src/HOLCF/Cfun.thy
author huffman
Tue Mar 23 10:07:39 2010 -0700 (2010-03-23)
changeset 35933 f135ebcc835c
parent 35914 91a7311177c4
child 36452 d37c6eed8117
permissions -rw-r--r--
remove continuous let-binding function CLet; add cont2cont rule ordinary Let
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(*  Title:      HOLCF/Cfun.thy
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    Author:     Franz Regensburger
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    Author:     Brian Huffman
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*)
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header {* The type of continuous functions *}
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theory Cfun
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imports Pcpodef Ffun Product_Cpo
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begin
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defaultsort cpo
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subsection {* Definition of continuous function type *}
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lemma Ex_cont: "\<exists>f. cont f"
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by (rule exI, rule cont_const)
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lemma adm_cont: "adm cont"
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by (rule admI, rule cont_lub_fun)
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cpodef (CFun)  ('a, 'b) cfun (infixr "->" 0) = "{f::'a => 'b. cont f}"
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by (simp_all add: Ex_cont adm_cont)
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type_notation (xsymbols)
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  cfun  ("(_ \<rightarrow>/ _)" [1, 0] 0)
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notation
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  Rep_CFun  ("(_$/_)" [999,1000] 999)
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notation (xsymbols)
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  Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
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notation (HTML output)
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  Rep_CFun  ("(_\<cdot>/_)" [999,1000] 999)
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subsection {* Syntax for continuous lambda abstraction *}
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syntax "_cabs" :: "'a"
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parse_translation {*
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(* rewrite (_cabs x t) => (Abs_CFun (%x. t)) *)
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  [mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_CFun})];
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*}
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text {* To avoid eta-contraction of body: *}
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typed_print_translation {*
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  let
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    fun cabs_tr' _ _ [Abs abs] = let
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          val (x,t) = atomic_abs_tr' abs
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        in Syntax.const @{syntax_const "_cabs"} $ x $ t end
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      | cabs_tr' _ T [t] = let
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          val xT = domain_type (domain_type T);
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          val abs' = ("x",xT,(incr_boundvars 1 t)$Bound 0);
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          val (x,t') = atomic_abs_tr' abs';
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        in Syntax.const @{syntax_const "_cabs"} $ x $ t' end;
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  in [(@{const_syntax Abs_CFun}, cabs_tr')] end;
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*}
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text {* Syntax for nested abstractions *}
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syntax
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  "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic"  ("(3LAM _./ _)" [1000, 10] 10)
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syntax (xsymbols)
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  "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic" ("(3\<Lambda> _./ _)" [1000, 10] 10)
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parse_ast_translation {*
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(* rewrite (LAM x y z. t) => (_cabs x (_cabs y (_cabs z t))) *)
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(* cf. Syntax.lambda_ast_tr from src/Pure/Syntax/syn_trans.ML *)
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  let
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    fun Lambda_ast_tr [pats, body] =
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          Syntax.fold_ast_p @{syntax_const "_cabs"}
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            (Syntax.unfold_ast @{syntax_const "_cargs"} pats, body)
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      | Lambda_ast_tr asts = raise Syntax.AST ("Lambda_ast_tr", asts);
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  in [(@{syntax_const "_Lambda"}, Lambda_ast_tr)] end;
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*}
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print_ast_translation {*
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(* rewrite (_cabs x (_cabs y (_cabs z t))) => (LAM x y z. t) *)
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(* cf. Syntax.abs_ast_tr' from src/Pure/Syntax/syn_trans.ML *)
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  let
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    fun cabs_ast_tr' asts =
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      (case Syntax.unfold_ast_p @{syntax_const "_cabs"}
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          (Syntax.Appl (Syntax.Constant @{syntax_const "_cabs"} :: asts)) of
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        ([], _) => raise Syntax.AST ("cabs_ast_tr'", asts)
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      | (xs, body) => Syntax.Appl
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          [Syntax.Constant @{syntax_const "_Lambda"},
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           Syntax.fold_ast @{syntax_const "_cargs"} xs, body]);
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  in [(@{syntax_const "_cabs"}, cabs_ast_tr')] end
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*}
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text {* Dummy patterns for continuous abstraction *}
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translations
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  "\<Lambda> _. t" => "CONST Abs_CFun (\<lambda> _. t)"
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subsection {* Continuous function space is pointed *}
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lemma UU_CFun: "\<bottom> \<in> CFun"
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by (simp add: CFun_def inst_fun_pcpo cont_const)
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instance cfun :: (finite_po, finite_po) finite_po
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by (rule typedef_finite_po [OF type_definition_CFun])
