body]);
wenzelm@35115
    93
  in [(@{syntax_const "_cabs"}, cabs_ast_tr')] end
huffman@17816
    94
*}
huffman@15641
    95
huffman@18087
    96
text {* Dummy patterns for continuous abstraction *}
huffman@18079
    97
translations
wenzelm@25131
    98
  "\<Lambda> _. t" => "CONST Abs_CFun (\<lambda> _. t)"
huffman@18087
    99
huffman@18079
   100
huffman@17832
   101
subsection {* Continuous function space is pointed *}
huffman@15589
   102
huffman@16098
   103
lemma UU_CFun: "\<bottom> \<in> CFun"
huffman@16098
   104
by (simp add: CFun_def inst_fun_pcpo cont_const)
huffman@16098
   105
huffman@35525
   106
instance cfun :: (finite_po, finite_po) finite_po
huffman@25827
   107
by (rule typedef_finite_po [OF type_definition_CFun])
huffman@25827
   108
huffman@35525
   109
instance cfun :: (finite_po, chfin) chfin
huffman@31076
   110
by (rule typedef_chfin [OF type_definition_CFun below_CFun_def])
huffman@25827
   111
huffman@35525
   112
instance cfun :: (cpo, discrete_cpo) discrete_cpo
huffman@31076
   113
by intro_classes (simp add: below_CFun_def Rep_CFun_inject)
huffman@26025
   114
huffman@35525
   115
instance cfun :: (cpo, pcpo) pcpo
huffman@31076
   116
by (rule typedef_pcpo [OF type_definition_CFun below_CFun_def UU_CFun])
huffman@16098
   117
huffman@16209
   118
lemmas Rep_CFun_strict =
huffman@31076
   119
  typedef_Rep_strict [OF type_definition_CFun below_CFun_def UU_CFun]
huffman@16209
   120
huffman@16209
   121
lemmas Abs_CFun_strict =
huffman@31076
   122
  typedef_Abs_strict [OF type_definition_CFun below_CFun_def UU_CFun]
huffman@16098
   123
huffman@17832
   124
text {* function application is strict in its first argument *}
huffman@17832
   125
huffman@17832
   126
lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
huffman@17832
   127
by (simp add: Rep_CFun_strict)
huffman@17832
   128
huffman@35641
   129
lemma LAM_strict [simp]: "(\<Lambda> x. \<bottom>) = \<bottom>"
huffman@35641
   130
by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)
huffman@35641
   131
huffman@17832
   132
text {* for compatibility with old HOLCF-Version *}
huffman@17832
   133
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
huffman@35641
   134
by simp
huffman@17832
   135
huffman@17832
   136
subsection {* Basic properties of continuous functions *}
huffman@17832
   137
huffman@17832
   138
text {* Beta-equality for continuous functions *}
huffman@16209
   139
huffman@16209
   140
lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f"
huffman@16209
   141
by (simp add: Abs_CFun_inverse CFun_def)
huffman@16098
   142
huffman@16209
   143
lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
huffman@16209
   144
by (simp add: Abs_CFun_inverse2)
huffman@16209
   145
huffman@16209
   146
text {* Eta-equality for continuous functions *}
huffman@16209
   147
huffman@16209
   148
lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
huffman@16209
   149
by (rule Rep_CFun_inverse)
huffman@16209
   150
huffman@16209
   151
text {* Extensionality for continuous functions *}
huffman@16209
   152
huffman@17832
   153
lemma expand_cfun_eq: "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
huffman@17832
   154
by (simp add: Rep_CFun_inject [symmetric] expand_fun_eq)
huffman@17832
   155
huffman@16209
   156
lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
huffman@17832
   157
by (simp add: expand_cfun_eq)
huffman@17832
   158
huffman@17832
   159
text {* Extensionality wrt. ordering for continuous functions *}
huffman@15576
   160
huffman@31076
   161
lemma expand_cfun_below: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
huffman@31076
   162
by (simp add: below_CFun_def expand_fun_below)
huffman@17832
   163
huffman@31076
   164
lemma below_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
huffman@31076
   165
by (simp add: expand_cfun_below)
huffman@17832
   166
huffman@17832
   167
text {* Congruence for continuous function application *}
huffman@15589
   168
huffman@16209
   169
lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y"
huffman@15589
   170
by simp
huffman@15589
   171
huffman@16209
   172
lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x"
huffman@15589
