src/HOL/NatDef.ML
author paulson
Mon May 26 12:39:57 1997 +0200 (1997-05-26)
changeset 3343 45986997f1fe
parent 3308 da002cef7090
child 3355 0d955bcf8e0a
permissions -rw-r--r--
Renamed lessD to Suc_leI
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(*  Title:      HOL/NatDef.ML
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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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*)
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goal 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 thy "Zero_Rep: Nat";
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by (stac Nat_unfold 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 thy "i: Nat ==> Suc_Rep(i) : Nat";
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by (stac Nat_unfold 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 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 (blast_tac (!claset addIs prems) 1);
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qed "Nat_induct";
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val prems = goalw thy [Zero_def,Suc_def]
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    "[| P(0);   \
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\       !!n. P(n) ==> P(Suc(n)) |]  ==> 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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          COND (Datatype.occs_in_prems a (i+1)) all_tac
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               (rename_last_tac a [""] (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 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 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 (!claset 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 (Blast_tac 1);
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qed "natE";
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(*Install 'automatic' induction tactic, pretending nat is a datatype *)
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(*except for induct_tac and exhaust_tac, everything is dummy*)
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datatypes := [("nat",{case_const = Bound 0, case_rewrites = [],
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  constructors = [], induct_tac = nat_ind_tac,
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  exhaustion = natE,
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  exhaust_tac = fn v => ALLNEWSUBGOALS (res_inst_tac [("n",v)] natE)
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                                       (rotate_tac ~1),
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  nchotomy = flexpair_def, case_cong = flexpair_def})];
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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 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 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 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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AddIffs [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 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 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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AddIffs [Suc_Suc_eq];
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goal thy "n ~= Suc(n)";
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_simp_tac);
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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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goal thy "!!n. n ~= 0 ==> EX m. n = Suc m";
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br natE 1;
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by (REPEAT (Blast_tac 1));
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qed "not0_implies_Suc";
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(*** nat_case -- the selection operator for nat ***)
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goalw thy [nat_case_def] "nat_case a f 0 = a";
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by (blast_tac (!claset addIs [select_equality]) 1);
