(* 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 1);
qed "nat_bnd_mono";
(* 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 (stac nat_unfold 1);
by (rtac (singletonI RS UnI1) 1);
qed "nat_0I";
val prems = goal Nat.thy "n : nat ==> succ(n) : nat";
by (stac nat_unfold 1);
by (rtac (RepFunI RS UnI2) 1);
by (resolve_tac prems 1);
qed "nat_succI";
goal Nat.thy "1 : nat";
by (rtac (nat_0I RS nat_succI) 1);
qed "nat_1I";
goal Nat.thy "2 : nat";
by (rtac (nat_1I RS nat_succI) 1);
qed "nat_2I";
Addsimps [nat_0I, nat_1I, nat_2I, nat_succI];
AddSIs [nat_0I, nat_1I, nat_2I];
goal Nat.thy "bool <= nat";
by (REPEAT (ares_tac [subsetI,nat_0I,nat_1I] 1
ORELSE eresolve_tac [boolE,ssubst] 1));
qed "bool_subset_nat";
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 (!claset addIs prems) 1);
qed "nat_induct";
(*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";
by (rtac (major RS (nat_unfold RS equalityD1 RS subsetD) RS UnE) 1);
by (DEPTH_SOLVE (eresolve_tac [singletonE,RepFunE] 1
ORELSE ares_tac prems 1));
qed "natE";
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));
qed "nat_into_Ord";
(* i: nat ==> 0 le i; same thing as 0<succ(i) *)
bind_thm ("nat_0_le", nat_into_Ord RS Ord_0_le);
(* i: nat ==> i le i; same thing as i<succ(i) *)
bind_thm ("nat_le_refl", nat_into_Ord RS le_refl);
goal Nat.thy "Ord(nat)";
by (rtac OrdI 1);
by (etac (nat_into_Ord 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));
qed "Ord_nat";
goalw Nat.thy [Limit_def] "Limit(nat)";
by (safe_tac (!claset addSIs [ltI, nat_succI, Ord_nat]));
by (etac ltD 1);
qed "Limit_nat";
Addsimps [Ord_nat, Limit_nat];
AddSIs [Ord_nat, Limit_nat];
goal Nat.thy "!!i. Limit(i) ==> nat le i";
by (rtac subset_imp_le 1);
by (rtac subsetI 1);
by (etac nat_induct 1);
by (blast_tac (!claset addIs [Limit_has_succ RS ltD, ltI, Limit_is_Ord]) 2);
by (REPEAT (ares_tac [Limit_has_0 RS ltD,
Ord_nat, Limit_is_Ord] 1));
qed "nat_le_Limit";
(* succ(i): nat ==> i: nat *)
val succ_natD = [succI1, asm_rl, Ord_nat] MRS Ord_trans;
(* [| succ(i): k; k: nat |] ==> i: k *)
val succ_in_naturalD = [succI1, asm_rl, nat_into_Ord] MRS Ord_trans;
goal Nat.thy "!!m n. [| m<n; n: nat |] ==> m: nat";
by (etac ltE 1);
by (etac (Ord_nat RSN (3,Ord_trans)) 1);
by (assume_tac 1);
qed "lt_nat_in_nat";
(** 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 le x; P(x) |] ==> P(succ(x)) \
\ |] ==> m le n --> P(m) --> P(n)";
by (nat_ind_tac "n" prems 1);
by (ALLGOALS
(asm_simp_tac
(!simpset addsimps (prems@distrib_simps@[le0_iff, le_succ_iff]))));
qed "nat_induct_from_lemma";
(*Induction starting from m rather than 0*)
val prems = goal Nat.thy
"[| m le n; m: nat; n: nat; \
\ P(m); \
\ !!x. [| x: nat; m le 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));
qed "nat_induct_from";
(*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));
qed "diff_induct";
(** Induction principle analogous to trancl_induct **)
goal Nat.thy
"!!m. m: nat ==> P(m,succ(m)) --> (ALL x: nat. P(m,x) --> P(m,succ(x))) --> \
\ (ALL n:nat. m<n --> P(m,n))";
by (etac nat_induct 1);
by (ALLGOALS
(EVERY' [rtac (impI RS impI), rtac (nat_induct RS ballI), assume_tac,
fast_tac le_cs, fast_tac le_cs]));
qed "succ_lt_induct_lemma";
val prems = goal Nat.thy
"[| m<n; n: nat; \
\ P(m,succ(m)); \
\ !!x. [| x: nat; P(m,x) |] ==> P(m,succ(x)) \
\ |] ==> P(m,n)";
by (res_inst_tac [("P4","P")]
(succ_lt_induct_lemma RS mp RS mp RS bspec RS mp) 1);
by (REPEAT (ares_tac (prems @ [ballI, impI, lt_nat_in_nat]) 1));
qed "succ_lt_induct";
(** nat_case **)
goalw Nat.thy [nat_case_def] "nat_case(a,b,0) = a";
by (blast_tac (!claset addIs [the_equality]) 1);
qed "nat_case_0";
goalw Nat.thy [nat_case_def] "nat_case(a,b,succ(m)) = b(m)";
by (blast_tac (!claset addIs [the_equality]) 1);
qed "nat_case_succ";
Addsimps [nat_case_0, nat_case_succ];
val major::prems = goal Nat.thy
"[| n: nat; a: C(0); !!m. m: nat ==> b(m): C(succ(m)) \
\ |] ==> nat_case(a,b,n) : C(n)";
by (rtac (major RS nat_induct) 1);
by (ALLGOALS (asm_simp_tac (!simpset addsimps prems)));
qed "nat_case_type";
(** nat_rec -- used to define eclose and transrec, then obsolete;
rec, from arith.ML, has fewer typing conditions **)
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);
qed "nat_rec_0";
val [prem] = goal Nat.thy
"m: nat ==> nat_rec(succ(m),a,b) = b(m, nat_rec(m,a,b))";
by (rtac nat_rec_trans 1);
by (simp_tac (!simpset addsimps [prem, Memrel_iff, vimage_singleton_iff]) 1);
qed "nat_rec_succ";
(** The union of two natural numbers is a natural number -- their maximum **)
goal Nat.thy "!!i j. [| i: nat; j: nat |] ==> i Un j: nat";
by (rtac (Un_least_lt RS ltD) 1);
by (REPEAT (ares_tac [ltI, Ord_nat] 1));
qed "Un_nat_type";
goal Nat.thy "!!i j. [| i: nat; j: nat |] ==> i Int j: nat";
by (rtac (Int_greatest_lt RS ltD) 1);
by (REPEAT (ares_tac [ltI, Ord_nat] 1));
qed "Int_nat_type";