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instance cfun :: (finite_po, chfin) chfin
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by (rule typedef_chfin [OF type_definition_CFun below_CFun_def])
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instance cfun :: (cpo, discrete_cpo) discrete_cpo
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by intro_classes (simp add: below_CFun_def Rep_CFun_inject)
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instance cfun :: (cpo, pcpo) pcpo
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by (rule typedef_pcpo [OF type_definition_CFun below_CFun_def UU_CFun])
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lemmas Rep_CFun_strict =
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  typedef_Rep_strict [OF type_definition_CFun below_CFun_def UU_CFun]
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lemmas Abs_CFun_strict =
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  typedef_Abs_strict [OF type_definition_CFun below_CFun_def UU_CFun]
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text {* function application is strict in its first argument *}
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lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
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by (simp add: Rep_CFun_strict)
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lemma LAM_strict [simp]: "(\<Lambda> x. \<bottom>) = \<bottom>"
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by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)
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text {* for compatibility with old HOLCF-Version *}
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lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
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by simp
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subsection {* Basic properties of continuous functions *}
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text {* Beta-equality for continuous functions *}
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lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f"
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by (simp add: Abs_CFun_inverse CFun_def)
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lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
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by (simp add: Abs_CFun_inverse2)
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text {* Eta-equality for continuous functions *}
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lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
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by (rule Rep_CFun_inverse)
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text {* Extensionality for continuous functions *}
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lemma expand_cfun_eq: "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
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by (simp add: Rep_CFun_inject [symmetric] expand_fun_eq)
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lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
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by (simp add: expand_cfun_eq)
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text {* Extensionality wrt. ordering for continuous functions *}
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lemma expand_cfun_below: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
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by (simp add: below_CFun_def expand_fun_below)
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lemma below_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
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by (simp add: expand_cfun_below)
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text {* Congruence for continuous function application *}
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lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y"
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by simp
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lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x"
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by simp
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lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
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by simp
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subsection {* Continuity of application *}
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lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
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by (rule cont_Rep_CFun [THEN cont2cont_fun])
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lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
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apply (cut_tac x=f in Rep_CFun)
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apply (simp add: CFun_def)
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done
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lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono]
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lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard]
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lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
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text {* contlub, cont properties of @{term Rep_CFun} in each argument *}
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lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))"
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by (rule cont_Rep_CFun2 [THEN cont2contlubE])
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lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(\<Squnion>i. Y i)"
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by (rule cont_Rep_CFun2 [THEN contE])
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lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
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by (rule cont_Rep_CFun1 [THEN cont2contlubE])
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lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| (\<Squnion>i. F i)\<cdot>x"
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by (rule cont_Rep_CFun1 [THEN contE])
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text {* monotonicity of application *}
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lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
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by (simp add: expand_cfun_below)
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lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
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by (rule monofun_Rep_CFun2 [THEN monofunE])
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lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
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by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg])
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text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
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lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
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by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
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lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
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by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
