src/HOL/Arith.ML
author nipkow
Wed Oct 16 10:37:17 1996 +0200 (1996-10-16)
changeset 2099 c5f004bfcbab
parent 2031 03a843f0f447
child 2498 7914881f47c0
permissions -rw-r--r--
Defined pred using nat_case rather than nat_rec.
Added expand_nat_case
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(*  Title:      HOL/Arith.ML
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    ID:         $Id$
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    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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    Copyright   1993  University of Cambridge
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Proofs about elementary arithmetic: addition, multiplication, etc.
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Tests definitions and simplifier.
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*)
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open Arith;
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(*** Basic rewrite rules for the arithmetic operators ***)
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goalw Arith.thy [pred_def] "pred 0 = 0";
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by(Simp_tac 1);
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qed "pred_0";
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goalw Arith.thy [pred_def] "pred(Suc n) = n";
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by(Simp_tac 1);
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qed "pred_Suc";
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val [add_0,add_Suc] = nat_recs add_def; 
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val [mult_0,mult_Suc] = nat_recs mult_def;
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store_thm("add_0",add_0);
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store_thm("add_Suc",add_Suc);
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store_thm("mult_0",mult_0);
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store_thm("mult_Suc",mult_Suc);
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Addsimps [pred_0,pred_Suc,add_0,add_Suc,mult_0,mult_Suc];
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(** pred **)
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val prems = goal Arith.thy "n ~= 0 ==> Suc(pred n) = n";
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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 "Suc_pred";
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Addsimps [Suc_pred];
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(** Difference **)
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bind_thm("diff_0", diff_def RS def_nat_rec_0);
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qed_goalw "diff_0_eq_0" Arith.thy [diff_def, pred_def]
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    "0 - n = 0"
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 (fn _ => [nat_ind_tac "n" 1,  ALLGOALS Asm_simp_tac]);
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(*Must simplify BEFORE the induction!!  (Else we get a critical pair)
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  Suc(m) - Suc(n)   rewrites to   pred(Suc(m) - n)  *)
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qed_goalw "diff_Suc_Suc" Arith.thy [diff_def, pred_def]
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    "Suc(m) - Suc(n) = m - n"
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 (fn _ =>
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  [Simp_tac 1, nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]);
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Addsimps [diff_0, diff_0_eq_0, diff_Suc_Suc];
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goal Arith.thy "!!k. 0<k ==> EX j. k = Suc(j)";
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by (etac rev_mp 1);
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by (nat_ind_tac "k" 1);
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by (Simp_tac 1);
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by (Fast_tac 1);
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val lemma = result();
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(* [| 0 < k; !!j. [| j: nat; k = succ(j) |] ==> Q |] ==> Q *)
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bind_thm ("zero_less_natE", lemma RS exE);
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(**** Inductive properties of the operators ****)
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(*** Addition ***)
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qed_goal "add_0_right" Arith.thy "m + 0 = m"
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 (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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qed_goal "add_Suc_right" Arith.thy "m + Suc(n) = Suc(m+n)"
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 (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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Addsimps [add_0_right,add_Suc_right];
