src/HOL/Library/Ramsey.thy
 author wenzelm Sun Oct 01 18:29:28 2006 +0200 (2006-10-01) changeset 20810 3377a830b727 parent 19954 e4c9f6946db3 child 21634 369e38e35686 permissions -rwxr-xr-x
moved theory Infinite_Set to Library;
tuned proofs;
```     1 (*  Title:      HOL/Library/Ramsey.thy
```
```     2     ID:         \$Id\$
```
```     3     Author:     Tom Ridge. Converted to structured Isar by L C Paulson
```
```     4 *)
```
```     5
```
```     6 header "Ramsey's Theorem"
```
```     7
```
```     8 theory Ramsey imports Main Infinite_Set begin
```
```     9
```
```    10
```
```    11 subsection{*Preliminaries*}
```
```    12
```
```    13 subsubsection{*``Axiom'' of Dependent Choice*}
```
```    14
```
```    15 consts choice :: "('a => bool) => ('a * 'a) set => nat => 'a"
```
```    16   --{*An integer-indexed chain of choices*}
```
```    17 primrec
```
```    18   choice_0:   "choice P r 0 = (SOME x. P x)"
```
```    19
```
```    20   choice_Suc: "choice P r (Suc n) = (SOME y. P y & (choice P r n, y) \<in> r)"
```
```    21
```
```    22
```
```    23 lemma choice_n:
```
```    24   assumes P0: "P x0"
```
```    25       and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
```
```    26   shows "P (choice P r n)"
```
```    27 proof (induct n)
```
```    28   case 0 show ?case by (force intro: someI P0)
```
```    29 next
```
```    30   case Suc thus ?case by (auto intro: someI2_ex [OF Pstep])
```
```    31 qed
```
```    32
```
```    33 lemma dependent_choice:
```
```    34   assumes trans: "trans r"
```
```    35       and P0: "P x0"
```
```    36       and Pstep: "!!x. P x ==> \<exists>y. P y & (x,y) \<in> r"
```
```    37   obtains f :: "nat => 'a" where
```
```    38     "!!n. P (f n)" and "!!n m. n < m ==> (f n, f m) \<in> r"
```
```    39 proof
```
```    40   fix n
```
```    41   show "P (choice P r n)" by (blast intro: choice_n [OF P0 Pstep])
```
```    42 next
```
```    43   have PSuc: "\<forall>n. (choice P r n, choice P r (Suc n)) \<in> r"
```
```    44     using Pstep [OF choice_n [OF P0 Pstep]]
```
```    45     by (auto intro: someI2_ex)
```
```    46   fix n m :: nat
```
```    47   assume less: "n < m"
```
```    48   show "(choice P r n, choice P r m) \<in> r" using PSuc
```
```    49     by (auto intro: less_Suc_induct [OF less] transD [OF trans])
```
```    50 qed
```
```    51
```
```    52
```
```    53 subsubsection {*Partitions of a Set*}
```
```    54
```
```    55 definition
```
```    56   part :: "nat => nat => 'a set => ('a set => nat) => bool"
```
```    57   --{*the function @{term f} partitions the @{term r}-subsets of the typically
```
```    58        infinite set @{term Y} into @{term s} distinct categories.*}
```
```    59   "part r s Y f = (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X < s)"
```
```    60
```
```    61 text{*For induction, we decrease the value of @{term r} in partitions.*}
```
```    62 lemma part_Suc_imp_part:
```
```    63      "[| infinite Y; part (Suc r) s Y f; y \<in> Y |]
```
```    64       ==> part r s (Y - {y}) (%u. f (insert y u))"
```
```    65   apply(simp add: part_def, clarify)
```
```    66   apply(drule_tac x="insert y X" in spec)
```
```    67   apply(force simp:card_Diff_singleton_if)
```
```    68   done
```
```    69
```
```    70 lemma part_subset: "part r s YY f ==> Y \<subseteq> YY ==> part r s Y f"
```
```    71   unfolding part_def by blast
```
```    72
```
```    73
```
```    74 subsection {*Ramsey's Theorem: Infinitary Version*}
```
```    75
```
```    76 lemma Ramsey_induction:
```
```    77   fixes s and r::nat
```
```    78   shows
```
```    79   "!!(YY::'a set) (f::'a set => nat).
