src/HOL/Nat.ML
author nipkow
Mon Mar 04 14:37:33 1996 +0100 (1996-03-04)
changeset 1531 e5eb247ad13c
parent 1485 240cc98b94a7
child 1552 6f71b5d46700
permissions -rw-r--r--
Added a constant UNIV == {x.True}
Added many new rewrite rules for sets.
Moved LEAST into Nat.
Added cardinality to Finite.
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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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val 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_anti_refl) 1);
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qed "less_not_refl";
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(* n(n ==> R *)
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bind_thm ("less_anti_refl", (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_anti_refl]) 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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(** 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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by (rtac impI 1);
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by (etac less_zeroE 1);
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by (fast_tac (HOL_cs addSIs [lessI RS less_SucI]
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                     addSDs [Suc_inject]
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                     addEs  [less_trans, lessE]) 1);
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qed "Suc_lessD";
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val [major,minor] = goal Nat.thy 
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   307
    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
clasohm@923
   308
\    |] ==> P";
clasohm@923
   309
by (rtac (major RS lessE) 1);
clasohm@923
   310
by (etac (lessI RS minor) 1);
clasohm@923
   311
by (etac (Suc_lessD RS minor) 1);
clasohm@923
   312
by (assume_tac 1);
clasohm@923
   313
qed "Suc_lessE";
clasohm@923
   314
clasohm@923
   315
val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n";
clasohm@923
   316
by (rtac (major RS lessE) 1);
clasohm@923
   317
by (REPEAT (rtac lessI 1
clasohm@923
   318
     ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1));
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   319
qed "Suc_less_SucD";
clasohm@923
   320
clasohm@923
   321
val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)";
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   322
by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1);
clasohm@923
   323
by (fast_tac (HOL_cs addIs prems) 1);
clasohm@923
   324
by (nat_ind_tac "n" 1);
clasohm@923
   325
by (rtac impI 1);
clasohm@923
   326
by (etac less_zeroE 1);
clasohm@923
   327
by (fast_tac (HOL_cs addSIs [lessI]
clasohm@1465
   328
                     addSDs [Suc_inject]
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   329
                     addEs  [less_trans, lessE]) 1);
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   330
qed "Suc_mono";
clasohm@923
   331
clasohm@923
   332
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)";
clasohm@923
   333
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
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   334
qed "Suc_less_eq";
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   335
Addsimps [Suc_less_eq];
clasohm@923
   336
clasohm@923
   337
goal Nat.thy "~(Suc(n) < n)";
clasohm@923
   338
by(fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1);
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   339
qed "not_Suc_n_less_n";
nipkow@1301
   340
Addsimps [not_Suc_n_less_n];
nipkow@1301
   341
nipkow@1301
   342
goal Nat.thy "!!i. i<j ==> j<k --> Suc i < k";
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   343
by(nat_ind_tac "k" 1);
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   344
by(ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
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   345
by(fast_tac (HOL_cs addDs [Suc_lessD]) 1);
nipkow@1485
   346
qed_spec_mp "less_trans_Suc";
clasohm@923
   347
clasohm@923
   348
(*"Less than" is a linear ordering*)
clasohm@923
