src/HOL/Nat.ML
author oheimb
Tue Apr 23 17:34:05 1996 +0200 (1996-04-23)
changeset 1679 6a82e122b337
parent 1672 2c109cd2fdd0
child 1760 6f41a494f3b1
permissions -rw-r--r--
*** empty log message ***
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(*  Title:      HOL/nat
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    ID:         $Id$
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    Author:     Tobias Nipkow, Cambridge University Computer Laboratory
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    Copyright   1991  University of Cambridge
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For nat.thy.  Type nat is defined as a set (Nat) over the type ind.
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*)
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open Nat;
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goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))";
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by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1));
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qed "Nat_fun_mono";
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val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski);
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(* Zero is a natural number -- this also justifies the type definition*)
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goal Nat.thy "Zero_Rep: Nat";
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by (rtac (Nat_unfold RS ssubst) 1);
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by (rtac (singletonI RS UnI1) 1);
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qed "Zero_RepI";
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val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat";
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by (rtac (Nat_unfold RS ssubst) 1);
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by (rtac (imageI RS UnI2) 1);
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by (resolve_tac prems 1);
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qed "Suc_RepI";
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(*** Induction ***)
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val major::prems = goal Nat.thy
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    "[| i: Nat;  P(Zero_Rep);   \
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\       !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |]  ==> P(i)";
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by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1);
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by (fast_tac (set_cs addIs prems) 1);
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qed "Nat_induct";
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val prems = goalw Nat.thy [Zero_def,Suc_def]
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    "[| P(0);   \
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\       !!k. P(k) ==> P(Suc(k)) |]  ==> P(n)";
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by (rtac (Rep_Nat_inverse RS subst) 1);   (*types force good instantiation*)
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by (rtac (Rep_Nat RS Nat_induct) 1);
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by (REPEAT (ares_tac prems 1
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     ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1));
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qed "nat_induct";
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(*Perform induction on n. *)
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fun nat_ind_tac a i = 
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    EVERY [res_inst_tac [("n",a)] nat_induct i,
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           rename_last_tac a ["1"] (i+1)];
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(*A special form of induction for reasoning about m<n and m-n*)
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val prems = goal Nat.thy
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    "[| !!x. P x 0;  \
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\       !!y. P 0 (Suc y);  \
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\       !!x y. [| P x y |] ==> P (Suc x) (Suc y)  \
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\    |] ==> P m n";
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by (res_inst_tac [("x","m")] spec 1);
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by (nat_ind_tac "n" 1);
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by (rtac allI 2);
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by (nat_ind_tac "x" 2);
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by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1));
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qed "diff_induct";
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(*Case analysis on the natural numbers*)
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val prems = goal Nat.thy 
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    "[| n=0 ==> P;  !!x. n = Suc(x) ==> P |] ==> P";
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by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1);
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by (fast_tac (HOL_cs addSEs prems) 1);
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by (nat_ind_tac "n" 1);
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by (rtac (refl RS disjI1) 1);
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by (fast_tac HOL_cs 1);
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qed "natE";
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(*** Isomorphisms: Abs_Nat and Rep_Nat ***)
