src/HOL/HOLCF/Cfun.thy
 author wenzelm Sat Aug 16 11:35:33 2014 +0200 (2014-08-16) changeset 57945 cacb00a569e0 parent 52143 36ffe23b25f8 child 57954 c52223cd1003 permissions -rw-r--r--
prefer 'named_theorems' over Named_Thms, with subtle change of semantics due to visual order vs. internal reverse order;
```     1 (*  Title:      HOL/HOLCF/Cfun.thy
```
```     2     Author:     Franz Regensburger
```
```     3     Author:     Brian Huffman
```
```     4 *)
```
```     5
```
```     6 header {* The type of continuous functions *}
```
```     7
```
```     8 theory Cfun
```
```     9 imports Cpodef Fun_Cpo Product_Cpo
```
```    10 begin
```
```    11
```
```    12 default_sort cpo
```
```    13
```
```    14 subsection {* Definition of continuous function type *}
```
```    15
```
```    16 definition "cfun = {f::'a => 'b. cont f}"
```
```    17
```
```    18 cpodef ('a, 'b) cfun (infixr "->" 0) = "cfun :: ('a => 'b) set"
```
```    19   unfolding cfun_def by (auto intro: cont_const adm_cont)
```
```    20
```
```    21 type_notation (xsymbols)
```
```    22   cfun  ("(_ \<rightarrow>/ _)" [1, 0] 0)
```
```    23
```
```    24 notation
```
```    25   Rep_cfun  ("(_\$/_)" [999,1000] 999)
```
```    26
```
```    27 notation (xsymbols)
```
```    28   Rep_cfun  ("(_\<cdot>/_)" [999,1000] 999)
```
```    29
```
```    30 notation (HTML output)
```
```    31   Rep_cfun  ("(_\<cdot>/_)" [999,1000] 999)
```
```    32
```
```    33 subsection {* Syntax for continuous lambda abstraction *}
```
```    34
```
```    35 syntax "_cabs" :: "[logic, logic] \<Rightarrow> logic"
```
```    36
```
```    37 parse_translation {*
```
```    38 (* rewrite (_cabs x t) => (Abs_cfun (%x. t)) *)
```
```    39   [Syntax_Trans.mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_cfun})];
```
```    40 *}
```
```    41
```
```    42 print_translation {*
```
```    43   [(@{const_syntax Abs_cfun}, fn _ => fn [Abs abs] =>
```
```    44       let val (x, t) = Syntax_Trans.atomic_abs_tr' abs
```
```    45       in Syntax.const @{syntax_const "_cabs"} \$ x \$ t end)]
```
```    46 *}  -- {* To avoid eta-contraction of body *}
```
```    47
```
```    48 text {* Syntax for nested abstractions *}
```
```    49
```
```    50 syntax
```
```    51   "_Lambda" :: "[cargs, logic] \<Rightarrow> logic"  ("(3LAM _./ _)" [1000, 10] 10)
```
```    52
```
```    53 syntax (xsymbols)
```
```    54   "_Lambda" :: "[cargs, logic] \<Rightarrow> logic" ("(3\<Lambda> _./ _)" [1000, 10] 10)
```
```    55
```
```    56 parse_ast_translation {*
```
```    57 (* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
```
```    58 (* cf. Syntax.lambda_ast_tr from src/Pure/Syntax/syn_trans.ML *)
```
```    59   let
```
```    60     fun Lambda_ast_tr [pats, body] =
```
```    61           Ast.fold_ast_p @{syntax_const "_cabs"}
```
```    62             (Ast.unfold_ast @{syntax_const "_cargs"} (Ast.strip_positions pats), body)
```
```    63       | Lambda_ast_tr asts = raise Ast.AST ("Lambda_ast_tr", asts);
```
```    64   in [(@{syntax_const "_Lambda"}, K Lambda_ast_tr)] end;
```
```    65 *}
```
```    66
```
```    67 print_ast_translation {*
```
```    68 (* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
```
```    69 (* cf. Syntax.abs_ast_tr' from src/Pure/Syntax/syn_trans.ML *)
```
```    70   let
```
```    71     fun cabs_ast_tr' asts =
```
```    72       (case Ast.unfold_ast_p @{syntax_const "_cabs"}