   349
goal Nat.thy "m<n | m=n | n<(m::nat)";
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   350
by (nat_ind_tac "m" 1);
clasohm@923
   351
by (nat_ind_tac "n" 1);
clasohm@923
   352
by (rtac (refl RS disjI1 RS disjI2) 1);
clasohm@923
   353
by (rtac (zero_less_Suc RS disjI1) 1);
clasohm@923
   354
by (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1);
clasohm@923
   355
qed "less_linear";
clasohm@923
   356
clasohm@923
   357
(*Can be used with less_Suc_eq to get n=m | n<m *)
clasohm@923
   358
goal Nat.thy "(~ m < n) = (n < Suc(m))";
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   359
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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   360
by(ALLGOALS Asm_simp_tac);
clasohm@923
   361
qed "not_less_eq";
clasohm@923
   362
clasohm@923
   363
(*Complete induction, aka course-of-values induction*)
clasohm@923
   364
val prems = goalw Nat.thy [less_def]
clasohm@923
   365
    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
clasohm@923
   366
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
clasohm@923
   367
by (eresolve_tac prems 1);
clasohm@923
   368
qed "less_induct";
clasohm@923
   369
clasohm@923
   370
clasohm@923
   371
(*** Properties of <= ***)
clasohm@923
   372
clasohm@923
   373
goalw Nat.thy [le_def] "0 <= n";
clasohm@923
   374
by (rtac not_less0 1);
clasohm@923
   375
qed "le0";
clasohm@923
   376
nipkow@1301
   377
goalw Nat.thy [le_def] "~ Suc n <= n";
nipkow@1301
   378
by(Simp_tac 1);
nipkow@1301
   379
qed "Suc_n_not_le_n";
nipkow@1301
   380
nipkow@1301
   381
goalw Nat.thy [le_def] "(i <= 0) = (i = 0)";
nipkow@1301
   382
by(nat_ind_tac "i" 1);
nipkow@1301
   383
by(ALLGOALS Asm_simp_tac);
nipkow@1301
   384
qed "le_0";
nipkow@1301
   385
nipkow@1301
   386
Addsimps [less_not_refl,
nipkow@1301
   387
          less_Suc_eq, le0, le_0,
nipkow@1301
   388
          Suc_Suc_eq, Suc_n_not_le_n,
clasohm@1264
   389
          n_not_Suc_n, Suc_n_not_n,
clasohm@1264
   390
          nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
clasohm@923
   391
clasohm@923
   392
(*Prevents simplification of f and g: much faster*)
clasohm@923
   393
qed_goal "nat_case_weak_cong" Nat.thy
clasohm@923
   394
  "m=n ==> nat_case a f m = nat_case a f n"
clasohm@923
   395
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
clasohm@923
   396
clasohm@923
   397
qed_goal "nat_rec_weak_cong" Nat.thy
clasohm@923
   398
  "m=n ==> nat_rec m a f = nat_rec n a f"
clasohm@923
   399
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
clasohm@923
   400
clasohm@923
   401
val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=(n::nat)";
clasohm@923
   402
by (resolve_tac prems 1);
clasohm@923
   403
qed "leI";
clasohm@923
   404
clasohm@923
   405
val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))";
clasohm@923
   406
by (resolve_tac prems 1);
clasohm@923
   407
qed "leD";
clasohm@923
   408
clasohm@923
   409
val leE = make_elim leD;
clasohm@923
   410
clasohm@923
   411
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
clasohm@923
   412
by (fast_tac HOL_cs 1);
clasohm@923
   413
qed "not_leE";
clasohm@923
   414
clasohm@923
   415
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n";
clasohm@1264
   416
by(Simp_tac 1);
clasohm@923
   417
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
clasohm@923
   418
qed "lessD";
clasohm@923
   419
clasohm@923
   420
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
clasohm@1264
   421
by(Asm_full_simp_tac 1);
clasohm@923
   422
by(fast_tac HOL_cs 1);
clasohm@923
   423
qed "Suc_leD";
clasohm@923
   424
nipkow@1327
   425
goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n";
nipkow@1327
   426
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1);
nipkow@1327
   427
qed "le_SucI";
nipkow@1327
   428
Addsimps[le_SucI];
nipkow@1327
   429
clasohm@923
   430
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)";
clasohm@923
   431
by (fast_tac (HOL_cs addEs [less_asym]) 1);
clasohm@923
   432
qed "less_imp_le";
clasohm@923
   433
clasohm@923
   434
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
clasohm@923
   435