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(*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat),
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  since we assume the isomorphism equations will one day be given by Isabelle*)
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goal Nat.thy "inj(Rep_Nat)";
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by (rtac inj_inverseI 1);
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by (rtac Rep_Nat_inverse 1);
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qed "inj_Rep_Nat";
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goal Nat.thy "inj_onto Abs_Nat Nat";
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by (rtac inj_onto_inverseI 1);
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by (etac Abs_Nat_inverse 1);
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qed "inj_onto_Abs_Nat";
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(*** Distinctness of constructors ***)
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goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0";
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by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1);
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by (rtac Suc_Rep_not_Zero_Rep 1);
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by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1));
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qed "Suc_not_Zero";
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bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym));
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Addsimps [Suc_not_Zero,Zero_not_Suc];
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bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE));
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val Zero_neq_Suc = sym RS Suc_neq_Zero;
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(** Injectiveness of Suc **)
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goalw Nat.thy [Suc_def] "inj(Suc)";
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by (rtac injI 1);
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by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1);
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by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1));
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by (dtac (inj_Suc_Rep RS injD) 1);
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by (etac (inj_Rep_Nat RS injD) 1);
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qed "inj_Suc";
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val Suc_inject = inj_Suc RS injD;
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goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)";
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by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); 
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qed "Suc_Suc_eq";
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goal Nat.thy "n ~= Suc(n)";
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by (nat_ind_tac "n" 1);
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by (ALLGOALS(asm_simp_tac (!simpset addsimps [Suc_Suc_eq])));
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qed "n_not_Suc_n";
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bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym);
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(*** nat_case -- the selection operator for nat ***)
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goalw Nat.thy [nat_case_def] "nat_case a f 0 = a";
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by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1);
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qed "nat_case_0";
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goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
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by (fast_tac (set_cs addIs [select_equality] 
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                       addEs [make_elim Suc_inject, Suc_neq_Zero]) 1);
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qed "nat_case_Suc";
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(** Introduction rules for 'pred_nat' **)
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goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat";
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by (fast_tac set_cs 1);
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qed "pred_natI";
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val major::prems = goalw Nat.thy [pred_nat_def]
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    "[| p : pred_nat;  !!x n. [| p = (n, Suc(n)) |] ==> R \
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\    |] ==> R";
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by (rtac (major RS CollectE) 1);
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by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1));
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qed "pred_natE";
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goalw Nat.thy [wf_def] "wf(pred_nat)";
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by (strip_tac 1);
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by (nat_ind_tac "x" 1);
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by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, 
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                             make_elim Suc_inject]) 2);
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by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1);
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qed "wf_pred_nat";
