--- a/src/HOL/Library/Ramsey.thy Fri Jun 23 10:48:34 2006 +0200
+++ b/src/HOL/Library/Ramsey.thy Fri Jun 23 13:42:19 2006 +0200
@@ -25,7 +25,7 @@
proof (induct n)
case 0 show ?case by (force intro: someI P0)
next
- case (Suc n) thus ?case by (auto intro: someI2_ex [OF Pstep])
+ case Suc thus ?case by (auto intro: someI2_ex [OF Pstep])
qed
lemma dependent_choice:
@@ -125,10 +125,8 @@
qed
qed
from dependent_choice [OF transr propr0 proprstep]
- obtain g where "(\<forall>n::nat. ?propr(g n)) & (\<forall>n m. n<m -->(g n, g m) \<in> ?ramr)"
- .. --{*for some reason, can't derive the following directly from dc*}
- hence pg: "!!n. ?propr (g n)"
- and rg: "!!n m. n<m ==> (g n, g m) \<in> ?ramr" by auto
+ obtain g where pg: "!!n::nat. ?propr (g n)"
+ and rg: "!!n m. n<m ==> (g n, g m) \<in> ?ramr" by force
let ?gy = "(\<lambda>n. let (y,Y,t) = g n in y)"
let ?gt = "(\<lambda>n. let (y,Y,t) = g n in t)"
have rangeg: "\<exists>k. range ?gt \<subseteq> {..<k}"
@@ -195,7 +193,7 @@
by (simp add: card_Diff_singleton_if cardX ya)
ultimately show ?thesis
using pg [of "LEAST x. x \<in> AA"] fields cardX
- by (clarify, drule_tac x="X-{ya}" in spec, simp)
+ by (clarsimp simp del:insert_Diff_single)
qed
also have "... = s'" using AA AAleast fields by auto
finally show ?thesis .
@@ -215,7 +213,7 @@
\<forall>X. X \<subseteq> Z & finite X & card X = r --> f X < s|]
==> \<exists>Y t. Y \<subseteq> Z & infinite Y & t < s
& (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X = t)"
-by (blast intro: ramsey_induction [unfolded part_def, rule_format])
+by (blast intro: ramsey_induction [unfolded part_def])
end