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lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
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by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])
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lemma ch2ch_Rep_CFun [simp]:
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  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
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by (simp add: chain_def monofun_cfun)
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lemma ch2ch_LAM [simp]:
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  "\<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)"
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by (simp add: chain_def expand_cfun_below)
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text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
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lemma contlub_cfun: 
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  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
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by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)
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lemma cont_cfun: 
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  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
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apply (rule thelubE)
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apply (simp only: ch2ch_Rep_CFun)
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apply (simp only: contlub_cfun)
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done
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lemma contlub_LAM:
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  "\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk>
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    \<Longrightarrow> (\<Lambda> x. \<Squnion>i. F i x) = (\<Squnion>i. \<Lambda> x. F i x)"
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apply (simp add: thelub_CFun)
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apply (simp add: Abs_CFun_inverse2)
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apply (simp add: thelub_fun ch2ch_lambda)
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done
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lemmas lub_distribs = 
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  contlub_cfun [symmetric]
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  contlub_LAM [symmetric]
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text {* strictness *}
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lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
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apply (rule UU_I)
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apply (erule subst)
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apply (rule minimal [THEN monofun_cfun_arg])
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done
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text {* the lub of a chain of continous functions is monotone *}
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lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
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apply (drule ch2ch_monofun [OF monofun_Rep_CFun])
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apply (simp add: thelub_fun [symmetric])
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apply (erule monofun_lub_fun)
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apply (simp add: monofun_Rep_CFun2)
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done
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text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *}
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lemma ex_lub_cfun:
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  "\<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))"
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by (simp add: diag_lub)
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text {* the lub of a chain of cont. functions is continuous *}
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lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
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apply (rule cont2cont_lub)
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apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
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apply (rule cont_Rep_CFun2)
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done
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text {* type @{typ "'a -> 'b"} is chain complete *}
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lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
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by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE)
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lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
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by (rule lub_cfun [THEN thelubI])
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subsection {* Continuity simplification procedure *}
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text {* cont2cont lemma for @{term Rep_CFun} *}
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lemma cont2cont_Rep_CFun [cont2cont]:
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  assumes f: "cont (\<lambda>x. f x)"
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  assumes t: "cont (\<lambda>x. t x)"
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  shows "cont (\<lambda>x. (f x)\<cdot>(t x))"
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proof -
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  have "cont (\<lambda>x. Rep_CFun (f x))"
huffman@29049
   310
    using cont_Rep_CFun f by (rule cont2cont_app3)
huffman@29049
   311
  thus "cont (\<lambda>x. (f x)\<cdot>(t x))"
huffman@29049
   312
    using cont_Rep_CFun2 t by (rule cont2cont_app2)
huffman@29049
   313
qed
huffman@15576
   314
huffman@15589
   315
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
huffman@15576
   316
huffman@15576
   317
lemma cont2mono_LAM:
huffman@29049
   318
  "\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk>
huffman@29049
   319
    \<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)"
huffman@31076
   320
  unfolding monofun_def expand_cfun_below by simp
huffman@15576
   321
huffman@29049
   322
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *}
huffman@15576
   323
huffman@29530
   324
text {*
huffman@29530
   325
  Not suitable as a cont2cont rule, because on nested lambdas
huffman@29530
   326
  it causes exponential blow-up in the number of subgoals.
huffman@29530
   327
*}
huffman@29530
   328
huffman@15576
   329
lemma cont2cont_LAM:
huffman@29049
   330
  assumes f1: "\<And>x. cont (\<lambda>y. f x y)"
huffman@29049
   331
  assumes f2: "\<And>y. cont (\<lambda>x. f x y)"
huffman@29049
   332
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
huffman@29049
   333
proof (rule cont_Abs_CFun)
huffman@29049
   334
  fix x
huffman@29049
   335
  from f1 show "f x \<in> CFun" by (simp add: CFun_def)
huffman@29049
   336
  from f2 show "cont f" by (rule cont2cont_lambda)
huffman@29049
   337
qed
huffman@15576
   338
huffman@29530
   339
text {*
huffman@29530
   340
  This version does work as a cont2cont rule, since it
huffman@29530
   341
  has only a single subgoal.