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(*** nat_rec -- by wf recursion on pred_nat ***)
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(* The unrolling rule for nat_rec *)
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goal Nat.thy
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   "(%n. nat_rec n c d) = wfrec pred_nat (%f. nat_case ?c (%m. ?d m (f m)))";
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  by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1);
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bind_thm("nat_rec_unfold", wf_pred_nat RS 
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                            ((result() RS eq_reflection) RS def_wfrec));
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(*---------------------------------------------------------------------------
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 * Old:
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 * bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec))); 
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 *---------------------------------------------------------------------------*)
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(** conversion rules **)
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goal Nat.thy "nat_rec 0 c h = c";
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by (rtac (nat_rec_unfold RS trans) 1);
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by (simp_tac (!simpset addsimps [nat_case_0]) 1);
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qed "nat_rec_0";
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goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)";
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by (rtac (nat_rec_unfold RS trans) 1);
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by (simp_tac (!simpset addsimps [nat_case_Suc, pred_natI, cut_apply]) 1);
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qed "nat_rec_Suc";
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(*These 2 rules ease the use of primitive recursion.  NOTE USE OF == *)
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val [rew] = goal Nat.thy
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    "[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c";
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by (rewtac rew);
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by (rtac nat_rec_0 1);
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qed "def_nat_rec_0";
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val [rew] = goal Nat.thy
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    "[| !!n. f(n) == nat_rec n c h |] ==> f(Suc(n)) = h n (f n)";
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by (rewtac rew);
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by (rtac nat_rec_Suc 1);
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qed "def_nat_rec_Suc";
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fun nat_recs def =
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      [standard (def RS def_nat_rec_0),
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       standard (def RS def_nat_rec_Suc)];
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(*** Basic properties of "less than" ***)
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(** Introduction properties **)
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val prems = goalw Nat.thy [less_def] "[| i<j;  j<k |] ==> i<(k::nat)";
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by (rtac (trans_trancl RS transD) 1);
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by (resolve_tac prems 1);
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by (resolve_tac prems 1);
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qed "less_trans";
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goalw Nat.thy [less_def] "n < Suc(n)";
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by (rtac (pred_natI RS r_into_trancl) 1);
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qed "lessI";
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Addsimps [lessI];
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(* i<j ==> i<Suc(j) *)
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val less_SucI = lessI RSN (2, less_trans);
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goal Nat.thy "0 < Suc(n)";
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by (nat_ind_tac "n" 1);
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by (rtac lessI 1);
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by (etac less_trans 1);
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by (rtac lessI 1);
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qed "zero_less_Suc";
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Addsimps [zero_less_Suc];
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(** Elimination properties **)
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val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)";
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by (fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1);
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qed "less_not_sym";
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(* [| n(m; m(n |] ==> R *)
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bind_thm ("less_asym", (less_not_sym RS notE));
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goalw Nat.thy [less_def] "~ n<(n::nat)";
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by (rtac notI 1);
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by (etac (wf_pred_nat RS wf_trancl RS wf_irrefl) 1);
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qed "less_not_refl";
