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permissions  rwrr 
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(* Title: HOLCF/Cfun.thy 
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Author: Franz Regensburger 
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Definition of the type > of continuous functions. 
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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) ">" (infixr ">" 0) = "{f::'a => 'b. cont f}" 
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by (simp_all add: Ex_cont adm_cont) 
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syntax (xsymbols) 
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">" :: "[type, type] => type" ("(_ \<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 etacontraction 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 ">" :: (finite_po, finite_po) finite_po 
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by (rule typedef_finite_po [OF type_definition_CFun]) 
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instance ">" :: (finite_po, chfin) chfin 
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by (rule typedef_chfin [OF type_definition_CFun below_CFun_def]) 
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instance ">" :: (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 ">" :: (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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text {* for compatibility with old HOLCFVersion *} 
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lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)" 
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by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict) 
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subsection {* Basic properties of continuous functions *} 
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text {* Betaequality 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 {* Etaequality 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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160 

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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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163 

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164 
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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167 
by simp 
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168 

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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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178 
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 contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub] 
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lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard] 
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lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard] 
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lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard] 
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lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard] 
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192 

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text {* contlub, cont properties of @{term Rep_CFun} in each argument *} 
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27413  195 
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 contlub_Rep_CFun2 [THEN contlubE]) 
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27413  198 
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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27413  201 
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 contlub_Rep_CFun1 [THEN contlubE]) 
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27413  204 
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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208 

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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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211 

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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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214 

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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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219 

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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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222 

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lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
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224 
by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun]) 
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225 

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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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18076  229 
lemma ch2ch_Rep_CFun [simp]: 
230 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" 

25884  231 
by (simp add: chain_def monofun_cfun) 
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25884  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)" 

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by (simp add: chain_def expand_cfun_below) 
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236 

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text {* contlub, cont properties of @{term Rep_CFun} in both arguments *} 
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238 

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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))" 
18076  241 
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub) 
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242 

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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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245 
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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249 

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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)" 
25884  253 
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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257 

25901  258 
lemmas lub_distribs = 
259 
contlub_cfun [symmetric] 

260 
contlub_LAM [symmetric] 

261 

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text {* strictness *} 
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263 

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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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269 

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text {* the lub of a chain of continous functions is monotone *} 
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271 

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lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)" 
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273 
apply (drule ch2ch_monofun [OF monofun_Rep_CFun]) 
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274 
apply (simp add: thelub_fun [symmetric]) 
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275 
apply (erule monofun_lub_fun) 
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apply (simp add: monofun_Rep_CFun2) 
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done 
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16386  279 
text {* a lemma about the exchange of lubs for type @{typ "'a > 'b"} *} 
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16699  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))" 

18076  283 
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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288 
apply (rule cont2cont_lub) 
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289 
apply (erule monofun_Rep_CFun [THEN ch2ch_monofun]) 
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290 
apply (rule cont_Rep_CFun2) 
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291 
done 
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292 

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text {* type @{typ "'a > 'b"} is chain complete *} 
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294 

16920  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) 

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27413  298 
lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" 
16920  299 
by (rule lub_cfun [THEN thelubI]) 
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301 
subsection {* Continuity simplification procedure *} 
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302 

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text {* cont2cont lemma for @{term Rep_CFun} *} 
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lemma cont2cont_Rep_CFun [cont2cont]: 
29049  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 

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text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
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lemma cont2mono_LAM: 
29049  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)" 

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unfolding monofun_def expand_cfun_below by simp 
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29049  323 
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *} 
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324 

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text {* 
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326 
Not suitable as a cont2cont rule, because on nested lambdas 
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327 
it causes exponential blowup in the number of subgoals. 
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328 
*} 
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329 

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lemma cont2cont_LAM: 
29049  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 

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text {* 
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341 
This version does work as a cont2cont rule, since it 
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342 
has only a single subgoal. 
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343 
*} 
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344 

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345 
lemma cont2cont_LAM' [cont2cont]: 
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346 
fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" 
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347 
assumes f: "cont (\<lambda>p. f (fst p) (snd p))" 
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348 
shows "cont (\<lambda>x. \<Lambda> y. f x y)" 
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349 
proof (rule cont2cont_LAM) 
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350 
fix x :: 'a show "cont (\<lambda>y. f x y)" 
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351 
using f by (rule cont_fst_snd_D2) 
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352 
next 
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353 
fix y :: 'b show "cont (\<lambda>x. f x y)" 
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354 
using f by (rule cont_fst_snd_D1) 
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355 
qed 
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356 