huffman@29530
   342
*}
huffman@29530
   343
huffman@29530
   344
lemma cont2cont_LAM' [cont2cont]:
huffman@29530
   345
  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo"
huffman@29530
   346
  assumes f: "cont (\<lambda>p. f (fst p) (snd p))"
huffman@29530
   347
  shows "cont (\<lambda>x. \<Lambda> y. f x y)"
huffman@29530
   348
proof (rule cont2cont_LAM)
huffman@31041
   349
  fix x :: 'a show "cont (\<lambda>y. f x y)"
huffman@31041
   350
    using f by (rule cont_fst_snd_D2)
huffman@29530
   351
next
huffman@31041
   352
  fix y :: 'b show "cont (\<lambda>x. f x y)"
huffman@31041
   353
    using f by (rule cont_fst_snd_D1)
huffman@29530
   354
qed
huffman@29530
   355
huffman@29530
   356
lemma cont2cont_LAM_discrete [cont2cont]:
huffman@29530
   357
  "(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)"
huffman@29530
   358
by (simp add: cont2cont_LAM)
huffman@15576
   359
huffman@16055
   360
lemmas cont_lemmas1 =
huffman@16055
   361
  cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM
huffman@16055
   362
huffman@17832
   363
subsection {* Miscellaneous *}
huffman@17832
   364
huffman@17832
   365
text {* Monotonicity of @{term Abs_CFun} *}
huffman@15576
   366
huffman@17832
   367
lemma semi_monofun_Abs_CFun:
huffman@17832
   368
  "\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
huffman@31076
   369
by (simp add: below_CFun_def Abs_CFun_inverse2)
huffman@15576
   370
huffman@15589
   371
text {* some lemmata for functions with flat/chfin domain/range types *}
huffman@15576
   372
huffman@15576
   373
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)  
huffman@27413
   374
      ==> !s. ? n. (LUB i. Y i)$s = Y n$s"
huffman@15576
   375
apply (rule allI)
huffman@15576
   376
apply (subst contlub_cfun_fun)
huffman@15576
   377
apply assumption
huffman@15576
   378
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
huffman@15576
   379
done
huffman@15576
   380
huffman@18089
   381
lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))"
huffman@18089
   382
by (rule adm_subst, simp, rule adm_chfin)
huffman@18089
   383
huffman@16085
   384
subsection {* Continuous injection-retraction pairs *}
huffman@15589
   385
huffman@16085
   386
text {* Continuous retractions are strict. *}
huffman@15576
   387
huffman@16085
   388
lemma retraction_strict:
huffman@16085
   389
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
huffman@15576
   390
apply (rule UU_I)
huffman@16085
   391
apply (drule_tac x="\<bottom>" in spec)
huffman@16085
   392
apply (erule subst)
huffman@16085
   393
apply (rule monofun_cfun_arg)
huffman@16085
   394
apply (rule minimal)
huffman@15576
   395
done
huffman@15576
   396
huffman@16085
   397
lemma injection_eq:
huffman@16085
   398
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)"
huffman@16085
   399
apply (rule iffI)
huffman@16085
   400
apply (drule_tac f=f in cfun_arg_cong)
huffman@16085
   401
apply simp
huffman@16085
   402
apply simp
huffman@15576
   403
done
huffman@15576
   404
huffman@31076
   405
lemma injection_below:
huffman@16314
   406
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
huffman@16314
   407
apply (rule iffI)
huffman@16314
   408
apply (drule_tac f=f in monofun_cfun_arg)
huffman@16314
   409
apply simp
huffman@16314
   410
apply (erule monofun_cfun_arg)
huffman@16314
   411
done
huffman@16314
   412
huffman@16085
   413
lemma injection_defined_rev:
huffman@16085
   414
  "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>"
huffman@16085
   415
apply (drule_tac f=f in cfun_arg_cong)
huffman@16085
   416
apply (simp add: retraction_strict)
huffman@15576
   417
done
huffman@15576
   418
huffman@16085
   419
lemma injection_defined:
huffman@16085
   420
  "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>"
huffman@16085
   421
by (erule contrapos_nn, rule injection_defined_rev)
huffman@16085
