src/HOL/NatDef.ML
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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 "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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bind_thm ("Nat_unfold", Nat_fun_mono RS (Nat_def RS def_lfp_unfold));
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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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Goal "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 (assume_tac 1);
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qed "Suc_RepI";
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(*** Induction ***)
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val major::prems = Goal
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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_lfp_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 [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
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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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bind_thm ("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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bind_thm ("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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Goalw [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 (assume_tac 1);
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by (assume_tac 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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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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Goalw [less_def] "n<m ==> ~ m<(n::nat)";
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by (blast_tac (claset() addIs [wf_pred_nat, wf_trancl RS wf_asym])1);
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qed "less_not_sym";
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(* [| n<m; ~P ==> m<n |] ==> P *)
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bind_thm ("less_asym", less_not_sym RS contrapos_np);
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Goalw [less_def] "~ n<(n::nat)";
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by (rtac (wf_pred_nat RS wf_trancl RS wf_not_refl) 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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AddSEs [less_irrefl];
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Goal "n<m ==> m ~= (n::nat)";
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by (Blast_tac 1);
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qed "less_not_refl2";
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(* s < t ==> s ~= t *)
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bind_thm ("less_not_refl3", less_not_refl2 RS not_sym);
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val major::prems = Goalw [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::nat)";
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by (blast_tac (claset() addEs [lessE]) 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
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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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   221
by (nat_ind_tac "n" 1);
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   222
by (rtac (refl RS disjI1 RS disjI2) 1);
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   223
by (rtac (zero_less_Suc RS disjI1) 1);
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   224
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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   229
by (Blast_tac 1);
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qed "nat_neq_iff";
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   231
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val [major,eqCase,lessCase] = Goal 
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   233
   "[| (m::nat)<n ==> P n m; m=n ==> P n m; n<m ==> P n m |] ==> P n m";
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   234
by (rtac (less_linear RS disjE) 1);
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   235
by (etac disjE 2);
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   236
by (etac lessCase 1);
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   237
by (etac (sym RS eqCase) 1);
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   238
by (etac major 1);
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   239
qed "nat_less_cases";
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   241
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   242
(** Inductive (?) properties **)
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   243
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   244
Goal "[| m<n; Suc m ~= n |] ==> Suc(m) < n";
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   245
by (full_simp_tac (simpset() addsimps [nat_neq_iff]) 1);
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   246
by (blast_tac (claset() addSEs [less_irrefl, less_SucE] addEs [less_asym]) 1);
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   247
qed "Suc_lessI";
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   248
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   249
Goal "Suc(m) < n ==> m<n";
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   250
by (etac rev_mp 1);
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   251
by (nat_ind_tac "n" 1);
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   252
by (ALLGOALS (fast_tac (claset() addSIs [lessI RS less_SucI]
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   253
                                 addEs  [less_trans, lessE])));
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   254
qed "Suc_lessD";
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   255
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   256
val [major,minor] = Goal 
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   257
    "[| Suc(i)<k;  !!j. [| i<j;  k=Suc(j) |] ==> P \
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   258
\    |] ==> P";
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   259
by (rtac (major RS lessE) 1);
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   260
by (etac (lessI RS minor) 1);
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   261
by (etac (Suc_lessD RS minor) 1);
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   262
by (assume_tac 1);
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   263
qed "Suc_lessE";
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   264
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   265
Goal "Suc(m) < Suc(n) ==> m<n";
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   266
by (blast_tac (claset() addEs [lessE, make_elim Suc_lessD]) 1);
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   267
qed "Suc_less_SucD";
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   268
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   269
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   270
Goal "(Suc(m) < Suc(n)) = (m<n)";
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   271
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]);
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   272
qed "Suc_less_eq";
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17325ee838ab Suc_less_eq now with AddIffs. How could this have been overlooked?
