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Commit 6689586a authored by paulson's avatar paulson
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merged

......@@ -90,10 +90,8 @@ lemma init_segment_insert_iff:
shows "init_segment (insert n S) T \<longleftrightarrow> init_segment S T \<and> n \<in> T"
proof safe
assume L: "init_segment (insert n S) T"
then have "init_segment ({n} \<union> S) T"
by auto
then show "init_segment S T"
by (metis (no_types) Sn init_segment_Un init_segment_trans sup.commute)
by (metis Sn init_segment_Un init_segment_trans insert_is_Un sup_commute)
show "n \<in> T"
using L by (auto simp: init_segment_def)
next
......@@ -111,10 +109,6 @@ lemma init_segment_insert:
assumes "init_segment S T" and T: "less_sets T {n}"
shows "init_segment S (insert n T)"
proof (cases "T={}")
case True
then show ?thesis
using assms(1) by blast
next
case False
obtain S' where S': "T = S \<union> S'" "less_sets S S'"
by (meson assms init_segment_def)
......@@ -122,7 +116,7 @@ next
using T False by (auto simp: less_sets_def)
then show ?thesis
using init_segment_Un by presburger
qed
qed (use assms in auto)
subsection \<open>Definitions and basic properties\<close>
......@@ -240,10 +234,8 @@ proof -
assume "T \<subseteq> S"
then obtain m where "f m = T" "m < card (Pow S)"
using f by (blast elim: equalityE)
then have "decides \<F> T (F m)"
using "*" by blast
show "decides \<F> T ?N"
by (meson INT_lower \<open>decides \<F> T (F m)\<close> \<open>m < card (Pow S)\<close> decides_subset lessThan_iff)
then show "decides \<F> T ?N"
by (metis "*" INT_lower decides_subset lessThan_iff)
qed
qed
......@@ -587,10 +579,6 @@ proof -
by (metis Inf_nat_def1 assms(2) finite.emptyI)
have F: "F n \<noteq> [] \<and> sorted_wrt (\<le>) (F n) \<and> list.set (F n) \<subseteq> Collect infinite \<and> set (F n) \<subseteq> Pow M \<and> Inf ` list.set (F n) \<subseteq> M" for n
proof (induction n)
case 0
then show ?case
by (auto simp: InfM \<open>infinite M\<close>)
next
case (Suc n)
have "hd (F n) \<subseteq> M"
by (meson Pow_iff Suc.IH hd_in_set subsetD)
......@@ -605,12 +593,12 @@ proof -
by (metis 2 3 Inf_nat_def1 finite.simps in_mono)
with 1 2 3 show ?case
using Suc by simp
qed
qed (auto simp: InfM \<open>infinite M\<close>)
have \<Phi>F: "\<Phi> (F n) (Eps (\<Phi> (F n)))" for n
by (metis Ball_Collect F Pow_iff Un_subset_iff \<Phi>_Eps hd_in_set subset_code(1))
then have insert_closed: "insert_closed (F n) (Eps (\<Phi> (F n)))" for n
using \<Phi>_def by blast
have Eps_subset_hd: "Eps (\<Phi> (F n)) \<subseteq> hd (F n) " for n
have Eps_subset_hd: "Eps (\<Phi> (F n)) \<subseteq> hd (F n)" for n
using \<Phi>F \<Phi>_def by blast
define mmap where "mmap \<equiv> \<lambda>n. Inf (hd (F (Suc n)))"
have "mmap n < mmap (Suc n)" for n
......@@ -626,7 +614,7 @@ proof -
have hd_Suc_eq_Eps: "hd (F (Suc n)) = Eps (\<Phi> (F n))" for n
by simp
have "Inf (hd (F n)) \<in> hd (F n)" for n
by (metis Inf_nat_def1 \<Phi>F \<Phi>_def finite.emptyI rev_finite_subset)
by (metis Inf_nat_def1 \<Phi>F \<Phi>_def finite.emptyI finite_subset)
then have Inf_hd_in_Eps: "Inf (hd (F m)) \<in> Eps (\<Phi> (F n))" if "m>n" for m n
by (metis Eps_\<Phi>_decreasing Nat.lessE hd_Suc_eq_Eps less_imp_le_nat subsetD that)
