diff -r 3a8d722fd3ff -r 16c4ea954511 Nat.ML --- a/Nat.ML Fri Nov 11 10:35:03 1994 +0100 +++ b/Nat.ML Mon Nov 21 17:50:34 1994 +0100 @@ -10,7 +10,7 @@ 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); @@ -18,13 +18,13 @@ 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 ***) @@ -33,7 +33,7 @@ \ !!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); \ @@ -42,7 +42,7 @@ 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 = @@ -60,7 +60,7 @@ 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 @@ -70,7 +70,7 @@ 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 ***) @@ -80,12 +80,12 @@ 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 ***) @@ -93,7 +93,7 @@ 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); @@ -108,18 +108,18 @@ 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; @@ -127,25 +127,25 @@ 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] " : 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 = |] ==> 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); @@ -153,7 +153,7 @@ 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 ***) @@ -165,25 +165,25 @@ 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), @@ -198,11 +198,11 @@ 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 i ~ 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 R *) val less_asym = standard (less_not_sym RS notE); @@ -226,14 +226,14 @@ 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 R *) val less_anti_refl = standard (less_not_refl RS notE); goal Nat.thy "!!m. n 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] @@ -242,14 +242,14 @@ 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); @@ -261,12 +261,12 @@ 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 **) @@ -279,7 +279,7 @@ 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) P \ @@ -288,13 +288,13 @@ 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 Suc(m) < Suc(n)"; by (subgoal_tac "m Suc(m) < Suc(n)" 1); @@ -305,15 +305,15 @@ 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 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, @@ -364,64 +364,64 @@ val prems = goalw Nat.thy [le_def] "~(n 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 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];