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(*Associative law for addition*)
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qed_goal "add_assoc" Arith.thy "(m + n) + k = m + ((n + k)::nat)"
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 (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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(*Commutative law for addition*)  
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qed_goal "add_commute" Arith.thy "m + n = n + (m::nat)"
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 (fn _ =>  [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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qed_goal "add_left_commute" Arith.thy "x+(y+z)=y+((x+z)::nat)"
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 (fn _ => [rtac (add_commute RS trans) 1, rtac (add_assoc RS trans) 1,
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           rtac (add_commute RS arg_cong) 1]);
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(*Addition is an AC-operator*)
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val add_ac = [add_assoc, add_commute, add_left_commute];
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goal Arith.thy "!!k::nat. (k + m = k + n) = (m=n)";
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by (nat_ind_tac "k" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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qed "add_left_cancel";
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goal Arith.thy "!!k::nat. (m + k = n + k) = (m=n)";
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by (nat_ind_tac "k" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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qed "add_right_cancel";
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goal Arith.thy "!!k::nat. (k + m <= k + n) = (m<=n)";
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by (nat_ind_tac "k" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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qed "add_left_cancel_le";
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goal Arith.thy "!!k::nat. (k + m < k + n) = (m<n)";
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by (nat_ind_tac "k" 1);
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by (Simp_tac 1);
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by (Asm_simp_tac 1);
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qed "add_left_cancel_less";
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Addsimps [add_left_cancel, add_right_cancel,
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          add_left_cancel_le, add_left_cancel_less];
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goal Arith.thy "(m+n = 0) = (m=0 & n=0)";
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by (nat_ind_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_is_0";
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Addsimps [add_is_0];
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goal Arith.thy "!!n. n ~= 0 ==> m + pred n = pred(m+n)";
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by (nat_ind_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_pred";
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Addsimps [add_pred];
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(*** Multiplication ***)
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(*right annihilation in product*)
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qed_goal "mult_0_right" Arith.thy "m * 0 = 0"
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 (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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(*right Sucessor law for multiplication*)
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qed_goal "mult_Suc_right" Arith.thy  "m * Suc(n) = m + (m * n)"
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 (fn _ => [nat_ind_tac "m" 1,
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           ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
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Addsimps [mult_0_right,mult_Suc_right];
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goal Arith.thy "1 * n = n";
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by (Asm_simp_tac 1);
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qed "mult_1";
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goal Arith.thy "n * 1 = n";
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by (Asm_simp_tac 1);
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qed "mult_1_right";
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(*Commutative law for multiplication*)
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qed_goal "mult_commute" Arith.thy "m * n = n * (m::nat)"
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 (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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(*addition distributes over multiplication*)
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qed_goal "add_mult_distrib" Arith.thy "(m + n)*k = (m*k) + ((n*k)::nat)"
