(*  Title:      HOLCF/Cont.thy

    Author:     Franz Regensburger

    Author:     Brian Huffman

*)

header {* Continuity and monotonicity *}

theory Cont

imports Pcpo

begin

text {*

   Now we change the default class! Form now on all untyped type variables are

   of default class po

*}

default_sort po

subsection {* Definitions *}

definition

  monofun :: "('a \<Rightarrow> 'b) \<Rightarrow> bool"  -- "monotonicity"  where

  "monofun f = (\<forall>x y. x \<sqsubseteq> y \<longrightarrow> f x \<sqsubseteq> f y)"

definition

  cont :: "('a::cpo \<Rightarrow> 'b::cpo) \<Rightarrow> bool"

where

  "cont f = (\<forall>Y. chain Y \<longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i))"

lemma contI:

  "\<lbrakk>\<And>Y. chain Y \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)\<rbrakk> \<Longrightarrow> cont f"

by (simp add: cont_def)

lemma contE:

  "\<lbrakk>cont f; chain Y\<rbrakk> \<Longrightarrow> range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)"

by (simp add: cont_def)

lemma monofunI: 

  "\<lbrakk>\<And>x y. x \<sqsubseteq> y \<Longrightarrow> f x \<sqsubseteq> f y\<rbrakk> \<Longrightarrow> monofun f"

by (simp add: monofun_def)

lemma monofunE: 

  "\<lbrakk>monofun f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> f x \<sqsubseteq> f y"

by (simp add: monofun_def)

subsection {* Equivalence of alternate definition *}

text {* monotone functions map chains to chains *}

lemma ch2ch_monofun: "\<lbrakk>monofun f; chain Y\<rbrakk> \<Longrightarrow> chain (\<lambda>i. f (Y i))"

apply (rule chainI)

apply (erule monofunE)

apply (erule chainE)

done

text {* monotone functions map upper bound to upper bounds *}

lemma ub2ub_monofun: 

  "\<lbrakk>monofun f; range Y <| u\<rbrakk> \<Longrightarrow> range (\<lambda>i. f (Y i)) <| f u"

apply (rule ub_rangeI)

apply (erule monofunE)

apply (erule ub_rangeD)

done

text {* a lemma about binary chains *}

lemma binchain_cont:

  "\<lbrakk>cont f; x \<sqsubseteq> y\<rbrakk> \<Longrightarrow> range (\<lambda>i::nat. f (if i = 0 then x else y)) <<| f y"

apply (subgoal_tac "f (\<Squnion>i::nat. if i = 0 then x else y) = f y")

apply (erule subst)

apply (erule contE)

apply (erule bin_chain)

apply (rule_tac f=f in arg_cong)

apply (erule lub_bin_chain [THEN thelubI])

done

text {* continuity implies monotonicity *}

lemma cont2mono: "cont f \<Longrightarrow> monofun f"

apply (rule monofunI)

apply (drule (1) binchain_cont)

apply (drule_tac i=0 in is_ub_lub)

apply simp

done

lemmas cont2monofunE = cont2mono [THEN monofunE]

lemmas ch2ch_cont = cont2mono [THEN ch2ch_monofun]

text {* continuity implies preservation of lubs *}

lemma cont2contlubE:

  "\<lbrakk>cont f; chain Y\<rbrakk> \<Longrightarrow> f (\<Squnion> i. Y i) = (\<Squnion> i. f (Y i))"

apply (rule thelubI [symmetric])

apply (erule (1) contE)

done

lemma contI2:

  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo"

  assumes mono: "monofun f"

  assumes below: "\<And>Y. \<lbrakk>chain Y; chain (\<lambda>i. f (Y i))\<rbrakk>

     \<Longrightarrow> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))"

  shows "cont f"

proof (rule contI)

  fix Y :: "nat \<Rightarrow> 'a"

  assume Y: "chain Y"

  with mono have fY: "chain (\<lambda>i. f (Y i))"

    by (rule ch2ch_monofun)

  have "(\<Squnion>i. f (Y i)) = f (\<Squnion>i. Y i)"

    apply (rule below_antisym)

    apply (rule lub_below [OF fY])

    apply (rule monofunE [OF mono])

    apply (rule is_ub_thelub [OF Y])

    apply (rule below [OF Y fY])

    done

  with fY show "range (\<lambda>i. f (Y i)) <<| f (\<Squnion>i. Y i)"

    by (rule thelubE)

qed

subsection {* Collection of continuity rules *}

ML {*

structure Cont2ContData = Named_Thms

(

  val name = "cont2cont"

  val description = "continuity intro rule"

)

