src/HOLCF/Cont.thy

 (*  Title:      HOLCF/cont.thy
     ID:         $Id$
     Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
 
     Results about continuity and monotonicity
 *)
 
-Cont = Fun3 +
+theory Cont = Fun3:
 
 (* 
 
    Now we change the default class! Form now on all untyped typevariables are
    of default class po
 
 *)
 
 
-default 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 *)
 
 defs 
 
-monofun         "monofun(f) == ! x y. x << y --> f(x) << f(y)"
-
-contlub         "contlub(f) == ! Y. chain(Y) --> 
+monofun:         "monofun(f) == ! x y. x << y --> f(x) << f(y)"
+
+contlub:         "contlub(f) == ! Y. chain(Y) --> 
                                 f(lub(range(Y))) = lub(range(% i. f(Y(i))))"
 
-cont            "cont(f)   == ! Y. chain(Y) --> 
+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)                       *)
 (* ------------------------------------------------------------------------ *)
 
+(*  Title:      HOLCF/Cont.ML
+    ID:         $Id$
+    Author:     Franz Regensburger
+    License:    GPL (GNU GENERAL PUBLIC LICENSE)
+
+Results about continuity and monotonicity
+*)
+
+(* ------------------------------------------------------------------------ *)
+(* 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
+
+lemma contlubE: 
+        " contlub(f)==> 
+          ! Y. chain(Y) --> f(lub(range(Y))) = lub(range(%i. f(Y(i))))"
+apply (unfold contlub)
+apply assumption
+done
+
+
+lemma contI: 
+ "! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y))) ==> cont(f)"
+
+apply (unfold cont)
+apply assumption
+done
+
+lemma contE: 
+ "cont(f) ==> ! Y. chain(Y) --> range(% i. f(Y(i))) <<| f(lub(range(Y)))"
+apply (unfold cont)
+apply assumption
+done
+
+
+lemma monofunI: 
+        "! x y. x << y --> f(x) << f(y) ==> monofun(f)"
+apply (unfold monofun)
+apply assumption
+done
+
+lemma monofunE: 
+        "monofun(f) ==> ! x y. x << y --> f(x) << f(y)"
+apply (unfold monofun)
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the main purpose of cont.thy is to show:                                 *)
+(*              monofun(f) & contlub(f)  <==> cont(f)                      *)
+(* ------------------------------------------------------------------------ *)
+
+(* ------------------------------------------------------------------------ *)
+(* 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 chainE)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* 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 ub_rangeD)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* left to right: monofun(f) & contlub(f)  ==> cont(f)                     *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monocontlub2cont: 
+        "[|monofun(f);contlub(f)|] ==> cont(f)"
+apply (unfold cont)
+apply (intro strip)
+apply (rule thelubE)
+apply (erule ch2ch_monofun)
+apply assumption
+apply (erule contlubE [THEN spec, THEN mp, symmetric])
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* 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])
+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                                    *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont2mono: "cont(f) ==> monofun(f)"
+apply (unfold monofun)
+apply (intro strip)
+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)                      *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont2contlub: "cont(f) ==> contlub(f)"
+apply (unfold contlub)
+apply (intro strip)
+apply (rule thelubI [symmetric])
+apply (erule contE [THEN spec, THEN mp])
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* monotone functions map finite chains to finite chains                    *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monofun_finch2finch: 
+  "[| monofun f; finite_chain Y |] ==> finite_chain (%n. f (Y n))"
+apply (unfold finite_chain_def)
+apply (force elim!: ch2ch_monofun simp add: max_in_chain_def)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* 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                                                        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma ch2ch_MF2L: 
+"[|monofun(MF2); chain(F)|] ==> chain(%i. MF2 (F i) x)"
+apply (erule ch2ch_monofun [THEN ch2ch_fun])