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qed "nat_case_0";
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goalw thy [nat_case_def] "nat_case a f (Suc k) = f(k)";
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by (blast_tac (!claset addIs [select_equality]) 1);
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qed "nat_case_Suc";
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goalw thy [wf_def, pred_nat_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 (ALLGOALS Blast_tac);
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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 thy
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   "(%n. nat_rec c d n) = 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 thy "nat_rec c h 0 = 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 thy "nat_rec c h (Suc n) = h n (nat_rec c h n)";
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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_nat_def, 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 thy
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    "[| !!n. f(n) == nat_rec c h n |] ==> 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 thy
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    "[| !!n. f(n) == nat_rec c h n |] ==> 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 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 thy [less_def, pred_nat_def] "n < Suc(n)";
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by (simp_tac (!simpset addsimps [r_into_trancl]) 1);
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qed "lessI";
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AddIffs [lessI];
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(* i<j ==> i<Suc(j) *)
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bind_thm("less_SucI", lessI RSN (2, less_trans));
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Addsimps [less_SucI];
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goal 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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AddIffs [zero_less_Suc];
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(** Elimination properties **)
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val prems = goalw thy [less_def] "n<m ==> ~ m<(n::nat)";
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by (blast_tac (!claset 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 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 thy "!!m. n<m ==> m ~= (n::nat)";
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by (blast_tac (!claset addSEs [less_irrefl]) 1);
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qed "less_not_refl2";
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val major::prems = goalw thy [less_def, pred_nat_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 (ALLGOALS Full_simp_tac); 
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by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE'
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                  eresolve_tac (prems@[asm_rl, Pair_inject])));
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qed "lessE";
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goal 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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AddIffs [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 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 (Blast_tac 1);
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by (rtac less 1);
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by (Blast_tac 1);
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qed "less_SucE";
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goal thy "(m < Suc(n)) = (m < n | m = n)";
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by (blast_tac (!claset addSEs [less_SucE] addIs [less_trans]) 1);
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qed "less_Suc_eq";