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   273
AddIffs [Suc_less_eq];
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   274
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   275
(*Goal "~(Suc(n) < n)";
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   276
by (blast_tac (claset() addEs [Suc_lessD RS less_irrefl]) 1);
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   277
qed "not_Suc_n_less_n";
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   278
Addsimps [not_Suc_n_less_n];*)
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   279
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   280
Goal "i<j ==> j<k --> Suc i < k";
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   281
by (nat_ind_tac "k" 1);
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   282
by (ALLGOALS (asm_simp_tac (simpset())));
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parents: 4737
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   283
by (asm_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
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parents: 4737
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   284
by (blast_tac (claset() addDs [Suc_lessD]) 1);
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   285
qed_spec_mp "less_trans_Suc";
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   286
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   287
(*Can be used with less_Suc_eq to get n=m | n<m *)
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   288
Goal "(~ m < n) = (n < Suc(m))";
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   289
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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parents:
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   290
by (ALLGOALS Asm_simp_tac);
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   291
qed "not_less_eq";
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   292
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   293
(*Complete induction, aka course-of-values induction*)
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   294
val prems = Goalw [less_def]
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   295
    "[| !!n. [| ALL m::nat. m<n --> P(m) |] ==> P(n) |]  ==>  P(n)";
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   296
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1);
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parents:
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   297
by (eresolve_tac prems 1);
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   298
qed "nat_less_induct";
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   299
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   300
(*** Properties of <= ***)
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   301
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   302
(*Was le_eq_less_Suc, but this orientation is more useful*)
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diff changeset
   303
Goalw [le_def] "(m < Suc n) = (m <= n)";
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parents: 5478
diff changeset
   304
by (rtac (not_less_eq RS sym) 1);
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parents: 5478
diff changeset
   305
qed "less_Suc_eq_le";
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   306
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45986997f1fe Renamed lessD to Suc_leI
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parents: 3308
diff changeset
   307
(*  m<=n ==> m < Suc n  *)
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parents: 5478
diff changeset
   308
bind_thm ("le_imp_less_Suc", less_Suc_eq_le RS iffD2);
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45986997f1fe Renamed lessD to Suc_leI
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parents: 3308
diff changeset
   309
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parents: 8741
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   310
Goalw [le_def] "(0::nat) <= n";
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parents:
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   311
by (rtac not_less0 1);
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parents:
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   312
qed "le0";
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diff changeset
   313
AddIffs [le0];
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parents:
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   314
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   315
Goalw [le_def] "~ Suc n <= n";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   316
by (Simp_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   317
qed "Suc_n_not_le_n";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   318
8942
6aad5381ba83 added type constraint ::nat because 0 is now overloaded
paulson
parents: 8741
diff changeset
   319
Goalw [le_def] "!!i::nat. (i <= 0) = (i = 0)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   320
by (nat_ind_tac "i" 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   321
by (ALLGOALS Asm_simp_tac);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   322
qed "le_0_eq";
4614
122015efd4e1 New AddIffs le_0_eq and neq0_conv
paulson
parents: 4599
diff changeset
   323
AddIffs [le_0_eq];
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   324
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   325
Goal "(m <= Suc(n)) = (m<=n | m = Suc n)";
5500
7e0ed3e31590 new theorem less_Suc_eq_le and le_simps
paulson
parents: 5478
diff changeset
   326
by (simp_tac (simpset() delsimps [less_Suc_eq_le]
7e0ed3e31590 new theorem less_Suc_eq_le and le_simps
paulson
parents: 5478
diff changeset
   327
			addsimps [less_Suc_eq_le RS sym, less_Suc_eq]) 1);
3355
0d955bcf8e0a New theorem le_Suc_eq
paulson
parents: 3343
diff changeset
   328
qed "le_Suc_eq";
0d955bcf8e0a New theorem le_Suc_eq
paulson
parents: 3343
diff changeset
