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(*  Title: 	HOL/nat
    ID:         $Id$
    Author: 	Tobias Nipkow, Cambridge University Computer Laboratory
    Copyright   1991  University of Cambridge

For nat.thy.  Type nat is defined as a set (Nat) over the type ind.
*)

open Nat;

goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
val Nat_fun_mono = result();

val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);

(* Zero is a natural number -- this also justifies the type definition*)
goal Nat.thy "Zero_Rep: Nat";
by (rtac (Nat_unfold RS ssubst) 1);
by (rtac (singletonI RS UnI1) 1);
val Zero_RepI = result();

val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat";
by (rtac (Nat_unfold RS ssubst) 1);
by (rtac (imageI RS UnI2) 1);
by (resolve_tac prems 1);
val Suc_RepI = result();

(*** Induction ***)

val major::prems = goal Nat.thy
    "[| i: Nat;  P(Zero_Rep);   \
\       !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |]  ==> P(i)";
by (rtac (major RS (Nat_def RS def_induct)) 1);
by (rtac Nat_fun_mono 1);
by (fast_tac (set_cs addIs prems) 1);
val Nat_induct = result();

val prems = goalw Nat.thy [Zero_def,Suc_def]
    "[| P(0);   \
\       !!k. P(k) ==> P(Suc(k)) |]  ==> P(n)";
by (rtac (Rep_Nat_inverse RS subst) 1);   (*types force good instantiation*)
by (rtac (Rep_Nat RS Nat_induct) 1);
by (REPEAT (ares_tac prems 1
     ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
val nat_induct = result();

(*Perform induction on n. *)
fun nat_ind_tac a i = 
    EVERY [res_inst_tac [("n",a)] nat_induct i,
	   rename_last_tac a ["1"] (i+1)];

(*A special form of induction for reasoning about m<n and m-n*)
val prems = goal Nat.thy
    "[| !!x. P(x,0);  \
\       !!y. P(0,Suc(y));  \
\       !!x y. [| P(x,y) |] ==> P(Suc(x),Suc(y))  \
\    |] ==> P(m,n)";
by (res_inst_tac [("x","m")] spec 1);
by (nat_ind_tac "n" 1);
by (rtac allI 2);
by (nat_ind_tac "x" 2);
by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
val diff_induct = result();

(*Case analysis on the natural numbers*)
val prems = goal Nat.thy 
    "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
by (fast_tac (HOL_cs addSEs prems) 1);
by (nat_ind_tac "n" 1);
by (rtac (refl RS disjI1) 1);
by (fast_tac HOL_cs 1);
val natE = result();

(*** Isomorphisms: Abs_Nat and Rep_Nat ***)

(*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
  since we assume the isomorphism equations will one day be given by Isabelle*)

goal Nat.thy "inj(Rep_Nat)";
by (rtac inj_inverseI 1);
by (rtac Rep_Nat_inverse 1);
val inj_Rep_Nat = result();

goal Nat.thy "inj_onto(Abs_Nat,Nat)";
by (rtac inj_onto_inverseI 1);
by (etac Abs_Nat_inverse 1);
val inj_onto_Abs_Nat = result();

(*** Distinctness of constructors ***)

goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0";
by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
by (rtac Suc_Rep_not_Zero_Rep 1);
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
val Suc_not_Zero = result();

val Zero_not_Suc = standard (Suc_not_Zero RS not_sym);

val Suc_neq_Zero = standard (Suc_not_Zero RS notE);
val Zero_neq_Suc = sym RS Suc_neq_Zero;

(** Injectiveness of Suc **)

goalw Nat.thy [Suc_def] "inj(Suc)";
by (rtac injI 1);
by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
by (dtac (inj_Suc_Rep RS injD) 1);
by (etac (inj_Rep_Nat RS injD) 1);
val inj_Suc = result();

val Suc_inject = inj_Suc RS injD;;

goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)";
by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); 
val Suc_Suc_eq = result();

goal Nat.thy "n ~= Suc(n)";
by (nat_ind_tac "n" 1);
by (ALLGOALS(asm_simp_tac (HOL_ss addsimps [Zero_not_Suc,Suc_Suc_eq])));
val n_not_Suc_n = result();

val Suc_n_not_n = n_not_Suc_n RS not_sym;

(*** nat_case -- the selection operator for nat ***)

goalw Nat.thy [nat_case_def] "nat_case(0, a, f) = a";
by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
val nat_case_0 = result();

goalw Nat.thy [nat_case_def] "nat_case(Suc(k), a, f) = f(k)";
by (fast_tac (set_cs addIs [select_equality] 
	               addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
val nat_case_Suc = result();

