paulson@19944: (*  Title:      HOL/Library/Ramsey.thy
wenzelm@32960:     Author:     Tom Ridge.  Converted to structured Isar by L C Paulson
paulson@19944: *)
paulson@19944: 
paulson@19944: header "Ramsey's Theorem"
paulson@19944: 
haftmann@25594: theory Ramsey
haftmann@30738: imports Main Infinite_Set
haftmann@25594: begin
paulson@19944: 
nipkow@40695: subsection{* Finite Ramsey theorem(s) *}
nipkow@40695: 
nipkow@40695: text{* To distinguish the finite and infinite ones, lower and upper case
nipkow@40695: names are used.
nipkow@40695: 
nipkow@40695: This is the most basic version in terms of cliques and independent
nipkow@40695: sets, i.e. the version for graphs and 2 colours. *}
nipkow@40695: 
nipkow@40695: definition "clique V E = (\<forall>v\<in>V. \<forall>w\<in>V. v\<noteq>w \<longrightarrow> {v,w} : E)"
nipkow@40695: definition "indep V E = (\<forall>v\<in>V. \<forall>w\<in>V. v\<noteq>w \<longrightarrow> \<not> {v,w} : E)"
nipkow@40695: 
nipkow@40695: lemma ramsey2:
nipkow@40695:   "\<exists>r\<ge>1. \<forall> (V::'a set) (E::'a set set). finite V \<and> card V \<ge> r \<longrightarrow>
nipkow@40695:   (\<exists> R \<subseteq> V. card R = m \<and> clique R E \<or> card R = n \<and> indep R E)"
nipkow@40695:   (is "\<exists>r\<ge>1. ?R m n r")
nipkow@40695: proof(induct k == "m+n" arbitrary: m n)
nipkow@40695:   case 0
nipkow@40695:   show ?case (is "EX r. ?R r")
nipkow@40695:   proof
nipkow@40695:     show "?R 1" using 0
nipkow@40695:       by (clarsimp simp: indep_def)(metis card.empty emptyE empty_subsetI)
nipkow@40695:   qed
nipkow@40695: next
nipkow@40695:   case (Suc k)
nipkow@40695:   { assume "m=0"
nipkow@40695:     have ?case (is "EX r. ?R r")
nipkow@40695:     proof
nipkow@40695:       show "?R 1" using `m=0`
nipkow@40695:         by (simp add:clique_def)(metis card.empty emptyE empty_subsetI)
nipkow@40695:     qed
nipkow@40695:   } moreover
nipkow@40695:   { assume "n=0"
nipkow@40695:     have ?case (is "EX r. ?R r")
nipkow@40695:     proof
nipkow@40695:       show "?R 1" using `n=0`
nipkow@40695:         by (simp add:indep_def)(metis card.empty emptyE empty_subsetI)
nipkow@40695:     qed
nipkow@40695:   } moreover
nipkow@40695:   { assume "m\<noteq>0" "n\<noteq>0"
wenzelm@46575:     then have "k = (m - 1) + n" "k = m + (n - 1)" using `Suc k = m+n` by auto
wenzelm@46575:     from Suc(1)[OF this(1)] Suc(1)[OF this(2)]
nipkow@40695:     obtain r1 r2 where "r1\<ge>1" "r2\<ge>1" "?R (m - 1) n r1" "?R m (n - 1) r2"
nipkow@40695:       by auto
wenzelm@46575:     then have "r1+r2 \<ge> 1" by arith
nipkow@40695:     moreover
nipkow@40695:     have "?R m n (r1+r2)" (is "ALL V E. _ \<longrightarrow> ?EX V E m n")
nipkow@40695:     proof clarify
nipkow@40695:       fix V :: "'a set" and E :: "'a set set"
nipkow@40695:       assume "finite V" "r1+r2 \<le> card V"
nipkow@40695:       with `r1\<ge>1` have "V \<noteq> {}" by auto
nipkow@40695:       then obtain v where "v : V" by blast
nipkow@40695:       let ?M = "{w : V. w\<noteq>v & {v,w} : E}"
nipkow@40695:       let ?N = "{w : V. w\<noteq>v & {v,w} ~: E}"
nipkow@40695:       have "V = insert v (?M \<union> ?N)" using `v : V` by auto
wenzelm@46575:       then have "card V = card(insert v (?M \<union> ?N))" by metis
nipkow@40695:       also have "\<dots> = card ?M + card ?N + 1" using `finite V`
nipkow@44890:         by(fastforce intro: card_Un_disjoint)
nipkow@40695:       finally have "card V = card ?M + card ?N + 1" .
