author | paulson |
Wed, 27 Mar 1996 18:45:17 +0100 | |
changeset 1618 | 372880456b5b |
parent 1552 | 6f71b5d46700 |
child 1660 | 8cb42cd97579 |
permissions | -rw-r--r-- |
1465 | 1 |
(* Title: HOL/nat |
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ID: $Id$ |
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Author: Tobias Nipkow, Cambridge University Computer Laboratory |
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Copyright 1991 University of Cambridge |
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For nat.thy. Type nat is defined as a set (Nat) over the type ind. |
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*) |
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open Nat; |
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goal Nat.thy "mono(%X. {Zero_Rep} Un (Suc_Rep``X))"; |
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by (REPEAT (ares_tac [monoI, subset_refl, image_mono, Un_mono] 1)); |
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qed "Nat_fun_mono"; |
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val Nat_unfold = Nat_fun_mono RS (Nat_def RS def_lfp_Tarski); |
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(* Zero is a natural number -- this also justifies the type definition*) |
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goal Nat.thy "Zero_Rep: Nat"; |
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by (rtac (Nat_unfold RS ssubst) 1); |
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by (rtac (singletonI RS UnI1) 1); |
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qed "Zero_RepI"; |
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val prems = goal Nat.thy "i: Nat ==> Suc_Rep(i) : Nat"; |
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by (rtac (Nat_unfold RS ssubst) 1); |
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by (rtac (imageI RS UnI2) 1); |
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by (resolve_tac prems 1); |
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qed "Suc_RepI"; |
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(*** Induction ***) |
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val major::prems = goal Nat.thy |
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"[| i: Nat; P(Zero_Rep); \ |
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\ !!j. [| j: Nat; P(j) |] ==> P(Suc_Rep(j)) |] ==> P(i)"; |
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by (rtac ([Nat_def, Nat_fun_mono, major] MRS def_induct) 1); |
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by (fast_tac (set_cs addIs prems) 1); |
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qed "Nat_induct"; |
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val prems = goalw Nat.thy [Zero_def,Suc_def] |
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"[| P(0); \ |
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\ !!k. P(k) ==> P(Suc(k)) |] ==> P(n)"; |
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by (rtac (Rep_Nat_inverse RS subst) 1); (*types force good instantiation*) |
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by (rtac (Rep_Nat RS Nat_induct) 1); |
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by (REPEAT (ares_tac prems 1 |
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ORELSE eresolve_tac [Abs_Nat_inverse RS subst] 1)); |
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qed "nat_induct"; |
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(*Perform induction on n. *) |
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fun nat_ind_tac a i = |
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EVERY [res_inst_tac [("n",a)] nat_induct i, |
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rename_last_tac a ["1"] (i+1)]; |
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(*A special form of induction for reasoning about m<n and m-n*) |
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val prems = goal Nat.thy |
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"[| !!x. P x 0; \ |
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\ !!y. P 0 (Suc y); \ |
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\ !!x y. [| P x y |] ==> P (Suc x) (Suc y) \ |
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\ |] ==> P m n"; |
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by (res_inst_tac [("x","m")] spec 1); |
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by (nat_ind_tac "n" 1); |
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by (rtac allI 2); |
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by (nat_ind_tac "x" 2); |
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by (REPEAT (ares_tac (prems@[allI]) 1 ORELSE etac spec 1)); |
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qed "diff_induct"; |
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(*Case analysis on the natural numbers*) |
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val prems = goal Nat.thy |
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"[| n=0 ==> P; !!x. n = Suc(x) ==> P |] ==> P"; |
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by (subgoal_tac "n=0 | (EX x. n = Suc(x))" 1); |
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by (fast_tac (HOL_cs addSEs prems) 1); |
