--- a/src/HOL/Infinite_Set.thy Thu Jun 22 18:48:25 2006 +0200
+++ b/src/HOL/Infinite_Set.thy Fri Jun 23 09:55:01 2006 +0200
@@ -221,7 +221,7 @@
*}
lemma linorder_injI:
- assumes hyp: "\<forall>x y. x < (y::'a::linorder) \<longrightarrow> f x \<noteq> f y"
+ assumes hyp: "!!x y. x < (y::'a::linorder) ==> f x \<noteq> f y"
shows "inj f"
proof (rule inj_onI)
fix x y
@@ -295,8 +295,8 @@
qed
have inj: "inj pick"
proof (rule linorder_injI)
- show "\<forall>i j. i<(j::nat) \<longrightarrow> pick i \<noteq> pick j"
- proof (clarify)
+ show "!!i j. i<(j::nat) ==> pick i \<noteq> pick j"
+ proof
fix i j
assume ij: "i<(j::nat)"
and eq: "pick i = pick j"
--- a/src/HOL/IsaMakefile Thu Jun 22 18:48:25 2006 +0200
+++ b/src/HOL/IsaMakefile Fri Jun 23 09:55:01 2006 +0200
@@ -199,7 +199,7 @@
HOL-Library: HOL $(LOG)/HOL-Library.gz
$(LOG)/HOL-Library.gz: $(OUT)/HOL \
- Library/SetsAndFunctions.thy Library/BigO.thy \
+ Library/SetsAndFunctions.thy Library/BigO.thy Library/Ramsey.thy \
Library/EfficientNat.thy Library/ExecutableSet.thy Library/ExecutableRat.thy \
Library/FuncSet.thy Library/Library.thy \
Library/List_Prefix.thy Library/Multiset.thy Library/NatPair.thy \
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Ramsey.thy Fri Jun 23 09:55:01 2006 +0200
@@ -0,0 +1,221 @@
+(* Title: HOL/Library/Ramsey.thy
+ ID: $Id$
+ Author: Tom Ridge. Converted to structured Isar by L C Paulson
+*)
+
+header "Ramsey's Theorem"
+
+theory Ramsey imports Main begin
+
+
+subsection{*``Axiom'' of Dependent Choice*}
+
+consts choice :: "('a => bool) => (('a * 'a) set) => nat => 'a"
+ --{*An integer-indexed chain of choices*}
+primrec
+ choice_0: "choice P r 0 = (SOME x. P x)"
+
+ choice_Suc: "choice P r (Suc n) = (SOME y. P y & (choice P r n, y) \<in> r)"
+
+
+lemma choice_n:
+ assumes P0: "P x0"
+ and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
+ shows "P (choice P r n)"
+ 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])
+ qed
+
+lemma dependent_choice:
+ assumes trans: "trans r"
+ and P0: "P x0"
+ and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
+ shows "\<exists>f::nat=>'a. (\<forall>n. P (f n)) & (\<forall>n m. n<m --> (f n, f m) \<in> r)"
+proof (intro exI conjI)
+ show "\<forall>n. P (choice P r n)" by (blast intro: choice_n [OF P0 Pstep])
+next
+ have PSuc: "\<forall>n. (choice P r n, choice P r (Suc n)) \<in> r"
+ using Pstep [OF choice_n [OF P0 Pstep]]
+ by (auto intro: someI2_ex)
+ show "\<forall>n m. n<m --> (choice P r n, choice P r m) \<in> r"
+ proof (intro strip)
+ fix n and m::nat
+ assume less: "n<m"
+ show "(choice P r n, choice P r m) \<in> r" using PSuc
+ by (auto intro: less_Suc_induct [OF less] transD [OF trans])
+ qed
+qed
+
+
+subsection {*Partitions of a Set*}
+
+constdefs part :: "nat => nat => 'a set => ('a set => nat) => bool"
+ --{*the function @{term f} partitions the @{term r}-subsets of the typically
+ infinite set @{term Y} into @{term s} distinct categories.*}
+ "part r s Y f == \<forall>X. X \<subseteq> Y & finite X & card X = r --> f X < s"
+
+text{*For induction, we decrease the value of @{term r} in partitions.*}
+lemma part_Suc_imp_part:
+ "[| infinite Y; part (Suc r) s Y f; y \<in> Y |]
+ ==> part r s (Y - {y}) (%u. f (insert y u))"
+ apply(simp add: part_def, clarify)
+ apply(drule_tac x="insert y X" in spec)
+ apply(force simp:card_Diff_singleton_if)
+ done
+
+lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
+ by (simp add: part_def, blast)
+
+
+subsection {*Ramsey's Theorem: Infinitary Version*}
+
+lemma ramsey_induction:
+ fixes s::nat and r::nat
+ shows
+ "!!(YY::'a set) (f::'a set => nat).
+ [|infinite YY; part r s YY f|]
+ ==> \<exists>Y' t'. Y' \<subseteq> YY & infinite Y' & t' < s &
+ (\<forall>X. X \<subseteq> Y' & finite X & card X = r --> f X = t')"
+proof (induct r)
+ case 0
+ thus ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
+next
+ case (Suc r)
+ show ?case
+ proof -
+ from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy \<in> YY" by blast
+ let ?ramr = "{((y,Y,t),(y',Y',t')). y' \<in> Y & Y' \<subseteq> Y}"
+ let ?propr = "%(y,Y,t).
