src/HOLCF/Cfun.thy

add translation for wildcard pattern

(*  Title:      HOLCF/Cfun.thy
    ID:         $Id$
    Author:     Franz Regensburger

Definition of the type ->  of continuous functions.
*)

header {* The type of continuous functions *}

theory Cfun
imports Pcpodef
uses ("cont_proc.ML")
begin

defaultsort cpo

subsection {* Definition of continuous function type *}

lemma Ex_cont: "\<exists>f. cont f"
by (rule exI, rule cont_const)

lemma adm_cont: "adm cont"
by (rule admI, rule cont_lub_fun)

cpodef (CFun)  ('a, 'b) "->" (infixr "->" 0) = "{f::'a => 'b. cont f}"
by (simp add: Ex_cont adm_cont)

syntax (xsymbols)
  "->"     :: "[type, type] => type"      ("(_ \<rightarrow>/ _)" [1,0]0)

syntax
  Rep_CFun :: "('a \<rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("_$_" [999,1000] 999)

syntax (xsymbols)
  Rep_CFun :: "('a \<rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("(_\<cdot>_)" [999,1000] 999)

syntax (HTML output)
  Rep_CFun :: "('a \<rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)" ("(_\<cdot>_)" [999,1000] 999)

subsection {* Syntax for continuous lambda abstraction *}

syntax "_cabs" :: "'a"

parse_translation {*
(* rewrites (_cabs x t) --> (Abs_CFun (%x. t)) *)
  [mk_binder_tr ("_cabs", "Abs_CFun")];
*}

text {* To avoid eta-contraction of body: *}
print_translation {*
  let
    fun cabs_tr' [Abs abs] =
      let val (x,t) = atomic_abs_tr' abs
      in Syntax.const "_cabs" $ x $ t end
  in [("Abs_CFun", cabs_tr')] end;
*}

text {* syntax for nested abstractions *}

syntax
  "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic"  ("(3LAM _./ _)" [1000, 10] 10)

syntax (xsymbols)
  "_Lambda" :: "[cargs, 'a] \<Rightarrow> logic" ("(3\<Lambda>_./ _)" [1000, 10] 10)

parse_ast_translation {*
(* rewrites (LAM x y z. t) --> (_cabs x (_cabs y (_cabs z t))) *)
(* cf. Syntax.lambda_ast_tr from Syntax/syn_trans.ML *)
  let
    fun Lambda_ast_tr [pats, body] =
          Syntax.fold_ast_p "_cabs" (Syntax.unfold_ast "_cargs" pats, body)
      | Lambda_ast_tr asts = raise Syntax.AST ("Lambda_ast_tr", asts);
  in [("_Lambda", Lambda_ast_tr)] end;
*}

print_ast_translation {*
(* rewrites (_cabs x (_cabs y (_cabs z t))) --> (LAM x y z. t) *)
(* cf. Syntax.abs_ast_tr' from Syntax/syn_trans.ML *)
  let
    fun cabs_ast_tr' asts =
      (case Syntax.unfold_ast_p "_cabs"
          (Syntax.Appl (Syntax.Constant "_cabs" :: asts)) of
        ([], _) => raise Syntax.AST ("cabs_ast_tr'", asts)
      | (xs, body) => Syntax.Appl
          [Syntax.Constant "_Lambda", Syntax.fold_ast "_cargs" xs, body]);
  in [("_cabs", cabs_ast_tr')] end;
*}

translations
  "\<Lambda> _. t" == "Abs_CFun (\<lambda> _. t)"

subsection {* Continuous function space is pointed *}

lemma UU_CFun: "\<bottom> \<in> CFun"
by (simp add: CFun_def inst_fun_pcpo cont_const)

instance "->" :: (cpo, pcpo) pcpo
by (rule typedef_pcpo [OF type_definition_CFun less_CFun_def UU_CFun])

lemmas Rep_CFun_strict =
  typedef_Rep_strict [OF type_definition_CFun less_CFun_def UU_CFun]

