src/HOL/Library/Ramsey.thy
 author traytel Mon Nov 25 12:27:03 2013 +0100 (2013-11-25) changeset 54580 7b9336176a1c parent 53374 a14d2a854c02 child 58622 aa99568f56de permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Ramsey.thy
```
```     2     Author:     Tom Ridge.  Converted to structured Isar by L C Paulson
```
```     3 *)
```
```     4
```
```     5 header "Ramsey's Theorem"
```
```     6
```
```     7 theory Ramsey
```
```     8 imports Main Infinite_Set
```
```     9 begin
```
```    10
```
```    11 subsection{* Finite Ramsey theorem(s) *}
```
```    12
```
```    13 text{* To distinguish the finite and infinite ones, lower and upper case
```
```    14 names are used.
```
```    15
```
```    16 This is the most basic version in terms of cliques and independent
```
```    17 sets, i.e. the version for graphs and 2 colours. *}
```
```    18
```
```    19 definition "clique V E = (\<forall>v\<in>V. \<forall>w\<in>V. v\<noteq>w \<longrightarrow> {v,w} : E)"
```
```    20 definition "indep V E = (\<forall>v\<in>V. \<forall>w\<in>V. v\<noteq>w \<longrightarrow> \<not> {v,w} : E)"
```
```    21
```
```    22 lemma ramsey2:
```
```    23   "\<exists>r\<ge>1. \<forall> (V::'a set) (E::'a set set). finite V \<and> card V \<ge> r \<longrightarrow>
```
```    24   (\<exists> R \<subseteq> V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E)"
```
```    25   (is "\<exists>r\<ge>1. ?R m n r")
```
```    26 proof(induct k == "m+n" arbitrary: m n)
```
```    27   case 0
```
```    28   show ?case (is "EX r. ?R r")
```
```    29   proof
```
```    30     show "?R 1" using 0
```
```    31       by (clarsimp simp: indep_def)(metis card.empty emptyE empty_subsetI)
```
```    32   qed
```
```    33 next
```
```    34   case (Suc k)
```
```    35   { assume "m=0"
```
```    36     have ?case (is "EX r. ?R r")
```
```    37     proof
```
```    38       show "?R 1" using `m=0`
```
```    39         by (simp add:clique_def)(metis card.empty emptyE empty_subsetI)
```
```    40     qed
```
```    41   } moreover
```
```    42   { assume "n=0"
```
```    43     have ?case (is "EX r. ?R r")
```
```    44     proof
```
```    45       show "?R 1" using `n=0`
```
```    46         by (simp add:indep_def)(metis card.empty emptyE empty_subsetI)
```
```    47     qed
```
```    48   } moreover
```
```    49   { assume "m\<noteq>0" "n\<noteq>0"
```
```    50     then have "k = (m - 1) + n" "k = m + (n - 1)" using `Suc k = m+n` by auto
```
```    51     from Suc(1)[OF this(1)] Suc(1)[OF this(2)]
```
```    52     obtain r1 r2 where "r1\<ge>1" "r2\<ge>1" "?R (m - 1) n r1" "?R m (n - 1) r2"
```
```    53       by auto
```
```    54     then have "r1+r2 \<ge> 1" by arith
```
```    55     moreover
```
```    56     have "?R m n (r1+r2)" (is "ALL V E. _ \<longrightarrow> ?EX V E m n")
```
```    57     proof clarify
```
```    58       fix V :: "'a set" and E :: "'a set set"
```
```    59       assume "finite V" "r1+r2 \<le> card V"
```
```    60       with `r1\<ge>1` have "V \<noteq> {}" by auto
```
```    61       then obtain v where "v : V" by blast
```
```    62       let ?M = "{w : V. w\<noteq>v & {v,w} : E}"
```
```    63       let ?N = "{w : V. w\<noteq>v & {v,w} ~: E}"
```
```    64       have "V = insert v (?M \<union> ?N)" using `v : V` by auto
```
```    65       then have "card V = card(insert v (?M \<union> ?N))" by metis
```
```    66       also have "\<dots> = card ?M + card ?N + 1" using `finite V`
```
```    67         by(fastforce intro: card_Un_disjoint)
```
```    68       finally have "card V = card ?M + card ?N + 1" .