```
```    73           (Ast.Appl (Ast.Constant @{syntax_const "_cabs"} :: asts)) of
```
```    74         ([], _) => raise Ast.AST ("cabs_ast_tr'", asts)
```
```    75       | (xs, body) => Ast.Appl
```
```    76           [Ast.Constant @{syntax_const "_Lambda"},
```
```    77            Ast.fold_ast @{syntax_const "_cargs"} xs, body]);
```
```    78   in [(@{syntax_const "_cabs"}, K cabs_ast_tr')] end
```
```    79 *}
```
```    80
```
```    81 text {* Dummy patterns for continuous abstraction *}
```
```    82 translations
```
```    83   "\<Lambda> _. t" => "CONST Abs_cfun (\<lambda> _. t)"
```
```    84
```
```    85 subsection {* Continuous function space is pointed *}
```
```    86
```
```    87 lemma bottom_cfun: "\<bottom> \<in> cfun"
```
```    88 by (simp add: cfun_def inst_fun_pcpo)
```
```    89
```
```    90 instance cfun :: (cpo, discrete_cpo) discrete_cpo
```
```    91 by intro_classes (simp add: below_cfun_def Rep_cfun_inject)
```
```    92
```
```    93 instance cfun :: (cpo, pcpo) pcpo
```
```    94 by (rule typedef_pcpo [OF type_definition_cfun below_cfun_def bottom_cfun])
```
```    95
```
```    96 lemmas Rep_cfun_strict =
```
```    97   typedef_Rep_strict [OF type_definition_cfun below_cfun_def bottom_cfun]
```
```    98
```
```    99 lemmas Abs_cfun_strict =
```
```   100   typedef_Abs_strict [OF type_definition_cfun below_cfun_def bottom_cfun]
```
```   101
```
```   102 text {* function application is strict in its first argument *}
```
```   103
```
```   104 lemma Rep_cfun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
```
```   105 by (simp add: Rep_cfun_strict)
```
```   106
```
```   107 lemma LAM_strict [simp]: "(\<Lambda> x. \<bottom>) = \<bottom>"
```
```   108 by (simp add: inst_fun_pcpo [symmetric] Abs_cfun_strict)
```
```   109
```
```   110 text {* for compatibility with old HOLCF-Version *}
```
```   111 lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
```
```   112 by simp
```
```   113
```
```   114 subsection {* Basic properties of continuous functions *}
```
```   115
```
```   116 text {* Beta-equality for continuous functions *}
```
```   117
```
```   118 lemma Abs_cfun_inverse2: "cont f \<Longrightarrow> Rep_cfun (Abs_cfun f) = f"
```
```   119 by (simp add: Abs_cfun_inverse cfun_def)
```
```   120
```
```   121 lemma beta_cfun: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
```
```   122 by (simp add: Abs_cfun_inverse2)
```
```   123
```
```   124 text {* Beta-reduction simproc *}
```
```   125
```
```   126 text {*
```
```   127   Given the term @{term "(\<Lambda> x. f x)\<cdot>y"}, the procedure tries to
```
```   128   construct the theorem @{term "(\<Lambda> x. f x)\<cdot>y == f y"}.  If this
```
```   129   theorem cannot be completely solved by the cont2cont rules, then
```
```   130   the procedure returns the ordinary conditional @{text beta_cfun}
```
```   131   rule.
```
```   132
```
```   133   The simproc does not solve any more goals that would be solved by
```
```   134   using @{text beta_cfun} as a simp rule.  The advantage of the
```
```   135   simproc is that it can avoid deeply-nested calls to the simplifier
```
```   136   that would otherwise be caused by large continuity side conditions.
```
```   137
```
```   138   Update: The simproc now uses rule @{text Abs_cfun_inverse2} instead
```
```   139   of @{text beta_cfun}, to avoid problems with eta-contraction.
