src/HOLCF/Cfun.thy
 author wenzelm Tue Mar 02 23:59:54 2010 +0100 (2010-03-02) changeset 35427 ad039d29e01c parent 35168 07b3112e464b child 35547 991a6af75978 permissions -rw-r--r--
proper (type_)notation;
```     1 (*  Title:      HOLCF/Cfun.thy
```
```     2     Author:     Franz Regensburger
```
```     3
```
```     4 Definition of the type ->  of continuous functions.
```
```     5 *)
```
```     6
```
```     7 header {* The type of continuous functions *}
```
```     8
```
```     9 theory Cfun
```
```    10 imports Pcpodef Ffun Product_Cpo
```
```    11 begin
```
```    12
```
```    13 defaultsort cpo
```
```    14
```
```    15 subsection {* Definition of continuous function type *}
```
```    16
```
```    17 lemma Ex_cont: "\<exists>f. cont f"
```
```    18 by (rule exI, rule cont_const)
```
```    19
```
```    20 lemma adm_cont: "adm cont"
```
```    21 by (rule admI, rule cont_lub_fun)
```
```    22
```
```    23 cpodef (CFun)  ('a, 'b) "->" (infixr "->" 0) = "{f::'a => 'b. cont f}"
```
```    24 by (simp_all add: Ex_cont adm_cont)
```
```    25
```
```    26 type_notation (xsymbols)
```
```    27   "->"  ("(_ \<rightarrow>/ _)" [1, 0] 0)
```
```    28
```
```    29 notation
```
```    30   Rep_CFun  ("(_\$/_)" [999,1000] 999)
```
```    31
```
```    32 notation (xsymbols)
```
```    33   Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
```
```    34
```
```    35 notation (HTML output)
```
```    36   Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
```
```    37
```
```    38 subsection {* Syntax for continuous lambda abstraction *}
```
```    39
```
```    40 syntax "_cabs" :: "'a"
```
```    41
```
```    42 parse_translation {*
```
```    43 (* rewrite (_cabs x t) => (Abs_CFun (%x. t)) *)
```
```    44   [mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_CFun})];
```
```    45 *}
```
```    46
```
```    47 text {* To avoid eta-contraction of body: *}
```
```    48 typed_print_translation {*
```
```    49   let
```
```    50     fun cabs_tr' _ _ [Abs abs] = let
```
```    51           val (x,t) = atomic_abs_tr' abs
```
```    52         in Syntax.const @{syntax_const "_cabs"} \$ x \$ t end
```
```    53
```
```    54       | cabs_tr' _ T [t] = let
```
```    55           val xT = domain_type (domain_type T);
```
```    56           val abs' = ("x",xT,(incr_boundvars 1 t)\$Bound 0);
```
```    57           val (x,t') = atomic_abs_tr' abs';
```
```    58         in Syntax.const @{syntax_const "_cabs"} \$ x \$ t' end;
```
```    59
```
```    60   in [(@{const_syntax Abs_CFun}, cabs_tr')] end;
```
```    61 *}
```
```    62
```
```    63 text {* Syntax for nested abstractions *}
```
```    64
```
```    65 syntax
```
```    66   "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic"  ("(3LAM _./ _)" [1000, 10] 10)
```
```    67
```
```    68 syntax (xsymbols)
```
```    69   "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic" ("(3\<Lambda> _./ _)" [1000, 10] 10)
```
```    70
```
```    71 parse_ast_translation {*
```
```    72 (* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
```
```    73 (* cf. Syntax.lambda_ast_tr from src/Pure/Syntax/syn_trans.ML *)
```
```    74   let
```
```    75     fun Lambda_ast_tr [pats, body] =
```
```    76           Syntax.fold_ast_p @{syntax_const "_cabs"}
```
```    77             (Syntax.unfold_ast @{syntax_const "_cargs"} pats, body)
```
```    78       | Lambda_ast_tr asts = raise Syntax.AST ("Lambda_ast_tr", asts);
```
```    79   in [(@{syntax_const "_Lambda"}, Lambda_ast_tr)] end;
```
```    80 *}
```
```    81
```
```    82 print_ast_translation {*
```
```    83 (* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
```
```    84 (* cf. Syntax.abs_ast_tr' from src/Pure/Syntax/syn_trans.ML *)
```
```    85   let
```
```    86     fun cabs_ast_tr' asts =
```
```    87       (case Syntax.unfold_ast_p @{syntax_const "_cabs"}
