author | nipkow |
Sun, 12 Nov 1995 16:29:12 +0100 | |
changeset 1327 | 6c29cfab679c |
parent 1301 | 42782316d510 |
child 1465 | 5d7a7e439cec |
permissions | -rw-r--r-- |
923 | 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; |
923 | 116 |
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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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val 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 \ |
923 | 147 |
\ |] ==> 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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bind_thm ("nat_rec_unfold", (wf_pred_nat RS (nat_rec_def RS def_wfrec))); |
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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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1301 | 208 |
Addsimps [lessI]; |
923 | 209 |
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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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1301 | 219 |
Addsimps [zero_less_Suc]; |
923 | 220 |
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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_anti_refl) 1); |
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qed "less_not_refl"; |
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(* n(n ==> R *) |
923 | 236 |
bind_thm ("less_anti_refl", (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_anti_refl]) 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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923 | 250 |
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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1301 | 258 |
Addsimps [not_less0]; |
923 | 259 |
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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 | 277 |
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 | 283 |
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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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297 |
"[| 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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305 |
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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307 |
by (REPEAT (rtac lessI 1 |
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308 |
ORELSE eresolve_tac [make_elim Suc_inject, ssubst, Suc_lessD] 1)); |
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309 |
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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316 |
by (etac less_zeroE 1); |
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317 |
by (fast_tac (HOL_cs addSIs [lessI] |
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318 |
addSDs [Suc_inject] |
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319 |
addEs [less_trans, lessE]) 1); |
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320 |
qed "Suc_mono"; |
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321 |
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322 |
goal Nat.thy "(Suc(m) < Suc(n)) = (m<n)"; |
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323 |
by (EVERY1 [rtac iffI, etac Suc_less_SucD, etac Suc_mono]); |
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324 |
qed "Suc_less_eq"; |
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1301 | 325 |
Addsimps [Suc_less_eq]; |
923 | 326 |
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327 |
goal Nat.thy "~(Suc(n) < n)"; |
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328 |
by(fast_tac (HOL_cs addEs [Suc_lessD RS less_anti_refl]) 1); |
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329 |
qed "not_Suc_n_less_n"; |
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1301 | 330 |
Addsimps [not_Suc_n_less_n]; |
331 |
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332 |
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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336 |
bind_thm("less_trans_Suc",result() RS mp); |
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923 | 337 |
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338 |
(*"Less than" is a linear ordering*) |
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339 |
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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342 |
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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(*Can be used with less_Suc_eq to get n=m | n<m *) |
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348 |
goal Nat.thy "(~ m < n) = (n < Suc(m))"; |
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by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
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1301 | 350 |
by(ALLGOALS Asm_simp_tac); |
923 | 351 |
qed "not_less_eq"; |
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353 |
(*Complete induction, aka course-of-values induction*) |
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354 |
val prems = goalw Nat.thy [less_def] |
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355 |