```
```    69       then have "r1+r2 \<le> card ?M + card ?N + 1" using `r1+r2 \<le> card V` by simp
```
```    70       then have "r1 \<le> card ?M \<or> r2 \<le> card ?N" by arith
```
```    71       moreover
```
```    72       { assume "r1 \<le> card ?M"
```
```    73         moreover have "finite ?M" using `finite V` by auto
```
```    74         ultimately have "?EX ?M E (m - 1) n" using `?R (m - 1) n r1` by blast
```
```    75         then obtain R where "R \<subseteq> ?M" "v ~: R" and
```
```    76           CI: "card R = m - 1 \<and> clique R E \<or>
```
```    77                card R = n \<and> indep R E" (is "?C \<or> ?I")
```
```    78           by blast
```
```    79         have "R <= V" using `R <= ?M` by auto
```
```    80         have "finite R" using `finite V` `R \<subseteq> V` by (metis finite_subset)
```
```    81         { assume "?I"
```
```    82           with `R <= V` have "?EX V E m n" by blast
```
```    83         } moreover
```
```    84         { assume "?C"
```
```    85           then have "clique (insert v R) E" using `R <= ?M`
```
```    86            by(auto simp:clique_def insert_commute)
```
```    87           moreover have "card(insert v R) = m"
```
```    88             using `?C` `finite R` `v ~: R` `m\<noteq>0` by simp
```
```    89           ultimately have "?EX V E m n" using `R <= V` `v : V` by blast
```
```    90         } ultimately have "?EX V E m n" using CI by blast
```
```    91       } moreover
```
```    92       { assume "r2 \<le> card ?N"
```
```    93         moreover have "finite ?N" using `finite V` by auto
```
```    94         ultimately have "?EX ?N E m (n - 1)" using `?R m (n - 1) r2` by blast
```
```    95         then obtain R where "R \<subseteq> ?N" "v ~: R" and
```
```    96           CI: "card R = m \<and> clique R E \<or>
```
```    97                card R = n - 1 \<and> indep R E" (is "?C \<or> ?I")
```
```    98           by blast
```
```    99         have "R <= V" using `R <= ?N` by auto
```
```   100         have "finite R" using `finite V` `R \<subseteq> V` by (metis finite_subset)
```
```   101         { assume "?C"
```
```   102           with `R <= V` have "?EX V E m n" by blast
```
```   103         } moreover
```
```   104         { assume "?I"
```
```   105           then have "indep (insert v R) E" using `R <= ?N`
```
```   106             by(auto simp:indep_def insert_commute)
```
```   107           moreover have "card(insert v R) = n"
```
```   108             using `?I` `finite R` `v ~: R` `n\<noteq>0` by simp
```
```   109           ultimately have "?EX V E m n" using `R <= V` `v : V` by blast
```
```   110         } ultimately have "?EX V E m n" using CI by blast
```
```   111       } ultimately show "?EX V E m n" by blast
```
```   112     qed
```
```   113     ultimately have ?case by blast
```
```   114   } ultimately show ?case by blast
```
```   115 qed
```
```   116
```
```   117
```
```   118 subsection {* Preliminaries *}
```
```   119
```
```   120 subsubsection {* ``Axiom'' of Dependent Choice *}
```
```   121
```
```   122 primrec choice :: "('a => bool) => ('a * 'a) set => nat => 'a" where
```
```   123   --{*An integer-indexed chain of choices*}
```
```   124     choice_0:   "choice P r 0 = (SOME x. P x)"
```
