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(* Title: ZF/ex/ramsey.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1992 University of Cambridge
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Ramsey's Theorem (finite exponent 2 version)
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Based upon the article
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D Basin and M Kaufmann,
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The Boyer-Moore Prover and Nuprl: An Experimental Comparison.
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In G Huet and G Plotkin, editors, Logical Frameworks.
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(CUP, 1991), pages 89-119
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See also
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M Kaufmann,
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An example in NQTHM: Ramsey's Theorem
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Internal Note, Computational Logic, Inc., Austin, Texas 78703
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Available from the author: kaufmann@cli.com
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This function compute Ramsey numbers according to the proof given below
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(which, does not constrain the base case values at all.
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fun ram 0 j = 1
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| ram i 0 = 1
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| ram i j = ram (i-1) j + ram i (j-1)
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*)
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theory Ramsey = Main:
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constdefs
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Symmetric :: "i=>o"
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"Symmetric(E) == (\<forall>x y. <x,y>:E --> <y,x>:E)"
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Atleast :: "[i,i]=>o" (*not really necessary: ZF defines cardinality*)
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"Atleast(n,S) == (\<exists>f. f \<in> inj(n,S))"
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Clique :: "[i,i,i]=>o"
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"Clique(C,V,E) == (C \<subseteq> V) & (\<forall>x \<in> C. \<forall>y \<in> C. x\<noteq>y --> <x,y> \<in> E)"
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Indept :: "[i,i,i]=>o"
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"Indept(I,V,E) == (I \<subseteq> V) & (\<forall>x \<in> I. \<forall>y \<in> I. x\<noteq>y --> <x,y> \<notin> E)"
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Ramsey :: "[i,i,i]=>o"
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"Ramsey(n,i,j) == \<forall>V E. Symmetric(E) & Atleast(n,V) -->
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(\<exists>C. Clique(C,V,E) & Atleast(i,C)) |
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(\<exists>I. Indept(I,V,E) & Atleast(j,I))"
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(*** Cliques and Independent sets ***)
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lemma Clique0 [intro]: "Clique(0,V,E)"
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by (unfold Clique_def, blast)
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lemma Clique_superset: "[| Clique(C,V',E); V'<=V |] ==> Clique(C,V,E)"
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by (unfold Clique_def, blast)
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lemma Indept0 [intro]: "Indept(0,V,E)"
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by (unfold Indept_def, blast)
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lemma Indept_superset: "[| Indept(I,V',E); V'<=V |] ==> Indept(I,V,E)"
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by (unfold Indept_def, blast)
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(*** Atleast ***)
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lemma Atleast0 [intro]: "Atleast(0,A)"
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by (unfold Atleast_def inj_def Pi_def function_def, blast)
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lemma Atleast_succD:
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"Atleast(succ(m),A) ==> \<exists>x \<in> A. Atleast(m, A-{x})"
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apply (unfold Atleast_def)
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apply (blast dest: inj_is_fun [THEN apply_type] inj_succ_restrict)
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done
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lemma Atleast_superset:
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"[| Atleast(n,A); A \<subseteq> B |] ==> Atleast(n,B)"
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by (unfold Atleast_def, blast intro: inj_weaken_type)
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lemma Atleast_succI:
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"[| Atleast(m,B); b\<notin> B |] ==> Atleast(succ(m), cons(b,B))"
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apply (unfold Atleast_def succ_def)
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apply (blast intro: inj_extend elim: mem_irrefl)
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done
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lemma Atleast_Diff_succI:
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"[| Atleast(m, B-{x}); x \<in> B |] ==> Atleast(succ(m), B)"
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by (blast intro: Atleast_succI [THEN Atleast_superset])
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(*** Main Cardinality Lemma ***)
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(*The #-succ(0) strengthens the original theorem statement, but precisely
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the same proof could be used!!*)
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lemma pigeon2 [rule_format]:
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"m \<in> nat ==>
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\<forall>n \<in> nat. \<forall>A B. Atleast((m#+n) #- succ(0), A Un B) -->
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Atleast(m,A) | Atleast(n,B)"
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apply (induct_tac "m")
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apply (blast intro!: Atleast0)
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apply (simp)
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apply (rule ballI)
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apply (rename_tac m' n) (*simplifier does NOT preserve bound names!*)
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apply (induct_tac "n")
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apply auto
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apply (erule Atleast_succD [THEN bexE])
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apply (rename_tac n' A B z)
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apply (erule UnE)