```
```    88           (Syntax.Appl (Syntax.Constant @{syntax_const "_cabs"} :: asts)) of
```
```    89         ([], _) => raise Syntax.AST ("cabs_ast_tr'", asts)
```
```    90       | (xs, body) => Syntax.Appl
```
```    91           [Syntax.Constant @{syntax_const "_Lambda"},
```
```    92            Syntax.fold_ast @{syntax_const "_cargs"} xs, body]);
```
```    93   in [(@{syntax_const "_cabs"}, cabs_ast_tr')] end
```
```    94 *}
```
```    95
```
```    96 text {* Dummy patterns for continuous abstraction *}
```
```    97 translations
```
```    98   "\<Lambda> _. t" => "CONST Abs_CFun (\<lambda> _. t)"
```
```    99
```
```   100
```
```   101 subsection {* Continuous function space is pointed *}
```
```   102
```
```   103 lemma UU_CFun: "\<bottom> \<in> CFun"
```
```   104 by (simp add: CFun_def inst_fun_pcpo cont_const)
```
```   105
```
```   106 instance "->" :: (finite_po, finite_po) finite_po
```
```   107 by (rule typedef_finite_po [OF type_definition_CFun])
```
```   108
```
```   109 instance "->" :: (finite_po, chfin) chfin
```
```   110 by (rule typedef_chfin [OF type_definition_CFun below_CFun_def])
```
```   111
```
```   112 instance "->" :: (cpo, discrete_cpo) discrete_cpo
```
```   113 by intro_classes (simp add: below_CFun_def Rep_CFun_inject)
```
```   114
```
```   115 instance "->" :: (cpo, pcpo) pcpo
```
```   116 by (rule typedef_pcpo [OF type_definition_CFun below_CFun_def UU_CFun])
```
```   117
```
```   118 lemmas Rep_CFun_strict =
```
```   119   typedef_Rep_strict [OF type_definition_CFun below_CFun_def UU_CFun]
```
```   120
```
```   121 lemmas Abs_CFun_strict =
```
```   122   typedef_Abs_strict [OF type_definition_CFun below_CFun_def UU_CFun]
```
```   123
```
```   124 text {* function application is strict in its first argument *}
```
```   125
```
```   126 lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
```
```   127 by (simp add: Rep_CFun_strict)
```
```   128
```
```   129 text {* for compatibility with old HOLCF-Version *}
```
```   130 lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
```
```   131 by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)
```
```   132
```
```   133 subsection {* Basic properties of continuous functions *}
```
```   134
```
```   135 text {* Beta-equality for continuous functions *}
```
```   136
```
```   137 lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f"
```
```   138 by (simp add: Abs_CFun_inverse CFun_def)
```
```   139
```
```   140 lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
```
```   141 by (simp add: Abs_CFun_inverse2)
```
```   142
```
```   143 text {* Eta-equality for continuous functions *}
```
```   144
```
```   145 lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
```
```   146 by (rule Rep_CFun_inverse)
```
```   147
```
```   148 text {* Extensionality for continuous functions *}
```
```   149
```
```   150 lemma expand_cfun_eq: "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
```
```   151 by (simp add: Rep_CFun_inject [symmetric] expand_fun_eq)
```
```   152
```
```   153 lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
```
```   154 by (simp add: expand_cfun_eq)
```
```   155
```
```   156 text {* Extensionality wrt. ordering for continuous functions *}
```
```   157
```
```   158 lemma expand_cfun_below: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
```
```   159 by (simp add: below_CFun_def expand_fun_below)
```
```   160
```
```   161 lemma below_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
```
```   162 by (simp add: expand_cfun_below)
```
```   163
```
```   164 text {* Congruence for continuous function application *}
```
```   165
```
```   166 lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y"
```
```   167 by simp
```
```   168
```
```   169 lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x"
```
```   170 by simp
