Theory Cont

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theory Cont
imports Pcpo
begin

(*  Title:      HOLCF/Cont.thy
    ID:         $Id: Cont.thy,v 1.38 2008/03/27 18:49:24 huffman Exp $
    Author:     Franz Regensburger

Results about continuity and monotonicity.
*)

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
*}

defaultsort po

subsection {* Definitions *}

definition
  monofun :: "('a => 'b) => bool"  -- "monotonicity"  where
  "monofun f = (∀x y. x \<sqsubseteq> y --> f x \<sqsubseteq> f y)"

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

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

lemma contlubI:
  "[|!!Y. chain Y ==> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))|] ==> contlub f"
by (simp add: contlub_def)

lemma contlubE: 
  "[|contlub f; chain Y|] ==> f (\<Squnion>i. Y i) = (\<Squnion>i. f (Y i))" 
by (simp add: contlub_def)

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 {* @{prop "monofun f ∧ contlub f ≡ cont f"} *}

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

lemma ub2ub_monofun':
  "[|monofun f; S <| u|] ==> f ` S <| f u"
apply (rule ub_imageI)
apply (erule monofunE)
apply (erule (1) is_ubD)
done

text {* monotone functions map directed sets to directed sets *}

lemma dir2dir_monofun:
  assumes f: "monofun f"
  assumes S: "directed S"
  shows "directed (f ` S)"
proof (rule directedI)
  from directedD1 [OF S]
  obtain x where "x ∈ S" ..
  hence "f x ∈ f ` S" by simp
  thus "∃x. x ∈ f ` S" ..
next
  fix x assume "x ∈ f ` S"
  then obtain a where x: "x = f a" and a: "a ∈ S" ..
  fix y assume "y ∈ f ` S"
  then obtain b where y: "y = f b" and b: "b ∈ S" ..
  from directedD2 [OF S a b]
  obtain c where "c ∈ S" and "a \<sqsubseteq> c ∧ b \<sqsubseteq> c" ..
  hence "f c ∈ f ` S" and "x \<sqsubseteq> f c ∧ y \<sqsubseteq> f c"
    using monofunE [OF f] x y by simp_all
  thus "∃z∈f ` S. x \<sqsubseteq> z ∧ y \<sqsubseteq> z" ..
qed

text {* left to right: @{prop "monofun f ∧ contlub f ==> cont f"} *}

lemma monocontlub2cont: "[|monofun f; contlub f|] ==> cont f"
apply (rule contI)
apply (rule thelubE)
apply (erule (1) ch2ch_monofun)
apply (erule (1) contlubE [symmetric])
done

text {* first 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 lub_bin_chain [THEN thelubI])
done

text {* right to left: @{prop "cont f ==> monofun f ∧ contlub f"} *}
text {* part1: @{prop "cont f ==> monofun f"} *}

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

lemmas ch2ch_cont = cont2mono [THEN ch2ch_monofun]

text {* right to left: @{prop "cont f ==> monofun f ∧ contlub f"} *}
text {* part2: @{prop "cont f ==> contlub f"} *}

lemma cont2contlub: "cont f ==> contlub f"
apply (rule contlubI)
apply (rule thelubI [symmetric])
apply (erule (1) contE)
done

lemmas cont2contlubE = cont2contlub [THEN contlubE]

lemma contI2:
  assumes mono: "monofun f"
  assumes less: "!!Y. [|chain Y; chain (λi. f (Y i))|]
     ==> f (lub (range Y)) \<sqsubseteq> (\<Squnion>i. f (Y i))"
  shows "cont f"
apply (rule monocontlub2cont)
apply (rule mono)
apply (rule contlubI)
apply (rule antisym_less)
apply (rule less, assumption)
apply (erule ch2ch_monofun [OF mono])
apply (rule is_lub_thelub)
apply (erule ch2ch_monofun [OF mono])
apply (rule ub2ub_monofun [OF mono])
apply (rule is_lubD1)
apply (erule cpo_lubI)
done

subsection {* Continuity of basic functions *}

text {* The identity function is continuous *}

lemma cont_id: "cont (λx. x)"
apply (rule contI)
apply (erule cpo_lubI)
done

text {* constant functions are continuous *}

lemma cont_const: "cont (λx. c)"
apply (rule contI)
apply (rule lub_const)
done

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

lemma cont_if [simp]:
  "[|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])

lemma chfindom_monofun2cont: "monofun f ==> cont (f::'a::chfin => 'b::cpo)"
apply (rule monocontlub2cont)
apply assumption
apply (rule contlubI)
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 {* some properties of flat *}

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 {* functions with discrete domain *}

lemma cont_discrete_cpo [simp]: "cont (f::'a::discrete_cpo => 'b::cpo)"
apply (rule contI)
apply (drule discrete_chain_const, clarify)
apply (simp add: lub_const)
done

end

Definitions

lemma contlubI:

  (!!Y. chain Y ==> f (LUB i. Y i) = (LUB i. f (Y i))) ==> contlub f

lemma contlubE:

  [| contlub f; chain Y |] ==> f (LUB i. Y i) = (LUB i. f (Y i))

lemma contI:

  (!!Y. chain Y ==> rangei. f (Y i)) <<| f (LUB i. Y i)) ==> cont f

lemma contE:

  [| cont f; chain Y |] ==> rangei. f (Y i)) <<| f (LUB i. Y i)

lemma monofunI:

  (!!x y. x << y ==> f x << f y) ==> monofun f

lemma monofunE:

  [| monofun f; x << y |] ==> f x << f y

@{prop "monofun f ∧ contlub f ≡ cont f"}

lemma ch2ch_monofun:

  [| monofun f; chain Y |] ==> chaini. f (Y i))

lemma ub2ub_monofun:

  [| monofun f; range Y <| u |] ==> rangei. f (Y i)) <| f u

lemma ub2ub_monofun':

  [| monofun f; S <| u |] ==> f ` S <| f u

lemma dir2dir_monofun:

  [| monofun f; directed S |] ==> directed (f ` S)

lemma monocontlub2cont:

  [| monofun f; contlub f |] ==> cont f

lemma binchain_cont:

  [| cont f; x << y |] ==> rangei. f (if i = 0 then x else y)) <<| f y

lemma cont2mono:

  cont f ==> monofun f

lemma ch2ch_cont:

  [| cont f; chain Y |] ==> chaini. f (Y i))

lemma cont2contlub:

  cont f ==> contlub f

lemma cont2contlubE:

  [| cont f; chain Y |] ==> f (LUB i. Y i) = (LUB i. f (Y i))

lemma contI2:

  [| monofun f;
     !!Y. [| chain Y; chaini. f (Y i)) |] ==> f (Lub Y) << (LUB i. f (Y i)) |]
  ==> cont f

Continuity of basic functions

lemma cont_id:

  contx. x)

lemma cont_const:

  contx. c)

lemma cont_if:

  [| cont f; cont g |] ==> contx. if b then f x else g x)

Finite chains and flat pcpos

lemma monofun_finch2finch:

  [| monofun f; finite_chain Y |] ==> finite_chainn. f (Y n))

lemma cont_finch2finch:

  [| cont f; finite_chain Y |] ==> finite_chainn. f (Y n))

lemma chfindom_monofun2cont:

  monofun f ==> cont f

lemma flatdom_strict2mono:

  f UU = UU ==> monofun f

lemma flatdom_strict2cont:

  f UU = UU ==> cont f

lemma cont_discrete_cpo:

  cont f