```
```   140 *}
```
```   141
```
```   142 simproc_setup beta_cfun_proc ("Rep_cfun (Abs_cfun f)") = {*
```
```   143   fn phi => fn ctxt => fn ct =>
```
```   144     let
```
```   145       val dest = Thm.dest_comb;
```
```   146       val f = (snd o dest o snd o dest) ct;
```
```   147       val [T, U] = Thm.dest_ctyp (ctyp_of_term f);
```
```   148       val tr = instantiate' [SOME T, SOME U] [SOME f]
```
```   149           (mk_meta_eq @{thm Abs_cfun_inverse2});
```
```   150       val rules = Named_Theorems.get ctxt @{named_theorems cont2cont};
```
```   151       val tac = SOLVED' (REPEAT_ALL_NEW (match_tac rules));
```
```   152     in SOME (perhaps (SINGLE (tac 1)) tr) end
```
```   153 *}
```
```   154
```
```   155 text {* Eta-equality for continuous functions *}
```
```   156
```
```   157 lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
```
```   158 by (rule Rep_cfun_inverse)
```
```   159
```
```   160 text {* Extensionality for continuous functions *}
```
```   161
```
```   162 lemma cfun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f\<cdot>x = g\<cdot>x)"
```
```   163 by (simp add: Rep_cfun_inject [symmetric] fun_eq_iff)
```
```   164
```
```   165 lemma cfun_eqI: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
```
```   166 by (simp add: cfun_eq_iff)
```
```   167
```
```   168 text {* Extensionality wrt. ordering for continuous functions *}
```
```   169
```
```   170 lemma cfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)"
```
```   171 by (simp add: below_cfun_def fun_below_iff)
```
```   172
```
```   173 lemma cfun_belowI: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
```
```   174 by (simp add: cfun_below_iff)
```
```   175
```
```   176 text {* Congruence for continuous function application *}
```
```   177
```
```   178 lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y"
```
```   179 by simp
```
```   180
```
```   181 lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x"
```
```   182 by simp
```
```   183
```
```   184 lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
```
```   185 by simp
```
```   186
```
```   187 subsection {* Continuity of application *}
```
```   188
```
```   189 lemma cont_Rep_cfun1: "cont (\<lambda>f. f\<cdot>x)"
```
```   190 by (rule cont_Rep_cfun [OF cont_id, THEN cont2cont_fun])
```
```   191
```
```   192 lemma cont_Rep_cfun2: "cont (\<lambda>x. f\<cdot>x)"
```
```   193 apply (cut_tac x=f in Rep_cfun)
```
```   194 apply (simp add: cfun_def)
```
```   195 done
```
```   196
```
```   197 lemmas monofun_Rep_cfun = cont_Rep_cfun [THEN cont2mono]
```
```   198
```
```   199 lemmas monofun_Rep_cfun1 = cont_Rep_cfun1 [THEN cont2mono]
```
```   200 lemmas monofun_Rep_cfun2 = cont_Rep_cfun2 [THEN cont2mono]
```
```   201
```
```   202 text {* contlub, cont properties of @{term Rep_cfun} in each argument *}
```
```   203
```
```   204 lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
```
```   205 by (rule cont_Rep_cfun2 [THEN cont2contlubE])
```
```   206
```
```   207 lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
```
```   208 by (rule cont_Rep_cfun1 [THEN cont2contlubE])
```
```   209
```
```   210 text {* monotonicity of application *}
```
```   211
```
```   212 lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
```
```   213 by (simp add: cfun_below_iff)
```
```   214
```
```   215 lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
```
```   216 by (rule monofun_Rep_cfun2 [THEN monofunE])
```
```   217
```
```   218 lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
```
```   219 by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])
```
```   220
```
```   221 text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
```
```   222
```
```   223 lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
```
```   224 by (erule monofun_Rep_cfun2 [THEN ch2ch_monofun])
```
```   225
```
```   226 lemma ch2ch_Rep_cfunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
```
```   227 by (rule monofun_Rep_cfun2 [THEN ch2ch_monofun])
```
```   228
```
```   229 lemma ch2ch_Rep_cfunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
```