lemmas Abs_CFun_strict =
  typedef_Abs_strict [OF type_definition_CFun less_CFun_def UU_CFun]

text {* function application is strict in its first argument *}

lemma Rep_CFun_strict1 [simp]: "\<bottom>\<cdot>x = \<bottom>"
by (simp add: Rep_CFun_strict)

text {* for compatibility with old HOLCF-Version *}
lemma inst_cfun_pcpo: "\<bottom> = (\<Lambda> x. \<bottom>)"
by (simp add: inst_fun_pcpo [symmetric] Abs_CFun_strict)

subsection {* Basic properties of continuous functions *}

text {* Beta-equality for continuous functions *}

lemma Abs_CFun_inverse2: "cont f \<Longrightarrow> Rep_CFun (Abs_CFun f) = f"
by (simp add: Abs_CFun_inverse CFun_def)

lemma beta_cfun [simp]: "cont f \<Longrightarrow> (\<Lambda> x. f x)\<cdot>u = f u"
by (simp add: Abs_CFun_inverse2)

text {* Eta-equality for continuous functions *}

lemma eta_cfun: "(\<Lambda> x. f\<cdot>x) = f"
by (rule Rep_CFun_inverse)

text {* Extensionality for continuous functions *}

lemma expand_cfun_eq: "(f = g) = (\<forall>x. f\<cdot>x = g\<cdot>x)"
by (simp add: Rep_CFun_inject [symmetric] expand_fun_eq)

lemma ext_cfun: "(\<And>x. f\<cdot>x = g\<cdot>x) \<Longrightarrow> f = g"
by (simp add: expand_cfun_eq)

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

lemma expand_cfun_less: "f \<sqsubseteq> g = (\<forall>x. f\<cdot>x \<sqsubseteq> g\<cdot>x)" 
by (simp add: less_CFun_def expand_fun_less)

lemma less_cfun_ext: "(\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x) \<Longrightarrow> f \<sqsubseteq> g"
by (simp add: expand_cfun_less)

text {* Congruence for continuous function application *}

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

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

lemma cfun_arg_cong: "x = y \<Longrightarrow> f\<cdot>x = f\<cdot>y"
by simp

subsection {* Continuity of application *}

lemma cont_Rep_CFun1: "cont (\<lambda>f. f\<cdot>x)"
by (rule cont_Rep_CFun [THEN cont2cont_CF1L])

lemma cont_Rep_CFun2: "cont (\<lambda>x. f\<cdot>x)"
apply (rule_tac P = "cont" in CollectD)
apply (fold CFun_def)
apply (rule Rep_CFun)
done

lemmas monofun_Rep_CFun = cont_Rep_CFun [THEN cont2mono]
lemmas contlub_Rep_CFun = cont_Rep_CFun [THEN cont2contlub]

lemmas monofun_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2mono, standard]
lemmas contlub_Rep_CFun1 = cont_Rep_CFun1 [THEN cont2contlub, standard]
lemmas monofun_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2mono, standard]
lemmas contlub_Rep_CFun2 = cont_Rep_CFun2 [THEN cont2contlub, standard]

text {* contlub, cont properties of @{term Rep_CFun} in each argument *}

lemma contlub_cfun_arg: "chain Y \<Longrightarrow> f\<cdot>(lub (range Y)) = (\<Squnion>i. f\<cdot>(Y i))"
by (rule contlub_Rep_CFun2 [THEN contlubE])

lemma cont_cfun_arg: "chain Y \<Longrightarrow> range (\<lambda>i. f\<cdot>(Y i)) <<| f\<cdot>(lub (range Y))"
by (rule cont_Rep_CFun2 [THEN contE])

lemma contlub_cfun_fun: "chain F \<Longrightarrow> lub (range F)\<cdot>x = (\<Squnion>i. F i\<cdot>x)"
by (rule contlub_Rep_CFun1 [THEN contlubE])