(** Introduction rules for 'pred_nat' **)

goalw Nat.thy [pred_nat_def] "<n, Suc(n)> : pred_nat";
by (fast_tac set_cs 1);
val pred_natI = result();

val major::prems = goalw Nat.thy [pred_nat_def]
    "[| p : pred_nat;  !!x n. [| p = <n, Suc(n)> |] ==> R \
\    |] ==> R";
by (rtac (major RS CollectE) 1);
by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1));
val pred_natE = result();

goalw Nat.thy [wf_def] "wf(pred_nat)";
by (strip_tac 1);
by (nat_ind_tac "x" 1);
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, 
			     make_elim Suc_inject]) 2);
by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);
val wf_pred_nat = result();


(*** nat_rec -- by wf recursion on pred_nat ***)

val nat_rec_unfold = standard (wf_pred_nat RS (nat_rec_def RS def_wfrec));

(** conversion rules **)

goal Nat.thy "nat_rec(0,c,h) = c";
by (rtac (nat_rec_unfold RS trans) 1);
by (rtac nat_case_0 1);
val nat_rec_0 = result();

goal Nat.thy "nat_rec(Suc(n), c, h) = h(n, nat_rec(n,c,h))";
by (rtac (nat_rec_unfold RS trans) 1);
by (rtac (nat_case_Suc RS trans) 1);
by(simp_tac (HOL_ss addsimps [pred_natI,cut_apply]) 1);
val nat_rec_Suc = result();

(*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
val [rew] = goal Nat.thy
    "[| !!n. f(n) == nat_rec(n,c,h) |] ==> f(0) = c";
by (rewtac rew);
by (rtac nat_rec_0 1);
val def_nat_rec_0 = result();

val [rew] = goal Nat.thy
    "[| !!n. f(n) == nat_rec(n,c,h) |] ==> f(Suc(n)) = h(n,f(n))";
by (rewtac rew);
by (rtac nat_rec_Suc 1);
val def_nat_rec_Suc = result();

fun nat_recs def =
      [standard (def RS def_nat_rec_0),
       standard (def RS def_nat_rec_Suc)];


(*** Basic properties of "less than" ***)

(** Introduction properties **)

val prems = goalw Nat.thy [less_def] "[| i<j;  j<k |] ==> i<k::nat";
by (rtac (trans_trancl RS transD) 1);
by (resolve_tac prems 1);
by (resolve_tac prems 1);
val less_trans = result();

goalw Nat.thy [less_def] "n < Suc(n)";
by (rtac (pred_natI RS r_into_trancl) 1);
val lessI = result();

(* i<j ==> i<Suc(j) *)
val less_SucI = lessI RSN (2, less_trans);

goal Nat.thy "0 < Suc(n)";
by (nat_ind_tac "n" 1);
by (rtac lessI 1);
by (etac less_trans 1);
by (rtac lessI 1);
val zero_less_Suc = result();

(** Elimination properties **)

val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<n::nat";
by(fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_anti_sym]@prems))1);
val less_not_sym = result();

(* [| n<m; m<n |] ==> R *)
val less_anti_sym = standard (less_not_sym RS notE);

goalw Nat.thy [less_def] "~ n<n::nat";
by (rtac notI 1);
by (etac (wf_pred_nat RS wf_trancl RS wf_anti_refl) 1);
val less_not_refl = result();

(* n<n ==> R *)
val less_anti_refl = standard (less_not_refl RS notE);

goal Nat.thy "!!m. n<m ==> m ~= n::nat";
by(fast_tac (HOL_cs addEs [less_anti_refl]) 1);
val less_not_refl2 = result();


val major::prems = goalw Nat.thy [less_def]
    "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
\    |] ==> P";
by (rtac (major RS tranclE) 1);
by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1);
by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1);
val lessE = result();

goal Nat.thy "~ n<0";
by (rtac notI 1);
by (etac lessE 1);
by (etac Zero_neq_Suc 1);
by (etac Zero_neq_Suc 1);
val not_less0 = result();

(* n<0 ==> R *)
val less_zeroE = standard (not_less0 RS notE);

val [major,less,eq] = goal Nat.thy
    "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
by (rtac (major RS lessE) 1);
by (rtac eq 1);
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
by (rtac less 1);
by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
val less_SucE = result();

goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";
by (fast_tac (HOL_cs addSIs [lessI]
		     addEs  [less_trans, less_SucE]) 1);
val less_Suc_eq = result();