```   230 by (rule monofun_Rep_cfun1 [THEN ch2ch_monofun])
```
```   231
```
```   232 lemma ch2ch_Rep_cfun [simp]:
```
```   233   "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
```
```   234 by (simp add: chain_def monofun_cfun)
```
```   235
```
```   236 lemma ch2ch_LAM [simp]:
```
```   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)"
```
```   238 by (simp add: chain_def cfun_below_iff)
```
```   239
```
```   240 text {* contlub, cont properties of @{term Rep_cfun} in both arguments *}
```
```   241
```
```   242 lemma lub_APP:
```
```   243   "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i\<cdot>(Y i)) = (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
```
```   244 by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)
```
```   245
```
```   246 lemma lub_LAM:
```
```   247   "\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk>
```
```   248     \<Longrightarrow> (\<Squnion>i. \<Lambda> x. F i x) = (\<Lambda> x. \<Squnion>i. F i x)"
```
```   249 by (simp add: lub_cfun lub_fun ch2ch_lambda)
```
```   250
```
```   251 lemmas lub_distribs = lub_APP lub_LAM
```
```   252
```
```   253 text {* strictness *}
```
```   254
```
```   255 lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
```
```   256 apply (rule bottomI)
```
```   257 apply (erule subst)
```
```   258 apply (rule minimal [THEN monofun_cfun_arg])
```
```   259 done
```
```   260
```
```   261 text {* type @{typ "'a -> 'b"} is chain complete *}
```
```   262
```
```   263 lemma lub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
```
```   264 by (simp add: lub_cfun lub_fun ch2ch_lambda)
```
```   265
```
```   266 subsection {* Continuity simplification procedure *}
```
```   267
```
```   268 text {* cont2cont lemma for @{term Rep_cfun} *}
```
```   269
```
```   270 lemma cont2cont_APP [simp, cont2cont]:
```
```   271   assumes f: "cont (\<lambda>x. f x)"
```
```   272   assumes t: "cont (\<lambda>x. t x)"
```
```   273   shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
```
```   274 proof -
```
```   275   have 1: "\<And>y. cont (\<lambda>x. (f x)\<cdot>y)"
```
```   276     using cont_Rep_cfun1 f by (rule cont_compose)
```
```   277   show "cont (\<lambda>x. (f x)\<cdot>(t x))"
```
```   278     using t cont_Rep_cfun2 1 by (rule cont_apply)
```
```   279 qed
```
```   280
```
```   281 text {*
```
```   282   Two specific lemmas for the combination of LCF and HOL terms.
```
```   283   These lemmas are needed in theories that use types like @{typ "'a \<rightarrow> 'b \<Rightarrow> 'c"}.
```
```   284 *}
```
```   285
```
```   286 lemma cont_APP_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s)"
```
```   287 by (rule cont2cont_APP [THEN cont2cont_fun])
```
```   288
```
```   289 lemma cont_APP_app_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s t)"
```
```   290 by (rule cont_APP_app [THEN cont2cont_fun])
```
```   291
```
```   292
```
```   293 text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
```
```   294
```
```   295 lemma cont2mono_LAM:
```
```   296   "\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk>
```
```   297     \<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)"
```
```   298   unfolding monofun_def cfun_below_iff by simp
```
```   299
```
```   300 text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
```
```   301
```
```   302 text {*
```
```   303   Not suitable as a cont2cont rule, because on nested lambdas
```
```   304   it causes exponential blow-up in the number of subgoals.
```
```   305 *}
```
```   306
```
```   307 lemma cont2cont_LAM:
```
```   308   assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
```
```   309   assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
```
```   310   shows "cont (\<lambda>x. \<Lambda> y. f x y)"
```
```   311 proof (rule cont_Abs_cfun)
```
```   312   fix x
```
```   313   from f1 show "f x \<in> cfun" by (simp add: cfun_def)
```
```   314   from f2 show "cont f" by (rule cont2cont_lambda)
```
```   315 qed
```
```   316
```
```   317 text {*
```
```   318   This version does work as a cont2cont rule, since it
```
```   319   has only a single subgoal.
