src/HOL/NatDef.ML
author paulson
Tue, 10 Feb 1998 10:26:58 +0100
changeset 4614 122015efd4e1
parent 4599 3a4348a3d6ff
child 4635 c448e09d0fca
permissions -rw-r--r--
New AddIffs le_0_eq and neq0_conv
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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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local fun raw_nat_ind_tac a i = 
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    res_inst_tac [("n",a)] nat_induct i  THEN  rename_last_tac a [""] (i+1)
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in
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val nat_ind_tac = Datatype.occs_in_prems raw_nat_ind_tac
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end;
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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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(*** 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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by (rtac 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 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 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 (Clarify_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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(*Used in TFL/post.sml*)
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goalw thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
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by (rtac refl 1);
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qed "less_eq";
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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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goal thy "(n<1) = (n=0)";
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by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
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qed "less_one";
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AddIffs [less_one];
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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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goal thy "(n ~= 0) = (0 < n)";
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by (rtac iffI 1);
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 by (etac gr_implies_not0 2);
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by (rtac natE 1);
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 by (contr_tac 1);
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by (etac ssubst 1);
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by (rtac zero_less_Suc 1);
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qed "neq0_conv";
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AddIffs [neq0_conv];
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bind_thm("gr0I", notI RS (neq0_conv RS iffD1));
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goal thy "(~(0 < n)) = (n=0)";
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by (rtac iffI 1);
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 by (etac swap 1);
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 by (ALLGOALS Asm_full_simp_tac);
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qed "not_gr0";
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Addsimps [not_gr0];
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goal thy "!!m. m<n ==> 0 < n";
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by (dtac gr_implies_not0 1);
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by (Asm_full_simp_tac 1);
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qed "gr_implies_gr0";
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Addsimps [gr_implies_gr0];
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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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   327
by (ALLGOALS (fast_tac (claset() addSIs [lessI RS less_SucI]
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diff changeset
   328
                                addEs  [less_trans, lessE])));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   329
qed "Suc_lessD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   330
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   331
val [major,minor] = goal thy 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   332
    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   333
\    |] ==> P";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   334
by (rtac (major RS lessE) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   335
by (etac (lessI RS minor) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   336
by (etac (Suc_lessD RS minor) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   337
by (assume_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   338
qed "Suc_lessE";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   339
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   340
goal thy "!!m n. Suc(m) < Suc(n) ==> m<n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   341
by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   342
qed "Suc_less_SucD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   343
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   344
goal thy "!!m n. m<n ==> Suc(m) < Suc(n)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   345
by (etac rev_mp 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   346
by (nat_ind_tac "n" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   347
by (ALLGOALS (fast_tac (claset() addEs  [less_trans, lessE])));
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   348
qed "Suc_mono";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   349
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   350
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   351
goal thy "(Suc(m) < Suc(n)) = (m<n)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   352
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   353
qed "Suc_less_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   354
Addsimps [Suc_less_eq];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   355
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   356
goal thy "~(Suc(n) < n)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   357
by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   358
qed "not_Suc_n_less_n";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   359
Addsimps [not_Suc_n_less_n];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   360
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   361
goal thy "!!i. i<j ==> j<k --> Suc i < k";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   362
by (nat_ind_tac "k" 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   363
by (ALLGOALS (asm_simp_tac (simpset())));
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   364
by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   365
by (blast_tac (claset() addDs [Suc_lessD]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   366
qed_spec_mp "less_trans_Suc";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   367
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   368
(*"Less than" is a linear ordering*)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   369
goal thy "m<n | m=n | n<(m::nat)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   370
by (nat_ind_tac "m" 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   371
by (nat_ind_tac "n" 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   372
by (rtac (refl RS disjI1 RS disjI2) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   373
by (rtac (zero_less_Suc RS disjI1) 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   374
by (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   375
qed "less_linear";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   376
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   377
qed_goal "nat_less_cases" thy 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   378
   "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2922
diff changeset
   379
( fn [major,eqCase,lessCase] =>
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   380
        [
