--- a/src/HOLCF/Cont.thy Mon Mar 07 23:54:01 2005 +0100
+++ b/src/HOLCF/Cont.thy Tue Mar 08 00:00:49 2005 +0100
@@ -12,20 +12,17 @@
imports FunCpo
begin
-(*
-
- Now we change the default class! Form now on all untyped typevariables are
+text {*
+ Now we change the default class! Form now on all untyped type variables are
of default class po
-
-*)
-
+*}
defaultsort po
consts
- monofun :: "('a => 'b) => bool" (* monotonicity *)
- contlub :: "('a::cpo => 'b::cpo) => bool" (* first cont. def *)
- cont :: "('a::cpo => 'b::cpo) => bool" (* secnd cont. def *)
+ monofun :: "('a => 'b) => bool" -- "monotonicity"
+ contlub :: "('a::cpo => 'b::cpo) => bool" -- "first cont. def"
+ cont :: "('a::cpo => 'b::cpo) => bool" -- "secnd cont. def"
defs
@@ -37,147 +34,100 @@
cont: "cont(f) == ! Y. chain(Y) -->
range(% i. f(Y(i))) <<| f(lub(range(Y)))"
-(* ------------------------------------------------------------------------ *)
-(* the main purpose of cont.thy is to show: *)
-(* monofun(f) & contlub(f) <==> cont(f) *)
-(* ------------------------------------------------------------------------ *)
+text {*
+ the main purpose of cont.thy is to show:
+ @{prop "monofun(f) & contlub(f) == cont(f)"}
+*}
-(* Title: HOLCF/Cont.ML
- ID: $Id$
- Author: Franz Regensburger
- License: GPL (GNU GENERAL PUBLIC LICENSE)
-
-Results about continuity and monotonicity
-*)
-
-(* ------------------------------------------------------------------------ *)
-(* access to definition *)
-(* ------------------------------------------------------------------------ *)
+text {* access to definition *}
lemma contlubI:
"! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))==>
contlub(f)"
-apply (unfold contlub)
-apply assumption
-done
+by (unfold contlub)
lemma contlubE:
" contlub(f)==>
! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))"
-apply (unfold contlub)
-apply assumption
-done
-
+by (unfold contlub)
lemma contI:
"! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)"
-
-apply (unfold cont)
-apply assumption
-done
+by (unfold cont)
lemma contE:
"cont(f) ==> ! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))"
-apply (unfold cont)
-apply assumption
-done
-
+by (unfold cont)
lemma monofunI:
"! x y. x << y --> f(x) << f(y) ==> monofun(f)"
-apply (unfold monofun)
-apply assumption
-done
+by (unfold monofun)
lemma monofunE:
"monofun(f) ==> ! x y. x << y --> f(x) << f(y)"
-apply (unfold monofun)
-apply assumption
-done
+by (unfold monofun)
-(* ------------------------------------------------------------------------ *)
-(* the main purpose of cont.thy is to show: *)
-(* monofun(f) & contlub(f) <==> cont(f) *)
-(* ------------------------------------------------------------------------ *)
-
-(* ------------------------------------------------------------------------ *)
-(* monotone functions map chains to chains *)
-(* ------------------------------------------------------------------------ *)
+text {* monotone functions map chains to chains *}
lemma ch2ch_monofun:
"[| monofun(f); chain(Y) |] ==> chain(%i. f(Y(i)))"
apply (rule chainI)
-apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule monofunE [rule_format])
apply (erule chainE)
done
-(* ------------------------------------------------------------------------ *)
-(* monotone functions map upper bound to upper bounds *)
-(* ------------------------------------------------------------------------ *)
+text {* monotone functions map upper bound to upper bounds *}
lemma ub2ub_monofun:
"[| monofun(f); range(Y) <| u|] ==> range(%i. f(Y(i))) <| f(u)"
apply (rule ub_rangeI)
-apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule monofunE [rule_format])
apply (erule ub_rangeD)
done
-(* ------------------------------------------------------------------------ *)
-(* left to right: monofun(f) & contlub(f) ==> cont(f) *)
-(* ------------------------------------------------------------------------ *)
+text {* left to right: @{prop "monofun(f) & contlub(f) ==> cont(f)"} *}
lemma monocontlub2cont:
"[|monofun(f);contlub(f)|] ==> cont(f)"
-apply (unfold cont)
-apply (intro strip)
+apply (rule contI [rule_format])
