--- a/src/HOL/Library/Ramsey.thy Wed Mar 01 15:16:06 2017 +0100
+++ b/src/HOL/Library/Ramsey.thy Wed Mar 01 17:09:54 2017 +0100
@@ -2,116 +2,112 @@
Author: Tom Ridge. Converted to structured Isar by L C Paulson
*)
-section "Ramsey's Theorem"
+section \<open>Ramsey's Theorem\<close>
theory Ramsey
-imports Main Infinite_Set
+ imports Infinite_Set
begin
-subsection\<open>Finite Ramsey theorem(s)\<close>
+subsection \<open>Finite Ramsey theorem(s)\<close>
-text\<open>To distinguish the finite and infinite ones, lower and upper case
-names are used.
+text \<open>
+ To distinguish the finite and infinite ones, lower and upper case
+ names are used.
-This is the most basic version in terms of cliques and independent
-sets, i.e. the version for graphs and 2 colours.\<close>
+ This is the most basic version in terms of cliques and independent
+ sets, i.e. the version for graphs and 2 colours.
+\<close>
-definition "clique V E = (\<forall>v\<in>V. \<forall>w\<in>V. v\<noteq>w \<longrightarrow> {v,w} : E)"
-definition "indep V E = (\<forall>v\<in>V. \<forall>w\<in>V. v\<noteq>w \<longrightarrow> \<not> {v,w} : E)"
+definition "clique V E \<longleftrightarrow> (\<forall>v\<in>V. \<forall>w\<in>V. v \<noteq> w \<longrightarrow> {v, w} \<in> E)"
+definition "indep V E \<longleftrightarrow> (\<forall>v\<in>V. \<forall>w\<in>V. v \<noteq> w \<longrightarrow> {v, w} \<notin> E)"
lemma ramsey2:
- "\<exists>r\<ge>1. \<forall> (V::'a set) (E::'a set set). finite V \<and> card V \<ge> r \<longrightarrow>
- (\<exists> R \<subseteq> V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E)"
+ "\<exists>r\<ge>1. \<forall>(V::'a set) (E::'a set set). finite V \<and> card V \<ge> r \<longrightarrow>
+ (\<exists>R \<subseteq> V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E)"
(is "\<exists>r\<ge>1. ?R m n r")
-proof(induct k == "m+n" arbitrary: m n)
+proof (induct k \<equiv> "m + n" arbitrary: m n)
case 0
- show ?case (is "EX r. ?R r")
+ show ?case (is "EX r. ?Q r")
proof
- show "?R 1" using 0
- by (clarsimp simp: indep_def)(metis card.empty emptyE empty_subsetI)
+ from 0 show "?Q 1"
+ by (clarsimp simp: indep_def) (metis card.empty emptyE empty_subsetI)
qed
next
case (Suc k)
- { assume "m=0"
- have ?case (is "EX r. ?R r")
- proof
- show "?R 1" using \<open>m=0\<close>
- by (simp add:clique_def)(metis card.empty emptyE empty_subsetI)
- qed
- } moreover
- { assume "n=0"
- have ?case (is "EX r. ?R r")
- proof
- show "?R 1" using \<open>n=0\<close>
- by (simp add:indep_def)(metis card.empty emptyE empty_subsetI)
- qed
- } moreover
- { assume "m\<noteq>0" "n\<noteq>0"
- then have "k = (m - 1) + n" "k = m + (n - 1)" using \<open>Suc k = m+n\<close> by auto
- from Suc(1)[OF this(1)] Suc(1)[OF this(2)]
- obtain r1 r2 where "r1\<ge>1" "r2\<ge>1" "?R (m - 1) n r1" "?R m (n - 1) r2"
- by auto
- then have "r1+r2 \<ge> 1" by arith
- moreover
- have "?R m n (r1+r2)" (is "ALL V E. _ \<longrightarrow> ?EX V E m n")
+ consider "m = 0 \<or> n = 0" | "m \<noteq> 0" "n \<noteq> 0" by auto
+ then show ?case (is "EX r. ?Q r")
+ proof cases
+ case 1
+ then have "?Q 1"
+ by (simp add: clique_def) (meson card_empty empty_iff empty_subsetI indep_def)
+ then show ?thesis ..