```
```   171
```
```   172 lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
```
```   173 by simp
```
```   174
```
```   175 subsection {* Continuity of application *}
```
```   176
```
```   177 lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
```
```   178 by (rule cont_Rep_CFun [THEN cont2cont_fun])
```
```   179
```
```   180 lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
```
```   181 apply (cut_tac x=f in Rep_CFun)
```
```   182 apply (simp add: CFun_def)
```
```   183 done
```
```   184
```
```   185 lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono]
```
```   186 lemmas contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub]
```
```   187
```
```   188 lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard]
```
```   189 lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard]
```
```   190 lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
```
```   191 lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard]
```
```   192
```
```   193 text {* contlub, cont properties of @{term Rep_CFun} in each argument *}
```
```   194
```
```   195 lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
```
```   196 by (rule contlub_Rep_CFun2 [THEN contlubE])
```
```   197
```
```   198 lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(\<Squnion>i. Y i)"
```
```   199 by (rule cont_Rep_CFun2 [THEN contE])
```
```   200
```
```   201 lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
```
```   202 by (rule contlub_Rep_CFun1 [THEN contlubE])
```
```   203
```
```   204 lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| (\<Squnion>i. F i)\<cdot>x"
```
```   205 by (rule cont_Rep_CFun1 [THEN contE])
```
```   206
```
```   207 text {* monotonicity of application *}
```
```   208
```
```   209 lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
```
```   210 by (simp add: expand_cfun_below)
```
```   211
```
```   212 lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
```
```   213 by (rule monofun_Rep_CFun2 [THEN monofunE])
```
```   214
```
```   215 lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
```
```   216 by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])
```
```   217
```
```   218 text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
```
```   219
```
```   220 lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
```
```   221 by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
```
```   222
```
```   223 lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
```
```   224 by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
```
```   225
```
```   226 lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
```
```   227 by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])
```
```   228
```
```   229 lemma ch2ch_Rep_CFun [simp]:
```
```   230   "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
```
```   231 by (simp add: chain_def monofun_cfun)
```
```   232
```
```   233 lemma ch2ch_LAM [simp]:
```
```   234   "\<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)"
```
```   235 by (simp add: chain_def expand_cfun_below)
```
```   236
```
```   237 text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
```
```   238
```
```   239 lemma contlub_cfun:
```
```   240   "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
```
```   241 by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)
```
```   242
```
```   243 lemma cont_cfun:
```
```   244   "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
```
```   245 apply (rule thelubE)
```
```   246 apply (simp only: ch2ch_Rep_CFun)
```
```   247 apply (simp only: contlub_cfun)
```
```   248 done
```
```   249
```
```   250 lemma contlub_LAM:
```
```   251   "\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk>
```
```   252     \<Longrightarrow> (\<Lambda> x. \<Squnion>i. F i x) = (\<Squnion>i. \<Lambda> x. F i x)"
```
