author | huffman |
Tue, 12 Oct 2010 09:08:27 -0700 | |
changeset 40008 | 58ead6f77f8e |
parent 40006 | 116e94f9543b |
child 40011 | b974cf829099 |
permissions | -rw-r--r-- |
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(* Title: HOLCF/Cfun.thy |
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Author: Franz Regensburger |
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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 Pcpodef 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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lemma Ex_cont: "\<exists>f. cont f" |
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by (rule exI, rule cont_const) |
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||
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lemma adm_cont: "adm cont" |
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by (rule admI, rule cont_lub_fun) |
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||
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cpodef (CFun) ('a, 'b) cfun (infixr "->" 0) = "{f::'a => 'b. cont f}" |
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by (simp_all add: Ex_cont 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 eta-contraction 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 :: (finite_po, finite_po) finite_po |
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by (rule typedef_finite_po [OF type_definition_CFun]) |
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instance cfun :: (finite_po, chfin) chfin |
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by (rule typedef_chfin [OF type_definition_CFun below_CFun_def]) |
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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 HOLCF-Version *} |
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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 {* Beta-equality 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 {* Beta-reduction 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 deeply-nested 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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|
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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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|
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text {* Eta-equality for continuous functions *} |
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|
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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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|
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text {* Extensionality for continuous functions *} |
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|
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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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181 |
|
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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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184 |
|
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text {* Extensionality wrt. ordering for continuous functions *} |
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|
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lemma cfun_below_iff: "f \<sqsubseteq> g \<longleftrightarrow> (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" |
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by (simp add: below_CFun_def fun_below_iff) |
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189 |
|
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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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|
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text {* Congruence for continuous function application *} |
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194 |
|
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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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197 |
|
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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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200 |
|
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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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|
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subsection {* Continuity of application *} |
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|
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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_fun]) |
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|
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lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)" |
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apply (cut_tac x=f in Rep_CFun) |
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apply (simp add: CFun_def) |
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done |
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|
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lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono] |
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|
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lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard] |
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lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard] |
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218 |
|
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text {* contlub, cont properties of @{term Rep_CFun} in each argument *} |
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|
27413 | 221 |
lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. f\<cdot>(Y i))" |
35914 | 222 |
by (rule cont_Rep_CFun2 [THEN cont2contlubE]) |
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|
27413 | 224 |
lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(\<Squnion>i. Y i)" |
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225 |
by (rule cont_Rep_CFun2 [THEN contE]) |
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226 |
|
27413 | 227 |
lemma contlub_cfun_fun: "chain F \<Longrightarrow> (\<Squnion>i. F i)\<cdot>x = (\<Squnion>i. F i\<cdot>x)" |
35914 | 228 |
by (rule cont_Rep_CFun1 [THEN cont2contlubE]) |
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|
27413 | 230 |
lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| (\<Squnion>i. F i)\<cdot>x" |
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231 |
by (rule cont_Rep_CFun1 [THEN contE]) |
