src/HOL/Library/SCT_Theorem.thy

moved lemma all_not_ex to HOL.thy ; tuned proofs

(*  Title:      HOL/Library/SCT_Theorem.thy
    ID:         $Id$
    Author:     Alexander Krauss, TU Muenchen
*)

header ""   (* FIXME proper header *)

theory SCT_Theorem
imports Main Ramsey SCT_Misc SCT_Definition
begin

subsection {* The size change criterion SCT *}

definition is_thread :: "nat \<Rightarrow> nat sequence \<Rightarrow> (nat, scg) ipath \<Rightarrow> bool"
where
  "is_thread n \<theta> p = (\<forall>i\<ge>n. eqlat p \<theta> i)"

definition is_fthread :: 
  "nat sequence \<Rightarrow> (nat, scg) ipath \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
where
  "is_fthread \<theta> mp i j = (\<forall>k\<in>{i..<j}. eqlat mp \<theta> k)"

definition is_desc_fthread ::
  "nat sequence \<Rightarrow> (nat, scg) ipath \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"
where
  "is_desc_fthread \<theta> mp i j = 
  (is_fthread \<theta> mp i j \<and>
  (\<exists>k\<in>{i..<j}. descat mp \<theta> k))"

definition
  "has_fth p i j n m = 
  (\<exists>\<theta>. is_fthread \<theta> p i j \<and> \<theta> i = n \<and> \<theta> j = m)"

definition
  "has_desc_fth p i j n m = 
  (\<exists>\<theta>. is_desc_fthread \<theta> p i j \<and> \<theta> i = n \<and> \<theta> j = m)"


subsection {* Bounded graphs and threads *}

definition 
  "bounded_scg (P::nat) (G::scg) = 
  (\<forall>p e p'. has_edge G p e p' \<longrightarrow> p < P \<and> p' < P)"

definition
  "bounded_acg P ACG = 
  (\<forall>n G n'. has_edge ACG n G n' \<longrightarrow> n < P \<and> n' < P \<and> bounded_scg P G)"
  
definition
  "bounded_mp P mp = (\<forall>i. bounded_scg P (snd (mp i)))"

definition (* = finite (range \<theta>) *)
  "bounded_th (P::nat) n \<theta> = (\<forall>i>n. \<theta> i < P)"

definition 
  "finite_scg (G::scg) = (\<exists>P. bounded_scg P G)"

definition
  "finite_acg (A::acg) = (\<exists>P. bounded_acg P A)"

lemma "finite (insert x A) = finite A"
by simp


lemma finite_scg_finite[simp]: "finite_scg (Graph G) = finite G"
proof
  assume "finite_scg (Graph G)"
  thm bounded_scg_def
  
  then obtain P where bounded: "bounded_scg P (Graph G)" 
    by (auto simp:finite_scg_def)

  show "finite G"
  proof (rule finite_subset)
    from bounded 
    show "G \<subseteq> {0 .. P - 1} \<times> { LESS, LEQ } \<times> { 0 .. P - 1}"
      apply (auto simp:bounded_scg_def has_edge_def)
      apply force
      apply (case_tac "aa", auto)
      apply force
      done

    show "finite \<dots>"
      by (auto intro: finite_cartesian_product finite_atLeastAtMost)
  qed
next
  assume "finite G"
  thus "finite_scg (Graph G)"
  proof induct
    show "finite_scg (Graph {})"
      unfolding finite_scg_def bounded_scg_def has_edge_def by auto
  next
    fix x G 
    assume "finite G" "x \<notin> G" "finite_scg (Graph G)"
    then obtain P
      where bG: "bounded_scg P (Graph G)" 
      by (auto simp:finite_scg_def)
    
    obtain p e p' where x: "x = (p, e, p')" by (cases x, auto)
    
    let ?Q = "max P (max (Suc p) (Suc p'))"
    have "P \<le> ?Q" "Suc p \<le> ?Q" "Suc p' \<le> ?Q" by auto
    with bG
    have "bounded_scg ?Q (Graph (insert x G))" 
      unfolding x bounded_scg_def has_edge_def
      apply simp
      apply (intro allI, elim allE)
      by auto
    thus "finite_scg (Graph (insert x G))"
      by (auto simp:finite_scg_def)
  qed
qed

lemma finite_acg_empty:
  "finite_acg (Graph {})"
unfolding finite_acg_def bounded_acg_def has_edge_def
by auto

lemma bounded_scg_up: "bounded_scg P G \<Longrightarrow> P \<le> Q \<Longrightarrow> bounded_scg Q G"
  unfolding bounded_scg_def
  by force

lemma bounded_up: "bounded_acg P G \<Longrightarrow> P \<le> Q \<Longrightarrow> bounded_acg Q G"
  unfolding bounded_acg_def 
  apply auto
  apply force+
  apply (rule bounded_scg_up)
  by auto

lemma bacg_insert:
  assumes "bounded_acg P (Graph A)"
  assumes "n < P" "m < P" "bounded_scg P G"
  shows "bounded_acg P (Graph (insert (n, G, m) A))"
  using prems
  unfolding bounded_acg_def has_edge_def 
  by auto

lemma finite_acg_ins:
  "finite_acg (Graph (insert (n,G,m) A)) = 
  (finite_scg G \<and> finite_acg (Graph A))" (is "?A = (?B \<and> ?C)")
proof
  assume "?A"
  thus "?B \<and> ?C"
    unfolding finite_acg_def bounded_acg_def finite_scg_def has_edge_def
    by auto
next
  assume "?B \<and> ?C"
  thus ?A
  proof
    assume "?B" "?C"

    from `?C`
    obtain P where bA: "bounded_acg P (Graph A)" by (auto simp:finite_acg_def)
    from `?B` 
    obtain P' where bG: "bounded_scg P' G" by (auto simp:finite_scg_def)

    let ?Q = "max (max P P') (max (Suc n) (Suc m))"
    have "P \<le> ?Q" "n < ?Q" "m < ?Q" by auto
    have "bounded_acg ?Q (Graph (insert (n, G, m) A))"
      apply (rule bacg_insert)
      apply (rule bounded_up)
      apply (rule bA)
      apply auto
      apply (rule bounded_scg_up)
      apply (rule bG)
      by auto
    thus "finite_acg (Graph (insert (n, G, m) A))"
      by (auto simp:finite_acg_def)
  qed
qed


lemma bounded_mpath:
  assumes "has_ipath acg mp"
  and "bounded_acg P acg"
  shows "bounded_mp P mp"
  using prems
  unfolding bounded_acg_def bounded_mp_def has_ipath_def
  by blast

lemma bounded_th: 
  assumes th: "is_thread n \<theta> mp"
  and mp: "bounded_mp P mp"
  shows "bounded_th P n \<theta>"
  unfolding bounded_th_def
proof (intro allI impI)
  fix i assume "n < i"
  
