--- a/src/HOL/SupInf.thy Sun May 09 14:21:44 2010 -0700
+++ b/src/HOL/SupInf.thy Sun May 09 17:47:43 2010 -0700
@@ -74,7 +74,7 @@
lemma Sup_upper2: (*REAL_IMP_LE_SUP in HOL4*)
fixes x :: real
shows "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y \<le> Sup X"
- by (metis Sup_upper real_le_trans)
+ by (metis Sup_upper order_trans)
lemma Sup_real_iff : (*REAL_SUP_LE in HOL4*)
fixes z :: real
@@ -86,7 +86,7 @@
shows "(!!x. x \<in> X \<Longrightarrow> x \<le> a)
\<Longrightarrow> (!!y. (!!x. x \<in> X \<Longrightarrow> x \<le> y) \<Longrightarrow> a \<le> y) \<Longrightarrow> Sup X = a"
by (metis Sup_least Sup_upper add_le_cancel_left diff_add_cancel insert_absorb
- insert_not_empty real_le_antisym)
+ insert_not_empty order_antisym)
lemma Sup_le:
fixes S :: "real set"
@@ -109,28 +109,28 @@
apply (simp add: max_def)
apply (rule Sup_eq_maximum)
apply (rule insertI1)
- apply (metis Sup_upper [OF _ z] insertE real_le_linear real_le_trans)
+ apply (metis Sup_upper [OF _ z] insertE linear order_trans)
done
next
case False
hence 1:"a < Sup X" by simp
have "Sup X \<le> Sup (insert a X)"
apply (rule Sup_least)
- apply (metis empty_psubset_nonempty psubset_eq x)
+ apply (metis ex_in_conv x)
apply (rule Sup_upper_EX)
apply blast
- apply (metis insert_iff real_le_linear real_le_refl real_le_trans z)
+ apply (metis insert_iff linear order_refl order_trans z)
done
moreover
have "Sup (insert a X) \<le> Sup X"
apply (rule Sup_least)
apply blast
- apply (metis False Sup_upper insertE real_le_linear z)
+ apply (metis False Sup_upper insertE linear z)
done
ultimately have "Sup (insert a X) = Sup X"
by (blast intro: antisym )
thus ?thesis
- by (metis 1 min_max.le_iff_sup real_less_def)
+ by (metis 1 min_max.le_iff_sup less_le)
qed
lemma Sup_insert_if:
@@ -177,7 +177,7 @@
fixes S :: "real set"
assumes fS: "finite S" and Se: "S \<noteq> {}"
shows "a \<ge> Sup S \<longleftrightarrow> (\<forall> x \<in> S. a \<ge> x)"
-by (metis Max_ge Se Sup_finite_Max Sup_finite_in fS real_le_trans)
+by (metis Max_ge Se Sup_finite_Max Sup_finite_in fS order_trans)
lemma Sup_finite_gt_iff:
fixes S :: "real set"
@@ -291,19 +291,19 @@
lemma Inf_lower2:
fixes x :: real
shows "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X \<le> y"
- by (metis Inf_lower real_le_trans)
+ by (metis Inf_lower order_trans)
lemma Inf_real_iff:
fixes z :: real
shows "X \<noteq> {} \<Longrightarrow> (!!x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> (\<exists>x\<in>X. x<y) \<longleftrightarrow> Inf X < y"
- by (metis Inf_greatest Inf_lower less_le_not_le real_le_linear
+ by (metis Inf_greatest Inf_lower less_le_not_le linear
order_less_le_trans)
lemma Inf_eq:
fixes a :: real
shows "(!!x. x \<in> X \<Longrightarrow> a \<le> x) \<Longrightarrow> (!!y. (!!x. x \<in> X \<Longrightarrow> y \<le> x) \<Longrightarrow> y \<le> a) \<Longrightarrow> Inf X = a"
by (metis Inf_greatest Inf_lower add_le_cancel_left diff_add_cancel
- insert_absorb insert_not_empty real_le_antisym)
+ insert_absorb insert_not_empty order_antisym)
lemma Inf_ge:
fixes S :: "real set"
@@ -324,27 +324,27 @@
case True
thus ?thesis
by (simp add: min_def)
- (blast intro: Inf_eq_minimum real_le_refl real_le_trans z)
+ (blast intro: Inf_eq_minimum order_trans z)
next
case False
hence 1:"Inf X < a" by simp
have "Inf (insert a X) \<le> Inf X"
apply (rule Inf_greatest)
- apply (metis empty_psubset_nonempty psubset_eq x)
+ apply (metis ex_in_conv x)
apply (rule Inf_lower_EX)
apply (erule insertI2)
- apply (metis insert_iff real_le_linear real_le_refl real_le_trans z)
+ apply (metis insert_iff linear order_refl order_trans z)
done
moreover
have "Inf X \<le> Inf (insert a X)"
apply (rule Inf_greatest)
apply blast
- apply (metis False Inf_lower insertE real_le_linear z)
+ apply (metis False Inf_lower insertE linear z)
done
ultimately have "Inf (insert a X) = Inf X"
by (blast intro: antisym )
thus ?thesis
- by (metis False min_max.inf_absorb2 real_le_linear)
+ by (metis False min_max.inf_absorb2 linear)
qed
lemma Inf_insert_if:
@@ -377,13 +377,13 @@
lemma Inf_finite_ge_iff:
fixes S :: "real set"
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall> x \<in> S. a \<le> x)"
-by (metis Inf_finite_Min Inf_finite_in Min_le real_le_trans)
+by (metis Inf_finite_Min Inf_finite_in Min_le order_trans)
lemma Inf_finite_le_iff:
fixes S :: "real set"
shows "finite S \<Longrightarrow> S \<noteq> {} \<Longrightarrow> a \<ge> Inf S \<longleftrightarrow> (\<exists> x \<in> S. a \<ge> x)"
by (metis Inf_finite_Min Inf_finite_ge_iff Inf_finite_in Min_le
- real_le_antisym real_le_linear)
+ order_antisym linear)
lemma Inf_finite_gt_iff:
fixes S :: "real set"
@@ -417,7 +417,7 @@
also have "... \<le> e"
apply (rule Sup_asclose)
apply (auto simp add: S)
- apply (metis abs_minus_add_cancel b add_commute real_diff_def)
+ apply (metis abs_minus_add_cancel b add_commute diff_def)
done
finally have "\<bar>- Sup (uminus ` S) - l\<bar> \<le> e" .
thus ?thesis
@@ -474,13 +474,13 @@
proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
by (rule SupInf.Sup_upper [where z=b], auto)
- (metis `a < b` `\<not> P b` real_le_linear real_less_def)
+ (metis `a < b` `\<not> P b` linear less_le)
next
show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
apply (rule SupInf.Sup_least)
apply (auto simp add: )
apply (metis less_le_not_le)
- apply (metis `a<b` `~ P b` real_le_linear real_less_def)
+ apply (metis `a<b` `~ P b` linear less_le)
done
next
fix x
@@ -495,7 +495,7 @@
assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
by (rule_tac z="b" in SupInf.Sup_upper, auto)
- (metis `a<b` `~ P b` real_le_linear real_less_def)
+ (metis `a<b` `~ P b` linear less_le)
qed
end