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val prems = goal 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" 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 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 (ALLGOALS (fast_tac (!claset addSIs [lessI RS less_SucI]
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                                addEs  [less_trans, lessE])));
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qed "Suc_lessD";
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val [major,minor] = goal thy 
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    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
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\    |] ==> P";
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by (rtac (major RS lessE) 1);
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by (etac (lessI RS minor) 1);
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by (etac (Suc_lessD RS minor) 1);
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by (assume_tac 1);
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qed "Suc_lessE";
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goal thy "!!m n. Suc(m) < Suc(n) ==> m<n";
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by (blast_tac (!claset addEs [lessE, make_elim Suc_lessD]) 1);
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qed "Suc_less_SucD";
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goal thy "!!m n. m<n ==> Suc(m) < Suc(n)";
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by (etac rev_mp 1);
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by (nat_ind_tac "n" 1);
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by (ALLGOALS (fast_tac (!claset addEs  [less_trans, lessE])));
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qed "Suc_mono";
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goal thy "(Suc(m) < Suc(n)) = (m<n)";
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by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
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qed "Suc_less_eq";
nipkow@2608
   334
Addsimps [Suc_less_eq];
nipkow@2608
   335
nipkow@2608
   336
goal thy "~(Suc(n) < n)";
paulson@2891
   337
by (blast_tac (!claset addEs [Suc_lessD RS less_irrefl]) 1);
nipkow@2608
   338
qed "not_Suc_n_less_n";
nipkow@2608
   339
Addsimps [not_Suc_n_less_n];
nipkow@2608
   340
nipkow@2608
   341
goal thy "!!i. i<j ==> j<k --> Suc i < k";
nipkow@2608
   342
by (nat_ind_tac "k" 1);
nipkow@2608
   343
by (ALLGOALS (asm_simp_tac (!simpset)));
nipkow@2608
   344
by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
paulson@2891
   345
by (blast_tac (!claset addDs [Suc_lessD]) 1);
nipkow@2608
   346
qed_spec_mp "less_trans_Suc";
nipkow@2608
   347
nipkow@2608
   348
(*"Less than" is a linear ordering*)
nipkow@2608
   349
goal thy "m<n | m=n | n<(m::nat)";
nipkow@2608
   350
by (nat_ind_tac "m" 1);
nipkow@2608
   351
by (nat_ind_tac "n" 1);
nipkow@2608
   352
by (rtac (refl RS disjI1 RS disjI2) 1);
nipkow@2608
   353
by (rtac (zero_less_Suc RS disjI1) 1);
paulson@2935
   354
by (blast_tac (!claset addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
nipkow@2608
   355
qed "less_linear";
nipkow@2608
   356
nipkow@2608
   357
qed_goal "nat_less_cases" thy 
nipkow@2608
   358
   "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
paulson@2935
   359
( fn [major,eqCase,lessCase] =>
nipkow@2608
   360
        [
paulson@2935
   361
        (rtac (less_linear RS disjE) 1),
nipkow@2608
   362
        (etac disjE 2),
paulson@2935
   363
        (etac lessCase 1),
paulson@2935
   364
        (etac (sym RS eqCase) 1),
paulson@2935
   365
        (etac major 1)
nipkow@2608
   366
        ]);
nipkow@2608
   367
nipkow@2608
   368
(*Can be used with less_Suc_eq to get n=m | n<m *)
nipkow@2608
   369
goal thy "(~ m < n) = (n < Suc(m))";
nipkow@2608
   370
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
nipkow@2608
   371
by (ALLGOALS Asm_simp_tac);
nipkow@2608
   372
qed "not_less_eq";
nipkow@2608
   373
nipkow@2608
   374
(*Complete induction, aka course-of-values induction*)
nipkow@2608
   375
val prems = goalw thy [less_def]
nipkow@2608
   376
    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
