author  wenzelm 
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parent 45695  b108b3d7c49e 
child 49759  ecf1d5d87e5e 
permissions  rwrr 
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(* Title: HOL/HOLCF/Cfun.thy 
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Author: Franz Regensburger 
35794  3 
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 Cpodef Fun_Cpo Product_Cpo 
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begin 
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default_sort cpo 
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subsection {* Definition of continuous function type *} 
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definition "cfun = {f::'a => 'b. cont f}" 
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cpodef (open) ('a, 'b) cfun (infixr ">" 0) = "cfun :: ('a => 'b) set" 

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unfolding cfun_def by (auto intro: cont_const 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" :: "[logic, logic] \<Rightarrow> logic" 
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parse_translation {* 
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(* rewrite (_cabs x t) => (Abs_cfun (%x. t)) *) 
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[Syntax_Trans.mk_binder_tr (@{syntax_const "_cabs"}, @{const_syntax Abs_cfun})]; 
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*} 
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print_translation {* 
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[(@{const_syntax Abs_cfun}, fn [Abs abs] => 
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let val (x, t) = Syntax_Trans.atomic_abs_tr' abs 
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in Syntax.const @{syntax_const "_cabs"} $ x $ t end)] 
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*}  {* To avoid etacontraction of body *} 
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text {* Syntax for nested abstractions *} 
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syntax 
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"_Lambda" :: "[cargs, logic] \<Rightarrow> logic" ("(3LAM _./ _)" [1000, 10] 10) 
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syntax (xsymbols) 
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"_Lambda" :: "[cargs, logic] \<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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Ast.fold_ast_p @{syntax_const "_cabs"} 
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(Ast.unfold_ast @{syntax_const "_cargs"} (Ast.strip_positions pats), body) 
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 Lambda_ast_tr asts = raise Ast.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 Ast.unfold_ast_p @{syntax_const "_cabs"} 
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(Ast.Appl (Ast.Constant @{syntax_const "_cabs"} :: asts)) of 
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([], _) => raise Ast.AST ("cabs_ast_tr'", asts) 
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 (xs, body) => Ast.Appl 
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[Ast.Constant @{syntax_const "_Lambda"}, 
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Ast.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 bottom_cfun: "\<bottom> \<in> cfun" 
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by (simp add: cfun_def inst_fun_pcpo) 
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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 bottom_cfun]) 
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lemmas Rep_cfun_strict = 
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typedef_Rep_strict [OF type_definition_cfun below_cfun_def bottom_cfun] 
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lemmas Abs_cfun_strict = 
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typedef_Abs_strict [OF type_definition_cfun below_cfun_def bottom_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 HOLCFVersion *} 
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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 {* 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: "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 {* Betareduction simproc *} 
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text {* 
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Given the term @{term "(\<Lambda> x. f x)\<cdot>y"}, the procedure tries to 
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construct the theorem @{term "(\<Lambda> x. f x)\<cdot>y == f y"}. If this 
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theorem cannot be completely solved by the cont2cont rules, then 
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the procedure returns the ordinary conditional @{text beta_cfun} 
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rule. 
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The simproc does not solve any more goals that would be solved by 
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using @{text beta_cfun} as a simp rule. The advantage of the 
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simproc is that it can avoid deeplynested calls to the simplifier 
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that would otherwise be caused by large continuity side conditions. 
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Update: The simproc now uses rule @{text Abs_cfun_inverse2} instead 
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of @{text beta_cfun}, to avoid problems with etacontraction. 
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*} 
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simproc_setup beta_cfun_proc ("Rep_cfun (Abs_cfun f)") = {* 
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fn phi => fn ss => fn ct => 
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let 
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val dest = Thm.dest_comb; 
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val f = (snd o dest o snd o dest) ct; 
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val [T, U] = Thm.dest_ctyp (ctyp_of_term f); 
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val tr = instantiate' [SOME T, SOME U] [SOME f] 
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(mk_meta_eq @{thm Abs_cfun_inverse2}); 
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val rules = Cont2ContData.get (Simplifier.the_context ss); 
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val tac = SOLVED' (REPEAT_ALL_NEW (match_tac rules)); 
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in SOME (perhaps (SINGLE (tac 1)) tr) end 
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*} 
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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 cfun_eq_iff: "f = g \<longleftrightarrow> (\<forall>x. f\<cdot>x = g\<cdot>x)" 
40327  163 
by (simp add: Rep_cfun_inject [symmetric] fun_eq_iff) 
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164 