*}

setup Cont2ContData.setup

subsection {* Continuity of basic functions *}

text {* The identity function is continuous *}

lemma cont_id [simp, cont2cont]: "cont (\<lambda>x. x)"

apply (rule contI)

apply (erule cpo_lubI)

done

text {* constant functions are continuous *}

lemma cont_const [simp, cont2cont]: "cont (\<lambda>x. c)"

apply (rule contI)

apply (rule lub_const)

done

text {* application of functions is continuous *}

lemma cont_apply:

  fixes f :: "'a::cpo \<Rightarrow> 'b::cpo \<Rightarrow> 'c::cpo" and t :: "'a \<Rightarrow> 'b"

  assumes 1: "cont (\<lambda>x. t x)"

  assumes 2: "\<And>x. cont (\<lambda>y. f x y)"

  assumes 3: "\<And>y. cont (\<lambda>x. f x y)"

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

proof (rule contI2 [OF monofunI])

  fix x y :: "'a" assume "x \<sqsubseteq> y"

  then show "f x (t x) \<sqsubseteq> f y (t y)"

    by (auto intro: cont2monofunE [OF 1]

                    cont2monofunE [OF 2]

                    cont2monofunE [OF 3]

                    below_trans)

next

  fix Y :: "nat \<Rightarrow> 'a" assume "chain Y"

  then show "f (\<Squnion>i. Y i) (t (\<Squnion>i. Y i)) \<sqsubseteq> (\<Squnion>i. f (Y i) (t (Y i)))"

    by (simp only: cont2contlubE [OF 1] ch2ch_cont [OF 1]

                   cont2contlubE [OF 2] ch2ch_cont [OF 2]

                   cont2contlubE [OF 3] ch2ch_cont [OF 3]

                   diag_lub below_refl)

qed

lemma cont_compose:

  "\<lbrakk>cont c; cont (\<lambda>x. f x)\<rbrakk> \<Longrightarrow> cont (\<lambda>x. c (f x))"

by (rule cont_apply [OF _ _ cont_const])

text {* Least upper bounds preserve continuity *}

lemma cont2cont_lub [simp]:

  assumes chain: "\<And>x. chain (\<lambda>i. F i x)" and cont: "\<And>i. cont (\<lambda>x. F i x)"

  shows "cont (\<lambda>x. \<Squnion>i. F i x)"

apply (rule contI2)

apply (simp add: monofunI cont2monofunE [OF cont] lub_mono chain)

apply (simp add: cont2contlubE [OF cont])

apply (simp add: diag_lub ch2ch_cont [OF cont] chain)

done

text {* if-then-else is continuous *}

lemma cont_if [simp, cont2cont]:

  "\<lbrakk>cont f; cont g\<rbrakk> \<Longrightarrow> cont (\<lambda>x. if b then f x else g x)"

by (induct b) simp_all

subsection {* Finite chains and flat pcpos *}

text {* Monotone functions map finite chains to finite chains. *}

lemma monofun_finch2finch:

  "\<lbrakk>monofun f; finite_chain Y\<rbrakk> \<Longrightarrow> finite_chain (\<lambda>n. f (Y n))"

apply (unfold finite_chain_def)

apply (simp add: ch2ch_monofun)

apply (force simp add: max_in_chain_def)

done

text {* The same holds for continuous functions. *}

lemma cont_finch2finch:

  "\<lbrakk>cont f; finite_chain Y\<rbrakk> \<Longrightarrow> finite_chain (\<lambda>n. f (Y n))"

by (rule cont2mono [THEN monofun_finch2finch])

text {* All monotone functions with chain-finite domain are continuous. *}

lemma chfindom_monofun2cont: "monofun f \<Longrightarrow> cont (f::'a::chfin \<Rightarrow> 'b::cpo)"

apply (erule contI2)

apply (frule chfin2finch)

apply (clarsimp simp add: finite_chain_def)

apply (subgoal_tac "max_in_chain i (\<lambda>i. f (Y i))")

apply (simp add: maxinch_is_thelub ch2ch_monofun)

apply (force simp add: max_in_chain_def)

done

text {* All strict functions with flat domain are continuous. *}

lemma flatdom_strict2mono: "f \<bottom> = \<bottom> \<Longrightarrow> monofun (f::'a::flat \<Rightarrow> 'b::pcpo)"

apply (rule monofunI)

apply (drule ax_flat)

apply auto

done

lemma flatdom_strict2cont: "f \<bottom> = \<bottom> \<Longrightarrow> cont (f::'a::flat \<Rightarrow> 'b::pcpo)"

by (rule flatdom_strict2mono [THEN chfindom_monofun2cont])

text {* All functions with discrete domain are continuous. *}

lemma cont_discrete_cpo [simp, cont2cont]: "cont (f::'a::discrete_cpo \<Rightarrow> 'b::cpo)"

apply (rule contI)

apply (drule discrete_chain_const, clarify)

apply (simp add: lub_const)

done

end