wenzelm@46575:       then have "r1+r2 \<le> card ?M + card ?N + 1" using `r1+r2 \<le> card V` by simp
wenzelm@46575:       then have "r1 \<le> card ?M \<or> r2 \<le> card ?N" by arith
nipkow@40695:       moreover
nipkow@40695:       { assume "r1 \<le> card ?M"
nipkow@40695:         moreover have "finite ?M" using `finite V` by auto
nipkow@40695:         ultimately have "?EX ?M E (m - 1) n" using `?R (m - 1) n r1` by blast
nipkow@40695:         then obtain R where "R \<subseteq> ?M" "v ~: R" and
nipkow@40695:           CI: "card R = m - 1 \<and> clique R E \<or>
nipkow@40695:                card R = n \<and> indep R E" (is "?C \<or> ?I")
nipkow@40695:           by blast
nipkow@40695:         have "R <= V" using `R <= ?M` by auto
nipkow@40695:         have "finite R" using `finite V` `R \<subseteq> V` by (metis finite_subset)
nipkow@40695:         { assume "?I"
nipkow@40695:           with `R <= V` have "?EX V E m n" by blast
nipkow@40695:         } moreover
nipkow@40695:         { assume "?C"
wenzelm@46575:           then have "clique (insert v R) E" using `R <= ?M`
nipkow@40695:            by(auto simp:clique_def insert_commute)
nipkow@40695:           moreover have "card(insert v R) = m"
nipkow@40695:             using `?C` `finite R` `v ~: R` `m\<noteq>0` by simp
nipkow@40695:           ultimately have "?EX V E m n" using `R <= V` `v : V` by blast
nipkow@40695:         } ultimately have "?EX V E m n" using CI by blast
nipkow@40695:       } moreover
nipkow@40695:       { assume "r2 \<le> card ?N"
nipkow@40695:         moreover have "finite ?N" using `finite V` by auto
nipkow@40695:         ultimately have "?EX ?N E m (n - 1)" using `?R m (n - 1) r2` by blast
nipkow@40695:         then obtain R where "R \<subseteq> ?N" "v ~: R" and
nipkow@40695:           CI: "card R = m \<and> clique R E \<or>
nipkow@40695:                card R = n - 1 \<and> indep R E" (is "?C \<or> ?I")
nipkow@40695:           by blast
nipkow@40695:         have "R <= V" using `R <= ?N` by auto
nipkow@40695:         have "finite R" using `finite V` `R \<subseteq> V` by (metis finite_subset)
nipkow@40695:         { assume "?C"
nipkow@40695:           with `R <= V` have "?EX V E m n" by blast
nipkow@40695:         } moreover
nipkow@40695:         { assume "?I"
wenzelm@46575:           then have "indep (insert v R) E" using `R <= ?N`
nipkow@40695:             by(auto simp:indep_def insert_commute)
nipkow@40695:           moreover have "card(insert v R) = n"
nipkow@40695:             using `?I` `finite R` `v ~: R` `n\<noteq>0` by simp
nipkow@40695:           ultimately have "?EX V E m n" using `R <= V` `v : V` by blast
nipkow@40695:         } ultimately have "?EX V E m n" using CI by blast
nipkow@40695:       } ultimately show "?EX V E m n" by blast
nipkow@40695:     qed
nipkow@40695:     ultimately have ?case by blast
nipkow@40695:   } ultimately show ?case by blast
nipkow@40695: qed
nipkow@40695: 
nipkow@40695: 
wenzelm@22665: subsection {* Preliminaries *}
paulson@19944: 