apply (rule thelubE)
apply (erule ch2ch_monofun)
apply assumption
-apply (erule contlubE [THEN spec, THEN mp, symmetric])
+apply (erule contlubE [rule_format, symmetric])
apply assumption
done
-(* ------------------------------------------------------------------------ *)
-(* first a lemma about binary chains *)
-(* ------------------------------------------------------------------------ *)
+text {* first a lemma about binary chains *}
lemma binchain_cont: "[| cont(f); x << y |]
==> range(%i::nat. f(if i = 0 then x else y)) <<| f(y)"
apply (rule subst)
-prefer 2 apply (erule contE [THEN spec, THEN mp])
+prefer 2 apply (erule contE [rule_format])
apply (erule bin_chain)
apply (rule_tac y = "y" in arg_cong)
apply (erule lub_bin_chain [THEN thelubI])
done
-(* ------------------------------------------------------------------------ *)
-(* right to left: cont(f) ==> monofun(f) & contlub(f) *)
-(* part1: cont(f) ==> monofun(f *)
-(* ------------------------------------------------------------------------ *)
+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 (unfold monofun)
-apply (intro strip)
+apply (rule monofunI [rule_format])
apply (drule binchain_cont [THEN is_ub_lub])
apply (auto split add: split_if_asm)
done
-(* ------------------------------------------------------------------------ *)
-(* right to left: cont(f) ==> monofun(f) & contlub(f) *)
-(* part2: cont(f) ==> contlub(f) *)
-(* ------------------------------------------------------------------------ *)
+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 (unfold contlub)
-apply (intro strip)
+apply (rule contlubI [rule_format])
apply (rule thelubI [symmetric])
-apply (erule contE [THEN spec, THEN mp])
+apply (erule contE [rule_format])
apply assumption
done
-(* ------------------------------------------------------------------------ *)
-(* monotone functions map finite chains to finite chains *)
-(* ------------------------------------------------------------------------ *)
+text {* monotone functions map finite chains to finite chains *}
lemma monofun_finch2finch:
"[| monofun f; finite_chain Y |] ==> finite_chain (%n. f (Y n))"
@@ -185,31 +135,23 @@
apply (force elim!: ch2ch_monofun simp add: max_in_chain_def)
done
-(* ------------------------------------------------------------------------ *)
-(* The same holds for continuous functions *)
-(* ------------------------------------------------------------------------ *)
+text {* The same holds for continuous functions *}
lemmas cont_finch2finch = cont2mono [THEN monofun_finch2finch, standard]
(* [| cont ?f; finite_chain ?Y |] ==> finite_chain (%n. ?f (?Y n)) *)
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about a curried function that is monotone *)
-(* in both arguments *)
-(* ------------------------------------------------------------------------ *)
+text {*
+ The following results are about a curried function that is monotone
+ in both arguments
+*}
lemma ch2ch_MF2L:
"[|monofun(MF2); chain(F)|] ==> chain(%i. MF2 (F i) x)"
-apply (erule ch2ch_monofun [THEN ch2ch_fun])
-apply assumption
-done
-
+by (erule ch2ch_monofun [THEN ch2ch_fun])
lemma ch2ch_MF2R:
"[|monofun(MF2(f)); chain(Y)|] ==> chain(%i. MF2 f (Y i))"
-apply (erule ch2ch_monofun)
-apply assumption
-done
+by (erule ch2ch_monofun)
lemma ch2ch_MF2LR:
"[|monofun(MF2); !f. monofun(MF2(f)); chain(F); chain(Y)|] ==>
@@ -218,11 +160,10 @@
apply (rule trans_less)
apply (erule ch2ch_MF2L [THEN chainE])
apply assumption
-apply (rule monofunE [THEN spec, THEN spec, THEN mp], erule spec)
+apply (rule monofunE [rule_format], erule spec)
apply (erule chainE)
done
-
lemma ch2ch_lubMF2R:
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));
!f. monofun(MF2(f)::('b::po=>'c::cpo));
@@ -233,13 +174,12 @@
apply assumption
apply (rule ch2ch_MF2R, erule spec)
apply assumption
-apply (intro strip)
+apply (rule allI)
apply (rule chainE)
apply (erule ch2ch_MF2L)
apply assumption
done
-
lemma ch2ch_lubMF2L:
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));
!f. monofun(MF2(f)::('b::po=>'c::cpo));
@@ -250,27 +190,25 @@
apply assumption
