diff -r 000000000000 -r a5a9c433f639 src/ZF/nat.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/ZF/nat.ML	Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,182 @@
+(*  Title: 	ZF/nat.ML
+    ID:         $Id$
+    Author: 	Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1992  University of Cambridge
+
+For nat.thy.  Natural numbers in Zermelo-Fraenkel Set Theory 
+*)
+
+open Nat;
+
+goal Nat.thy "bnd_mono(Inf, %X. {0} Un {succ(i). i:X})";
+by (rtac bnd_monoI 1);
+by (REPEAT (ares_tac [subset_refl, RepFun_mono, Un_mono] 2)); 
+by (cut_facts_tac [infinity] 1);
+by (fast_tac ZF_cs 1);
+val nat_bnd_mono = result();
+
+(* nat = {0} Un {succ(x). x:nat} *)
+val nat_unfold = nat_bnd_mono RS (nat_def RS def_lfp_Tarski);
+
+(** Type checking of 0 and successor **)
+
+goal Nat.thy "0 : nat";
+by (rtac (nat_unfold RS ssubst) 1);
+by (rtac (singletonI RS UnI1) 1);
+val nat_0I = result();
+
+val prems = goal Nat.thy "n : nat ==> succ(n) : nat";
+by (rtac (nat_unfold RS ssubst) 1);
+by (rtac (RepFunI RS UnI2) 1);
+by (resolve_tac prems 1);
+val nat_succI = result();
+
+goalw Nat.thy [one_def] "1 : nat";
+by (rtac (nat_0I RS nat_succI) 1);
+val nat_1I = result();
+
+goal Nat.thy "bool <= nat";
+by (REPEAT (ares_tac [subsetI,nat_0I,nat_1I] 1 ORELSE etac boolE 1));
+val bool_subset_nat = result();
+
+val bool_into_nat = bool_subset_nat RS subsetD;
+
+
+(** Injectivity properties and induction **)
+
+(*Mathematical induction*)
+val major::prems = goal Nat.thy
+    "[| n: nat;  P(0);  !!x. [| x: nat;  P(x) |] ==> P(succ(x)) |] ==> P(n)";
+by (rtac ([nat_def, nat_bnd_mono, major] MRS def_induct) 1);
+by (fast_tac (ZF_cs addIs prems) 1);
+val nat_induct = result();
+
+(*Perform induction on n, then prove the n:nat subgoal using prems. *)
+fun nat_ind_tac a prems i = 
+    EVERY [res_inst_tac [("n",a)] nat_induct i,
+	   rename_last_tac a ["1"] (i+2),
+	   ares_tac prems i];
+
+val major::prems = goal Nat.thy
+    "[| n: nat;  n=0 ==> P;  !!x. [| x: nat; n=succ(x) |] ==> P |] ==> P";
+br (major RS (nat_unfold RS equalityD1 RS subsetD) RS UnE) 1;
+by (DEPTH_SOLVE (eresolve_tac [singletonE,RepFunE] 1
+          ORELSE ares_tac prems 1));
+val natE = result();
+
+val prems = goal Nat.thy "n: nat ==> Ord(n)";
+by (nat_ind_tac "n" prems 1);
+by (REPEAT (ares_tac [Ord_0, Ord_succ] 1));
+val naturals_are_ordinals = result();
+
+goal Nat.thy "!!n. n: nat ==> n=0 | 0:n";
+by (etac nat_induct 1);
+by (fast_tac ZF_cs 1);
+by (fast_tac (ZF_cs addIs [naturals_are_ordinals RS Ord_0_mem_succ]) 1);
+val natE0 = result();
+
+goal Nat.thy "Ord(nat)";
+by (rtac OrdI 1);
+by (etac (naturals_are_ordinals RS Ord_is_Transset) 2);
+by (rewtac Transset_def);
+by (rtac ballI 1);
+by (etac nat_induct 1);
+by (REPEAT (ares_tac [empty_subsetI,succ_subsetI] 1));
+val Ord_nat = result();
+
+(** Variations on mathematical induction **)
+
+(*complete induction*)
+val complete_induct = Ord_nat RSN (2, Ord_induct);
+
+val prems = goal Nat.thy
+    "[| m: nat;  n: nat;  \
+\       !!x. [| x: nat;  m<=x;  P(x) |] ==> P(succ(x)) \