  from mp have "bounded_scg P (snd (mp i))" unfolding bounded_mp_def ..
  moreover
  from th `n < i` have "eqlat mp \<theta> i" unfolding is_thread_def by auto
  ultimately
  show "\<theta> i < P" unfolding bounded_scg_def by auto
qed


lemma finite_range:
  fixes f :: "nat \<Rightarrow> 'a"
  assumes fin: "finite (range f)"
  shows "\<exists>x. \<exists>\<^sub>\<infinity>i. f i = x"
proof (rule classical)
  assume "\<not>(\<exists>x. \<exists>\<^sub>\<infinity>i. f i = x)"
  hence "\<forall>x. \<exists>j. \<forall>i>j. f i \<noteq> x"
    unfolding INF_nat by blast
  with choice
  have "\<exists>j. \<forall>x. \<forall>i>(j x). f i \<noteq> x" .
  then obtain j where 
    neq: "\<And>x i. j x < i \<Longrightarrow> f i \<noteq> x" by blast

  from fin have "finite (range (j o f))" 
    by (auto simp:comp_def)
  with finite_nat_bounded
  obtain m where "range (j o f) \<subseteq> {..<m}" by blast
  hence "j (f m) < m" unfolding comp_def by auto

  with neq[of "f m" m] show ?thesis by blast
qed


lemma bounded_th_infinite_visit:
  fixes \<theta> :: "nat sequence"
  assumes b: "bounded_th P n \<theta>"
  shows "\<exists>p. \<exists>\<^sub>\<infinity>i. \<theta> i = p" 
proof -
  have split: "range \<theta> = (\<theta> ` {0 .. n}) \<union> (\<theta> ` {i. n < i})"
    (is "\<dots> = ?A \<union> ?B")
    unfolding image_Un[symmetric] by auto

  have "finite ?A" by (rule finite_imageI) auto
  moreover
  have "finite ?B"
  proof (rule finite_subset)
    from b
    show "?B \<subseteq> { 0 ..< P }"
      unfolding bounded_th_def
      by auto
    show "finite \<dots>" by auto
  qed

  ultimately have "finite (range \<theta>)"
    unfolding split by auto

  with finite_range show ?thesis .
qed


lemma bounded_scgcomp:
  "bounded_scg P A 
  \<Longrightarrow> bounded_scg P B 
  \<Longrightarrow> bounded_scg P (A * B)"
  unfolding bounded_scg_def
  by (auto simp:in_grcomp)

lemma bounded_acgcomp:
  "bounded_acg P A 
  \<Longrightarrow> bounded_acg P B 
  \<Longrightarrow> bounded_acg P (A * B)"
  unfolding bounded_acg_def
  by (auto simp:in_grcomp intro!:bounded_scgcomp) 

lemma bounded_acgpow:
  "bounded_acg P A
  \<Longrightarrow> bounded_acg P (A ^ (Suc n))"
  by (induct n, simp add:power_Suc) 
   (subst power_Suc, blast intro:bounded_acgcomp)

lemma bounded_plus:
  assumes b: "bounded_acg P acg"
  shows "bounded_acg P (tcl acg)"
  unfolding bounded_acg_def
proof (intro allI impI conjI)
  fix n G m
  assume "tcl acg \<turnstile> n \<leadsto>\<^bsup>G\<^esup> m"
  then obtain i where "0 < i" and i: "acg ^ i \<turnstile> n \<leadsto>\<^bsup>G\<^esup> m"
    unfolding in_tcl by auto (* FIXME: Disambiguate \<turnstile> Grammar *)
  from b have "bounded_acg P (acg ^ (Suc (i - 1)))"
    by (rule bounded_acgpow)
  with `0 < i` have "bounded_acg P (acg ^ i)" by simp
  with i
  show "n < P" "m < P" "bounded_scg P G"
    unfolding bounded_acg_def
    by auto
qed


lemma bounded_scg_def':
  "bounded_scg P G = (\<forall>(p,e,p')\<in>dest_graph G. p < P \<and> p' < P)"
  unfolding bounded_scg_def has_edge_def
  by auto


lemma finite_bounded_scg: "finite { G. bounded_scg P G }"
proof (rule finite_subset)
  show "{G. bounded_scg P G} \<subseteq> 
    Graph ` (Pow ({0 .. P - 1} \<times> UNIV \<times> {0 .. P - 1}))"
  proof (rule, simp)
    fix G 
    
    assume b: "bounded_scg P G"

    show "G \<in> Graph ` Pow ({0..P - Suc 0} \<times> UNIV \<times> {0..P - Suc 0})"
    proof (cases G)
      fix G' assume [simp]: "G = Graph G'"
      
      from b show ?thesis
        by (auto simp:bounded_scg_def' image_def)
    qed
  qed

  show "finite \<dots>"
    apply (rule finite_imageI)
    apply (unfold finite_Pow_iff)
    by (intro finite_cartesian_product finite_atLeastAtMost 
      finite_class.finite)
qed

lemma bounded_finite:
  assumes bounded: "bounded_acg P A"
  shows "finite (dest_graph A)"
proof (rule finite_subset)
  
  from bounded
  show "dest_graph A \<subseteq> {0 .. P - 1} \<times> { G. bounded_scg P G } \<times> { 0 .. P - 1}"
    by (auto simp:bounded_acg_def has_edge_def) force+

  show "finite \<dots>"
    by (intro finite_cartesian_product finite_atLeastAtMost finite_bounded_scg)
qed


subsection {* Contraction and more *}

abbreviation 
  "pdesc P == (fst P, prod P, end_node P)"

lemma pdesc_acgplus: 
  assumes "has_ipath \<A> p"
  and "i < j"
  shows "has_edge (tcl \<A>) (fst (p\<langle>i,j\<rangle>)) (prod (p\<langle>i,j\<rangle>)) (end_node (p\<langle>i,j\<rangle>))"
  unfolding plus_paths
  apply (rule exI)
  apply (insert prems)  
  by (auto intro:sub_path_is_path[of "\<A>" p i j] simp:sub_path_def)


lemma combine_fthreads: 
  assumes range: "i < j" "j \<le> k"
  shows 
  "has_fth p i k m r =
  (\<exists>n. has_fth p i j m n \<and> has_fth p j k n r)" (is "?L = ?R")
proof (intro iffI)
  assume "?L"
  then obtain \<theta>
    where "is_fthread \<theta> p i k" 
    and [simp]: "\<theta> i = m" "\<theta> k = r"
    by (auto simp:has_fth_def)