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232 |
|
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text {* monotonicity of application *} |
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234 |
|
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lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x" |
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236 |
by (simp add: cfun_below_iff) |
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237 |
|
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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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240 |
|
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241 |
lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y" |
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242 |
by (rule below_trans [OF monofun_cfun_fun monofun_cfun_arg]) |
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243 |
|
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244 |
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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247 |
by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun]) |
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248 |
|
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lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))" |
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250 |
by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun]) |
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251 |
|
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lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)" |
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253 |
by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun]) |
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254 |
|
18076 | 255 |
lemma ch2ch_Rep_CFun [simp]: |
256 |
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))" |
|
25884 | 257 |
by (simp add: chain_def monofun_cfun) |
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258 |
|
25884 | 259 |
lemma ch2ch_LAM [simp]: |
260 |
"\<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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|
261 |
by (simp add: chain_def cfun_below_iff) |
18092
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parents:
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changeset
|
262 |
|
16209
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huffman
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|
263 |
text {* contlub, cont properties of @{term Rep_CFun} in both arguments *} |
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|
264 |
|
16209
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huffman
parents:
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|
265 |
lemma contlub_cfun: |
36ee7f6af79f
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huffman
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|
266 |
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))" |
18076 | 267 |
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub) |
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|
268 |
|
16209
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|
269 |
lemma cont_cfun: |
36ee7f6af79f
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|
270 |
"\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)" |
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
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|
271 |
apply (rule thelubE) |
36ee7f6af79f
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huffman
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changeset
|
272 |
apply (simp only: ch2ch_Rep_CFun) |
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
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|
273 |
apply (simp only: contlub_cfun) |
36ee7f6af79f
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huffman
parents:
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changeset
|
274 |
done |
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
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changeset
|
275 |
|
18092
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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diff
changeset
|
276 |
lemma contlub_LAM: |
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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diff
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|
277 |
"\<lbrakk>\<And>x. chain (\<lambda>i. F i x); \<And>i. cont (\<lambda>x. F i x)\<rbrakk> |
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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diff
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|
278 |
\<Longrightarrow> (\<Lambda> x. \<Squnion>i. F i x) = (\<Squnion>i. \<Lambda> x. F i x)" |
25884 | 279 |
apply (simp add: thelub_CFun) |
18092
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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diff
changeset
|
280 |
apply (simp add: Abs_CFun_inverse2) |
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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diff
changeset
|
281 |
apply (simp add: thelub_fun ch2ch_lambda) |
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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changeset
|
282 |
done |
2c5d5da79a1e
renamed and added ch2ch, cont2cont, mono2mono theorems ending in _fun, _lambda, _LAM
huffman
parents:
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diff
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|
283 |
|
25901 | 284 |
lemmas lub_distribs = |
285 |
contlub_cfun [symmetric] |
|
286 |
contlub_LAM [symmetric] |
|
287 |
||
16209
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huffman
parents:
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changeset
|
288 |
text {* strictness *} |
36ee7f6af79f
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parents:
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changeset
|
289 |
|
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
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changeset
|
290 |
lemma strictI: "f\<cdot>x = \<bottom> \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" |
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
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diff
changeset
|
291 |
apply (rule UU_I) |
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changeset
|
292 |
apply (erule subst) |
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|
293 |
apply (rule minimal [THEN monofun_cfun_arg]) |
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changeset
|
294 |
done |
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changeset