   329
4614
122015efd4e1 New AddIffs le_0_eq and neq0_conv
paulson
parents: 4599
diff changeset
   330
(* [| 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
   331
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
   332
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   333
Goalw [le_def] "~n<m ==> m<=(n::nat)";
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   334
by (assume_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   335
qed "leI";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   336
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   337
Goalw [le_def] "m<=n ==> ~ n < (m::nat)";
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   338
by (assume_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   339
qed "leD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   340
9108
9fff97d29837 bind_thm(s);
wenzelm
parents: 8942
diff changeset
   341
bind_thm ("leE", make_elim leD);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   342
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   343
Goal "(~n<m) = (m<=(n::nat))";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   344
by (blast_tac (claset() addIs [leI] addEs [leE]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   345
qed "not_less_iff_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   346
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   347
Goalw [le_def] "~ m <= n ==> n<(m::nat)";
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   348
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   349
qed "not_leE";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   350
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   351
Goalw [le_def] "(~n<=m) = (m<(n::nat))";
4599
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   352
by (Simp_tac 1);
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   353
qed "not_le_iff_less";
3a4348a3d6ff New max, min theorems
paulson
parents: 4535
diff changeset
   354
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   355
Goalw [le_def] "m < n ==> Suc(m) <= n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   356
by (simp_tac (simpset() addsimps [less_Suc_eq]) 1);
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   357
by (blast_tac (claset() addSEs [less_irrefl,less_asym]) 1);
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   358
qed "Suc_leI";  (*formerly called lessD*)
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   359
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   360
Goalw [le_def] "Suc(m) <= n ==> m <= n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   361
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   362
qed "Suc_leD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   363
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   364
(* stronger version of Suc_leD *)
5148
74919e8f221c More tidying and removal of "\!\!... from Goal commands
paulson
parents: 5143
diff changeset
   365
Goalw [le_def] "Suc m <= n ==> m < n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   366
by (asm_full_simp_tac (simpset() addsimps [less_Suc_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   367
by (cut_facts_tac [less_linear] 1);
2891
d8f254ad1ab9 Calls Blast_tac
paulson
parents: 2718
diff changeset
   368
by (Blast_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   369
qed "Suc_le_lessD";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   370
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   371
Goal "(Suc m <= n) = (m < n)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   372
by (blast_tac (claset() addIs [Suc_leI, Suc_le_lessD]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   373
qed "Suc_le_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   374
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   375
Goalw [le_def] "m <= n ==> m <= Suc n";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   376
by (blast_tac (claset() addDs [Suc_lessD]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   377
qed "le_SucI";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   378
6109
82b50115564c Simplified arithmetic.
nipkow
parents: 6075
diff changeset
   379
(*bind_thm ("le_Suc", not_Suc_n_less_n RS leI);*)
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   380
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   381
Goalw [le_def] "m < n ==> m <= (n::nat)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   382
by (blast_tac (claset() addEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   383
qed "less_imp_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   384
5591
paulson
parents: 5500
diff changeset
   385
(*For instance, (Suc m < Suc n)  =   (Suc m <= n)  =  (m<n) *)
9108
9fff97d29837 bind_thm(s);
wenzelm
parents: 8942
diff changeset
   386
bind_thms ("le_simps", [less_imp_le, less_Suc_eq_le, Suc_le_eq]);
5591
paulson
parents: 5500
diff changeset
   387
5354
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   388
3343
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   389
(** Equivalence of m<=n and  m<n | m=n **)
45986997f1fe Renamed lessD to Suc_leI
paulson
parents: 3308
diff changeset
   390
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   391
Goalw [le_def] "m <= n ==> m < n | m=(n::nat)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   392
by (cut_facts_tac [less_linear] 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   393
by (blast_tac (claset() addEs [less_irrefl,less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   394
qed "le_imp_less_or_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   395
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   396
Goalw [le_def] "m<n | m=n ==> m <=(n::nat)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   397
by (cut_facts_tac [less_linear] 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   398
by (blast_tac (claset() addSEs [less_irrefl] addEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   399
qed "less_or_eq_imp_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   400