lemma cont_cfun_fun: "chain F \<Longrightarrow> range (\<lambda>i. F i\<cdot>x) <<| lub (range F)\<cdot>x"
by (rule cont_Rep_CFun1 [THEN contE])

text {* monotonicity of application *}

lemma monofun_cfun_fun: "f \<sqsubseteq> g \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>x"
by (simp add: expand_cfun_less)

lemma monofun_cfun_arg: "x \<sqsubseteq> y \<Longrightarrow> f\<cdot>x \<sqsubseteq> f\<cdot>y"
by (rule monofun_Rep_CFun2 [THEN monofunE])

lemma monofun_cfun: "\<lbrakk>f \<sqsubseteq> g; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f\<cdot>x \<sqsubseteq> g\<cdot>y"
by (rule trans_less [OF monofun_cfun_fun monofun_cfun_arg])

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

lemma chain_monofun: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
by (erule monofun_Rep_CFun2 [THEN ch2ch_monofun])

lemma ch2ch_Rep_CFunR: "chain Y \<Longrightarrow> chain (\<lambda>i. f\<cdot>(Y i))"
by (rule monofun_Rep_CFun2 [THEN ch2ch_monofun])

lemma ch2ch_Rep_CFunL: "chain F \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>x)"
by (rule monofun_Rep_CFun1 [THEN ch2ch_monofun])

lemma ch2ch_Rep_CFun [simp]:
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. (F i)\<cdot>(Y i))"
apply (rule chainI)
apply (rule monofun_cfun)
apply (erule chainE)
apply (erule chainE)
done

text {* contlub, cont properties of @{term Rep_CFun} in both arguments *}

lemma contlub_cfun: 
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i) = (\<Squnion>i. F i\<cdot>(Y i))"
by (simp add: contlub_cfun_fun contlub_cfun_arg diag_lub)

lemma cont_cfun: 
  "\<lbrakk>chain F; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. F i\<cdot>(Y i)) <<| (\<Squnion>i. F i)\<cdot>(\<Squnion>i. Y i)"
apply (rule thelubE)
apply (simp only: ch2ch_Rep_CFun)
apply (simp only: contlub_cfun)
done

text {* strictness *}

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

text {* the lub of a chain of continous functions is monotone *}

lemma lub_cfun_mono: "chain F \<Longrightarrow> monofun (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
apply (drule ch2ch_monofun [OF monofun_Rep_CFun])
apply (simp add: thelub_fun [symmetric])
apply (erule monofun_lub_fun)
apply (simp add: monofun_Rep_CFun2)
done

text {* a lemma about the exchange of lubs for type @{typ "'a -> 'b"} *}

lemma ex_lub_cfun:
  "\<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))"
by (simp add: diag_lub)

text {* the lub of a chain of cont. functions is continuous *}

lemma cont_lub_cfun: "chain F \<Longrightarrow> cont (\<lambda>x. \<Squnion>i. F i\<cdot>x)"
apply (rule cont2cont_lub)
apply (erule monofun_Rep_CFun [THEN ch2ch_monofun])
apply (rule cont_Rep_CFun2)
done

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

lemma lub_cfun: "chain F \<Longrightarrow> range F <<| (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
by (simp only: contlub_cfun_fun [symmetric] eta_cfun thelubE)

lemma thelub_cfun: "chain F \<Longrightarrow> lub (range F) = (\<Lambda> x. \<Squnion>i. F i\<cdot>x)"
by (rule lub_cfun [THEN thelubI])

subsection {* Continuity simplification procedure *}

text {* cont2cont lemma for @{term Rep_CFun} *}

lemma cont2cont_Rep_CFun:
  "\<lbrakk>cont f; cont t\<rbrakk> \<Longrightarrow> cont (\<lambda>x. (f x)\<cdot>(t x))"
by (best intro: cont2cont_app2 cont_const cont_Rep_CFun cont_Rep_CFun2)

