--- a/src/HOLCF/Pcpo.thy Fri Jul 01 01:50:46 2005 +0200
+++ b/src/HOLCF/Pcpo.thy Fri Jul 01 02:30:59 2005 +0200
@@ -17,31 +17,35 @@
axclass cpo < po
-- {* class axiom: *}
- cpo: "chain S ==> ? x. range S <<| x"
+ cpo: "chain S \<Longrightarrow> \<exists>x. range S <<| x"
text {* 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"
+lemma thelubE: "\<lbrakk>chain S; (\<Squnion>i. S i) = (l::'a::cpo)\<rbrakk> \<Longrightarrow> range S <<| l"
by (blast dest: cpo intro: lubI)
text {* Properties of the lub *}
-lemma is_ub_thelub: "chain (S::nat => 'a::cpo) ==> S(x) << lub(range(S))"
+lemma is_ub_thelub: "chain (S::nat \<Rightarrow> 'a::cpo) \<Longrightarrow> S x \<sqsubseteq> (\<Squnion>i. S i)"
by (blast dest: cpo intro: lubI [THEN is_ub_lub])
-lemma is_lub_thelub: "[| chain (S::nat => 'a::cpo); range(S) <| x |] ==> lub(range S) << x"
+lemma is_lub_thelub:
+ "\<lbrakk>chain (S::nat \<Rightarrow> 'a::cpo); range S <| x\<rbrakk> \<Longrightarrow> (\<Squnion>i. S i) \<sqsubseteq> x"
by (blast dest: cpo intro: lubI [THEN is_lub_lub])
-lemma lub_range_mono: "[| range X <= range Y; chain Y; chain (X::nat=>'a::cpo) |] ==> lub(range X) << lub(range Y)"
+lemma lub_range_mono:
+ "\<lbrakk>range X \<subseteq> range Y; chain Y; chain (X::nat \<Rightarrow> 'a::cpo)\<rbrakk>
+ \<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)"
apply (erule is_lub_thelub)
apply (rule ub_rangeI)
-apply (subgoal_tac "? j. X i = Y j")
+apply (subgoal_tac "\<exists>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)"
+lemma lub_range_shift:
+ "chain (Y::nat \<Rightarrow> 'a::cpo) \<Longrightarrow> (\<Squnion>i. Y (i + j)) = (\<Squnion>i. Y i)"
apply (rule antisym_less)
apply (rule lub_range_mono)
apply fast
@@ -57,7 +61,8 @@
apply (rule le_add1)
done
-lemma maxinch_is_thelub: "chain Y ==> max_in_chain i Y = (lub(range(Y)) = ((Y i)::'a::cpo))"
+lemma maxinch_is_thelub:
+ "chain Y \<Longrightarrow> max_in_chain i Y = ((\<Squnion>i. Y i) = ((Y i)::'a::cpo))"
apply (rule iffI)
apply (fast intro!: thelubI lub_finch1)
apply (unfold max_in_chain_def)
@@ -67,10 +72,11 @@
apply (force elim!: is_ub_thelub)
done
-text {* the @{text "<<"} relation between two chains is preserved by their lubs *}
+text {* the @{text "\<sqsubseteq>"} 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))"
+lemma lub_mono:
+ "\<lbrakk>chain (X::nat \<Rightarrow> 'a::cpo); chain Y; \<forall>k. X k \<sqsubseteq> Y k\<rbrakk>
+ \<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)"
apply (erule is_lub_thelub)
apply (rule ub_rangeI)
apply (rule trans_less)
@@ -80,49 +86,47 @@
text {* 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))"
+lemma lub_equal:
+ "\<lbrakk>chain (X::nat \<Rightarrow> 'a::cpo); chain Y; \<forall>k. X k = Y k\<rbrakk>
+ \<Longrightarrow> (\<Squnion>i. X i) = (\<Squnion>i. Y i)"
by (simp only: expand_fun_eq [symmetric])
text {* 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))"
+lemma lub_mono2:
+ "\<lbrakk>\<exists>j::nat. \<forall>i>j. X i = Y i; chain (X::nat=>'a::cpo); chain Y\<rbrakk>
+ \<Longrightarrow> (\<Squnion>i. X i) \<sqsubseteq> (\<Squnion>i. Y i)"
apply (erule exE)
apply (rule is_lub_thelub)
apply assumption
apply (rule ub_rangeI)
-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)