```   253 apply (simp add: thelub_CFun)
```
```   254 apply (simp add: Abs_CFun_inverse2)
```
```   255 apply (simp add: thelub_fun ch2ch_lambda)
```
```   256 done
```
```   257
```
```   258 lemmas lub_distribs =
```
```   259   contlub_cfun [symmetric]
```
```   260   contlub_LAM [symmetric]
```
```   261
```
```   262 text {* strictness *}
```
```   263
```
```   264 lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
```
```   265 apply (rule UU_I)
```
```   266 apply (erule subst)
```
```   267 apply (rule minimal [THEN monofun_cfun_arg])
```
```   268 done
```
```   269
```
```   270 text {* the lub of a chain of continous functions is monotone *}
```
```   271
```
```   272 lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
```
```   273 apply (drule ch2ch_monofun [OF monofun_Rep_CFun])
```
```   274 apply (simp add: thelub_fun [symmetric])
```
```   275 apply (erule monofun_lub_fun)
```
```   276 apply (simp add: monofun_Rep_CFun2)
```
```   277 done
```
```   278
```
```   279 text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *}
```
```   280
```
```   281 lemma ex_lub_cfun:
```
```   282   "\<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))"
```
```   283 by (simp add: diag_lub)
```
```   284
```
```   285 text {* the lub of a chain of cont. functions is continuous *}
```
```   286
```
```   287 lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
```
```   288 apply (rule cont2cont_lub)
```
```   289 apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
```
```   290 apply (rule cont_Rep_CFun2)
```
```   291 done
```
```   292
```
```   293 text {* type @{typ "'a -> 'b"} is chain complete *}
```
```   294
```
```   295 lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
```
```   296 by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE)
```
```   297
```
```   298 lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
```
```   299 by (rule lub_cfun [THEN thelubI])
```
```   300
```
```   301 subsection {* Continuity simplification procedure *}
```
```   302
```
```   303 text {* cont2cont lemma for @{term Rep_CFun} *}
```
```   304
```
```   305 lemma cont2cont_Rep_CFun [cont2cont]:
```
```   306   assumes f: "cont (\<lambda>x. f x)"
```
```   307   assumes t: "cont (\<lambda>x. t x)"
```
```   308   shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
```
```   309 proof -
```
```   310   have "cont (\<lambda>x. Rep_CFun (f x))"
```
```   311     using cont_Rep_CFun f by (rule cont2cont_app3)
```
```   312   thus "cont (\<lambda>x. (f x)\<cdot>(t x))"
```
```   313     using cont_Rep_CFun2 t by (rule cont2cont_app2)
```
```   314 qed
```
```   315
```
```   316 text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
```
```   317
```
```   318 lemma cont2mono_LAM:
```
```   319   "\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk>
```
```   320     \<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)"
```
```   321   unfolding monofun_def expand_cfun_below by simp
```
```   322
```
```   323 text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
```
```   324
```
```   325 text {*
```
```   326   Not suitable as a cont2cont rule, because on nested lambdas
```
```   327   it causes exponential blow-up in the number of subgoals.
```
```   328 *}
```
```   329
```
```   330 lemma cont2cont_LAM:
```
```   331   assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
```
```   332   assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
```
```   333   shows "cont (\<lambda>x. \<Lambda> y. f x y)"
```
```   334 proof (rule cont_Abs_CFun)
```
```   335   fix x
```
```   336   from f1 show "f x \<in> CFun" by (simp add: CFun_def)
```
```   337   from f2 show "cont f" by (rule cont2cont_lambda)
```
```   338 qed
```
```   339
```
```   340 text {*
```
```   341   This version does work as a cont2cont rule, since it
```
```   342   has only a single subgoal.