wenzelm@22665: subsubsection {* ``Axiom'' of Dependent Choice *}
paulson@19944: 
haftmann@34941: primrec choice :: "('a => bool) => ('a * 'a) set => nat => 'a" where
paulson@19944:   --{*An integer-indexed chain of choices*}
haftmann@34941:     choice_0:   "choice P r 0 = (SOME x. P x)"
haftmann@34941:   | choice_Suc: "choice P r (Suc n) = (SOME y. P y & (choice P r n, y) \<in> r)"
paulson@19944: 
wenzelm@46575: lemma choice_n:
paulson@19944:   assumes P0: "P x0"
paulson@19944:       and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
paulson@19944:   shows "P (choice P r n)"
wenzelm@19948: proof (induct n)
wenzelm@46575:   case 0 show ?case by (force intro: someI P0)
wenzelm@19948: next
wenzelm@46575:   case Suc then show ?case by (auto intro: someI2_ex [OF Pstep])
wenzelm@19948: qed
paulson@19944: 
wenzelm@46575: lemma dependent_choice:
paulson@19944:   assumes trans: "trans r"
paulson@19944:       and P0: "P x0"
paulson@19944:       and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
paulson@19954:   obtains f :: "nat => 'a" where
paulson@19954:     "!!n. P (f n)" and "!!n m. n < m ==> (f n, f m) \<in> r"
paulson@19954: proof
paulson@19954:   fix n
paulson@19954:   show "P (choice P r n)" by (blast intro: choice_n [OF P0 Pstep])
paulson@19944: next
wenzelm@46575:   have PSuc: "\<forall>n. (choice P r n, choice P r (Suc n)) \<in> r"
paulson@19944:     using Pstep [OF choice_n [OF P0 Pstep]]
paulson@19944:     by (auto intro: someI2_ex)
paulson@19954:   fix n m :: nat
paulson@19954:   assume less: "n < m"
paulson@19954:   show "(choice P r n, choice P r m) \<in> r" using PSuc
paulson@19954:     by (auto intro: less_Suc_induct [OF less] transD [OF trans])
paulson@19954: qed
paulson@19944: 
paulson@19944: 
wenzelm@22665: subsubsection {* Partitions of a Set *}
paulson@19944: 
wenzelm@46575: definition part :: "nat => nat => 'a set => ('a set => nat) => bool"
paulson@19944:   --{*the function @{term f} partitions the @{term r}-subsets of the typically
paulson@19944:        infinite set @{term Y} into @{term s} distinct categories.*}
krauss@21634: where
wenzelm@19948:   "part r s Y f = (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X < s)"
paulson@19944: 
paulson@19944: text{*For induction, we decrease the value of @{term r} in partitions.*}
paulson@19944: lemma part_Suc_imp_part:
wenzelm@46575:      "[| infinite Y; part (Suc r) s Y f; y \<in> Y |]
paulson@19944:       ==> part r s (Y - {y}) (%u. f (insert y u))"
paulson@19944:   apply(simp add: part_def, clarify)
paulson@19944:   apply(drule_tac x="insert y X" in spec)
nipkow@24853:   apply(force)
paulson@19944:   done
paulson@19944: 
wenzelm@46575: lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
wenzelm@19948:   unfolding part_def by blast
wenzelm@46575: 
paulson@19944: 
wenzelm@22665: subsection {* Ramsey's Theorem: Infinitary Version *}
paulson@19944: 
wenzelm@46575: lemma Ramsey_induction:
paulson@19954:   fixes s and r::nat
paulson@19944:   shows
wenzelm@46575:   "!!(YY::'a set) (f::'a set => nat).
