--- a/src/HOLCF/Pcpo.thy Wed Mar 02 22:30:00 2005 +0100
+++ b/src/HOLCF/Pcpo.thy Wed Mar 02 22:57:08 2005 +0100
@@ -1,40 +1,321 @@
(* Title: HOLCF/Pcpo.thy
ID: $Id$
Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
introduction of the classes cpo and pcpo
*)
-Pcpo = Porder +
+theory Pcpo = Porder:
(* The class cpo of chain complete partial orders *)
(* ********************************************** *)
axclass cpo < po
(* class axiom: *)
- cpo "chain S ==> ? x. range S <<| x"
+ cpo: "chain S ==> ? x. range S <<| x"
(* The class pcpo of pointed cpos *)
(* ****************************** *)
axclass pcpo < cpo
- least "? x.!y. x<<y"
+ least: "? x.!y. x<<y"
consts
UU :: "'a::pcpo"
syntax (xsymbols)
- UU :: "'a::pcpo" ("\\<bottom>")
+ UU :: "'a::pcpo" ("\<bottom>")
defs
- UU_def "UU == @x.!y. x<<y"
+ UU_def: "UU == @x.!y. x<<y"
(* further useful classes for HOLCF domains *)
axclass chfin<cpo
-chfin "!Y. chain Y-->(? n. max_in_chain n Y)"
+chfin: "!Y. chain Y-->(? n. max_in_chain n Y)"
axclass flat<pcpo
-ax_flat "! x y. x << y --> (x = UU) | (x=y)"
+ax_flat: "! x y. x << y --> (x = UU) | (x=y)"
+
+(* Title: HOLCF/Pcpo.ML
+ ID: $Id$
+ Author: Franz Regensburger
+ License: GPL (GNU GENERAL PUBLIC LICENSE)
+
+introduction of the classes cpo and pcpo
+*)
+
+
+(* ------------------------------------------------------------------------ *)
+(* derive the old rule minimal *)
+(* ------------------------------------------------------------------------ *)
+
+lemma UU_least: "ALL z. UU << z"
+apply (unfold UU_def)
+apply (rule some_eq_ex [THEN iffD2])
+apply (rule least)
+done
+
+lemmas minimal = UU_least [THEN spec, standard]
+
+declare minimal [iff]
+
+(* ------------------------------------------------------------------------ *)
+(* in cpo's everthing equal to THE lub has lub properties for every chain *)
+(* ------------------------------------------------------------------------ *)
+
+lemma thelubE: "[| chain(S); lub(range(S)) = (l::'a::cpo) |] ==> range(S) <<| l "
+apply (blast dest: cpo intro: lubI)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* Properties of the lub *)
+(* ------------------------------------------------------------------------ *)
+
+
+lemma is_ub_thelub: "chain (S::nat => 'a::cpo) ==> S(x) << lub(range(S))"
+apply (blast dest: cpo intro: lubI [THEN is_ub_lub])
+done
+
+lemma is_lub_thelub: "[| chain (S::nat => 'a::cpo); range(S) <| x |] ==> lub(range S) << x"
+apply (blast dest: cpo intro: lubI [THEN is_lub_lub])
+done
+
+lemma lub_range_mono: "[| range X <= range Y; chain Y; chain (X::nat=>'a::cpo) |] ==> lub(range X) << lub(range Y)"
+apply (erule is_lub_thelub)
+apply (rule ub_rangeI)
+apply (subgoal_tac "? j. X i = Y j")
+apply clarsimp
+apply (erule is_ub_thelub)
+apply auto
+done
+
+lemma lub_range_shift: "chain (Y::nat=>'a::cpo) ==> lub(range (%i. Y(i + j))) = lub(range Y)"
+apply (rule antisym_less)
+apply (rule lub_range_mono)
+apply fast
+apply assumption
+apply (erule chain_shift)
+apply (rule is_lub_thelub)
+apply assumption
+apply (rule ub_rangeI)
+apply (rule trans_less)
+apply (rule_tac [2] is_ub_thelub)
+apply (erule_tac [2] chain_shift)
+apply (erule chain_mono3)
+apply (rule le_add1)
+done
+