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lemma cfun_eqI: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g" 
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by (simp add: cfun_eq_iff) 
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167 

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text {* Extensionality wrt. ordering for continuous functions *} 
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169 

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lemma cfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
40327  171 
by (simp add: below_cfun_def fun_below_iff) 
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172 

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lemma cfun_belowI: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g" 
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by (simp add: cfun_below_iff) 
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175 

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176 
text {* Congruence for continuous function application *} 
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177 

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lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y" 
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179 
by simp 
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180 

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lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x" 
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182 
by simp 
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183 

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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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186 

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subsection {* Continuity of application *} 
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40327  189 
lemma cont_Rep_cfun1: "cont (\<lambda>f. f\<cdot>x)" 
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by (rule cont_Rep_cfun [OF cont_id, THEN cont2cont_fun]) 
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40327  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) 

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done 
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40327  197 
lemmas monofun_Rep_cfun = cont_Rep_cfun [THEN cont2mono] 
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198 

45606  199 
lemmas monofun_Rep_cfun1 = cont_Rep_cfun1 [THEN cont2mono] 
200 
lemmas monofun_Rep_cfun2 = cont_Rep_cfun2 [THEN cont2mono] 

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40327  202 
text {* contlub, cont properties of @{term Rep_cfun} in each argument *} 
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203 

27413  204 
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))" 
40327  205 
by (rule cont_Rep_cfun2 [THEN cont2contlubE]) 
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27413  207 
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)" 
40327  208 
by (rule cont_Rep_cfun1 [THEN cont2contlubE]) 
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text {* monotonicity of application *} 
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211 

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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: cfun_below_iff) 
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214 

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lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y" 
40327  216 
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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222 

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lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
40327  224 
by (erule monofun_Rep_cfun2 [THEN ch2ch_monofun]) 
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225 

40327  226 
lemma ch2ch_Rep_cfunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
227 
by (rule monofun_Rep_cfun2 [THEN ch2ch_monofun]) 

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40327  229 
lemma ch2ch_Rep_cfunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)" 
230 
by (rule monofun_Rep_cfun1 [THEN ch2ch_monofun]) 

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40327  232 
lemma ch2ch_Rep_cfun [simp]: 
18076  233 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" 
25884  234 
by (simp add: chain_def monofun_cfun) 
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25884  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)" 

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by (simp add: chain_def cfun_below_iff) 
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239 

40327  240 
text {* contlub, cont properties of @{term Rep_cfun} in both arguments *} 
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241 

41027  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)" 

18076  244 
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub) 
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41027  246 
lemma lub_LAM: 
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"\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk> 
41027  248 
\<Longrightarrow> (\<Squnion>i. \<Lambda> x. F i x) = (\<Lambda> x. \<Squnion>i. F i x)" 
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by (simp add: lub_cfun lub_fun ch2ch_lambda) 
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250 

41027  251 
lemmas lub_distribs = lub_APP lub_LAM 
25901  252 

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text {* strictness *} 
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254 

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lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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256 
apply (rule bottomI) 
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257 
apply (erule subst) 
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apply (rule minimal [THEN monofun_cfun_arg]) 
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done 
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260 

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text {* type @{typ "'a > 'b"} is chain complete *} 
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262 

41031  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) 

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265 

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266 
subsection {* Continuity simplification procedure *} 
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267 

40327  268 
text {* cont2cont lemma for @{term Rep_cfun} *} 
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269 

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lemma cont2cont_APP [simp, cont2cont]: 
29049  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  

40006  275 
have 1: "\<And>y. cont (\<lambda>x. (f x)\<cdot>y)" 
40327  276 
using cont_Rep_cfun1 f by (rule cont_compose) 
40006  277 
show "cont (\<lambda>x. (f x)\<cdot>(t x))" 
40327  278 
using t cont_Rep_cfun2 1 by (rule cont_apply) 
29049  279 
qed 
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40008  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 

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lemma cont_APP_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s)" 
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287 
by (rule cont2cont_APP [THEN cont2cont_fun]) 
40008  288 

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289 
lemma cont_APP_app_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s t)" 
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290 
by (rule cont_APP_app [THEN cont2cont_fun]) 
40008  291 

292 

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text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
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294 

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lemma cont2mono_LAM: 
29049  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)" 