paulson@19944:       [|infinite YY; part r s YY f|]
wenzelm@46575:       ==> \<exists>Y' t'. Y' \<subseteq> YY & infinite Y' & t' < s &
paulson@19944:                   (\<forall>X. X \<subseteq> Y' & finite X & card X = r --> f X = t')"
paulson@19944: proof (induct r)
paulson@19944:   case 0
wenzelm@46575:   then show ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
paulson@19944: next
wenzelm@46575:   case (Suc r)
paulson@19944:   show ?case
paulson@19944:   proof -
paulson@19944:     from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy \<in> YY" by blast
paulson@19944:     let ?ramr = "{((y,Y,t),(y',Y',t')). y' \<in> Y & Y' \<subseteq> Y}"
wenzelm@46575:     let ?propr = "%(y,Y,t).
wenzelm@32960:                  y \<in> YY & y \<notin> Y & Y \<subseteq> YY & infinite Y & t < s
wenzelm@32960:                  & (\<forall>X. X\<subseteq>Y & finite X & card X = r --> (f o insert y) X = t)"
paulson@19944:     have infYY': "infinite (YY-{yy})" using Suc.prems by auto
paulson@19944:     have partf': "part r s (YY - {yy}) (f \<circ> insert yy)"
paulson@19944:       by (simp add: o_def part_Suc_imp_part yy Suc.prems)
wenzelm@46575:     have transr: "trans ?ramr" by (force simp add: trans_def)
paulson@19944:     from Suc.hyps [OF infYY' partf']
paulson@19944:     obtain Y0 and t0
paulson@19944:     where "Y0 \<subseteq> YY - {yy}"  "infinite Y0"  "t0 < s"
paulson@19944:           "\<forall>X. X\<subseteq>Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0"
wenzelm@46575:         by blast
paulson@19944:     with yy have propr0: "?propr(yy,Y0,t0)" by blast
wenzelm@46575:     have proprstep: "\<And>x. ?propr x \<Longrightarrow> \<exists>y. ?propr y \<and> (x, y) \<in> ?ramr"
paulson@19944:     proof -
paulson@19944:       fix x
wenzelm@46575:       assume px: "?propr x" then show "?thesis x"
paulson@19944:       proof (cases x)
paulson@19944:         case (fields yx Yx tx)
paulson@19944:         then obtain yx' where yx': "yx' \<in> Yx" using px
paulson@19944:                by (blast dest: infinite_imp_nonempty)
paulson@19944:         have infYx': "infinite (Yx-{yx'})" using fields px by auto
paulson@19944:         with fields px yx' Suc.prems
paulson@19944:         have partfx': "part r s (Yx - {yx'}) (f \<circ> insert yx')"
huffman@35175:           by (simp add: o_def part_Suc_imp_part part_subset [where YY=YY and Y=Yx])
wenzelm@32960:         from Suc.hyps [OF infYx' partfx']
wenzelm@32960:         obtain Y' and t'
wenzelm@32960:         where Y': "Y' \<subseteq> Yx - {yx'}"  "infinite Y'"  "t' < s"
wenzelm@32960:                "\<forall>X. X\<subseteq>Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'"
wenzelm@46575:             by blast
wenzelm@32960:         show ?thesis
wenzelm@32960:         proof
wenzelm@32960:           show "?propr (yx',Y',t') & (x, (yx',Y',t')) \<in> ?ramr"
wenzelm@32960:             using fields Y' yx' px by blast
wenzelm@32960:         qed
paulson@19944:       qed
paulson@19944:     qed
paulson@19944:     from dependent_choice [OF transr propr0 proprstep]
nipkow@19946:     obtain g where pg: "!!n::nat.  ?propr (g n)"
paulson@19954:       and rg: "!!n m. n<m ==> (g n, g m) \<in> ?ramr" by blast
haftmann@28741:     let ?gy = "fst o g"
haftmann@28741:     let ?gt = "snd o snd o g"
paulson@19944:     have rangeg: "\<exists>k. range ?gt \<subseteq> {..<k}"
paulson@19944:     proof (intro exI subsetI)
paulson@19944:       fix x
paulson@19944:       assume "x \<in> range ?gt"
paulson@19944:       then obtain n where "x = ?gt n" ..
