author | huffman |
Mon, 10 Oct 2005 05:30:02 +0200 | |
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parent 17815 | ccf54e3cabfa |
child 17817 | 405fb812e738 |
permissions | -rw-r--r-- |
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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 (xsymbols) |
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"->" :: "[type, type] => type" ("(_ \<rightarrow>/ _)" [1,0]0) |
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syntax |
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Rep_CFun :: "('a \<rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("_$_" [999,1000] 999) |
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"_Lambda" :: "[pttrns, 'a] \<Rightarrow> logic" ("(3LAM _./ _)" [0, 10] 10) |
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syntax (xsymbols) |
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"_Lambda" :: "[pttrns, 'a] \<Rightarrow> logic" ("(3\<Lambda>_./ _)" [0, 10] 10) |
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Rep_CFun :: "('a \<rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("(_\<cdot>_)" [999,1000] 999) |
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syntax (HTML output) |
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Rep_CFun :: "('a \<rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("(_\<cdot>_)" [999,1000] 999) |
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syntax |
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"_cabs" :: "[pttrn, 'a] \<Rightarrow> logic" |
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translations |
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"_cabs x t" == "Abs_CFun (%x. t)" |
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(* To avoid eta-contraction of body: *) |
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print_translation {* |
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let |
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fun cabs_tr' [Abs abs] = |
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let val (x,t) = atomic_abs_tr' abs |
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in Syntax.const "_cabs" $ x $ t end |
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in [("Abs_CFun", cabs_tr')] end |
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*} |
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parse_ast_translation {* |
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(* rewrites (LAM x y z. t) --> (LAM x. LAM y. LAM z. t) *) |
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(* c.f. Syntax.lambda_ast_tr from 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 "_cabs" (Syntax.unfold_ast "_pttrns" pats, body) |
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| Lambda_ast_tr asts = raise Syntax.AST ("lambda_ast_tr", asts); |
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in [("_Lambda", Lambda_ast_tr)] end |
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*} |
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print_ast_translation {* |
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(* rewrites (LAM x. LAM y. LAM z. t) --> (LAM x y z. t) *) |
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(* c.f. Syntax.abs_ast_tr' from 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 "_cabs" |
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(Syntax.Appl (Syntax.Constant "_cabs" :: asts)) of |
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([], _) => raise Syntax.AST ("abs_ast_tr'", asts) |
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| (xs, body) => Syntax.Appl |
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[Syntax.Constant "_Lambda", Syntax.fold_ast "_pttrns" xs, body]); |
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in [("_cabs", cabs_ast_tr')] end |
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*} |
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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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|
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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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|
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text {* contlub, cont properties of @{term Rep_CFun} in each argument *} |
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|
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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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|
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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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|
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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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|
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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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|
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text {* Extensionality wrt. @{term "op <<"} in @{typ "'a -> 'b"} *} |
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|
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lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g" |
16699 | 161 |
by (simp add: less_CFun_def less_fun_def) |
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162 |
|
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text {* monotonicity of application *} |
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|
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lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x" |
16699 | 166 |
by (simp add: less_CFun_def less_fun_def) |
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|
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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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|
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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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|
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text {* ch2ch - rules for the type @{typ "'a -> 'b"} *} |
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|
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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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|
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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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|
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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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|
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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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190 |
done |
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191 |
|
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192 |
text {* contlub, cont properties of @{term Rep_CFun} in both arguments *} |
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|
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194 |
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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201 |
done |
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|
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203 |
lemma cont_cfun: |
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204 |
"\<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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205 |
apply (rule thelubE) |
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206 |