```
```   343 *}
```
```   344
```
```   345 lemma cont2cont_LAM' [cont2cont]:
```
```   346   fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
```
```   347   assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
```
```   348   shows "cont (\<lambda>x. \<Lambda> y. f x y)"
```
```   349 proof (rule cont2cont_LAM)
```
```   350   fix x :: 'a show "cont (\<lambda>y. f x y)"
```
```   351     using f by (rule cont_fst_snd_D2)
```
```   352 next
```
```   353   fix y :: 'b show "cont (\<lambda>x. f x y)"
```
```   354     using f by (rule cont_fst_snd_D1)
```
```   355 qed
```
```   356
```
```   357 lemma cont2cont_LAM_discrete [cont2cont]:
```
```   358   "(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)"
```
```   359 by (simp add: cont2cont_LAM)
```
```   360
```
```   361 lemmas cont_lemmas1 =
```
```   362   cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM
```
```   363
```
```   364 subsection {* Miscellaneous *}
```
```   365
```
```   366 text {* Monotonicity of @{term Abs_CFun} *}
```
```   367
```
```   368 lemma semi_monofun_Abs_CFun:
```
```   369   "\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
```
```   370 by (simp add: below_CFun_def Abs_CFun_inverse2)
```
```   371
```
```   372 text {* some lemmata for functions with flat/chfin domain/range types *}
```
```   373
```
```   374 lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)
```
```   375       ==> !s. ? n. (LUB i. Y i)\$s = Y n\$s"
```
```   376 apply (rule allI)
```
```   377 apply (subst contlub_cfun_fun)
```
```   378 apply assumption
```
```   379 apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
```
```   380 done
```
```   381
```
```   382 lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))"
```
```   383 by (rule adm_subst, simp, rule adm_chfin)
```
```   384
```
```   385 subsection {* Continuous injection-retraction pairs *}
```
```   386
```
```   387 text {* Continuous retractions are strict. *}
```
```   388
```
```   389 lemma retraction_strict:
```
```   390   "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
```
```   391 apply (rule UU_I)
```
```   392 apply (drule_tac x="\<bottom>" in spec)
```
```   393 apply (erule subst)
```
```   394 apply (rule monofun_cfun_arg)
```
```   395 apply (rule minimal)
```
```   396 done
```
```   397
```
```   398 lemma injection_eq:
```
```   399   "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)"
```
```   400 apply (rule iffI)
```
```   401 apply (drule_tac f=f in cfun_arg_cong)
```
```   402 apply simp
```
```   403 apply simp
```
```   404 done
```
```   405
```
```   406 lemma injection_below:
```
```   407   "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
```
```   408 apply (rule iffI)
```
```   409 apply (drule_tac f=f in monofun_cfun_arg)
```
```   410 apply simp
```
```   411 apply (erule monofun_cfun_arg)
```
```   412 done
```
```   413
```
```   414 lemma injection_defined_rev:
```
```   415   "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>"
```
```   416 apply (drule_tac f=f in cfun_arg_cong)
```
```   417 apply (simp add: retraction_strict)
```
```   418 done
```
```   419
```
```   420 lemma injection_defined:
```
```   421   "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>"
```
```   422 by (erule contrapos_nn, rule injection_defined_rev)
```
```   423
```
```   424 text {* propagation of flatness and chain-finiteness by retractions *}
```
```   425
```
```   426 lemma chfin2chfin:
```
```   427   "\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y
```
```   428     \<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
```
```   429 apply clarify
```
```   430 apply (drule_tac f=g in chain_monofun)
```
```   431 apply (drule chfin)
```
```   432 apply (unfold max_in_chain_def)
```
```   433 apply (simp add: injection_eq)
```
```   434 done
```
```   435
```
```   436 lemma flat2flat:
```
```   437   "\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y
```
```   438     \<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y"
```
```   439 apply clarify
```
```   440 apply (drule_tac f=g in monofun_cfun_arg)
```
```   441 apply (drule ax_flat)
```
```   442 apply (erule disjE)
```
```   443 apply (simp add: injection_defined_rev)
```
```   444 apply (simp add: injection_eq)
```
```   445 done
```
```   446
```
```   447 text {* a result about functions with flat codomain *}
```
```   448
```
```   449 lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
```
```   450 by (drule ax_flat, simp)
```
```   451
```
```   452 lemma flat_codom:
```
```   453   "f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
```
```   454 apply (case_tac "f\<cdot>x = \<bottom>")
```
```   455 apply (rule disjI1)
```
```   456 apply (rule UU_I)
```
```   457 apply (erule_tac t="\<bottom>" in subst)
```
```   458 apply (rule minimal [THEN monofun_cfun_arg])
```
```   459 apply clarify
```
```   460 apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
```
```   461 apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
```
```   462 apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
```
```   463 done
```
```   464
```
```   465
```
```   466 subsection {* Identity and composition *}
```
```   467
```
```   468 definition
```
```   469   ID :: "'a \<rightarrow> 'a" where
```
```   470   "ID = (\<Lambda> x. x)"
```
```   471
```
```   472 definition
```
```   473   cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where
```
```   474   oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))"
```
```   475
```
```   476 abbreviation
```
```   477   cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c"  (infixr "oo" 100)  where
```
```   478   "f oo g == cfcomp\<cdot>f\<cdot>g"
```
```   479
```
```   480 lemma ID1 [simp]: "ID\<cdot>x = x"
```
```   481 by (simp add: ID_def)
```
```   482
```
```   483 lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
```
```   484 by (simp add: oo_def)
```
```   485
```
```   486 lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
```
```   487 by (simp add: cfcomp1)
```
```   488
```
```   489 lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
```
```   490 by (simp add: cfcomp1)
```
```   491
```
```   492 lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>"
```
```   493 by (simp add: expand_cfun_eq)
```
```   494
```
```   495 text {*
```
```   496   Show that interpretation of (pcpo,@{text "_->_"}) is a category.