then have image_mmap_subset_hd_F: "mmap ` {n..} \<subseteq> hd (F (Suc n))" for n
......@@ -687,9 +675,7 @@ proof -
ultimately show False
by (meson range_mmap_subset rejects_subset)
qed
show "less_sets S {a}"
by (auto simp: Sn)
qed
qed (auto simp: Sn)
then show "strongly_accepts \<F> (insert a S) (range mmap)"
using range_mmap_subset strongly_accepts_subset by auto
qed
......@@ -750,7 +736,7 @@ proof clarify
have "S \<noteq> {}"
using Suc.hyps(2) by auto
have "less_sets (S - {Sup S}) {Sup S}"
using Suc.prems(1) finite_Sup_less_iff nat_neq_iff by (auto simp: less_sets_def)
by (simp add: Suc.prems(1) Sup_nat_def \<open>S \<noteq> {}\<close> dual_order.strict_iff_order less_sets_def)
then have "strongly_accepts (?\<F> 0) (insert (Sup S) (S - {Sup S})) P"
by (metis P Seq Suc.prems finite_Diff insert_subset sacc)
then show ?case
......@@ -775,14 +761,8 @@ proof (induction r)
show ?case
proof (cases "r<2")
case True
then consider "Suc r = 1" | "Suc r = 2"
by linarith
then show ?thesis
proof cases
case 1 with \<F> show ?thesis by (auto simp: Ramsey_def)
next
case 2 with \<F> show ?thesis by (metis Nash_Williams_2)
qed
by (metis Nash_Williams_2 One_nat_def Ramsey_def assms(1) less_2_cases less_one numeral_2_eq_2 order_refl)
next
case False
with Suc.IH have Ram: "Ramsey \<F> r"
......@@ -808,9 +788,7 @@ proof (induction r)
apply (clarsimp simp add: disjoint_iff g_def less_Suc_eq)
by (metis True diff_less less_nat_zero_code nat_neq_iff zero_less_one)
then show ?thesis
apply (rule_tac x=N in exI)
apply (rule_tac x=i in exI)
by (simp add: \<open>N \<subseteq> M\<close> \<open>i < r\<close> \<open>infinite N\<close> less_SucI)
by (metis \<open>N \<subseteq> M\<close> \<open>infinite N\<close> less_Suc_eq)
next
case False
then have "i = r-1"
......@@ -822,22 +800,20 @@ proof (induction r)
using fim i False \<open>i<r\<close> by (force simp: h_def)
then obtain P j where "P \<subseteq> N" "infinite P" "j<r" and j: "\<forall>k<r. j\<noteq>k \<longrightarrow> h -` {k} \<inter> \<F> \<inter> Pow P = {}"
by (metis Ramsey_def Ram \<open>infinite N\<close>)
show ?thesis
have "\<exists>i. \<forall>j<Suc r. j \<noteq> i \<longrightarrow> f -` {j} \<inter> \<F> \<inter> Pow P = {}"
proof (cases "j=0")
case True
then show ?thesis
apply (rule_tac x=P in exI)
apply (rule_tac x=r in exI)
using null [of 0] \<open>r \<ge> 2\<close> \<open>P \<subseteq> N\<close> \<open>N \<subseteq> M\<close> j \<open>infinite P\<close>
by (force simp: h_def disjoint_iff less_Suc_eq intro: subset_trans)
using null [of 0] \<open>r \<ge> 2\<close> \<open>P \<subseteq> N\<close> j by (force simp: h_def disjoint_iff less_Suc_eq)
next
case False
then show ?thesis
apply (rule_tac x=P in exI)
apply (rule_tac x=j in exI)
using j \<open>P \<subseteq> N\<close> \<open>N \<subseteq> M\<close> \<open>j < r\<close> \<open>infinite P\<close> False
by (auto simp: h_def less_Suc_eq disjoint_iff intro: less_trans)
using j \<open>j < r\<close> by (auto simp: h_def less_Suc_eq disjoint_iff intro: less_trans)
qed
then show ?thesis
by (metis Suc.prems \<open>N \<subseteq> M\<close> \<open>P \<subseteq> N\<close> \<open>infinite P\<close> subset_trans)
qed
qed
qed
......
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