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298 
unfolding monofun_def cfun_below_iff by simp 
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29049  300 
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *} 
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301 

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text {* 
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Not suitable as a cont2cont rule, because on nested lambdas 
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304 
it causes exponential blowup in the number of subgoals. 
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305 
*} 
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306 

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lemma cont2cont_LAM: 
29049  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)" 

40327  311 
proof (rule cont_Abs_cfun) 
29049  312 
fix x 
40327  313 
from f1 show "f x \<in> cfun" by (simp add: cfun_def) 
29049  314 
from f2 show "cont f" by (rule cont2cont_lambda) 
315 
qed 

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316 

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text {* 
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This version does work as a cont2cont rule, since it 
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319 
has only a single subgoal. 
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320 
*} 
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321 

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322 
lemma cont2cont_LAM' [simp, cont2cont]: 
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323 
fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" 
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324 
assumes f: "cont (\<lambda>p. f (fst p) (snd p))" 
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325 
shows "cont (\<lambda>x. \<Lambda> y. f x y)" 
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326 
using assms by (simp add: cont2cont_LAM prod_cont_iff) 
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327 

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328 
lemma cont2cont_LAM_discrete [simp, cont2cont]: 
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329 
"(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)" 
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330 
by (simp add: cont2cont_LAM) 
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331 

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332 
subsection {* Miscellaneous *} 
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333 

40327  334 
text {* Monotonicity of @{term Abs_cfun} *} 
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335 

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336 
lemma monofun_LAM: 
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337 
"\<lbrakk>cont f; cont g; \<And>x. f x \<sqsubseteq> g x\<rbrakk> \<Longrightarrow> (\<Lambda> x. f x) \<sqsubseteq> (\<Lambda> x. g x)" 
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338 
by (simp add: cfun_below_iff) 
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339 

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340 
text {* some lemmata for functions with flat/chfin domain/range types *} 
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341 

40327  342 
lemma chfin_Rep_cfunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
27413  343 
==> !s. ? n. (LUB i. Y i)$s = Y n$s" 
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344 
apply (rule allI) 
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345 
apply (subst contlub_cfun_fun) 
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346 
apply assumption 
40771  347 
apply (fast intro!: lub_eqI chfin lub_finch2 chfin2finch ch2ch_Rep_cfunL) 
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348 
done 
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349 

18089  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 

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353 
subsection {* Continuous injectionretraction pairs *} 
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354 

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355 
text {* Continuous retractions are strict. *} 
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356 

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357 
lemma retraction_strict: 
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358 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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359 
apply (rule bottomI) 
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360 
apply (drule_tac x="\<bottom>" in spec) 
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361 
apply (erule subst) 
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362 
apply (rule monofun_cfun_arg) 
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363 
apply (rule minimal) 
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364 
done 
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365 

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366 
lemma injection_eq: 
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367 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" 
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368 
apply (rule iffI) 
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369 
apply (drule_tac f=f in cfun_arg_cong) 
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370 
apply simp 
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371 
apply simp 
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372 
done 
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373 

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374 
lemma injection_below: 
16314  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 

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382 
lemma injection_defined_rev: 
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383 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" 
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384 
apply (drule_tac f=f in cfun_arg_cong) 
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385 
apply (simp add: retraction_strict) 
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386 
done 
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387 

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388 
lemma injection_defined: 
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389 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" 
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390 
by (erule contrapos_nn, rule injection_defined_rev) 
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391 

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392 
text {* a result about functions with flat codomain *} 
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393 

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394 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
25920  395 
by (drule ax_flat, simp) 
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396 

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397 
lemma flat_codom: 
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398 
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" 
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399 
apply (case_tac "f\<cdot>x = \<bottom>") 
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400 
apply (rule disjI1) 
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401 
apply (rule bottomI) 
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402 
apply (erule_tac t="\<bottom>" in subst) 
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403 
apply (rule minimal [THEN monofun_cfun_arg]) 
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404 
apply clarify 
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405 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
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406 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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407 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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408 
done 
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409 

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410 
subsection {* Identity and composition *} 
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411 

25135
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412 
definition 
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413 
ID :: "'a \<rightarrow> 'a" where 
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414 
"ID = (\<Lambda> x. x)" 
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415 

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416 
definition 
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417 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where 
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418 
oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
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419 

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420 
abbreviation 
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421 
cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) where 
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422 
"f oo g == cfcomp\<cdot>f\<cdot>g" 
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423 