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(* n(n ==> R *)
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bind_thm ("less_irrefl", (less_not_refl RS notE));
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goal Nat.thy "!!m. n<m ==> m ~= (n::nat)";
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by (fast_tac (HOL_cs addEs [less_irrefl]) 1);
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qed "less_not_refl2";
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val major::prems = goalw Nat.thy [less_def]
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    "[| i<k;  k=Suc(i) ==> P;  !!j. [| i<j;  k=Suc(j) |] ==> P \
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\    |] ==> P";
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by (rtac (major RS tranclE) 1);
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by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
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                  eresolve_tac (prems@[pred_natE, Pair_inject])));
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by (rtac refl 1);
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qed "lessE";
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goal Nat.thy "~ n<0";
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by (rtac notI 1);
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by (etac lessE 1);
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by (etac Zero_neq_Suc 1);
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by (etac Zero_neq_Suc 1);
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qed "not_less0";
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Addsimps [not_less0];
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(* n<0 ==> R *)
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bind_thm ("less_zeroE", (not_less0 RS notE));
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val [major,less,eq] = goal Nat.thy
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    "[| m < Suc(n);  m<n ==> P;  m=n ==> P |] ==> P";
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by (rtac (major RS lessE) 1);
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by (rtac eq 1);
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by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
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by (rtac less 1);
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by (fast_tac (HOL_cs addSDs [Suc_inject]) 1);
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qed "less_SucE";
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goal Nat.thy "(m < Suc(n)) = (m < n | m = n)";
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by (fast_tac (HOL_cs addSIs [lessI]
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                     addEs  [less_trans, less_SucE]) 1);
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qed "less_Suc_eq";
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val prems = goal Nat.thy "m<n ==> n ~= 0";
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by (res_inst_tac [("n","n")] natE 1);
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by (cut_facts_tac prems 1);
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by (ALLGOALS Asm_full_simp_tac);
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qed "gr_implies_not0";
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Addsimps [gr_implies_not0];
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qed_goal "zero_less_eq" Nat.thy "0 < n = (n ~= 0)" (fn _ => [
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	rtac iffI 1,
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	etac gr_implies_not0 1,
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	rtac natE 1,
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	contr_tac 1,
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	etac ssubst 1,
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	rtac zero_less_Suc 1]);
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(** Inductive (?) properties **)
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val [prem] = goal Nat.thy "Suc(m) < n ==> m<n";
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by (rtac (prem RS rev_mp) 1);
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by (nat_ind_tac "n" 1);
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   307
by (rtac impI 1);
clasohm@923
   308
by (etac less_zeroE 1);
clasohm@923
   309
by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
clasohm@1465
   310
                     addSDs [Suc_inject]
clasohm@1465
   311
                     addEs  [less_trans, lessE]) 1);
clasohm@923
   312
qed "Suc_lessD";
clasohm@923
   313
clasohm@923
   314
val [major,minor] = goal Nat.thy 
clasohm@923
   315
    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
clasohm@923
   316
\    |] ==> P";
clasohm@923
   317
by (rtac (major RS lessE) 1);
clasohm@923
   318
by (etac (lessI RS minor) 1);
clasohm@923
   319
by (etac (Suc_lessD RS minor) 1);
clasohm@923
   320
by (assume_tac 1);
clasohm@923
   321
qed "Suc_lessE";
clasohm@923
   322
clasohm@923
   323
val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
clasohm@923
   324
by (rtac (major RS lessE) 1);
clasohm@923
   325
by (REPEAT (rtac lessI 1
clasohm@923
   326
     ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
clasohm@923
   327
qed "Suc_less_SucD";
clasohm@923
   328
clasohm@923
   329
val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
clasohm@923
   330
by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
clasohm@923
   331
by (fast_tac (HOL_cs addIs prems) 1);
clasohm@923
   332
by (nat_ind_tac "n" 1);
clasohm@923
   333
by (rtac impI 1);
clasohm@923
   334
by (etac less_zeroE 1);
clasohm@923
   335
by (fast_tac (HOL_cs addSIs [lessI]
clasohm@1465
