--- a/src/HOL/NatArith.ML Tue Jul 24 11:25:54 2001 +0200
+++ b/src/HOL/NatArith.ML Wed Jul 25 13:13:01 2001 +0200
@@ -135,54 +135,5 @@
-(** GREATEST theorems for type "nat": not dual to LEAST, since there is
- no greatest natural number. [The LEAST proofs are in Nat.ML, but the
- GREATEST proofs require more infrastructure.] **)
-
-Goal "[|P k; ALL x. P x --> (EX y. P y & ~ ((m y::nat) <= m x))|] \
-\ ==> EX y. P y & ~ (m y < m k + n)";
-by (induct_tac "n" 1);
-by (Force_tac 1);
-(*ind step*)
-by (force_tac (claset(), simpset() addsimps [le_Suc_eq]) 1);
-qed "ex_has_greatest_nat_lemma";
-
-Goal "[|P k; ! y. P y --> m y < b|] \
-\ ==> ? x. P x & (! y. P y --> (m y::nat) <= m x)";
-by (rtac ccontr 1);
-by (cut_inst_tac [("P","P"),("n","b - m k")] ex_has_greatest_nat_lemma 1);
-by (subgoal_tac "m k <= b" 3);
-by Auto_tac;
-qed "ex_has_greatest_nat";
-
-Goalw [GreatestM_def]
- "[|P k; ! y. P y --> m y < b|] \
-\ ==> P (GreatestM m P) & (!y. P y --> (m y::nat) <= m (GreatestM m P))";
-by (rtac someI_ex 1);
-by (etac ex_has_greatest_nat 1);
-by (assume_tac 1);
-qed "GreatestM_nat_lemma";
-
-bind_thm ("GreatestM_natI", GreatestM_nat_lemma RS conjunct1);
-
-Goal "[|P x; ! y. P y --> m y < b|] \
-\ ==> (m x::nat) <= m (GreatestM m P)";
-by (blast_tac (claset() addDs [GreatestM_nat_lemma RS conjunct2 RS spec]) 1);
-qed "GreatestM_nat_le";
-
-(** Specialization to GREATEST **)
-
-Goalw [Greatest_def]
- "[|P (k::nat); ! y. P y --> y < b|] ==> P (GREATEST x. P x)";
-by (rtac GreatestM_natI 1);
-by Auto_tac;
-qed "GreatestI";
-
-Goalw [Greatest_def]
- "[|P x; ! y. P y --> y < b|] ==> (x::nat) <= (GREATEST x. P x)";
-by (rtac GreatestM_nat_le 1);
-by Auto_tac;
-qed "Greatest_le";
-
(*No analogue of not_less_Least or Least_Suc yet, since it isn't used much*)