author  huffman 
Mon, 06 Dec 2010 11:22:42 0800  
changeset 41031  9883d1417ce1 
parent 41030  ff7d177128ef 
child 41322  43a5b9a0ee8a 
permissions  rwrr 
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(* Title: 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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cpodef ('a, 'b) cfun (infixr ">" 0) = "{f::'a => 'b. cont f}" 
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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" :: "'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) 

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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 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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*} 
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simproc_setup beta_cfun_proc ("Abs_cfun f\<cdot>x") = {* 
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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, x) = (apfst (snd o dest o snd o dest) 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, SOME x] 
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(mk_meta_eq @{thm beta_cfun}); 
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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)" 
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by (simp add: Rep_cfun_inject [symmetric] fun_eq_iff) 
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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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text {* Extensionality wrt. ordering for continuous functions *} 
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lemma cfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
40327  176 
by (simp add: below_cfun_def fun_below_iff) 
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177 

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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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180 

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181 
text {* Congruence for continuous function application *} 
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182 

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lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y" 
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184 
by simp 
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185 

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lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x" 
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187 
by simp 
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188 

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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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191 

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subsection {* Continuity of application *} 
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40327  194 
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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196 

40327  197 
lemma cont_Rep_cfun2: "cont (\<lambda>x. f\<cdot>x)" 
198 
apply (cut_tac x=f in Rep_cfun) 

199 
apply (simp add: cfun_def) 

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done 
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40327  202 
lemmas monofun_Rep_cfun = cont_Rep_cfun [THEN cont2mono] 
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203 

40327  204 
lemmas monofun_Rep_cfun1 = cont_Rep_cfun1 [THEN cont2mono, standard] 
205 
lemmas monofun_Rep_cfun2 = cont_Rep_cfun2 [THEN cont2mono, standard] 

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40327  207 
text {* contlub, cont properties of @{term Rep_cfun} in each argument *} 
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208 

27413  209 
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))" 
40327  210 
by (rule cont_Rep_cfun2 [THEN cont2contlubE]) 
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27413  212 
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)" 
40327  213 
by (rule cont_Rep_cfun1 [THEN cont2contlubE]) 
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214 

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text {* monotonicity of application *} 
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216 

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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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219 

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lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y" 
40327  221 
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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224 
by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg]) 
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225 

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text {* ch2ch  rules for the type @{typ "'a > 'b"} *} 
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227 

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lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
40327  229 
by (erule monofun_Rep_cfun2 [THEN ch2ch_monofun]) 
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230 

40327  231 
lemma ch2ch_Rep_cfunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" 
232 
by (rule monofun_Rep_cfun2 [THEN ch2ch_monofun]) 

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40327  234 
lemma ch2ch_Rep_cfunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)" 
235 
by (rule monofun_Rep_cfun1 [THEN ch2ch_monofun]) 

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40327  237 
lemma ch2ch_Rep_cfun [simp]: 
18076  238 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" 
25884  239 
by (simp add: chain_def monofun_cfun) 
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25884  241 
lemma ch2ch_LAM [simp]: 
242 
"\<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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243 
by (simp add: chain_def cfun_below_iff) 
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244 

40327  245 
text {* contlub, cont properties of @{term Rep_cfun} in both arguments *} 
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246 

41027  247 
lemma lub_APP: 
248 
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i\<cdot>(Y i)) = (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)" 

18076  249 
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub) 
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41027  251 
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  253 
\<Longrightarrow> (\<Squnion>i. \<Lambda> x. F i x) = (\<Lambda> x. \<Squnion>i. F i x)" 
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apply (simp add: lub_cfun) 
40327  255 
apply (simp add: Abs_cfun_inverse2) 
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apply (simp add: lub_fun ch2ch_lambda) 
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done 
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258 

41027  259 
lemmas lub_distribs = lub_APP lub_LAM 
25901  260 

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261 
text {* strictness *} 
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262 