  with range
  have "is_fthread \<theta> p i j" and "is_fthread \<theta> p j k"
    by (auto simp:is_fthread_def)
  hence "has_fth p i j m (\<theta> j)" and "has_fth p j k (\<theta> j) r"
    by (auto simp:has_fth_def)
  thus "?R" by auto
next
  assume "?R"
  then obtain n \<theta>1 \<theta>2
    where ths: "is_fthread \<theta>1 p i j" "is_fthread \<theta>2 p j k"
    and [simp]: "\<theta>1 i = m" "\<theta>1 j = n" "\<theta>2 j = n" "\<theta>2 k = r"
    by (auto simp:has_fth_def)

  let ?\<theta> = "(\<lambda>i. if i < j then \<theta>1 i else \<theta>2 i)"
  have "is_fthread ?\<theta> p i k"
    unfolding is_fthread_def
  proof
    fix l assume range: "l \<in> {i..<k}"
    
    show "eqlat p ?\<theta> l"
    proof (cases rule:three_cases)
      assume "Suc l < j"
      with ths range show ?thesis 
	unfolding is_fthread_def Ball_def
	by simp
    next
      assume "Suc l = j"
      hence "l < j" "\<theta>2 (Suc l) = \<theta>1 (Suc l)" by auto
      with ths range show ?thesis 
	unfolding is_fthread_def Ball_def
	by simp
    next
      assume "j \<le> l"
      with ths range show ?thesis 
	unfolding is_fthread_def Ball_def
	by simp
    qed arith
  qed
  moreover 
  have "?\<theta> i = m" "?\<theta> k = r" using range by auto
  ultimately show "has_fth p i k m r" 
    by (auto simp:has_fth_def)
qed 


lemma desc_is_fthread:
  "is_desc_fthread \<theta> p i k \<Longrightarrow> is_fthread \<theta> p i k"
  unfolding is_desc_fthread_def
  by simp


lemma combine_dfthreads: 
  assumes range: "i < j" "j \<le> k"
  shows 
  "has_desc_fth p i k m r =
  (\<exists>n. (has_desc_fth p i j m n \<and> has_fth p j k n r)
  \<or> (has_fth p i j m n \<and> has_desc_fth p j k n r))" (is "?L = ?R")
proof 
  assume "?L"
  then obtain \<theta>
    where desc: "is_desc_fthread \<theta> p i k" 
    and [simp]: "\<theta> i = m" "\<theta> k = r"
    by (auto simp:has_desc_fth_def)

  hence "is_fthread \<theta> p i k"
    by (simp add: desc_is_fthread)
  with range have fths: "is_fthread \<theta> p i j" "is_fthread \<theta> p j k"
    unfolding is_fthread_def
    by auto
  hence hfths: "has_fth p i j m (\<theta> j)" "has_fth p j k (\<theta> j) r"
    by (auto simp:has_fth_def)

  from desc obtain l 
    where "i \<le> l" "l < k"
    and "descat p \<theta> l"
    by (auto simp:is_desc_fthread_def)

  with fths
  have "is_desc_fthread \<theta> p i j \<or> is_desc_fthread \<theta> p j k"
    unfolding is_desc_fthread_def
    by (cases "l < j") auto
  hence "has_desc_fth p i j m (\<theta> j) \<or> has_desc_fth p j k (\<theta> j) r"
    by (auto simp:has_desc_fth_def)
  with hfths show ?R
    by auto
next
  assume "?R"
  then obtain n \<theta>1 \<theta>2
    where "(is_desc_fthread \<theta>1 p i j \<and> is_fthread \<theta>2 p j k)
    \<or> (is_fthread \<theta>1 p i j \<and> is_desc_fthread \<theta>2 p j k)"
    and [simp]: "\<theta>1 i = m" "\<theta>1 j = n" "\<theta>2 j = n" "\<theta>2 k = r"
    by (auto simp:has_fth_def has_desc_fth_def)

  hence ths2: "is_fthread \<theta>1 p i j" "is_fthread \<theta>2 p j k"
    and dths: "is_desc_fthread \<theta>1 p i j \<or> is_desc_fthread \<theta>2 p j k"
    by (auto simp:desc_is_fthread)

  let ?\<theta> = "(\<lambda>i. if i < j then \<theta>1 i else \<theta>2 i)"
  have "is_fthread ?\<theta> p i k"
    unfolding is_fthread_def
  proof
    fix l assume range: "l \<in> {i..<k}"
    
    show "eqlat p ?\<theta> l"
    proof (cases rule:three_cases)
      assume "Suc l < j"
      with ths2 range show ?thesis 
	unfolding is_fthread_def Ball_def
	by simp
    next
      assume "Suc l = j"
      hence "l < j" "\<theta>2 (Suc l) = \<theta>1 (Suc l)" by auto
      with ths2 range show ?thesis 
	unfolding is_fthread_def Ball_def
	by simp
    next
      assume "j \<le> l"
      with ths2 range show ?thesis 
	unfolding is_fthread_def Ball_def
	by simp
    qed arith
  qed
  moreover
  from dths
  have "\<exists>l. i \<le> l \<and> l < k \<and> descat p ?\<theta> l"
  proof
    assume "is_desc_fthread \<theta>1 p i j"

    then obtain l where range: "i \<le> l" "l < j" and "descat p \<theta>1 l"
      unfolding is_desc_fthread_def Bex_def by auto
    hence "descat p ?\<theta> l" 
      by (cases "Suc l = j", auto)
    with `j \<le> k` and range show ?thesis 
      by (rule_tac x="l" in exI, auto)
  next
    assume "is_desc_fthread \<theta>2 p j k"
    then obtain l where range: "j \<le> l" "l < k" and "descat p \<theta>2 l"
      unfolding is_desc_fthread_def Bex_def by auto
    with `i < j` have "descat p ?\<theta> l" "i \<le> l"
      by auto
    with range show ?thesis 
      by (rule_tac x="l" in exI, auto)
  qed
  ultimately have "is_desc_fthread ?\<theta> p i k"
    by (simp add: is_desc_fthread_def Bex_def)

  moreover 
  have "?\<theta> i = m" "?\<theta> k = r" using range by auto

  ultimately show "has_desc_fth p i k m r" 
    by (auto simp:has_desc_fth_def)
qed 

