
diff -r c00df7765656 -r 5c900a821a7c src/ZF/ex/Ramsey.thy
--- a/src/ZF/ex/Ramsey.thy	Sat Feb 02 13:26:51 2002 +0100
+++ b/src/ZF/ex/Ramsey.thy	Mon Feb 04 13:16:54 2002 +0100
@@ -9,38 +9,191 @@
     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
+    (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: kaufmann@cli.com
+
+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)
+
 *)
 
-Ramsey = Main +
-consts
-  Symmetric             :: i=>o
-  Atleast               :: [i,i]=>o
-  Clique,Indept,Ramsey  :: [i,i,i]=>o
+theory Ramsey = Main:
+constdefs
+  Symmetric :: "i=>o"
+    "Symmetric(E) == (\<forall>x y. <x,y>:E --> <y,x>:E)"
+
+  Atleast :: "[i,i]=>o"  (*not really necessary: ZF defines cardinality*)
+    "Atleast(n,S) == (\<exists>f. f \<in> inj(n,S))"
+
+  Clique  :: "[i,i,i]=>o"
+    "Clique(C,V,E) == (C \<subseteq> V) & (\<forall>x \<in> C. \<forall>y \<in> C. x\<noteq>y --> <x,y> \<in> E)"
+
+  Indept  :: "[i,i,i]=>o"
+    "Indept(I,V,E) == (I \<subseteq> V) & (\<forall>x \<in> I. \<forall>y \<in> I. x\<noteq>y --> <x,y> \<notin> E)"
+  
+  Ramsey  :: "[i,i,i]=>o"
+    "Ramsey(n,i,j) == \<forall>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)
 
-defs
+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) ==> \<exists>x \<in> A. Atleast(m, A-{x})"
+apply (unfold Atleast_def)
+apply (blast dest: inj_is_fun [THEN apply_type] inj_succ_restrict)
+done
 
-  Symmetric_def
-    "Symmetric(E) == (\\<forall>x y. <x,y>:E --> <y,x>:E)"
+lemma Atleast_superset: 
+    "[| Atleast(n,A);  A \<subseteq> B |] ==> Atleast(n,B)"
+by (unfold Atleast_def, blast intro: inj_weaken_type)
+
+lemma Atleast_succI: 
+    "[| Atleast(m,B);  b\<notin> 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 \<in> B |] ==> Atleast(succ(m), B)"
+by (blast intro: Atleast_succI [THEN Atleast_superset]) 
+
+(*** Main Cardinality Lemma ***)
 
-  Clique_def
-    "Clique(C,V,E) == (C \\<subseteq> V) & (\\<forall>x \\<in> C. \\<forall>y \\<in> C. x\\<noteq>y --> <x,y> \\<in> E)"
+(*The #-succ(0) strengthens the original theorem statement, but precisely
+  the same proof could be used!!*)
+lemma pigeon2 [rule_format]:
+     "m \<in> nat ==>  
+          \<forall>n \<in> nat. \<forall>A B. Atleast((m#+n) #- succ(0), A Un B) -->    
+                           Atleast(m,A) | Atleast(n,B)"
+apply (induct_tac "m")
+apply (blast intro!: Atleast0)
+apply (simp)
+apply (rule ballI)
+apply (rename_tac m' n) (*simplifier does NOT preserve bound names!*)
+apply (induct_tac "n")
+apply 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
 
-  Indept_def
-    "Indept(I,V,E) == (I \\<subseteq> V) & (\\<forall>x \\<in> I. \\<forall>y \\<in> I. x\\<noteq>y --> <x,y> \\<notin> E)"
+
+(**** 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 **)
 
-  Atleast_def
-    "Atleast(n,S) == (\\<exists>f. f \\<in> inj(n,S))"
+(*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 \<in> nat;  n \<in> nat |]   
+      ==> Atleast(succ(m), {x \<in> A. ~P(x)}) | Atleast(n, {x \<in> A. P(x)})"
+apply (rule nat_succI [THEN pigeon2])
+apply assumption+
+apply (rule Atleast_superset)
+apply auto
+done
+
+(*For the Atleast part, proves ~(a \<in> I) from the second premise!*)
+lemma Indept_succ: 
+    "[| Indept(I, {z \<in> V-{a}. <a,z> \<notin> E}, E);  Symmetric(E);  a \<in> 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 \<in> V-{a}. <a,z>:E}, E);  Symmetric(E);  a \<in> 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 **)
 
-  Ramsey_def
-    "Ramsey(n,i,j) == \\<forall>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))"
+(*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)
+apply clarify
+apply (erule Atleast_succD [THEN bexE])
+apply (erule_tac P1 = "%z.<x,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 \<in> nat ==> \<forall>j \<in> nat. \<exists>n \<in> 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 \<in> nat;  j \<in> nat |] ==> \<exists>n \<in> nat. Ramsey(n,i,j)"
+by (blast dest: ramsey_lemma)
 
 end