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by (nat_ind_tac "n" 1); |
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by (rtac (refl RS disjI1) 1); |
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by (fast_tac HOL_cs 1); |
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qed "natE"; |
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(*** Isomorphisms: Abs_Nat and Rep_Nat ***) |
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(*We can't take these properties as axioms, or take Abs_Nat==Inv(Rep_Nat), |
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since we assume the isomorphism equations will one day be given by Isabelle*) |
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goal Nat.thy "inj(Rep_Nat)"; |
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by (rtac inj_inverseI 1); |
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by (rtac Rep_Nat_inverse 1); |
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qed "inj_Rep_Nat"; |
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goal Nat.thy "inj_onto Abs_Nat Nat"; |
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by (rtac inj_onto_inverseI 1); |
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by (etac Abs_Nat_inverse 1); |
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qed "inj_onto_Abs_Nat"; |
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(*** Distinctness of constructors ***) |
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goalw Nat.thy [Zero_def,Suc_def] "Suc(m) ~= 0"; |
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by (rtac (inj_onto_Abs_Nat RS inj_onto_contraD) 1); |
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by (rtac Suc_Rep_not_Zero_Rep 1); |
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by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI, Zero_RepI] 1)); |
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qed "Suc_not_Zero"; |
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bind_thm ("Zero_not_Suc", (Suc_not_Zero RS not_sym)); |
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Addsimps [Suc_not_Zero,Zero_not_Suc]; |
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bind_thm ("Suc_neq_Zero", (Suc_not_Zero RS notE)); |
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val Zero_neq_Suc = sym RS Suc_neq_Zero; |
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(** Injectiveness of Suc **) |
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goalw Nat.thy [Suc_def] "inj(Suc)"; |
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by (rtac injI 1); |
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by (dtac (inj_onto_Abs_Nat RS inj_ontoD) 1); |
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by (REPEAT (resolve_tac [Rep_Nat, Suc_RepI] 1)); |
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by (dtac (inj_Suc_Rep RS injD) 1); |
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by (etac (inj_Rep_Nat RS injD) 1); |
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qed "inj_Suc"; |
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val Suc_inject = inj_Suc RS injD; |
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goal Nat.thy "(Suc(m)=Suc(n)) = (m=n)"; |
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by (EVERY1 [rtac iffI, etac Suc_inject, etac arg_cong]); |
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qed "Suc_Suc_eq"; |
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goal Nat.thy "n ~= Suc(n)"; |
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by (nat_ind_tac "n" 1); |
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by (ALLGOALS(asm_simp_tac (!simpset addsimps [Suc_Suc_eq]))); |
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qed "n_not_Suc_n"; |
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bind_thm ("Suc_n_not_n", n_not_Suc_n RS not_sym); |
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(*** nat_case -- the selection operator for nat ***) |
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goalw Nat.thy [nat_case_def] "nat_case a f 0 = a"; |
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by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1); |
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qed "nat_case_0"; |
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goalw Nat.thy [nat_case_def] "nat_case a f (Suc k) = f(k)"; |
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by (fast_tac (set_cs addIs [select_equality] |
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addEs [make_elim Suc_inject, Suc_neq_Zero]) 1); |
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qed "nat_case_Suc"; |
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(** Introduction rules for 'pred_nat' **) |
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goalw Nat.thy [pred_nat_def] "(n, Suc(n)) : pred_nat"; |
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by (fast_tac set_cs 1); |
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qed "pred_natI"; |
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val major::prems = goalw Nat.thy [pred_nat_def] |
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"[| p : pred_nat; !!x n. [| p = (n, Suc(n)) |] ==> R \ |