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424 
lemma ID1 [simp]: "ID\<cdot>x = x" 
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425 
by (simp add: ID_def) 
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426 

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427 
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" 
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428 
by (simp add: oo_def) 
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429 

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430 
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" 
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431 
by (simp add: cfcomp1) 
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432 

27274  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 

19709  436 
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>" 
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437 
by (simp add: cfun_eq_iff) 
19709  438 

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439 
text {* 
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440 
Show that interpretation of (pcpo,@{text "_>_"}) is a category. 
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441 
The class of objects is interpretation of syntactical class pcpo. 
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442 
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a > 'b"}. 
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443 
The identity arrow is interpretation of @{term ID}. 
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444 
The composition of f and g is interpretation of @{text "oo"}. 
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445 
*} 
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446 

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447 
lemma ID2 [simp]: "f oo ID = f" 
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448 
by (rule cfun_eqI, simp) 
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449 

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450 
lemma ID3 [simp]: "ID oo f = f" 
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451 
by (rule cfun_eqI, simp) 
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452 

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453 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
40002
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454 
by (rule cfun_eqI, simp) 
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455 

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456 
subsection {* Strictified functions *} 
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457 

36452  458 
default_sort pcpo 
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459 

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460 
definition 
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461 
seq :: "'a \<rightarrow> 'b \<rightarrow> 'b" where 
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462 
"seq = (\<Lambda> x. if x = \<bottom> then \<bottom> else ID)" 
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463 

40794  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 

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474 

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475 
lemma seq_conv_if: "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)" 
40794  476 
unfolding seq_def by simp 
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477 

41400
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diff
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478 
lemma seq_simps [simp]: 
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479 
"seq\<cdot>\<bottom> = \<bottom>" 
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480 
"seq\<cdot>x\<cdot>\<bottom> = \<bottom>" 
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diff
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481 
"x \<noteq> \<bottom> \<Longrightarrow> seq\<cdot>x = ID" 
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482 
by (simp_all add: seq_conv_if) 
40093  483 

484 
definition 

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485 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where 
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486 
"strictify = (\<Lambda> f x. seq\<cdot>x\<cdot>(f\<cdot>x))" 
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487 

17815  488 
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" 
40046
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diff
changeset

489 
unfolding strictify_def by simp 
16085
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rewrote continuous isomorphism section, cleaned up
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diff
changeset

490 

c004b9bc970e
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491 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
17815  492 
by (simp add: strictify_conv_if) 
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changeset

493 

c004b9bc970e
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494 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
17815  495 
by (simp add: strictify_conv_if) 
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changeset

496 

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497 
subsection {* Continuity of letbindings *} 
17816
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diff
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498 

35933
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499 
lemma cont2cont_Let: 
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500 
assumes f: "cont (\<lambda>x. f x)" 
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changeset

501 
assumes g1: "\<And>y. cont (\<lambda>x. g x y)" 
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remove continuous letbinding function CLet; add cont2cont rule ordinary Let
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502 
assumes g2: "\<And>x. cont (\<lambda>y. g x y)" 
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remove continuous letbinding function CLet; add cont2cont rule ordinary Let
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diff
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503 
shows "cont (\<lambda>x. let y = f x in g x y)" 
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diff
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504 
unfolding Let_def using f g2 g1 by (rule cont_apply) 
17816
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new syntax translations for continuous lambda abstraction
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diff
changeset

505 

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506 
lemma cont2cont_Let' [simp, cont2cont]: 
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507 
assumes f: "cont (\<lambda>x. f x)" 
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508 
assumes g: "cont (\<lambda>p. g (fst p) (snd p))" 
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509 
shows "cont (\<lambda>x. let y = f x in g x y)" 
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remove continuous letbinding function CLet; add cont2cont rule ordinary Let
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diff
changeset

510 
using f 
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511 
proof (rule cont2cont_Let) 
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512 
fix x show "cont (\<lambda>y. g x y)" 
40003
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remove unused lemmas cont_fst_snd_D1, cont_fst_snd_D2
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diff
changeset

513 
using g by (simp add: prod_cont_iff) 
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514 
next 
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515 
fix y show "cont (\<lambda>x. g x y)" 
40003
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diff
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516 
using g by (simp add: prod_cont_iff) 
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517 
qed 
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518 

39145  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 

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527 
end 