   173
by simp
huffman@15589
   174
huffman@16209
   175
lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
huffman@15589
   176
by simp
huffman@15589
   177
huffman@16209
   178
subsection {* Continuity of application *}
huffman@15576
   179
huffman@16209
   180
lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
huffman@18092
   181
by (rule cont_Rep_CFun [THEN cont2cont_fun])
huffman@15576
   182
huffman@16209
   183
lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
huffman@18092
   184
apply (cut_tac x=f in Rep_CFun)
huffman@18092
   185
apply (simp add: CFun_def)
huffman@15576
   186
done
huffman@15576
   187
huffman@16209
   188
lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono]
huffman@16209
   189
lemmas contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub]
huffman@15589
   190
huffman@16209
   191
lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard]
huffman@16209
   192
lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard]
huffman@16209
   193
lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
huffman@16209
   194
lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard]
huffman@16209
   195
huffman@16209
   196
text {* contlub, cont properties of @{term Rep_CFun} in each argument *}
huffman@16209
   197
huffman@27413
   198
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
huffman@16209
   199
by (rule contlub_Rep_CFun2 [THEN contlubE])
huffman@15576
   200
huffman@27413
   201
lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(\<Squnion>i. Y i)"
huffman@16209
   202
by (rule cont_Rep_CFun2 [THEN contE])
huffman@16209
   203
huffman@27413
   204
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
huffman@16209
   205
by (rule contlub_Rep_CFun1 [THEN contlubE])
huffman@15576
   206
huffman@27413
   207
lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| (\<Squnion>i. F i)\<cdot>x"
huffman@16209
   208
by (rule cont_Rep_CFun1 [THEN contE])
huffman@15576
   209
huffman@16209
   210
text {* monotonicity of application *}
huffman@16209
   211
huffman@16209
   212
lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
huffman@31076
   213
by (simp add: expand_cfun_below)
huffman@15576
   214
huffman@16209
   215
lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
huffman@16209
   216
by (rule monofun_Rep_CFun2 [THEN monofunE])
huffman@15576
   217
huffman@16209
   218
lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
huffman@31076
   219
by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])
huffman@15576
   220
huffman@16209
   221
text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
huffman@15576
   222
huffman@16209
   223
lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
huffman@16209
   224
by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
huffman@16209
   225
huffman@16209
   226
lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
huffman@16209
   227
by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
huffman@15576
   228
huffman@16209
   229
lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
huffman@16209
   230
by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])
huffman@15576
   231
huffman@18076
   232
lemma ch2ch_Rep_CFun [simp]:
huffman@18076
   233
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
huffman@25884
   234
by (simp add: chain_def monofun_cfun)
huffman@15576
   235
huffman@25884
   236
lemma ch2ch_LAM [simp]:
huffman@25884
   237
  "\<lbrakk>\<And>x. chain (\<lambda>i. S i x); \<And>i. cont (\<lambda>x. S i x)\<rbrakk> \<Longrightarrow> chain (\<lambda>i. \<Lambda> x. S i x)"
huffman@31076
   238
by (simp add: chain_def expand_cfun_below)
huffman@18092
   239
huffman@16209
   240
text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
huffman@15576
   241
huffman@16209
   242
lemma contlub_cfun: 
huffman@16209
   243
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
huffman@18076
   244
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)
huffman@15576
   245
huffman@16209
   246
lemma cont_cfun: 
huffman@16209
   247
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
huffman@16209
   248
apply (rule thelubE)
huffman@16209
   249
apply (simp only: ch2ch_Rep_CFun)