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357 
lemma cont2cont_LAM_discrete [cont2cont]: 
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358 
"(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)" 
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359 
by (simp add: cont2cont_LAM) 
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360 

16055  361 
lemmas cont_lemmas1 = 
362 
cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM 

363 

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364 
subsection {* Miscellaneous *} 
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365 

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366 
text {* Monotonicity of @{term Abs_CFun} *} 
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367 

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368 
lemma semi_monofun_Abs_CFun: 
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369 
"\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g" 
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370 
by (simp add: below_CFun_def Abs_CFun_inverse2) 
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371 

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372 
text {* some lemmata for functions with flat/chfin domain/range types *} 
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373 

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374 
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
27413  375 
==> !s. ? n. (LUB i. Y i)$s = Y n$s" 
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376 
apply (rule allI) 
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377 
apply (subst contlub_cfun_fun) 
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378 
apply assumption 
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379 
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) 
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380 
done 
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381 

18089  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 

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385 
subsection {* Continuous injectionretraction pairs *} 
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386 

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387 
text {* Continuous retractions are strict. *} 
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388 

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389 
lemma retraction_strict: 
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390 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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391 
apply (rule UU_I) 
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392 
apply (drule_tac x="\<bottom>" in spec) 
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393 
apply (erule subst) 
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394 
apply (rule monofun_cfun_arg) 
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395 
apply (rule minimal) 
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396 
done 
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397 

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398 
lemma injection_eq: 
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399 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" 
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400 
apply (rule iffI) 
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401 
apply (drule_tac f=f in cfun_arg_cong) 
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402 
apply simp 
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403 
apply simp 
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404 
done 
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405 

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406 
lemma injection_below: 
16314  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 

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414 
lemma injection_defined_rev: 
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415 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" 
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416 
apply (drule_tac f=f in cfun_arg_cong) 
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417 
apply (simp add: retraction_strict) 
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418 
done 
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419 

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420 
lemma injection_defined: 
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421 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" 
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422 
by (erule contrapos_nn, rule injection_defined_rev) 
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423 

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424 
text {* propagation of flatness and chainfiniteness by retractions *} 
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425 

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426 
lemma chfin2chfin: 
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427 
"\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y 
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428 
\<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" 
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429 
apply clarify 
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430 
apply (drule_tac f=g in chain_monofun) 
25921  431 
apply (drule chfin) 
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432 
apply (unfold max_in_chain_def) 
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433 
apply (simp add: injection_eq) 
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434 
done 
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435 

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436 
lemma flat2flat: 
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437 
"\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y 
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438 
\<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y" 
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439 
apply clarify 
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440 
apply (drule_tac f=g in monofun_cfun_arg) 
25920  441 
apply (drule ax_flat) 
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442 
apply (erule disjE) 
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443 
apply (simp add: injection_defined_rev) 
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444 
apply (simp add: injection_eq) 
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445 
done 
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446 

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447 
text {* a result about functions with flat codomain *} 
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448 

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449 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
25920  450 
by (drule ax_flat, simp) 
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451 

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452 
lemma flat_codom: 
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453 
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" 
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454 
apply (case_tac "f\<cdot>x = \<bottom>") 
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changeset

455 
apply (rule disjI1) 
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changeset

456 
apply (rule UU_I) 
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changeset

457 
apply (erule_tac t="\<bottom>" in subst) 
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changeset

458 
apply (rule minimal [THEN monofun_cfun_arg]) 
16085
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changeset

459 
apply clarify 
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changeset

460 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
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461 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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462 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
15589
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463 
done 
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464 

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465 

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466 
subsection {* Identity and composition *} 
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467 

25135
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468 
definition 
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469 
ID :: "'a \<rightarrow> 'a" where 
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470 
"ID = (\<Lambda> x. x)" 
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471 

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472 
definition 
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changeset

473 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where 
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474 
oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
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475 

25131
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476 
abbreviation 
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477 
cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) where 
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478 
"f oo g == cfcomp\<cdot>f\<cdot>g" 
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changeset

479 

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480 
lemma ID1 [simp]: "ID\<cdot>x = x" 
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481 
by (simp add: ID_def) 
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changeset

482 

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483 
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" 
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484 
by (simp add: oo_def) 
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changeset

485 

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486 
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" 
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487 
by (simp add: cfcomp1) 
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changeset

488 

27274  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 

19709  492 
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>" 
493 
by (simp add: expand_cfun_eq) 

494 

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495 
text {* 
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496 
Show that interpretation of (pcpo,@{text "_>_"}) is a category. 
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497 
The class of objects is interpretation of syntactical class pcpo. 
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changeset