    

lemma fth_single:
  "has_fth p i (Suc i) m n = eql (snd (p i)) m n" (is "?L = ?R")
proof 
  assume "?L" thus "?R"
    unfolding is_fthread_def Ball_def has_fth_def
    by auto
next
  let ?\<theta> = "\<lambda>k. if k = i then m else n"
  assume edge: "?R"
  hence "is_fthread ?\<theta> p i (Suc i) \<and> ?\<theta> i = m \<and> ?\<theta> (Suc i) = n"
    unfolding is_fthread_def Ball_def
    by auto

  thus "?L"
    unfolding has_fth_def 
    by auto
qed

lemma desc_fth_single:
  "has_desc_fth p i (Suc i) m n = 
  dsc (snd (p i)) m n" (is "?L = ?R")
proof 
  assume "?L" thus "?R"
    unfolding is_desc_fthread_def has_desc_fth_def is_fthread_def
    Bex_def 
    by (elim exE conjE) (case_tac "k = i", auto)
next
  let ?\<theta> = "\<lambda>k. if k = i then m else n"
  assume edge: "?R"
  hence "is_desc_fthread ?\<theta> p i (Suc i) \<and> ?\<theta> i = m \<and> ?\<theta> (Suc i) = n"
    unfolding is_desc_fthread_def is_fthread_def Ball_def Bex_def
    by auto
  thus "?L"
    unfolding has_desc_fth_def 
    by auto
qed

lemma mk_eql: "(G \<turnstile> m \<leadsto>\<^bsup>e\<^esup> n) \<Longrightarrow> eql G m n"
  by (cases e, auto)

lemma eql_scgcomp:
  "eql (G * H) m r =
  (\<exists>n. eql G m n \<and> eql H n r)" (is "?L = ?R")
proof
  show "?L \<Longrightarrow> ?R"
    by (auto simp:in_grcomp intro!:mk_eql)

  assume "?R"
  then obtain n where l: "eql G m n" and r:"eql H n r" by auto
  thus ?L
    by (cases "dsc G m n") (auto simp:in_grcomp mult_sedge_def)
qed

lemma desc_scgcomp:
  "dsc (G * H) m r =
  (\<exists>n. (dsc G m n \<and> eql H n r) \<or> (eq G m n \<and> dsc H n r))" (is "?L = ?R")
proof
  show "?R \<Longrightarrow> ?L" by (auto simp:in_grcomp mult_sedge_def)

  assume "?L"
  thus ?R
    by (auto simp:in_grcomp mult_sedge_def)
  (case_tac "e", auto, case_tac "e'", auto)
qed


lemma has_fth_unfold:
  assumes "i < j"
  shows "has_fth p i j m n = 
    (\<exists>r. has_fth p i (Suc i) m r \<and> has_fth p (Suc i) j r n)"
    by (rule combine_fthreads) (insert `i < j`, auto)

lemma has_dfth_unfold:
  assumes range: "i < j"
  shows 
  "has_desc_fth p i j m r =
  (\<exists>n. (has_desc_fth p i (Suc i) m n \<and> has_fth p (Suc i) j n r)
  \<or> (has_fth p i (Suc i) m n \<and> has_desc_fth p (Suc i) j n r))"
  by (rule combine_dfthreads) (insert `i < j`, auto)


lemma Lemma7a:
 "i \<le> j \<Longrightarrow> has_fth p i j m n = eql (prod (p\<langle>i,j\<rangle>)) m n"
proof (induct i arbitrary: m rule:inc_induct)
  case base show ?case
    unfolding has_fth_def is_fthread_def sub_path_def
    by (auto simp:in_grunit one_sedge_def)
next
  case (step i)
  note IH = `\<And>m. has_fth p (Suc i) j m n = 
  eql (prod (p\<langle>Suc i,j\<rangle>)) m n`

  have "has_fth p i j m n 
    = (\<exists>r. has_fth p i (Suc i) m r \<and> has_fth p (Suc i) j r n)"
    by (rule has_fth_unfold[OF `i < j`])
  also have "\<dots> = (\<exists>r. has_fth p i (Suc i) m r 
    \<and> eql (prod (p\<langle>Suc i,j\<rangle>)) r n)"
    by (simp only:IH)
  also have "\<dots> = (\<exists>r. eql (snd (p i)) m r
    \<and> eql (prod (p\<langle>Suc i,j\<rangle>)) r n)"
    by (simp only:fth_single)
  also have "\<dots> = eql (snd (p i) * prod (p\<langle>Suc i,j\<rangle>)) m n"
    by (simp only:eql_scgcomp)
  also have "\<dots> = eql (prod (p\<langle>i,j\<rangle>)) m n"
    by (simp only:prod_unfold[OF `i < j`])
  finally show ?case .
qed


lemma Lemma7b:
assumes "i \<le> j"
shows
  "has_desc_fth p i j m n = 
  dsc (prod (p\<langle>i,j\<rangle>)) m n"
using prems
proof (induct i arbitrary: m rule:inc_induct)
  case base show ?case
    unfolding has_desc_fth_def is_desc_fthread_def sub_path_def 
    by (auto simp:in_grunit one_sedge_def)
next
  case (step i)
  thus ?case 
    by (simp only:prod_unfold desc_scgcomp desc_fth_single
    has_dfth_unfold fth_single Lemma7a) auto
qed


lemma descat_contract:
  assumes [simp]: "increasing s"
  shows
  "descat (contract s p) \<theta> i = 
  has_desc_fth p (s i) (s (Suc i)) (\<theta> i) (\<theta> (Suc i))"
  by (simp add:Lemma7b increasing_weak contract_def)

lemma eqlat_contract:
  assumes [simp]: "increasing s"
  shows
  "eqlat (contract s p) \<theta> i = 
  has_fth p (s i) (s (Suc i)) (\<theta> i) (\<theta> (Suc i))"
  by (auto simp:Lemma7a increasing_weak contract_def)


subsubsection {* Connecting threads *}

definition
  "connect s \<theta>s = (\<lambda>k. \<theta>s (section_of s k) k)"


lemma next_in_range:
  assumes [simp]: "increasing s"
  assumes a: "k \<in> section s i"
  shows "(Suc k \<in> section s i) \<or> (Suc k \<in> section s (Suc i))"
proof -
  from a have "k < s (Suc i)" by simp
  
  hence "Suc k < s (Suc i) \<or> Suc k = s (Suc i)" by arith
  thus ?thesis
  proof
    assume "Suc k < s (Suc i)"
    with a have "Suc k \<in> section s i" by simp
    thus ?thesis ..
  next
    assume eq: "Suc k = s (Suc i)"
    with increasing_strict have "Suc k < s (Suc (Suc i))" by simp
    with eq have "Suc k \<in> section s (Suc i)" by simp
    thus ?thesis ..
  qed
qed


lemma connect_threads:
  assumes [simp]: "increasing s"
  assumes connected: "\<theta>s i (s (Suc i)) = \<theta>s (Suc i) (s (Suc i))"
  assumes fth: "is_fthread (\<theta>s i) p (s i) (s (Suc i))"

  shows
  "is_fthread (connect s \<theta>s) p (s i) (s (Suc i))"
  unfolding is_fthread_def
proof 
  fix k assume krng: "k \<in> section s i"

  with fth have eqlat: "eqlat p (\<theta>s i) k" 
    unfolding is_fthread_def by simp

