src/HOLCF/Cfun.thy
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(*  Title:      HOLCF/Cfun.thy
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    ID:         $Id$
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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
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uses ("cont_proc.ML")
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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 add: Ex_cont adm_cont)
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syntax
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  Rep_CFun :: "('a -> 'b) => ('a => 'b)" ("_$_" [999,1000] 999)
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                                                (* application *)
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  Abs_CFun :: "('a => 'b) => ('a -> 'b)" (binder "LAM " 10)
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                                                (* abstraction *)
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syntax (xsymbols)
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  "->"     :: "[type, type] => type"      ("(_ \<rightarrow>/ _)" [1,0]0)
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  "LAM "   :: "[idts, 'a => 'b] => ('a -> 'b)"
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					("(3\<Lambda>_./ _)" [0, 10] 10)
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  Rep_CFun :: "('a -> 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999)
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syntax (HTML output)
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  Rep_CFun :: "('a -> 'b) => ('a => 'b)" ("(_\<cdot>_)" [999,1000] 999)
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subsection {* Class instances *}
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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 "->" :: (cpo, pcpo) pcpo
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by (rule typedef_pcpo [OF type_definition_CFun less_CFun_def UU_CFun])
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lemmas Rep_CFun_strict =
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  typedef_Rep_strict [OF type_definition_CFun less_CFun_def UU_CFun]
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lemmas Abs_CFun_strict =
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  typedef_Abs_strict [OF type_definition_CFun less_CFun_def UU_CFun]
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text {* Additional lemma about the isomorphism between
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        @{typ "'a -> 'b"} and @{term CFun} *}
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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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text {* Beta-equality for continuous functions *}
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lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
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by (simp add: Abs_CFun_inverse2)
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text {* Eta-equality for continuous functions *}
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lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
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by (rule Rep_CFun_inverse)
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text {* Extensionality for continuous functions *}
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lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
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by (simp add: Rep_CFun_inject [symmetric] ext)
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text {* lemmas about application of continuous functions *}
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lemma cfun_cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f\<cdot>x = g\<cdot>y"
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by simp
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lemma cfun_fun_cong: "f = g \<Longrightarrow> f\<cdot>x = g\<cdot>x"
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by simp
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lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
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by simp
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subsection {* Continuity of application *}
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lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
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by (rule cont_Rep_CFun [THEN cont2cont_CF1L])
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lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
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apply (rule_tac P = "cont" in CollectD)
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apply (fold CFun_def)
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apply (rule Rep_CFun)
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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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text {* contlub, cont properties of @{term Rep_CFun} in each argument *}
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lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(lub (range Y)) = (\<Squnion>i. f\<cdot>(Y i))"
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by (rule contlub_Rep_CFun2 [THEN contlubE])
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lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(lub (range Y))"
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by (rule cont_Rep_CFun2 [THEN contE])
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lemma contlub_cfun_fun: "chain F \<Longrightarrow> lub (range F)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
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by (rule contlub_Rep_CFun1 [THEN contlubE])
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lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| lub (range F)\<cdot>x"
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by (rule cont_Rep_CFun1 [THEN contE])
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text {* Extensionality wrt. @{term "op <<"} in @{typ "'a -> 'b"} *}
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lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
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by (simp add: less_CFun_def less_fun_def)
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text {* monotonicity of application *}
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lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
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by (simp add: less_CFun_def less_fun_def)
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lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
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by (rule monofun_Rep_CFun2 [THEN monofunE])
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lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
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by (rule trans_less [OF monofun_cfun_fun monofun_cfun_arg])
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text {* ch2ch - rules for the type @{typ "'a -> 'b"} *}
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lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
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by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
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lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
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by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])
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lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
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by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])
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lemma ch2ch_Rep_CFun: "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
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apply (rule chainI)
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apply (rule monofun_cfun)
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apply (erule chainE)
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apply (erule chainE)
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done
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text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}
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lemma contlub_cfun: 
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  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
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apply (simp only: contlub_cfun_fun)
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apply (simp only: contlub_cfun_arg)
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apply (rule diag_lub)
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apply (erule monofun_Rep_CFun1 [THEN ch2ch_monofun])
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apply (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])
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done
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lemma cont_cfun: 
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  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
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apply (rule thelubE)
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apply (simp only: ch2ch_Rep_CFun)
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apply (simp only: contlub_cfun)
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done
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text {* strictness *}
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lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
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apply (rule UU_I)
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apply (erule subst)