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

lemma cont2mono_LAM:
assumes p1: "!!x. cont(c1 x)"
assumes p2: "!!y. monofun(%x. c1 x y)"
shows "monofun(%x. LAM y. c1 x y)"
apply (rule monofunI)
apply (rule less_cfun_ext)
apply (simp add: p1)
apply (erule p2 [THEN monofunE])
done

text {* cont2cont Lemma for @{term "%x. LAM y. c1 x y"} *}

lemma cont2cont_LAM:
assumes p1: "!!x. cont(c1 x)"
assumes p2: "!!y. cont(%x. c1 x y)"
shows "cont(%x. LAM y. c1 x y)"
apply (rule cont_Abs_CFun)
apply (simp add: p1 CFun_def)
apply (simp add: p2 cont2cont_lambda)
done

text {* continuity simplification procedure *}

lemmas cont_lemmas1 =
  cont_const cont_id cont_Rep_CFun2 cont2cont_Rep_CFun cont2cont_LAM

use "cont_proc.ML";
setup ContProc.setup;

(*val cont_tac = (fn i => (resolve_tac cont_lemmas i));*)
(*val cont_tacR = (fn i => (REPEAT (cont_tac i)));*)

subsection {* Miscellaneous *}

text {* Monotonicity of @{term Abs_CFun} *}

lemma semi_monofun_Abs_CFun:
  "\<lbrakk>cont f; cont g; f \<sqsubseteq> g\<rbrakk> \<Longrightarrow> Abs_CFun f \<sqsubseteq> Abs_CFun g"
by (simp add: less_CFun_def Abs_CFun_inverse2)

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

lemma chfin_Rep_CFunR: "chain (Y::nat => 'a::cpo->'b::chfin)  
      ==> !s. ? n. lub(range(Y))$s = Y n$s"
apply (rule allI)
apply (subst contlub_cfun_fun)
apply assumption
apply (fast intro!: thelubI chfin lub_finch2 chfin2finch ch2ch_Rep_CFunL)
done

subsection {* Continuous injection-retraction pairs *}

text {* Continuous retractions are strict. *}

lemma retraction_strict:
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> f\<cdot>\<bottom> = \<bottom>"
apply (rule UU_I)
apply (drule_tac x="\<bottom>" in spec)
apply (erule subst)
apply (rule monofun_cfun_arg)
apply (rule minimal)
done

lemma injection_eq:
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x = g\<cdot>y) = (x = y)"
apply (rule iffI)
apply (drule_tac f=f in cfun_arg_cong)
apply simp
apply simp
done

lemma injection_less:
  "\<forall>x. f\<cdot>(g\<cdot>x) = x \<Longrightarrow> (g\<cdot>x \<sqsubseteq> g\<cdot>y) = (x \<sqsubseteq> y)"
apply (rule iffI)
apply (drule_tac f=f in monofun_cfun_arg)
apply simp
apply (erule monofun_cfun_arg)
done

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

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

text {* propagation of flatness and chain-finiteness by retractions *}

lemma chfin2chfin:
  "\<forall>y. (f::'a::chfin \<rightarrow> 'b)\<cdot>(g\<cdot>y) = y
    \<Longrightarrow> \<forall>Y::nat \<Rightarrow> 'b. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
apply clarify
apply (drule_tac f=g in chain_monofun)
apply (drule chfin [rule_format])
apply (unfold max_in_chain_def)
apply (simp add: injection_eq)
done

lemma flat2flat:
  "\<forall>y. (f::'a::flat \<rightarrow> 'b::pcpo)\<cdot>(g\<cdot>y) = y
    \<Longrightarrow> \<forall>x y::'b. x \<sqsubseteq> y \<longrightarrow> x = \<bottom> \<or> x = y"
apply clarify
apply (drule_tac f=g in monofun_cfun_arg)
apply (drule ax_flat [rule_format])
apply (erule disjE)
apply (simp add: injection_defined_rev)
apply (simp add: injection_eq)
done