  from krng and next_in_range 
  have "(Suc k \<in> section s i) \<or> (Suc k \<in> section s (Suc i))" 
    by simp
  thus "eqlat p (connect s \<theta>s) k"
  proof
    assume "Suc k \<in> section s i"
    with krng eqlat show ?thesis
      unfolding connect_def
      by (simp only:section_of_known `increasing s`)
  next
    assume skrng: "Suc k \<in> section s (Suc i)"
    with krng have "Suc k = s (Suc i)" by auto

    with krng skrng eqlat show ?thesis
      unfolding connect_def
      by (simp only:section_of_known connected[symmetric] `increasing s`)
  qed
qed


lemma connect_dthreads:
  assumes inc[simp]: "increasing s"
  assumes connected: "\<theta>s i (s (Suc i)) = \<theta>s (Suc i) (s (Suc i))"
  assumes fth: "is_desc_fthread (\<theta>s i) p (s i) (s (Suc i))"

  shows
  "is_desc_fthread (connect s \<theta>s) p (s i) (s (Suc i))"
  unfolding is_desc_fthread_def
proof 
  show "is_fthread (connect s \<theta>s) p (s i) (s (Suc i))"
    apply (rule connect_threads)
    apply (insert fth)
    by (auto simp:connected is_desc_fthread_def)

  from fth
  obtain k where dsc: "descat p (\<theta>s i) k" and krng: "k \<in> section s i"
    unfolding is_desc_fthread_def by blast
  
  from krng and next_in_range 
  have "(Suc k \<in> section s i) \<or> (Suc k \<in> section s (Suc i))" 
    by simp
  hence "descat p (connect s \<theta>s) k"
  proof
    assume "Suc k \<in> section s i"
    with krng dsc show ?thesis unfolding connect_def
      by (simp only:section_of_known inc)
  next
    assume skrng: "Suc k \<in> section s (Suc i)"
    with krng have "Suc k = s (Suc i)" by auto

    with krng skrng dsc show ?thesis unfolding connect_def
      by (simp only:section_of_known connected[symmetric] inc)
  qed
  with krng show "\<exists>k\<in>section s i. descat p (connect s \<theta>s) k" ..
qed



lemma mk_inf_thread:
  assumes [simp]: "increasing s"
  assumes fths: "\<And>i. i > n \<Longrightarrow> is_fthread \<theta> p (s i) (s (Suc i))"
  shows "is_thread (s (Suc n)) \<theta> p"
  unfolding is_thread_def 
proof (intro allI impI)
  fix j assume st: "s (Suc n) \<le> j"

  let ?k = "section_of s j"
  from in_section_of st
  have rs: "j \<in> section s ?k" by simp

  with st have "s (Suc n) < s (Suc ?k)" by simp
  with increasing_bij have "n < ?k" by simp
  with rs and fths[of ?k]
  show "eqlat p \<theta> j" by (simp add:is_fthread_def)
qed


lemma mk_inf_desc_thread:
  assumes [simp]: "increasing s"
  assumes fths: "\<And>i. i > n \<Longrightarrow> is_fthread \<theta> p (s i) (s (Suc i))"
  assumes fdths: "\<exists>\<^sub>\<infinity>i. is_desc_fthread \<theta> p (s i) (s (Suc i))"
  shows "is_desc_thread \<theta> p"
  unfolding is_desc_thread_def 
proof (intro exI conjI)

  from mk_inf_thread[of s n] is_thread_def fths
  show "\<forall>i\<ge>s (Suc n). eqlat p \<theta> i" by simp

  show "\<exists>\<^sub>\<infinity>l. descat p \<theta> l"
    unfolding INF_nat
  proof
    fix i 
    
    let ?k = "section_of s i"
    from fdths obtain j
      where "?k < j" "is_desc_fthread \<theta> p (s j) (s (Suc j))"
      unfolding INF_nat by auto
    then obtain l where "s j \<le> l" and desc: "descat p \<theta> l"
      unfolding is_desc_fthread_def
      by auto

    have "i < s (Suc ?k)" by (rule section_of2) simp
    also have "\<dots> \<le> s j"
      by (rule increasing_weak [OF `increasing s`]) (insert `?k < j`, arith)
    also note `\<dots> \<le> l`
    finally have "i < l" .
    with desc
    show "\<exists>l. i < l \<and> descat p \<theta> l" by blast
  qed
qed


lemma desc_ex_choice:
  assumes A: "((\<exists>n.\<forall>i\<ge>n. \<exists>x. P x i) \<and> (\<exists>\<^sub>\<infinity>i. \<exists>x. Q x i))"
  and imp: "\<And>x i. Q x i \<Longrightarrow> P x i"
  shows "\<exists>xs. ((\<exists>n.\<forall>i\<ge>n. P (xs i) i) \<and> (\<exists>\<^sub>\<infinity>i. Q (xs i) i))"
  (is "\<exists>xs. ?Ps xs \<and> ?Qs xs")
proof
  let ?w = "\<lambda>i. (if (\<exists>x. Q x i) then (SOME x. Q x i)
                                 else (SOME x. P x i))"

  from A
  obtain n where P: "\<And>i. n \<le> i \<Longrightarrow> \<exists>x. P x i"
    by auto
  {
    fix i::'a assume "n \<le> i"

    have "P (?w i) i"
    proof (cases "\<exists>x. Q x i")
      case True
      hence "Q (?w i) i" by (auto intro:someI)
      with imp show "P (?w i) i" .
    next
      case False
      with P[OF `n \<le> i`] show "P (?w i) i" 
        by (auto intro:someI)
    qed
  }

  hence "?Ps ?w" by (rule_tac x=n in exI) auto

  moreover
  from A have "\<exists>\<^sub>\<infinity>i. (\<exists>x. Q x i)" ..
  hence "?Qs ?w" by (rule INF_mono) (auto intro:someI)
  ultimately
  show "?Ps ?w \<and> ?Qs ?w" ..
qed



lemma dthreads_join:
  assumes [simp]: "increasing s"
  assumes dthread: "is_desc_thread \<theta> (contract s p)"
  shows "\<exists>\<theta>s. desc (\<lambda>i. is_fthread (\<theta>s i) p (s i) (s (Suc i))
                           \<and> \<theta>s i (s i) = \<theta> i 
                           \<and> \<theta>s i (s (Suc i)) = \<theta> (Suc i))
                   (\<lambda>i. is_desc_fthread (\<theta>s i) p (s i) (s (Suc i))
                           \<and> \<theta>s i (s i) = \<theta> i 
                           \<and> \<theta>s i (s (Suc i)) = \<theta> (Suc i))"
    apply (rule desc_ex_choice)
    apply (insert dthread)
    apply (simp only:is_desc_thread_def)
    apply (simp add:eqlat_contract)
    apply (simp add:descat_contract)
    apply (simp only:has_fth_def has_desc_fth_def)
    by (auto simp:is_desc_fthread_def)