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 (fn _ => [nat_ind_tac "m" 1,
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           ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
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qed_goal "add_mult_distrib2" Arith.thy "k*(m + n) = (k*m) + ((k*n)::nat)"
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 (fn _ => [nat_ind_tac "m" 1,
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           ALLGOALS(asm_simp_tac (!simpset addsimps add_ac))]);
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(*Associative law for multiplication*)
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qed_goal "mult_assoc" Arith.thy "(m * n) * k = m * ((n * k)::nat)"
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  (fn _ => [nat_ind_tac "m" 1, 
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            ALLGOALS (asm_simp_tac (!simpset addsimps [add_mult_distrib]))]);
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qed_goal "mult_left_commute" Arith.thy "x*(y*z) = y*((x*z)::nat)"
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 (fn _ => [rtac trans 1, rtac mult_commute 1, rtac trans 1,
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           rtac mult_assoc 1, rtac (mult_commute RS arg_cong) 1]);
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val mult_ac = [mult_assoc,mult_commute,mult_left_commute];
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(*** Difference ***)
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qed_goal "diff_self_eq_0" Arith.thy "m - m = 0"
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 (fn _ => [nat_ind_tac "m" 1, ALLGOALS Asm_simp_tac]);
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Addsimps [diff_self_eq_0];
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(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
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val [prem] = goal Arith.thy "[| ~ m<n |] ==> n+(m-n) = (m::nat)";
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by (rtac (prem RS rev_mp) 1);
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by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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by (ALLGOALS (Asm_simp_tac));
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qed "add_diff_inverse";
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(*** Remainder ***)
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goal Arith.thy "m - n < Suc(m)";
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by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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by (etac less_SucE 3);
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by (ALLGOALS (asm_simp_tac (!simpset addsimps [less_Suc_eq])));
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qed "diff_less_Suc";
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goal Arith.thy "!!m::nat. m - n <= m";
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by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_le_self";
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goal Arith.thy "!!n::nat. (n+m) - n = m";
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_add_inverse";
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goal Arith.thy "!!n::nat.(m+n) - n = m";
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by (res_inst_tac [("m1","m")] (add_commute RS ssubst) 1);
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by (REPEAT (ares_tac [diff_add_inverse] 1));
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qed "diff_add_inverse2";
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goal Arith.thy "!!k::nat. (k+m) - (k+n) = m - n";
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by (nat_ind_tac "k" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_cancel";
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Addsimps [diff_cancel];
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goal Arith.thy "!!m::nat. (m+k) - (n+k) = m - n";
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val add_commute_k = read_instantiate [("n","k")] add_commute;
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by (asm_simp_tac (!simpset addsimps ([add_commute_k])) 1);
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qed "diff_cancel2";
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Addsimps [diff_cancel2];
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goal Arith.thy "!!n::nat. n - (n+m) = 0";
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by (nat_ind_tac "n" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_add_0";
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Addsimps [diff_add_0];
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(** Difference distributes over multiplication **)
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goal Arith.thy "!!m::nat. (m - n) * k = (m * k) - (n * k)";