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2922
diff changeset
   381
        (rtac (less_linear RS disjE) 1),
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   382
        (etac disjE 2),
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2922
diff changeset
   383
        (etac lessCase 1),
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2922
diff changeset
   384
        (etac (sym RS eqCase) 1),
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2922
diff changeset
   385
        (etac major 1)
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   386
        ]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   387
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   388
(*Can be used with less_Suc_eq to get n=m | n<m *)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   389
goal thy "(~ m < n) = (n < Suc(m))";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   390
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   391
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   392
qed "not_less_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   393
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   394
(*Complete induction, aka course-of-values induction*)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   395
val prems = goalw thy [less_def]
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   396
    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   397
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   398
by (eresolve_tac prems 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   399
qed "less_induct";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   400
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   401
qed_goal "nat_induct2" thy 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   402
"[| P 0; P 1; !!k. P k ==> P (Suc (Suc k)) |] ==> P n" (fn prems => [
3023
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   403
        cut_facts_tac prems 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   404
        rtac less_induct 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   405
        res_inst_tac [("n","n")] natE 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   406
         hyp_subst_tac 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   407
         atac 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   408
        hyp_subst_tac 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   409
        res_inst_tac [("n","x")] natE 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   410
         hyp_subst_tac 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   411
         atac 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   412
        hyp_subst_tac 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   413
        resolve_tac prems 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   414
        dtac spec 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   415
        etac mp 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   416
        rtac (lessI RS less_trans) 1,
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   417
        rtac (lessI RS Suc_mono) 1]);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   418
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   419
(*** Properties of <= ***)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   420
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   421
goalw thy [le_def] "(m <= n) = (m < Suc n)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   422
by (rtac not_less_eq 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   423
qed "le_eq_less_Suc";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   424
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   425
(*  m<=n ==> m < Suc n  *)
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   426
bind_thm ("le_imp_less_Suc", le_eq_less_Suc RS iffD1);
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   427
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   428
goalw thy [le_def] "0 <= n";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   429
by (rtac not_less0 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   430
qed "le0";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   431
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   432
goalw thy [le_def] "~ Suc n <= n";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   433
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   434
qed "Suc_n_not_le_n";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   435
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   436
goalw thy [le_def] "(i <= 0) = (i = 0)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   437
by (nat_ind_tac "i" 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   438
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   439
qed "le_0_eq";
4614
122015efd4e1 New AddIffs le_0_eq and neq0_conv
paulson
parents: 4599
diff changeset
   440
AddIffs [le_0_eq];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   441
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   442
Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   443
          Suc_n_not_le_n,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   444
          n_not_Suc_n, Suc_n_not_n,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   445
          nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   446
3355
0d955bcf8e0a New theorem le_Suc_eq
paulson
parents: 3343
diff changeset
   447
goal thy "!!m. (m <= Suc(n)) = (m<=n | m = Suc n)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   448
by (simp_tac (simpset() addsimps [le_eq_less_Suc]) 1);
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   449
by (blast_tac (claset() addSEs [less_SucE] addIs [less_SucI]) 1);
3355
0d955bcf8e0a New theorem le_Suc_eq
paulson
parents: 3343
diff changeset
   450
qed "le_Suc_eq";
0d955bcf8e0a New theorem le_Suc_eq
paulson
parents: 3343
diff changeset
   451
4614
122015efd4e1 New AddIffs le_0_eq and neq0_conv
paulson
parents: 4599
diff changeset
   452
(* [| m <= Suc n;  m <= n ==> R;  m = Suc n ==> R |] ==> R *)
122015efd4e1 New AddIffs le_0_eq and neq0_conv
paulson
parents: 4599
diff changeset
   453
bind_thm ("le_SucE", le_Suc_eq RS iffD1 RS disjE);
122015efd4e1 New AddIffs le_0_eq and neq0_conv
paulson
parents: 4599
diff changeset
   454
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   455
(*
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   456
goal thy "(Suc m < n | Suc m = n) = (m < n)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   457
by (stac (less_Suc_eq RS sym) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   458
by (rtac Suc_less_eq 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   459
qed "Suc_le_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   460
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   461
this could make the simpset (with less_Suc_eq added again) more confluent,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   462
but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   463
*)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   464
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   465
(*Prevents simplification of f and g: much faster*)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   466
qed_goal "nat_case_weak_cong" thy
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   467
  "m=n ==> nat_case a f m = nat_case a f n"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   468
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   469
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   470
qed_goal "nat_rec_weak_cong" thy
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   471
  "m=n ==> nat_rec a f m = nat_rec a f n"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   472
  (fn [prem] => [rtac (prem RS arg_cong) 1]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   473
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   474
qed_goal "expand_nat_case" thy
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   475
  "P(nat_case z s n) = ((n=0 --> P z) & (!m. n = Suc m --> P(s m)))"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   476
  (fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   477
