src/HOL/NatDef.ML
author berghofe
Fri Jul 24 13:30:28 1998 +0200 (1998-07-24)
changeset 5187 55f07169cf5f
parent 5148 74919e8f221c
child 5316 7a8975451a89
permissions -rw-r--r--
Removed nat_case, nat_rec, and natE (now provided by datatype
package).
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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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Blast_tac proofs here can get PROOF FAILED of Ord theorems like order_refl
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and order_less_irrefl.  We do not add the "nat" versions to the basic claset
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because the type will be promoted to type class "order".
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*)
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Goal "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 "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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  res_inst_tac [("n",a)] nat_induct i  THEN  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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(*** 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 "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 "inj_on Abs_Nat Nat";
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by (rtac inj_on_inverseI 1);
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by (etac Abs_Nat_inverse 1);
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qed "inj_on_Abs_Nat";
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(*** Distinctness of constructors ***)
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Goalw [Zero_def,Suc_def] "Suc(m) ~= 0";
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by (rtac (inj_on_Abs_Nat RS inj_on_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 [Suc_def] "inj(Suc)";
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by (rtac injI 1);
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by (dtac (inj_on_Abs_Nat RS inj_onD) 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 "(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 "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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(*** Basic properties of "less than" ***)
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Goalw [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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(*Used in TFL/post.sml*)
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Goalw [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 [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 "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 [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 "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 "~ 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 "(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 "(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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Goal "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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(*"Less than" is a linear ordering*)
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Goal "m<n | m=n | n<(m::nat)";
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by (nat_ind_tac "m" 1);
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by (nat_ind_tac "n" 1);
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by (rtac (refl RS disjI1 RS disjI2) 1);
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by (rtac (zero_less_Suc RS disjI1) 1);
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by (blast_tac (claset() addIs [Suc_mono, less_SucI] addEs [lessE]) 1);
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qed "less_linear";
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Goal "!!m::nat. (m ~= n) = (m<n | n<m)";
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by (cut_facts_tac [less_linear] 1);
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by (blast_tac (claset() addSEs [less_irrefl]) 1);
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qed "nat_neq_iff";
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qed_goal "nat_less_cases" thy 
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   "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m"
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( fn [major,eqCase,lessCase] =>
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        [
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        (rtac (less_linear RS disjE) 1),
