(* Title: HOL/Library/Ramsey.thy
Author: Tom Ridge. Converted to structured Isar by L C Paulson
*)
header "Ramsey's Theorem"
theory Ramsey
imports Main Infinite_Set
begin
subsection {* Preliminaries *}
subsubsection {* ``Axiom'' of Dependent Choice *}
primrec choice :: "('a => bool) => ('a * 'a) set => nat => 'a" where
--{*An integer-indexed chain of choices*}
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 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"
obtains f :: "nat => 'a" where
"!!n. P (f n)" and "!!n m. n < m ==> (f n, f m) \<in> r"
proof
fix n
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]]
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])
qed
subsubsection {* Partitions of a Set *}
definition
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.*}
where
"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)
done
lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
unfolding part_def by blast
subsection {* Ramsey's Theorem: Infinitary Version *}
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)
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 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
qed
from dependent_choice [OF transr propr0 proprstep]
obtain g where pg: "!!n::nat. ?propr (g n)"
and rg: "!!n m. n<m ==> (g n, g m) \<in> ?ramr" by blast
let ?gy = "fst o g"
let ?gt = "snd o snd o g"
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 "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'}"
by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: nat_infinite)
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
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)
show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
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'"
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: 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}
thus ?thesis by blast
qed
qed
qed
qed
theorem Ramsey:
fixes s 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])
corollary Ramsey2:
fixes s::nat and Z::"'a set" and f::"'a set => 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)"
proof -
have part2: "\<forall>X. X \<subseteq> Z & finite X & card X = 2 --> f X < s"
using part by (fastsimp simp add: nat_number 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)"
by (insert Ramsey [OF infZ part2]) auto
moreover from this have "\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t" by auto
ultimately show ?thesis by iprover
qed
subsection {* Disjunctive Well-Foundedness *}
text {*
An application of Ramsey's theorem to program termination. See
\cite{Podelski-Rybalchenko}.
*}
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
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)"
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
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
text{*To be equal to the union of some well-founded relations is equivalent
to being the subset of such a union.*}
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
theorem trans_disj_wf_implies_wf:
assumes transr: "trans r"
and dwf: "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"
proof -
fix i and j::nat
assume less: "i<j"
thus "(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)
qed
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"
hence "(s j, s i) \<in> r" by (rule s [of i j])
thus "\<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"
apply (auto simp add: linorder_neq_iff)
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)"
by (rule Ramsey2) (auto intro: trless nat_infinite)
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"
by auto
have "\<forall>m. (s (enumerate K (Suc m)), s(enumerate K m)) \<in> T k"
proof
fix m::nat
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
thus "(s ?j, s ?i) \<in> T k"
by (simp add: ijk [symmetric] transition_idx_in ij)
qed
hence "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
thus False using wfT less by blast
qed
end