+apply (case_tac "j < i")
+apply (rule_tac s="Y i" and t="X i" in subst)
+apply simp
+apply (erule is_ub_thelub)
+apply (rule_tac y = "X (Suc j)" in trans_less)
+apply (erule chain_mono)
+apply (erule not_less_eq [THEN iffD1])
+apply (rule_tac s="Y (Suc j)" and t="X (Suc j)" in subst)
+apply 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))"
+lemma lub_equal2:
+ "\<lbrakk>\<exists>j. \<forall>i>j. X i = Y i; chain (X::nat \<Rightarrow> 'a::cpo); chain Y\<rbrakk>
+ \<Longrightarrow> (\<Squnion>i. X i) = (\<Squnion>i. Y i)"
by (blast intro: antisym_less lub_mono2 sym)
-lemma lub_mono3: "[|chain(Y::nat=>'a::cpo);chain(X);
- ALL i. EX j. Y(i)<< X(j)|]==> lub(range(Y))<<lub(range(X))"
+lemma lub_mono3:
+ "\<lbrakk>chain (Y::nat \<Rightarrow> 'a::cpo); chain X; \<forall>i. \<exists>j. Y i \<sqsubseteq> X j\<rbrakk>
+ \<Longrightarrow> (\<Squnion>i. Y i) \<sqsubseteq> (\<Squnion>i. X i)"
apply (rule is_lub_thelub)
apply assumption
apply (rule ub_rangeI)
apply (erule allE)
apply (erule exE)
-apply (rule trans_less)
-apply (rule_tac [2] is_ub_thelub)
-prefer 2 apply (assumption)
-apply assumption
+apply (erule trans_less)
+apply (erule is_ub_thelub)
done
lemma ch2ch_lub:
@@ -177,20 +181,18 @@
text {* The class pcpo of pointed cpos *}
axclass pcpo < cpo
- least: "? x.!y. x<<y"
+ least: "\<exists>x. \<forall>y. x \<sqsubseteq> y"
-consts
- UU :: "'a::pcpo"
+constdefs
+ UU :: "'a::pcpo"
+ "UU \<equiv> THE x. ALL y. x \<sqsubseteq> y"
syntax (xsymbols)
- UU :: "'a::pcpo" ("\<bottom>")
-
-defs
- UU_def: "UU == THE x. ALL y. x<<y"
+ UU :: "'a::pcpo" ("\<bottom>")
text {* derive the old rule minimal *}
-lemma UU_least: "ALL z. UU << z"
+lemma UU_least: "\<forall>z. \<bottom> \<sqsubseteq> z"
apply (unfold UU_def)
apply (rule theI')
apply (rule ex_ex1I)
@@ -198,13 +200,12 @@
apply (blast intro: antisym_less)
done
-lemmas minimal = UU_least [THEN spec, standard]
-
-declare minimal [iff]
+lemma minimal [iff]: "\<bottom> \<sqsubseteq> x"
+by (rule UU_least [THEN spec])
-text {* useful lemmas about @{term UU} *}
+text {* useful lemmas about @{term \<bottom>} *}
-lemma eq_UU_iff: "(x=UU)=(x<<UU)"
+lemma eq_UU_iff: "(x = \<bottom>) = (x \<sqsubseteq> \<bottom>)"
apply (rule iffI)
apply (erule ssubst)
apply (rule refl_less)
@@ -213,34 +214,33 @@
apply (rule minimal)
done
-lemma UU_I: "x << UU ==> x = UU"
+lemma UU_I: "x \<sqsubseteq> \<bottom> \<Longrightarrow> x = \<bottom>"
by (subst eq_UU_iff)
-lemma not_less2not_eq: "~(x::'a::po)<<y ==> ~x=y"
+lemma not_less2not_eq: "\<not> (x::'a::po) \<sqsubseteq> y \<Longrightarrow> x \<noteq> y"
by auto
-lemma chain_UU_I: "[|chain(Y);lub(range(Y))=UU|] ==> ALL i. Y(i)=UU"
+lemma chain_UU_I: "\<lbrakk>chain Y; (\<Squnion>i. Y i) = \<bottom>\<rbrakk> \<Longrightarrow> \<forall>i. Y i = \<bottom>"
apply (rule allI)
-apply (rule antisym_less)
-apply (rule_tac [2] minimal)
+apply (rule UU_I)
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"
+lemma chain_UU_I_inverse: "\<forall>i::nat. Y i = \<bottom> \<Longrightarrow> (\<Squnion>i. Y i) = \<bottom>"
apply (rule lub_chain_maxelem)
apply (erule spec)
apply simp
done
-lemma chain_UU_I_inverse2: "~lub(range(Y::(nat=>'a::pcpo)))=UU ==> EX i.~ Y(i)=UU"
