Theory Cont

(*  Title:      HOL/HOLCF/Cont.thy
    Author:     Franz Regensburger
    Author:     Brian Huffman
*)

section ‹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 ⊑ y ⟶ f x ⊑ f y)"

definition cont :: "('a::cpo ⇒ 'b::cpo) ⇒ bool"
  where "cont f = (∀Y. chain Y ⟶ range (λi. f (Y i)) <<| f (⨆i. Y i))"

lemma contI: "(⋀Y. chain Y ⟹ range (λi. f (Y i)) <<| f (⨆i. Y i)) ⟹ cont f"
  by (simp add: cont_def)

lemma contE: "cont f ⟹ chain Y ⟹ range (λi. f (Y i)) <<| f (⨆i. Y i)"
  by (simp add: cont_def)

lemma monofunI: "(⋀x y. x ⊑ y ⟹ f x ⊑ f y) ⟹ monofun f"
  by (simp add: monofun_def)

lemma monofunE: "monofun f ⟹ x ⊑ y ⟹ f x ⊑ 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 ⊑ y ⟹ range (λi::nat. f (if i = 0 then x else y)) <<| f y"
  apply (subgoal_tac "f (⨆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 (⨆i. Y i) = (⨆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 (⨆i. Y i) ⊑ (⨆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 "(⨆i. f (Y i)) = f (⨆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 (⨆i. Y i)"
    by (rule thelubE)
qed


subsection ‹Collection of continuity rules›

named_theorems cont2cont "continuity intro rule"


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 ⊑ y"
  then show "f x (t x) ⊑ 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 (⨆i. Y i) (t (⨆i. Y i)) ⊑ (⨆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. ⨆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))"
  by (force simp add: finite_chain_def ch2ch_monofun max_in_chain_def)

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"
  for 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"
  for f :: "'a::flat ⇒ 'b::pcpo"
  apply (rule monofunI)
  apply (drule ax_flat)
  apply auto
  done

lemma flatdom_strict2cont: "f ⊥ = ⊥ ⟹ cont f"
  for 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"
  for f :: "'a::discrete_cpo ⇒ 'b::cpo"
  apply (rule contI)
  apply (drule discrete_chain_const, clarify)
  apply simp
  done

end