```
```   497   The class of objects is interpretation of syntactical class pcpo.
```
```   498   The class of arrows  between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}.
```
```   499   The identity arrow is interpretation of @{term ID}.
```
```   500   The composition of f and g is interpretation of @{text "oo"}.
```
```   501 *}
```
```   502
```
```   503 lemma ID2 [simp]: "f oo ID = f"
```
```   504 by (rule ext_cfun, simp)
```
```   505
```
```   506 lemma ID3 [simp]: "ID oo f = f"
```
```   507 by (rule ext_cfun, simp)
```
```   508
```
```   509 lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
```
```   510 by (rule ext_cfun, simp)
```
```   511
```
```   512
```
```   513 subsection {* Strictified functions *}
```
```   514
```
```   515 defaultsort pcpo
```
```   516
```
```   517 definition
```
```   518   strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
```
```   519   "strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
```
```   520
```
```   521 text {* results about strictify *}
```
```   522
```
```   523 lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
```
```   524 by simp
```
```   525
```
```   526 lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
```
```   527 apply (rule monofunI)
```
```   528 apply (auto simp add: monofun_cfun_arg)
```
```   529 done
```
```   530
```
```   531 (*FIXME: long proof*)
```
```   532 lemma contlub_strictify2: "contlub (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
```
```   533 apply (rule contlubI)
```
```   534 apply (case_tac "(\<Squnion>i. Y i) = \<bottom>")
```
```   535 apply (drule (1) chain_UU_I)
```
```   536 apply simp
```
```   537 apply (simp del: if_image_distrib)
```
```   538 apply (simp only: contlub_cfun_arg)
```
```   539 apply (rule lub_equal2)
```
```   540 apply (rule chain_mono2 [THEN exE])
```
```   541 apply (erule chain_UU_I_inverse2)
```
```   542 apply (assumption)
```
```   543 apply (rule_tac x=x in exI, clarsimp)
```
```   544 apply (erule chain_monofun)
```
```   545 apply (erule monofun_strictify2 [THEN ch2ch_monofun])
```
```   546 done
```
```   547
```
```   548 lemmas cont_strictify2 =
```
```   549   monocontlub2cont [OF monofun_strictify2 contlub_strictify2, standard]
```
```   550
```
```   551 lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
```
```   552   unfolding strictify_def
```
```   553   by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM)
```
```   554
```
```   555 lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
```
```   556 by (simp add: strictify_conv_if)
```
```   557
```
```   558 lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x"
```
```   559 by (simp add: strictify_conv_if)
```
```   560
```
```   561 subsection {* Continuous let-bindings *}
```
```   562
```
```   563 definition
```
```   564   CLet :: "'a \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'b" where
```
```   565   "CLet = (\<Lambda> s f. f\<cdot>s)"
```
```   566
```
```   567 syntax
```
```   568   "_CLet" :: "[letbinds, 'a] => 'a" ("(Let (_)/ in (_))" 10)
```
```   569
```
```   570 translations
```
```   571   "_CLet (_binds b bs) e" == "_CLet b (_CLet bs e)"
```
```   572   "Let x = a in e" == "CONST CLet\<cdot>a\<cdot>(\<Lambda> x. e)"
```
```   573
```
```   574 end
```