+lemma chain_UU_I_inverse2: "(\<Squnion>i. Y i) \<noteq> \<bottom> \<Longrightarrow> \<exists>i::nat. Y i \<noteq> \<bottom>"
by (blast intro: chain_UU_I_inverse)
-lemma notUU_I: "[| x<<y; ~x=UU |] ==> ~y=UU"
+lemma notUU_I: "\<lbrakk>x \<sqsubseteq> y; x \<noteq> \<bottom>\<rbrakk> \<Longrightarrow> y \<noteq> \<bottom>"
by (blast intro: UU_I)
lemma chain_mono2:
- "[|EX j. ~Y(j)=UU;chain(Y::nat=>'a::pcpo)|] ==> EX j. ALL i. j<i-->~Y(i)=UU"
+ "\<lbrakk>\<exists>j::nat. Y j \<noteq> \<bottom>; chain Y\<rbrakk> \<Longrightarrow> \<exists>j. \<forall>i>j. Y i \<noteq> \<bottom>"
by (blast dest: notUU_I chain_mono)
subsection {* Chain-finite and flat cpos *}
@@ -248,17 +248,17 @@
text {* further useful classes for HOLCF domains *}
axclass chfin < po
- chfin: "!Y. chain Y-->(? n. max_in_chain n Y)"
+ chfin: "\<forall>Y. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
axclass flat < pcpo
- ax_flat: "! x y. x << y --> (x = UU) | (x=y)"
+ ax_flat: "\<forall>x y. x \<sqsubseteq> y \<longrightarrow> (x = \<bottom>) \<or> (x = y)"
text {* some properties for chfin and flat *}
text {* chfin types are cpo *}
lemma chfin_imp_cpo:
- "chain (S::nat=>'a::chfin) ==> EX x. range S <<| x"
+ "chain (S::nat \<Rightarrow> 'a::chfin) \<Longrightarrow> \<exists>x. range S <<| x"
apply (frule chfin [rule_format])
apply (blast intro: lub_finch1)
done
@@ -269,14 +269,14 @@
text {* flat types are chfin *}
lemma flat_imp_chfin:
- "ALL Y::nat=>'a::flat. chain Y --> (EX n. max_in_chain n Y)"
+ "\<forall>Y::nat \<Rightarrow> 'a::flat. chain Y \<longrightarrow> (\<exists>n. max_in_chain n Y)"
apply (unfold max_in_chain_def)
apply clarify
-apply (case_tac "ALL i. Y (i) =UU")
+apply (case_tac "\<forall>i. Y i = \<bottom>")
apply simp
apply simp
apply (erule exE)
-apply (rule_tac x = "i" in exI)
+apply (rule_tac x="i" in exI)
apply clarify
apply (erule le_imp_less_or_eq [THEN disjE])
apply safe
@@ -288,18 +288,18 @@
text {* flat subclass of chfin @{text "-->"} @{text adm_flat} not needed *}
-lemma flat_eq: "(a::'a::flat) ~= UU ==> a << b = (a = b)"
+lemma flat_eq: "(a::'a::flat) \<noteq> \<bottom> \<Longrightarrow> a \<sqsubseteq> b = (a = b)"
by (safe dest!: ax_flat [rule_format])
-lemma chfin2finch: "chain (Y::nat=>'a::chfin) ==> finite_chain Y"
+lemma chfin2finch: "chain (Y::nat \<Rightarrow> 'a::chfin) \<Longrightarrow> finite_chain Y"
by (simp add: chfin finite_chain_def)
text {* 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))"
+ "\<lbrakk>chain Y; \<forall>i. P (Y i);
+ \<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i); \<not> finite_chain Y\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
+ \<Longrightarrow> P (\<Squnion>i. Y i)"
apply (case_tac "finite_chain Y")
prefer 2 apply fast
apply (unfold finite_chain_def)
@@ -310,10 +310,9 @@
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))"
+ "\<lbrakk>chain Y; \<forall>i. P (Y i); \<And>Y. \<lbrakk>chain Y; \<forall>i. P (Y i);
+ \<forall>i. \<exists>j>i. Y i \<noteq> Y j \<and> Y i \<sqsubseteq> Y j\<rbrakk> \<Longrightarrow> P (\<Squnion>i. Y i)\<rbrakk>
+ \<Longrightarrow> P (\<Squnion>i. Y i)"
apply (erule infinite_chain_adm_lemma)
apply assumption
apply (erule thin_rl)
@@ -322,4 +321,4 @@
apply (fast dest: le_imp_less_or_eq elim: chain_mono)
done
-end
+end