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   478
val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   479
by (resolve_tac prems 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   480
qed "leI";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   481
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   482
val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   483
by (resolve_tac prems 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   484
qed "leD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   485
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   486
val leE = make_elim leD;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   487
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   488
goal thy "(~n<m) = (m<=(n::nat))";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   489
by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   490
qed "not_less_iff_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   491
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   492
goalw thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   493
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   494
qed "not_leE";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   495
4599
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   496
goalw thy [le_def] "(~n<=m) = (m<(n::nat))";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   497
by (Simp_tac 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   498
qed "not_le_iff_less";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   499
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   500
goalw thy [le_def] "!!m. m < n ==> Suc(m) <= n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   501
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   502
by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   503
qed "Suc_leI";  (*formerly called lessD*)
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   504
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   505
goalw thy [le_def] "!!m. Suc(m) <= n ==> m <= n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   506
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   507
qed "Suc_leD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   508
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   509
(* stronger version of Suc_leD *)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   510
goalw thy [le_def] 
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   511
        "!!m. Suc m <= n ==> m < n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   512
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   513
by (cut_facts_tac [less_linear] 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   514
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   515
qed "Suc_le_lessD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   516
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   517
goal thy "(Suc m <= n) = (m < n)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   518
by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   519
qed "Suc_le_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   520
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   521
goalw thy [le_def] "!!m. m <= n ==> m <= Suc n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   522
by (blast_tac (claset() addDs [Suc_lessD]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   523
qed "le_SucI";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   524
Addsimps[le_SucI];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   525
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   526
bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   527
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   528
goalw thy [le_def] "!!m. m < n ==> m <= (n::nat)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   529
by (blast_tac (claset() addEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   530
qed "less_imp_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   531
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   532
(** Equivalence of m<=n and  m<n | m=n **)
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   533
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   534
goalw thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   535
by (cut_facts_tac [less_linear] 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   536
by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   537
qed "le_imp_less_or_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   538
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   539
goalw thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   540
by (cut_facts_tac [less_linear] 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   541
by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   542
qed "less_or_eq_imp_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   543
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   544
goal thy "(m <= (n::nat)) = (m < n | m=n)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   545
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   546
qed "le_eq_less_or_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   547
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   548
goal thy "n <= (n::nat)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   549
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   550
qed "le_refl";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   551
4468
paulson
parents: 4423
diff changeset
   552
goal thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)";
paulson
parents: 4423
diff changeset
   553
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   554
	                addIs [less_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   555
qed "le_less_trans";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   556
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   557
goal thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)";
4468
paulson
parents: 4423
diff changeset
   558
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   559
	                addIs [less_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   560
qed "less_le_trans";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   561
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   562
goal thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)";
4468
paulson
parents: 4423
diff changeset
   563
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   564
	                addIs [less_or_eq_imp_le, less_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   565
qed "le_trans";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   566
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   567
goal thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)";
4468
paulson
parents: 4423
diff changeset
   568
(*order_less_irrefl could make this proof fail*)
paulson
parents: 4423
diff changeset
   569
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   570
	                addSEs [less_irrefl] addEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   571
qed "le_anti_sym";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   572
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   573
goal thy "(Suc(n) <= Suc(m)) = (n <= m)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   574
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   575
qed "Suc_le_mono";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   576
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   577
AddIffs [Suc_le_mono];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   578
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   579
(* Axiom 'order_le_less' of class 'order': *)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   580
goal thy "(m::nat) < n = (m <= n & m ~= n)";
3023
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   581
by (rtac iffI 1);
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   582
 by (rtac conjI 1);
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   583
  by (etac less_imp_le 1);
01364e2f30ad Ran expandshort
paulson
parents: 2935
diff changeset
   584
 by (etac (less_not_refl2 RS not_sym) 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   585