+ next
+ case 2
+ with Suc(2) have "k = (m - 1) + n" "k = m + (n - 1)" by auto
+ from this [THEN Suc(1)]
+ obtain r1 r2 where "r1 \<ge> 1" "r2 \<ge> 1" "?R (m - 1) n r1" "?R m (n - 1) r2" by auto
+ then have "r1 + r2 \<ge> 1" by arith
+ moreover have "?R m n (r1 + r2)" (is "\<forall>V E. _ \<longrightarrow> ?EX V E m n")
proof clarify
- fix V :: "'a set" and E :: "'a set set"
- assume "finite V" "r1+r2 \<le> card V"
- with \<open>r1\<ge>1\<close> have "V \<noteq> {}" by auto
- then obtain v where "v : V" by blast
- let ?M = "{w : V. w\<noteq>v & {v,w} : E}"
- let ?N = "{w : V. w\<noteq>v & {v,w} ~: E}"
- have "V = insert v (?M \<union> ?N)" using \<open>v : V\<close> by auto
- then have "card V = card(insert v (?M \<union> ?N))" by metis
- also have "\<dots> = card ?M + card ?N + 1" using \<open>finite V\<close>
- by(fastforce intro: card_Un_disjoint)
+ fix V :: "'a set"
+ fix E :: "'a set set"
+ assume "finite V" "r1 + r2 \<le> card V"
+ with \<open>r1 \<ge> 1\<close> have "V \<noteq> {}" by auto
+ then obtain v where "v \<in> V" by blast
+ let ?M = "{w \<in> V. w \<noteq> v \<and> {v, w} \<in> E}"
+ let ?N = "{w \<in> V. w \<noteq> v \<and> {v, w} \<notin> E}"
+ from \<open>v \<in> V\<close> have "V = insert v (?M \<union> ?N)" by auto
+ then have "card V = card (insert v (?M \<union> ?N))" by metis
+ also from \<open>finite V\<close> have "\<dots> = card ?M + card ?N + 1"
+ by (fastforce intro: card_Un_disjoint)
finally have "card V = card ?M + card ?N + 1" .
- then have "r1+r2 \<le> card ?M + card ?N + 1" using \<open>r1+r2 \<le> card V\<close> by simp
- then have "r1 \<le> card ?M \<or> r2 \<le> card ?N" by arith
- moreover
- { assume "r1 \<le> card ?M"
- moreover have "finite ?M" using \<open>finite V\<close> by auto
- ultimately have "?EX ?M E (m - 1) n" using \<open>?R (m - 1) n r1\<close> by blast
- then obtain R where "R \<subseteq> ?M" "v ~: R" and
- CI: "card R = m - 1 \<and> clique R E \<or>
- card R = n \<and> indep R E" (is "?C \<or> ?I")
+ with \<open>r1 + r2 \<le> card V\<close> have "r1 + r2 \<le> card ?M + card ?N + 1" by simp
+ then consider "r1 \<le> card ?M" | "r2 \<le> card ?N" by arith
+ then show "?EX V E m n"
+ proof cases
+ case 1
+ from \<open>finite V\<close> have "finite ?M" by auto
+ with \<open>?R (m - 1) n r1\<close> and 1 have "?EX ?M E (m - 1) n" by blast
+ then obtain R where "R \<subseteq> ?M" "v \<notin> R"
+ and CI: "card R = m - 1 \<and> clique R E \<or> card R = n \<and> indep R E" (is "?C \<or> ?I")
by blast
- have "R <= V" using \<open>R <= ?M\<close> by auto
- have "finite R" using \<open>finite V\<close> \<open>R \<subseteq> V\<close> by (metis finite_subset)
- { assume "?I"
- with \<open>R <= V\<close> have "?EX V E m n" by blast
- } moreover
- { assume "?C"
- then have "clique (insert v R) E" using \<open>R <= ?M\<close>
- by(auto simp:clique_def insert_commute)
- moreover have "card(insert v R) = m"
- using \<open>?C\<close> \<open>finite R\<close> \<open>v ~: R\<close> \<open>m\<noteq>0\<close> by simp
- ultimately have "?EX V E m n" using \<open>R <= V\<close> \<open>v : V\<close> by (metis insert_subset)
- } ultimately have "?EX V E m n" using CI by blast
- } moreover
- { assume "r2 \<le> card ?N"
- moreover have "finite ?N" using \<open>finite V\<close> by auto