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        (etac disjE 2),
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        (etac lessCase 1),
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        (etac (sym RS eqCase) 1),
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        (etac major 1)
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        ]);
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(** Inductive (?) properties **)
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Goal "[| m<n; Suc m ~= n |] ==> Suc(m) < n";
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by (full_simp_tac (simpset() addsimps [nat_neq_iff]) 1);
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by (blast_tac (claset() addSEs [less_irrefl, less_SucE] addEs [less_asym]) 1);
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qed "Suc_lessI";
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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 "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 "(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";
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Addsimps [Suc_less_eq];
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Goal "~(Suc(n) < n)";
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by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
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qed "not_Suc_n_less_n";
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Addsimps [not_Suc_n_less_n];
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Goal "i<j ==> j<k --> Suc i < k";
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by (nat_ind_tac "k" 1);
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by (ALLGOALS (asm_simp_tac (simpset())));
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by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
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by (blast_tac (claset() addDs [Suc_lessD]) 1);
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qed_spec_mp "less_trans_Suc";
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(*Can be used with less_Suc_eq to get n=m | n<m *)
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Goal "(~ m < n) = (n < Suc(m))";
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by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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by (ALLGOALS Asm_simp_tac);
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qed "not_less_eq";
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(*Complete induction, aka course-of-values induction*)
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val prems = goalw thy [less_def]
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    "[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
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by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
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by (eresolve_tac prems 1);
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qed "less_induct";
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(*** Properties of <= ***)
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Goalw [le_def] "(m <= n) = (m < Suc n)";
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by (rtac not_less_eq 1);
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qed "le_eq_less_Suc";
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(*  m<=n ==> m < Suc n  *)
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bind_thm ("le_imp_less_Suc", le_eq_less_Suc RS iffD1);
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Goalw [le_def] "0 <= n";
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by (rtac not_less0 1);
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qed "le0";
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Goalw [le_def] "~ Suc n <= n";
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by (Simp_tac 1);
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qed "Suc_n_not_le_n";
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wenzelm@5069
   325
Goalw [le_def] "(i <= 0) = (i = 0)";
nipkow@2608
   326
by (nat_ind_tac "i" 1);
nipkow@2608
   327
by (ALLGOALS Asm_simp_tac);
nipkow@2608
   328
qed "le_0_eq";
paulson@4614
   329
AddIffs [le_0_eq];
nipkow@2608
   330
nipkow@2608
   331
Addsimps [(*less_Suc_eq, makes simpset non-confluent*) le0, le_0_eq,
nipkow@2608
   332
          Suc_n_not_le_n,
berghofe@5187
   333
          n_not_Suc_n, Suc_n_not_n];
