Theory Cont

theory Cont
imports Pcpo
(*  Title:      HOL/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 => 'b) => bool"  -- "monotonicity"  where
  "monofun f = (∀x y. x \<sqsubseteq> y --> f x \<sqsubseteq> f y)"

definition
  cont :: "('a::cpo => 'b::cpo) => bool"
where
  "cont f = (∀Y. chain Y --> range (λi. f (Y i)) <<| f (\<Squnion>i. Y i))"

lemma contI:
  "[|!!Y. chain Y ==> range (λi. f (Y i)) <<| f (\<Squnion>i. Y i)|] ==> cont f"
by (simp add: cont_def)

lemma contE:
  "[|cont f; chain Y|] ==> range (λi. f (Y i)) <<| f (\<Squnion>i. Y i)"
by (simp add: cont_def)

lemma monofunI: 
  "[|!!x y. x \<sqsubseteq> y ==> f x \<sqsubseteq> f y|] ==> monofun f"
by (simp add: monofun_def)

lemma monofunE: 
  "[|monofun f; x \<sqsubseteq> y|] ==> 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: "[|monofun f; chain Y|] ==> chain (λi. 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: 
  "[|monofun f; range Y <| u|] ==> range (λi. 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:
  "[|cont f; x \<sqsubseteq> y|] ==> range (λi::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 is_lub_bin_chain [THEN lub_eqI])
done

text {* continuity implies monotonicity *}

lemma cont2mono: "cont f ==> monofun f"
apply (rule monofunI)
apply (drule (1) binchain_cont)
apply (drule_tac i=0 in is_lub_rangeD1)
apply simp
done

lemmas cont2monofunE = cont2mono [THEN monofunE]

lemmas ch2ch_cont = cont2mono [THEN ch2ch_monofun]

text {* continuity implies preservation of lubs *}

lemma cont2contlubE:
  "[|cont f; chain Y|] ==> f (\<Squnion> i. Y i) = (\<Squnion> i. f (Y i))"
apply (rule lub_eqI [symmetric])
apply (erule (1) contE)
done

lemma contI2:
  fixes f :: "'a::cpo => 'b::cpo"
  assumes mono: "monofun f"
  assumes below: "!!Y. [|chain Y; chain (λi. f (Y i))|]
     ==> f (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. f (Y i))"
  shows "cont f"
proof (rule contI)
  fix Y :: "nat => 'a"
  assume Y: "chain Y"
  with mono have fY: "chain (λi. 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 (λi. f (Y i)) <<| f (\<Squnion>i. Y i)"
    by (rule thelubE)
qed

subsection {* Collection of continuity rules *}

ML {*
structure Cont2ContData = Named_Thms
(
  val name = @{binding 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 (λx. x)"
apply (rule contI)
apply (erule cpo_lubI)
done

text {* constant functions are continuous *}

lemma cont_const [simp, cont2cont]: "cont (λx. c)"
  using is_lub_const by (rule contI)

text {* application of functions is continuous *}

lemma cont_apply:
  fixes f :: "'a::cpo => 'b::cpo => 'c::cpo" and t :: "'a => 'b"
  assumes 1: "cont (λx. t x)"
  assumes 2: "!!x. cont (λy. f x y)"
  assumes 3: "!!y. cont (λx. f x y)"
  shows "cont (λx. (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 => '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:
  "[|cont c; cont (λx. f x)|] ==> cont (λx. c (f x))"
by (rule cont_apply [OF _ _ cont_const])

text {* Least upper bounds preserve continuity *}

lemma cont2cont_lub [simp]:
  assumes chain: "!!x. chain (λi. F i x)" and cont: "!!i. cont (λx. F i x)"
  shows "cont (λx. \<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]:
  "[|cont f; cont g|] ==> cont (λx. 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:
  "[|monofun f; finite_chain Y|] ==> finite_chain (λn. 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:
  "[|cont f; finite_chain Y|] ==> finite_chain (λn. f (Y n))"
by (rule cont2mono [THEN monofun_finch2finch])

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

lemma chfindom_monofun2cont: "monofun f ==> cont (f::'a::chfin => 'b::cpo)"
apply (erule contI2)
apply (frule chfin2finch)
apply (clarsimp simp add: finite_chain_def)
apply (subgoal_tac "max_in_chain i (λi. 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 ⊥ = ⊥ ==> monofun (f::'a::flat => 'b::pcpo)"
apply (rule monofunI)
apply (drule ax_flat)
apply auto
done

lemma flatdom_strict2cont: "f ⊥ = ⊥ ==> cont (f::'a::flat => '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 => 'b::cpo)"
apply (rule contI)
apply (drule discrete_chain_const, clarify)
apply (simp add: is_lub_const)
done

end