```   125   | choice_Suc: "choice P r (Suc n) = (SOME y. P y & (choice P r n, y) \<in> r)"
```
```   126
```
```   127 lemma choice_n:
```
```   128   assumes P0: "P x0"
```
```   129       and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
```
```   130   shows "P (choice P r n)"
```
```   131 proof (induct n)
```
```   132   case 0 show ?case by (force intro: someI P0)
```
```   133 next
```
```   134   case Suc then show ?case by (auto intro: someI2_ex [OF Pstep])
```
```   135 qed
```
```   136
```
```   137 lemma dependent_choice:
```
```   138   assumes trans: "trans r"
```
```   139       and P0: "P x0"
```
```   140       and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
```
```   141   obtains f :: "nat => 'a" where
```
```   142     "!!n. P (f n)" and "!!n m. n < m ==> (f n, f m) \<in> r"
```
```   143 proof
```
```   144   fix n
```
```   145   show "P (choice P r n)" by (blast intro: choice_n [OF P0 Pstep])
```
```   146 next
```
```   147   have PSuc: "\<forall>n. (choice P r n, choice P r (Suc n)) \<in> r"
```
```   148     using Pstep [OF choice_n [OF P0 Pstep]]
```
```   149     by (auto intro: someI2_ex)
```
```   150   fix n m :: nat
```
```   151   assume less: "n < m"
```
```   152   show "(choice P r n, choice P r m) \<in> r" using PSuc
```
```   153     by (auto intro: less_Suc_induct [OF less] transD [OF trans])
```
```   154 qed
```
```   155
```
```   156
```
```   157 subsubsection {* Partitions of a Set *}
```
```   158
```
```   159 definition part :: "nat => nat => 'a set => ('a set => nat) => bool"
```
```   160   --{*the function @{term f} partitions the @{term r}-subsets of the typically
```
```   161        infinite set @{term Y} into @{term s} distinct categories.*}
```
```   162 where
```
```   163   "part r s Y f = (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X < s)"
```
```   164
```
```   165 text{*For induction, we decrease the value of @{term r} in partitions.*}
```
```   166 lemma part_Suc_imp_part:
```
```   167      "[| infinite Y; part (Suc r) s Y f; y \<in> Y |]
```
```   168       ==> part r s (Y - {y}) (%u. f (insert y u))"
```
```   169   apply(simp add: part_def, clarify)
```
```   170   apply(drule_tac x="insert y X" in spec)
```
```   171   apply(force)
```
```   172   done
```
```   173
```
```   174 lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
```
```   175   unfolding part_def by blast
```
```   176
```
```   177
```
```   178 subsection {* Ramsey's Theorem: Infinitary Version *}
```
```   179
```
```   180 lemma Ramsey_induction:
```
```   181   fixes s and r::nat
```
```   182   shows
```
```   183   "!!(YY::'a set) (f::'a set => nat).
```
```   184       [|infinite YY; part r s YY f|]
```
```   185       ==> \<exists>Y' t'. Y' \<subseteq> YY & infinite Y' & t' < s &
```
```   186                   (\<forall>X. X \<subseteq> Y' & finite X & card X = r --> f X = t')"
```
```   187 proof (induct r)
```
```   188   case 0
```
```   189   then show ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
```
```   190 next
```
```   191   case (Suc r)
```
```   192   show ?case
```
```   193   proof -
```
```   194     from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy \<in> YY" by blast
```
```   195     let ?ramr = "{((y,Y,t),(y',Y',t')). y' \<in> Y & Y' \<subseteq> Y}"
```
```   196     let ?propr = "%(y,Y,t).