nipkow@2608
   377
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
nipkow@2608
   378
by (eresolve_tac prems 1);
nipkow@2608
   379
qed "less_induct";
nipkow@2608
   380
nipkow@2608
   381
qed_goal "nat_induct2" thy 
nipkow@2608
   382
"[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
paulson@3023
   383
        cut_facts_tac prems 1,
paulson@3023
   384
        rtac less_induct 1,
paulson@3023
   385
        res_inst_tac [("n","n")] natE 1,
paulson@3023
   386
         hyp_subst_tac 1,
paulson@3023
   387
         atac 1,
paulson@3023
   388
        hyp_subst_tac 1,
paulson@3023
   389
        res_inst_tac [("n","x")] natE 1,
paulson@3023
   390
         hyp_subst_tac 1,
paulson@3023
   391
         atac 1,
paulson@3023
   392
        hyp_subst_tac 1,
paulson@3023
   393
        resolve_tac prems 1,
paulson@3023
   394
        dtac spec 1,
paulson@3023
   395
        etac mp 1,
paulson@3023
   396
        rtac (lessI RS less_trans) 1,
paulson@3023
   397
        rtac (lessI RS Suc_mono) 1]);
nipkow@2608
   398
nipkow@2608
   399
(*** Properties of <= ***)
nipkow@2608
   400
nipkow@2608
   401
goalw thy [le_def] "(m <= n) = (m < Suc n)";
nipkow@2608
   402
by (rtac not_less_eq 1);
nipkow@2608
   403
qed "le_eq_less_Suc";
nipkow@2608
   404
paulson@3343
   405
(*  m<=n ==> m < Suc n  *)
paulson@3343
   406
bind_thm ("le_imp_less_Suc", le_eq_less_Suc RS iffD1);
paulson@3343
   407
nipkow@2608
   408
goalw thy [le_def] "0 <= n";
nipkow@2608
   409
by (rtac not_less0 1);
nipkow@2608
   410
qed "le0";
nipkow@2608
   411
nipkow@2608
   412
goalw thy [le_def] "~ Suc n <= n";
nipkow@2608
   413
by (Simp_tac 1);
nipkow@2608
   414
qed "Suc_n_not_le_n";
nipkow@2608
   415
nipkow@2608
   416
goalw thy [le_def] "(i <= 0) = (i = 0)";
nipkow@2608
   417
by (nat_ind_tac "i" 1);
nipkow@2608
   418
by (ALLGOALS Asm_simp_tac);
nipkow@2608
   419
qed "le_0_eq";
nipkow@2608
   420
nipkow@2608
   421
Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
nipkow@2608
   422
          Suc_n_not_le_n,
nipkow@2608
   423
          n_not_Suc_n, Suc_n_not_n,
nipkow@2608
   424
          nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
nipkow@2608
   425
nipkow@2608
   426
(*
nipkow@2608
   427
goal thy "(Suc m < n | Suc m = n) = (m < n)";
nipkow@2608
   428
by (stac (less_Suc_eq RS sym) 1);
nipkow@2608
   429
by (rtac Suc_less_eq 1);
nipkow@2608
   430
qed "Suc_le_eq";
nipkow@2608
   431
nipkow@2608
   432
this could make the simpset (with less_Suc_eq added again) more confluent,
nipkow@2608
   433
but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
nipkow@2608
   434
*)
nipkow@2608
   435
nipkow@2608
   436
(*Prevents simplification of f and g: much faster*)
nipkow@2608
   437
qed_goal "nat_case_weak_cong" thy
nipkow@2608
   438
  "m=n ==> nat_case a f m = nat_case a f n"
nipkow@2608
   439
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
nipkow@2608
   440
nipkow@2608
   441
qed_goal "nat_rec_weak_cong" thy
nipkow@2608
   442
  "m=n ==> nat_rec a f m = nat_rec a f n"
nipkow@2608
   443
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
nipkow@2608
   444
nipkow@2608
   445
qed_goal "expand_nat_case" thy
nipkow@2608
   446
  "P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))"
nipkow@2608
   447
  (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
nipkow@2608
   448
nipkow@2608
   449
val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
nipkow@2608
   450
by (resolve_tac prems 1);
nipkow@2608
   451
qed "leI";
nipkow@2608
   452
nipkow@2608
   453
val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
nipkow@2608
   454
by (resolve_tac prems 1);
nipkow@2608
   455
qed "leD";
nipkow@2608
   456
nipkow@2608
   457
val leE = make_elim leD;
nipkow@2608
   458
nipkow@2608
   459
goal thy "(~n<m) = (m<=(n::nat))";
paulson@2891
   460
by (blast_tac (!claset addIs [leI] addEs [leE]) 1);
nipkow@2608
   461
qed "not_less_iff_le";
nipkow@2608
   462
nipkow@2608
   463
goalw thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
paulson@2891
   464
by (Blast_tac 1);
nipkow@2608
   465
qed "not_leE";
nipkow@2608
   466
nipkow@2608