+ y \<in> YY & y \<notin> Y & Y \<subseteq> YY & infinite Y & t < s
+ & (\<forall>X. X\<subseteq>Y & finite X & card X = r --> (f o insert y) X = t)"
+ have infYY': "infinite (YY-{yy})" using Suc.prems by auto
+ have partf': "part r s (YY - {yy}) (f \<circ> insert yy)"
+ by (simp add: o_def part_Suc_imp_part yy Suc.prems)
+ have transr: "trans ?ramr" by (force simp add: trans_def)
+ from Suc.hyps [OF infYY' partf']
+ obtain Y0 and t0
+ where "Y0 \<subseteq> YY - {yy}" "infinite Y0" "t0 < s"
+ "\<forall>X. X\<subseteq>Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0"
+ by blast
+ with yy have propr0: "?propr(yy,Y0,t0)" by blast
+ have proprstep: "\<And>x. ?propr x \<Longrightarrow> \<exists>y. ?propr y \<and> (x, y) \<in> ?ramr"
+ proof -
+ fix x
+ assume px: "?propr x" thus "?thesis x"
+ proof (cases x)
+ case (fields yx Yx tx)
+ then obtain yx' where yx': "yx' \<in> Yx" using px
+ by (blast dest: infinite_imp_nonempty)
+ have infYx': "infinite (Yx-{yx'})" using fields px by auto
+ with fields px yx' Suc.prems
+ have partfx': "part r s (Yx - {yx'}) (f \<circ> insert yx')"
+ by (simp add: o_def part_Suc_imp_part part_subset [where ?YY=YY])
+ from Suc.hyps [OF infYx' partfx']
+ obtain Y' and t'
+ where Y': "Y' \<subseteq> Yx - {yx'}" "infinite Y'" "t' < s"
+ "\<forall>X. X\<subseteq>Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'"
+ by blast
+ show ?thesis
+ proof
+ show "?propr (yx',Y',t') & (x, (yx',Y',t')) \<in> ?ramr"
+ using fields Y' yx' px by blast
+ qed
+ 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
+ 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}"
+ proof (intro exI subsetI)
+ fix x
+ assume "x \<in> range ?gt"
+ then obtain n where "x = ?gt n" ..
+ with pg [of n] show "x \<in> {..<s}" by (cases "g n") auto
+ qed
+ have "\<exists>s' \<in> range ?gt. infinite (?gt -` {s'})"
+ by (rule inf_img_fin_dom [OF _ nat_infinite])
+ (simp add: finite_nat_iff_bounded rangeg)
+ then obtain s' and n'
+ where s': "s' = ?gt n'"
+ and infeqs': "infinite {n. ?gt n = s'}"
+ by (auto simp add: vimage_def)
+ with pg [of n'] have less': "s'<s" by (cases "g n'") auto
+ have inj_gy: "inj ?gy"
+ proof (rule linorder_injI)
+ fix m and m'::nat assume less: "m < m'" show "?gy m \<noteq> ?gy m'"
+ using rg [OF less] pg [of m] by (cases "g m", cases "g m'", auto)
+ qed
+ show ?thesis
+ proof (intro exI conjI)
+ show "?gy ` {n. ?gt n = s'} \<subseteq> YY" using pg
+ by (auto simp add: Let_def split_beta)
+ next
+ show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
+ by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
+ next
+ show "s' < s" by (rule less')
+ next
+ show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
+ --> f X = s'"
+ proof -
+ {fix X
+ assume "X \<subseteq> ?gy ` {n. ?gt n = s'}"
+ and cardX: "finite X" "card X = Suc r"
+ then obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
+ by (auto simp add: subset_image_iff)
+ with cardX have "AA\<noteq>{}" by auto
+ hence AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
+ have "f X = s'"
+ proof (cases "g (LEAST x. x \<in> AA)")
+ case (fields ya Ya ta)
+ with AAleast Xeq
+ have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
+ hence "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
+ also have "... = ta"
+ proof -
+ have "X - {ya} \<subseteq> Ya"
+ proof
+ fix x
+ assume x: "x \<in> X - {ya}"
+ then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
+ by (auto simp add: Xeq)
+ hence "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
+ hence lessa': "(LEAST x. x \<in> AA) < a'"
+ using Least_le [of "%x. x \<in> AA", OF a'] by arith
+ show "x \<in> Ya" using xeq fields rg [OF lessa'] by auto
+ qed
+ moreover
+ have "card (X - {ya}) = r"
+ 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)
+ qed
+ also have "... = s'" using AA AAleast fields by auto
+ finally show ?thesis .
+ qed}
+ thus ?thesis by blast
+ qed
+ qed
+ qed
+qed
+
+
+text{*Repackaging of Tom Ridge's final result*}
+theorem Ramsey:
+ fixes s::nat and r::nat and Z::"'a set" and f::"'a set => nat"
+ shows
+ "[|infinite Z;
+ \<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])
+
+end
+