apply (simp only: ch2ch_Rep_CFun) |
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207 |
apply (simp only: contlub_cfun) |
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208 |
done |
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|
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210 |
text {* strictness *} |
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|
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212 |
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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216 |
done |
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|
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text {* the lub of a chain of continous functions is monotone *} |
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|
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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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223 |
apply (erule monofun_lub_fun) |
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apply (simp add: monofun_Rep_CFun2) |
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done |
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|
16386 | 227 |
text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *} |
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|
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lemma ex_lub_cfun: |
230 |
"\<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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|
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text {* the lub of a chain of cont. functions is continuous *} |
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|
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lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)" |
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236 |
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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240 |
|
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text {* type @{typ "'a -> 'b"} is chain complete *} |
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|
16920 | 243 |
lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" |
244 |
by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE) |
|
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245 |
|
16920 | 246 |
lemma thelub_cfun: "chain F \<Longrightarrow> lub (range F) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" |
247 |
by (rule lub_cfun [THEN thelubI]) |
|
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248 |
|
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249 |
subsection {* Miscellaneous *} |
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250 |
|
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text {* Monotonicity of @{term Abs_CFun} *} |
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252 |
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16699 | 253 |
lemma semi_monofun_Abs_CFun: |
254 |
"\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g" |
|
255 |
by (simp add: less_CFun_def Abs_CFun_inverse2) |
|
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256 |
|
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text {* for compatibility with old HOLCF-Version *} |
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258 |
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)" |
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259 |
by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict) |
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260 |
|
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261 |
subsection {* Continuity of application *} |
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262 |
|
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text {* cont2cont lemma for @{term Rep_CFun} *} |
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264 |
|
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265 |
lemma cont2cont_Rep_CFun: |
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266 |
"\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x)\<cdot>(t x))" |
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267 |
by (best intro: cont2cont_app2 cont_const cont_Rep_CFun cont_Rep_CFun2) |
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268 |
|
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text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} |
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270 |
|
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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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275 |
apply (rule monofunI) |
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276 |
apply (rule less_cfun_ext) |
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277 |
apply (simp add: p1) |
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apply (erule p2 [THEN monofunE]) |
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279 |
done |
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280 |
|
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text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *} |
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282 |
|
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283 |
lemma cont2cont_LAM: |
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284 |
assumes p1: "!!x. cont(c1 x)" |
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assumes p2: "!!y. cont(%x. c1 x y)" |
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286 |
shows "cont(%x. LAM y. c1 x y)" |
16098 | 287 |
apply (rule cont_Abs_CFun) |
288 |
apply (simp add: p1 CFun_def) |
|
289 |
apply (simp add: p2 cont2cont_CF1L_rev) |
|
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done |
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291 |
|
16386 | 292 |
text {* continuity simplification procedure *} |
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293 |
|
16055 | 294 |
lemmas cont_lemmas1 = |
295 |
cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM |
|
296 |
||
16386 | 297 |
use "cont_proc.ML"; |
298 |
setup ContProc.setup; |
|
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299 |
|
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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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302 |
|
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text {* function application is strict in its first argument *} |
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304 |
|
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|
305 |
lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>" |
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306 |
by (simp add: Rep_CFun_strict) |
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307 |
|
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|
308 |
text {* some lemmata for functions with flat/chfin domain/range types *} |
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309 |
|
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310 |
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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312 |
apply (rule allI) |
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313 |
apply (subst contlub_cfun_fun) |
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314 |
apply assumption |
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315 |
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) |
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316 |
done |
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|
317 |
|
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|
318 |
subsection {* Continuous injection-retraction pairs *} |
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319 |