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(**case z \<in> B. Instantiate the '\<forall>A B' induction hypothesis. **)
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apply (drule_tac [2] x1 = "A" and x = "B-{z}" in spec [THEN spec])
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apply (erule_tac [2] mp [THEN disjE])
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(*cases Atleast(succ(m1),A) and Atleast(succ(k),B)*)
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apply (erule_tac [3] asm_rl notE Atleast_Diff_succI)+
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(*proving the condition*)
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prefer 2 apply (blast intro: Atleast_superset)
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(**case z \<in> A. Instantiate the '\<forall>n \<in> nat. \<forall>A B' induction hypothesis. **)
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apply (drule_tac x2="succ(n')" and x1="A-{z}" and x="B"
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in bspec [THEN spec, THEN spec])
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apply (erule nat_succI)
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apply (erule mp [THEN disjE])
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(*cases Atleast(succ(m1),A) and Atleast(succ(k),B)*)
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apply (erule_tac [2] asm_rl Atleast_Diff_succI notE)+;
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(*proving the condition*)
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apply simp
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apply (blast intro: Atleast_superset)
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done
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(**** Ramsey's Theorem ****)
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(** Base cases of induction; they now admit ANY Ramsey number **)
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lemma Ramsey0j: "Ramsey(n,0,j)"
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by (unfold Ramsey_def, blast)
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lemma Ramseyi0: "Ramsey(n,i,0)"
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by (unfold Ramsey_def, blast)
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(** Lemmas for induction step **)
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(*The use of succ(m) here, rather than #-succ(0), simplifies the proof of
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Ramsey_step_lemma.*)
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lemma Atleast_partition: "[| Atleast(m #+ n, A); m \<in> nat; n \<in> nat |]
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==> Atleast(succ(m), {x \<in> A. ~P(x)}) | Atleast(n, {x \<in> A. P(x)})"
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apply (rule nat_succI [THEN pigeon2])
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apply assumption+
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apply (rule Atleast_superset)
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apply auto
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done
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(*For the Atleast part, proves ~(a \<in> I) from the second premise!*)
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lemma Indept_succ:
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"[| Indept(I, {z \<in> V-{a}. <a,z> \<notin> E}, E); Symmetric(E); a \<in> V;
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Atleast(j,I) |] ==>
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Indept(cons(a,I), V, E) & Atleast(succ(j), cons(a,I))"
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apply (unfold Symmetric_def Indept_def)
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apply (blast intro!: Atleast_succI)
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done
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lemma Clique_succ:
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"[| Clique(C, {z \<in> V-{a}. <a,z>:E}, E); Symmetric(E); a \<in> V;
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Atleast(j,C) |] ==>
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Clique(cons(a,C), V, E) & Atleast(succ(j), cons(a,C))"
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apply (unfold Symmetric_def Clique_def)
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apply (blast intro!: Atleast_succI)
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done
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(** Induction step **)
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(*Published proofs gloss over the need for Ramsey numbers to be POSITIVE.*)
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lemma Ramsey_step_lemma:
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"[| Ramsey(succ(m), succ(i), j); Ramsey(n, i, succ(j));
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m \<in> nat; n \<in> nat |] ==> Ramsey(succ(m#+n), succ(i), succ(j))"
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apply (unfold Ramsey_def)
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apply clarify
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apply (erule Atleast_succD [THEN bexE])
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apply (erule_tac P1 = "%z.<x,z>:E" in Atleast_partition [THEN disjE],
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assumption+)
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(*case m*)
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apply (fast dest!: Indept_succ elim: Clique_superset)
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(*case n*)
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apply (fast dest!: Clique_succ elim: Indept_superset)
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done
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(** The actual proof **)
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(*Again, the induction requires Ramsey numbers to be positive.*)
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lemma ramsey_lemma: "i \<in> nat ==> \<forall>j \<in> nat. \<exists>n \<in> nat. Ramsey(succ(n), i, j)"
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apply (induct_tac "i")
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apply (blast intro!: Ramsey0j)
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apply (rule ballI)
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apply (induct_tac "j")
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apply (blast intro!: Ramseyi0)
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apply (blast intro!: add_type Ramsey_step_lemma)
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done
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(*Final statement in a tidy form, without succ(...) *)
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lemma ramsey: "[| i \<in> nat; j \<in> nat |] ==> \<exists>n \<in> nat. Ramsey(n,i,j)"
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by (blast dest: ramsey_lemma)
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end
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