|
295 |
|
15589
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|
296 |
text {* type @{typ "'a -> 'b"} is chain complete *} |
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changeset
|
297 |
|
16920 | 298 |
lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" |
299 |
by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE) |
|
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|
300 |
|
27413 | 301 |
lemma thelub_cfun: "chain F \<Longrightarrow> (\<Squnion>i. F i) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)" |
16920 | 302 |
by (rule lub_cfun [THEN thelubI]) |
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changeset
|
303 |
|
17832
e18fc1a9a0e0
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huffman
parents:
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diff
changeset
|
304 |
subsection {* Continuity simplification procedure *} |
15589
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parents:
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changeset
|
305 |
|
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|
306 |
text {* cont2cont lemma for @{term Rep_CFun} *} |
15576
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|
307 |
|
37079
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huffman
parents:
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diff
changeset
|
308 |
lemma cont2cont_Rep_CFun [simp, cont2cont]: |
29049 | 309 |
assumes f: "cont (\<lambda>x. f x)" |
310 |
assumes t: "cont (\<lambda>x. t x)" |
|
311 |
shows "cont (\<lambda>x. (f x)\<cdot>(t x))" |
|
312 |
proof - |
|
40006 | 313 |
have 1: "\<And>y. cont (\<lambda>x. (f x)\<cdot>y)" |
314 |
using cont_Rep_CFun1 f by (rule cont_compose) |
|
315 |
show "cont (\<lambda>x. (f x)\<cdot>(t x))" |
|
316 |
using t cont_Rep_CFun2 1 by (rule cont_apply) |
|
29049 | 317 |
qed |
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|
318 |
|
40008 | 319 |
text {* |
320 |
Two specific lemmas for the combination of LCF and HOL terms. |
|
321 |
These lemmas are needed in theories that use types like @{typ "'a \<rightarrow> 'b \<Rightarrow> 'c"}. |
|
322 |
*} |
|
323 |
||
324 |
lemma cont_Rep_CFun_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s)" |
|
325 |
by (rule cont2cont_Rep_CFun [THEN cont2cont_fun]) |
|
326 |
||
327 |
lemma cont_Rep_CFun_app_app [simp]: "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. ((f x)\<cdot>(g x)) s t)" |
|
328 |
by (rule cont_Rep_CFun_app [THEN cont2cont_fun]) |
|
329 |
||
330 |
||
15589
69bea57212ef
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huffman
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15577
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changeset
|
331 |
text {* cont2mono Lemma for @{term "%x. LAM y. c1(x)(y)"} *} |
15576
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diff
changeset
|
332 |
|
efb95d0d01f7
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parents:
diff
changeset
|
333 |
lemma cont2mono_LAM: |
29049 | 334 |
"\<lbrakk>\<And>x. cont (\<lambda>y. f x y); \<And>y. monofun (\<lambda>x. f x y)\<rbrakk> |
335 |
\<Longrightarrow> monofun (\<lambda>x. \<Lambda> y. f x y)" |
|
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
336 |
unfolding monofun_def cfun_below_iff by simp |
15576
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huffman
parents:
diff
changeset
|
337 |
|
29049 | 338 |
text {* cont2cont Lemma for @{term "%x. LAM y. f x y"} *} |
15576
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changeset
|
339 |
|
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
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diff
changeset
|
340 |
text {* |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
341 |
Not suitable as a cont2cont rule, because on nested lambdas |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
342 |
it causes exponential blow-up in the number of subgoals. |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
343 |
*} |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
344 |
|
15576
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changeset
|
345 |
lemma cont2cont_LAM: |
29049 | 346 |
assumes f1: "\<And>x. cont (\<lambda>y. f x y)" |
347 |
assumes f2: "\<And>y. cont (\<lambda>x. f x y)" |
|
348 |
shows "cont (\<lambda>x. \<Lambda> y. f x y)" |
|
349 |
proof (rule cont_Abs_CFun) |
|
350 |
fix x |
|
351 |
from f1 show "f x \<in> CFun" by (simp add: CFun_def) |
|
352 |
from f2 show "cont f" by (rule cont2cont_lambda) |
|
353 |
qed |
|
15576
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changeset
|
354 |
|
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
355 |
text {* |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
356 |
This version does work as a cont2cont rule, since it |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
357 |
has only a single subgoal. |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
358 |
*} |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
359 |
|
37079
0cd15d8c90a0
remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents:
36452
diff
changeset
|
360 |
lemma cont2cont_LAM' [simp, cont2cont]: |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
361 |
fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
362 |
assumes f: "cont (\<lambda>p. f (fst p) (snd p))" |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
363 |
shows "cont (\<lambda>x. \<Lambda> y. f x y)" |
39808
1410c84013b9
rename cont2cont_split to cont2cont_prod_case; add lemmas prod_contI, prod_cont_iff; simplify some proofs
huffman
parents:
39302
diff
changeset
|
364 |
using assms by (simp add: cont2cont_LAM prod_cont_iff) |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
365 |
|
37079
0cd15d8c90a0
remove cont2cont simproc; instead declare cont2cont rules as simp rules
huffman
parents:
36452
diff
changeset
|
366 |
lemma cont2cont_LAM_discrete [simp, cont2cont]: |
29530
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
367 |
"(\<And>y::'a::discrete_cpo. cont (\<lambda>x. f x y)) \<Longrightarrow> cont (\<lambda>x. \<Lambda> y. f x y)" |
9905b660612b
change to simpler, more extensible continuity simproc
huffman
parents:
29138
diff
changeset
|
368 |
by (simp add: cont2cont_LAM) |
15576
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huffman
parents:
diff
changeset
|
369 |
|
16055 | 370 |
lemmas cont_lemmas1 = |
371 |
cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM |
|
372 |
||
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset
|
373 |
subsection {* Miscellaneous *} |
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset
|
374 |
|
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset
|
375 |
text {* Monotonicity of @{term Abs_CFun} *} |
15576