|
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320 |
text {* Continuous retractions are strict. *} |
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321 |
|
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322 |
lemma retraction_strict: |
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323 |
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" |
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324 |
apply (rule UU_I) |
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325 |
apply (drule_tac x="\<bottom>" in spec) |
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326 |
apply (erule subst) |
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|
327 |
apply (rule monofun_cfun_arg) |
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|
328 |
apply (rule minimal) |
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329 |
done |
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|
330 |
|
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|
331 |
lemma injection_eq: |
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|
332 |
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" |
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|
333 |
apply (rule iffI) |
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|
334 |
apply (drule_tac f=f in cfun_arg_cong) |
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|
335 |
apply simp |
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|
336 |
apply simp |
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337 |
done |
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|
338 |
|
16314 | 339 |
lemma injection_less: |
340 |
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)" |
|
341 |
apply (rule iffI) |
|
342 |
apply (drule_tac f=f in monofun_cfun_arg) |
|
343 |
apply simp |
|
344 |
apply (erule monofun_cfun_arg) |
|
345 |
done |
|
346 |
||
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|
347 |
lemma injection_defined_rev: |
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|
348 |
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" |
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|
349 |
apply (drule_tac f=f in cfun_arg_cong) |
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|
350 |
apply (simp add: retraction_strict) |
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351 |
done |
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|
352 |
|
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|
353 |
lemma injection_defined: |
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|
354 |
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" |
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|
355 |
by (erule contrapos_nn, rule injection_defined_rev) |
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|
356 |
|
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|
357 |
text {* propagation of flatness and chain-finiteness by retractions *} |
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|
358 |
|
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|
359 |
lemma chfin2chfin: |
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|
360 |
"\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y |
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|
361 |
\<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" |
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|
362 |
apply clarify |
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|
363 |
apply (drule_tac f=g in chain_monofun) |
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|
364 |
apply (drule chfin [rule_format]) |
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|
365 |
apply (unfold max_in_chain_def) |
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|
366 |
apply (simp add: injection_eq) |
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|
367 |
done |
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|
368 |
|
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|
369 |
lemma flat2flat: |
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|
370 |
"\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y |
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|
371 |
\<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y" |
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|
372 |
apply clarify |
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|
373 |
apply (drule_tac f=g in monofun_cfun_arg) |
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|
374 |
apply (drule ax_flat [rule_format]) |
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|
375 |
apply (erule disjE) |
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|
376 |
apply (simp add: injection_defined_rev) |
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|
377 |
apply (simp add: injection_eq) |
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|
378 |
done |
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379 |
|
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380 |
text {* a result about functions with flat codomain *} |
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|
381 |
|
16085
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|
382 |
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" |
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|
383 |
by (drule ax_flat [rule_format], simp) |
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|
384 |
|
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|
385 |
lemma flat_codom: |
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|
386 |
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" |
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387 |
apply (case_tac "f\<cdot>x = \<bottom>") |
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|
388 |
apply (rule disjI1) |
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389 |
apply (rule UU_I) |
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390 |
apply (erule_tac t="\<bottom>" in subst) |
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|
391 |
apply (rule minimal [THEN monofun_cfun_arg]) |
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|
392 |
apply clarify |
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|
393 |
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) |
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|
394 |
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) |
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|
395 |
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) |
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|
396 |
done |
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|
397 |
|
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|
398 |
|
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|
399 |
subsection {* Identity and composition *} |
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|
400 |
|
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401 |
consts |
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|
402 |
ID :: "'a \<rightarrow> 'a" |
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|
403 |
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" |
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|
404 |
|
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|
405 |
syntax "@oo" :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) |
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|
406 |
|
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|
407 |
translations "f1 oo f2" == "cfcomp$f1$f2" |
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|
408 |
|
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|
409 |
defs |
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|
410 |
ID_def: "ID \<equiv> (\<Lambda> x. x)" |
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|