- ultimately have "?EX ?N E m (n - 1)" using \<open>?R m (n - 1) r2\<close> by blast
- then obtain R where "R \<subseteq> ?N" "v ~: R" and
- CI: "card R = m \<and> clique R E \<or>
- card R = n - 1 \<and> indep R E" (is "?C \<or> ?I")
+ from \<open>R \<subseteq> ?M\<close> have "R \<subseteq> V" by auto
+ with \<open>finite V\<close> have "finite R" by (metis finite_subset)
+ from CI show ?thesis
+ proof
+ assume "?I"
+ with \<open>R \<subseteq> V\<close> show ?thesis by blast
+ next
+ assume "?C"
+ with \<open>R \<subseteq> ?M\<close> have *: "clique (insert v R) E"
+ by (auto simp: clique_def insert_commute)
+ from \<open>?C\<close> \<open>finite R\<close> \<open>v \<notin> R\<close> \<open>m \<noteq> 0\<close> have "card (insert v R) = m" by simp
+ with \<open>R \<subseteq> V\<close> \<open>v \<in> V\<close> * show ?thesis by (metis insert_subset)
+ qed
+ next
+ case 2
+ from \<open>finite V\<close> have "finite ?N" by auto
+ with \<open>?R m (n - 1) r2\<close> 2 have "?EX ?N E m (n - 1)" by blast
+ then obtain R where "R \<subseteq> ?N" "v \<notin> R"
+ and CI: "card R = m \<and> clique R E \<or> card R = n - 1 \<and> indep R E" (is "?C \<or> ?I")
by blast
- have "R <= V" using \<open>R <= ?N\<close> by auto
- have "finite R" using \<open>finite V\<close> \<open>R \<subseteq> V\<close> by (metis finite_subset)
- { assume "?C"
- with \<open>R <= V\<close> have "?EX V E m n" by blast
- } moreover
- { assume "?I"
- then have "indep (insert v R) E" using \<open>R <= ?N\<close>
- by(auto simp:indep_def insert_commute)
- moreover have "card(insert v R) = n"
- using \<open>?I\<close> \<open>finite R\<close> \<open>v ~: R\<close> \<open>n\<noteq>0\<close> by simp
- ultimately have "?EX V E m n" using \<open>R <= V\<close> \<open>v : V\<close> by (metis insert_subset)
- } ultimately have "?EX V E m n" using CI by blast
- } ultimately show "?EX V E m n" by blast
+ from \<open>R \<subseteq> ?N\<close> have "R \<subseteq> V" by auto
+ with \<open>finite V\<close> have "finite R" by (metis finite_subset)
+ from CI show ?thesis
+ proof
+ assume "?C"
+ with \<open>R \<subseteq> V\<close> show ?thesis by blast
+ next
+ assume "?I"
+ with \<open>R \<subseteq> ?N\<close> have *: "indep (insert v R) E"
+ by (auto simp: indep_def insert_commute)
+ from \<open>?I\<close> \<open>finite R\<close> \<open>v \<notin> R\<close> \<open>n \<noteq> 0\<close> have "card (insert v R) = n" by simp
+ with \<open>R \<subseteq> V\<close> \<open>v \<in> V\<close> * show ?thesis by (metis insert_subset)
+ qed
+ qed
qed
- ultimately have ?case by blast
- } ultimately show ?case by blast
+ ultimately show ?thesis by blast
+ qed
qed
@@ -119,122 +115,115 @@
subsubsection \<open>``Axiom'' of Dependent Choice\<close>
-primrec choice :: "('a => bool) => ('a * 'a) set => nat => 'a" where
- \<comment>\<open>An integer-indexed chain of choices\<close>
- 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)"
+primrec choice :: "('a \<Rightarrow> bool) \<Rightarrow> ('a \<times> 'a) set \<Rightarrow> nat \<Rightarrow> 'a"
+ where \<comment> \<open>An integer-indexed chain of choices\<close>
+ choice_0: "choice P r 0 = (SOME x. P x)"
+ | choice_Suc: "choice P r (Suc n) = (SOME y. P y \<and> (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"
+ and Pstep: "\<And>x. P x \<Longrightarrow> \<exists>y. P y \<and> (x, y) \<in> r"