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\ |] ==> R"; |
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by (rtac (major RS CollectE) 1); |
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by (REPEAT (eresolve_tac ([asm_rl,exE]@prems) 1)); |
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qed "pred_natE"; |
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goalw Nat.thy [wf_def] "wf(pred_nat)"; |
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by (strip_tac 1); |
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by (nat_ind_tac "x" 1); |
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by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, |
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make_elim Suc_inject]) 2); |
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by (fast_tac (HOL_cs addSEs [mp, pred_natE, Pair_inject, Zero_neq_Suc]) 1); |
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qed "wf_pred_nat"; |
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(*** nat_rec -- by wf recursion on pred_nat ***) |
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(* The unrolling rule for nat_rec *) |
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goal Nat.thy |
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"(%n. nat_rec n c d) = wfrec pred_nat (%f. nat_case ?c (%m. ?d m (f m)))"; |
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by (simp_tac (HOL_ss addsimps [nat_rec_def]) 1); |
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bind_thm("nat_rec_unfold", wf_pred_nat RS |
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((result() RS eq_reflection) RS def_wfrec)); |
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(*--------------------------------------------------------------------------- |
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* Old: |
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* bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec))); |
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*---------------------------------------------------------------------------*) |
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(** conversion rules **) |
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goal Nat.thy "nat_rec 0 c h = c"; |
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by (rtac (nat_rec_unfold RS trans) 1); |
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by (simp_tac (!simpset addsimps [nat_case_0]) 1); |
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qed "nat_rec_0"; |
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goal Nat.thy "nat_rec (Suc n) c h = h n (nat_rec n c h)"; |
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by (rtac (nat_rec_unfold RS trans) 1); |
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by (simp_tac (!simpset addsimps [nat_case_Suc, pred_natI, cut_apply]) 1); |
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qed "nat_rec_Suc"; |
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(*These 2 rules ease the use of primitive recursion. NOTE USE OF == *) |
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val [rew] = goal Nat.thy |
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"[| !!n. f(n) == nat_rec n c h |] ==> f(0) = c"; |
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by (rewtac rew); |
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by (rtac nat_rec_0 1); |
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qed "def_nat_rec_0"; |
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val [rew] = goal Nat.thy |
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"[| !!n. f(n) == nat_rec n c h |] ==> f(Suc(n)) = h n (f n)"; |
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by (rewtac rew); |
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by (rtac nat_rec_Suc 1); |
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qed "def_nat_rec_Suc"; |
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fun nat_recs def = |
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[standard (def RS def_nat_rec_0), |
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standard (def RS def_nat_rec_Suc)]; |
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(*** Basic properties of "less than" ***) |
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(** Introduction properties **) |
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val prems = goalw Nat.thy [less_def] "[| i<j; j<k |] ==> i<(k::nat)"; |
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by (rtac (trans_trancl RS transD) 1); |
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by (resolve_tac prems 1); |
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by (resolve_tac prems 1); |
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qed "less_trans"; |
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goalw Nat.thy [less_def] "n < Suc(n)"; |
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by (rtac (pred_natI RS r_into_trancl) 1); |
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qed "lessI"; |
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Addsimps [lessI]; |
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(* i<j ==> i<Suc(j) *) |