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by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_mult_distrib" ;
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goal Arith.thy "!!m::nat. k * (m - n) = (k * m) - (k * n)";
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val mult_commute_k = read_instantiate [("m","k")] mult_commute;
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by (simp_tac (!simpset addsimps [diff_mult_distrib, mult_commute_k]) 1);
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qed "diff_mult_distrib2" ;
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(*NOT added as rewrites, since sometimes they are used from right-to-left*)
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(** Less-then properties **)
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(*In ordinary notation: if 0<n and n<=m then m-n < m *)
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goal Arith.thy "!!m. [| 0<n; ~ m<n |] ==> m - n < m";
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by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
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by (Fast_tac 1);
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by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
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by (ALLGOALS(asm_simp_tac(!simpset addsimps [diff_less_Suc])));
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qed "diff_less";
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val wf_less_trans = wf_pred_nat RS wf_trancl RSN (2, def_wfrec RS trans);
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goalw Nat.thy [less_def] "(m,n) : pred_nat^+ = (m<n)";
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by (rtac refl 1);
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qed "less_eq";
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goal Arith.thy "(%m. m mod n) = wfrec (trancl pred_nat) \
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             \                      (%f j. if j<n then j else f (j-n))";
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by (simp_tac (HOL_ss addsimps [mod_def]) 1);
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val mod_def1 = result() RS eq_reflection;
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goal Arith.thy "!!m. m<n ==> m mod n = m";
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by (rtac (mod_def1 RS wf_less_trans) 1);
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by (Asm_simp_tac 1);
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qed "mod_less";
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goal Arith.thy "!!m. [| 0<n;  ~m<n |] ==> m mod n = (m-n) mod n";
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by (rtac (mod_def1 RS wf_less_trans) 1);
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by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
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qed "mod_geq";
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(*** Quotient ***)
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goal Arith.thy "(%m. m div n) = wfrec (trancl pred_nat) \
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                        \            (%f j. if j<n then 0 else Suc (f (j-n)))";
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by (simp_tac (HOL_ss addsimps [div_def]) 1);
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val div_def1 = result() RS eq_reflection;
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goal Arith.thy "!!m. m<n ==> m div n = 0";
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by (rtac (div_def1 RS wf_less_trans) 1);
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by (Asm_simp_tac 1);
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qed "div_less";
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goal Arith.thy "!!M. [| 0<n;  ~m<n |] ==> m div n = Suc((m-n) div n)";
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by (rtac (div_def1 RS wf_less_trans) 1);
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by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1);
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qed "div_geq";
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(*Main Result about quotient and remainder.*)
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goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m";
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   300
by (res_inst_tac [("n","m")] less_induct 1);
clasohm@923
   301
by (rename_tac "k" 1);    (*Variable name used in line below*)
clasohm@923
   302
by (case_tac "k<n" 1);
oheimb@1660
   303
by (ALLGOALS (asm_simp_tac(!simpset addsimps ([add_assoc] @
clasohm@923
   304
                       [mod_less, mod_geq, div_less, div_geq,
clasohm@1465
   305
                        add_diff_inverse, diff_less]))));
clasohm@923
   306
qed "mod_div_equality";
clasohm@923
   307
clasohm@923
   308
clasohm@923
   309
(*** More results about difference ***)
clasohm@923
   310
clasohm@923
   311
val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