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huffman
parents:
diff
changeset
|
376 |
|
17832
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset
|
377 |
lemma semi_monofun_Abs_CFun: |
e18fc1a9a0e0
rearranged subsections; added theorems expand_cfun_eq, expand_cfun_less
huffman
parents:
17817
diff
changeset
|
378 |
"\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g" |
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
31041
diff
changeset
|
379 |
by (simp add: below_CFun_def Abs_CFun_inverse2) |
15576
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huffman
parents:
diff
changeset
|
380 |
|
15589
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huffman
parents:
15577
diff
changeset
|
381 |
text {* some lemmata for functions with flat/chfin domain/range types *} |
15576
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huffman
parents:
diff
changeset
|
382 |
|
efb95d0d01f7
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huffman
parents:
diff
changeset
|
383 |
lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin) |
27413 | 384 |
==> !s. ? n. (LUB i. Y i)$s = Y n$s" |
15576
efb95d0d01f7
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huffman
parents:
diff
changeset
|
385 |
apply (rule allI) |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
386 |
apply (subst contlub_cfun_fun) |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
387 |
apply assumption |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
388 |
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL) |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
389 |
done |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
390 |
|
18089 | 391 |
lemma adm_chfindom: "adm (\<lambda>(u::'a::cpo \<rightarrow> 'b::chfin). P(u\<cdot>s))" |
392 |
by (rule adm_subst, simp, rule adm_chfin) |
|
393 |
||
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
394 |
subsection {* Continuous injection-retraction pairs *} |
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
395 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
396 |
text {* Continuous retractions are strict. *} |
15576
efb95d0d01f7
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huffman
parents:
diff
changeset
|
397 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
398 |
lemma retraction_strict: |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
399 |
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>" |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
400 |
apply (rule UU_I) |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
401 |
apply (drule_tac x="\<bottom>" in spec) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
402 |
apply (erule subst) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
403 |
apply (rule monofun_cfun_arg) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
404 |
apply (rule minimal) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
405 |
done |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
406 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
407 |
lemma injection_eq: |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
408 |
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
409 |
apply (rule iffI) |
c004b9bc970e
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huffman
parents:
16070
diff
changeset
|
410 |
apply (drule_tac f=f in cfun_arg_cong) |
c004b9bc970e
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huffman
parents:
16070
diff
changeset
|
411 |
apply simp |
c004b9bc970e
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huffman
parents:
16070
diff
changeset
|
412 |
apply simp |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
413 |
done |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
414 |
|
31076
99fe356cbbc2
rename constant sq_le to below; rename class sq_ord to below; less->below in many lemma names
huffman
parents:
31041
diff
changeset
|
415 |
lemma injection_below: |
16314 | 416 |
"\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)" |
417 |
apply (rule iffI) |
|
418 |
apply (drule_tac f=f in monofun_cfun_arg) |
|
419 |
apply simp |
|
420 |
apply (erule monofun_cfun_arg) |
|
421 |
done |
|
422 |
||
16085
c004b9bc970e
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parents:
16070
diff
changeset
|
423 |
lemma injection_defined_rev: |
c004b9bc970e
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huffman
parents:
16070
diff
changeset
|
424 |
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; g\<cdot>z = \<bottom>\<rbrakk> \<Longrightarrow> z = \<bottom>" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
425 |
apply (drule_tac f=f in cfun_arg_cong) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
426 |
apply (simp add: retraction_strict) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
427 |
done |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
428 |
|
16085
c004b9bc970e
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huffman
parents:
16070
diff
changeset
|
429 |
lemma injection_defined: |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
430 |
"\<lbrakk>\<forall>x. f\<cdot>(g\<cdot>x) = x; z \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> g\<cdot>z \<noteq> \<bottom>" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
431 |
by (erule contrapos_nn, rule injection_defined_rev) |
c004b9bc970e
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huffman
parents:
16070
diff
changeset
|
432 |
|
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
433 |
text {* propagation of flatness and chain-finiteness by retractions *} |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
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diff
changeset
|
434 |
|
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
435 |
lemma chfin2chfin: |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
436 |
"\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
437 |
\<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
438 |
apply clarify |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
439 |
apply (drule_tac f=g in chain_monofun) |
25921 | 440 |
apply (drule chfin) |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
441 |
apply (unfold max_in_chain_def) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
442 |
apply (simp add: injection_eq) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
443 |
done |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
444 |
|
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
445 |
lemma flat2flat: |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
446 |
"\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
447 |
\<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
448 |
apply clarify |
16209
36ee7f6af79f