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lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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264 
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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267 
done 
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268 

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text {* type @{typ "'a > 'b"} is chain complete *} 
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270 

41031  271 
lemma lub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" 
272 
by (simp add: lub_cfun lub_fun ch2ch_lambda) 

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273 

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274 
subsection {* Continuity simplification procedure *} 
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275 

40327  276 
text {* cont2cont lemma for @{term Rep_cfun} *} 
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277 

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lemma cont2cont_APP [simp, cont2cont]: 
29049  279 
assumes f: "cont (\<lambda>x. f x)" 
280 
assumes t: "cont (\<lambda>x. t x)" 

281 
shows "cont (\<lambda>x. (f x)\<cdot>(t x))" 

282 
proof  

40006  283 
have 1: "\<And>y. cont (\<lambda>x. (f x)\<cdot>y)" 
40327  284 
using cont_Rep_cfun1 f by (rule cont_compose) 
40006  285 
show "cont (\<lambda>x. (f x)\<cdot>(t x))" 
40327  286 
using t cont_Rep_cfun2 1 by (rule cont_apply) 
29049  287 
qed 
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40008  289 
text {* 
290 
Two specific lemmas for the combination of LCF and HOL terms. 

291 
These lemmas are needed in theories that use types like @{typ "'a \<rightarrow> 'b \<Rightarrow> 'c"}. 

292 
*} 

293 

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294 
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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295 
by (rule cont2cont_APP [THEN cont2cont_fun]) 
40008  296 

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297 
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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298 
by (rule cont_APP_app [THEN cont2cont_fun]) 
40008  299 

300 

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text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} 
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302 

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lemma cont2mono_LAM: 
29049  304 
"\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk> 
305 
\<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)" 

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306 
unfolding monofun_def cfun_below_iff by simp 
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307 

29049  308 
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *} 
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309 

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text {* 
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311 
Not suitable as a cont2cont rule, because on nested lambdas 
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312 
it causes exponential blowup in the number of subgoals. 
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313 
*} 
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314 

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lemma cont2cont_LAM: 
29049  316 
assumes f1: "\<And>x. cont (\<lambda>y. f x y)" 
317 
assumes f2: "\<And>y. cont (\<lambda>x. f x y)" 

318 
shows "cont (\<lambda>x. \<Lambda> y. f x y)" 

40327  319 
proof (rule cont_Abs_cfun) 
29049  320 
fix x 
40327  321 
from f1 show "f x \<in> cfun" by (simp add: cfun_def) 
29049  322 
from f2 show "cont f" by (rule cont2cont_lambda) 
323 
qed 

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324 

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text {* 
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This version does work as a cont2cont rule, since it 
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327 
has only a single subgoal. 
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328 
*} 
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329 

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330 
lemma cont2cont_LAM' [simp, cont2cont]: 
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331 
fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" 
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332 
assumes f: "cont (\<lambda>p. f (fst p) (snd p))" 
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333 
shows "cont (\<lambda>x. \<Lambda> y. f x y)" 
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334 
using assms by (simp add: cont2cont_LAM prod_cont_iff) 
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335 

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336 
lemma cont2cont_LAM_discrete [simp, cont2cont]: 
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337 
"(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)" 
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338 
by (simp add: cont2cont_LAM) 
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339 

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340 
subsection {* Miscellaneous *} 
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341 

40327  342 
text {* Monotonicity of @{term Abs_cfun} *} 
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343 

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344 
lemma monofun_LAM: 
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345 
"\<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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346 
by (simp add: cfun_below_iff) 
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347 

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348 
text {* some lemmata for functions with flat/chfin domain/range types *} 
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349 

40327  350 
lemma chfin_Rep_cfunR: "chain (Y::nat => 'a::cpo>'b::chfin) 
27413  351 
==> !s. ? n. (LUB i. Y i)$s = Y n$s" 
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352 
apply (rule allI) 
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353 
apply (subst contlub_cfun_fun) 
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354 
apply assumption 
40771  355 
apply (fast intro!: lub_eqI chfin lub_finch2 chfin2finch ch2ch_Rep_cfunL) 
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356 
done 
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357 