clasohm@923
   312
by (rtac (prem RS rev_mp) 1);
clasohm@923
   313
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
oheimb@1660
   314
by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1);
oheimb@1660
   315
by (ALLGOALS (Asm_simp_tac));
clasohm@923
   316
qed "less_imp_diff_is_0";
clasohm@923
   317
clasohm@923
   318
val prems = goal Arith.thy "m-n = 0  -->  n-m = 0  -->  m=n";
clasohm@923
   319
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
clasohm@1264
   320
by (REPEAT(Simp_tac 1 THEN TRY(atac 1)));
nipkow@1485
   321
qed_spec_mp "diffs0_imp_equal";
clasohm@923
   322
clasohm@923
   323
val [prem] = goal Arith.thy "m<n ==> 0<n-m";
clasohm@923
   324
by (rtac (prem RS rev_mp) 1);
clasohm@923
   325
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
oheimb@1660
   326
by (ALLGOALS (Asm_simp_tac));
clasohm@923
   327
qed "less_imp_diff_positive";
clasohm@923
   328
clasohm@923
   329
val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
clasohm@923
   330
by (rtac (prem RS rev_mp) 1);
clasohm@923
   331
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
oheimb@1660
   332
by (ALLGOALS (Asm_simp_tac));
clasohm@923
   333
qed "Suc_diff_n";
clasohm@923
   334
nipkow@1398
   335
goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
paulson@1552
   336
by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
clasohm@923
   337
                    setloop (split_tac [expand_if])) 1);
clasohm@923
   338
qed "if_Suc_diff_n";
clasohm@923
   339
clasohm@923
   340
goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
clasohm@923
   341
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
berghofe@1760
   342
by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o Fast_tac));
clasohm@923
   343
qed "zero_induct_lemma";
clasohm@923
   344
clasohm@923
   345
val prems = goal Arith.thy "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
clasohm@923
   346
by (rtac (diff_self_eq_0 RS subst) 1);
clasohm@923
   347
by (rtac (zero_induct_lemma RS mp RS mp) 1);
clasohm@923
   348
by (REPEAT (ares_tac ([impI,allI]@prems) 1));
clasohm@923
   349
qed "zero_induct";
clasohm@923
   350
clasohm@923
   351
(*13 July 1992: loaded in 105.7s*)
clasohm@923
   352
paulson@1618
   353
paulson@1618
   354
(*** Further facts about mod (mainly for mutilated checkerboard ***)
paulson@1618
   355
paulson@1618
   356
goal Arith.thy
paulson@1618
   357
    "!!m n. 0<n ==> \
paulson@1618
   358
\           Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))";
paulson@1618
   359
by (res_inst_tac [("n","m")] less_induct 1);
paulson@1618
   360
by (excluded_middle_tac "Suc(na)<n" 1);
paulson@1618
   361
(* case Suc(na) < n *)
paulson@1618
   362
by (forward_tac [lessI RS less_trans] 2);
paulson@1618
   363
by (asm_simp_tac (!simpset addsimps [mod_less, less_not_refl2 RS not_sym]) 2);
paulson@1618
   364
(* case n <= Suc(na) *)
paulson@1618
   365
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, mod_geq]) 1);
paulson@1618
   366
by (etac (le_imp_less_or_eq RS disjE) 1);
paulson@1618
   367
by (asm_simp_tac (!simpset addsimps [Suc_diff_n]) 1);
paulson@1618
   368
by (asm_full_simp_tac (!simpset addsimps [not_less_eq RS sym, 
paulson@1618
   369
                                          diff_less, mod_geq]) 1);
paulson@1618
   370
by (asm_simp_tac (!simpset addsimps [mod_less]) 1);
paulson@1618
   371
qed "mod_Suc";
paulson@1618
   372
paulson@1618
   373
goal Arith.thy "!!m n. 0<n ==> m mod n < n";
paulson@1618
   374
by (res_inst_tac [("n","m")] less_induct 1);
paulson@1618
   375
by (excluded_middle_tac "na<n" 1);
paulson@1618
   376
(*case na<n*)
paulson@1618
   377
by (asm_simp_tac (!simpset addsimps [mod_less]) 2);
paulson@1618
   378
(*case n le na*)
paulson@1618
   379
by (asm_full_simp_tac (!simpset addsimps [mod_geq, diff_less]) 1);
paulson@1618
   380
qed "mod_less_divisor";
paulson@1618
   381
paulson@1618
   382
paulson@1626
   383
(** Evens and Odds **)
paulson@1626
   384
paulson@1909
   385
(*With less_zeroE, causes case analysis on b<2*)
paulson@1909
   386
AddSEs [less_SucE];
berghofe@1760
   387
paulson@1626
   388
goal thy "!!k b. b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)";
paulson@1626
   389
by (subgoal_tac "k mod 2 < 2" 1);
paulson@1626
   390
by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2);
paulson@1626
   391
by (asm_simp_tac (!simpset setloop split_tac [expand_if]) 1);
berghofe@1760
   392
by (Fast_tac 1);
paulson@1626
   393
qed "mod2_cases";
paulson@1626
   394
paulson@1626
   395
goal thy "Suc(Suc(m)) mod 2 = m mod 2";
paulson@1626
   396
by (subgoal_tac "m mod 2 < 2" 1);
paulson@1626
   397
by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2);
paulson@1909
   398
by (Step_tac 1);
paulson@1626
   399
by (ALLGOALS (asm_simp_tac (!simpset addsimps [mod_Suc])));
paulson@1626
   400
qed "mod2_Suc_Suc";
paulson@1626
   401
Addsimps [mod2_Suc_Suc];
paulson@1626
   402
paulson@1626
   403
goal thy "(m+m) mod 2 = 0";
paulson@1626
   404
by (nat_ind_tac "m" 1);
paulson@1626
   405
by (simp_tac (!simpset addsimps [mod_less]) 1);