   422
huffman@16085
   423
text {* propagation of flatness and chain-finiteness by retractions *}
huffman@16085
   424
huffman@16085
   425
lemma chfin2chfin:
huffman@16085
   426
  "\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y
huffman@16085
   427
    \<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
huffman@16085
   428
apply clarify
huffman@16085
   429
apply (drule_tac f=g in chain_monofun)
huffman@25921
   430
apply (drule chfin)
huffman@16085
   431
apply (unfold max_in_chain_def)
huffman@16085
   432
apply (simp add: injection_eq)
huffman@16085
   433
done
huffman@16085
   434
huffman@16085
   435
lemma flat2flat:
huffman@16085
   436
  "\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y
huffman@16085
   437
    \<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y"
huffman@16085
   438
apply clarify
huffman@16209
   439
apply (drule_tac f=g in monofun_cfun_arg)
huffman@25920
   440
apply (drule ax_flat)
huffman@16085
   441
apply (erule disjE)
huffman@16085
   442
apply (simp add: injection_defined_rev)
huffman@16085
   443
apply (simp add: injection_eq)
huffman@15576
   444
done
huffman@15576
   445
huffman@15589
   446
text {* a result about functions with flat codomain *}
huffman@15576
   447
huffman@16085
   448
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
huffman@25920
   449
by (drule ax_flat, simp)
huffman@16085
   450
huffman@16085
   451
lemma flat_codom:
huffman@16085
   452
  "f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
huffman@16085
   453
apply (case_tac "f\<cdot>x = \<bottom>")
huffman@15576
   454
apply (rule disjI1)
huffman@15576
   455
apply (rule UU_I)
huffman@16085
   456
apply (erule_tac t="\<bottom>" in subst)
huffman@15576
   457
apply (rule minimal [THEN monofun_cfun_arg])
huffman@16085
   458
apply clarify
huffman@16085
   459
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
huffman@16085
   460
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
huffman@16085
   461
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
huffman@15589
   462
done
huffman@15589
   463
huffman@15589
   464
huffman@15589
   465
subsection {* Identity and composition *}
huffman@15589
   466
wenzelm@25135
   467
definition
wenzelm@25135
   468
  ID :: "'a \<rightarrow> 'a" where
wenzelm@25135
   469
  "ID = (\<Lambda> x. x)"
wenzelm@25135
   470
wenzelm@25135
   471
definition
wenzelm@25135
   472
  cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where
wenzelm@25135
   473
  oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))"
huffman@15589
   474
wenzelm@25131
   475
abbreviation
wenzelm@25131
   476
  cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c"  (infixr "oo" 100)  where
wenzelm@25131
   477
  "f oo g == cfcomp\<cdot>f\<cdot>g"
huffman@15589
   478
huffman@16085
   479
lemma ID1 [simp]: "ID\<cdot>x = x"
huffman@16085
   480
by (simp add: ID_def)
huffman@15576
   481
huffman@16085
   482
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
huffman@15589
   483
by (simp add: oo_def)
huffman@15576
   484
huffman@16085
   485
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
huffman@15589
   486
by (simp add: cfcomp1)
huffman@15576
   487
huffman@27274
   488
lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))"
huffman@27274
   489
by (simp add: cfcomp1)
huffman@27274
   490
huffman@19709
   491
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>"
huffman@19709
   492
by (simp add: expand_cfun_eq)
huffman@19709
   493
huffman@15589
   494
text {*
huffman@15589
   495
  Show that interpretation of (pcpo,@{text "_->_"}) is a category.
huffman@15589
   496
  The class of objects is interpretation of syntactical class pcpo.
huffman@15589
   497
  The class of arrows  between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}.
huffman@15589
   498
  The identity arrow is interpretation of @{term ID}.