   467
goalw thy [le_def] "!!m. m < n ==> Suc(m) <= n";
nipkow@2608
   468
by (simp_tac (!simpset addsimps [less_Suc_eq]) 1);
paulson@3085
   469
by (blast_tac (!claset addSEs [less_irrefl,less_asym]) 1);
paulson@3343
   470
qed "Suc_leI";  (*formerly called lessD*)
nipkow@2608
   471
nipkow@2608
   472
goalw thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
nipkow@2608
   473
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
nipkow@2608
   474
qed "Suc_leD";
nipkow@2608
   475
nipkow@2608
   476
(* stronger version of Suc_leD *)
nipkow@2608
   477
goalw thy [le_def] 
nipkow@2608
   478
        "!!m. Suc m <= n ==> m < n";
nipkow@2608
   479
by (asm_full_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
nipkow@2608
   480
by (cut_facts_tac [less_linear] 1);
paulson@2891
   481
by (Blast_tac 1);
nipkow@2608
   482
qed "Suc_le_lessD";
nipkow@2608
   483
nipkow@2608
   484
goal thy "(Suc m <= n) = (m < n)";
paulson@3343
   485
by (blast_tac (!claset addIs [Suc_leI, Suc_le_lessD]) 1);
nipkow@2608
   486
qed "Suc_le_eq";
nipkow@2608
   487
nipkow@2608
   488
goalw thy [le_def] "!!m. m <= n ==> m <= Suc n";
paulson@2891
   489
by (blast_tac (!claset addDs [Suc_lessD]) 1);
nipkow@2608
   490
qed "le_SucI";
nipkow@2608
   491
Addsimps[le_SucI];
nipkow@2608
   492
nipkow@2608
   493
bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
nipkow@2608
   494
nipkow@2608
   495
goalw thy [le_def] "!!m. m < n ==> m <= (n::nat)";
paulson@2891
   496
by (blast_tac (!claset addEs [less_asym]) 1);
nipkow@2608
   497
qed "less_imp_le";
nipkow@2608
   498
paulson@3343
   499
(** Equivalence of m<=n and  m<n | m=n **)
paulson@3343
   500
nipkow@2608
   501
goalw thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
nipkow@2608
   502
by (cut_facts_tac [less_linear] 1);
paulson@2891
   503
by (blast_tac (!claset addEs [less_irrefl,less_asym]) 1);
nipkow@2608
   504
qed "le_imp_less_or_eq";
nipkow@2608
   505
nipkow@2608
   506
goalw thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
nipkow@2608
   507
by (cut_facts_tac [less_linear] 1);
paulson@3085
   508
by (blast_tac (!claset addSEs [less_irrefl] addEs [less_asym]) 1);
nipkow@2608
   509
qed "less_or_eq_imp_le";
nipkow@2608
   510
paulson@3343
   511
goal thy "(m <= (n::nat)) = (m < n | m=n)";
nipkow@2608
   512
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
nipkow@2608
   513
qed "le_eq_less_or_eq";
nipkow@2608
   514
nipkow@2608
   515
goal thy "n <= (n::nat)";
nipkow@2608
   516
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
nipkow@2608
   517
qed "le_refl";
nipkow@2608
   518
nipkow@2608
   519
val prems = goal thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
nipkow@2608
   520
by (dtac le_imp_less_or_eq 1);
paulson@2935
   521
by (blast_tac (!claset addIs [less_trans]) 1);
nipkow@2608
   522
qed "le_less_trans";
nipkow@2608
   523
nipkow@2608
   524
goal thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
nipkow@2608
   525
by (dtac le_imp_less_or_eq 1);
paulson@2935
   526
by (blast_tac (!claset addIs [less_trans]) 1);
nipkow@2608
   527
qed "less_le_trans";
nipkow@2608
   528
nipkow@2608
   529
goal thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
paulson@2891
   530
by (EVERY1[dtac le_imp_less_or_eq, 
paulson@3023
   531
           dtac le_imp_less_or_eq,
paulson@3023
   532
           rtac less_or_eq_imp_le, 
paulson@3023
   533
           blast_tac (!claset addIs [less_trans])]);
nipkow@2608
   534
qed "le_trans";
nipkow@2608
   535
paulson@2891
   536
goal thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
paulson@2891
   537
by (EVERY1[dtac le_imp_less_or_eq, 
paulson@3023
   538
           dtac le_imp_less_or_eq,
paulson@3023
   539
           blast_tac (!claset addEs [less_irrefl,less_asym])]);
nipkow@2608
   540
qed "le_anti_sym";
nipkow@2608
   541
nipkow@2608
   542
goal thy "(Suc(n) <= Suc(m)) = (n <= m)";
nipkow@2608
   543
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
nipkow@2608
   544
qed "Suc_le_mono";
nipkow@2608
   545
nipkow@2608
   546
AddIffs [Suc_le_mono];
nipkow@2608
   547
nipkow@2608
   548
(* Axiom 'order_le_less' of class 'order': *)