apply (erule ch2ch_MF2L)
apply assumption
-apply (intro strip)
+apply (rule allI)
apply (rule chainE)
apply (rule ch2ch_MF2R, erule spec)
apply assumption
done
-
lemma lub_MF2_mono:
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));
!f. monofun(MF2(f)::('b::po=>'c::cpo));
chain(F)|] ==>
monofun(% x. lub(range(% j. MF2 (F j) (x))))"
-apply (rule monofunI)
-apply (intro strip)
+apply (rule monofunI [rule_format])
apply (rule lub_mono)
apply (erule ch2ch_MF2L)
apply assumption
apply (erule ch2ch_MF2L)
apply assumption
-apply (intro strip)
-apply (rule monofunE [THEN spec, THEN spec, THEN mp], erule spec)
+apply (rule allI)
+apply (rule monofunE [rule_format], erule spec)
apply assumption
done
@@ -289,7 +227,7 @@
apply assumption
apply (erule ch2ch_lubMF2L)
apply (assumption+)
-apply (intro strip)
+apply (rule allI)
apply (rule is_ub_thelub)
apply (erule ch2ch_MF2L)
apply assumption
@@ -301,13 +239,12 @@
apply assumption
apply (erule ch2ch_lubMF2R)
apply (assumption+)
-apply (intro strip)
+apply (rule allI)
apply (rule is_ub_thelub)
apply (rule ch2ch_MF2R, erule spec)
apply assumption
done
-
lemma diag_lubMF2_1:
"[|monofun(MF2::('a::po=>'b::po=>'c::cpo));
!f. monofun(MF2(f)::('b::po=>'c::cpo));
@@ -342,7 +279,7 @@
apply (assumption+)
apply (erule ch2ch_lubMF2L)
apply (assumption+)
-apply (intro strip)
+apply (rule allI)
apply (rule is_ub_thelub)
apply (erule ch2ch_MF2L)
apply assumption
@@ -361,11 +298,10 @@
apply (assumption+)
done
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about a curried function that is continuous *)
-(* in both arguments *)
-(* ------------------------------------------------------------------------ *)
+text {*
+ The following results are about a curried function that is continuous
+ in both arguments
+*}
lemma contlub_CF2:
assumes prem1: "cont(CF2)"
@@ -385,18 +321,15 @@
apply (auto simp add: cont2mono prems)
done
-(* ------------------------------------------------------------------------ *)
-(* The following results are about application for functions in 'a=>'b *)
-(* ------------------------------------------------------------------------ *)
+text {*
+ The following results are about application for functions in @{typ "'a=>'b"}
+*}
lemma monofun_fun_fun: "f1 << f2 ==> f1(x) << f2(x)"
-apply (erule less_fun [THEN iffD1, THEN spec])
-done
+by (erule less_fun [THEN iffD1, THEN spec])
lemma monofun_fun_arg: "[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)"
-apply (erule monofunE [THEN spec, THEN spec, THEN mp])
-apply assumption
-done
+by (erule monofunE [THEN spec, THEN spec, THEN mp])
lemma monofun_fun: "[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)"
apply (rule trans_less)
@@ -405,15 +338,13 @@
apply (erule monofun_fun_fun)
done
-
-(* ------------------------------------------------------------------------ *)
-(* The following results are about the propagation of monotonicity and *)
-(* continuity *)
-(* ------------------------------------------------------------------------ *)
+text {*
+ The following results are about the propagation of monotonicity and
+ continuity
+*}
lemma mono2mono_MF1L: "[|monofun(c1)|] ==> monofun(%x. c1 x y)"
-apply (rule monofunI)
-apply (intro strip)
+apply (rule monofunI [rule_format])
apply (erule monofun_fun_arg [THEN monofun_fun_fun])
apply assumption
done
@@ -421,8 +352,7 @@
lemma cont2cont_CF1L: "[|cont(c1)|] ==> cont(%x. c1 x y)"
apply (rule monocontlub2cont)
apply (erule cont2mono [THEN mono2mono_MF1L])
-apply (rule contlubI)
-apply (intro strip)
+apply (rule contlubI [rule_format])
apply (frule asm_rl)
apply (erule cont2contlub [THEN contlubE, THEN spec, THEN mp, THEN ssubst])
apply assumption
@@ -436,8 +366,7 @@
(********* Note "(%x.%y.c1 x y) = c1" ***********)
lemma mono2mono_MF1L_rev: "!y. monofun(%x. c1 x y) ==> monofun(c1)"
-apply (rule monofunI)
-apply (intro strip)
+apply (rule monofunI [rule_format])
apply (rule less_fun [THEN iffD2])
apply (blast dest: monofunE)
done
@@ -446,8 +375,7 @@
apply (rule monocontlub2cont)
apply (rule cont2mono [THEN allI, THEN mono2mono_MF1L_rev])
apply (erule spec)