498 
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a > 'b"}. 
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499 
The identity arrow is interpretation of @{term ID}. 
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500 
The composition of f and g is interpretation of @{text "oo"}. 
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501 
*} 
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changeset

502 

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changeset

503 
lemma ID2 [simp]: "f oo ID = f" 
15589
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changeset

504 
by (rule ext_cfun, simp) 
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diff
changeset

505 

16085
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changeset

506 
lemma ID3 [simp]: "ID oo f = f" 
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changeset

507 
by (rule ext_cfun, simp) 
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changeset

508 

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changeset

509 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
15589
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changeset

510 
by (rule ext_cfun, simp) 
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changeset

511 

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changeset

512 

c004b9bc970e
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513 
subsection {* Strictified functions *} 
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changeset

514 

c004b9bc970e
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changeset

515 
defaultsort pcpo 
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changeset

516 

25131
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changeset

517 
definition 
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changeset

518 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
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diff
changeset

519 
"strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)" 
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changeset

520 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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changeset

521 
text {* results about strictify *} 
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changeset

522 

17815  523 
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)" 
524 
by (simp add: cont_if) 

16085
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changeset

525 

17815  526 
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)" 
527 
apply (rule monofunI) 

25786  528 
apply (auto simp add: monofun_cfun_arg) 
16085
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rewrote continuous isomorphism section, cleaned up
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changeset

529 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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changeset

530 

17815  531 
(*FIXME: long proof*) 
25723  532 
lemma contlub_strictify2: "contlub (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)" 
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removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
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changeset

533 
apply (rule contlubI) 
27413  534 
apply (case_tac "(\<Squnion>i. Y i) = \<bottom>") 
16699  535 
apply (drule (1) chain_UU_I) 
18076  536 
apply simp 
17815  537 
apply (simp del: if_image_distrib) 
538 
apply (simp only: contlub_cfun_arg) 

16085
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rewrote continuous isomorphism section, cleaned up
huffman
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diff
changeset

539 
apply (rule lub_equal2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

540 
apply (rule chain_mono2 [THEN exE]) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
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16070
diff
changeset

541 
apply (erule chain_UU_I_inverse2) 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset

542 
apply (assumption) 
17815  543 
apply (rule_tac x=x in exI, clarsimp) 
16085
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rewrote continuous isomorphism section, cleaned up
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changeset

544 
apply (erule chain_monofun) 
17815  545 
apply (erule monofun_strictify2 [THEN ch2ch_monofun]) 
16085
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changeset

546 
done 
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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changeset

547 

17815  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)" 

29530
9905b660612b
change to simpler, more extensible continuity simproc
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diff
changeset

552 
unfolding strictify_def 
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
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diff
changeset

553 
by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM) 
16085
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rewrote continuous isomorphism section, cleaned up
huffman
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diff
changeset

554 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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diff
changeset

555 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
17815  556 
by (simp add: strictify_conv_if) 
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rewrote continuous isomorphism section, cleaned up
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changeset

557 

c004b9bc970e
rewrote continuous isomorphism section, cleaned up
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diff
changeset

558 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
17815  559 
by (simp add: strictify_conv_if) 
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rewrote continuous isomorphism section, cleaned up
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changeset

560 

17816
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huffman
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diff
changeset

561 
subsection {* Continuous letbindings *} 
9942c5ed866a
new syntax translations for continuous lambda abstraction
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diff
changeset

562 

25131
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563 
definition 
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modernized specifications ('definition', 'abbreviation', 'notation');
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changeset

564 
CLet :: "'a \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'b" where 
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
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parents:
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diff
changeset

565 
"CLet = (\<Lambda> s f. f\<cdot>s)" 
17816
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new syntax translations for continuous lambda abstraction
huffman
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17815
diff
changeset

566 

9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
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17815
diff
changeset

567 
syntax 
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new syntax translations for continuous lambda abstraction
huffman
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17815
diff
changeset

568 
"_CLet" :: "[letbinds, 'a] => 'a" ("(Let (_)/ in (_))" 10) 
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

569 

9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

570 
translations 
9942c5ed866a
new syntax translations for continuous lambda abstraction
huffman
parents:
17815
diff
changeset

571 
"_CLet (_binds b bs) e" == "_CLet b (_CLet bs e)" 
25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
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changeset

572 
"Let x = a in e" == "CONST CLet\<cdot>a\<cdot>(\<Lambda> x. e)" 
17816
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new syntax translations for continuous lambda abstraction
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diff
changeset

573 

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574 
end 