+lemma maxinch_is_thelub: "chain Y ==> max_in_chain i Y = (lub(range(Y)) = ((Y i)::'a::cpo))"
+apply (rule iffI)
+apply (fast intro!: thelubI lub_finch1)
+apply (unfold max_in_chain_def)
+apply (safe intro!: antisym_less)
+apply (fast elim!: chain_mono3)
+apply (drule sym)
+apply (force elim!: is_ub_thelub)
+done
+
+
+(* ------------------------------------------------------------------------ *)
+(* the << relation between two chains is preserved by their lubs *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_mono: "[|chain(C1::(nat=>'a::cpo));chain(C2); ALL k. C1(k) << C2(k)|]
+ ==> lub(range(C1)) << lub(range(C2))"
+apply (erule is_lub_thelub)
+apply (rule ub_rangeI)
+apply (rule trans_less)
+apply (erule spec)
+apply (erule is_ub_thelub)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* the = relation between two chains is preserved by their lubs *)
+(* ------------------------------------------------------------------------ *)
+
+lemma lub_equal: "[| chain(C1::(nat=>'a::cpo));chain(C2);ALL k. C1(k)=C2(k)|]
+ ==> lub(range(C1))=lub(range(C2))"
+apply (rule antisym_less)
+apply (rule lub_mono)
+apply assumption
+apply assumption
+apply (intro strip)
+apply (rule antisym_less_inverse [THEN conjunct1])
+apply (erule spec)
+apply (rule lub_mono)
+apply assumption
+apply assumption
+apply (intro strip)
+apply (rule antisym_less_inverse [THEN conjunct2])
+apply (erule spec)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* more results about mono and = of lubs of chains *)
+(* ------------------------------------------------------------------------ *)
+lemma lub_mono2: "[|EX j. ALL i. j<i --> X(i::nat)=Y(i);chain(X::nat=>'a::cpo);chain(Y)|]
+ ==> lub(range(X))<<lub(range(Y))"
+apply (erule exE)
+apply (rule is_lub_thelub)
+apply assumption
+apply (rule ub_rangeI)
+(* apply (intro strip) *)
+apply (case_tac "j<i")
+apply (rule_tac s = "Y (i) " and t = "X (i) " in subst)
+apply (rule sym)
+apply fast
+apply (rule is_ub_thelub)
+apply assumption
+apply (rule_tac y = "X (Suc (j))" in trans_less)
+apply (rule chain_mono)
+apply assumption
+apply (rule not_less_eq [THEN subst])
+apply assumption
+apply (rule_tac s = "Y (Suc (j))" and t = "X (Suc (j))" in subst)
+apply (simp (no_asm_simp))
+apply (erule is_ub_thelub)
+done
+
+lemma lub_equal2: "[|EX j. ALL i. j<i --> X(i)=Y(i); chain(X::nat=>'a::cpo); chain(Y)|]
+ ==> lub(range(X))=lub(range(Y))"
+apply (blast intro: antisym_less lub_mono2 sym)
+done
+
+lemma lub_mono3: "[|chain(Y::nat=>'a::cpo);chain(X);
+ ALL i. EX j. Y(i)<< X(j)|]==> lub(range(Y))<<lub(range(X))"
+apply (rule is_lub_thelub)
+apply assumption
+apply (rule ub_rangeI)
+(* apply (intro strip) *)
+apply (erule allE)
+apply (erule exE)
+apply (rule trans_less)
+apply (rule_tac [2] is_ub_thelub)
+prefer 2 apply (assumption)
+apply assumption
+done
+
+(* ------------------------------------------------------------------------ *)
+(* usefull lemmas about UU *)
+(* ------------------------------------------------------------------------ *)
+
+lemma eq_UU_iff: "(x=UU)=(x<<UU)"
+apply (rule iffI)
+apply (erule ssubst)
+apply (rule refl_less)
+apply (rule antisym_less)
+apply assumption
+apply (rule minimal)
+done
+
+lemma UU_I: "x << UU ==> x = UU"