paulson@19944:       with pg [of n] show "x \<in> {..<s}" by (cases "g n") auto
paulson@19944:     qed
wenzelm@20810:     have "finite (range ?gt)"
wenzelm@20810:       by (simp add: finite_nat_iff_bounded rangeg)
paulson@19944:     then obtain s' and n'
wenzelm@20810:       where s': "s' = ?gt n'"
wenzelm@20810:         and infeqs': "infinite {n. ?gt n = s'}"
traytel@54580:       by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: infinite_UNIV_nat)
paulson@19944:     with pg [of n'] have less': "s'<s" by (cases "g n'") auto
paulson@19944:     have inj_gy: "inj ?gy"
paulson@19944:     proof (rule linorder_injI)
wenzelm@19949:       fix m m' :: nat assume less: "m < m'" show "?gy m \<noteq> ?gy m'"
wenzelm@19948:         using rg [OF less] pg [of m] by (cases "g m", cases "g m'") auto
paulson@19944:     qed
paulson@19944:     show ?thesis
paulson@19944:     proof (intro exI conjI)
paulson@19944:       show "?gy ` {n. ?gt n = s'} \<subseteq> YY" using pg
wenzelm@46575:         by (auto simp add: Let_def split_beta)
paulson@19944:       show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
wenzelm@46575:         by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
paulson@19944:       show "s' < s" by (rule less')
wenzelm@46575:       show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
paulson@19944:           --> f X = s'"
paulson@19944:       proof -
wenzelm@46575:         {fix X
paulson@19944:          assume "X \<subseteq> ?gy ` {n. ?gt n = s'}"
paulson@19944:             and cardX: "finite X" "card X = Suc r"
wenzelm@46575:          then obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
wenzelm@46575:              by (auto simp add: subset_image_iff)
paulson@19944:          with cardX have "AA\<noteq>{}" by auto
wenzelm@46575:          then have AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
paulson@19944:          have "f X = s'"
wenzelm@46575:          proof (cases "g (LEAST x. x \<in> AA)")
paulson@19944:            case (fields ya Ya ta)
wenzelm@46575:            with AAleast Xeq
wenzelm@46575:            have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
wenzelm@46575:            then have "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
wenzelm@46575:            also have "... = ta"
paulson@19944:            proof -
paulson@19944:              have "X - {ya} \<subseteq> Ya"
wenzelm@46575:              proof
paulson@19954:                fix x assume x: "x \<in> X - {ya}"
wenzelm@46575:                then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
wenzelm@46575:                  by (auto simp add: Xeq)
wenzelm@46575:                then have "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
wenzelm@46575:                then have lessa': "(LEAST x. x \<in> AA) < a'"
paulson@19944:                  using Least_le [of "%x. x \<in> AA", OF a'] by arith
paulson@19944:                show "x \<in> Ya" using xeq fields rg [OF lessa'] by auto
paulson@19944:              qed
paulson@19944:              moreover
paulson@19944:              have "card (X - {ya}) = r"
nipkow@24853:                by (simp add: cardX ya)
wenzelm@46575:              ultimately show ?thesis
paulson@19944:                using pg [of "LEAST x. x \<in> AA"] fields cardX
wenzelm@32960:                by (clarsimp simp del:insert_Diff_single)
paulson@19944:            qed
paulson@19944:            also have "... = s'" using AA AAleast fields by auto
paulson@19944:            finally show ?thesis .