18089  358 
lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))" 
359 
by (rule adm_subst, simp, rule adm_chfin) 

360 

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361 
subsection {* Continuous injectionretraction pairs *} 
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362 

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363 
text {* Continuous retractions are strict. *} 
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364 

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365 
lemma retraction_strict: 
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366 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" 
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367 
apply (rule UU_I) 
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368 
apply (drule_tac x="\<bottom>" in spec) 
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369 
apply (erule subst) 
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370 
apply (rule monofun_cfun_arg) 
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371 
apply (rule minimal) 
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372 
done 
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373 

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374 
lemma injection_eq: 
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375 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" 
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376 
apply (rule iffI) 
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377 
apply (drule_tac f=f in cfun_arg_cong) 
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378 
apply simp 
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379 
apply simp 
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380 
done 
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381 

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382 
lemma injection_below: 
16314  383 
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)" 
384 
apply (rule iffI) 

385 
apply (drule_tac f=f in monofun_cfun_arg) 

386 
apply simp 

387 
apply (erule monofun_cfun_arg) 

388 
done 

389 

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390 
lemma injection_defined_rev: 
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391 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" 
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392 
apply (drule_tac f=f in cfun_arg_cong) 
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393 
apply (simp add: retraction_strict) 
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394 
done 
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395 

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396 
lemma injection_defined: 
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397 
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" 
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398 
by (erule contrapos_nn, rule injection_defined_rev) 
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399 

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400 
text {* a result about functions with flat codomain *} 
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401 

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402 
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" 
25920  403 
by (drule ax_flat, simp) 
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404 

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405 
lemma flat_codom: 
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406 
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" 
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407 
apply (case_tac "f\<cdot>x = \<bottom>") 
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408 
apply (rule disjI1) 
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409 
apply (rule UU_I) 
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410 
apply (erule_tac t="\<bottom>" in subst) 
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411 
apply (rule minimal [THEN monofun_cfun_arg]) 
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412 
apply clarify 
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413 
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) 
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414 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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415 
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) 
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416 
done 
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417 

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418 
subsection {* Identity and composition *} 
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419 

25135
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420 
definition 
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421 
ID :: "'a \<rightarrow> 'a" where 
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422 
"ID = (\<Lambda> x. x)" 
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423 

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424 
definition 
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425 
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where 
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426 
oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
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427 

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428 
abbreviation 
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429 
cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) where 
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430 
"f oo g == cfcomp\<cdot>f\<cdot>g" 
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431 

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432 
lemma ID1 [simp]: "ID\<cdot>x = x" 
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433 
by (simp add: ID_def) 
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434 

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435 
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" 
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436 
by (simp add: oo_def) 
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437 

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438 
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" 
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439 
by (simp add: cfcomp1) 
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440 

27274  441 
lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))" 
442 
by (simp add: cfcomp1) 

443 

19709  444 
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>" 
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445 
by (simp add: cfun_eq_iff) 
19709  446 

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447 
text {* 
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448 
Show that interpretation of (pcpo,@{text "_>_"}) is a category. 
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449 
The class of objects is interpretation of syntactical class pcpo. 
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450 
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a > 'b"}. 
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451 
The identity arrow is interpretation of @{term ID}. 
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452 
The composition of f and g is interpretation of @{text "oo"}. 
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453 
*} 
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454 

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455 
lemma ID2 [simp]: "f oo ID = f" 
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456 
by (rule cfun_eqI, simp) 
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457 

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458 
lemma ID3 [simp]: "ID oo f = f" 
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459 
by (rule cfun_eqI, simp) 
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460 

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461 
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" 
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462 
by (rule cfun_eqI, simp) 
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463 