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   401
Goal "(m <= (n::nat)) = (m < n | m=n)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   402
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
   403
qed "le_eq_less_or_eq";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   404
4635
c448e09d0fca New theorem eq_imp_le
paulson
parents: 4614
diff changeset
   405
(*Useful with Blast_tac.   m=n ==> m<=n *)
c448e09d0fca New theorem eq_imp_le
paulson
parents: 4614
diff changeset
   406
bind_thm ("eq_imp_le", disjI2 RS less_or_eq_imp_le);
c448e09d0fca New theorem eq_imp_le
paulson
parents: 4614
diff changeset
   407
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   408
Goal "n <= (n::nat)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   409
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   410
qed "le_refl";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   411
5354
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   412
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   413
Goal "[| i <= j; j < k |] ==> i < (k::nat)";
4468
paulson
parents: 4423
diff changeset
   414
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   415
	                addIs [less_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   416
qed "le_less_trans";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   417
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   418
Goal "[| i < j; j <= k |] ==> i < (k::nat)";
4468
paulson
parents: 4423
diff changeset
   419
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   420
	                addIs [less_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   421
qed "less_le_trans";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   422
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   423
Goal "[| i <= j; j <= k |] ==> i <= (k::nat)";
4468
paulson
parents: 4423
diff changeset
   424
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   425
	                addIs [less_or_eq_imp_le, less_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   426
qed "le_trans";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   427
5143
b94cd208f073 Removal of leading "\!\!..." from most Goal commands
paulson
parents: 5132
diff changeset
   428
Goal "[| m <= n; n <= m |] ==> m = (n::nat)";
4468
paulson
parents: 4423
diff changeset
   429
(*order_less_irrefl could make this proof fail*)
paulson
parents: 4423
diff changeset
   430
by (blast_tac (claset() addSDs [le_imp_less_or_eq]
paulson
parents: 4423
diff changeset
   431
	                addSEs [less_irrefl] addEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   432
qed "le_anti_sym";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   433
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   434
Goal "(Suc(n) <= Suc(m)) = (n <= m)";
5500
7e0ed3e31590 new theorem less_Suc_eq_le and le_simps
paulson
parents: 5478
diff changeset
   435
by (simp_tac (simpset() addsimps le_simps) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   436
qed "Suc_le_mono";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   437
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   438
AddIffs [Suc_le_mono];
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   439
5500
7e0ed3e31590 new theorem less_Suc_eq_le and le_simps
paulson
parents: 5478
diff changeset
   440
(* Axiom 'order_less_le' of class 'order': *)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   441
Goal "(m::nat) < n = (m <= n & m ~= n)";
4737
4544290d5a6b The theorem nat_neqE, and some tidying
paulson
parents: 4686
diff changeset
   442
by (simp_tac (simpset() addsimps [le_def, nat_neq_iff]) 1);
4544290d5a6b The theorem nat_neqE, and some tidying
paulson
parents: 4686
diff changeset
   443
by (blast_tac (claset() addSEs [less_asym]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   444
qed "nat_less_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   445
5354
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   446
(* [| m <= n; m ~= n |] ==> m < n *)
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   447
bind_thm ("le_neq_implies_less", [nat_less_le, conjI] MRS iffD2);
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   448
4640
ac6cf9f18653 Congruence rules use == in premises now.
nipkow
parents: 4635
diff changeset
   449
(* Axiom 'linorder_linear' of class 'linorder': *)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   450
Goal "(m::nat) <= n | n <= m";
4640
ac6cf9f18653 Congruence rules use == in premises now.
nipkow
parents: 4635
diff changeset
   451
by (simp_tac (simpset() addsimps [le_eq_less_or_eq]) 1);
ac6cf9f18653 Congruence rules use == in premises now.
nipkow
parents: 4635
diff changeset
   452
by (cut_facts_tac [less_linear] 1);
5132
24f992a25adc isatool expandshort;
wenzelm
parents: 5069
diff changeset
   453
by (Blast_tac 1);
4640
ac6cf9f18653 Congruence rules use == in premises now.
nipkow
parents: 4635
diff changeset
   454
qed "nat_le_linear";
ac6cf9f18653 Congruence rules use == in premises now.
nipkow
parents: 4635
diff changeset
   455
5354
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   456
Goal "~ n < m ==> (n < Suc m) = (n = m)";
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   457
by (blast_tac (claset() addSEs [less_SucE]) 1);
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   458
qed "not_less_less_Suc_eq";
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   459
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   460
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   461
(*Rewrite (n < Suc m) to (n=m) if  ~ n<m or m<=n hold.
da63d9b35caf new theorems; adds [le_refl, less_imp_le] as simprules
paulson
parents: 5316
diff changeset
   462
  Not suitable as default simprules because they often lead to looping*)
9108
9fff97d29837 bind_thm(s);
wenzelm
parents: 8942
diff changeset
   463
bind_thms ("not_less_simps", [not_less_less_Suc_eq, leD RS not_less_less_Suc_eq]);
4640
ac6cf9f18653 Congruence rules use == in premises now.
