--- a/src/HOL/Finite_Set.thy Thu Jun 18 15:08:57 2009 +0200
+++ b/src/HOL/Finite_Set.thy Thu Jun 18 17:14:08 2009 +0100
@@ -524,7 +524,7 @@
subsection {* A fold functional for finite sets *}
text {* The intended behaviour is
-@{text "fold f z {x\<^isub>1, ..., x\<^isub>n} = f x\<^isub>1 (\<dots> (f x\<^isub>n z)\<dots>)"}
+@{text "fold f z {x1, ..., xn} = f x1 (\<dots> (f xn z)\<dots>)"}
if @{text f} is ``left-commutative'':
*}
@@ -1879,14 +1879,14 @@
(if a:A then setprod f A / f a else setprod f A)"
by (erule finite_induct) (auto simp add: insert_Diff_if)
-lemma setprod_inversef: "finite A ==>
- ALL x: A. f x \<noteq> (0::'a::{field,division_by_zero}) ==>
- setprod (inverse \<circ> f) A = inverse (setprod f A)"
+lemma setprod_inversef:
+ fixes f :: "'b \<Rightarrow> 'a::{field,division_by_zero}"
+ shows "finite A ==> setprod (inverse \<circ> f) A = inverse (setprod f A)"
by (erule finite_induct) auto
lemma setprod_dividef:
- "[|finite A;
- \<forall>x \<in> A. g x \<noteq> (0::'a::{field,division_by_zero})|]
+ fixes f :: "'b \<Rightarrow> 'a::{field,division_by_zero}"
+ shows "finite A
==> setprod (%x. f x / g x) A = setprod f A / setprod g A"
apply (subgoal_tac
"setprod (%x. f x / g x) A = setprod (%x. f x * (inverse \<circ> g) x) A")
@@ -2721,16 +2721,16 @@
begin
definition
- Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>\<^bsub>fin\<^esub>_" [900] 900)
+ Inf_fin :: "'a set \<Rightarrow> 'a" ("\<Sqinter>fin_" [900] 900)
where
"Inf_fin = fold1 inf"
definition
- Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>\<^bsub>fin\<^esub>_" [900] 900)
+ Sup_fin :: "'a set \<Rightarrow> 'a" ("\<Squnion>fin_" [900] 900)
where
"Sup_fin = fold1 sup"
-lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>\<^bsub>fin\<^esub>A \<le> \<Squnion>\<^bsub>fin\<^esub>A"
+lemma Inf_le_Sup [simp]: "\<lbrakk> finite A; A \<noteq> {} \<rbrakk> \<Longrightarrow> \<Sqinter>finA \<le> \<Squnion>finA"
apply(unfold Sup_fin_def Inf_fin_def)
apply(subgoal_tac "EX a. a:A")
prefer 2 apply blast
@@ -2741,13 +2741,13 @@
done
lemma sup_Inf_absorb [simp]:
- "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>\<^bsub>fin\<^esub>A) = a"
+ "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> sup a (\<Sqinter>finA) = a"
apply(subst sup_commute)
apply(simp add: Inf_fin_def sup_absorb2 fold1_belowI)
done
lemma inf_Sup_absorb [simp]:
- "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>\<^bsub>fin\<^esub>A) = a"
+ "finite A \<Longrightarrow> a \<in> A \<Longrightarrow> inf a (\<Squnion>finA) = a"
by (simp add: Sup_fin_def inf_absorb1
lower_semilattice.fold1_belowI [OF dual_lattice])
@@ -2759,7 +2759,7 @@
lemma sup_Inf1_distrib:
assumes "finite A"
and "A \<noteq> {}"
- shows "sup x (\<Sqinter>\<^bsub>fin\<^esub>A) = \<Sqinter>\<^bsub>fin\<^esub>{sup x a|a. a \<in> A}"
+ shows "sup x (\<Sqinter>finA) = \<Sqinter>fin{sup x a|a. a \<in> A}"
proof -
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
@@ -2771,7 +2771,7 @@
lemma sup_Inf2_distrib:
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
- shows "sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B) = \<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B}"
+ shows "sup (\<Sqinter>finA) (\<Sqinter>finB) = \<Sqinter>fin{sup a b|a b. a \<in> A \<and> b \<in> B}"
using A proof (induct rule: finite_ne_induct)
case singleton thus ?case
by (simp add: sup_Inf1_distrib [OF B] fold1_singleton_def [OF Inf_fin_def])
@@ -2788,13 +2788,13 @@
thus ?thesis by(simp add: insert(1) B(1))
qed
have ne: "{sup a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
- have "sup (\<Sqinter>\<^bsub>fin\<^esub>(insert x A)) (\<Sqinter>\<^bsub>fin\<^esub>B) = sup (inf x (\<Sqinter>\<^bsub>fin\<^esub>A)) (\<Sqinter>\<^bsub>fin\<^esub>B)"
+ have "sup (\<Sqinter>fin(insert x A)) (\<Sqinter>finB) = sup (inf x (\<Sqinter>finA)) (\<Sqinter>finB)"
using insert by (simp add: fold1_insert_idem_def [OF Inf_fin_def])