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

lemma flat_eqI: "\<lbrakk>(x::'a::flat) \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> x = y"
by (drule ax_flat [rule_format], simp)

lemma flat_codom:
  "f\<cdot>x = (c::'b::flat) \<Longrightarrow> f\<cdot>\<bottom> = \<bottom> \<or> (\<forall>z. f\<cdot>z = c)"
apply (case_tac "f\<cdot>x = \<bottom>")
apply (rule disjI1)
apply (rule UU_I)
apply (erule_tac t="\<bottom>" in subst)
apply (rule minimal [THEN monofun_cfun_arg])
apply clarify
apply (rule_tac a = "f\<cdot>\<bottom>" in refl [THEN box_equals])
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
apply (erule minimal [THEN monofun_cfun_arg, THEN flat_eqI])
done


subsection {* Identity and composition *}

consts
  ID      :: "'a \<rightarrow> 'a"
  cfcomp  :: "('b \<rightarrow> 'c) \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'c"

syntax  "@oo" :: "['b \<rightarrow> 'c, 'a \<rightarrow> 'b] \<Rightarrow> 'a \<rightarrow> 'c" (infixr "oo" 100)
     
translations  "f oo g" == "cfcomp\<cdot>f\<cdot>g"

defs
  ID_def: "ID \<equiv> (\<Lambda> x. x)"
  oo_def: "cfcomp \<equiv> (\<Lambda> f g x. f\<cdot>(g\<cdot>x))" 

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

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

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

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

lemma ID2 [simp]: "f oo ID = f"
by (rule ext_cfun, simp)

lemma ID3 [simp]: "ID oo f = f"
by (rule ext_cfun, simp)

lemma assoc_oo: "f oo (g oo h) = (f oo g) oo h"
by (rule ext_cfun, simp)


subsection {* Strictified functions *}

defaultsort pcpo

constdefs
  strictify  :: "('a \<rightarrow> 'b) \<rightarrow> 'a \<rightarrow> 'b"
  "strictify \<equiv> (\<Lambda> f x. if x = \<bottom> then \<bottom> else f\<cdot>x)"

text {* results about strictify *}

lemma cont_strictify1: "cont (\<lambda>f. if x = \<bottom> then \<bottom> else f\<cdot>x)"
by (simp add: cont_if)

lemma monofun_strictify2: "monofun (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
apply (rule monofunI)
apply (auto simp add: monofun_cfun_arg eq_UU_iff [symmetric])
done

(*FIXME: long proof*)
lemma contlub_strictify2: "contlub (\<lambda>x. if x = \<bottom> then \<bottom> else f\<cdot>x)"
apply (rule contlubI)
apply (case_tac "lub (range Y) = \<bottom>")
apply (drule (1) chain_UU_I)
apply simp
apply (simp del: if_image_distrib)
apply (simp only: contlub_cfun_arg)
apply (rule lub_equal2)
apply (rule chain_mono2 [THEN exE])
apply (erule chain_UU_I_inverse2)
apply (assumption)
apply (rule_tac x=x in exI, clarsimp)
apply (erule chain_monofun)
apply (erule monofun_strictify2 [THEN ch2ch_monofun])
done

lemmas cont_strictify2 =
  monocontlub2cont [OF monofun_strictify2 contlub_strictify2, standard]

lemma strictify_conv_if: "strictify\<cdot>f\<cdot>x = (if x = \<bottom> then \<bottom> else f\<cdot>x)"
by (unfold strictify_def, simp add: cont_strictify1 cont_strictify2)

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

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

subsection {* Continuous let-bindings *}

constdefs
  CLet :: "'a \<rightarrow> ('a \<rightarrow> 'b) \<rightarrow> 'b"
  "CLet \<equiv> \<Lambda> s f. f\<cdot>s"

syntax
  "_CLet" :: "[letbinds, 'a] => 'a" ("(Let (_)/ in (_))" 10)

translations
  "_CLet (_binds b bs) e" == "_CLet b (_CLet bs e)"
  "Let x = a in e" == "CLet\<cdot>a\<cdot>(\<Lambda> x. e)"

end