by (cut_facts_tac [less_linear] 1);
clasohm@923
   436
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
clasohm@923
   437
qed "le_imp_less_or_eq";
clasohm@923
   438
clasohm@923
   439
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
clasohm@923
   440
by (cut_facts_tac [less_linear] 1);
clasohm@923
   441
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1);
clasohm@923
   442
by (flexflex_tac);
clasohm@923
   443
qed "less_or_eq_imp_le";
clasohm@923
   444
clasohm@923
   445
goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)";
clasohm@923
   446
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
clasohm@923
   447
qed "le_eq_less_or_eq";
clasohm@923
   448
clasohm@923
   449
goal Nat.thy "n <= (n::nat)";
clasohm@1264
   450
by(simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
clasohm@923
   451
qed "le_refl";
clasohm@923
   452
clasohm@923
   453
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
clasohm@923
   454
by (dtac le_imp_less_or_eq 1);
clasohm@923
   455
by (fast_tac (HOL_cs addIs [less_trans]) 1);
clasohm@923
   456
qed "le_less_trans";
clasohm@923
   457
clasohm@923
   458
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
clasohm@923
   459
by (dtac le_imp_less_or_eq 1);
clasohm@923
   460
by (fast_tac (HOL_cs addIs [less_trans]) 1);
clasohm@923
   461
qed "less_le_trans";
clasohm@923
   462
clasohm@923
   463
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
clasohm@923
   464
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
clasohm@923
   465
          rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]);
clasohm@923
   466
qed "le_trans";
clasohm@923
   467
clasohm@923
   468
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
clasohm@923
   469
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq,
clasohm@923
   470
          fast_tac (HOL_cs addEs [less_anti_refl,less_asym])]);
clasohm@923
   471
qed "le_anti_sym";
clasohm@923
   472
clasohm@923
   473
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)";
clasohm@1264
   474
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
clasohm@923
   475
qed "Suc_le_mono";
clasohm@923
   476
clasohm@1264
   477
Addsimps [le_refl,Suc_le_mono];
nipkow@1531
   478
nipkow@1531
   479
nipkow@1531
   480
(** LEAST -- the least number operator **)
nipkow@1531
   481
nipkow@1531
   482
val [prem1,prem2] = goalw Nat.thy [Least_def]
nipkow@1531
   483
    "[| P(k);  !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
nipkow@1531
   484
by (rtac select_equality 1);
nipkow@1531
   485
by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1);
nipkow@1531
   486
by (cut_facts_tac [less_linear] 1);
nipkow@1531
   487
by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1);
nipkow@1531
   488
qed "Least_equality";
nipkow@1531
   489
nipkow@1531
   490
val [prem] = goal Nat.thy "P(k) ==> P(LEAST x.P(x))";
nipkow@1531
   491
by (rtac (prem RS rev_mp) 1);
nipkow@1531
   492
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@1531
   493
by (rtac impI 1);
nipkow@1531
   494
by (rtac classical 1);
nipkow@1531
   495
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@1531
   496
by (assume_tac 1);
nipkow@1531
   497
by (assume_tac 2);
nipkow@1531
   498
by (fast_tac HOL_cs 1);
nipkow@1531
   499
qed "LeastI";
nipkow@1531
   500
nipkow@1531
   501
(*Proof is almost identical to the one above!*)
nipkow@1531
   502
val [prem] = goal Nat.thy "P(k) ==> (LEAST x.P(x)) <= k";
nipkow@1531
   503
by (rtac (prem RS rev_mp) 1);
nipkow@1531
   504
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@1531
   505
by (rtac impI 1);
nipkow@1531
   506
by (rtac classical 1);
nipkow@1531
   507
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@1531
   508
by (assume_tac 1);
nipkow@1531
   509
by (rtac le_refl 2);
nipkow@1531
   510
by (fast_tac (HOL_cs addIs [less_imp_le,le_trans]) 1);
nipkow@1531
   511
qed "Least_le";
nipkow@1531
   512
nipkow@1531
   513
val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)";
nipkow@1531
   514
by (rtac notI 1);
nipkow@1531
   515
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
nipkow@1531
   516
by (rtac prem 1);
nipkow@1531
   517
qed "not_less_Least";