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Cont.thy
Sprache: Isabelle
(* Title: HOL/HOLCF/Cont.thy
Author: Franz Regensburger
Author: Brian Huffman
*)
section \<open>Continuity and monotonicity\<close>
theory Cont
imports Pcpo
begin
text \<open>
Now we change the default class! Form now on all untyped type variables are
of default class po
\<close>
default_sort po
subsection \<open>Definitions\<close>
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 (\i. f (Y i)) <<| f (\i. Y i))"
lemma contI: "(\Y. chain Y \ range (\i. f (Y i)) <<| f (\i. Y i)) \ cont f"
by (simp add: cont_def)
lemma contE: "cont f \ chain Y \ range (\i. 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 \<open>Equivalence of alternate definition\<close>
text \<open>monotone functions map chains to chains\<close>
lemma ch2ch_monofun: "monofun f \ chain Y \ chain (\i. f (Y i))"
apply (rule chainI)
apply (erule monofunE)
apply (erule chainE)
done
text \<open>monotone functions map upper bound to upper bounds\<close>
lemma ub2ub_monofun: "monofun f \ range Y <| u \ range (\i. f (Y i)) <| f u"
apply (rule ub_rangeI)
apply (erule monofunE)
apply (erule ub_rangeD)
done
text \<open>a lemma about binary chains\<close>
lemma binchain_cont: "cont f \ x \ y \ range (\i::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 \<open>continuity implies monotonicity\<close>
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 \<open>continuity implies preservation of lubs\<close>
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 (\i. 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 (\i. 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 (\i. f (Y i)) <<| f (\i. Y i)"
by (rule thelubE)
qed
subsection \<open>Collection of continuity rules\<close>
named_theorems cont2cont "continuity intro rule"
subsection \<open>Continuity of basic functions\<close>
text \<open>The identity function is continuous\<close>
lemma cont_id [simp, cont2cont]: "cont (\x. x)"
apply (rule contI)
apply (erule cpo_lubI)
done
text \<open>constant functions are continuous\<close>
lemma cont_const [simp, cont2cont]: "cont (\x. c)"
using is_lub_const by (rule contI)
text \<open>application of functions is continuous\<close>
lemma cont_apply:
fixes f :: "'a::cpo \ 'b::cpo \ 'c::cpo" and t :: "'a \ 'b"
assumes 1: "cont (\x. t x)"
assumes 2: "\x. cont (\y. f x y)"
assumes 3: "\y. cont (\x. f x y)"
shows "cont (\x. (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 (\x. f x) \ cont (\x. c (f x))"
by (rule cont_apply [OF _ _ cont_const])
text \<open>Least upper bounds preserve continuity\<close>
lemma cont2cont_lub [simp]:
assumes chain: "\x. chain (\i. F i x)"
and cont: "\i. cont (\x. F i x)"
shows "cont (\x. \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 \<open>if-then-else is continuous\<close>
lemma cont_if [simp, cont2cont]: "cont f \ cont g \ cont (\x. if b then f x else g x)"
by (induct b) simp_all
subsection \<open>Finite chains and flat pcpos\<close>
text \<open>Monotone functions map finite chains to finite chains.\<close>
lemma monofun_finch2finch: "monofun f \ finite_chain Y \ finite_chain (\n. f (Y n))"
by (force simp add: finite_chain_def ch2ch_monofun max_in_chain_def)
text \<open>The same holds for continuous functions.\<close>
lemma cont_finch2finch: "cont f \ finite_chain Y \ finite_chain (\n. f (Y n))"
by (rule cont2mono [THEN monofun_finch2finch])
text \<open>All monotone functions with chain-finite domain are continuous.\<close>
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 (\i. f (Y i))")
apply (simp add: maxinch_is_thelub ch2ch_monofun)
apply (force simp add: max_in_chain_def)
done
text \<open>All strict functions with flat domain are continuous.\<close>
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 \<open>All functions with discrete domain are continuous.\<close>
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
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