lemma INF_drop_prefix:
  "(\<exists>\<^sub>\<infinity>i::nat. i > n \<and> P i) = (\<exists>\<^sub>\<infinity>i. P i)"
  apply (auto simp:INF_nat)
  apply (drule_tac x = "max m n" in spec)
  apply (elim exE conjE)
  apply (rule_tac x = "na" in exI)
  by auto



lemma contract_keeps_threads:
  assumes inc[simp]: "increasing s"
  shows "(\<exists>\<theta>. is_desc_thread \<theta> p) 
  \<longleftrightarrow> (\<exists>\<theta>. is_desc_thread \<theta> (contract s p))"
  (is "?A \<longleftrightarrow> ?B")
proof 
  assume "?A"
  then obtain \<theta> n 
    where fr: "\<forall>i\<ge>n. eqlat p \<theta> i" 
      and ds: "\<exists>\<^sub>\<infinity>i. descat p \<theta> i"
    unfolding is_desc_thread_def 
    by auto

  let ?c\<theta> = "\<lambda>i. \<theta> (s i)"

  have "is_desc_thread ?c\<theta> (contract s p)"
    unfolding is_desc_thread_def
  proof (intro exI conjI)
    
    show "\<forall>i\<ge>n. eqlat (contract s p) ?c\<theta> i"
    proof (intro allI impI)
      fix i assume "n \<le> i"
      also have "i \<le> s i" 
	using increasing_inc by auto
      finally have "n \<le> s i" .

      with fr have "is_fthread \<theta> p (s i) (s (Suc i))"
	unfolding is_fthread_def by auto
      hence "has_fth p (s i) (s (Suc i)) (\<theta> (s i)) (\<theta> (s (Suc i)))"
	unfolding has_fth_def by auto
      with less_imp_le[OF increasing_strict]
      have "eql (prod (p\<langle>s i,s (Suc i)\<rangle>)) (\<theta> (s i)) (\<theta> (s (Suc i)))"
	by (simp add:Lemma7a)
      thus "eqlat (contract s p) ?c\<theta> i" unfolding contract_def
	by auto
    qed

    show "\<exists>\<^sub>\<infinity>i. descat (contract s p) ?c\<theta> i"
      unfolding INF_nat 
    proof 
      fix i

      let ?K = "section_of s (max (s (Suc i)) n)"
      from `\<exists>\<^sub>\<infinity>i. descat p \<theta> i` obtain j
	  where "s (Suc ?K) < j" "descat p \<theta> j"
	unfolding INF_nat by blast
      
      let ?L = "section_of s j"
      {
	fix x assume r: "x \<in> section s ?L"
	
	have e1: "max (s (Suc i)) n < s (Suc ?K)" by (rule section_of2) simp
        note `s (Suc ?K) < j`
        also have "j < s (Suc ?L)"
          by (rule section_of2) simp
        finally have "Suc ?K \<le> ?L"
          by (simp add:increasing_bij)          
	with increasing_weak have "s (Suc ?K) \<le> s ?L" by simp
	with e1 r have "max (s (Suc i)) n < x" by simp
        
	hence "(s (Suc i)) < x" "n < x" by auto
      }
      note range_est = this
      
      have "is_desc_fthread \<theta> p (s ?L) (s (Suc ?L))"
	unfolding is_desc_fthread_def is_fthread_def
      proof
	show "\<forall>m\<in>section s ?L. eqlat p \<theta> m"
        proof 
          fix m assume "m\<in>section s ?L"
          with range_est(2) have "n < m" . 
          with fr show "eqlat p \<theta> m" by simp
        qed

        from in_section_of inc less_imp_le[OF `s (Suc ?K) < j`]
	have "j \<in> section s ?L" .

	with `descat p \<theta> j`
	show "\<exists>m\<in>section s ?L. descat p \<theta> m" ..
      qed
      
      with less_imp_le[OF increasing_strict]
      have a: "descat (contract s p) ?c\<theta> ?L"
	unfolding contract_def Lemma7b[symmetric]
	by (auto simp:Lemma7b[symmetric] has_desc_fth_def)
      
      have "i < ?L"
      proof (rule classical)
	assume "\<not> i < ?L" 
	hence "s ?L < s (Suc i)" 
          by (simp add:increasing_bij)
	also have "\<dots> < s ?L"
	  by (rule range_est(1)) (simp add:increasing_strict)
	finally show ?thesis .
      qed
      with a show "\<exists>l. i < l \<and> descat (contract s p) ?c\<theta> l"
        by blast
    qed
  qed
  with exI show "?B" .
next
  assume "?B"
  then obtain \<theta> 
    where dthread: "is_desc_thread \<theta> (contract s p)" ..

  with dthreads_join inc 
  obtain \<theta>s where ths_spec:
    "desc (\<lambda>i. is_fthread (\<theta>s i) p (s i) (s (Suc i))
                  \<and> \<theta>s i (s i) = \<theta> i 
                  \<and> \<theta>s i (s (Suc i)) = \<theta> (Suc i))
          (\<lambda>i. is_desc_fthread (\<theta>s i) p (s i) (s (Suc i))
                  \<and> \<theta>s i (s i) = \<theta> i 
                  \<and> \<theta>s i (s (Suc i)) = \<theta> (Suc i))" 
      (is "desc ?alw ?inf") 
    by blast

  then obtain n where fr: "\<forall>i\<ge>n. ?alw i" by blast
  hence connected: "\<And>i. n < i \<Longrightarrow> \<theta>s i (s (Suc i)) = \<theta>s (Suc i) (s (Suc i))"
    by auto
  
  let ?j\<theta> = "connect s \<theta>s"
  
  from fr ths_spec have ths_spec2:
      "\<And>i. i > n \<Longrightarrow> is_fthread (\<theta>s i) p (s i) (s (Suc i))"
      "\<exists>\<^sub>\<infinity>i. is_desc_fthread (\<theta>s i) p (s i) (s (Suc i))"
    by (auto intro:INF_mono)
  
  have p1: "\<And>i. i > n \<Longrightarrow> is_fthread ?j\<theta> p (s i) (s (Suc i))"
    by (rule connect_threads) (auto simp:connected ths_spec2)
  
  from ths_spec2(2)
  have "\<exists>\<^sub>\<infinity>i. n < i \<and> is_desc_fthread (\<theta>s i) p (s i) (s (Suc i))"
    unfolding INF_drop_prefix .
  