nipkow@2608
   334
paulson@5143
   335
Goal "(m <= Suc(n)) = (m<=n | m = Suc n)";
wenzelm@4089
   336
by (simp_tac (simpset() addsimps [le_eq_less_Suc]) 1);
wenzelm@4089
   337
by (blast_tac (claset() addSEs [less_SucE] addIs [less_SucI]) 1);
paulson@3355
   338
qed "le_Suc_eq";
paulson@3355
   339
paulson@4614
   340
(* [| m <= Suc n;  m <= n ==> R;  m = Suc n ==> R |] ==> R *)
paulson@4614
   341
bind_thm ("le_SucE", le_Suc_eq RS iffD1 RS disjE);
paulson@4614
   342
nipkow@2608
   343
(*
wenzelm@5069
   344
Goal "(Suc m < n | Suc m = n) = (m < n)";
nipkow@2608
   345
by (stac (less_Suc_eq RS sym) 1);
nipkow@2608
   346
by (rtac Suc_less_eq 1);
nipkow@2608
   347
qed "Suc_le_eq";
nipkow@2608
   348
nipkow@2608
   349
this could make the simpset (with less_Suc_eq added again) more confluent,
nipkow@2608
   350
but less_Suc_eq makes additional problems with terms of the form 0 < Suc (...)
nipkow@2608
   351
*)
nipkow@2608
   352
nipkow@2608
   353
val prems = goalw thy [le_def] "~n<m ==> m<=(n::nat)";
nipkow@2608
   354
by (resolve_tac prems 1);
nipkow@2608
   355
qed "leI";
nipkow@2608
   356
nipkow@2608
   357
val prems = goalw thy [le_def] "m<=n ==> ~ n < (m::nat)";
nipkow@2608
   358
by (resolve_tac prems 1);
nipkow@2608
   359
qed "leD";
nipkow@2608
   360
nipkow@2608
   361
val leE = make_elim leD;
nipkow@2608
   362
wenzelm@5069
   363
Goal "(~n<m) = (m<=(n::nat))";
wenzelm@4089
   364
by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
nipkow@2608
   365
qed "not_less_iff_le";
nipkow@2608
   366
paulson@5143
   367
Goalw [le_def] "~ m <= n ==> n<(m::nat)";
paulson@2891
   368
by (Blast_tac 1);
nipkow@2608
   369
qed "not_leE";
nipkow@2608
   370
wenzelm@5069
   371
Goalw [le_def] "(~n<=m) = (m<(n::nat))";
paulson@4599
   372
by (Simp_tac 1);
paulson@4599
   373
qed "not_le_iff_less";
paulson@4599
   374
paulson@5143
   375
Goalw [le_def] "m < n ==> Suc(m) <= n";
wenzelm@4089
   376
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
wenzelm@4089
   377
by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
paulson@3343
   378
qed "Suc_leI";  (*formerly called lessD*)
nipkow@2608
   379
paulson@5143
   380
Goalw [le_def] "Suc(m) <= n ==> m <= n";
wenzelm@4089
   381
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
nipkow@2608
   382
qed "Suc_leD";
nipkow@2608
   383
nipkow@2608
   384
(* stronger version of Suc_leD *)
paulson@5148
   385
Goalw [le_def] "Suc m <= n ==> m < n";
wenzelm@4089
   386
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
nipkow@2608
   387
by (cut_facts_tac [less_linear] 1);
paulson@2891
   388
by (Blast_tac 1);
nipkow@2608
   389
qed "Suc_le_lessD";
nipkow@2608
   390
wenzelm@5069
   391
Goal "(Suc m <= n) = (m < n)";
wenzelm@4089
   392
by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
nipkow@2608
   393
qed "Suc_le_eq";
nipkow@2608
   394
paulson@5143
   395
Goalw [le_def] "m <= n ==> m <= Suc n";
wenzelm@4089
   396
by (blast_tac (claset() addDs [Suc_lessD]) 1);
nipkow@2608
   397
qed "le_SucI";
nipkow@2608
   398
Addsimps[le_SucI];
nipkow@2608
   399
nipkow@2608
   400
bind_thm ("le_Suc", not_Suc_n_less_n RS leI);
nipkow@2608
   401
paulson@5143
   402
Goalw [le_def] "m < n ==> m <= (n::nat)";
wenzelm@4089
   403
by (blast_tac (claset() addEs [less_asym]) 1);
nipkow@2608
   404
qed "less_imp_le";
nipkow@2608
   405
paulson@3343
   406
(** Equivalence of m<=n and  m<n | m=n **)
paulson@3343
   407
paulson@5143
   408
Goalw [le_def] "m <= n ==> m < n | m=(n::nat)";
nipkow@2608
   409
by (cut_facts_tac [less_linear] 1);
wenzelm@4089
   410
by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
nipkow@2608
   411
qed "le_imp_less_or_eq";
nipkow@2608
   412
paulson@5143
   413
Goalw [le_def] "m<n | m=n ==> m <=(n::nat)";
nipkow@2608
   414
by (cut_facts_tac [less_linear] 1);
wenzelm@4089
   415
by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
nipkow@2608
   416
qed "less_or_eq_imp_le";
nipkow@2608
   417
wenzelm@5069
   418
Goal "(m <= (n::nat)) = (m < n | m=n)";
nipkow@2608
   419
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1));
nipkow@2608
   420
qed "le_eq_less_or_eq";
nipkow@2608
   421
paulson@4635
   422
(*Useful with Blast_tac.   m=n ==> m<=n *)
paulson@4635
   423
bind_thm ("eq_imp_le", disjI2 RS less_or_eq_imp_le);
paulson@4635
   424
wenzelm@5069
   425
Goal "n <= (n::nat)";
wenzelm@4089
   426
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
nipkow@2608
   427
qed "le_refl";
nipkow@2608
   428
paulson@5143
   429
Goal "[| i <= j; j < k |] ==> i < (k::nat)";
paulson@4468
   430
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson@4468