```
```    80       [|infinite YY; part r s YY f|]
```
```    81       ==> \<exists>Y' t'. Y' \<subseteq> YY & infinite Y' & t' < s &
```
```    82                   (\<forall>X. X \<subseteq> Y' & finite X & card X = r --> f X = t')"
```
```    83 proof (induct r)
```
```    84   case 0
```
```    85   thus ?case by (auto simp add: part_def card_eq_0_iff cong: conj_cong)
```
```    86 next
```
```    87   case (Suc r)
```
```    88   show ?case
```
```    89   proof -
```
```    90     from Suc.prems infinite_imp_nonempty obtain yy where yy: "yy \<in> YY" by blast
```
```    91     let ?ramr = "{((y,Y,t),(y',Y',t')). y' \<in> Y & Y' \<subseteq> Y}"
```
```    92     let ?propr = "%(y,Y,t).
```
```    93 		 y \<in> YY & y \<notin> Y & Y \<subseteq> YY & infinite Y & t < s
```
```    94 		 & (\<forall>X. X\<subseteq>Y & finite X & card X = r --> (f o insert y) X = t)"
```
```    95     have infYY': "infinite (YY-{yy})" using Suc.prems by auto
```
```    96     have partf': "part r s (YY - {yy}) (f \<circ> insert yy)"
```
```    97       by (simp add: o_def part_Suc_imp_part yy Suc.prems)
```
```    98     have transr: "trans ?ramr" by (force simp add: trans_def)
```
```    99     from Suc.hyps [OF infYY' partf']
```
```   100     obtain Y0 and t0
```
```   101     where "Y0 \<subseteq> YY - {yy}"  "infinite Y0"  "t0 < s"
```
```   102           "\<forall>X. X\<subseteq>Y0 \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yy) X = t0"
```
```   103         by blast
```
```   104     with yy have propr0: "?propr(yy,Y0,t0)" by blast
```
```   105     have proprstep: "\<And>x. ?propr x \<Longrightarrow> \<exists>y. ?propr y \<and> (x, y) \<in> ?ramr"
```
```   106     proof -
```
```   107       fix x
```
```   108       assume px: "?propr x" thus "?thesis x"
```
```   109       proof (cases x)
```
```   110         case (fields yx Yx tx)
```
```   111         then obtain yx' where yx': "yx' \<in> Yx" using px
```
```   112                by (blast dest: infinite_imp_nonempty)
```
```   113         have infYx': "infinite (Yx-{yx'})" using fields px by auto
```
```   114         with fields px yx' Suc.prems
```
```   115         have partfx': "part r s (Yx - {yx'}) (f \<circ> insert yx')"
```
```   116           by (simp add: o_def part_Suc_imp_part part_subset [where ?YY=YY])
```
```   117 	from Suc.hyps [OF infYx' partfx']
```
```   118 	obtain Y' and t'
```
```   119 	where Y': "Y' \<subseteq> Yx - {yx'}"  "infinite Y'"  "t' < s"
```
```   120 	       "\<forall>X. X\<subseteq>Y' \<and> finite X \<and> card X = r \<longrightarrow> (f \<circ> insert yx') X = t'"
```
```   121 	    by blast
```
```   122 	show ?thesis
```
```   123 	proof
```
```   124 	  show "?propr (yx',Y',t') & (x, (yx',Y',t')) \<in> ?ramr"
```
```   125   	    using fields Y' yx' px by blast
```
```   126 	qed
```
```   127       qed
```
```   128     qed
```
```   129     from dependent_choice [OF transr propr0 proprstep]
```
```   130     obtain g where pg: "!!n::nat.  ?propr (g n)"
```
```   131       and rg: "!!n m. n<m ==> (g n, g m) \<in> ?ramr" by blast
```
```   132     let ?gy = "(\<lambda>n. let (y,Y,t) = g n in y)"
```
```   133     let ?gt = "(\<lambda>n. let (y,Y,t) = g n in t)"
```
```   134     have rangeg: "\<exists>k. range ?gt \<subseteq> {..<k}"
```
```   135     proof (intro exI subsetI)
```
```   136       fix x
```
```   137       assume "x \<in> range ?gt"
```
```   138       then obtain n where "x = ?gt n" ..