huffman@15589
   499
  The composition of f and g is interpretation of @{text "oo"}.
huffman@15589
   500
*}
huffman@15576
   501
huffman@16085
   502
lemma ID2 [simp]: "f oo ID = f"
huffman@15589
   503
by (rule ext_cfun, simp)
huffman@15576
   504
huffman@16085
   505
lemma ID3 [simp]: "ID oo f = f"
huffman@15589
   506
by (rule ext_cfun, simp)
huffman@15576
   507
huffman@15576
   508
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
huffman@15589
   509
by (rule ext_cfun, simp)
huffman@15576
   510
huffman@16085
   511
huffman@16085
   512
subsection {* Strictified functions *}
huffman@16085
   513
huffman@16085
   514
defaultsort pcpo
huffman@16085
   515
wenzelm@25131
   516
definition
wenzelm@25131
   517
  strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where
wenzelm@25131
   518
  "strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@16085
   519
huffman@16085
   520
text {* results about strictify *}
huffman@16085
   521
huffman@17815
   522
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@35168
   523
by simp
huffman@16085
   524
huffman@17815
   525
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@17815
   526
apply (rule monofunI)
huffman@25786
   527
apply (auto simp add: monofun_cfun_arg)
huffman@16085
   528
done
huffman@16085
   529
huffman@35914
   530
lemma cont_strictify2: "cont (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@35914
   531
apply (rule contI2)
huffman@35914
   532
apply (rule monofun_strictify2)
huffman@35914
   533
apply (case_tac "(\<Squnion>i. Y i) = \<bottom>", simp)
huffman@35914
   534
apply (simp add: contlub_cfun_arg del: if_image_distrib)
huffman@35914
   535
apply (drule chain_UU_I_inverse2, clarify, rename_tac j)
huffman@35914
   536
apply (rule lub_mono2, rule_tac x=j in exI, simp_all)
huffman@35914
   537
apply (auto dest!: chain_mono_less)
huffman@16085
   538
done
huffman@16085
   539
huffman@17815
   540
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
huffman@29530
   541
  unfolding strictify_def
huffman@29530
   542
  by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM)
huffman@16085
   543
huffman@16085
   544
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
huffman@17815
   545
by (simp add: strictify_conv_if)
huffman@16085
   546
huffman@16085
   547
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x"
huffman@17815
   548
by (simp add: strictify_conv_if)
huffman@16085
   549
huffman@35933
   550
subsection {* Continuity of let-bindings *}
huffman@17816
   551
huffman@35933
   552
lemma cont2cont_Let:
huffman@35933
   553
  assumes f: "cont (\<lambda>x. f x)"
huffman@35933
   554
  assumes g1: "\<And>y. cont (\<lambda>x. g x y)"
huffman@35933
   555
  assumes g2: "\<And>x. cont (\<lambda>y. g x y)"
huffman@35933
   556
  shows "cont (\<lambda>x. let y = f x in g x y)"
huffman@35933
   557
unfolding Let_def using f g2 g1 by (rule cont_apply)
huffman@17816
   558
huffman@35933
   559
lemma cont2cont_Let' [cont2cont]:
huffman@35933
   560
  assumes f: "cont (\<lambda>x. f x)"
huffman@35933
   561
  assumes g: "cont (\<lambda>p. g (fst p) (snd p))"
huffman@35933
   562
  shows "cont (\<lambda>x. let y = f x in g x y)"
huffman@35933
   563
using f
huffman@35933
   564
proof (rule cont2cont_Let)
huffman@35933
   565
  fix x show "cont (\<lambda>y. g x y)"
huffman@35933
   566
    using g by (rule cont_fst_snd_D2)
huffman@35933
   567
next
huffman@35933
   568
  fix y show "cont (\<lambda>x. g x y)"
huffman@35933
   569
    using g by (rule cont_fst_snd_D1)
huffman@35933
   570
qed
huffman@17816
   571
huffman@15576
   572
end