```
```   197                  y \<in> YY & y \<notin> Y & Y \<subseteq> YY & infinite Y & t < s
```
```   198                  & (\<forall>X. X\<subseteq>Y & finite X & card X = r --> (f o insert y) X = t)"
```
```   199     have infYY': "infinite (YY-{yy})" using Suc.prems by auto
```
```   200     have partf': "part r s (YY - {yy}) (f \<circ> insert yy)"
```
```   201       by (simp add: o_def part_Suc_imp_part yy Suc.prems)
```
```   202     have transr: "trans ?ramr" by (force simp add: trans_def)
```
```   203     from Suc.hyps [OF infYY' partf']
```
```   204     obtain Y0 and t0
```
```   205     where "Y0 \<subseteq> YY - {yy}"  "infinite Y0"  "t0 < s"
```
```   206           "\<forall>X. X\<subseteq>Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0"
```
```   207         by blast
```
```   208     with yy have propr0: "?propr(yy,Y0,t0)" by blast
```
```   209     have proprstep: "\<And>x. ?propr x \<Longrightarrow> \<exists>y. ?propr y \<and> (x, y) \<in> ?ramr"
```
```   210     proof -
```
```   211       fix x
```
```   212       assume px: "?propr x" then show "?thesis x"
```
```   213       proof (cases x)
```
```   214         case (fields yx Yx tx)
```
```   215         then obtain yx' where yx': "yx' \<in> Yx" using px
```
```   216                by (blast dest: infinite_imp_nonempty)
```
```   217         have infYx': "infinite (Yx-{yx'})" using fields px by auto
```
```   218         with fields px yx' Suc.prems
```
```   219         have partfx': "part r s (Yx - {yx'}) (f \<circ> insert yx')"
```
```   220           by (simp add: o_def part_Suc_imp_part part_subset [where YY=YY and Y=Yx])
```
```   221         from Suc.hyps [OF infYx' partfx']
```
```   222         obtain Y' and t'
```
```   223         where Y': "Y' \<subseteq> Yx - {yx'}"  "infinite Y'"  "t' < s"
```
```   224                "\<forall>X. X\<subseteq>Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'"
```
```   225             by blast
```
```   226         show ?thesis
```
```   227         proof
```
```   228           show "?propr (yx',Y',t') & (x, (yx',Y',t')) \<in> ?ramr"
```
```   229             using fields Y' yx' px by blast
```
```   230         qed
```
```   231       qed
```
```   232     qed
```
```   233     from dependent_choice [OF transr propr0 proprstep]
```
```   234     obtain g where pg: "!!n::nat.  ?propr (g n)"
```
```   235       and rg: "!!n m. n<m ==> (g n, g m) \<in> ?ramr" by blast
```
```   236     let ?gy = "fst o g"
```
```   237     let ?gt = "snd o snd o g"
```
```   238     have rangeg: "\<exists>k. range ?gt \<subseteq> {..<k}"
```
```   239     proof (intro exI subsetI)
```
```   240       fix x
```
```   241       assume "x \<in> range ?gt"
```
```   242       then obtain n where "x = ?gt n" ..
```
```   243       with pg [of n] show "x \<in> {..<s}" by (cases "g n") auto
```
```   244     qed
```
```   245     have "finite (range ?gt)"
```
```   246       by (simp add: finite_nat_iff_bounded rangeg)
```
```   247     then obtain s' and n'
```
```   248       where s': "s' = ?gt n'"
```
```   249         and infeqs': "infinite {n. ?gt n = s'}"
```
```   250       by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: infinite_UNIV_nat)
```
```   251     with pg [of n'] have less': "s'<s" by (cases "g n'") auto
```
```   252     have inj_gy: "inj ?gy"
```
```   253     proof (rule linorder_injI)
```
```   254       fix m m' :: nat assume less: "m < m'" show "?gy m \<noteq> ?gy m'"
```
```   255         using rg [OF less] pg [of m] by (cases "g m", cases "g m'") auto
```
```   256     qed
```
```   257     show ?thesis
```
```   258     proof (intro exI conjI)
```
```   259       show "?gy ` {n. ?gt n = s'} \<subseteq> YY" using pg
```
```   260         by (auto simp add: Let_def split_beta)
```
```   261       show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
```
```   262         by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
```
```   263       show "s' < s" by (rule less')
```
```   264       show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
```
```   265           --> f X = s'"
```
```   266       proof -
```
```   267         {fix X
```
```   268          assume "X \<subseteq> ?gy ` {n. ?gt n = s'}"
```
```   269             and cardX: "finite X" "card X = Suc r"
```
```   270          then obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
```
```   271              by (auto simp add: subset_image_iff)
```
```   272          with cardX have "AA\<noteq>{}" by auto
```
```   273          then have AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
```
```   274          have "f X = s'"
```
```   275          proof (cases "g (LEAST x. x \<in> AA)")
```
```   276            case (fields ya Ya ta)
```
```   277            with AAleast Xeq
```
```   278            have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
```
```   279            then have "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
```
```   280            also have "... = ta"
```
```   281            proof -
```
```   282              have "X - {ya} \<subseteq> Ya"
```
```   283              proof
```
```   284                fix x assume x: "x \<in> X - {ya}"
```
```   285                then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
```
```   286                  by (auto simp add: Xeq)
```
```   287                then have "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
```
```   288                then have lessa': "(LEAST x. x \<in> AA) < a'"
```
```   289                  using Least_le [of "%x. x \<in> AA", OF a'] by arith
```
```   290                show "x \<in> Ya" using xeq fields rg [OF lessa'] by auto
```
```   291              qed
```
```   292              moreover
```
```   293              have "card (X - {ya}) = r"
```
```   294                by (simp add: cardX ya)
```
```   295              ultimately show ?thesis
```
```   296                using pg [of "LEAST x. x \<in> AA"] fields cardX
```
```   297                by (clarsimp simp del:insert_Diff_single)
```
```   298            qed
```
```   299            also have "... = s'" using AA AAleast fields by auto
```
```   300            finally show ?thesis .