```
```   320 *}
```
```   321
```
```   322 lemma cont2cont_LAM' [simp, cont2cont]:
```
```   323   fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
```
```   324   assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
```
```   325   shows "cont (\<lambda>x. \<Lambda> y. f x y)"
```
```   326 using assms by (simp add: cont2cont_LAM prod_cont_iff)
```
```   327
```
```   328 lemma cont2cont_LAM_discrete [simp, cont2cont]:
```
```   329   "(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)"
```
```   330 by (simp add: cont2cont_LAM)
```
```   331
```
```   332 subsection {* Miscellaneous *}
```
```   333
```
```   334 text {* Monotonicity of @{term Abs_cfun} *}
```
```   335
```
```   336 lemma monofun_LAM:
```
```   337   "\<lbrakk>cont f; cont g; \<And>x. f x \<sqsubseteq> g x\<rbrakk> \<Longrightarrow> (\<Lambda> x. f x) \<sqsubseteq> (\<Lambda> x. g x)"
```
```   338 by (simp add: cfun_below_iff)
```
```   339
```
```   340 text {* some lemmata for functions with flat/chfin domain/range types *}
```
```   341
```
```   342 lemma chfin_Rep_cfunR: "chain (Y::nat => 'a::cpo->'b::chfin)
```
```   343       ==> !s. ? n. (LUB i. Y i)\$s = Y n\$s"
```
```   344 apply (rule allI)
```
```   345 apply (subst contlub_cfun_fun)
```
```   346 apply assumption
```
```   347 apply (fast intro!: lub_eqI chfin lub_finch2 chfin2finch ch2ch_Rep_cfunL)
```
```   348 done
```
```   349
```
```   350 lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))"
```
```   351 by (rule adm_subst, simp, rule adm_chfin)
```
```   352
```
```   353 subsection {* Continuous injection-retraction pairs *}
```
```   354
```
```   355 text {* Continuous retractions are strict. *}
```
```   356
```
```   357 lemma retraction_strict:
```
```   358   "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
```
```   359 apply (rule bottomI)
```
```   360 apply (drule_tac x="\<bottom>" in spec)
```
```   361 apply (erule subst)
```
```   362 apply (rule monofun_cfun_arg)
```
```   363 apply (rule minimal)
```
```   364 done
```
```   365
```
```   366 lemma injection_eq:
```
```   367   "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)"
```
```   368 apply (rule iffI)
```
```   369 apply (drule_tac f=f in cfun_arg_cong)
```
```   370 apply simp
```
```   371 apply simp
```
```   372 done
```
```   373
```
```   374 lemma injection_below:
```
```   375   "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
```
```   376 apply (rule iffI)
```
```   377 apply (drule_tac f=f in monofun_cfun_arg)
```
```   378 apply simp
```
```   379 apply (erule monofun_cfun_arg)
```
```   380 done
```
```   381
```
```   382 lemma injection_defined_rev:
```
```   383   "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>"
```
```   384 apply (drule_tac f=f in cfun_arg_cong)
```
```   385 apply (simp add: retraction_strict)
```
```   386 done
```
```   387
```
```   388 lemma injection_defined:
```
```   389   "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>"
```
```   390 by (erule contrapos_nn, rule injection_defined_rev)
```
```   391
```
```   392 text {* a result about functions with flat codomain *}
```
```   393
```
```   394 lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
```
```   395 by (drule ax_flat, simp)
```
```   396
```
```   397 lemma flat_codom:
```
```   398   "f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
```
```   399 apply (case_tac "f\<cdot>x = \<bottom>")
```
```   400 apply (rule disjI1)
```
```   401 apply (rule bottomI)
```
```   402 apply (erule_tac t="\<bottom>" in subst)
```
```   403 apply (rule minimal [THEN monofun_cfun_arg])
```
```   404 apply clarify
```
```   405 apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
```
```   406 apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
```
```   407 apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
```
```   408 done
```
```   409
```
```   410 subsection {* Identity and composition *}
```
```   411
```
```   412 definition
```
```   413   ID :: "'a \<rightarrow> 'a" where
```
```   414   "ID = (\<Lambda> x. x)"
```
```   415
```
```   416 definition
```
```   417   cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where
```
```   418   oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))"
```
```   419
```
```   420 abbreviation
```
```   421   cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c"  (infixr "oo" 100)  where
```
```   422   "f oo g == cfcomp\<cdot>f\<cdot>g"
```
```   423
```
```   424 lemma ID1 [simp]: "ID\<cdot>x = x"
```
```   425 by (simp add: ID_def)
```
```   426
```
```   427 lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
```
```   428 by (simp add: oo_def)
```
```   429
```
```   430 lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
```
```   431 by (simp add: cfcomp1)
```
```   432
```
```   433 lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
```
```   434 by (simp add: cfcomp1)
```
```   435
```
```   436 lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>"
```
```   437 by (simp add: cfun_eq_iff)
```
```   438
```
```   439 text {*
```
```   440   Show that interpretation of (pcpo,@{text "_->_"}) is a category.
```
```   441   The class of objects is interpretation of syntactical class pcpo.
```
```   442   The class of arrows  between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}.
```
```   443   The identity arrow is interpretation of @{term ID}.
```
```   444   The composition of f and g is interpretation of @{text "oo"}.
