author | lcp |
Thu, 07 Oct 1993 10:20:30 +0100 | |
changeset 5 | 968d2dccf2de |
parent 2 | befa4e9f7c90 |
child 20 | f4f9946ad741 |
permissions | -rw-r--r-- |
0 | 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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val Nat_fun_mono = result(); |
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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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val Zero_RepI = result(); |
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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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val Suc_RepI = result(); |
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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 (major RS (Nat_def RS def_induct)) 1); |
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by (rtac Nat_fun_mono 1); |
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by (fast_tac (set_cs addIs prems) 1); |
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val Nat_induct = result(); |
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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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val nat_induct = result(); |
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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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val diff_induct = result(); |
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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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val natE = result(); |
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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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val inj_Rep_Nat = result(); |
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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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val inj_onto_Abs_Nat = result(); |
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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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val Suc_not_Zero = result(); |
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val Zero_not_Suc = standard (Suc_not_Zero RS not_sym); |
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val Suc_neq_Zero = standard (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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val inj_Suc = result(); |
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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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val Suc_Suc_eq = result(); |
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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 (HOL_ss addsimps [Zero_not_Suc,Suc_Suc_eq]))); |
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val n_not_Suc_n = result(); |
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(*** nat_case -- the selection operator for nat ***) |
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goalw Nat.thy [nat_case_def] "nat_case(0, a, f) = a"; |
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by (fast_tac (set_cs addIs [select_equality] addEs [Zero_neq_Suc]) 1); |
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val nat_case_0 = result(); |
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goalw Nat.thy [nat_case_def] "nat_case(Suc(k), a, f) = 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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val nat_case_Suc = result(); |
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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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val pred_natI = result(); |
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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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val pred_natE = result(); |
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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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val wf_pred_nat = result(); |
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(*** nat_rec -- by wf recursion on pred_nat ***) |
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val nat_rec_unfold = standard (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 (rtac nat_case_0 1); |
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val nat_rec_0 = result(); |
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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 (rtac (nat_case_Suc RS trans) 1); |
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by(simp_tac (HOL_ss addsimps [pred_natI,cut_apply]) 1); |
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val nat_rec_Suc = result(); |
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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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val def_nat_rec_0 = result(); |
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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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val def_nat_rec_Suc = result(); |
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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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val less_trans = result(); |
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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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val lessI = result(); |
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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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val zero_less_Suc = result(); |
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(** Elimination properties **) |
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goalw Nat.thy [less_def] "n<m --> ~ m<n::nat"; |
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by (rtac (wf_pred_nat RS wf_trancl RS wf_anti_sym RS notI RS impI) 1); |
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by (assume_tac 1); |
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by (assume_tac 1); |
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val less_not_sym = result(); |
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(* [| n<m; m<n |] ==> R *) |
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val less_anti_sym = standard (less_not_sym RS mp 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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val less_not_refl = result(); |
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(* n<n ==> R *) |
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val less_anti_refl = standard (less_not_refl RS notE); |
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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 (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1); |
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by (fast_tac (HOL_cs addSEs (prems@[pred_natE, Pair_inject])) 1); |
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val lessE = result(); |
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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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val not_less0 = result(); |