-apply (rule contlubI)
-apply (intro strip)
+apply (rule contlubI [rule_format])
apply (rule ext)
apply (subst thelub_fun)
apply (rule cont2mono [THEN allI, THEN mono2mono_MF1L_rev, THEN ch2ch_monofun])
@@ -456,10 +384,10 @@
apply (blast dest: cont2contlub [THEN contlubE])
done
-(* ------------------------------------------------------------------------ *)
-(* What D.A.Schmidt calls continuity of abstraction *)
-(* never used here *)
-(* ------------------------------------------------------------------------ *)
+text {*
+ What D.A.Schmidt calls continuity of abstraction
+ never used here
+*}
lemma contlub_abstraction:
"[|chain(Y::nat=>'a);!y. cont(%x.(c::'a::cpo=>'b::cpo=>'c::cpo) x y)|] ==>
@@ -476,21 +404,18 @@
lemma mono2mono_app: "[|monofun(ft);!x. monofun(ft(x));monofun(tt)|] ==>
monofun(%x.(ft(x))(tt(x)))"
-apply (rule monofunI)
-apply (intro strip)
+apply (rule monofunI [rule_format])
apply (rule_tac ?f1.0 = "ft(x)" and ?f2.0 = "ft(y)" in monofun_fun)
apply (erule spec)
apply (erule spec)
-apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule monofunE [rule_format])
apply assumption
-apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule monofunE [rule_format])
apply assumption
done
-
lemma cont2contlub_app: "[|cont(ft);!x. cont(ft(x));cont(tt)|] ==> contlub(%x.(ft(x))(tt(x)))"
-apply (rule contlubI)
-apply (intro strip)
+apply (rule contlubI [rule_format])
apply (rule_tac f3 = "tt" in contlubE [THEN spec, THEN mp, THEN ssubst])
apply (erule cont2contlub)
apply assumption
@@ -500,70 +425,54 @@
apply assumption
done
-
lemma cont2cont_app: "[|cont(ft); !x. cont(ft(x)); cont(tt)|] ==> cont(%x.(ft(x))(tt(x)))"
apply (blast intro: monocontlub2cont mono2mono_app cont2mono cont2contlub_app)
done
-
lemmas cont2cont_app2 = cont2cont_app[OF _ allI]
(* [| cont ?ft; !!x. cont (?ft x); cont ?tt |] ==> *)
(* cont (%x. ?ft x (?tt x)) *)
-(* ------------------------------------------------------------------------ *)
-(* The identity function is continuous *)
-(* ------------------------------------------------------------------------ *)
+text {* The identity function is continuous *}
lemma cont_id: "cont(% x. x)"
-apply (rule contI)
-apply (intro strip)
+apply (rule contI [rule_format])
apply (erule thelubE)
apply (rule refl)
done
-(* ------------------------------------------------------------------------ *)
-(* constant functions are continuous *)
-(* ------------------------------------------------------------------------ *)
+text {* constant functions are continuous *}
lemma cont_const: "cont(%x. c)"
-apply (unfold cont)
-apply (intro strip)
+apply (rule contI [rule_format])
apply (blast intro: is_lubI ub_rangeI dest: ub_rangeD)
done
+lemma cont2cont_app3: "[|cont(f); cont(t) |] ==> cont(%x. f(t(x)))"
+by (best intro: cont2cont_app2 cont_const)
-lemma cont2cont_app3: "[|cont(f); cont(t) |] ==> cont(%x. f(t(x)))"
-apply (best intro: cont2cont_app2 cont_const)
-done
-
-(* ------------------------------------------------------------------------ *)
-(* A non-emptyness result for Cfun *)
-(* ------------------------------------------------------------------------ *)
+text {* A non-emptiness result for Cfun *}
lemma CfunI: "?x:Collect cont"
apply (rule CollectI)
apply (rule cont_const)
done
-(* ------------------------------------------------------------------------ *)
-(* some properties of flat *)
-(* ------------------------------------------------------------------------ *)
+text {* some properties of flat *}
lemma flatdom2monofun: "f UU = UU ==> monofun (f::'a::flat=>'b::pcpo)"
-
-apply (unfold monofun)
-apply (intro strip)
-apply (drule ax_flat [THEN spec, THEN spec, THEN mp])
+apply (rule monofunI [rule_format])
+apply (drule ax_flat [rule_format])
apply auto
done
declare range_composition [simp del]
+
lemma chfindom_monofun2cont: "monofun f ==> cont(f::'a::chfin=>'b::pcpo)"
apply (rule monocontlub2cont)
apply assumption
-apply (rule contlubI)
-apply (intro strip)
+apply (rule contlubI [rule_format])
apply (frule chfin2finch)
apply (rule antisym_less)
apply (clarsimp simp add: finite_chain_def maxinch_is_thelub)