411 |
oo_def: "cfcomp \<equiv> (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" |
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|
412 |
|
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|
413 |
lemma ID1 [simp]: "ID\<cdot>x = x" |
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|
414 |
by (simp add: ID_def) |
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changeset
|
415 |
|
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|
416 |
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" |
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417 |
by (simp add: oo_def) |
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changeset
|
418 |
|
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|
419 |
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" |
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420 |
by (simp add: cfcomp1) |
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|
421 |
|
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|
422 |
text {* |
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|
423 |
Show that interpretation of (pcpo,@{text "_->_"}) is a category. |
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|
424 |
The class of objects is interpretation of syntactical class pcpo. |
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|
425 |
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}. |
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|
426 |
The identity arrow is interpretation of @{term ID}. |
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|
427 |
The composition of f and g is interpretation of @{text "oo"}. |
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|
428 |
*} |
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|
429 |
|
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|
430 |
lemma ID2 [simp]: "f oo ID = f" |
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431 |
by (rule ext_cfun, simp) |
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|
432 |
|
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|
433 |
lemma ID3 [simp]: "ID oo f = f" |
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|
434 |
by (rule ext_cfun, simp) |
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changeset
|
435 |
|
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|
436 |
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" |
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437 |
by (rule ext_cfun, simp) |
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changeset
|
438 |
|
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|
439 |
|
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|
440 |
subsection {* Strictified functions *} |
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|
441 |
|
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|
442 |
defaultsort pcpo |
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|
443 |
|
17815 | 444 |
constdefs |
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|
445 |
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" |
17815 | 446 |
"strictify \<equiv> (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
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|
447 |
|
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|
448 |
text {* results about strictify *} |
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|
449 |
|
17815 | 450 |
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
451 |
by (simp add: cont_if) |
|
16085
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|
452 |
|
17815 | 453 |
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
454 |
apply (rule monofunI) |
|
455 |
apply (auto simp add: monofun_cfun_arg eq_UU_iff [symmetric]) |
|
16085
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|
456 |
done |
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|
457 |
|
17815 | 458 |
(*FIXME: long proof*) |
459 |
lemma contlub_strictify2: "contlub (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
|
16209
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changeset
|
460 |
apply (rule contlubI) |
16085
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changeset
|
461 |
apply (case_tac "lub (range Y) = \<bottom>") |
16699 | 462 |
apply (drule (1) chain_UU_I) |
17815 | 463 |
apply (simp add: thelub_const) |
464 |
apply (simp del: if_image_distrib) |
|
465 |
apply (simp only: contlub_cfun_arg) |
|
16085
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changeset
|
466 |
apply (rule lub_equal2) |
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changeset
|
467 |
apply (rule chain_mono2 [THEN exE]) |
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changeset
|
468 |
apply (erule chain_UU_I_inverse2) |
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changeset
|
469 |
apply (assumption) |
17815 | 470 |
apply (rule_tac x=x in exI, clarsimp) |
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changeset
|
471 |
apply (erule chain_monofun) |
17815 | 472 |
apply (erule monofun_strictify2 [THEN ch2ch_monofun]) |
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changeset
|
473 |
done |
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changeset
|
474 |
|
17815 | 475 |
lemmas cont_strictify2 = |
476 |
monocontlub2cont [OF monofun_strictify2 contlub_strictify2, standard] |
|
477 |
||
478 |
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" |
|
479 |
by (unfold strictify_def, simp add: cont_strictify1 cont_strictify2) |
|
16085
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changeset
|
480 |
|
c004b9bc970e
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changeset
|
481 |
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" |
17815 | 482 |
by (simp add: strictify_conv_if) |
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changeset
|
483 |
|
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changeset
|
484 |
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" |
17815 | 485 |
by (simp add: strictify_conv_if) |
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changeset
|
486 |
|
17816
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changeset
|
487 |
subsection {* Continuous let-bindings *} |
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changeset
|
488 |
|
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changeset
|
489 |
constdefs |
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parents:
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changeset
|
490 |
CLet :: "'a \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'b" |
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parents:
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changeset
|
491 |
"CLet \<equiv> \<Lambda> s f. f\<cdot>s" |
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changeset
|
492 |
|
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parents:
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changeset
|
493 |
syntax |
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changeset
|
494 |
"_CLet" :: "[letbinds, 'a] => 'a" ("(Let (_)/ in (_))" 10) |
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changeset
|
495 |
|
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parents:
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changeset
|
496 |
translations |
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parents:
17815
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changeset
|
497 |
"_CLet (_binds b bs) e" == "_CLet b (_CLet bs e)" |
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parents:
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changeset
|
498 |
"Let x = a in e" == "CLet$a$(LAM x. e)" |
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changeset
|
499 |
|
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changeset
|
500 |
end |