shows "P (choice P r n)"
proof (induct n)
- case 0 show ?case by (force intro: someI P0)
+ case 0
+ show ?case by (force intro: someI P0)
next
- case Suc then show ?case by (auto intro: someI2_ex [OF Pstep])
+ case Suc
+ then show ?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"
- obtains f :: "nat => 'a" where
- "!!n. P (f n)" and "!!n m. n < m ==> (f n, f m) \<in> r"
+ and P0: "P x0"
+ and Pstep: "\<And>x. P x \<Longrightarrow> \<exists>y. P y \<and> (x, y) \<in> r"
+ obtains f :: "nat \<Rightarrow> 'a" where "\<And>n. P (f n)" and "\<And>n m. n < m \<Longrightarrow> (f n, f m) \<in> r"
proof
fix n
- show "P (choice P r n)" by (blast intro: choice_n [OF P0 Pstep])
+ show "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]]
+ fix n m :: nat
+ assume "n < m"
+ from Pstep [OF choice_n [OF P0 Pstep]] have "(choice P r k, choice P r (Suc k)) \<in> r" for k
by (auto intro: someI2_ex)
- fix n 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])
+ then show "(choice P r n, choice P r m) \<in> r"
+ by (auto intro: less_Suc_induct [OF \<open>n < m\<close>] transD [OF trans])
qed
subsubsection \<open>Partitions of a Set\<close>
-definition part :: "nat => nat => 'a set => ('a set => nat) => bool"
- \<comment>\<open>the function @{term f} partitions the @{term r}-subsets of the typically
- infinite set @{term Y} into @{term s} distinct categories.\<close>
-where
- "part r s Y f = (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X < s)"
+definition part :: "nat \<Rightarrow> nat \<Rightarrow> 'a set \<Rightarrow> ('a set \<Rightarrow> nat) \<Rightarrow> bool"
+ \<comment> \<open>the function @{term f} partitions the @{term r}-subsets of the typically
+ infinite set @{term Y} into @{term s} distinct categories.\<close>
+ where "part r s Y f \<longleftrightarrow> (\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = r \<longrightarrow> f X < s)"
-text\<open>For induction, we decrease the value of @{term r} in partitions.\<close>
+text \<open>For induction, we decrease the value of @{term r} in partitions.\<close>
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)
+ "\<lbrakk>infinite Y; part (Suc r) s Y f; y \<in> Y\<rbrakk> \<Longrightarrow> part r s (Y - {y}) (\<lambda>u. f (insert y u))"
+ apply (simp add: part_def)
+ apply clarify
+ apply (drule_tac x="insert y X" in spec)
+ apply force
done
-lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
+lemma part_subset: "part r s YY f \<Longrightarrow> Y \<subseteq> YY \<Longrightarrow> part r s Y f"
unfolding part_def by blast
subsection \<open>Ramsey's Theorem: Infinitary Version\<close>
lemma Ramsey_induction:
- fixes s 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)
+ fixes s r :: nat
+ and YY :: "'a set"
+ and f :: "'a set \<Rightarrow> nat"
+ assumes "infinite YY" "part r s YY f"
+ shows "\<exists>Y' t'. Y' \<subseteq> YY \<and> infinite Y' \<and> t' < s \<and> (\<forall>X. X \<subseteq> Y' \<and> finite X \<and> card X = r \<longrightarrow> f X = t')"
+ using assms
+proof (induct r arbitrary: YY f)
case 0
- then show ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
+ then show ?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)
+ 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 \<and> Y' \<subseteq> Y}"
+ let ?propr = "\<lambda>(y, Y, t).