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val less_SucI = lessI RSN (2, less_trans); |
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goal Nat.thy "0 < Suc(n)"; |
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by (nat_ind_tac "n" 1); |
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by (rtac lessI 1); |
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by (etac less_trans 1); |
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by (rtac lessI 1); |
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qed "zero_less_Suc"; |
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Addsimps [zero_less_Suc]; |
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(** Elimination properties **) |
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val prems = goalw Nat.thy [less_def] "n<m ==> ~ m<(n::nat)"; |
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by (fast_tac (HOL_cs addIs ([wf_pred_nat, wf_trancl RS wf_asym]@prems))1); |
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qed "less_not_sym"; |
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(* [| n(m; m(n |] ==> R *) |
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bind_thm ("less_asym", (less_not_sym RS notE)); |
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goalw Nat.thy [less_def] "~ n<(n::nat)"; |
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by (rtac notI 1); |
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by (etac (wf_pred_nat RS wf_trancl RS wf_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 Nat.thy "!!m. n<m ==> m ~= (n::nat)"; |
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by (fast_tac (HOL_cs addEs [less_irrefl]) 1); |
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qed "less_not_refl2"; |
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val major::prems = goalw Nat.thy [less_def] |
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"[| i<k; k=Suc(i) ==> P; !!j. [| i<j; k=Suc(j) |] ==> P \ |
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\ |] ==> P"; |
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by (rtac (major RS tranclE) 1); |
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by (REPEAT_FIRST (bound_hyp_subst_tac ORELSE' |
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eresolve_tac (prems@[pred_natE, Pair_inject]))); |
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by (rtac refl 1); |
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qed "lessE"; |
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goal Nat.thy "~ n<0"; |
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by (rtac notI 1); |
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by (etac lessE 1); |
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by (etac Zero_neq_Suc 1); |
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by (etac Zero_neq_Suc 1); |
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qed "not_less0"; |
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Addsimps [not_less0]; |
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(* n<0 ==> R *) |
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bind_thm ("less_zeroE", (not_less0 RS notE)); |
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val [major,less,eq] = goal Nat.thy |
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"[| m < Suc(n); m<n ==> P; m=n ==> P |] ==> P"; |
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by (rtac (major RS lessE) 1); |
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by (rtac eq 1); |
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by (fast_tac (HOL_cs addSDs [Suc_inject]) 1); |
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by (rtac less 1); |
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by (fast_tac (HOL_cs addSDs [Suc_inject]) 1); |
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qed "less_SucE"; |
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goal Nat.thy "(m < Suc(n)) = (m < n | m = n)"; |
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by (fast_tac (HOL_cs addSIs [lessI] |
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addEs [less_trans, less_SucE]) 1); |
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qed "less_Suc_eq"; |
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||
1301 | 287 |
val prems = goal Nat.thy "m<n ==> n ~= 0"; |
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by (res_inst_tac [("n","n")] natE 1); |
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by (cut_facts_tac prems 1); |
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by (ALLGOALS Asm_full_simp_tac); |
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qed "gr_implies_not0"; |
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Addsimps [gr_implies_not0]; |
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923 | 293 |
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(** Inductive (?) properties **) |
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val [prem] = goal Nat.thy "Suc(m) < n ==> m<n"; |
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by (rtac (prem RS rev_mp) 1); |
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by (nat_ind_tac "n" 1); |
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by (rtac impI 1); |