+apply assumption
+done
+
+
+lemma ch2ch_MF2R: 
+"[|monofun(MF2(f)); chain(Y)|] ==> chain(%i. MF2 f (Y i))"
+apply (erule ch2ch_monofun)
+apply assumption
+done
+
+lemma ch2ch_MF2LR: 
+"[|monofun(MF2); !f. monofun(MF2(f)); chain(F); chain(Y)|] ==>  
+   chain(%i. MF2(F(i))(Y(i)))"
+apply (rule chainI)
+apply (rule trans_less)
+apply (erule ch2ch_MF2L [THEN chainE])
+apply assumption
+apply (rule monofunE [THEN spec, THEN spec, THEN mp], 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)); 
+        chain(F);chain(Y)|] ==>  
+        chain(%j. lub(range(%i. MF2 (F j) (Y i))))"
+apply (rule lub_mono [THEN chainI])
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+apply (intro strip)
+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)); 
+        chain(F);chain(Y)|] ==>  
+        chain(%i. lub(range(%j. MF2 (F j) (Y i))))"
+apply (rule lub_mono [THEN chainI])
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (intro strip)
+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 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 assumption
+done
+
+lemma ex_lubMF2: 
+"[|monofun(MF2::('a::po=>'b::po=>'c::cpo)); 
+   !f. monofun(MF2(f)::('b::po=>'c::cpo)); 
+        chain(F); chain(Y)|] ==>  
+                lub(range(%j. lub(range(%i. MF2(F j) (Y i))))) = 
+                lub(range(%i. lub(range(%j. MF2(F j) (Y i)))))"
+apply (rule antisym_less)
+apply (rule is_lub_thelub[OF _ ub_rangeI])
+apply (erule ch2ch_lubMF2R)
+apply (assumption+)
+apply (rule lub_mono)
+apply (rule ch2ch_MF2R, erule spec)
+apply assumption
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (intro strip)
+apply (rule is_ub_thelub)
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (rule is_lub_thelub[OF _ ub_rangeI])
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (rule lub_mono)
+apply (erule ch2ch_MF2L)
+apply assumption
+apply (erule ch2ch_lubMF2R)
+apply (assumption+)
+apply (intro strip)
+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)); 
+   chain(FY);chain(TY)|] ==> 
+  lub(range(%i. lub(range(%j. MF2(FY(j))(TY(i)))))) = 
+  lub(range(%i. MF2(FY(i))(TY(i))))"
+apply (rule antisym_less)
+apply (rule is_lub_thelub[OF _ ub_rangeI])
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (rule lub_mono3)
+apply (erule ch2ch_MF2L)
+apply (assumption+)
+apply (erule ch2ch_MF2LR)
+apply (assumption+)
+apply (rule allI)
+apply (rule_tac m = "i" and n = "ia" in nat_less_cases)
+apply (rule_tac x = "ia" in exI)
+apply (rule chain_mono)
+apply (erule allE)
+apply (erule ch2ch_MF2R)
+apply (assumption+)
+apply (erule ssubst)
+apply (rule_tac x = "ia" in exI)
+apply (rule refl_less)
+apply (rule_tac x = "i" in exI)
+apply (rule chain_mono)
+apply (erule ch2ch_MF2L)
+apply (assumption+)
+apply (rule lub_mono)
+apply (erule ch2ch_MF2LR)
+apply (assumption+)
+apply (erule ch2ch_lubMF2L)
+apply (assumption+)
+apply (intro strip)
+apply (rule is_ub_thelub)
+apply (erule ch2ch_MF2L)
+apply assumption
+done
+
+lemma diag_lubMF2_2: 
+"[|monofun(MF2::('a::po=>'b::po=>'c::cpo)); 
+   !f. monofun(MF2(f)::('b::po=>'c::cpo)); 
+   chain(FY);chain(TY)|] ==> 
+  lub(range(%j. lub(range(%i. MF2(FY(j))(TY(i)))))) = 
+  lub(range(%i. MF2(FY(i))(TY(i))))"
+apply (rule trans)
+apply (rule ex_lubMF2)
+apply (assumption+)
+apply (erule diag_lubMF2_1)
+apply (assumption+)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about a curried function that is continuous    *)
+(* in both arguments                                                        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma contlub_CF2:
+assumes prem1: "cont(CF2)"
+assumes prem2: "!f. cont(CF2(f))"
+assumes prem3: "chain(FY)"
+assumes prem4: "chain(TY)"
+shows "CF2(lub(range(FY)))(lub(range(TY))) = lub(range(%i. CF2(FY(i))(TY(i))))"
+apply (subst prem1 [THEN cont2contlub, THEN contlubE, THEN spec, THEN mp])
+apply assumption
+apply (subst thelub_fun)
+apply (rule prem1 [THEN cont2mono [THEN ch2ch_monofun]])
+apply assumption
+apply (rule trans)
+apply (rule prem2 [THEN spec, THEN cont2contlub, THEN contlubE, THEN spec, THEN mp, THEN ext, THEN arg_cong, THEN arg_cong])
+apply (rule prem4)
+apply (rule diag_lubMF2_2)