+ y \<in> YY \<and> y \<notin> Y \<and> Y \<subseteq> YY \<and> infinite Y \<and> t < s
+ \<and> (\<forall>X. X\<subseteq>Y \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert y) X = t)"
+ from Suc.prems have infYY': "infinite (YY - {yy})" by auto
+ from Suc.prems have partf': "part r s (YY - {yy}) (f \<circ> insert yy)"
+ by (simp add: o_def part_Suc_imp_part yy)
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"
+ obtain Y0 and t0 where "Y0 \<subseteq> YY - {yy}" "infinite Y0" "t0 < s"
+ "X \<subseteq> Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0" for X
+ by blast
+ with yy have propr0: "?propr(yy, Y0, t0)" by blast
+ have proprstep: "\<exists>y. ?propr y \<and> (x, y) \<in> ?ramr" if x: "?propr x" for x
+ proof (cases x)
+ case (fields yx Yx tx)
+ with x obtain yx' where yx': "yx' \<in> Yx"
+ by (blast dest: infinite_imp_nonempty)
+ from fields x have infYx': "infinite (Yx - {yx'})" by auto
+ with fields x 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 and Y=Yx])
+ from Suc.hyps [OF infYx' partfx'] obtain Y' and t'
+ where Y': "Y' \<subseteq> Yx - {yx'}" "infinite Y'" "t' < s"
+ "X \<subseteq> Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'" for X
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" then show "?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 and Y=Yx])
- 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
+ from fields x Y' yx' have "?propr (yx', Y', t') \<and> (x, (yx', Y', t')) \<in> ?ramr"
+ by blast
+ then show ?thesis ..
qed
from dependent_choice [OF transr propr0 proprstep]
- obtain g where pg: "?propr (g n)" and rg: "n<m ==> (g n, g m) \<in> ?ramr" for n m :: nat
+ obtain g where pg: "?propr (g n)" and rg: "n < m \<Longrightarrow> (g n, g m) \<in> ?ramr" for n m :: nat
by blast
- let ?gy = "fst o g"
- let ?gt = "snd o snd o g"
+ let ?gy = "fst \<circ> g"
+ let ?gt = "snd \<circ> snd \<circ> g"
have rangeg: "\<exists>k. range ?gt \<subseteq> {..<k}"
proof (intro exI subsetI)
fix x
@@ -244,61 +233,60 @@
qed
have "finite (range ?gt)"
by (simp add: finite_nat_iff_bounded rangeg)
- then obtain s' and n'
- where s': "s' = ?gt n'"
- and infeqs': "infinite {n. ?gt n = s'}"
+ then obtain s' and n' where s': "s' = ?gt n'" and infeqs': "infinite {n. ?gt n = s'}"
by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: infinite_UNIV_nat)
with pg [of n'] have less': "s'<s" by (cases "g n'") auto
have inj_gy: "inj ?gy"
proof (rule linorder_injI)
- fix m 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
+ fix m m' :: nat
+ assume "m < m'"
+ from rg [OF this] pg [of m] show "?gy m \<noteq> ?gy 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
+ from pg show "?gy ` {n. ?gt n = s'} \<subseteq> YY"
by (auto simp add: Let_def split_beta)
- show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
+ from infeqs' show "infinite (?gy ` {n. ?gt n = s'})"
by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
show "s' < s" by (rule less')
- show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
- --> f X = s'"
+ show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} \<and> finite X \<and> card X = Suc r \<longrightarrow> 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
- then have 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)
- then have "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)
- then have "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
- then have 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: cardX ya)
- ultimately show ?thesis
- using pg [of "LEAST x. x \<in> AA"] fields cardX
- by (clarsimp simp del:insert_Diff_single)
- qed
- also have "... = s'" using AA AAleast fields by auto
- finally show ?thesis .