```
```   445 *}
```
```   446
```
```   447 lemma ID2 [simp]: "f oo ID = f"
```
```   448 by (rule cfun_eqI, simp)
```
```   449
```
```   450 lemma ID3 [simp]: "ID oo f = f"
```
```   451 by (rule cfun_eqI, simp)
```
```   452
```
```   453 lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
```
```   454 by (rule cfun_eqI, simp)
```
```   455
```
```   456 subsection {* Strictified functions *}
```
```   457
```
```   458 default_sort pcpo
```
```   459
```
```   460 definition
```
```   461   seq :: "'a \<rightarrow> 'b \<rightarrow> 'b" where
```
```   462   "seq = (\<Lambda> x. if x = \<bottom> then \<bottom> else ID)"
```
```   463
```
```   464 lemma cont2cont_if_bottom [cont2cont, simp]:
```
```   465   assumes f: "cont (\<lambda>x. f x)" and g: "cont (\<lambda>x. g x)"
```
```   466   shows "cont (\<lambda>x. if f x = \<bottom> then \<bottom> else g x)"
```
```   467 proof (rule cont_apply [OF f])
```
```   468   show "\<And>x. cont (\<lambda>y. if y = \<bottom> then \<bottom> else g x)"
```
```   469     unfolding cont_def is_lub_def is_ub_def ball_simps
```
```   470     by (simp add: lub_eq_bottom_iff)
```
```   471   show "\<And>y. cont (\<lambda>x. if y = \<bottom> then \<bottom> else g x)"
```
```   472     by (simp add: g)
```
```   473 qed
```
```   474
```
```   475 lemma seq_conv_if: "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)"
```
```   476 unfolding seq_def by simp
```
```   477
```
```   478 lemma seq_simps [simp]:
```
```   479   "seq\<cdot>\<bottom> = \<bottom>"
```
```   480   "seq\<cdot>x\<cdot>\<bottom> = \<bottom>"
```
```   481   "x \<noteq> \<bottom> \<Longrightarrow> seq\<cdot>x = ID"
```
```   482 by (simp_all add: seq_conv_if)
```
```   483
```
```   484 definition
```
```   485   strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
```
```   486   "strictify = (\<Lambda> f x. seq\<cdot>x\<cdot>(f\<cdot>x))"
```
```   487
```
```   488 lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
```
```   489 unfolding strictify_def by simp
```
```   490
```
```   491 lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
```
```   492 by (simp add: strictify_conv_if)
```
```   493
```
```   494 lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x"
```
```   495 by (simp add: strictify_conv_if)
```
```   496
```
```   497 subsection {* Continuity of let-bindings *}
```
```   498
```
```   499 lemma cont2cont_Let:
```
```   500   assumes f: "cont (\<lambda>x. f x)"
```
```   501   assumes g1: "\<And>y. cont (\<lambda>x. g x y)"
```
```   502   assumes g2: "\<And>x. cont (\<lambda>y. g x y)"
```
```   503   shows "cont (\<lambda>x. let y = f x in g x y)"
```
```   504 unfolding Let_def using f g2 g1 by (rule cont_apply)
```
```   505
```
```   506 lemma cont2cont_Let' [simp, cont2cont]:
```
```   507   assumes f: "cont (\<lambda>x. f x)"
```
```   508   assumes g: "cont (\<lambda>p. g (fst p) (snd p))"
```
```   509   shows "cont (\<lambda>x. let y = f x in g x y)"
```
```   510 using f
```
```   511 proof (rule cont2cont_Let)
```
```   512   fix x show "cont (\<lambda>y. g x y)"
```
```   513     using g by (simp add: prod_cont_iff)
```
```   514 next
```
```   515   fix y show "cont (\<lambda>x. g x y)"
```
```   516     using g by (simp add: prod_cont_iff)
```
```   517 qed
```
```   518
```
```   519 text {* The simple version (suggested by Joachim Breitner) is needed if
```
```   520   the type of the defined term is not a cpo. *}
```
```   521
```
```   522 lemma cont2cont_Let_simple [simp, cont2cont]:
```
```   523   assumes "\<And>y. cont (\<lambda>x. g x y)"
```
```   524   shows "cont (\<lambda>x. let y = t in g x y)"
```
```   525 unfolding Let_def using assms .
```
```   526
```
```   527 end
```