```
```   301          qed}
```
```   302         then show ?thesis by blast
```
```   303       qed
```
```   304     qed
```
```   305   qed
```
```   306 qed
```
```   307
```
```   308
```
```   309 theorem Ramsey:
```
```   310   fixes s r :: nat and Z::"'a set" and f::"'a set => nat"
```
```   311   shows
```
```   312    "[|infinite Z;
```
```   313       \<forall>X. X \<subseteq> Z & finite X & card X = r --> f X < s|]
```
```   314   ==> \<exists>Y t. Y \<subseteq> Z & infinite Y & t < s
```
```   315             & (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X = t)"
```
```   316 by (blast intro: Ramsey_induction [unfolded part_def])
```
```   317
```
```   318
```
```   319 corollary Ramsey2:
```
```   320   fixes s::nat and Z::"'a set" and f::"'a set => nat"
```
```   321   assumes infZ: "infinite Z"
```
```   322       and part: "\<forall>x\<in>Z. \<forall>y\<in>Z. x\<noteq>y --> f{x,y} < s"
```
```   323   shows
```
```   324    "\<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)"
```
```   325 proof -
```
```   326   have part2: "\<forall>X. X \<subseteq> Z & finite X & card X = 2 --> f X < s"
```
```   327     using part by (fastforce simp add: eval_nat_numeral card_Suc_eq)
```
```   328   obtain Y t
```
```   329     where *: "Y \<subseteq> Z" "infinite Y" "t < s"
```
```   330           "(\<forall>X. X \<subseteq> Y & finite X & card X = 2 --> f X = t)"
```
```   331     by (insert Ramsey [OF infZ part2]) auto
```
```   332   then have "\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t" by auto
```
```   333   with * show ?thesis by iprover
```
```   334 qed
```
```   335
```
```   336
```
```   337 subsection {* Disjunctive Well-Foundedness *}
```
```   338
```
```   339 text {*
```
```   340   An application of Ramsey's theorem to program termination. See
```
```   341   \cite{Podelski-Rybalchenko}.
```
```   342 *}
```
```   343
```
```   344 definition disj_wf :: "('a * 'a)set => bool"
```
```   345   where "disj_wf r = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r = (\<Union>i<n. T i))"
```
```   346
```
```   347 definition transition_idx :: "[nat => 'a, nat => ('a*'a)set, nat set] => nat"
```
```   348   where
```
```   349     "transition_idx s T A =
```
```   350       (LEAST k. \<exists>i j. A = {i,j} & i<j & (s j, s i) \<in> T k)"
```
```   351
```
```   352
```
```   353 lemma transition_idx_less:
```
```   354     "[|i<j; (s j, s i) \<in> T k; k<n|] ==> transition_idx s T {i,j} < n"
```
```   355 apply (subgoal_tac "transition_idx s T {i, j} \<le> k", simp)
```
```   356 apply (simp add: transition_idx_def, blast intro: Least_le)
```
```   357 done
```
```   358
```
```   359 lemma transition_idx_in:
```
```   360     "[|i<j; (s j, s i) \<in> T k|] ==> (s j, s i) \<in> T (transition_idx s T {i,j})"
```
```   361 apply (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR
```
```   362             cong: conj_cong)
```
```   363 apply (erule LeastI)
```
```   364 done
```
```   365
```
```   366 text{*To be equal to the union of some well-founded relations is equivalent
```
```   367 to being the subset of such a union.*}
```
```   368 lemma disj_wf:
```
```   369      "disj_wf(r) = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r \<subseteq> (\<Union>i<n. T i))"
```
```   370 apply (auto simp add: disj_wf_def)
```
```   371 apply (rule_tac x="%i. T i Int r" in exI)
```
```   372 apply (rule_tac x=n in exI)
```
```   373 apply (force simp add: wf_Int1)
```
```   374 done
```
```   375
```
```   376 theorem trans_disj_wf_implies_wf:
```
```   377   assumes transr: "trans r"
```
```   378       and dwf:    "disj_wf(r)"
```
```   379   shows "wf r"
```
```   380 proof (simp only: wf_iff_no_infinite_down_chain, rule notI)
```
```   381   assume "\<exists>s. \<forall>i. (s (Suc i), s i) \<in> r"
```
```   382   then obtain s where sSuc: "\<forall>i. (s (Suc i), s i) \<in> r" ..