paulson@1626
   406
by (asm_simp_tac (!simpset addsimps [mod2_Suc_Suc, add_Suc_right]) 1);
paulson@1626
   407
qed "mod2_add_self";
paulson@1626
   408
Addsimps [mod2_add_self];
paulson@1626
   409
paulson@1909
   410
Delrules [less_SucE];
paulson@1909
   411
paulson@1626
   412
clasohm@923
   413
(**** Additional theorems about "less than" ****)
clasohm@923
   414
paulson@1909
   415
goal Arith.thy "? k::nat. n = n+k";
paulson@1909
   416
by (res_inst_tac [("x","0")] exI 1);
paulson@1909
   417
by (Simp_tac 1);
paulson@1909
   418
val lemma = result();
paulson@1909
   419
clasohm@923
   420
goal Arith.thy "!!m. m<n --> (? k. n=Suc(m+k))";
clasohm@923
   421
by (nat_ind_tac "n" 1);
paulson@1909
   422
by (ALLGOALS (simp_tac (!simpset addsimps [less_Suc_eq])));
paulson@1909
   423
by (step_tac (!claset addSIs [lemma]) 1);
clasohm@923
   424
by (res_inst_tac [("x","Suc(k)")] exI 1);
clasohm@1264
   425
by (Simp_tac 1);
nipkow@1485
   426
qed_spec_mp "less_eq_Suc_add";
clasohm@923
   427
clasohm@923
   428
goal Arith.thy "n <= ((m + n)::nat)";
clasohm@923
   429
by (nat_ind_tac "m" 1);
clasohm@1264
   430
by (ALLGOALS Simp_tac);
clasohm@923
   431
by (etac le_trans 1);
clasohm@923
   432
by (rtac (lessI RS less_imp_le) 1);
clasohm@923
   433
qed "le_add2";
clasohm@923
   434
clasohm@923
   435
goal Arith.thy "n <= ((n + m)::nat)";
clasohm@1264
   436
by (simp_tac (!simpset addsimps add_ac) 1);
clasohm@923
   437
by (rtac le_add2 1);
clasohm@923
   438
qed "le_add1";
clasohm@923
   439
clasohm@923
   440
bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
clasohm@923
   441
bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
clasohm@923
   442
clasohm@923
   443
(*"i <= j ==> i <= j+m"*)
clasohm@923
   444
bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
clasohm@923
   445
clasohm@923
   446
(*"i <= j ==> i <= m+j"*)
clasohm@923
   447
bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
clasohm@923
   448
clasohm@923
   449
(*"i < j ==> i < j+m"*)
clasohm@923
   450
bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
clasohm@923
   451
clasohm@923
   452
(*"i < j ==> i < m+j"*)
clasohm@923
   453
bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
clasohm@923
   454
nipkow@1152
   455
goal Arith.thy "!!i. i+j < (k::nat) ==> i<k";
paulson@1552
   456
by (etac rev_mp 1);
paulson@1552
   457
by (nat_ind_tac "j" 1);
clasohm@1264
   458
by (ALLGOALS Asm_simp_tac);
berghofe@1760
   459
by (fast_tac (!claset addDs [Suc_lessD]) 1);
nipkow@1152
   460
qed "add_lessD1";
nipkow@1152
   461
clasohm@923
   462
goal Arith.thy "!!k::nat. m <= n ==> m <= n+k";
paulson@1552
   463
by (etac le_trans 1);
paulson@1552
   464
by (rtac le_add1 1);
clasohm@923
   465
qed "le_imp_add_le";
clasohm@923
   466
clasohm@923
   467
goal Arith.thy "!!k::nat. m < n ==> m < n+k";
paulson@1552
   468
by (etac less_le_trans 1);
paulson@1552
   469
by (rtac le_add1 1);
clasohm@923
   470
qed "less_imp_add_less";
clasohm@923
   471
clasohm@923
   472
goal Arith.thy "m+k<=n --> m<=(n::nat)";
clasohm@923
   473
by (nat_ind_tac "k" 1);
clasohm@1264
   474
by (ALLGOALS Asm_simp_tac);
berghofe@1760
   475
by (fast_tac (!claset addDs [Suc_leD]) 1);
nipkow@1485
   476
qed_spec_mp "add_leD1";
clasohm@923
   477
clasohm@923
   478
goal Arith.thy "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n";
berghofe@1786
   479
by (safe_tac (!claset addSDs [less_eq_Suc_add]));
clasohm@923
   480
by (asm_full_simp_tac
clasohm@1264
   481
    (!simpset delsimps [add_Suc_right]
clasohm@1264
   482
                addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1);
paulson@1552
   483
by (etac subst 1);
clasohm@1264
   484
by (simp_tac (!simpset addsimps [less_add_Suc1]) 1);
clasohm@923
   485
qed "less_add_eq_less";
clasohm@923
   486
clasohm@923
   487
paulson@1713
   488
(*** Monotonicity of Addition ***)
clasohm@923
   489
clasohm@923
   490
(*strict, in 1st argument*)
clasohm@923
   491
goal Arith.thy "!!i j k::nat. i < j ==> i + k < j + k";
clasohm@923
   492
by (nat_ind_tac "k" 1);
clasohm@1264
   493
by (ALLGOALS Asm_simp_tac);
clasohm@923
   494
qed "add_less_mono1";
clasohm@923
   495
clasohm@923
   496
(*strict, in both arguments*)
clasohm@923
   497
goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l";
clasohm@923
   498
by (rtac (add_less_mono1 RS less_trans) 1);
lcp@1198
   499
by (REPEAT (assume_tac 1));
clasohm@923
   500
by (nat_ind_tac "j" 1);
clasohm@1264
   501
by (ALLGOALS Asm_simp_tac);
clasohm@923
   502
qed "add_less_mono";
clasohm@923
   503
clasohm@923
   504
(*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
clasohm@923
   505
val [lt_mono,le] = goal Arith.thy
clasohm@1465
   506
     "[| !!i j::nat. i<j ==> f(i) < f(j);       \
clasohm@1465
   507
\        i <= j                                 \
clasohm@923
   508
\     |] ==> f(i) <= (f(j)::nat)";
clasohm@923
   509
by (cut_facts_tac [le] 1);
clasohm@1264
   510
by (asm_full_simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1);
berghofe@1760
   511
by (fast_tac (!claset addSIs [lt_mono]) 1);
clasohm@923
   512