- qed}
+ have "f X = s'"
+ if X: "X \<subseteq> ?gy ` {n. ?gt n = s'}"
+ and cardX: "finite X" "card X = Suc r"
+ for X
+ proof -
+ from X 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
+ then have AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
+ show ?thesis
+ 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)
+ then have "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
+ also have "\<dots> = 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)
+ with fields x have "a' \<noteq> (LEAST x. x \<in> AA)" by auto
+ with Least_le [of "\<lambda>x. x \<in> AA", OF a'] have "(LEAST x. x \<in> AA) < a'"
+ by arith
+ from xeq fields rg [OF this] show "x \<in> Ya" by auto
+ qed
+ have "card (X - {ya}) = r"
+ by (simp add: cardX ya)
+ with pg [of "LEAST x. x \<in> AA"] fields cardX * show ?thesis
+ by (auto simp del: insert_Diff_single)
+ qed
+ also from AA AAleast fields have "\<dots> = s'" by auto
+ finally show ?thesis .
+ qed
+ qed
then show ?thesis by blast
qed
qed
@@ -307,27 +295,29 @@
theorem Ramsey:
- fixes s r :: nat and Z::"'a set" and f::"'a set => nat"
+ fixes s r :: nat
+ and Z :: "'a set"
+ and f :: "'a set \<Rightarrow> 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])
+ "\<lbrakk>infinite Z;
+ \<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = r \<longrightarrow> f X < s\<rbrakk>
+ \<Longrightarrow> \<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s
+ \<and> (\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = r \<longrightarrow> f X = t)"
+ by (blast intro: Ramsey_induction [unfolded part_def])
corollary Ramsey2:
- fixes s::nat and Z::"'a set" and f::"'a set => nat"
+ fixes s :: nat
+ and Z :: "'a set"
+ and f :: "'a set \<Rightarrow> nat"
assumes infZ: "infinite Z"
- and part: "\<forall>x\<in>Z. \<forall>y\<in>Z. x\<noteq>y --> f{x,y} < s"
- shows
- "\<exists>Y t. Y \<subseteq> Z & infinite Y & t < s & (\<forall>x\<in>Y. \<forall>y\<in>Y. x\<noteq>y --> f{x,y} = t)"
+ and part: "\<forall>x\<in>Z. \<forall>y\<in>Z. x \<noteq> y \<longrightarrow> f {x, y} < s"
+ shows "\<exists>Y t. Y \<subseteq> Z \<and> infinite Y \<and> t < s \<and> (\<forall>x\<in>Y. \<forall>y\<in>Y. x\<noteq>y \<longrightarrow> f {x, y} = t)"
proof -
- have part2: "\<forall>X. X \<subseteq> Z & finite X & card X = 2 --> f X < s"
- using part by (fastforce simp add: eval_nat_numeral card_Suc_eq)
- obtain Y t
- where *: "Y \<subseteq> Z" "infinite Y" "t < s"
- "(\<forall>X. X \<subseteq> Y & finite X & card X = 2 --> f X = t)"
+ from part have part2: "\<forall>X. X \<subseteq> Z \<and> finite X \<and> card X = 2 \<longrightarrow> f X < s"
+ by (fastforce simp add: eval_nat_numeral card_Suc_eq)
+ obtain Y t where *:
+ "Y \<subseteq> Z" "infinite Y" "t < s" "(\<forall>X. X \<subseteq> Y \<and> finite X \<and> card X = 2 \<longrightarrow> f X = t)"
by (insert Ramsey [OF infZ part2]) auto
then have "\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t" by auto
with * show ?thesis by iprover
@@ -341,97 +331,84 @@
@{cite "Podelski-Rybalchenko"}.