huffman@16209
   250
apply (simp only: contlub_cfun)
huffman@16209
   251
done
huffman@16209
   252
huffman@18092
   253
lemma contlub_LAM:
huffman@18092
   254
  "\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk>
huffman@18092
   255
    \<Longrightarrow> (\<Lambda> x. \<Squnion>i. F i x) = (\<Squnion>i. \<Lambda> x. F i x)"
huffman@25884
   256
apply (simp add: thelub_CFun)
huffman@18092
   257
apply (simp add: Abs_CFun_inverse2)
huffman@18092
   258
apply (simp add: thelub_fun ch2ch_lambda)
huffman@18092
   259
done
huffman@18092
   260
huffman@25901
   261
lemmas lub_distribs = 
huffman@25901
   262
  contlub_cfun [symmetric]
huffman@25901
   263
  contlub_LAM [symmetric]
huffman@25901
   264
huffman@16209
   265
text {* strictness *}
huffman@16209
   266
huffman@16209
   267
lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
huffman@16209
   268
apply (rule UU_I)
huffman@15576
   269
apply (erule subst)
huffman@15576
   270
apply (rule minimal [THEN monofun_cfun_arg])
huffman@15576
   271
done
huffman@15576
   272
huffman@16209
   273
text {* the lub of a chain of continous functions is monotone *}
huffman@15576
   274
huffman@16209
   275
lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
huffman@16209
   276
apply (drule ch2ch_monofun [OF monofun_Rep_CFun])
huffman@16209
   277
apply (simp add: thelub_fun [symmetric])
huffman@16209
   278
apply (erule monofun_lub_fun)
huffman@16209
   279
apply (simp add: monofun_Rep_CFun2)
huffman@15576
   280
done
huffman@15576
   281
huffman@16386
   282
text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *}
huffman@15576
   283
huffman@16699
   284
lemma ex_lub_cfun:
huffman@16699
   285
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>j. \<Squnion>i. F j\<cdot>(Y i)) = (\<Squnion>i. \<Squnion>j. F j\<cdot>(Y i))"
huffman@18076
   286
by (simp add: diag_lub)
huffman@15576
   287
huffman@15589
   288
text {* the lub of a chain of cont. functions is continuous *}
huffman@15576
   289
huffman@16209
   290
lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
huffman@16209
   291
apply (rule cont2cont_lub)
huffman@16209
   292
apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
huffman@16209
   293
apply (rule cont_Rep_CFun2)
huffman@15576
   294
done
huffman@15576
   295
huffman@15589
   296
text {* type @{typ "'a -> 'b"} is chain complete *}
huffman@15576
   297
huffman@16920
   298
lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
huffman@16920
   299
by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE)
huffman@15576
   300
huffman@27413
   301
lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
huffman@16920
   302
by (rule lub_cfun [THEN thelubI])
huffman@15576
   303
huffman@17832
   304
subsection {* Continuity simplification procedure *}
huffman@15589
   305
huffman@15589
   306
text {* cont2cont lemma for @{term Rep_CFun} *}
huffman@15576
   307
huffman@29530
   308
lemma cont2cont_Rep_CFun [cont2cont]:
huffman@29049
   309
  assumes f: "cont (\<lambda>x. f x)"
huffman@29049
   310
  assumes t: "cont (\<lambda>x. t x)"
huffman@29049
   311
  shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
huffman@29049
   312
proof -
huffman@29049
   313
  have "cont (\<lambda>x. Rep_CFun (f x))"
huffman@29049
   314
    using cont_Rep_CFun f by (rule cont2cont_app3)
huffman@29049
   315
  thus "cont (\<lambda>x. (f x)\<cdot>(t x))"
huffman@29049
   316
    using cont_Rep_CFun2 t by (rule cont2cont_app2)
huffman@29049
   317
qed
huffman@15576
   318
huffman@15589
   319
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
huffman@15576
   320
huffman@15576
   321
lemma cont2mono_LAM:
huffman@29049
   322
  "\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk>
huffman@29049
   323
    \<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)"
huffman@31076
   324
  unfolding monofun_def expand_cfun_below by simp
huffman@15576
   325
huffman@29049
   326
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
huffman@15576
   327
huffman@29530
   328
text {*
huffman@29530
   329
  Not suitable as a cont2cont rule, because on nested lambdas
huffman@29530
   330
  it causes exponential blow-up in the number of subgoals.
