Nat.ML

 
 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();
+qed "Nat_fun_mono";
 
 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();
+qed "Zero_RepI";
 
 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();
+qed "Suc_RepI";
 
 (*** 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 ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
 by (fast_tac (set_cs addIs prems) 1);
-val Nat_induct = result();
+qed "Nat_induct";
 
 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();
+qed "nat_induct";
 
 (*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)];

 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();
+qed "diff_induct";
 
 (*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();
+qed "natE";
 
 (*** 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();
+qed "inj_Rep_Nat";
 
 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();
+qed "inj_onto_Abs_Nat";
 
 (*** 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();
+qed "Suc_not_Zero";
 
 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;

 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();
+qed "inj_Suc";
 
 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();
+qed "Suc_Suc_eq";
 
 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();
+qed "n_not_Suc_n";
 
 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(a, f, 0) = a";
 by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
-val nat_case_0 = result();
+qed "nat_case_0";
 
 goalw Nat.thy [nat_case_def] "nat_case(a, f, Suc(k)) = f(k)";
 by (fast_tac (set_cs addIs [select_equality] 
 	               addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
-val nat_case_Suc = result();
+qed "nat_case_Suc";
 
 (** 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();
+qed "pred_natI";
 
 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();
+qed "pred_natE";
 
 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();
+qed "wf_pred_nat";
 
 
 (*** 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 (simp_tac (HOL_ss addsimps [nat_case_0]) 1);
-val nat_rec_0 = result();
+qed "nat_rec_0";
 
 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 (simp_tac (HOL_ss addsimps [nat_case_Suc, pred_natI, cut_apply]) 1);
-val nat_rec_Suc = result();
+qed "nat_rec_Suc";
 
 (*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();
+qed "def_nat_rec_0";
 
 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();
+qed "def_nat_rec_Suc";
 
 fun nat_recs def =
       [standard (def RS def_nat_rec_0),
        standard (def RS def_nat_rec_Suc)];
 

 
 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();
+qed "less_trans";
 
 goalw Nat.thy [less_def] "n < Suc(n)";
 by (rtac (pred_natI RS r_into_trancl) 1);
-val lessI = result();
+qed "lessI";
 
 (* 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();
+qed "zero_less_Suc";
 
 (** 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_asym]@prems))1);
-val less_not_sym = result();
+qed "less_not_sym";
 
 (* [| n<m; m<n |] ==> R *)
 val less_asym = 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();
+qed "less_not_refl";
 
 (* 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();
+qed "less_not_refl2";
 
 
 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();
+qed "lessE";
 
 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();
+qed "not_less0";
 
 (* n<0 ==> R *)
 val less_zeroE = standard (not_less0 RS notE);
 
 val [major,less,eq] = goal Nat.thy

 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();
+qed "less_SucE";
 
 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();
+qed "less_Suc_eq";
 
 
 (** Inductive (?) properties **)
 
 val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";

 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();
+qed "Suc_lessD";
 
 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();
+qed "Suc_lessE";
 
 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();
+qed "Suc_less_SucD";
 
 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();
+qed "Suc_mono";
 
 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();
+qed "Suc_less_eq";
 
 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();
+qed "not_Suc_n_less_n";
 
 (*"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();
+qed "less_linear";
 
 (*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();
+qed "not_less_eq";
 
 (*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();
+qed "less_induct";
 
 
 (*** Properties of <= ***)
 
 goalw Nat.thy [le_def] "0 <= n";
 by (rtac not_less0 1);
-val le0 = result();
+qed "le0";
 
 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,

   "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();
+qed "leI";
 
 val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))";
 by (resolve_tac prems 1);
-val leD = result();
+qed "leD";
 
 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();
+qed "not_leE";
 
 goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
 by(simp_tac nat_ss0 1);
 by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
-val lessD = result();
+qed "lessD";
 
 goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
 by(asm_full_simp_tac nat_ss0 1);
 by(fast_tac HOL_cs 1);
-val Suc_leD = result();
+qed "Suc_leD";
 
 goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
 by (fast_tac (HOL_cs addEs [less_asym]) 1);
-val less_imp_le = result();
+qed "less_imp_le";
 
 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_asym]) 1);
-val le_imp_less_or_eq = result();
+qed "le_imp_less_or_eq";
 
 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_asym]) 1);
 by (flexflex_tac);
-val less_or_eq_imp_le = result();
+qed "less_or_eq_imp_le";
 
 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();
+qed "le_eq_less_or_eq";
 
 goal Nat.thy "n <= (n::nat)";
 by(simp_tac (HOL_ss addsimps [le_eq_less_or_eq]) 1);
-val le_refl = result();
+qed "le_refl";
 
 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();
+qed "le_less_trans";
 
 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();
+qed "le_trans";
 
 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_asym])]);
-val le_anti_sym = result();
+qed "le_anti_sym";
 
 val nat_ss = nat_ss0 addsimps [le_refl];