\<close>
-definition disj_wf :: "('a * 'a)set => bool"
- where "disj_wf r = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r = (\<Union>i<n. T i))"
+definition disj_wf :: "('a \<times> 'a) set \<Rightarrow> bool"
+ where "disj_wf r \<longleftrightarrow> (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf (T i)) \<and> r = (\<Union>i<n. T i))"
-definition transition_idx :: "[nat => 'a, nat => ('a*'a)set, nat set] => nat"
- where
- "transition_idx s T A =
- (LEAST k. \<exists>i j. A = {i,j} & i<j & (s j, s i) \<in> T k)"
+definition transition_idx :: "(nat \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> ('a \<times> 'a) set) \<Rightarrow> nat set \<Rightarrow> nat"
+ where "transition_idx s T A = (LEAST k. \<exists>i j. A = {i, j} \<and> i < j \<and> (s j, s i) \<in> T k)"
lemma transition_idx_less:
- "[|i<j; (s j, s i) \<in> T k; k<n|] ==> transition_idx s T {i,j} < n"
-apply (subgoal_tac "transition_idx s T {i, j} \<le> k", simp)
-apply (simp add: transition_idx_def, blast intro: Least_le)
-done
+ assumes "i < j" "(s j, s i) \<in> T k" "k < n"
+ shows "transition_idx s T {i, j} < n"
+proof -
+ from assms(1,2) have "transition_idx s T {i, j} \<le> k"
+ by (simp add: transition_idx_def, blast intro: Least_le)
+ with assms(3) show ?thesis by simp
+qed
lemma transition_idx_in:
- "[|i<j; (s j, s i) \<in> T k|] ==> (s j, s i) \<in> T (transition_idx s T {i,j})"
-apply (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR
- cong: conj_cong)
-apply (erule LeastI)
-done
+ assumes "i < j" "(s j, s i) \<in> T k"
+ shows "(s j, s i) \<in> T (transition_idx s T {i, j})"
+ using assms
+ by (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR cong: conj_cong) (erule LeastI)
-text\<open>To be equal to the union of some well-founded relations is equivalent
-to being the subset of such a union.\<close>
-lemma disj_wf:
- "disj_wf(r) = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r \<subseteq> (\<Union>i<n. T i))"
-apply (auto simp add: disj_wf_def)
-apply (rule_tac x="%i. T i Int r" in exI)
-apply (rule_tac x=n in exI)
-apply (force simp add: wf_Int1)
-done
+text \<open>To be equal to the union of some well-founded relations is equivalent
+ to being the subset of such a union.\<close>
+lemma disj_wf: "disj_wf r \<longleftrightarrow> (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) \<and> r \<subseteq> (\<Union>i<n. T i))"
+ apply (auto simp add: disj_wf_def)
+ apply (rule_tac x="\<lambda>i. T i Int r" in exI)
+ apply (rule_tac x=n in exI)
+ apply (force simp add: wf_Int1)
+ done
theorem trans_disj_wf_implies_wf:
- assumes transr: "trans r"
- and dwf: "disj_wf(r)"
+ assumes "trans r"
+ and "disj_wf r"
shows "wf r"
proof (simp only: wf_iff_no_infinite_down_chain, rule notI)
assume "\<exists>s. \<forall>i. (s (Suc i), s i) \<in> r"
then obtain s where sSuc: "\<forall>i. (s (Suc i), s i) \<in> r" ..