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(* n<0 ==> R *) |
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val less_zeroE = standard (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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val less_SucE = result(); |
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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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val less_Suc_eq = result(); |
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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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val Suc_lessD = result(); |
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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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val Suc_lessE = result(); |
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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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val Suc_less_SucD = result(); |
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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] |
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addEs [less_trans, lessE]) 1); |
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val Suc_mono = result(); |
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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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val Suc_less_eq = result(); |
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val [major] = goal Nat.thy "Suc(n)<n ==> P"; |
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by (rtac (major RS Suc_lessD RS less_anti_refl) 1); |
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val not_Suc_n_less_n = result(); |
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(*"Less than" is a linear ordering*) |
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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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val less_linear = result(); |
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(*Can be used with less_Suc_eq to get n=m | n<m *) |
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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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by(ALLGOALS(asm_simp_tac (HOL_ss addsimps |
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[not_less0,zero_less_Suc,Suc_less_eq]))); |
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val not_less_eq = result(); |
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(*Complete induction, aka course-of-values induction*) |
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val prems = goalw Nat.thy [less_def] |
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"[| !!n. [| ! m::nat. m<n --> P(m) |] ==> P(n) |] ==> P(n)"; |
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by (wf_ind_tac "n" [wf_pred_nat RS wf_trancl] 1); |
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by (eresolve_tac prems 1); |
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val less_induct = result(); |
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(*** Properties of <= ***) |
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goalw Nat.thy [le_def] "0 <= n"; |
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by (rtac not_less0 1); |
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val le0 = result(); |
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val nat_simps = [not_less0, less_not_refl, zero_less_Suc, lessI, |
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Suc_less_eq, less_Suc_eq, le0, |
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Suc_not_Zero, Zero_not_Suc, Suc_Suc_eq, |
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nat_case_0, nat_case_Suc, nat_rec_0, nat_rec_Suc]; |
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val nat_ss = pair_ss addsimps nat_simps; |
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||
2
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
354 |
(*Prevents simplification of f and g: much faster*) |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
355 |
val nat_case_weak_cong = prove_goal Nat.thy |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
356 |
"m=n ==> nat_case(m,a,f) = nat_case(n,a,f)" |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
357 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
358 |
|
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
359 |
val nat_rec_weak_cong = prove_goal Nat.thy |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
360 |
"m=n ==> nat_rec(m,a,f) = nat_rec(n,a,f)" |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
361 |
(fn [prem] => [rtac (prem RS arg_cong) 1]); |
befa4e9f7c90
Added weak congruence rules to HOL: if_weak_cong, case_weak_cong,
lcp
parents:
0
diff
changeset
|
362 |
|
0 | 363 |
val prems = goalw Nat.thy [le_def] "~(n<m) ==> m<=n::nat"; |
364 |
by (resolve_tac prems 1); |
|
365 |
val leI = result(); |
|
366 |
||
367 |
val prems = goalw Nat.thy [le_def] "m<=n ==> ~(n<m::nat)"; |
|
368 |
by (resolve_tac prems 1); |
|
369 |
val leD = result(); |
|
370 |
||
371 |
val leE = make_elim leD; |
|
372 |
||
373 |
goalw Nat.thy [le_def] "!!m. ~ m <= n ==> n<m::nat"; |
|
374 |
by (fast_tac HOL_cs 1); |
|
375 |
val not_leE = result(); |
|
376 |
||
377 |
goalw Nat.thy [le_def] "!!m. m < n ==> Suc(m) <= n"; |
|
378 |
by(simp_tac (HOL_ss addsimps [less_Suc_eq]) 1); |
|
379 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1); |
|
380 |
val lessD = result(); |
|
381 |
||
382 |
goalw Nat.thy [le_def] "!!m. m < n ==> m <= n::nat"; |
|
383 |
by (fast_tac (HOL_cs addEs [less_anti_sym]) 1); |
|
384 |
val less_imp_le = result(); |
|
385 |
||
386 |
goalw Nat.thy [le_def] "!!m. m <= n ==> m < n | m=n::nat"; |
|
387 |
by (cut_facts_tac [less_linear] 1); |
|
388 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1); |
|
389 |
val le_imp_less_or_eq = result(); |
|
390 |
||
391 |
goalw Nat.thy [le_def] "!!m. m<n | m=n ==> m <= n::nat"; |
|
392 |
by (cut_facts_tac [less_linear] 1); |
|
393 |
by (fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym]) 1); |
|
394 |
by (flexflex_tac); |
|
395 |
val less_or_eq_imp_le = result(); |
|
396 |
||
397 |
goal Nat.thy "(x <= y::nat) = (x < y | x=y)"; |
|
398 |
by (REPEAT(ares_tac [iffI,less_or_eq_imp_le,le_imp_less_or_eq] 1)); |
|
399 |
val le_eq_less_or_eq = result(); |
|
400 |
||
401 |
goal Nat.thy "n <= n::nat"; |
|
402 |
by(simp_tac (HOL_ss addsimps [le_eq_less_or_eq]) 1); |
|
403 |
val le_refl = result(); |
|
404 |
||
405 |
val prems = goal Nat.thy "!!i. [| i <= j; j < k |] ==> i < k::nat"; |
|
406 |
by (dtac le_imp_less_or_eq 1); |
|
407 |
by (fast_tac (HOL_cs addIs [less_trans]) 1); |
|
408 |
val le_less_trans = result(); |
|
409 |
||
410 |
goal Nat.thy "!!i. [| i <= j; j <= k |] ==> i <= k::nat"; |
|
411 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
412 |
rtac less_or_eq_imp_le, fast_tac (HOL_cs addIs [less_trans])]); |
|
413 |
val le_trans = result(); |
|
414 |
||
415 |
val prems = goal Nat.thy "!!m. [| m <= n; n <= m |] ==> m = n::nat"; |
|
416 |
by (EVERY1[dtac le_imp_less_or_eq, dtac le_imp_less_or_eq, |
|
417 |
fast_tac (HOL_cs addEs [less_anti_refl,less_anti_sym])]); |
|
418 |
val le_anti_sym = result(); |