+apply (auto simp add: cont2mono prems)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* The following results are about application for functions in 'a=>'b      *)
+(* ------------------------------------------------------------------------ *)
+
+lemma monofun_fun_fun: "f1 << f2 ==> f1(x) << f2(x)"
+apply (erule less_fun [THEN iffD1, THEN spec])
+done
+
+lemma monofun_fun_arg: "[|monofun(f); x1 << x2|] ==> f(x1) << f(x2)"
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply assumption
+done
+
+lemma monofun_fun: "[|monofun(f1); monofun(f2); f1 << f2; x1 << x2|] ==> f1(x1) << f2(x2)"
+apply (rule trans_less)
+apply (erule monofun_fun_arg)
+apply assumption
+apply (erule monofun_fun_fun)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* 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 (erule monofun_fun_arg [THEN monofun_fun_fun])
+apply assumption
+done
+
+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 (frule asm_rl)
+apply (erule cont2contlub [THEN contlubE, THEN spec, THEN mp, THEN ssubst])
+apply assumption
+apply (subst thelub_fun)
+apply (rule ch2ch_monofun)
+apply (erule cont2mono)
+apply assumption
+apply (rule refl)
+done
+
+(*********  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 less_fun [THEN iffD2])
+apply (blast dest: monofunE)
+done
+
+lemma cont2cont_CF1L_rev: "!y. cont(%x. c1 x y) ==> cont(c1)"
+apply (rule monocontlub2cont)
+apply (rule cont2mono [THEN allI, THEN mono2mono_MF1L_rev])
+apply (erule spec)
+apply (rule contlubI)
+apply (intro strip)
+apply (rule ext)
+apply (subst thelub_fun)
+apply (rule cont2mono [THEN allI, THEN mono2mono_MF1L_rev, THEN ch2ch_monofun])
+apply (erule spec)
+apply assumption
+apply (blast dest: cont2contlub [THEN contlubE])
+done
+
+(* ------------------------------------------------------------------------ *)
+(* 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)|] ==> 
+  (%y. lub(range(%i. c (Y i) y))) = (lub(range(%i.%y. c (Y i) y)))"
+apply (rule trans)
+prefer 2 apply (rule cont2contlub [THEN contlubE, THEN spec, THEN mp])
+prefer 2 apply (assumption)
+apply (erule cont2cont_CF1L_rev)
+apply (rule ext)
+apply (rule cont2contlub [THEN contlubE, THEN spec, THEN mp, symmetric])
+apply (erule spec)
+apply assumption
+done
+
+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_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 assumption
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+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_tac f3 = "tt" in contlubE [THEN spec, THEN mp, THEN ssubst])
+apply (erule cont2contlub)
+apply assumption
+apply (rule contlub_CF2)
+apply (assumption+)
+apply (erule cont2mono [THEN ch2ch_monofun])
+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                                      *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_id: "cont(% x. x)"
+apply (rule contI)
+apply (intro strip)
+apply (erule thelubE)
+apply (rule refl)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* constant functions are continuous                                        *)
+(* ------------------------------------------------------------------------ *)
+
+lemma cont_const: "cont(%x. c)"
+apply (unfold cont)
+apply (intro strip)
+apply (blast intro: is_lubI ub_rangeI dest: ub_rangeD)
+done
+
+
+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                                          *)
+(* ------------------------------------------------------------------------ *)
+
+lemma CfunI: "?x:Collect cont"
+apply (rule CollectI)
+apply (rule cont_const)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* 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 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 (frule chfin2finch)
+apply (rule antisym_less)
+apply (clarsimp simp add: finite_chain_def maxinch_is_thelub)
+apply (rule is_ub_thelub)
+apply (erule ch2ch_monofun)
+apply assumption
+apply (drule monofun_finch2finch[COMP swap_prems_rl])
+apply assumption
+apply (simp add: finite_chain_def)
+apply (erule conjE)
+apply (erule exE)
+apply (simp add: maxinch_is_thelub)
+apply (erule monofunE [THEN spec, THEN spec, THEN mp])
+apply (erule is_ub_thelub)
+done
+
+lemmas flatdom_strict2cont = flatdom2monofun [THEN chfindom_monofun2cont, standard]
+(* f UU = UU ==> cont (f::'a=>'b::pcpo)" *)
+
 end