by (blast_tac (claset() addSDs [le_imp_less_or_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   586
qed "nat_less_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   587
4599
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   588
(** max **)
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   589
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   590
goalw thy [max_def] "!!z::nat. (z <= max x y) = (z <= x | z <= y)";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   591
by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits [expand_if]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   592
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   593
qed "le_max_iff_disj";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   594
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   595
goalw thy [max_def] "!!z::nat. (max x y <= z) = (x <= z & y <= z)";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   596
by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits [expand_if]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   597
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   598
qed "max_le_iff_conj";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   599
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   600
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   601
(** min **)
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   602
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   603
goalw thy [min_def] "!!z::nat. (z <= min x y) = (z <= x & z <= y)";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   604
by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits [expand_if]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   605
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   606
qed "le_min_iff_conj";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   607
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   608
goalw thy [min_def] "!!z::nat. (min x y <= z) = (x <= z | y <= z)";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   609
by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits [expand_if]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   610
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   611
qed "min_le_iff_disj";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   612
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   613
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   614
(** LEAST -- the least number operator **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   615
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   616
goal thy "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   617
by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   618
val lemma = result();
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   619
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   620
(* This is an old def of Least for nat, which is derived for compatibility *)
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   621
goalw thy [Least_def]
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   622
  "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   623
by (simp_tac (simpset() addsimps [lemma]) 1);
3457
a8ab7c64817c Ran expandshort
paulson
parents: 3378
diff changeset
   624
by (rtac eq_reflection 1);
a8ab7c64817c Ran expandshort
paulson
parents: 3378
diff changeset
   625
by (rtac refl 1);
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   626
qed "Least_nat_def";
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   627
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   628
val [prem1,prem2] = goalw thy [Least_nat_def]
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3768
diff changeset
   629
    "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   630
by (rtac select_equality 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   631
by (blast_tac (claset() addSIs [prem1,prem2]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   632
by (cut_facts_tac [less_linear] 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   633
by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   634
qed "Least_equality";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   635
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3768
diff changeset
   636
val [prem] = goal thy "P(k::nat) ==> P(LEAST x. P(x))";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   637
by (rtac (prem RS rev_mp) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   638
by (res_inst_tac [("n","k")] less_induct 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   639
by (rtac impI 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   640
by (rtac classical 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   641
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   642
by (assume_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   643
by (assume_tac 2);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   644
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   645
qed "LeastI";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   646
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   647
(*Proof is almost identical to the one above!*)
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3768
diff changeset
   648
val [prem] = goal thy "P(k::nat) ==> (LEAST x. P(x)) <= k";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   649
by (rtac (prem RS rev_mp) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   650
by (res_inst_tac [("n","k")] less_induct 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   651
by (rtac impI 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   652
by (rtac classical 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   653
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   654
by (assume_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   655
by (rtac le_refl 2);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   656
by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   657
qed "Least_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   658
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3768
diff changeset
   659
val [prem] = goal thy "k < (LEAST x. P(x)) ==> ~P(k::nat)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   660
by (rtac notI 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   661
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   662
by (rtac prem 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   663
qed "not_less_Least";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   664
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   665
qed_goalw "Least_Suc" thy [Least_nat_def]
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   666
 "!!P. [| ? n. P(Suc n); ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   667
 (fn _ => [
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   668
        rtac select_equality 1,
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   669
        fold_goals_tac [Least_nat_def],
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   670
        safe_tac (claset() addSEs [LeastI]),
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   671
        rename_tac "j" 1,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   672
        res_inst_tac [("n","j")] natE 1,
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   673
        Blast_tac 1,
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   674
        blast_tac (claset() addDs [Suc_less_SucD, not_less_Least]) 1,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   675
        rename_tac "k n" 1,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   676
        res_inst_tac [("n","k")] natE 1,
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   677
        Blast_tac 1,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   678
        hyp_subst_tac 1,
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   679
        rewtac Least_nat_def,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   680
        rtac (select_equality RS arg_cong RS sym) 1,
4153
e534c4c32d54 Ran expandshort, especially to introduce Safe_tac
paulson
parents: 4104
diff changeset
   681