   431
	                addIs [less_trans]) 1);
nipkow@2608
   432
qed "le_less_trans";
nipkow@2608
   433
paulson@5143
   434
Goal "[| i < j; j <= k |] ==> i < (k::nat)";
paulson@4468
   435
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson@4468
   436
	                addIs [less_trans]) 1);
nipkow@2608
   437
qed "less_le_trans";
nipkow@2608
   438
paulson@5143
   439
Goal "[| i <= j; j <= k |] ==> i <= (k::nat)";
paulson@4468
   440
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson@4468
   441
	                addIs [less_or_eq_imp_le, less_trans]) 1);
nipkow@2608
   442
qed "le_trans";
nipkow@2608
   443
paulson@5143
   444
Goal "[| m <= n; n <= m |] ==> m = (n::nat)";
paulson@4468
   445
(*order_less_irrefl could make this proof fail*)
paulson@4468
   446
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson@4468
   447
	                addSEs [less_irrefl] addEs [less_asym]) 1);
nipkow@2608
   448
qed "le_anti_sym";
nipkow@2608
   449
wenzelm@5069
   450
Goal "(Suc(n) <= Suc(m)) = (n <= m)";
wenzelm@4089
   451
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
nipkow@2608
   452
qed "Suc_le_mono";
nipkow@2608
   453
nipkow@2608
   454
AddIffs [Suc_le_mono];
nipkow@2608
   455
nipkow@2608
   456
(* Axiom 'order_le_less' of class 'order': *)
wenzelm@5069
   457
Goal "(m::nat) < n = (m <= n & m ~= n)";
paulson@4737
   458
by (simp_tac (simpset() addsimps [le_def, nat_neq_iff]) 1);
paulson@4737
   459
by (blast_tac (claset() addSEs [less_asym]) 1);
nipkow@2608
   460
qed "nat_less_le";
nipkow@2608
   461
nipkow@4640
   462
(* Axiom 'linorder_linear' of class 'linorder': *)
wenzelm@5069
   463
Goal "(m::nat) <= n | n <= m";
nipkow@4640
   464
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
nipkow@4640
   465
by (cut_facts_tac [less_linear] 1);
wenzelm@5132
   466
by (Blast_tac 1);
nipkow@4640
   467
qed "nat_le_linear";
nipkow@4640
   468
nipkow@4640
   469
nipkow@4640
   470
(** max
paulson@4599
   471
wenzelm@5069
   472
Goalw [max_def] "!!z::nat. (z <= max x y) = (z <= x | z <= y)";
nipkow@4686
   473
by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
paulson@4599
   474
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
paulson@4599
   475
qed "le_max_iff_disj";
paulson@4599
   476
wenzelm@5069
   477
Goalw [max_def] "!!z::nat. (max x y <= z) = (x <= z & y <= z)";
nipkow@4686
   478
by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
paulson@4599
   479
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
paulson@4599
   480
qed "max_le_iff_conj";
paulson@4599
   481
paulson@4599
   482
paulson@4599
   483
(** min **)
paulson@4599
   484
wenzelm@5069
   485
Goalw [min_def] "!!z::nat. (z <= min x y) = (z <= x & z <= y)";
nipkow@4686
   486
by (simp_tac (simpset() addsimps [not_le_iff_less]) 1);
paulson@4599
   487
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
paulson@4599
   488
qed "le_min_iff_conj";
paulson@4599
   489
wenzelm@5069
   490
Goalw [min_def] "!!z::nat. (min x y <= z) = (x <= z | y <= z)";
nipkow@4686
   491
by (simp_tac (simpset() addsimps [not_le_iff_less] addsplits) 1);
paulson@4599
   492
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 1);
paulson@4599
   493
qed "min_le_iff_disj";
nipkow@4640
   494
 **)
paulson@4599
   495
nipkow@2608
   496
(** LEAST -- the least number operator **)
nipkow@2608
   497
wenzelm@5069
   498
Goal "(! m::nat. P m --> n <= m) = (! m. m < n --> ~ P m)";
wenzelm@4089
   499
by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
nipkow@3143
   500
val lemma = result();
nipkow@3143
   501
nipkow@3143
   502
(* This is an old def of Least for nat, which is derived for compatibility *)
wenzelm@5069
   503
Goalw [Least_def]
nipkow@3143
   504
  "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
wenzelm@4089
   505
by (simp_tac (simpset() addsimps [lemma]) 1);
nipkow@3143
   506
qed "Least_nat_def";
nipkow@3143
   507
nipkow@3143
   508
val [prem1,prem2] = goalw thy [Least_nat_def]
wenzelm@3842
   509
    "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
nipkow@2608
   510
by (rtac select_equality 1);
wenzelm@4089
   511
by (blast_tac (claset() addSIs [prem1,prem2]) 1);
nipkow@2608
   512
by (cut_facts_tac [less_linear] 1);
wenzelm@4089
   513
by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
nipkow@2608
   514
qed "Least_equality";
nipkow@2608
   515
wenzelm@3842
   516
val [prem] = goal thy "P(k::nat) ==> P(LEAST x. P(x))";
nipkow@2608
   517
by (rtac (prem RS rev_mp) 1);
nipkow@2608
   518
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@2608