"[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)"; |
|
356 |
by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1); |
|
357 |
by (eresolve_tac prems 1); |
|
358 |
qed "less_induct"; |
|
359 |
||
360 |
||
361 |
(*** Properties of <= ***) |
|
362 |
||
363 |
goalw Nat.thy [le_def] "0 <= n"; |
|
364 |
by (rtac not_less0 1); |
|
365 |
qed "le0"; |
|
366 |
||
1301 | 367 |
goalw Nat.thy [le_def] "~ Suc n <= n"; |
368 |
by(Simp_tac 1); |
|
369 |
qed "Suc_n_not_le_n"; |
|
370 |
||
371 |
goalw Nat.thy [le_def] "(i <= 0) = (i = 0)"; |
|
372 |
by(nat_ind_tac "i" 1); |
|
373 |
by(ALLGOALS Asm_simp_tac); |
|
374 |
qed "le_0"; |
|
375 |
||
376 |
Addsimps [less_not_refl, |
|
377 |
less_Suc_eq, le0, le_0, |
|
378 |
Suc_Suc_eq, Suc_n_not_le_n, |
|
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parents:
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diff
changeset
|
379 |
n_not_Suc_n, Suc_n_not_n, |
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parents:
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diff
changeset
|
380 |
nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc]; |
923 | 381 |
|
382 |
(*Prevents simplification of f and g: much faster*) |
|
383 |
qed_goal "nat_case_weak_cong" Nat.thy |
|
384 |
"m=n ==> nat_case a f m = nat_case a f n" |
|
385 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
386 |
||
387 |
qed_goal "nat_rec_weak_cong" Nat.thy |
|
388 |
"m=n ==> nat_rec m a f = nat_rec n a f" |
|
389 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
|
390 |
||
391 |
val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=(n::nat)"; |
|
392 |
by (resolve_tac prems 1); |
|
393 |
qed "leI"; |
|
394 |
||
395 |
val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<(m::nat))"; |
|
396 |
by (resolve_tac prems 1); |
|
397 |
qed "leD"; |
|
398 |
||
399 |
val leE = make_elim leD; |
|
400 |
||
401 |
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<(m::nat)"; |
|
402 |
by (fast_tac HOL_cs 1); |
|
403 |
qed "not_leE"; |
|
404 |
||
405 |
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n"; |
|
1264
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clasohm
parents:
1024
diff
changeset
|
406 |
by(Simp_tac 1); |
923 | 407 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1); |
408 |
qed "lessD"; |
|
409 |
||
410 |
goalw Nat.thy [le_def] "!!m. Suc(m) <= n ==> m <= n"; |
|
1264
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clasohm
parents:
1024
diff
changeset
|
411 |
by(Asm_full_simp_tac 1); |
923 | 412 |
by(fast_tac HOL_cs 1); |
413 |
qed "Suc_leD"; |
|
414 |
||
1327
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
415 |
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
|
416 |
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
|
417 |
qed "le_SucI"; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
418 |
Addsimps[le_SucI]; |
6c29cfab679c
added new arithmetic lemmas and the functions take and drop.
nipkow
parents:
1301
diff
changeset
|
419 |
|
923 | 420 |
goalw Nat.thy [le_def] "!!m. m < n ==> m <= (n::nat)"; |
421 |
by (fast_tac (HOL_cs addEs [less_asym]) 1); |
|
422 |
qed "less_imp_le"; |
|
423 |
||
424 |
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=(n::nat)"; |
|
425 |
by (cut_facts_tac [less_linear] 1); |
|
426 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1); |
|
427 |
qed "le_imp_less_or_eq"; |
|
428 |
||
429 |
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <=(n::nat)"; |
|
430 |
by (cut_facts_tac [less_linear] 1); |
|
431 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_asym]) 1); |
|
432 |
by (flexflex_tac); |
|
433 |
qed "less_or_eq_imp_le"; |
|
434 |
||
435 |
goal Nat.thy "(x <= (y::nat)) = (x < y | x=y)"; |
|
436 |
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1)); |
|
437 |
qed "le_eq_less_or_eq"; |
|
438 |
||
439 |
goal Nat.thy "n <= (n::nat)"; |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
440 |
by(simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
923 | 441 |
qed "le_refl"; |
442 |
||
443 |
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < (k::nat)"; |
|
444 |
by (dtac le_imp_less_or_eq 1); |
|
445 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
446 |
qed "le_less_trans"; |
|
447 |
||
448 |
goal Nat.thy "!!i. [| i < j; j <= k |] ==> i < (k::nat)"; |
|
449 |
by (dtac le_imp_less_or_eq 1); |
|
450 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
451 |
qed "less_le_trans"; |
|
452 |
||
453 |
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= (k::nat)"; |
|
454 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
455 |
rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]); |
|
456 |
qed "le_trans"; |
|
457 |
||
458 |
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = (n::nat)"; |
|
459 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
460 |
fast_tac (HOL_cs addEs [less_anti_refl,less_asym])]); |
|
461 |
qed "le_anti_sym"; |
|
462 |
||
463 |
goal Nat.thy "(Suc(n) <= Suc(m)) = (n <= m)"; |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
464 |
by (simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
923 | 465 |
qed "Suc_le_mono"; |
466 |
||
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1024
diff
changeset
|
467 |
Addsimps [le_refl,Suc_le_mono]; |