removed dependencies on MF2 lemmas; removed some obsolete theorems; cleaned up many proofs; renamed less_cfun2 to less_cfun_ext
huffman
parents:
16098
diff
changeset
|
449 |
apply (drule_tac f=g in monofun_cfun_arg) |
25920 | 450 |
apply (drule ax_flat) |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
451 |
apply (erule disjE) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
452 |
apply (simp add: injection_defined_rev) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
453 |
apply (simp add: injection_eq) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
454 |
done |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
455 |
|
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
456 |
text {* a result about functions with flat codomain *} |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
457 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
458 |
lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y" |
25920 | 459 |
by (drule ax_flat, simp) |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
460 |
|
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
461 |
lemma flat_codom: |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
462 |
"f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
463 |
apply (case_tac "f\<cdot>x = \<bottom>") |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
464 |
apply (rule disjI1) |
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
465 |
apply (rule UU_I) |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
466 |
apply (erule_tac t="\<bottom>" in subst) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
467 |
apply (rule minimal [THEN monofun_cfun_arg]) |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
468 |
apply clarify |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
469 |
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals]) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
470 |
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
471 |
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI]) |
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
472 |
done |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
473 |
|
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
474 |
subsection {* Identity and composition *} |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
475 |
|
25135
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
476 |
definition |
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
477 |
ID :: "'a \<rightarrow> 'a" where |
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
478 |
"ID = (\<Lambda> x. x)" |
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
479 |
|
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
480 |
definition |
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
481 |
cfcomp :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c" where |
4f8176c940cf
modernized specifications ('definition', 'axiomatization');
wenzelm
parents:
25131
diff
changeset
|
482 |
oo_def: "cfcomp = (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" |
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
483 |
|
25131
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset
|
484 |
abbreviation |
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset
|
485 |
cfcomp_syn :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100) where |
2c8caac48ade
modernized specifications ('definition', 'abbreviation', 'notation');
wenzelm
parents:
23152
diff
changeset
|
486 |
"f oo g == cfcomp\<cdot>f\<cdot>g" |
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
487 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
488 |
lemma ID1 [simp]: "ID\<cdot>x = x" |
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
489 |
by (simp add: ID_def) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
490 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
491 |
lemma cfcomp1: "(f oo g) = (\<Lambda> x. f\<cdot>(g\<cdot>x))" |
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
492 |
by (simp add: oo_def) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
493 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
494 |
lemma cfcomp2 [simp]: "(f oo g)\<cdot>x = f\<cdot>(g\<cdot>x)" |
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
495 |
by (simp add: cfcomp1) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
496 |
|
27274 | 497 |
lemma cfcomp_LAM: "cont g \<Longrightarrow> f oo (\<Lambda> x. g x) = (\<Lambda> x. f\<cdot>(g x))" |
498 |
by (simp add: cfcomp1) |
|
499 |
||
19709 | 500 |
lemma cfcomp_strict [simp]: "\<bottom> oo f = \<bottom>" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
501 |
by (simp add: cfun_eq_iff) |
19709 | 502 |
|
15589
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
503 |
text {* |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
504 |
Show that interpretation of (pcpo,@{text "_->_"}) is a category. |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
505 |
The class of objects is interpretation of syntactical class pcpo. |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
506 |
The class of arrows between objects @{typ 'a} and @{typ 'b} is interpret. of @{typ "'a -> 'b"}. |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
507 |
The identity arrow is interpretation of @{term ID}. |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
508 |
The composition of f and g is interpretation of @{text "oo"}. |
69bea57212ef
reordered and arranged for document generation, cleaned up some proofs
huffman
parents:
15577
diff
changeset
|
509 |
*} |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
510 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
511 |
lemma ID2 [simp]: "f oo ID = f" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
512 |
by (rule cfun_eqI, simp) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
513 |
|
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
514 |
lemma ID3 [simp]: "ID oo f = f" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
515 |
by (rule cfun_eqI, simp) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
516 |
|
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
517 |
lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
518 |
by (rule cfun_eqI, simp) |
15576
efb95d0d01f7
converted to new-style theories, and combined numbered files
huffman
parents:
diff
changeset
|
519 |
|
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
520 |
subsection {* Map operator for continuous function space *} |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
521 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
522 |
definition |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
523 |
cfun_map :: "('b \<rightarrow> 'a) \<rightarrow> ('c \<rightarrow> 'd) \<rightarrow> ('a \<rightarrow> 'c) \<rightarrow> ('b \<rightarrow> 'd)" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