   519
by (rtac impI 1);
nipkow@2608
   520
by (rtac classical 1);
nipkow@2608
   521
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@2608
   522
by (assume_tac 1);
nipkow@2608
   523
by (assume_tac 2);
paulson@2891
   524
by (Blast_tac 1);
nipkow@2608
   525
qed "LeastI";
nipkow@2608
   526
nipkow@2608
   527
(*Proof is almost identical to the one above!*)
wenzelm@3842
   528
val [prem] = goal thy "P(k::nat) ==> (LEAST x. P(x)) <= k";
nipkow@2608
   529
by (rtac (prem RS rev_mp) 1);
nipkow@2608
   530
by (res_inst_tac [("n","k")] less_induct 1);
nipkow@2608
   531
by (rtac impI 1);
nipkow@2608
   532
by (rtac classical 1);
nipkow@2608
   533
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
nipkow@2608
   534
by (assume_tac 1);
nipkow@2608
   535
by (rtac le_refl 2);
wenzelm@4089
   536
by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
nipkow@2608
   537
qed "Least_le";
nipkow@2608
   538
wenzelm@3842
   539
val [prem] = goal thy "k < (LEAST x. P(x)) ==> ~P(k::nat)";
nipkow@2608
   540
by (rtac notI 1);
nipkow@2608
   541
by (etac (rewrite_rule [le_def] Least_le RS notE) 1);
nipkow@2608
   542
by (rtac prem 1);
nipkow@2608
   543
qed "not_less_Least";
nipkow@2608
   544
nipkow@2608
   545
(*** Instantiation of transitivity prover ***)
nipkow@2608
   546
nipkow@2608
   547
structure Less_Arith =
nipkow@2608
   548
struct
nipkow@2608
   549
val nat_leI = leI;
nipkow@2608
   550
val nat_leD = leD;
nipkow@2608
   551
val lessI = lessI;
nipkow@2608
   552
val zero_less_Suc = zero_less_Suc;
nipkow@2608
   553
val less_reflE = less_irrefl;
nipkow@2608
   554
val less_zeroE = less_zeroE;
nipkow@2608
   555
val less_incr = Suc_mono;
nipkow@2608
   556
val less_decr = Suc_less_SucD;
nipkow@2608
   557
val less_incr_rhs = Suc_mono RS Suc_lessD;
nipkow@2608
   558
val less_decr_lhs = Suc_lessD;
nipkow@2608
   559
val less_trans_Suc = less_trans_Suc;
paulson@3343
   560
val leI = Suc_leI RS (Suc_le_mono RS iffD1);
nipkow@2608
   561
val not_lessI = leI RS leD
nipkow@2608
   562
val not_leI = prove_goal thy "!!m::nat. n < m ==> ~ m <= n"
nipkow@2608
   563
  (fn _ => [etac swap2 1, etac leD 1]);
nipkow@2608
   564
val eqI = prove_goal thy "!!m. [| m < Suc n; n < Suc m |] ==> m=n"
nipkow@2608
   565
  (fn _ => [etac less_SucE 1,
wenzelm@4089
   566
            blast_tac (claset() addSDs [Suc_less_SucD] addSEs [less_irrefl]
paulson@2891
   567
                              addDs [less_trans_Suc]) 1,
paulson@2935
   568
            assume_tac 1]);
nipkow@2608
   569
val leD = le_eq_less_Suc RS iffD1;
nipkow@2608
   570
val not_lessD = nat_leI RS leD;
nipkow@2608
   571
val not_leD = not_leE
nipkow@2608
   572
val eqD1 = prove_goal thy  "!!n. m = n ==> m < Suc n"
nipkow@2608
   573
 (fn _ => [etac subst 1, rtac lessI 1]);
nipkow@2608
   574
val eqD2 = sym RS eqD1;
nipkow@2608
   575
nipkow@2608
   576
fun is_zero(t) =  t = Const("0",Type("nat",[]));
nipkow@2608
   577
nipkow@2608
   578
fun nnb T = T = Type("fun",[Type("nat",[]),
nipkow@2608
   579
                            Type("fun",[Type("nat",[]),
nipkow@2608
   580
                                        Type("bool",[])])])
nipkow@2608
   581
nipkow@2608
   582
fun decomp_Suc(Const("Suc",_)$t) = let val (a,i) = decomp_Suc t in (a,i+1) end
nipkow@2608
   583
  | decomp_Suc t = (t,0);
nipkow@2608
   584
nipkow@2608
   585
fun decomp2(rel,T,lhs,rhs) =
nipkow@2608
   586
  if not(nnb T) then None else
nipkow@2608
   587
  let val (x,i) = decomp_Suc lhs
nipkow@2608
   588
      val (y,j) = decomp_Suc rhs
nipkow@2608
   589
  in case rel of
nipkow@2608
   590
       "op <"  => Some(x,i,"<",y,j)
nipkow@2608
   591
     | "op <=" => Some(x,i,"<=",y,j)
nipkow@2608
   592
     | "op ="  => Some(x,i,"=",y,j)
nipkow@2608
   593
     | _       => None
nipkow@2608
   594
  end;
nipkow@2608
   595
nipkow@2608
   596
fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
nipkow@2608
   597
  | negate None = None;
nipkow@2608
   598
nipkow@2608
   599
fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs)
paulson@2718
   600
  | decomp(_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
nipkow@2608
   601
      negate(decomp2(rel,T,lhs,rhs))
nipkow@2608
   602
  | decomp _ = None
nipkow@2608
   603
nipkow@2608
   604
end;
nipkow@2608
   605
nipkow@2608
   606
structure Trans_Tac = Trans_Tac_Fun(Less_Arith);
nipkow@2608
   607
nipkow@2608
   608
open Trans_Tac;
nipkow@2608
   609
nipkow@2608
   610
(*** eliminates ~= in premises, which trans_tac cannot deal with ***)
paulson@4737
   611
bind_thm("nat_neqE", nat_neq_iff RS iffD1 RS disjE);