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