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by (etac less_zeroE 1); |
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by (fast_tac (HOL_cs addSIs [lessI RS less_SucI] |
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addSDs [Suc_inject] |
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addEs [less_trans, lessE]) 1); |
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qed "Suc_lessD"; |
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val [major,minor] = goal Nat.thy |
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"[| 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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val [major] = goal Nat.thy "Suc(m) < Suc(n) ==> m<n"; |
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by (rtac (major RS lessE) 1); |
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by (REPEAT (rtac lessI 1 |
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ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1)); |
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qed "Suc_less_SucD"; |
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val prems = goal Nat.thy "m<n ==> Suc(m) < Suc(n)"; |
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by (subgoal_tac "m<n --> Suc(m) < Suc(n)" 1); |
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by (fast_tac (HOL_cs addIs prems) 1); |
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by (nat_ind_tac "n" 1); |
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by (rtac impI 1); |
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by (etac less_zeroE 1); |
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by (fast_tac (HOL_cs addSIs [lessI] |
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addSDs [Suc_inject] |
329 |
addEs [less_trans, lessE]) 1); |
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qed "Suc_mono"; |
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332 |
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)"; |
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by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]); |
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qed "Suc_less_eq"; |
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1301 | 335 |
Addsimps [Suc_less_eq]; |
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goal Nat.thy "~(Suc(n) < n)"; |
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1618 | 338 |
by (fast_tac (HOL_cs addEs [Suc_lessD RS less_irrefl]) 1); |
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qed "not_Suc_n_less_n"; |
1301 | 340 |
Addsimps [not_Suc_n_less_n]; |
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342 |
goal Nat.thy "!!i. 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 addsimps [less_Suc_eq]))); |
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by (fast_tac (HOL_cs addDs [Suc_lessD]) 1); |
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240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
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346 |
qed_spec_mp "less_trans_Suc"; |
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348 |
(*"Less than" is a linear ordering*) |
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349 |
goal Nat.thy "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 (fast_tac (HOL_cs addIs [lessI, Suc_mono, less_SucI] addEs [lessE]) 1); |
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qed "less_linear"; |
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||
357 |
(*Can be used with less_Suc_eq to get n=m | n<m *) |
|
358 |
goal Nat.thy "(~ m < n) = (n < Suc(m))"; |
|
359 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
1552 | 360 |
by (ALLGOALS Asm_simp_tac); |
923 | 361 |
qed "not_less_eq"; |
362 |
||
363 |
(*Complete induction, aka course-of-values induction*) |
|
364 |
val prems = goalw Nat.thy [less_def] |
|
365 |
"[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)"; |
|
366 |
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1); |
|
367 |
by (eresolve_tac prems 1); |
|
368 |
qed "less_induct"; |
|
369 |
||
370 |
||
371 |
(*** Properties of <= ***) |
|
372 |
||
373 |
goalw Nat.thy [le_def] "0 <= n"; |
|
374 |
by (rtac not_less0 1); |
|
375 |
qed "le0"; |
|
376 |
||
1301 | 377 |
goalw Nat.thy [le_def] "~ Suc n <= n"; |
1552 | 378 |
by (Simp_tac 1); |
1301 | 379 |
qed "Suc_n_not_le_n"; |
380 |
||
381 |
goalw Nat.thy [le_def] "(i <= 0) = (i = 0)"; |
|
1552 | 382 |
by (nat_ind_tac "i" 1); |
383 |
by (ALLGOALS Asm_simp_tac); |
|
1301 | 384 |
qed "le_0"; |
385 |
||
386 |
Addsimps [less_not_refl, |
|
387 |
less_Suc_eq, le0, le_0, |
|
388 |
Suc_Suc_eq, Suc_n_not_le_n, |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
389 |
n_not_Suc_n, Suc_n_not_n, |
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
390 |
nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc]; |
923 | 391 |
|
392 |
(*Prevents simplification of f and g: much faster*) |
|
393 |
qed_goal "nat_case_weak_cong" Nat.thy |
|
394 |
"m=n ==> nat_case a f m = nat_case a f n" |
|
395 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
396 |
||
397 |
qed_goal "nat_rec_weak_cong" Nat.thy |
|
398 |
"m=n ==> nat_rec m a f = nat_rec n a f" |
|