qed "less_mono_imp_le_mono";
clasohm@923
   513
clasohm@923
   514
(*non-strict, in 1st argument*)
clasohm@923
   515
goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k";
clasohm@923
   516
by (res_inst_tac [("f", "%j.j+k")] less_mono_imp_le_mono 1);
paulson@1552
   517
by (etac add_less_mono1 1);
clasohm@923
   518
by (assume_tac 1);
clasohm@923
   519
qed "add_le_mono1";
clasohm@923
   520
clasohm@923
   521
(*non-strict, in both arguments*)
clasohm@923
   522
goal Arith.thy "!!k l::nat. [|i<=j;  k<=l |] ==> i + k <= j + l";
clasohm@923
   523
by (etac (add_le_mono1 RS le_trans) 1);
clasohm@1264
   524
by (simp_tac (!simpset addsimps [add_commute]) 1);
clasohm@923
   525
(*j moves to the end because it is free while k, l are bound*)
paulson@1552
   526
by (etac add_le_mono1 1);
clasohm@923
   527
qed "add_le_mono";
paulson@1713
   528
paulson@1713
   529
(*** Monotonicity of Multiplication ***)
paulson@1713
   530
paulson@1713
   531
goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k";
paulson@1713
   532
by (nat_ind_tac "k" 1);
paulson@1713
   533
by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_le_mono])));
paulson@1713
   534
qed "mult_le_mono1";
paulson@1713
   535
paulson@1713
   536
(*<=monotonicity, BOTH arguments*)
paulson@1713
   537
goal Arith.thy "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l";
paulson@2007
   538
by (etac (mult_le_mono1 RS le_trans) 1);
paulson@1713
   539
by (rtac le_trans 1);
paulson@2007
   540
by (stac mult_commute 2);
paulson@2007
   541
by (etac mult_le_mono1 2);
paulson@2007
   542
by (simp_tac (!simpset addsimps [mult_commute]) 1);
paulson@1713
   543
qed "mult_le_mono";
paulson@1713
   544
paulson@1713
   545
(*strict, in 1st argument; proof is by induction on k>0*)
paulson@1713
   546
goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j";
paulson@2031
   547
by (etac zero_less_natE 1);
paulson@1713
   548
by (Asm_simp_tac 1);
paulson@1713
   549
by (nat_ind_tac "x" 1);
paulson@1713
   550
by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_less_mono])));
paulson@1713
   551
qed "mult_less_mono2";
paulson@1713
   552
paulson@1713
   553
goal Arith.thy "(0 < m*n) = (0<m & 0<n)";
paulson@1713
   554
by (nat_ind_tac "m" 1);
paulson@1713
   555
by (nat_ind_tac "n" 2);
paulson@1713
   556
by (ALLGOALS Asm_simp_tac);
paulson@1713
   557
qed "zero_less_mult_iff";
paulson@1713
   558
paulson@1795
   559
goal Arith.thy "(m*n = 1) = (m=1 & n=1)";
paulson@1795
   560
by (nat_ind_tac "m" 1);
paulson@1795
   561
by (Simp_tac 1);
paulson@1795
   562
by (nat_ind_tac "n" 1);
paulson@1795
   563
by (Simp_tac 1);
paulson@1795
   564
by (fast_tac (!claset addss !simpset) 1);
paulson@1795
   565
qed "mult_eq_1_iff";
paulson@1795
   566
paulson@1713
   567
(*Cancellation law for division*)
paulson@1713
   568
goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n";
paulson@1713
   569
by (res_inst_tac [("n","m")] less_induct 1);
paulson@1713
   570
by (case_tac "na<n" 1);
paulson@1713
   571
by (asm_simp_tac (!simpset addsimps [div_less, zero_less_mult_iff, 
paulson@2031
   572
                                     mult_less_mono2]) 1);
paulson@1713
   573
by (subgoal_tac "~ k*na < k*n" 1);
paulson@1713
   574
by (asm_simp_tac
paulson@1713
   575
     (!simpset addsimps [zero_less_mult_iff, div_geq,
paulson@2031
   576
                         diff_mult_distrib2 RS sym, diff_less]) 1);
paulson@1713
   577
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, 
paulson@2031
   578
                                          le_refl RS mult_le_mono]) 1);
paulson@1713
   579
qed "div_cancel";
paulson@1713
   580
paulson@1713
   581
goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)";
paulson@1713
   582
by (res_inst_tac [("n","m")] less_induct 1);
paulson@1713
   583
by (case_tac "na<n" 1);
paulson@1713
   584
by (asm_simp_tac (!simpset addsimps [mod_less, zero_less_mult_iff, 
paulson@2031
   585
                                     mult_less_mono2]) 1);
paulson@1713
   586
by (subgoal_tac "~ k*na < k*n" 1);
paulson@1713
   587
by (asm_simp_tac
paulson@1713
   588
     (!simpset addsimps [zero_less_mult_iff, mod_geq,
paulson@2031
   589
                         diff_mult_distrib2 RS sym, diff_less]) 1);
paulson@1713
   590
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, 
paulson@2031
   591
                                          le_refl RS mult_le_mono]) 1);
paulson@1713
   592
qed "mult_mod_distrib";
paulson@1713
   593
paulson@1713
   594
paulson@1795
   595
(** Lemma for gcd **)
paulson@1795
   596
paulson@1795
   597
goal Arith.thy "!!m n. m = m*n ==> n=1 | m=0";
paulson@1795
   598
by (dtac sym 1);
paulson@1795
   599
by (rtac disjCI 1);
paulson@1795
   600
by (rtac nat_less_cases 1 THEN assume_tac 2);
paulson@1909
   601
by (fast_tac (!claset addSEs [less_SucE] addss !simpset) 1);
paulson@1979
   602
by (best_tac (!claset addDs [mult_less_mono2] 
paulson@1795
   603
                      addss (!simpset addsimps [zero_less_eq RS sym])) 1);
paulson@1795
   604
qed "mult_eq_self_implies_10";
paulson@1795
   605
paulson@1795
   606