nipkow@2608
   549
goal thy "(m::nat) < n = (m <= n & m ~= n)";
paulson@3023
   550
by (rtac iffI 1);
paulson@3023
   551
 by (rtac conjI 1);
paulson@3023
   552
  by (etac less_imp_le 1);
paulson@3023
   553
 by (etac (less_not_refl2 RS not_sym) 1);
paulson@3023
   554
by (blast_tac (!claset addSDs [le_imp_less_or_eq]) 1);
nipkow@2608
   555
qed "nat_less_le";
nipkow@2608
   556
nipkow@2608
   557
(** LEAST -- the least number operator **)
nipkow@2608
   558
nipkow@3143
   559
goal thy "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
nipkow@3143
   560
by(blast_tac (!claset addIs [leI] addEs [leE]) 1);
nipkow@3143
   561
val lemma = result();
nipkow@3143
   562
nipkow@3143
   563
(* This is an old def of Least for nat, which is derived for compatibility *)
nipkow@3143
   564
goalw thy [Least_def]
nipkow@3143
   565
  "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
nipkow@3143
   566
by(simp_tac (!simpset addsimps [lemma]) 1);
nipkow@3143
   567
br eq_reflection 1;
nipkow@3143
   568
br refl 1;
nipkow@3143
   569
qed "Least_nat_def";
nipkow@3143
   570
nipkow@3143
   571
val [prem1,prem2] = goalw thy [Least_nat_def]
nipkow@2608
   572
    "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k";
nipkow@2608
   573
by (rtac select_equality 1);
paulson@2891
   574
by (blast_tac (!claset addSIs [prem1,prem2]) 1);
nipkow@2608
   575
by (cut_facts_tac [less_linear] 1);
paulson@2891
   576
by (blast_tac (!claset addSIs [prem1] addSDs [prem2]) 1);
nipkow@2608
   577
qed "Least_equality";
nipkow@2608
   578
nipkow@2608
   579
val [prem] = goal thy "P(k::nat) ==> P(LEAST x.P(x))";
nipkow@2608
   580
by (rtac (prem RS rev_mp) 1);
nipkow@2608
   581
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@2608
   582
by (rtac impI 1);
nipkow@2608
   583
by (rtac classical 1);
nipkow@2608
   584
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@2608
   585
by (assume_tac 1);
nipkow@2608
   586
by (assume_tac 2);
paulson@2891
   587
by (Blast_tac 1);
nipkow@2608
   588
qed "LeastI";
nipkow@2608
   589
nipkow@2608
   590
(*Proof is almost identical to the one above!*)
nipkow@2608
   591
val [prem] = goal thy "P(k::nat) ==> (LEAST x.P(x)) <= k";
nipkow@2608
   592
by (rtac (prem RS rev_mp) 1);
nipkow@2608
   593
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@2608
   594
by (rtac impI 1);
nipkow@2608
   595
by (rtac classical 1);
nipkow@2608
   596
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@2608
   597
by (assume_tac 1);
nipkow@2608
   598
by (rtac le_refl 2);
paulson@2891
   599
by (blast_tac (!claset addIs [less_imp_le,le_trans]) 1);
nipkow@2608
   600
qed "Least_le";
nipkow@2608
   601
nipkow@2608
   602
val [prem] = goal thy "k < (LEAST x.P(x)) ==> ~P(k::nat)";
nipkow@2608
   603
by (rtac notI 1);
nipkow@2608
   604
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
nipkow@2608
   605
by (rtac prem 1);
nipkow@2608
   606
qed "not_less_Least";
nipkow@2608
   607
nipkow@3143
   608
qed_goalw "Least_Suc" thy [Least_nat_def]
nipkow@2608
   609
 "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
nipkow@2608
   610
 (fn _ => [
nipkow@2608
   611
        rtac select_equality 1,
nipkow@3143
   612
        fold_goals_tac [Least_nat_def],
nipkow@2608
   613
        safe_tac (!claset addSEs [LeastI]),
nipkow@2608
   614
        rename_tac "j" 1,
nipkow@2608
   615
        res_inst_tac [("n","j")] natE 1,
paulson@2891
   616
        Blast_tac 1,
paulson@2891
   617
        blast_tac (!claset addDs [Suc_less_SucD, not_less_Least]) 1,
nipkow@2608
   618
        rename_tac "k n" 1,
nipkow@2608
   619
        res_inst_tac [("n","k")] natE 1,
paulson@2891
   620
        Blast_tac 1,
nipkow@2608
   621
        hyp_subst_tac 1,
nipkow@3143
   622
        rewtac Least_nat_def,
nipkow@2608
   623
        rtac (select_equality RS arg_cong RS sym) 1,
nipkow@2608
   624
        safe_tac (!claset),
nipkow@2608
   625
        dtac Suc_mono 1,
paulson@2891
   626
        Blast_tac 1,
nipkow@2608
   627
        cut_facts_tac [less_linear] 1,
nipkow@2608
   628
        safe_tac (!claset),
nipkow@2608
   629
        atac 2,
paulson@2891
   630
        Blast_tac 2,