huffman@29530
   331
*}
huffman@29530
   332
huffman@15576
   333
lemma cont2cont_LAM:
huffman@29049
   334
  assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
huffman@29049
   335
  assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
huffman@29049
   336
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
huffman@29049
   337
proof (rule cont_Abs_CFun)
huffman@29049
   338
  fix x
huffman@29049
   339
  from f1 show "f x \<in> CFun" by (simp add: CFun_def)
huffman@29049
   340
  from f2 show "cont f" by (rule cont2cont_lambda)
huffman@29049
   341
qed
huffman@15576
   342
huffman@29530
   343
text {*
huffman@29530
   344
  This version does work as a cont2cont rule, since it
huffman@29530
   345
  has only a single subgoal.
huffman@29530
   346
*}
huffman@29530
   347
huffman@29530
   348
lemma cont2cont_LAM' [cont2cont]:
huffman@29530
   349
  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
huffman@29530
   350
  assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
huffman@29530
   351
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
huffman@29530
   352
proof (rule cont2cont_LAM)
huffman@31041
   353
  fix x :: 'a show "cont (\<lambda>y. f x y)"
huffman@31041
   354
    using f by (rule cont_fst_snd_D2)
huffman@29530
   355
next
huffman@31041
   356
  fix y :: 'b show "cont (\<lambda>x. f x y)"
huffman@31041
   357
    using f by (rule cont_fst_snd_D1)
huffman@29530
   358
qed
huffman@29530
   359
huffman@29530
   360
lemma cont2cont_LAM_discrete [cont2cont]:
huffman@29530
   361
  "(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)"
huffman@29530
   362
by (simp add: cont2cont_LAM)
huffman@15576
   363
huffman@16055
   364
lemmas cont_lemmas1 =
huffman@16055
   365
  cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM
huffman@16055
   366
huffman@17832
   367
subsection {* Miscellaneous *}
huffman@17832
   368
huffman@17832
   369
text {* Monotonicity of @{term Abs_CFun} *}
huffman@15576
   370
huffman@17832
   371
lemma semi_monofun_Abs_CFun:
huffman@17832
   372
  "\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
huffman@31076
   373
by (simp add: below_CFun_def Abs_CFun_inverse2)
huffman@15576
   374
huffman@15589
   375
text {* some lemmata for functions with flat/chfin domain/range types *}
huffman@15576
   376
huffman@15576
   377
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)  
huffman@27413
   378
      ==> !s. ? n. (LUB i. Y i)$s = Y n$s"
huffman@15576
   379
apply (rule allI)
huffman@15576
   380
apply (subst contlub_cfun_fun)
huffman@15576
   381
apply assumption
huffman@15576
   382
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
huffman@15576
   383
done
huffman@15576
   384
huffman@18089
   385
lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))"
huffman@18089
   386
by (rule adm_subst, simp, rule adm_chfin)
huffman@18089
   387
huffman@16085
   388
subsection {* Continuous injection-retraction pairs *}
huffman@15589
   389
huffman@16085
   390
text {* Continuous retractions are strict. *}
huffman@15576
   391
huffman@16085
   392
lemma retraction_strict:
huffman@16085
   393
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
huffman@15576
   394
apply (rule UU_I)
huffman@16085
   395
apply (drule_tac x="\<bottom>" in spec)
huffman@16085
   396
apply (erule subst)
huffman@16085
   397
apply (rule monofun_cfun_arg)
huffman@16085
   398
apply (rule minimal)
huffman@15576
   399
done
huffman@15576
   400
huffman@16085
   401
lemma injection_eq:
huffman@16085
   402
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)"
huffman@16085
   403
apply (rule iffI)
huffman@16085
   404
apply (drule_tac f=f in cfun_arg_cong)
huffman@16085
   405
apply simp
huffman@16085
   406
apply simp
huffman@15576
   407
done
huffman@15576
   408
huffman@31076
   409
lemma injection_below:
huffman@16314
   410
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
huffman@16314
   411
apply (rule iffI)
huffman@16314
   412
apply (drule_tac f=f in monofun_cfun_arg)
huffman@16314
   413
apply simp
huffman@16314
   414
apply (erule monofun_cfun_arg)
huffman@16314
   415
done