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apply (rule minimal [THEN monofun_cfun_arg])
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done
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text {* the lub of a chain of continous functions is monotone *}
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lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
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apply (drule ch2ch_monofun [OF monofun_Rep_CFun])
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apply (simp add: thelub_fun [symmetric])
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apply (erule monofun_lub_fun)
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apply (simp add: monofun_Rep_CFun2)
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done
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text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *}
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lemma ex_lub_cfun:
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  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>j. \<Squnion>i. F j\<cdot>(Y i)) = (\<Squnion>i. \<Squnion>j. F j\<cdot>(Y i))"
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by (simp add: diag_lub ch2ch_Rep_CFunL ch2ch_Rep_CFunR)
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text {* the lub of a chain of cont. functions is continuous *}
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lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
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apply (rule cont2cont_lub)
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apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
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apply (rule cont_Rep_CFun2)
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done
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text {* type @{typ "'a -> 'b"} is chain complete *}
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lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
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by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE)
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lemma thelub_cfun: "chain F \<Longrightarrow> lub (range F) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
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by (rule lub_cfun [THEN thelubI])
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subsection {* Miscellaneous *}
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text {* Monotonicity of @{term Abs_CFun} *}
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lemma semi_monofun_Abs_CFun:
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  "\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
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by (simp add: less_CFun_def Abs_CFun_inverse2)
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text {* for compatibility with old HOLCF-Version *}
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lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
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by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)
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subsection {* Continuity of application *}
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text {* cont2cont lemma for @{term Rep_CFun} *}
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lemma cont2cont_Rep_CFun:
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  "\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x)\<cdot>(t x))"
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by (best intro: cont2cont_app2 cont_const cont_Rep_CFun cont_Rep_CFun2)
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text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *}
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lemma cont2mono_LAM:
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assumes p1: "!!x. cont(c1 x)"
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assumes p2: "!!y. monofun(%x. c1 x y)"
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shows "monofun(%x. LAM y. c1 x y)"
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apply (rule monofunI)
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apply (rule less_cfun_ext)
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apply (simp add: p1)
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apply (erule p2 [THEN monofunE])
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done
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text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *}
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lemma cont2cont_LAM:
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assumes p1: "!!x. cont(c1 x)"
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assumes p2: "!!y. cont(%x. c1 x y)"
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shows "cont(%x. LAM y. c1 x y)"
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apply (rule cont_Abs_CFun)
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apply (simp add: p1 CFun_def)
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apply (simp add: p2 cont2cont_CF1L_rev)
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done
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text {* continuity simplification procedure *}
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lemmas cont_lemmas1 =
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  cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM
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use "cont_proc.ML";
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setup ContProc.setup;
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(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*)
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(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*)
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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 {* some lemmata for functions with flat/chfin domain/range types *}
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lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)  
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      ==> !s. ? n. lub(range(Y))$s = Y n$s"
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apply (rule allI)
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apply (subst contlub_cfun_fun)
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apply assumption
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apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
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done
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subsection {* Continuous injection-retraction pairs *}
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text {* Continuous retractions are strict. *}
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lemma retraction_strict:
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  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
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apply (rule UU_I)
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apply (drule_tac x="\<bottom>" in spec)
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apply (erule subst)
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apply (rule monofun_cfun_arg)
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apply (rule minimal)
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done
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lemma injection_eq:
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  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)"
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apply (rule iffI)
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apply (drule_tac f=f in cfun_arg_cong)
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apply simp
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apply simp
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done
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lemma injection_less:
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  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
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apply (rule iffI)
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apply (drule_tac f=f in monofun_cfun_arg)
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apply simp
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apply (erule monofun_cfun_arg)
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done
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lemma injection_defined_rev:
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  "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>"
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apply (drule_tac f=f in cfun_arg_cong)
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apply (simp add: retraction_strict)
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done
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lemma injection_defined:
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  "\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>"
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by (erule contrapos_nn, rule injection_defined_rev)
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text {* propagation of flatness and chain-finiteness by retractions *}
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lemma chfin2chfin:
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  "\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y
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    \<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
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apply clarify
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apply (drule_tac f=g in chain_monofun)
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apply (drule chfin [rule_format])
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apply (unfold max_in_chain_def)