nipkow
parents: 4635
diff changeset
   464
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   465
(** LEAST -- the least number operator **)
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   466
9160
7a98dbf3e579 using the new theorem wf_not_refl; tidied
paulson
parents: 9108
diff changeset
   467
Goal "(ALL m::nat. P m --> n <= m) = (ALL m. m < n --> ~ P m)";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   468
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
   469
val lemma = result();
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   470
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   471
(* This is an old def of Least for nat, which is derived for compatibility *)
5069
3ea049f7979d isatool fixgoal;
wenzelm
parents: 4830
diff changeset
   472
Goalw [Least_def]
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   473
  "(LEAST n::nat. P n) == (@n. P(n) & (ALL m. m < n --> ~P(m)))";
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   474
by (simp_tac (simpset() addsimps [lemma]) 1);
3143
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   475
qed "Least_nat_def";
d60e49b86c6a Modified def of Least, which, as Markus correctly complained, looked like
nipkow
parents: 3085
diff changeset
   476
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   477
val [prem1,prem2] = Goalw [Least_nat_def]
3842
b55686a7b22c fixed dots;
wenzelm
parents: 3768
diff changeset
   478
    "[| P(k::nat);  !!x. x<k ==> ~P(x) |] ==> (LEAST x. P(x)) = k";
9969
4753185f1dd2 renamed (most of...) the select rules
paulson
parents: 9870
diff changeset
   479
by (rtac some_equality 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   480
by (blast_tac (claset() addSIs [prem1,prem2]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   481
by (cut_facts_tac [less_linear] 1);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   482
by (blast_tac (claset() addSIs [prem1] addSDs [prem2]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   483
qed "Least_equality";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   484
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   485
Goal "P(k::nat) ==> P(LEAST x. P(x))";
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   486
by (etac rev_mp 1);
9870
2374ba026fc6 less_induct -> nat_less_induct
nipkow
parents: 9160
diff changeset
   487
by (res_inst_tac [("n","k")] nat_less_induct 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   488
by (rtac impI 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   489
by (rtac classical 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   490
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   491
by (assume_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   492
by (assume_tac 2);
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 "LeastI";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   495
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   496
(*Proof is almost identical to the one above!*)
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   497
Goal "P(k::nat) ==> (LEAST x. P(x)) <= k";
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   498
by (etac rev_mp 1);
9870
2374ba026fc6 less_induct -> nat_less_induct
nipkow
parents: 9160
diff changeset
   499
by (res_inst_tac [("n","k")] nat_less_induct 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   500
by (rtac impI 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   501
by (rtac classical 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   502
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   503
by (assume_tac 1);
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   504
by (rtac le_refl 2);
4089
96fba19bcbe2 isatool fixclasimp;
wenzelm
parents: 4032
diff changeset
   505
by (blast_tac (claset() addIs [less_imp_le,le_trans]) 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   506
qed "Least_le";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   507
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   508
Goal "k < (LEAST x. P(x)) ==> ~P(k::nat)";
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   509
by (rtac notI 1);
5316
7a8975451a89 even more tidying of Goal commands
paulson
parents: 5187
diff changeset
   510
by (etac (rewrite_rule [le_def] Least_le RS notE) 1 THEN assume_tac 1);
2608
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   511
qed "not_less_Least";
450c9b682a92 New class "order" and accompanying changes.
nipkow
parents:
diff changeset
   512
5983
79e301a6a51b At last: linear arithmetic for nat!
nipkow
parents: 5654
diff changeset
   513
(* [| m ~= n; m < n ==> P; n < m ==> P |] ==> P *)
4737
4544290d5a6b The theorem nat_neqE, and some tidying
paulson
parents: 4686
diff changeset
   514
bind_thm("nat_neqE", nat_neq_iff RS iffD1 RS disjE);
7064
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   515
9160
7a98dbf3e579 using the new theorem wf_not_refl; tidied
paulson
parents: 9108
diff changeset
   516
Goal "(S::nat set) ~= {} ==> EX x:S. ALL y:S. x <= y";
7064
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   517
by (cut_facts_tac [wf_pred_nat RS wf_trancl RS (wf_eq_minimal RS iffD1)] 1);
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   518
by (dres_inst_tac [("x","S")] spec 1);
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   519
by (Asm_full_simp_tac 1);
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   520
by (etac impE 1);
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   521
by (Force_tac 1);
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   522
by (force_tac (claset(), simpset() addsimps [less_eq,not_le_iff_less]) 1);
b053e0ab9f60 New lemmas by Stefan Merz.
nipkow
parents: 7030
diff changeset
   523
qed "nonempty_has_least";