paulson@19944:          qed}
wenzelm@46575:         then show ?thesis by blast
wenzelm@46575:       qed
wenzelm@46575:     qed
paulson@19944:   qed
paulson@19944: qed
paulson@19944: 
paulson@19944: 
paulson@19944: theorem Ramsey:
wenzelm@19949:   fixes s r :: nat and Z::"'a set" and f::"'a set => nat"
paulson@19944:   shows
paulson@19944:    "[|infinite Z;
paulson@19944:       \<forall>X. X \<subseteq> Z & finite X & card X = r --> f X < s|]
wenzelm@46575:   ==> \<exists>Y t. Y \<subseteq> Z & infinite Y & t < s
paulson@19944:             & (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X = t)"
paulson@19954: by (blast intro: Ramsey_induction [unfolded part_def])
paulson@19954: 
paulson@19954: 
paulson@19954: corollary Ramsey2:
paulson@19954:   fixes s::nat and Z::"'a set" and f::"'a set => nat"
paulson@19954:   assumes infZ: "infinite Z"
paulson@19954:       and part: "\<forall>x\<in>Z. \<forall>y\<in>Z. x\<noteq>y --> f{x,y} < s"
paulson@19954:   shows
paulson@19954:    "\<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)"
paulson@19954: proof -
paulson@19954:   have part2: "\<forall>X. X \<subseteq> Z & finite X & card X = 2 --> f X < s"
nipkow@44890:     using part by (fastforce simp add: eval_nat_numeral card_Suc_eq)
wenzelm@46575:   obtain Y t
wenzelm@53374:     where *: "Y \<subseteq> Z" "infinite Y" "t < s"
paulson@19954:           "(\<forall>X. X \<subseteq> Y & finite X & card X = 2 --> f X = t)"
paulson@19954:     by (insert Ramsey [OF infZ part2]) auto
wenzelm@53374:   then have "\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t" by auto
wenzelm@53374:   with * show ?thesis by iprover
paulson@19954: qed
paulson@19954: 
paulson@19954: 
wenzelm@22665: subsection {* Disjunctive Well-Foundedness *}
paulson@19954: 
wenzelm@22367: text {*
wenzelm@22367:   An application of Ramsey's theorem to program termination. See
wenzelm@22367:   \cite{Podelski-Rybalchenko}.
paulson@19954: *}
paulson@19954: 
wenzelm@46575: definition disj_wf :: "('a * 'a)set => bool"
wenzelm@46575:   where "disj_wf r = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r = (\<Union>i<n. T i))"
paulson@19954: 
wenzelm@46575: definition transition_idx :: "[nat => 'a, nat => ('a*'a)set, nat set] => nat"
wenzelm@46575:   where
wenzelm@46575:     "transition_idx s T A =
wenzelm@46575:       (LEAST k. \<exists>i j. A = {i,j} & i<j & (s j, s i) \<in> T k)"
paulson@19954: 
paulson@19954: 
paulson@19954: lemma transition_idx_less:
paulson@19954:     "[|i<j; (s j, s i) \<in> T k; k<n|] ==> transition_idx s T {i,j} < n"
wenzelm@46575: apply (subgoal_tac "transition_idx s T {i, j} \<le> k", simp)
wenzelm@46575: apply (simp add: transition_idx_def, blast intro: Least_le)
paulson@19954: done
paulson@19954: 
paulson@19954: lemma transition_idx_in:
paulson@19954:     "[|i<j; (s j, s i) \<in> T k|] ==> (s j, s i) \<in> T (transition_idx s T {i,j})"
wenzelm@46575: apply (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR
wenzelm@46575:             cong: conj_cong)
wenzelm@46575: apply (erule LeastI)
paulson@19954: done
paulson@19954: 
paulson@19954: text{*To be equal to the union of some well-founded relations is equivalent
paulson@19954: to being the subset of such a union.*}
paulson@19954: lemma disj_wf:
paulson@19954:      "disj_wf(r) = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r \<subseteq> (\<Union>i<n. T i))"
wenzelm@46575: apply (auto simp add: disj_wf_def)
wenzelm@46575: apply (rule_tac x="%i. T i Int r" in exI)
wenzelm@46575: apply (rule_tac x=n in exI)
wenzelm@46575: apply (force simp add: wf_Int1)
paulson@19954: done
paulson@19954: 
paulson@19954: theorem trans_disj_wf_implies_wf:
paulson@19954:   assumes transr: "trans r"
paulson@19954:       and dwf:    "disj_wf(r)"
paulson@19954:   shows "wf r"
paulson@19954: proof (simp only: wf_iff_no_infinite_down_chain, rule notI)
paulson@19954:   assume "\<exists>s. \<forall>i. (s (Suc i), s i) \<in> r"
paulson@19954:   then obtain s where sSuc: "\<forall>i. (s (Suc i), s i) \<in> r" ..