huffman@16314
   416
huffman@16085
   417
lemma injection_defined_rev:
huffman@16085
   418
  "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>"
huffman@16085
   419
apply (drule_tac f=f in cfun_arg_cong)
huffman@16085
   420
apply (simp add: retraction_strict)
huffman@15576
   421
done
huffman@15576
   422
huffman@16085
   423
lemma injection_defined:
huffman@16085
   424
  "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>"
huffman@16085
   425
by (erule contrapos_nn, rule injection_defined_rev)
huffman@16085
   426
huffman@16085
   427
text {* propagation of flatness and chain-finiteness by retractions *}
huffman@16085
   428
huffman@16085
   429
lemma chfin2chfin:
huffman@16085
   430
  "\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y
huffman@16085
   431
    \<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
huffman@16085
   432
apply clarify
huffman@16085
   433
apply (drule_tac f=g in chain_monofun)
huffman@25921
   434
apply (drule chfin)
huffman@16085
   435
apply (unfold max_in_chain_def)
huffman@16085
   436
apply (simp add: injection_eq)
huffman@16085
   437
done
huffman@16085
   438
huffman@16085
   439
lemma flat2flat:
huffman@16085
   440
  "\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y
huffman@16085
   441
    \<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y"
huffman@16085
   442
apply clarify
huffman@16209
   443
apply (drule_tac f=g in monofun_cfun_arg)
huffman@25920
   444
apply (drule ax_flat)
huffman@16085
   445
apply (erule disjE)
huffman@16085
   446
apply (simp add: injection_defined_rev)
huffman@16085
   447
apply (simp add: injection_eq)
huffman@15576
   448
done
huffman@15576
   449
huffman@15589
   450
text {* a result about functions with flat codomain *}
huffman@15576
   451
huffman@16085
   452
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
huffman@25920
   453
by (drule ax_flat, simp)
huffman@16085
   454
huffman@16085
   455
lemma flat_codom:
huffman@16085
   456
  "f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
huffman@16085
   457
apply (case_tac "f\<cdot>x = \<bottom>")
huffman@15576
   458
apply (rule disjI1)
huffman@15576
   459
apply (rule UU_I)
huffman@16085
   460
apply (erule_tac t="\<bottom>" in subst)
huffman@15576
   461
apply (rule minimal [THEN monofun_cfun_arg])
huffman@16085
   462
apply clarify
huffman@16085
   463
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
huffman@16085
   464
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
huffman@16085
   465
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
huffman@15589
   466
done
huffman@15589
   467
huffman@15589
   468
huffman@15589
   469
subsection {* Identity and composition *}
huffman@15589
   470
wenzelm@25135
   471
definition
wenzelm@25135
   472
  ID :: "'a \<rightarrow> 'a" where
wenzelm@25135
   473
  "ID = (\<Lambda> x. x)"
wenzelm@25135
   474
wenzelm@25135
   475
definition
wenzelm@25135
   476
  cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where
wenzelm@25135
   477
  oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))"
huffman@15589
   478
wenzelm@25131
   479
abbreviation
wenzelm@25131
   480
  cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c"  (infixr "oo" 100)  where
wenzelm@25131
   481
  "f oo g == cfcomp\<cdot>f\<cdot>g"
huffman@15589
   482
huffman@16085
   483
lemma ID1 [simp]: "ID\<cdot>x = x"
huffman@16085
   484
by (simp add: ID_def)
huffman@15576
   485
huffman@16085
   486
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
huffman@15589
   487
by (simp add: oo_def)
huffman@15576
   488
huffman@16085
   489
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
huffman@15589
   490
by (simp add: cfcomp1)
huffman@15576
   491
huffman@27274
   492
lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
huffman@27274
   493
by (simp add: cfcomp1)
huffman@27274
   494
huffman@19709
   495
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>"
huffman@19709
   496
by (simp add: expand_cfun_eq)
huffman@19709
   497
huffman@15589
   498
text {*
huffman@15589
   499
  Show that interpretation of (pcpo,@{text "_->_"}) is a category.