  hence p2: "\<exists>\<^sub>\<infinity>i. is_desc_fthread ?j\<theta> p (s i) (s (Suc i))"
    apply (rule INF_mono)
    apply (rule connect_dthreads)
    by (auto simp:connected)
  
  with `increasing s` p1
  have "is_desc_thread ?j\<theta> p" 
    by (rule mk_inf_desc_thread)
  with exI show "?A" .
qed


lemma repeated_edge:
  assumes "\<And>i. i > n \<Longrightarrow> dsc (snd (p i)) k k"
  shows "is_desc_thread (\<lambda>i. k) p"
proof-
  have th: "\<forall> m. \<exists>na>m. n < na" by arith
  show ?thesis using prems
  unfolding is_desc_thread_def 
  apply (auto)
  apply (rule_tac x="Suc n" in exI, auto)
  apply (rule INF_mono[where P="\<lambda>i. n < i"])
  apply (simp only:INF_nat)
  by (auto simp add: th)
qed

lemma fin_from_inf:
  assumes "is_thread n \<theta> p"
  assumes "n < i"
  assumes "i < j"
  shows "is_fthread \<theta> p i j"
  using prems
  unfolding is_thread_def is_fthread_def 
  by auto


subsection {* Ramsey's Theorem *}

definition 
  "set2pair S = (THE (x,y). x < y \<and> S = {x,y})"

lemma set2pair_conv: 
  fixes x y :: nat
  assumes "x < y"
  shows "set2pair {x, y} = (x, y)"
  unfolding set2pair_def
proof (rule the_equality, simp_all only:split_conv split_paired_all)
  from `x < y` show "x < y \<and> {x,y}={x,y}" by simp
next
  fix a b
  assume a: "a < b \<and> {x, y} = {a, b}"
  hence "{a, b} = {x, y}" by simp_all
  hence "(a, b) = (x, y) \<or> (a, b) = (y, x)" 
    by (cases "x = y") auto
  thus "(a, b) = (x, y)"
  proof 
    assume "(a, b) = (y, x)"
    with a and `x < y`
    show ?thesis by auto (* contradiction *)
  qed
qed  

definition 
  "set2list = inv set"

lemma finite_set2list: 
  assumes "finite S" 
  shows "set (set2list S) = S"
  unfolding set2list_def 
proof (rule f_inv_f)
  from `finite S` have "\<exists>l. set l = S"
    by (rule finite_list)
  thus "S \<in> range set"
    unfolding image_def
    by auto
qed


corollary RamseyNatpairs:
  fixes S :: "'a set"
    and f :: "nat \<times> nat \<Rightarrow> 'a"

  assumes "finite S"
  and inS: "\<And>x y. x < y \<Longrightarrow> f (x, y) \<in> S"

  obtains T :: "nat set" and s :: "'a"
  where "infinite T"
    and "s \<in> S"
    and "\<And>x y. \<lbrakk>x \<in> T; y \<in> T; x < y\<rbrakk> \<Longrightarrow> f (x, y) = s"
proof -
  from `finite S`
  have "set (set2list S) = S" by (rule finite_set2list)
  then 
  obtain l where S: "S = set l" by auto
  also from set_conv_nth have "\<dots> = {l ! i |i. i < length l}" .
  finally have "S = {l ! i |i. i < length l}" .

  let ?s = "length l"

  from inS 
  have index_less: "\<And>x y. x \<noteq> y \<Longrightarrow> index_of l (f (set2pair {x, y})) < ?s"
  proof -
    fix x y :: nat
    assume neq: "x \<noteq> y"
    have "f (set2pair {x, y}) \<in> S"
    proof (cases "x < y")
      case True hence "set2pair {x, y} = (x, y)"
	by (rule set2pair_conv)
      with True inS
      show ?thesis by simp
    next
      case False 
      with neq have y_less: "y < x" by simp
      have x:"{x,y} = {y,x}" by auto
      with y_less have "set2pair {x, y} = (y, x)"
	by (simp add:set2pair_conv)
      with y_less inS
      show ?thesis by simp
    qed

    thus "index_of l (f (set2pair {x, y})) < length l"
      by (simp add: S index_of_length)
  qed

  have "\<exists>Y. infinite Y \<and>
    (\<exists>t. t < ?s \<and>
         (\<forall>x\<in>Y. \<forall>y\<in>Y. x \<noteq> y \<longrightarrow>
                      index_of l (f (set2pair {x, y})) = t))"
    by (rule Ramsey2[of "UNIV::nat set", simplified])
       (auto simp:index_less)
  then obtain T i
    where inf: "infinite T"
    and i: "i < length l"
    and d: "\<And>x y. \<lbrakk>x \<in> T; y\<in>T; x \<noteq> y\<rbrakk>
    \<Longrightarrow> index_of l (f (set2pair {x, y})) = i"
    by auto

  have "l ! i \<in> S" unfolding S using i
    by (rule nth_mem)
  moreover
  have "\<And>x y. x \<in> T \<Longrightarrow> y\<in>T \<Longrightarrow> x < y
    \<Longrightarrow> f (x, y) = l ! i"
  proof -
    fix x y assume "x \<in> T" "y \<in> T" "x < y"
    with d have 
      "index_of l (f (set2pair {x, y})) = i" by auto
    with `x < y`
    have "i = index_of l (f (x, y))" 
      by (simp add:set2pair_conv)
    with `i < length l`
    show "f (x, y) = l ! i" 
      by (auto intro:index_of_member[symmetric] iff:index_of_length)
  qed
  moreover note inf
  ultimately
  show ?thesis using prems
    by blast
qed


subsection {* Main Result *}

theorem LJA_Theorem4: 
  assumes "bounded_acg P \<A>"
  shows "SCT \<A> \<longleftrightarrow> SCT' \<A>"
proof
  assume "SCT \<A>"
  
  show "SCT' \<A>"
  proof (rule classical)
    assume "\<not> SCT' \<A>"
    
    then obtain n G
      where in_closure: "(tcl \<A>) \<turnstile> n \<leadsto>\<^bsup>G\<^esup> n"
      and idemp: "G * G = G"
      and no_strict_arc: "\<forall>p. \<not>(G \<turnstile> p \<leadsto>\<^bsup>\<down>\<^esup> p)"
      unfolding SCT'_def no_bad_graphs_def by auto
    
    from in_closure obtain k
      where k_pow: "\<A> ^ k \<turnstile> n \<leadsto>\<^bsup>G\<^esup> n"
      and "0 < k"
      unfolding in_tcl by auto
	
    from power_induces_path k_pow
    obtain loop where loop_props:
      "has_fpath \<A> loop"
      "n = fst loop" "n = end_node loop"
      "G = prod loop" "k = length (snd loop)" . 