+apply (subst eq_UU_iff)
+apply assumption
+done
+
+lemma not_less2not_eq: "~(x::'a::po)<<y ==> ~x=y"
+apply auto
+done
+
+lemma chain_UU_I: "[|chain(Y);lub(range(Y))=UU|] ==> ALL i. Y(i)=UU"
+apply (rule allI)
+apply (rule antisym_less)
+apply (rule_tac [2] minimal)
+apply (erule subst)
+apply (erule is_ub_thelub)
+done
+
+
+lemma chain_UU_I_inverse: "ALL i. Y(i::nat)=UU ==> lub(range(Y::(nat=>'a::pcpo)))=UU"
+apply (rule lub_chain_maxelem)
+apply (erule spec)
+apply (rule allI)
+apply (rule antisym_less_inverse [THEN conjunct1])
+apply (erule spec)
+done
+
+lemma chain_UU_I_inverse2: "~lub(range(Y::(nat=>'a::pcpo)))=UU ==> EX i.~ Y(i)=UU"
+apply (blast intro: chain_UU_I_inverse)
+done
+
+lemma notUU_I: "[| x<<y; ~x=UU |] ==> ~y=UU"
+apply (blast intro: UU_I)
+done
+
+lemma chain_mono2:
+ "[|EX j. ~Y(j)=UU;chain(Y::nat=>'a::pcpo)|] ==> EX j. ALL i. j<i-->~Y(i)=UU"
+apply (blast dest: notUU_I chain_mono)
+done
+
+(**************************************)
+(* some properties for chfin and flat *)
+(**************************************)
+
+(* ------------------------------------------------------------------------ *)
+(* flat types are chfin *)
+(* ------------------------------------------------------------------------ *)
+
+(*Used only in an "instance" declaration (Fun1.thy)*)
+lemma flat_imp_chfin:
+ "ALL Y::nat=>'a::flat. chain Y --> (EX n. max_in_chain n Y)"
+apply (unfold max_in_chain_def)
+apply clarify
+apply (case_tac "ALL i. Y (i) =UU")
+apply (rule_tac x = "0" in exI)
+apply (simp (no_asm_simp))
+apply simp
+apply (erule exE)
+apply (rule_tac x = "i" in exI)
+apply (intro strip)
+apply (erule le_imp_less_or_eq [THEN disjE])
+apply safe
+apply (blast dest: chain_mono ax_flat [THEN spec, THEN spec, THEN mp])
+done
+
+(* flat subclass of chfin --> adm_flat not needed *)
+
+lemma flat_eq: "(a::'a::flat) ~= UU ==> a << b = (a = b)"
+apply (safe intro!: refl_less)
+apply (drule ax_flat [THEN spec, THEN spec, THEN mp])
+apply (fast intro!: refl_less ax_flat [THEN spec, THEN spec, THEN mp])
+done
+
+lemma chfin2finch: "chain (Y::nat=>'a::chfin) ==> finite_chain Y"
+apply (force simp add: chfin finite_chain_def)
+done
+
+(* ------------------------------------------------------------------------ *)
+(* lemmata for improved admissibility introdution rule *)
+(* ------------------------------------------------------------------------ *)
+
+lemma infinite_chain_adm_lemma:
+"[|chain Y; ALL i. P (Y i);
+ (!!Y. [| chain Y; ALL i. P (Y i); ~ finite_chain Y |] ==> P (lub(range Y)))
+ |] ==> P (lub (range Y))"
+(* apply (cut_tac prems) *)
+apply (case_tac "finite_chain Y")
+prefer 2 apply fast
+apply (unfold finite_chain_def)
+apply safe
+apply (erule lub_finch1 [THEN thelubI, THEN ssubst])
+apply assumption
+apply (erule spec)
+done
+
+lemma increasing_chain_adm_lemma:
+"[|chain Y; ALL i. P (Y i);
+ (!!Y. [| chain Y; ALL i. P (Y i);
+ ALL i. EX j. i < j & Y i ~= Y j & Y i << Y j|]
+ ==> P (lub (range Y))) |] ==> P (lub (range Y))"
+(* apply (cut_tac prems) *)
+apply (erule infinite_chain_adm_lemma)
+apply assumption
+apply (erule thin_rl)
+apply (unfold finite_chain_def)
+apply (unfold max_in_chain_def)
+apply (fast dest: le_imp_less_or_eq elim: chain_mono)
+done
end