- have s: "!!i j. i < j ==> (s j, s i) \<in> r"
+ from \<open>disj_wf r\<close> obtain T and n :: nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
+ by (auto simp add: disj_wf_def)
+ have s_in_T: "\<exists>k. (s j, s i) \<in> T k \<and> k<n" if "i < j" for i j
proof -
- fix i and j::nat
- assume less: "i<j"
- then show "(s j, s i) \<in> r"
- proof (rule less_Suc_induct)
- show "\<And>i. (s (Suc i), s i) \<in> r" by (simp add: sSuc)
- show "\<And>i j k. \<lbrakk>(s j, s i) \<in> r; (s k, s j) \<in> r\<rbrakk> \<Longrightarrow> (s k, s i) \<in> r"
- using transr by (unfold trans_def, blast)
+ from \<open>i < j\<close> have "(s j, s i) \<in> r"
+ proof (induct rule: less_Suc_induct)
+ case 1
+ then show ?case by (simp add: sSuc)
+ next
+ case 2
+ with \<open>trans r\<close> show ?case
+ unfolding trans_def by blast
qed
+ then show ?thesis by (auto simp add: r)
qed
- from dwf
- obtain T and n::nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
- by (auto simp add: disj_wf_def)
- have s_in_T: "\<And>i j. i<j ==> \<exists>k. (s j, s i) \<in> T k & k<n"
- proof -
- fix i and j::nat
- assume less: "i<j"
- then have "(s j, s i) \<in> r" by (rule s [of i j])
- then show "\<exists>k. (s j, s i) \<in> T k & k<n" by (auto simp add: r)
- qed
- have trless: "!!i j. i\<noteq>j ==> transition_idx s T {i,j} < n"
+ have trless: "i \<noteq> j \<Longrightarrow> transition_idx s T {i, j} < n" for i j
apply (auto simp add: linorder_neq_iff)
- apply (blast dest: s_in_T transition_idx_less)
+ apply (blast dest: s_in_T transition_idx_less)
apply (subst insert_commute)
apply (blast dest: s_in_T transition_idx_less)
done
- have
- "\<exists>K k. K \<subseteq> UNIV & infinite K & k < n &
- (\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k)"
+ have "\<exists>K k. K \<subseteq> UNIV \<and> infinite K \<and> k < n \<and>
+ (\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> transition_idx s T {i, j} = k)"
by (rule Ramsey2) (auto intro: trless infinite_UNIV_nat)
- then obtain K and k
- where infK: "infinite K" and less: "k < n" and
- allk: "\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k"
+ then obtain K and k where infK: "infinite K" and "k < n"
+ and allk: "\<forall>i\<in>K. \<forall>j\<in>K. i \<noteq> j \<longrightarrow> transition_idx s T {i, j} = k"
by auto
- have "\<forall>m. (s (enumerate K (Suc m)), s(enumerate K m)) \<in> T k"
- proof
- fix m::nat
+ have "(s (enumerate K (Suc m)), s (enumerate K m)) \<in> T k" for m :: nat
+ proof -
let ?j = "enumerate K (Suc m)"
let ?i = "enumerate K m"
- have jK: "?j \<in> K" by (simp add: enumerate_in_set infK)
- have iK: "?i \<in> K" by (simp add: enumerate_in_set infK)
have ij: "?i < ?j" by (simp add: enumerate_step infK)
- have ijk: "transition_idx s T {?i,?j} = k" using iK jK ij
- by (simp add: allk)
- obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n"
- using s_in_T [OF ij] by blast
- then show "(s ?j, s ?i) \<in> T k"
- by (simp add: ijk [symmetric] transition_idx_in ij)
+ have "?j \<in> K" "?i \<in> K" by (simp_all add: enumerate_in_set infK)
+ with ij have k: "k = transition_idx s T {?i, ?j}" by (simp add: allk)
+ from s_in_T [OF ij] obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n" by blast
+ then show "(s ?j, s ?i) \<in> T k" by (simp add: k transition_idx_in ij)
qed
- then have "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
- then show False using wfT less by blast
+ then have "\<not> wf (T k)"
+ unfolding wf_iff_no_infinite_down_chain by fast
+ with wfT \<open>k < n\<close> show False by blast
qed
end