+\    |] ==> m <= n --> P(m) --> P(n)";
+by (nat_ind_tac "n" prems 1);
+by (ALLGOALS
+    (ASM_SIMP_TAC
+     (ZF_ss addrews (prems@distrib_rews@[subset_empty_iff, subset_succ_iff, 
+					 Ord_nat RS Ord_in_Ord]))));
+val nat_induct_from_lemma = result();
+
+(*Induction starting from m rather than 0*)
+val prems = goal Nat.thy
+    "[| m <= n;  m: nat;  n: nat;  \
+\       P(m);  \
+\       !!x. [| x: nat;  m<=x;  P(x) |] ==> P(succ(x)) \
+\    |] ==> P(n)";
+by (rtac (nat_induct_from_lemma RS mp RS mp) 1);
+by (REPEAT (ares_tac prems 1));
+val nat_induct_from = result();
+
+(*Induction suitable for subtraction and less-than*)
+val prems = goal Nat.thy
+    "[| m: nat;  n: nat;  \
+\       !!x. [| x: nat |] ==> P(x,0);  \
+\       !!y. [| y: nat |] ==> P(0,succ(y));  \
+\       !!x y. [| x: nat;  y: nat;  P(x,y) |] ==> P(succ(x),succ(y))  \
+\    |] ==> P(m,n)";
+by (res_inst_tac [("x","m")] bspec 1);
+by (resolve_tac prems 2);
+by (nat_ind_tac "n" prems 1);
+by (rtac ballI 2);
+by (nat_ind_tac "x" [] 2);
+by (REPEAT (ares_tac (prems@[ballI]) 1 ORELSE etac bspec 1));
+val diff_induct = result();
+
+(** nat_case **)
+
+goalw Nat.thy [nat_case_def] "nat_case(0,a,b) = a";
+by (fast_tac (ZF_cs addIs [the_equality]) 1);
+val nat_case_0 = result();
+
+goalw Nat.thy [nat_case_def] "nat_case(succ(m),a,b) = b(m)";
+by (fast_tac (ZF_cs addIs [the_equality]) 1);
+val nat_case_succ = result();
+
+val major::prems = goal Nat.thy
+    "[| n: nat;  a: C(0);  !!m. m: nat ==> b(m): C(succ(m))  \
+\    |] ==> nat_case(n,a,b) : C(n)";
+by (rtac (major RS nat_induct) 1);
+by (REPEAT (resolve_tac [nat_case_0 RS ssubst,
+			 nat_case_succ RS ssubst] 1 
+       THEN resolve_tac prems 1));
+by (assume_tac 1);
+val nat_case_type = result();
+
+val prems = goalw Nat.thy [nat_case_def]
+    "[| n=n';  a=a';  !!m z. b(m)=b'(m)  \
+\    |] ==> nat_case(n,a,b)=nat_case(n',a',b')";
+by (REPEAT (resolve_tac [the_cong,disj_cong,ex_cong] 1
+     ORELSE EVERY1 (map rtac ((prems RL [ssubst]) @ [iff_refl]))));
+val nat_case_cong = result();
+
+
+(** nat_rec -- used to define eclose and transrec, then obsolete **)
+
+val nat_rec_trans = wf_Memrel RS (nat_rec_def RS def_wfrec RS trans);
+
+goal Nat.thy "nat_rec(0,a,b) = a";
+by (rtac nat_rec_trans 1);
+by (rtac nat_case_0 1);
+val nat_rec_0 = result();
+
+val [prem] = goal Nat.thy 
+    "m: nat ==> nat_rec(succ(m),a,b) = b(m, nat_rec(m,a,b))";
+val nat_rec_ss = ZF_ss 
+      addcongs (mk_typed_congs Nat.thy [("b", "[i,i]=>i")])
+      addrews [prem, nat_case_succ, nat_succI, Memrel_iff, 
+	       vimage_singleton_iff];
+by (rtac nat_rec_trans 1);
+by (SIMP_TAC nat_rec_ss 1);
+val nat_rec_succ = result();
+
+(** The union of two natural numbers is a natural number -- their maximum **)
+
+(*  [| ?i : nat; ?j : nat |] ==> ?i Un ?j : nat  *)
+val Un_nat_type = standard (Ord_nat RSN (3,Ord_member_UnI));
+
+(*  [| ?i : nat; ?j : nat |] ==> ?i Int ?j : nat  *)
+val Int_nat_type = standard (Ord_nat RSN (3,Ord_member_IntI));
+