399 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
400 |
||
1618 | 401 |
val prems = goalw Nat.thy [le_def] "~n<m ==> m<=(n::nat)"; |
923 | 402 |
by (resolve_tac prems 1); |
403 |
qed "leI"; |
|
404 |
||
1618 | 405 |
val prems = goalw Nat.thy [le_def] "m<=n ==> ~ n < (m::nat)"; |
923 | 406 |
by (resolve_tac prems 1); |
407 |
qed "leD"; |
|
408 |
||
409 |
val leE = make_elim leD; |
|
410 |
||
1618 | 411 |
goal Nat.thy "(~n<m) = (m<=(n::nat))"; |
412 |
by (fast_tac (HOL_cs addIs [leI] addEs [leE]) 1); |
|
413 |
qed "not_less_iff_le"; |
|
414 |
||
923 | 415 |
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)"; |
416 |
by (fast_tac HOL_cs 1); |
|
417 |
qed "not_leE"; |
|
418 |
||
419 |
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n"; |
|
1552 | 420 |
by (Simp_tac 1); |
1618 | 421 |
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1); |
923 | 422 |
qed "lessD"; |
423 |
||
424 |
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n"; |
|
1552 | 425 |
by (Asm_full_simp_tac 1); |
426 |
by (fast_tac HOL_cs 1); |
|
923 | 427 |
qed "Suc_leD"; |
428 |
||
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
429 |
goalw Nat.thy [le_def] "!!m. m <= n ==> m <= Suc n"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
430 |
by (fast_tac (HOL_cs addDs [Suc_lessD]) 1); |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
431 |
qed "le_SucI"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
432 |
Addsimps[le_SucI]; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
433 |
|
923 | 434 |
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)"; |
435 |
by (fast_tac (HOL_cs addEs [less_asym]) 1); |
|
436 |
qed "less_imp_le"; |
|
437 |
||
438 |
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)"; |
|
439 |
by (cut_facts_tac [less_linear] 1); |
|
1618 | 440 |
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1); |
923 | 441 |
qed "le_imp_less_or_eq"; |
442 |
||
443 |
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)"; |
|
444 |
by (cut_facts_tac [less_linear] 1); |
|
1618 | 445 |
by (fast_tac (HOL_cs addEs [less_irrefl,less_asym]) 1); |
923 | 446 |
by (flexflex_tac); |
447 |
qed "less_or_eq_imp_le"; |
|
448 |
||
449 |
goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)"; |
|
450 |
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1)); |
|
451 |
qed "le_eq_less_or_eq"; |
|
452 |
||
453 |
goal Nat.thy "n <= (n::nat)"; |
|
1552 | 454 |
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
923 | 455 |
qed "le_refl"; |
456 |
||
457 |
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)"; |
|
458 |
by (dtac le_imp_less_or_eq 1); |
|
459 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
460 |
qed "le_less_trans"; |
|
461 |
||
462 |
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)"; |
|
463 |
by (dtac le_imp_less_or_eq 1); |
|
464 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
465 |
qed "less_le_trans"; |
|
466 |
||
467 |
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)"; |
|
468 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
469 |
rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]); |
|
470 |
qed "le_trans"; |
|
471 |
||
472 |
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)"; |
|
473 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
1618 | 474 |
fast_tac (HOL_cs addEs [less_irrefl,less_asym])]); |
923 | 475 |
qed "le_anti_sym"; |
476 |
||
477 |
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)"; |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
478 |
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
923 | 479 |
qed "Suc_le_mono"; |
480 |
||
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
481 |
Addsimps [le_refl,Suc_le_mono]; |
1531 | 482 |
|
483 |
||
484 |
(** LEAST -- the least number operator **) |
|
485 |
||
486 |
val [prem1,prem2] = goalw Nat.thy [Least_def] |
|
487 |
"[| P(k); !!x. x<k ==> ~P(x) |] ==> (LEAST x.P(x)) = k"; |
|
488 |
by (rtac select_equality 1); |
|
489 |
by (fast_tac (HOL_cs addSIs [prem1,prem2]) 1); |
|
490 |
by (cut_facts_tac [less_linear] 1); |
|
491 |
by (fast_tac (HOL_cs addSIs [prem1] addSDs [prem2]) 1); |
|
492 |
qed "Least_equality"; |
|
493 |
||
494 |
val [prem] = goal Nat.thy "P(k) ==> P(LEAST x.P(x))"; |
|
495 |
by (rtac (prem RS rev_mp) 1); |
|
496 |
by (res_inst_tac [("n","k")] less_induct 1); |
|
497 |
by (rtac impI 1); |
|
498 |
by (rtac classical 1); |
|
499 |
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1); |
|
500 |
by (assume_tac 1); |
|
501 |
by (assume_tac 2); |
|
502 |
by (fast_tac HOL_cs 1); |
|
503 |
qed "LeastI"; |
|
504 |
||
505 |
(*Proof is almost identical to the one above!*) |
|
506 |
val [prem] = goal Nat.thy "P(k) ==> (LEAST x.P(x)) <= k"; |
|
507 |
by (rtac (prem RS rev_mp) 1); |
|
508 |
by (res_inst_tac [("n","k")] less_induct 1); |
|
509 |
by (rtac impI 1); |
|
510 |
by (rtac classical 1); |
|
511 |
by (res_inst_tac [("s","n")] (Least_equality RS ssubst) 1); |
|
512 |
by (assume_tac 1); |
|
513 |
by (rtac le_refl 2); |
|
514 |
by (fast_tac (HOL_cs addIs [less_imp_le,le_trans]) 1); |
|
515 |
qed "Least_le"; |
|
516 |
||
517 |
val [prem] = goal Nat.thy "k < (LEAST x.P(x)) ==> ~P(k)"; |
|
518 |
by (rtac notI 1); |
|
519 |
by (etac (rewrite_rule [le_def] Least_le RS notE) 1); |
|
520 |
by (rtac prem 1); |
|
521 |
qed "not_less_Least"; |