524 |
where |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
525 |
"cfun_map = (\<Lambda> a b f x. b\<cdot>(f\<cdot>(a\<cdot>x)))" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
526 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
527 |
lemma cfun_map_beta [simp]: "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))" |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
528 |
unfolding cfun_map_def by simp |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
529 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
530 |
lemma cfun_map_ID: "cfun_map\<cdot>ID\<cdot>ID = ID" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
531 |
unfolding cfun_eq_iff by simp |
39985
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
532 |
|
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
533 |
lemma cfun_map_map: |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
534 |
"cfun_map\<cdot>f1\<cdot>g1\<cdot>(cfun_map\<cdot>f2\<cdot>g2\<cdot>p) = |
310f98585107
move stuff from Algebraic.thy to Bifinite.thy and elsewhere
huffman
parents:
39808
diff
changeset
|
535 |
cfun_map\<cdot>(\<Lambda> x. f2\<cdot>(f1\<cdot>x))\<cdot>(\<Lambda> x. g1\<cdot>(g2\<cdot>x))\<cdot>p" |
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
40001
diff
changeset
|
536 |
by (rule cfun_eqI) simp |
16085
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
537 |
|
c004b9bc970e
rewrote continuous isomorphism section, cleaned up
huffman
parents:
16070
diff
changeset
|
538 |
subsection {* Strictified functions *} |
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539 |
|
36452 | 540 |
default_sort pcpo |
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541 |
|
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definition |
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543 |
strictify :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b" where |
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544 |
"strictify = (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
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545 |
|
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546 |
text {* results about strictify *} |
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547 |
|
17815 | 548 |
lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
35168 | 549 |
by simp |
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550 |
|
17815 | 551 |
lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
552 |
apply (rule monofunI) |
|
25786 | 553 |
apply (auto simp add: monofun_cfun_arg) |
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554 |
done |
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|
555 |
|
35914 | 556 |
lemma cont_strictify2: "cont (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)" |
557 |
apply (rule contI2) |
|
558 |
apply (rule monofun_strictify2) |
|
559 |
apply (case_tac "(\<Squnion>i. Y i) = \<bottom>", simp) |
|
560 |
apply (simp add: contlub_cfun_arg del: if_image_distrib) |
|
561 |
apply (drule chain_UU_I_inverse2, clarify, rename_tac j) |
|
562 |
apply (rule lub_mono2, rule_tac x=j in exI, simp_all) |
|
563 |
apply (auto dest!: chain_mono_less) |
|
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564 |
done |
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565 |
|
17815 | 566 |
lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)" |
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567 |
unfolding strictify_def |
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568 |
by (simp add: cont_strictify1 cont_strictify2 cont2cont_LAM) |
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|
569 |
|
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|
570 |
lemma strictify1 [simp]: "strictify\<cdot>f\<cdot>\<bottom> = \<bottom>" |
17815 | 571 |
by (simp add: strictify_conv_if) |
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572 |
|
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|
573 |
lemma strictify2 [simp]: "x \<noteq> \<bottom> \<Longrightarrow> strictify\<cdot>f\<cdot>x = f\<cdot>x" |
17815 | 574 |
by (simp add: strictify_conv_if) |
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|
575 |
|
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576 |
subsection {* Continuity of let-bindings *} |
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|
577 |
|
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578 |
lemma cont2cont_Let: |
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579 |
assumes f: "cont (\<lambda>x. f x)" |
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580 |
assumes g1: "\<And>y. cont (\<lambda>x. g x y)" |
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581 |
assumes g2: "\<And>x. cont (\<lambda>y. g x y)" |
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|
582 |
shows "cont (\<lambda>x. let y = f x in g x y)" |
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|
583 |
unfolding Let_def using f g2 g1 by (rule cont_apply) |
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|
584 |
|
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585 |
lemma cont2cont_Let' [simp, cont2cont]: |
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586 |
assumes f: "cont (\<lambda>x. f x)" |
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|
587 |
assumes g: "cont (\<lambda>p. g (fst p) (snd p))" |
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|
588 |
shows "cont (\<lambda>x. let y = f x in g x y)" |
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|
589 |
using f |
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|
590 |
proof (rule cont2cont_Let) |
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|
591 |
fix x show "cont (\<lambda>y. g x y)" |
40003
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|
592 |
using g by (simp add: prod_cont_iff) |
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|
593 |
next |
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|
594 |
fix y show "cont (\<lambda>x. g x y)" |
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|
595 |
using g by (simp add: prod_cont_iff) |
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596 |
qed |
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|
597 |
|
39145 | 598 |
text {* The simple version (suggested by Joachim Breitner) is needed if |
599 |
the type of the defined term is not a cpo. *} |
|
600 |
||
601 |
lemma cont2cont_Let_simple [simp, cont2cont]: |
|
602 |
assumes "\<And>y. cont (\<lambda>x. g x y)" |
|
603 |
shows "cont (\<lambda>x. let y = t in g x y)" |
|
604 |
unfolding Let_def using assms . |
|
605 |
||
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606 |
end |