```
```   383   have s: "!!i j. i < j ==> (s j, s i) \<in> r"
```
```   384   proof -
```
```   385     fix i and j::nat
```
```   386     assume less: "i<j"
```
```   387     then show "(s j, s i) \<in> r"
```
```   388     proof (rule less_Suc_induct)
```
```   389       show "\<And>i. (s (Suc i), s i) \<in> r" by (simp add: sSuc)
```
```   390       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"
```
```   391         using transr by (unfold trans_def, blast)
```
```   392     qed
```
```   393   qed
```
```   394   from dwf
```
```   395   obtain T and n::nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
```
```   396     by (auto simp add: disj_wf_def)
```
```   397   have s_in_T: "\<And>i j. i<j ==> \<exists>k. (s j, s i) \<in> T k & k<n"
```
```   398   proof -
```
```   399     fix i and j::nat
```
```   400     assume less: "i<j"
```
```   401     then have "(s j, s i) \<in> r" by (rule s [of i j])
```
```   402     then show "\<exists>k. (s j, s i) \<in> T k & k<n" by (auto simp add: r)
```
```   403   qed
```
```   404   have trless: "!!i j. i\<noteq>j ==> transition_idx s T {i,j} < n"
```
```   405     apply (auto simp add: linorder_neq_iff)
```
```   406     apply (blast dest: s_in_T transition_idx_less)
```
```   407     apply (subst insert_commute)
```
```   408     apply (blast dest: s_in_T transition_idx_less)
```
```   409     done
```
```   410   have
```
```   411    "\<exists>K k. K \<subseteq> UNIV & infinite K & k < n &
```
```   412           (\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k)"
```
```   413     by (rule Ramsey2) (auto intro: trless infinite_UNIV_nat)
```
```   414   then obtain K and k
```
```   415     where infK: "infinite K" and less: "k < n" and
```
```   416           allk: "\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k"
```
```   417     by auto
```
```   418   have "\<forall>m. (s (enumerate K (Suc m)), s(enumerate K m)) \<in> T k"
```
```   419   proof
```
```   420     fix m::nat
```
```   421     let ?j = "enumerate K (Suc m)"
```
```   422     let ?i = "enumerate K m"
```
```   423     have jK: "?j \<in> K" by (simp add: enumerate_in_set infK)
```
```   424     have iK: "?i \<in> K" by (simp add: enumerate_in_set infK)
```
```   425     have ij: "?i < ?j" by (simp add: enumerate_step infK)
```
```   426     have ijk: "transition_idx s T {?i,?j} = k" using iK jK ij
```
```   427       by (simp add: allk)
```
```   428     obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n"
```
```   429       using s_in_T [OF ij] by blast
```
```   430     then show "(s ?j, s ?i) \<in> T k"
```
```   431       by (simp add: ijk [symmetric] transition_idx_in ij)
```
```   432   qed
```
```   433   then have "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
```
```   434   then show False using wfT less by blast
```
```   435 qed
```
```   436
```
```   437 end
```