   336
                     addSDs [Suc_inject]
clasohm@1465
   337
                     addEs  [less_trans, lessE]) 1);
clasohm@923
   338
qed "Suc_mono";
clasohm@923
   339
oheimb@1672
   340
oheimb@1679
   341
(*
oheimb@1672
   342
goal Nat.thy "(Suc m < n | Suc m = n) = (m < n)";
oheimb@1672
   343
oheimb@1672
   344
oheimb@1672
   345
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
oheimb@1672
   346
by(stac less_Suc_eq 1);
oheimb@1672
   347
by(rtac 
oheimb@1679
   348
*)
oheimb@1672
   349
clasohm@923
   350
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
clasohm@923
   351
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
clasohm@923
   352
qed "Suc_less_eq";
nipkow@1301
   353
Addsimps [Suc_less_eq];
clasohm@923
   354
clasohm@923
   355
goal Nat.thy "~(Suc(n) < n)";
paulson@1618
   356
by (fast_tac (HOL_cs addEs [Suc_lessD RS less_irrefl]) 1);
clasohm@923
   357
qed "not_Suc_n_less_n";
nipkow@1301
   358
Addsimps [not_Suc_n_less_n];
nipkow@1301
   359
nipkow@1301
   360
goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k";
paulson@1552
   361
by (nat_ind_tac "k" 1);
oheimb@1660
   362
by (ALLGOALS (asm_simp_tac (!simpset)));
oheimb@1660
   363
by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
paulson@1552
   364
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
nipkow@1485
   365
qed_spec_mp "less_trans_Suc";
clasohm@923
   366
clasohm@923
   367
(*"Less than" is a linear ordering*)
clasohm@923
   368
goal Nat.thy "m<n | m=n | n<(m::nat)";
clasohm@923
   369
by (nat_ind_tac "m" 1);
clasohm@923
   370
by (nat_ind_tac "n" 1);
clasohm@923
   371
by (rtac (refl RS disjI1 RS disjI2) 1);
clasohm@923
   372
by (rtac (zero_less_Suc RS disjI1) 1);
clasohm@923
   373
by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
clasohm@923
   374
qed "less_linear";
clasohm@923
   375
oheimb@1660
   376
qed_goal "nat_less_cases" Nat.thy 
oheimb@1660
   377
   "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
oheimb@1660
   378
( fn prems =>
oheimb@1660
   379
        [
oheimb@1660
   380
        (res_inst_tac [("m1","n"),("n1","m")] (less_linear RS disjE) 1),
oheimb@1660
   381
        (etac disjE 2),
oheimb@1660
   382
        (etac (hd (tl (tl prems))) 1),
oheimb@1660
   383
        (etac (sym RS hd (tl prems)) 1),
oheimb@1660
   384
        (etac (hd prems) 1)
oheimb@1660
   385
        ]);
oheimb@1660
   386
clasohm@923
   387
(*Can be used with less_Suc_eq to get n=m | n<m *)
clasohm@923
   388
goal Nat.thy "(~ m < n) = (n < Suc(m))";
clasohm@923
   389
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@1552
   390
by (ALLGOALS Asm_simp_tac);
clasohm@923
   391
qed "not_less_eq";
clasohm@923
   392
clasohm@923
   393
(*Complete induction, aka course-of-values induction*)
clasohm@923
   394
val prems = goalw Nat.thy [less_def]
clasohm@923
   395
    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
clasohm@923
   396
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
clasohm@923
   397
by (eresolve_tac prems 1);
clasohm@923
   398
qed "less_induct";
clasohm@923
   399
clasohm@923
   400
clasohm@923
   401
(*** Properties of <= ***)
clasohm@923
   402
clasohm@923
   403
goalw Nat.thy [le_def] "0 <= n";
clasohm@923
   404
by (rtac not_less0 1);
clasohm@923
   405
qed "le0";
clasohm@923
   406
nipkow@1301
   407
goalw Nat.thy [le_def] "~ Suc n <= n";
paulson@1552
   408
by (Simp_tac 1);
nipkow@1301
   409
qed "Suc_n_not_le_n";
nipkow@1301
   410
nipkow@1301
   411
goalw Nat.thy [le_def] "(i <= 0) = (i = 0)";
paulson@1552
   412
by (nat_ind_tac "i" 1);
paulson@1552
   413
by (ALLGOALS Asm_simp_tac);
nipkow@1301
   414
qed "le_0";
nipkow@1301
   415
nipkow@1301
   416
Addsimps [less_not_refl,
oheimb@1660
   417
          (*less_Suc_eq,*) le0, le_0,
nipkow@1301
   418
          Suc_Suc_eq, Suc_n_not_le_n,
clasohm@1264
   419
          n_not_Suc_n, Suc_n_not_n,
clasohm@1264
   420
          nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
clasohm@923
   421
clasohm@923
   422
(*Prevents simplification of f and g: much faster*)
clasohm@923
   423
qed_goal "nat_case_weak_cong" Nat.thy
clasohm@923
   424
  "m=n ==> nat_case a f m = nat_case a f n"
clasohm@923
   425
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
clasohm@923
   426
clasohm@923
   427
qed_goal "nat_rec_weak_cong" Nat.thy
clasohm@923
   428
  "m=n ==> nat_rec m a f = nat_rec n a f"
clasohm@923
   429
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
clasohm@923
   430
paulson@1618
   431
val prems = goalw Nat.thy [le_def] "~n<m ==> m<=(n::nat)";
clasohm@923
   432
by (resolve_tac prems 1);
clasohm@923
   433
qed "leI";
clasohm@923
   434
paulson@1618
   435
val prems = goalw Nat.thy [le_def] "m<=n ==> ~ n < (m::nat)";
clasohm@923
   436
by (resolve_tac prems 1);
clasohm@923
   437
qed "leD";
clasohm@923
   438
clasohm@923
   439
val leE = make_elim leD;
clasohm@923
   440
paulson@1618
   441
goal Nat.thy "(~n<m) = (m<=(n::nat))";
paulson@1618
   442
by (fast_tac (HOL_cs addIs [leI] addEs [leE]) 1);
paulson@1618
   443
qed "not_less_iff_le";
paulson@1618
   444
clasohm@923
   445
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
clasohm@923
   446
by (fast_tac HOL_cs 1);
clasohm@923
   447
qed "not_leE";
clasohm@923
   448
clasohm@923
   449
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
oheimb@1660
   450
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1);
paulson@1618
   451
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1);
clasohm@923
   452
qed "lessD";
clasohm@923
   453
clasohm@923
   454
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
oheimb@1660
   455
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
paulson@1552
   456
by (fast_tac HOL_cs 1);
clasohm@923
   457
qed "Suc_leD";
clasohm@923
   458
nipkow@1327
   459
goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n";
nipkow@1327
   460
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
nipkow@1327
   461
qed "le_SucI";
nipkow@1327
   462
Addsimps[le_SucI];
nipkow@1327
   463
clasohm@923
   464
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
clasohm@923
   465
by (fast_tac (HOL_cs addEs [less_asym]) 1);
clasohm@923
   466
qed "less_imp_le";
clasohm@923
   467
clasohm@923
   468
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
clasohm@923
   469
by (cut_facts_tac [less_linear] 1);
paulson@1618
   470
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1);
clasohm@923
   471
qed "le_imp_less_or_eq";
clasohm@923
   472
clasohm@923
   473
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
clasohm@923
   474
by (cut_facts_tac [less_linear] 1);
paulson@1618
   475
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1);
clasohm@923
   476
by (flexflex_tac);
clasohm@923
   477
qed "less_or_eq_imp_le";
clasohm@923
   478
clasohm@923
   479
goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)";
clasohm@923
   480
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
clasohm@923
   481
qed "le_eq_less_or_eq";
clasohm@923
   482
clasohm@923
   483
goal Nat.thy "n <= (n::nat)";
paulson@1552
   484
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
clasohm@923
   485
qed "le_refl";
clasohm@923
   486
clasohm@923
   487
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
clasohm@923
   488
by (dtac le_imp_less_or_eq 1);
clasohm@923
   489
by (fast_tac (HOL_cs addIs [less_trans]) 1);
clasohm@923
   490
qed "le_less_trans";
clasohm@923
   491
clasohm@923
   492
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
clasohm@923
   493
by (dtac le_imp_less_or_eq 1);
clasohm@923
   494
by (fast_tac (HOL_cs addIs [less_trans]) 1);
clasohm@923
   495
qed "less_le_trans";
clasohm@923
   496
clasohm@923
   497
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
clasohm@923
   498
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
clasohm@923
   499
          rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]);
clasohm@923
   500
qed "le_trans";
clasohm@923
   501
clasohm@923
   502
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
clasohm@923
   503
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
paulson@1618
   504
          fast_tac (HOL_cs addEs [less_irrefl,less_asym])]);
clasohm@923
   505
qed "le_anti_sym";
clasohm@923
   506
clasohm@923
   507
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";
clasohm@1264
   508
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
clasohm@923
   509
qed "Suc_le_mono";
clasohm@923
   510
clasohm@1264
   511
Addsimps [le_refl,Suc_le_mono];
nipkow@1531
   512
nipkow@1531
   513
nipkow@1531
   514
(** LEAST -- the least number operator **)
nipkow@1531
   515
nipkow@1531
   516
val [prem1,prem2] = goalw Nat.thy [Least_def]
nipkow@1531
   517
    "[| P(k);  !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
nipkow@1531
   518
by (rtac select_equality 1);
nipkow@1531
   519
by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1);
nipkow@1531
   520
by (cut_facts_tac [less_linear] 1);
nipkow@1531
   521
by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1);
nipkow@1531
   522
qed "Least_equality";
nipkow@1531
   523
nipkow@1531
   524
val [prem] = goal Nat.thy "P(k) ==> P(LEAST x.P(x))";
nipkow@1531
   525
by (rtac (prem RS rev_mp) 1);
nipkow@1531
   526
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@1531
   527
by (rtac impI 1);
nipkow@1531
   528
by (rtac classical 1);
nipkow@1531
   529
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@1531
   530
by (assume_tac 1);
nipkow@1531
   531
by (assume_tac 2);
nipkow@1531
   532
by (fast_tac HOL_cs 1);
nipkow@1531
   533
qed "LeastI";
nipkow@1531
   534
nipkow@1531
   535
(*Proof is almost identical to the one above!*)
nipkow@1531
   536
val [prem] = goal Nat.thy "P(k) ==> (LEAST x.P(x)) <= k";
nipkow@1531
   537
by (rtac (prem RS rev_mp) 1);
nipkow@1531
   538
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@1531
   539
by (rtac impI 1);
nipkow@1531
   540
by (rtac classical 1);
nipkow@1531
   541
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@1531
   542
by (assume_tac 1);
nipkow@1531
   543
by (rtac le_refl 2);
nipkow@1531
   544
by (fast_tac (HOL_cs addIs [less_imp_le,le_trans]) 1);
nipkow@1531
   545
qed "Least_le";
nipkow@1531
   546
nipkow@1531
   547
val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)";
nipkow@1531
   548
by (rtac notI 1);
nipkow@1531
   549
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
nipkow@1531
   550
by (rtac prem 1);
nipkow@1531
   551
qed "not_less_Least";
oheimb@1660
   552
oheimb@1660
   553
qed_goalw "Least_Suc" Nat.thy [Least_def]
oheimb@1660
   554
 "[| ? n. P (Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"
oheimb@1660
   555
 (fn prems => [
oheimb@1660
   556
	cut_facts_tac prems 1,
oheimb@1660
   557
	rtac select_equality 1,
oheimb@1660
   558
	fold_goals_tac [Least_def],
oheimb@1660
   559
	safe_tac (HOL_cs addSEs [LeastI]),
oheimb@1660
   560
	res_inst_tac [("n","j")] natE 1,
oheimb@1660
   561
	fast_tac HOL_cs 1,
oheimb@1660
   562
	fast_tac (HOL_cs addDs [Suc_less_SucD] addDs [not_less_Least]) 1,	
oheimb@1660
   563
	res_inst_tac [("n","k")] natE 1,
oheimb@1660
   564
	fast_tac HOL_cs 1,
oheimb@1660
   565
	hyp_subst_tac 1,
oheimb@1660
   566
	rewtac Least_def,
oheimb@1660
   567
	rtac (select_equality RS arg_cong RS sym) 1,
oheimb@1660
   568
	safe_tac HOL_cs,
oheimb@1660
   569
	dtac Suc_mono 1,
oheimb@1660
   570
	fast_tac HOL_cs 1,
oheimb@1660
   571
	cut_facts_tac [less_linear] 1,
oheimb@1660
   572
	safe_tac HOL_cs,
oheimb@1660
   573
	atac 2,
oheimb@1660
   574
	fast_tac HOL_cs 2,
oheimb@1660
   575
	dtac Suc_mono 1,
oheimb@1660
   576
	fast_tac HOL_cs 1]);