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apply (simp add: injection_eq)
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done
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lemma flat2flat:
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  "\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y
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    \<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y"
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apply clarify
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apply (drule_tac f=g in monofun_cfun_arg)
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apply (drule ax_flat [rule_format])
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apply (erule disjE)
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apply (simp add: injection_defined_rev)
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apply (simp add: injection_eq)
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done
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text {* a result about functions with flat codomain *}
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lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
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by (drule ax_flat [rule_format], simp)
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lemma flat_codom:
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  "f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
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apply (case_tac "f\<cdot>x = \<bottom>")
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apply (rule disjI1)
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apply (rule UU_I)
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apply (erule_tac t="\<bottom>" in subst)
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apply (rule minimal [THEN monofun_cfun_arg])
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apply clarify
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apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
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apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
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apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
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done
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subsection {* Identity and composition *}
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consts
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  ID      :: "'a \<rightarrow> 'a"
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  cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c"
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syntax  "@oo" :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100)
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translations  "f1 oo f2" == "cfcomp$f1$f2"
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defs
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  ID_def: "ID \<equiv> (\<Lambda> x. x)"
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  oo_def: "cfcomp \<equiv> (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 
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lemma ID1 [simp]: "ID\<cdot>x = x"
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by (simp add: ID_def)
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lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))"
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by (simp add: oo_def)
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lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)"
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by (simp add: cfcomp1)
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text {*
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  Show that interpretation of (pcpo,@{text "_->_"}) is a category.
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  The class of objects is interpretation of syntactical class pcpo.
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  The class of arrows  between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}.
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  The identity arrow is interpretation of @{term ID}.
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  The composition of f and g is interpretation of @{text "oo"}.
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*}
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lemma ID2 [simp]: "f oo ID = f"
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by (rule ext_cfun, simp)
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lemma ID3 [simp]: "ID oo f = f"
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   398
by (rule ext_cfun, simp)
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   399
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   400
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
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   401
by (rule ext_cfun, simp)
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parents:
diff changeset
   402
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diff changeset
   403
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   404
subsection {* Strictified functions *}
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   405
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   406
defaultsort pcpo
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   407
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   408
constdefs
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   409
  strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b"
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   410
  "strictify \<equiv> (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
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diff changeset
   411
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   412
text {* results about strictify *}
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diff changeset
   413
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diff changeset
   414
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
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   415
by (simp add: cont_if)
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diff changeset
   416
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   417
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
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   418
apply (rule monofunI)
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   419
apply (auto simp add: monofun_cfun_arg eq_UU_iff [symmetric])
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   420
done
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diff changeset
   421
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   422
(*FIXME: long proof*)
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   423
lemma contlub_strictify2: "contlub (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
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   424
apply (rule contlubI)
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   425
apply (case_tac "lub (range Y) = \<bottom>")
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   426
apply (drule (1) chain_UU_I)
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diff changeset
   427
apply (simp add: thelub_const)
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diff changeset
   428
apply (simp del: if_image_distrib)
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diff changeset
   429
apply (simp only: contlub_cfun_arg)
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diff changeset
   430
apply (rule lub_equal2)
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diff changeset
   431
apply (rule chain_mono2 [THEN exE])
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diff changeset
   432
apply (erule chain_UU_I_inverse2)
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diff changeset
   433
apply (assumption)
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diff changeset
   434
apply (rule_tac x=x in exI, clarsimp)
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diff changeset
   435
apply (erule chain_monofun)
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diff changeset
   436
apply (erule monofun_strictify2 [THEN ch2ch_monofun])
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diff changeset
   437
done
c004b9bc970e rewrote continuous isomorphism section, cleaned up
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diff changeset
   438
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diff changeset
   439
lemmas cont_strictify2 =
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diff changeset
   440
  monocontlub2cont [OF monofun_strictify2 contlub_strictify2, standard]
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diff changeset
   441
ccf54e3cabfa removed Istrictify; simplified some proofs
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diff changeset
   442
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
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diff changeset
   443
by (unfold strictify_def, simp add: cont_strictify1 cont_strictify2)
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diff changeset
   444
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diff changeset
   445
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>"
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diff changeset
   446
by (simp add: strictify_conv_if)
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diff changeset
   447
c004b9bc970e rewrote continuous isomorphism section, cleaned up
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diff changeset
   448
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x"
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diff changeset
   449
by (simp add: strictify_conv_if)
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diff changeset
   450
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diff changeset
   451
end