(** Inductive (?) properties **)

val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";
by (rtac (prem RS rev_mp) 1);
by (nat_ind_tac "n" 1);
by (rtac impI 1);
by (etac less_zeroE 1);
by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
	 	     addSDs [Suc_inject]
		     addEs  [less_trans, lessE]) 1);
val Suc_lessD = result();

val [major,minor] = goal Nat.thy 
    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
\    |] ==> P";
by (rtac (major RS lessE) 1);
by (etac (lessI RS minor) 1);
by (etac (Suc_lessD RS minor) 1);
by (assume_tac 1);
val Suc_lessE = result();

val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
by (rtac (major RS lessE) 1);
by (REPEAT (rtac lessI 1
     ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
val Suc_less_SucD = result();

val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
by (fast_tac (HOL_cs addIs prems) 1);
by (nat_ind_tac "n" 1);
by (rtac impI 1);
by (etac less_zeroE 1);
by (fast_tac (HOL_cs addSIs [lessI]
	 	     addSDs [Suc_inject]
		     addEs  [less_trans, lessE]) 1);
val Suc_mono = result();

goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
val Suc_less_eq = result();

goal Nat.thy "~(Suc(n) < n)";
by(fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1);
val not_Suc_n_less_n = result();

(*"Less than" is a linear ordering*)
goal Nat.thy "m<n | m=n | n<m::nat";
by (nat_ind_tac "m" 1);
by (nat_ind_tac "n" 1);
by (rtac (refl RS disjI1 RS disjI2) 1);
by (rtac (zero_less_Suc RS disjI1) 1);
by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
val less_linear = result();

(*Can be used with less_Suc_eq to get n=m | n<m *)
goal Nat.thy "(~ m < n) = (n < Suc(m))";
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by(ALLGOALS(asm_simp_tac (HOL_ss addsimps
                          [not_less0,zero_less_Suc,Suc_less_eq])));
val not_less_eq = result();

(*Complete induction, aka course-of-values induction*)
val prems = goalw Nat.thy [less_def]
    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
by (eresolve_tac prems 1);
val less_induct = result();


(*** Properties of <= ***)

goalw Nat.thy [le_def] "0 <= n";
by (rtac not_less0 1);
val le0 = result();

val nat_simps = [not_less0, less_not_refl, zero_less_Suc, lessI, 
	     Suc_less_eq, less_Suc_eq, le0, not_Suc_n_less_n,
	     Suc_not_Zero, Zero_not_Suc, Suc_Suc_eq,
             n_not_Suc_n, Suc_n_not_n,
	     nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];

val nat_ss = sum_ss  addsimps  nat_simps;

(*Prevents simplification of f and g: much faster*)
val nat_case_weak_cong = prove_goal Nat.thy
  "m=n ==> nat_case(m,a,f) = nat_case(n,a,f)"
  (fn [prem] => [rtac (prem RS arg_cong) 1]);

val nat_rec_weak_cong = prove_goal Nat.thy
  "m=n ==> nat_rec(m,a,f) = nat_rec(n,a,f)"
  (fn [prem] => [rtac (prem RS arg_cong) 1]);

val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=n::nat";
by (resolve_tac prems 1);
val leI = result();

val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<m::nat)";
by (resolve_tac prems 1);
val leD = result();

val leE = make_elim leD;

goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<m::nat";
by (fast_tac HOL_cs 1);
val not_leE = result();

goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
by(simp_tac nat_ss 1);
by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1);
val lessD = result();

goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
by(asm_full_simp_tac nat_ss 1);
by(fast_tac HOL_cs 1);
val Suc_leD = result();

goalw Nat.thy [le_def] "!!m. m < n ==> m <= n::nat";
by (fast_tac (HOL_cs addEs [less_anti_sym]) 1);
val less_imp_le = result();

goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=n::nat";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1);
val le_imp_less_or_eq = result();

goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <= n::nat";
by (cut_facts_tac [less_linear] 1);
by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1);
by (flexflex_tac);
val less_or_eq_imp_le = result();

goal Nat.thy "(x <= y::nat) = (x < y | x=y)";
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
val le_eq_less_or_eq = result();

goal Nat.thy "n <= n::nat";
by(simp_tac (HOL_ss addsimps [le_eq_less_or_eq]) 1);
val le_refl = result();

val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < k::nat";
by (dtac le_imp_less_or_eq 1);
by (fast_tac (HOL_cs addIs [less_trans]) 1);
val le_less_trans = result();

goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= k::nat";
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
          rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]);
val le_trans = result();

val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = n::nat";
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
          fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym])]);
val le_anti_sym = result();

val nat_ss = nat_ss addsimps [le_refl];