    with `0 < k` and path_loop_graph
    have "has_ipath \<A> (omega loop)" by blast
    with `SCT \<A>` 
    have thread: "\<exists>\<theta>. is_desc_thread \<theta> (omega loop)" by (auto simp:SCT_def)

    let ?s = "\<lambda>i. k * i"
    let ?cp = "\<lambda>i. (n, G)"

    from loop_props have "fst loop = end_node loop" by auto
    with `0 < k` `k = length (snd loop)`
    have "\<And>i. (omega loop)\<langle>?s i,?s (Suc i)\<rangle> = loop"
      by (rule sub_path_loop)

    with `n = fst loop` `G = prod loop` `k = length (snd loop)`
    have a: "contract ?s (omega loop) = ?cp"
      unfolding contract_def
      by (simp add:path_loop_def split_def fst_p0)

    from `0 < k` have "increasing ?s"
      by (auto simp:increasing_def)
    with thread have "\<exists>\<theta>. is_desc_thread \<theta> ?cp"
      unfolding a[symmetric] 
      by (unfold contract_keeps_threads[symmetric])

    then obtain \<theta> where desc: "is_desc_thread \<theta> ?cp" by auto

    then obtain n where thr: "is_thread n \<theta> ?cp" 
      unfolding is_desc_thread_def is_thread_def 
      by auto

    have "bounded_th P n \<theta>" 
    proof -
      from `bounded_acg P \<A>`
      have "bounded_acg P (tcl \<A>)" by (rule bounded_plus)
      with in_closure have "bounded_scg P G"
        unfolding bounded_acg_def by simp
      hence "bounded_mp P ?cp"
        unfolding bounded_mp_def by simp
      with thr bounded_th 
      show ?thesis by auto
    qed
    with bounded_th_infinite_visit
    obtain p where inf_visit: "\<exists>\<^sub>\<infinity>i. \<theta> i = p" by blast

    then obtain i where "n < i" "\<theta> i = p" 
      by (auto simp:INF_nat)

    from desc
    have "\<exists>\<^sub>\<infinity>i. descat ?cp \<theta> i"
      unfolding is_desc_thread_def by auto
    then obtain j 
      where "i < j" and "descat ?cp \<theta> j"
      unfolding INF_nat by auto
    from inf_visit obtain k where "j < k" "\<theta> k = p"
      by (auto simp:INF_nat)

    from `i < j` `j < k` `n < i` thr fin_from_inf `descat ?cp \<theta> j`
    have "is_desc_fthread \<theta> ?cp i k"
      unfolding is_desc_fthread_def
      by auto

    with `\<theta> k = p` `\<theta> i = p`
    have dfth: "has_desc_fth ?cp i k p p"
      unfolding has_desc_fth_def
      by auto

    from `i < j` `j < k` have "i < k" by auto
    hence "prod (?cp\<langle>i, k\<rangle>) = G"
    proof (induct i rule:strict_inc_induct)
      case base thus ?case by (simp add:sub_path_def)
    next
      case (step i) thus ?case
	by (simp add:sub_path_def upt_rec[of i k] idemp)
    qed

    with `i < j` `j < k` dfth Lemma7b
    have "dsc G p p" by auto
    with no_strict_arc have False by auto
    thus ?thesis ..
  qed
next
  assume "SCT' \<A>"

  show "SCT \<A>"
  proof (rule classical)
    assume "\<not> SCT \<A>"

    with SCT_def
    obtain p 
      where ipath: "has_ipath \<A> p"
      and no_desc_th: "\<not> (\<exists>\<theta>. is_desc_thread \<theta> p)"
      by auto

    from `bounded_acg P \<A>`
    have "finite (dest_graph (tcl \<A>))" (is "finite ?AG")
      by (intro bounded_finite bounded_plus)

    from pdesc_acgplus[OF ipath]
    have a: "\<And>x y. x<y \<Longrightarrow> pdesc p\<langle>x,y\<rangle> \<in> dest_graph (tcl \<A>)"
      unfolding has_edge_def .
      
    obtain S G
      where "infinite S" "G \<in> dest_graph (tcl \<A>)" 
      and all_G: "\<And>x y. \<lbrakk> x \<in> S; y \<in> S; x < y\<rbrakk> \<Longrightarrow> 
      pdesc (p\<langle>x,y\<rangle>) = G"
      apply (rule RamseyNatpairs[of ?AG "\<lambda>(x,y). pdesc p\<langle>x, y\<rangle>"])
      apply (rule `finite (dest_graph (tcl \<A>))`)
      by (simp only:split_conv, rule a, auto)

    obtain n H m where
      G_struct: "G = (n, H, m)" by (cases G)

    let ?s = "enumerate S"
    let ?q = "contract ?s p"

    note all_in_S[simp] = enumerate_in_set[OF `infinite S`]
	from `infinite S` 
    have inc[simp]: "increasing ?s" 
      unfolding increasing_def by (simp add:enumerate_mono)
    note increasing_bij[OF this, simp]
      
    from ipath_contract inc ipath
    have "has_ipath (tcl \<A>) ?q" .

    from all_G G_struct 
    have all_H: "\<And>i. (snd (?q i)) = H"
	  unfolding contract_def 
      by simp
    
    have loop: "(tcl \<A>) \<turnstile> n \<leadsto>\<^bsup>H\<^esup> n"
      and idemp: "H * H = H"
    proof - 
      let ?i = "?s 0" and ?j = "?s (Suc 0)" and ?k = "?s (Suc (Suc 0))"
      
      have "pdesc (p\<langle>?i,?j\<rangle>) = G"
		and "pdesc (p\<langle>?j,?k\<rangle>) = G"
		and "pdesc (p\<langle>?i,?k\<rangle>) = G"
		using all_G 
		by auto
	  
      with G_struct 
      have "m = end_node (p\<langle>?i,?j\<rangle>)"
				"n = fst (p\<langle>?j,?k\<rangle>)"
				and Hs:	"prod (p\<langle>?i,?j\<rangle>) = H"
				"prod (p\<langle>?j,?k\<rangle>) = H"
				"prod (p\<langle>?i,?k\<rangle>) = H"
		by auto
			
      hence "m = n" by simp
      thus "tcl \<A> \<turnstile> n \<leadsto>\<^bsup>H\<^esup> n"
		using G_struct `G \<in> dest_graph (tcl \<A>)`
		by (simp add:has_edge_def)
	  
      from sub_path_prod[of ?i ?j ?k p]      
      show "H * H = H"
		unfolding Hs by simp
    qed
    moreover have "\<And>k. \<not>dsc H k k"
    proof
      fix k :: nat assume "dsc H k k"
      
      with all_H repeated_edge
      have "\<exists>\<theta>. is_desc_thread \<theta> ?q" by fast
	  with inc have "\<exists>\<theta>. is_desc_thread \<theta> p"
		by (subst contract_keeps_threads) 
      with no_desc_th
      show False ..
    qed
    ultimately 
    have False
      using `SCT' \<A>`[unfolded SCT'_def no_bad_graphs_def]
      by blast
    thus ?thesis ..
  qed
qed


lemma LJA_apply:
  assumes fin: "finite_acg A"
  assumes "SCT' A"
  shows "SCT A"
proof -
  from fin obtain P where b: "bounded_acg P A"
    unfolding finite_acg_def ..
  show ?thesis using LJA_Theorem4[OF b] and `SCT' A`
    by simp
qed

end