        Safe_tac,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   682
        dtac Suc_mono 1,
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   683
        Blast_tac 1,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   684
        cut_facts_tac [less_linear] 1,
4153
e534c4c32d54 Ran expandshort, especially to introduce Safe_tac
paulson
parents: 4104
diff changeset
   685
        Safe_tac,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   686
        atac 2,
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   687
        Blast_tac 2,
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   688
        dtac Suc_mono 1,
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   689
        Blast_tac 1]);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   690
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   691
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   692
(*** Instantiation of transitivity prover ***)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   693
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   694
structure Less_Arith =
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   695
struct
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   696
val nat_leI = leI;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   697
val nat_leD = leD;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   698
val lessI = lessI;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   699
val zero_less_Suc = zero_less_Suc;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   700
val less_reflE = less_irrefl;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   701
val less_zeroE = less_zeroE;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   702
val less_incr = Suc_mono;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   703
val less_decr = Suc_less_SucD;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   704
val less_incr_rhs = Suc_mono RS Suc_lessD;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   705
val less_decr_lhs = Suc_lessD;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   706
val less_trans_Suc = less_trans_Suc;
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   707
val leI = Suc_leI RS (Suc_le_mono RS iffD1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   708
val not_lessI = leI RS leD
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   709
val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   710
  (fn _ => [etac swap2 1, etac leD 1]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   711
val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   712
  (fn _ => [etac less_SucE 1,
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   713
            blast_tac (claset() addSDs [Suc_less_SucD] addSEs [less_irrefl]
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   714
                              addDs [less_trans_Suc]) 1,
2935
998cb95fdd43 Yet more fast_tac->blast_tac, and other tidying
paulson
parents: 2922
diff changeset
   715
            assume_tac 1]);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   716
val leD = le_eq_less_Suc RS iffD1;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   717
val not_lessD = nat_leI RS leD;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   718
val not_leD = not_leE
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   719
val eqD1 = prove_goal thy  "!!n. m = n ==> m < Suc n"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   720
 (fn _ => [etac subst 1, rtac lessI 1]);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   721
val eqD2 = sym RS eqD1;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   722
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   723
fun is_zero(t) =  t = Const("0",Type("nat",[]));
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   724
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   725
fun nnb T = T = Type("fun",[Type("nat",[]),
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   726
                            Type("fun",[Type("nat",[]),
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   727
                                        Type("bool",[])])])
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   728
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   729
fun decomp_Suc(Const("Suc",_)$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   730
  | decomp_Suc t = (t,0);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   731
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   732
fun decomp2(rel,T,lhs,rhs) =
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   733
  if not(nnb T) then None else
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   734
  let val (x,i) = decomp_Suc lhs
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   735
      val (y,j) = decomp_Suc rhs
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   736
  in case rel of
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   737
       "op <"  => Some(x,i,"<",y,j)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   738
     | "op <=" => Some(x,i,"<=",y,j)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   739
     | "op ="  => Some(x,i,"=",y,j)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   740
     | _       => None
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   741
  end;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   742
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   743
fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   744
  | negate None = None;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   745
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   746
fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs)
2718
460fd0f8d478 Renamed constant "not" to "Not"
paulson
parents: 2680
diff changeset
   747
  | decomp(_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   748
      negate(decomp2(rel,T,lhs,rhs))
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   749
  | decomp _ = None
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   750
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   751
end;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   752
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   753
structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   754
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   755
open Trans_Tac;
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   756
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   757
(*** eliminates ~= in premises, which trans_tac cannot deal with ***)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   758
qed_goal "nat_neqE" thy
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   759
  "[| (m::nat) ~= n; m < n ==> P; n < m ==> P |] ==> P"
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   760
  (fn major::prems => [cut_facts_tac [less_linear] 1,
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   761
                       REPEAT(eresolve_tac ([disjE,major RS notE]@prems) 1)]);
2680
20fa49e610ca function nat_add_primrec added to allow primrec definitions over nat
pusch
parents: 2608
diff changeset
   762
20fa49e610ca function nat_add_primrec added to allow primrec definitions over nat
pusch
parents: 2608
diff changeset
   763
20fa49e610ca function nat_add_primrec added to allow primrec definitions over nat
pusch
parents: 2608
diff changeset
   764
20fa49e610ca function nat_add_primrec added to allow primrec definitions over nat
pusch
parents: 2608
diff changeset
   765
(* add function nat_add_primrec *) 
4032
4b1c69d8b767 For each datatype `t' there is now a theorem `split_t_case' of the form
nipkow
parents: 4000
diff changeset
   766
val (_, nat_add_primrec, _, _) = Datatype.add_datatype
3308
da002cef7090 Added overloaded function `size' for all datatypes.
nipkow
parents: 3292
diff changeset
   767
([], "nat", [("0", [], Mixfix ("0", [], max_pri)), ("Suc", [dtTyp ([],
3768
67f4ac759100 fully qualified names: Theory.add_XXX;
wenzelm
parents: 3718
diff changeset
   768
"nat")], NoSyn)]) (Theory.add_name "Arith" HOL.thy);
67f4ac759100 fully qualified names: Theory.add_XXX;
wenzelm
parents: 3718
diff changeset
   769
(*pretend Arith is part of the basic theory to fool package*)