nipkow@2608
   631
        dtac Suc_mono 1,
paulson@2891
   632
        Blast_tac 1]);
nipkow@2608
   633
nipkow@2608
   634
nipkow@2608
   635
(*** Instantiation of transitivity prover ***)
nipkow@2608
   636
nipkow@2608
   637
structure Less_Arith =
nipkow@2608
   638
struct
nipkow@2608
   639
val nat_leI = leI;
nipkow@2608
   640
val nat_leD = leD;
nipkow@2608
   641
val lessI = lessI;
nipkow@2608
   642
val zero_less_Suc = zero_less_Suc;
nipkow@2608
   643
val less_reflE = less_irrefl;
nipkow@2608
   644
val less_zeroE = less_zeroE;
nipkow@2608
   645
val less_incr = Suc_mono;
nipkow@2608
   646
val less_decr = Suc_less_SucD;
nipkow@2608
   647
val less_incr_rhs = Suc_mono RS Suc_lessD;
nipkow@2608
   648
val less_decr_lhs = Suc_lessD;
nipkow@2608
   649
val less_trans_Suc = less_trans_Suc;
paulson@3343
   650
val leI = Suc_leI RS (Suc_le_mono RS iffD1);
nipkow@2608
   651
val not_lessI = leI RS leD
nipkow@2608
   652
val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
nipkow@2608
   653
  (fn _ => [etac swap2 1, etac leD 1]);
nipkow@2608
   654
val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
nipkow@2608
   655
  (fn _ => [etac less_SucE 1,
paulson@2935
   656
            blast_tac (!claset addSDs [Suc_less_SucD] addSEs [less_irrefl]
paulson@2891
   657
                              addDs [less_trans_Suc]) 1,
paulson@2935
   658
            assume_tac 1]);
nipkow@2608
   659
val leD = le_eq_less_Suc RS iffD1;
nipkow@2608
   660
val not_lessD = nat_leI RS leD;
nipkow@2608
   661
val not_leD = not_leE
nipkow@2608
   662
val eqD1 = prove_goal thy  "!!n. m = n ==> m < Suc n"
nipkow@2608
   663
 (fn _ => [etac subst 1, rtac lessI 1]);
nipkow@2608
   664
val eqD2 = sym RS eqD1;
nipkow@2608
   665
nipkow@2608
   666
fun is_zero(t) =  t = Const("0",Type("nat",[]));
nipkow@2608
   667
nipkow@2608
   668
fun nnb T = T = Type("fun",[Type("nat",[]),
nipkow@2608
   669
                            Type("fun",[Type("nat",[]),
nipkow@2608
   670
                                        Type("bool",[])])])
nipkow@2608
   671
nipkow@2608
   672
fun decomp_Suc(Const("Suc",_)$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
nipkow@2608
   673
  | decomp_Suc t = (t,0);
nipkow@2608
   674
nipkow@2608
   675
fun decomp2(rel,T,lhs,rhs) =
nipkow@2608
   676
  if not(nnb T) then None else
nipkow@2608
   677
  let val (x,i) = decomp_Suc lhs
nipkow@2608
   678
      val (y,j) = decomp_Suc rhs
nipkow@2608
   679
  in case rel of
nipkow@2608
   680
       "op <"  => Some(x,i,"<",y,j)
nipkow@2608
   681
     | "op <=" => Some(x,i,"<=",y,j)
nipkow@2608
   682
     | "op ="  => Some(x,i,"=",y,j)
nipkow@2608
   683
     | _       => None
nipkow@2608
   684
  end;
nipkow@2608
   685
nipkow@2608
   686
fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
nipkow@2608
   687
  | negate None = None;
nipkow@2608
   688
nipkow@2608
   689
fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs)
paulson@2718
   690
  | decomp(_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
nipkow@2608
   691
      negate(decomp2(rel,T,lhs,rhs))
nipkow@2608
   692
  | decomp _ = None
nipkow@2608
   693
nipkow@2608
   694
end;
nipkow@2608
   695
nipkow@2608
   696
structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
nipkow@2608
   697
nipkow@2608
   698
open Trans_Tac;
nipkow@2608
   699
nipkow@2608
   700
(*** eliminates ~= in premises, which trans_tac cannot deal with ***)
nipkow@2608
   701
qed_goal "nat_neqE" thy
nipkow@2608
   702
  "[| (m::nat) ~= n; m < n ==> P; n < m ==> P |] ==> P"
nipkow@2608
   703
  (fn major::prems => [cut_facts_tac [less_linear] 1,
nipkow@2608
   704
                       REPEAT(eresolve_tac ([disjE,major RS notE]@prems) 1)]);
pusch@2680
   705
pusch@2680
   706
pusch@2680
   707
pusch@2680
   708
(* add function nat_add_primrec *) 
nipkow@3308
   709
val (_, nat_add_primrec, _) = Datatype.add_datatype
nipkow@3308
   710
([], "nat", [("0", [], Mixfix ("0", [], max_pri)), ("Suc", [dtTyp ([],
nipkow@3308
   711
"nat")], NoSyn)]) (add_thyname "Arith" HOL.thy);
nipkow@3308
   712
(* pretend Arith is part of the basic theory to fool package *)
pusch@2680
   713