- also have "\<dots> = inf (sup x (\<Sqinter>\<^bsub>fin\<^esub>B)) (sup (\<Sqinter>\<^bsub>fin\<^esub>A) (\<Sqinter>\<^bsub>fin\<^esub>B))" by(rule sup_inf_distrib2)
- also have "\<dots> = inf (\<Sqinter>\<^bsub>fin\<^esub>{sup x b|b. b \<in> B}) (\<Sqinter>\<^bsub>fin\<^esub>{sup a b|a b. a \<in> A \<and> b \<in> B})"
+ also have "\<dots> = inf (sup x (\<Sqinter>finB)) (sup (\<Sqinter>finA) (\<Sqinter>finB))" by(rule sup_inf_distrib2)
+ also have "\<dots> = inf (\<Sqinter>fin{sup x b|b. b \<in> B}) (\<Sqinter>fin{sup a b|a b. a \<in> A \<and> b \<in> B})"
using insert by(simp add:sup_Inf1_distrib[OF B])
- also have "\<dots> = \<Sqinter>\<^bsub>fin\<^esub>({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
- (is "_ = \<Sqinter>\<^bsub>fin\<^esub>?M")
+ also have "\<dots> = \<Sqinter>fin({sup x b |b. b \<in> B} \<union> {sup a b |a b. a \<in> A \<and> b \<in> B})"
+ (is "_ = \<Sqinter>fin?M")
using B insert
by (simp add: Inf_fin_def fold1_Un2 [OF finB _ finAB ne])
also have "?M = {sup a b |a b. a \<in> insert x A \<and> b \<in> B}"
@@ -2804,7 +2804,7 @@
lemma inf_Sup1_distrib:
assumes "finite A" and "A \<noteq> {}"
- shows "inf x (\<Squnion>\<^bsub>fin\<^esub>A) = \<Squnion>\<^bsub>fin\<^esub>{inf x a|a. a \<in> A}"
+ shows "inf x (\<Squnion>finA) = \<Squnion>fin{inf x a|a. a \<in> A}"
proof -
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)
@@ -2815,7 +2815,7 @@
lemma inf_Sup2_distrib:
assumes A: "finite A" "A \<noteq> {}" and B: "finite B" "B \<noteq> {}"
- shows "inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B) = \<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B}"
+ shows "inf (\<Squnion>finA) (\<Squnion>finB) = \<Squnion>fin{inf a b|a b. a \<in> A \<and> b \<in> B}"
using A proof (induct rule: finite_ne_induct)
case singleton thus ?case
by(simp add: inf_Sup1_distrib [OF B] fold1_singleton_def [OF Sup_fin_def])
@@ -2832,13 +2832,13 @@
have ne: "{inf a b |a b. a \<in> A \<and> b \<in> B} \<noteq> {}" using insert B by blast
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)
- have "inf (\<Squnion>\<^bsub>fin\<^esub>(insert x A)) (\<Squnion>\<^bsub>fin\<^esub>B) = inf (sup x (\<Squnion>\<^bsub>fin\<^esub>A)) (\<Squnion>\<^bsub>fin\<^esub>B)"
+ have "inf (\<Squnion>fin(insert x A)) (\<Squnion>finB) = inf (sup x (\<Squnion>finA)) (\<Squnion>finB)"
using insert by (simp add: fold1_insert_idem_def [OF Sup_fin_def])
- also have "\<dots> = sup (inf x (\<Squnion>\<^bsub>fin\<^esub>B)) (inf (\<Squnion>\<^bsub>fin\<^esub>A) (\<Squnion>\<^bsub>fin\<^esub>B))" by(rule inf_sup_distrib2)
- also have "\<dots> = sup (\<Squnion>\<^bsub>fin\<^esub>{inf x b|b. b \<in> B}) (\<Squnion>\<^bsub>fin\<^esub>{inf a b|a b. a \<in> A \<and> b \<in> B})"
+ also have "\<dots> = sup (inf x (\<Squnion>finB)) (inf (\<Squnion>finA) (\<Squnion>finB))" by(rule inf_sup_distrib2)
+ also have "\<dots> = sup (\<Squnion>fin{inf x b|b. b \<in> B}) (\<Squnion>fin{inf a b|a b. a \<in> A \<and> b \<in> B})"
using insert by(simp add:inf_Sup1_distrib[OF B])
- also have "\<dots> = \<Squnion>\<^bsub>fin\<^esub>({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
- (is "_ = \<Squnion>\<^bsub>fin\<^esub>?M")
+ also have "\<dots> = \<Squnion>fin({inf x b |b. b \<in> B} \<union> {inf a b |a b. a \<in> A \<and> b \<in> B})"
+ (is "_ = \<Squnion>fin?M")
using B insert
by (simp add: Sup_fin_def fold1_Un2 [OF finB _ finAB ne])
also have "?M = {inf a b |a b. a \<in> insert x A \<and> b \<in> B}"
@@ -2857,7 +2857,7 @@
lemma Inf_fin_Inf:
assumes "finite A" and "A \<noteq> {}"
- shows "\<Sqinter>\<^bsub>fin\<^esub>A = Inf A"
+ shows "\<Sqinter>finA = Inf A"
proof -
interpret ab_semigroup_idem_mult inf
by (rule ab_semigroup_idem_mult_inf)
@@ -2868,7 +2868,7 @@
lemma Sup_fin_Sup:
assumes "finite A" and "A \<noteq> {}"
- shows "\<Squnion>\<^bsub>fin\<^esub>A = Sup A"
+ shows "\<Squnion>finA = Sup A"
proof -
interpret ab_semigroup_idem_mult sup
by (rule ab_semigroup_idem_mult_sup)