```
```   139       with pg [of n] show "x \<in> {..<s}" by (cases "g n") auto
```
```   140     qed
```
```   141     have "finite (range ?gt)"
```
```   142       by (simp add: finite_nat_iff_bounded rangeg)
```
```   143     then obtain s' and n'
```
```   144       where s': "s' = ?gt n'"
```
```   145         and infeqs': "infinite {n. ?gt n = s'}"
```
```   146       by (rule inf_img_fin_domE) (auto simp add: vimage_def intro: nat_infinite)
```
```   147     with pg [of n'] have less': "s'<s" by (cases "g n'") auto
```
```   148     have inj_gy: "inj ?gy"
```
```   149     proof (rule linorder_injI)
```
```   150       fix m m' :: nat assume less: "m < m'" show "?gy m \<noteq> ?gy m'"
```
```   151         using rg [OF less] pg [of m] by (cases "g m", cases "g m'") auto
```
```   152     qed
```
```   153     show ?thesis
```
```   154     proof (intro exI conjI)
```
```   155       show "?gy ` {n. ?gt n = s'} \<subseteq> YY" using pg
```
```   156         by (auto simp add: Let_def split_beta)
```
```   157       show "infinite (?gy ` {n. ?gt n = s'})" using infeqs'
```
```   158         by (blast intro: inj_gy [THEN subset_inj_on] dest: finite_imageD)
```
```   159       show "s' < s" by (rule less')
```
```   160       show "\<forall>X. X \<subseteq> ?gy ` {n. ?gt n = s'} & finite X & card X = Suc r
```
```   161           --> f X = s'"
```
```   162       proof -
```
```   163         {fix X
```
```   164          assume "X \<subseteq> ?gy ` {n. ?gt n = s'}"
```
```   165             and cardX: "finite X" "card X = Suc r"
```
```   166          then obtain AA where AA: "AA \<subseteq> {n. ?gt n = s'}" and Xeq: "X = ?gy`AA"
```
```   167              by (auto simp add: subset_image_iff)
```
```   168          with cardX have "AA\<noteq>{}" by auto
```
```   169          hence AAleast: "(LEAST x. x \<in> AA) \<in> AA" by (auto intro: LeastI_ex)
```
```   170          have "f X = s'"
```
```   171          proof (cases "g (LEAST x. x \<in> AA)")
```
```   172            case (fields ya Ya ta)
```
```   173            with AAleast Xeq
```
```   174            have ya: "ya \<in> X" by (force intro!: rev_image_eqI)
```
```   175            hence "f X = f (insert ya (X - {ya}))" by (simp add: insert_absorb)
```
```   176            also have "... = ta"
```
```   177            proof -
```
```   178              have "X - {ya} \<subseteq> Ya"
```
```   179              proof
```
```   180                fix x assume x: "x \<in> X - {ya}"
```
```   181                then obtain a' where xeq: "x = ?gy a'" and a': "a' \<in> AA"
```
```   182                  by (auto simp add: Xeq)
```
```   183                hence "a' \<noteq> (LEAST x. x \<in> AA)" using x fields by auto
```
```   184                hence lessa': "(LEAST x. x \<in> AA) < a'"
```
```   185                  using Least_le [of "%x. x \<in> AA", OF a'] by arith
```
```   186                show "x \<in> Ya" using xeq fields rg [OF lessa'] by auto
```
```   187              qed
```
```   188              moreover
```
```   189              have "card (X - {ya}) = r"
```
```   190                by (simp add: card_Diff_singleton_if cardX ya)
```
```   191              ultimately show ?thesis
```
```   192                using pg [of "LEAST x. x \<in> AA"] fields cardX
```
```   193 	       by (clarsimp simp del:insert_Diff_single)
```
```   194            qed
```
```   195            also have "... = s'" using AA AAleast fields by auto
```
```   196            finally show ?thesis .