paulson@19954:   have s: "!!i j. i < j ==> (s j, s i) \<in> r"
paulson@19954:   proof -
paulson@19954:     fix i and j::nat
paulson@19954:     assume less: "i<j"
wenzelm@46575:     then show "(s j, s i) \<in> r"
paulson@19954:     proof (rule less_Suc_induct)
wenzelm@46575:       show "\<And>i. (s (Suc i), s i) \<in> r" by (simp add: sSuc)
paulson@19954:       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"
wenzelm@46575:         using transr by (unfold trans_def, blast)
paulson@19954:     qed
wenzelm@46575:   qed
paulson@19954:   from dwf
paulson@19954:   obtain T and n::nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
paulson@19954:     by (auto simp add: disj_wf_def)
paulson@19954:   have s_in_T: "\<And>i j. i<j ==> \<exists>k. (s j, s i) \<in> T k & k<n"
paulson@19954:   proof -
paulson@19954:     fix i and j::nat
paulson@19954:     assume less: "i<j"
wenzelm@46575:     then have "(s j, s i) \<in> r" by (rule s [of i j])
wenzelm@46575:     then show "\<exists>k. (s j, s i) \<in> T k & k<n" by (auto simp add: r)
wenzelm@46575:   qed
paulson@19954:   have trless: "!!i j. i\<noteq>j ==> transition_idx s T {i,j} < n"
paulson@19954:     apply (auto simp add: linorder_neq_iff)
wenzelm@46575:     apply (blast dest: s_in_T transition_idx_less)
wenzelm@46575:     apply (subst insert_commute)
wenzelm@46575:     apply (blast dest: s_in_T transition_idx_less)
paulson@19954:     done
paulson@19954:   have
wenzelm@46575:    "\<exists>K k. K \<subseteq> UNIV & infinite K & k < n &
paulson@19954:           (\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k)"
traytel@54580:     by (rule Ramsey2) (auto intro: trless infinite_UNIV_nat)
wenzelm@46575:   then obtain K and k
paulson@19954:     where infK: "infinite K" and less: "k < n" and
paulson@19954:           allk: "\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k"
paulson@19954:     by auto
paulson@19954:   have "\<forall>m. (s (enumerate K (Suc m)), s(enumerate K m)) \<in> T k"
paulson@19954:   proof
paulson@19954:     fix m::nat
paulson@19954:     let ?j = "enumerate K (Suc m)"
paulson@19954:     let ?i = "enumerate K m"
wenzelm@46575:     have jK: "?j \<in> K" by (simp add: enumerate_in_set infK)
wenzelm@46575:     have iK: "?i \<in> K" by (simp add: enumerate_in_set infK)
wenzelm@46575:     have ij: "?i < ?j" by (simp add: enumerate_step infK)
wenzelm@46575:     have ijk: "transition_idx s T {?i,?j} = k" using iK jK ij
paulson@19954:       by (simp add: allk)
wenzelm@46575:     obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n"
paulson@19954:       using s_in_T [OF ij] by blast
wenzelm@46575:     then show "(s ?j, s ?i) \<in> T k"
wenzelm@46575:       by (simp add: ijk [symmetric] transition_idx_in ij)
paulson@19954:   qed
wenzelm@46575:   then have "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
wenzelm@46575:   then show False using wfT less by blast
paulson@19954: qed
paulson@19954: 
paulson@19944: end