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464 
subsection {* Strictified functions *} 
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465 

36452  466 
default_sort pcpo 
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467 

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468 
definition 
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469 
seq :: "'a \<rightarrow> 'b \<rightarrow> 'b" where 
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470 
"seq = (\<Lambda> x. if x = \<bottom> then \<bottom> else ID)" 
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471 

40794  472 
lemma cont2cont_if_bottom [cont2cont, simp]: 
473 
assumes f: "cont (\<lambda>x. f x)" and g: "cont (\<lambda>x. g x)" 

474 
shows "cont (\<lambda>x. if f x = \<bottom> then \<bottom> else g x)" 

475 
proof (rule cont_apply [OF f]) 

476 
show "\<And>x. cont (\<lambda>y. if y = \<bottom> then \<bottom> else g x)" 

477 
unfolding cont_def is_lub_def is_ub_def ball_simps 

478 
by (simp add: lub_eq_bottom_iff) 

479 
show "\<And>y. cont (\<lambda>x. if y = \<bottom> then \<bottom> else g x)" 

480 
by (simp add: g) 

481 
qed 

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482 

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483 
lemma seq_conv_if: "seq\<cdot>x = (if x = \<bottom> then \<bottom> else ID)" 
40794  484 
unfolding seq_def by simp 
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485 

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486 
lemma seq1 [simp]: "seq\<cdot>\<bottom> = \<bottom>" 
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487 
by (simp add: seq_conv_if) 
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488 

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489 
lemma seq2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> seq\<cdot>x = ID" 
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490 
by (simp add: seq_conv_if) 
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491 

40767
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492 
lemma seq3 [simp]: "seq\<cdot>x\<cdot>\<bottom> = \<bottom>" 
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493 
by (simp add: seq_conv_if) 
40093  494 

495 
definition 

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496 
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where 
40767
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497 
"strictify = (\<Lambda> f x. seq\<cdot>x\<cdot>(f\<cdot>x))" 
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498 

17815  499 
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" 
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500 
unfolding strictify_def by simp 
16085
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501 

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502 
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" 
17815  503 
by (simp add: strictify_conv_if) 
16085
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504 

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505 
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" 
17815  506 
by (simp add: strictify_conv_if) 
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507 

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508 
subsection {* Continuity of letbindings *} 
17816
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509 

35933
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510 
lemma cont2cont_Let: 
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511 
assumes f: "cont (\<lambda>x. f x)" 
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512 
assumes g1: "\<And>y. cont (\<lambda>x. g x y)" 
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513 
assumes g2: "\<And>x. cont (\<lambda>y. g x y)" 
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514 
shows "cont (\<lambda>x. let y = f x in g x y)" 
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515 
unfolding Let_def using f g2 g1 by (rule cont_apply) 
17816
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516 

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517 
lemma cont2cont_Let' [simp, cont2cont]: 
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518 
assumes f: "cont (\<lambda>x. f x)" 
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519 
assumes g: "cont (\<lambda>p. g (fst p) (snd p))" 
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520 
shows "cont (\<lambda>x. let y = f x in g x y)" 
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521 
using f 
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522 
proof (rule cont2cont_Let) 
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523 
fix x show "cont (\<lambda>y. g x y)" 
40003
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diff
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524 
using g by (simp add: prod_cont_iff) 
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525 
next 
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526 
fix y show "cont (\<lambda>x. g x y)" 
40003
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527 
using g by (simp add: prod_cont_iff) 
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528 
qed 
17816
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529 

39145  530 
text {* The simple version (suggested by Joachim Breitner) is needed if 
531 
the type of the defined term is not a cpo. *} 

532 

533 
lemma cont2cont_Let_simple [simp, cont2cont]: 

534 
assumes "\<And>y. cont (\<lambda>x. g x y)" 

535 
shows "cont (\<lambda>x. let y = t in g x y)" 

536 
unfolding Let_def using assms . 

537 

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538 
end 