```
```   197          qed}
```
```   198         thus ?thesis by blast
```
```   199       qed
```
```   200     qed
```
```   201   qed
```
```   202 qed
```
```   203
```
```   204
```
```   205 theorem Ramsey:
```
```   206   fixes s r :: nat and Z::"'a set" and f::"'a set => nat"
```
```   207   shows
```
```   208    "[|infinite Z;
```
```   209       \<forall>X. X \<subseteq> Z & finite X & card X = r --> f X < s|]
```
```   210   ==> \<exists>Y t. Y \<subseteq> Z & infinite Y & t < s
```
```   211             & (\<forall>X. X \<subseteq> Y & finite X & card X = r --> f X = t)"
```
```   212 by (blast intro: Ramsey_induction [unfolded part_def])
```
```   213
```
```   214
```
```   215 corollary Ramsey2:
```
```   216   fixes s::nat and Z::"'a set" and f::"'a set => nat"
```
```   217   assumes infZ: "infinite Z"
```
```   218       and part: "\<forall>x\<in>Z. \<forall>y\<in>Z. x\<noteq>y --> f{x,y} < s"
```
```   219   shows
```
```   220    "\<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)"
```
```   221 proof -
```
```   222   have part2: "\<forall>X. X \<subseteq> Z & finite X & card X = 2 --> f X < s"
```
```   223     by (auto simp add: numeral_2_eq_2 card_2_eq part)
```
```   224   obtain Y t
```
```   225     where "Y \<subseteq> Z" "infinite Y" "t < s"
```
```   226           "(\<forall>X. X \<subseteq> Y & finite X & card X = 2 --> f X = t)"
```
```   227     by (insert Ramsey [OF infZ part2]) auto
```
```   228   moreover from this have  "\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow> f {x, y} = t" by auto
```
```   229   ultimately show ?thesis by iprover
```
```   230 qed
```
```   231
```
```   232
```
```   233
```
```   234
```
```   235 subsection {*Disjunctive Well-Foundedness*}
```
```   236
```
```   237 text{*An application of Ramsey's theorem to program termination. See
```
```   238
```
```   239 Andreas Podelski and Andrey Rybalchenko, Transition Invariants, 19th Annual
```
```   240 IEEE Symposium on Logic in Computer Science (LICS'04), pages 32--41 (2004).
```
```   241 *}
```
```   242
```
```   243 definition
```
```   244   disj_wf         :: "('a * 'a)set => bool"
```
```   245   "disj_wf r = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r = (\<Union>i<n. T i))"
```
```   246
```
```   247   transition_idx :: "[nat => 'a, nat => ('a*'a)set, nat set] => nat"
```
```   248   "transition_idx s T A =
```
```   249     (LEAST k. \<exists>i j. A = {i,j} & i<j & (s j, s i) \<in> T k)"
```
```   250
```
```   251
```
```   252 lemma transition_idx_less:
```
```   253     "[|i<j; (s j, s i) \<in> T k; k<n|] ==> transition_idx s T {i,j} < n"
```
```   254 apply (subgoal_tac "transition_idx s T {i, j} \<le> k", simp)
```
```   255 apply (simp add: transition_idx_def, blast intro: Least_le)
```
```   256 done
```
```   257
```
```   258 lemma transition_idx_in:
```
```   259     "[|i<j; (s j, s i) \<in> T k|] ==> (s j, s i) \<in> T (transition_idx s T {i,j})"
```
```   260 apply (simp add: transition_idx_def doubleton_eq_iff conj_disj_distribR
```
```   261             cong: conj_cong)
```
```   262 apply (erule LeastI)
```
```   263 done
```
```   264
```
```   265 text{*To be equal to the union of some well-founded relations is equivalent
```
```   266 to being the subset of such a union.*}
```
```   267 lemma disj_wf:
```
```   268      "disj_wf(r) = (\<exists>T. \<exists>n::nat. (\<forall>i<n. wf(T i)) & r \<subseteq> (\<Union>i<n. T i))"
```
```   269 apply (auto simp add: disj_wf_def)
```
```   270 apply (rule_tac x="%i. T i Int r" in exI)
```
```   271 apply (rule_tac x=n in exI)
```
```   272 apply (force simp add: wf_Int1)
```
```   273 done
```
```   274
```
```   275 theorem trans_disj_wf_implies_wf:
```
```   276   assumes transr: "trans r"
```
```   277       and dwf:    "disj_wf(r)"
```
```   278   shows "wf r"
```
```   279 proof (simp only: wf_iff_no_infinite_down_chain, rule notI)
```
```   280   assume "\<exists>s. \<forall>i. (s (Suc i), s i) \<in> r"
```
```   281   then obtain s where sSuc: "\<forall>i. (s (Suc i), s i) \<in> r" ..