huffman@15589
   500
  The class of objects is interpretation of syntactical class pcpo.
huffman@15589
   501
  The class of arrows  between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}.
huffman@15589
   502
  The identity arrow is interpretation of @{term ID}.
huffman@15589
   503
  The composition of f and g is interpretation of @{text "oo"}.
huffman@15589
   504
*}
huffman@15576
   505
huffman@16085
   506
lemma ID2 [simp]: "f oo ID = f"
huffman@15589
   507
by (rule ext_cfun, simp)
huffman@15576
   508
huffman@16085
   509
lemma ID3 [simp]: "ID oo f = f"
huffman@15589
   510
by (rule ext_cfun, simp)
huffman@15576
   511
huffman@15576
   512
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
huffman@15589
   513
by (rule ext_cfun, simp)
huffman@15576
   514
huffman@16085
   515
huffman@16085
   516
subsection {* Strictified functions *}
huffman@16085
   517
huffman@16085
   518
defaultsort pcpo
huffman@16085
   519
wenzelm@25131
   520
definition
wenzelm@25131
   521
  strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
wenzelm@25131
   522
  "strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@16085
   523
huffman@16085
   524
text {* results about strictify *}
huffman@16085
   525
huffman@17815
   526
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@35168
   527
by simp
huffman@16085
   528
huffman@17815
   529
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@17815
   530
apply (rule monofunI)
huffman@25786
   531
apply (auto simp add: monofun_cfun_arg)
huffman@16085
   532
done
huffman@16085
   533
huffman@17815
   534
(*FIXME: long proof*)
huffman@25723
   535
lemma contlub_strictify2: "contlub (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@16209
   536
apply (rule contlubI)
huffman@27413
   537
apply (case_tac "(\<Squnion>i. Y i) = \<bottom>")
huffman@16699
   538
apply (drule (1) chain_UU_I)
huffman@18076
   539
apply simp
huffman@17815
   540
apply (simp del: if_image_distrib)
huffman@17815
   541
apply (simp only: contlub_cfun_arg)
huffman@16085
   542
apply (rule lub_equal2)
huffman@16085
   543
apply (rule chain_mono2 [THEN exE])
huffman@16085
   544
apply (erule chain_UU_I_inverse2)
huffman@16085
   545
apply (assumption)
huffman@17815
   546
apply (rule_tac x=x in exI, clarsimp)
huffman@16085
   547
apply (erule chain_monofun)
huffman@17815
   548
apply (erule monofun_strictify2 [THEN ch2ch_monofun])
huffman@16085
   549
done
huffman@16085
   550
huffman@17815
   551
lemmas cont_strictify2 =
huffman@17815
   552
  monocontlub2cont [OF monofun_strictify2 contlub_strictify2, standard]
huffman@17815
   553
huffman@17815
   554
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@29530
   555
  unfolding strictify_def
huffman@29530
   556
  by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM)
huffman@16085
   557
huffman@16085
   558
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
huffman@17815
   559
by (simp add: strictify_conv_if)
huffman@16085
   560
huffman@16085
   561
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x"
huffman@17815
   562
by (simp add: strictify_conv_if)
huffman@16085
   563
huffman@17816
   564
subsection {* Continuous let-bindings *}
huffman@17816
   565
wenzelm@25131
   566
definition
wenzelm@25131
   567
  CLet :: "'a \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'b" where
wenzelm@25131
   568
  "CLet = (\<Lambda> s f. f\<cdot>s)"
huffman@17816
   569
huffman@17816
   570
syntax
huffman@17816
   571
  "_CLet" :: "[letbinds, 'a] => 'a" ("(Let (_)/ in (_))" 10)
huffman@17816
   572
huffman@17816
   573
translations
huffman@17816
   574
  "_CLet (_binds b bs) e" == "_CLet b (_CLet bs e)"
wenzelm@25131
   575
  "Let x = a in e" == "CONST CLet\<cdot>a\<cdot>(\<Lambda> x. e)"
huffman@17816
   576
huffman@15576
   577
end