```
```   282   have s: "!!i j. i < j ==> (s j, s i) \<in> r"
```
```   283   proof -
```
```   284     fix i and j::nat
```
```   285     assume less: "i<j"
```
```   286     thus "(s j, s i) \<in> r"
```
```   287     proof (rule less_Suc_induct)
```
```   288       show "\<And>i. (s (Suc i), s i) \<in> r" by (simp add: sSuc)
```
```   289       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"
```
```   290         using transr by (unfold trans_def, blast)
```
```   291     qed
```
```   292   qed
```
```   293   from dwf
```
```   294   obtain T and n::nat where wfT: "\<forall>k<n. wf(T k)" and r: "r = (\<Union>k<n. T k)"
```
```   295     by (auto simp add: disj_wf_def)
```
```   296   have s_in_T: "\<And>i j. i<j ==> \<exists>k. (s j, s i) \<in> T k & k<n"
```
```   297   proof -
```
```   298     fix i and j::nat
```
```   299     assume less: "i<j"
```
```   300     hence "(s j, s i) \<in> r" by (rule s [of i j])
```
```   301     thus "\<exists>k. (s j, s i) \<in> T k & k<n" by (auto simp add: r)
```
```   302   qed
```
```   303   have trless: "!!i j. i\<noteq>j ==> transition_idx s T {i,j} < n"
```
```   304     apply (auto simp add: linorder_neq_iff)
```
```   305     apply (blast dest: s_in_T transition_idx_less)
```
```   306     apply (subst insert_commute)
```
```   307     apply (blast dest: s_in_T transition_idx_less)
```
```   308     done
```
```   309   have
```
```   310    "\<exists>K k. K \<subseteq> UNIV & infinite K & k < n &
```
```   311           (\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k)"
```
```   312     by (rule Ramsey2) (auto intro: trless nat_infinite)
```
```   313   then obtain K and k
```
```   314     where infK: "infinite K" and less: "k < n" and
```
```   315           allk: "\<forall>i\<in>K. \<forall>j\<in>K. i\<noteq>j --> transition_idx s T {i,j} = k"
```
```   316     by auto
```
```   317   have "\<forall>m. (s (enumerate K (Suc m)), s(enumerate K m)) \<in> T k"
```
```   318   proof
```
```   319     fix m::nat
```
```   320     let ?j = "enumerate K (Suc m)"
```
```   321     let ?i = "enumerate K m"
```
```   322     have jK: "?j \<in> K" by (simp add: enumerate_in_set infK)
```
```   323     have iK: "?i \<in> K" by (simp add: enumerate_in_set infK)
```
```   324     have ij: "?i < ?j" by (simp add: enumerate_step infK)
```
```   325     have ijk: "transition_idx s T {?i,?j} = k" using iK jK ij
```
```   326       by (simp add: allk)
```
```   327     obtain k' where "(s ?j, s ?i) \<in> T k'" "k'<n"
```
```   328       using s_in_T [OF ij] by blast
```
```   329     thus "(s ?j, s ?i) \<in> T k"
```
```   330       by (simp add: ijk [symmetric] transition_idx_in ij)
```
```   331   qed
```
```   332   hence "~ wf(T k)" by (force simp add: wf_iff_no_infinite_down_chain)
```
```   333   thus False using wfT less by blast
```
```   334 qed
```
```   335
```
```   336
```
```   337 end
```