author | wenzelm |
Fri, 07 Mar 1997 11:48:46 +0100 | |
changeset 2754 | 59bd96046ad6 |
parent 2682 | 13cdbf95ed92 |
child 2922 | 580647a879cf |
permissions | -rw-r--r-- |
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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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Addsimps [pred_0,pred_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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qed_goalw "diff_0_eq_0" Arith.thy [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 [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_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 "pred_Suc_diff" Arith.thy "pred(Suc m - n) = m - n" |
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(fn _ => [nat_ind_tac "n" 1, ALLGOALS Asm_simp_tac]); |
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Addsimps [pred_Suc_diff]; |
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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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|
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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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|
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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); |
259 |
qed "less_eq"; |
|
260 |
||
1475 | 261 |
goal Arith.thy "(%m. m mod n) = wfrec (trancl pred_nat) \ |
262 |
\ (%f j. if j<n then j else f (j-n))"; |
|
263 |
by (simp_tac (HOL_ss addsimps [mod_def]) 1); |
|
264 |
val mod_def1 = result() RS eq_reflection; |
|
265 |
||
923 | 266 |
goal Arith.thy "!!m. m<n ==> m mod n = m"; |
1475 | 267 |
by (rtac (mod_def1 RS wf_less_trans) 1); |
1552 | 268 |
by (Asm_simp_tac 1); |
923 | 269 |
qed "mod_less"; |
270 |
||
271 |
goal Arith.thy "!!m. [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n"; |
|
1475 | 272 |
by (rtac (mod_def1 RS wf_less_trans) 1); |
1552 | 273 |
by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1); |
923 | 274 |
qed "mod_geq"; |
275 |
||
276 |
||
277 |
(*** Quotient ***) |
|
278 |
||
1475 | 279 |
goal Arith.thy "(%m. m div n) = wfrec (trancl pred_nat) \ |
280 |
\ (%f j. if j<n then 0 else Suc (f (j-n)))"; |
|
281 |
by (simp_tac (HOL_ss addsimps [div_def]) 1); |
|
282 |
val div_def1 = result() RS eq_reflection; |
|
283 |
||
923 | 284 |
goal Arith.thy "!!m. m<n ==> m div n = 0"; |
1475 | 285 |
by (rtac (div_def1 RS wf_less_trans) 1); |
1552 | 286 |
by (Asm_simp_tac 1); |
923 | 287 |
qed "div_less"; |
288 |
||
289 |
goal Arith.thy "!!M. [| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)"; |
|
1475 | 290 |
by (rtac (div_def1 RS wf_less_trans) 1); |
1552 | 291 |
by (asm_simp_tac (!simpset addsimps [diff_less, cut_apply, less_eq]) 1); |
923 | 292 |
qed "div_geq"; |
293 |
||
294 |
(*Main Result about quotient and remainder.*) |
|
295 |
goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m"; |
|
296 |
by (res_inst_tac [("n","m")] less_induct 1); |
|
297 |
by (rename_tac "k" 1); (*Variable name used in line below*) |
|
298 |
by (case_tac "k<n" 1); |
|
1660 | 299 |
by (ALLGOALS (asm_simp_tac(!simpset addsimps ([add_assoc] @ |
923 | 300 |
[mod_less, mod_geq, div_less, div_geq, |
1465 | 301 |
add_diff_inverse, diff_less])))); |
923 | 302 |
qed "mod_div_equality"; |
303 |
||
304 |
||
305 |
(*** More results about difference ***) |
|
306 |
||
307 |
val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0"; |
|
308 |
by (rtac (prem RS rev_mp) 1); |
|
309 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
1660 | 310 |
by (asm_simp_tac (!simpset addsimps [less_Suc_eq]) 1); |
311 |
by (ALLGOALS (Asm_simp_tac)); |
|
923 | 312 |
qed "less_imp_diff_is_0"; |
313 |
||
314 |
val prems = goal Arith.thy "m-n = 0 --> n-m = 0 --> m=n"; |
|
315 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
316 |
by (REPEAT(Simp_tac 1 THEN TRY(atac 1))); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1475
diff
changeset
|
317 |
qed_spec_mp "diffs0_imp_equal"; |
923 | 318 |
|
319 |
val [prem] = goal Arith.thy "m<n ==> 0<n-m"; |
|
320 |
by (rtac (prem RS rev_mp) 1); |
|
321 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
1660 | 322 |
by (ALLGOALS (Asm_simp_tac)); |
923 | 323 |
qed "less_imp_diff_positive"; |
324 |
||
325 |
val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)"; |
|
326 |
by (rtac (prem RS rev_mp) 1); |
|
327 |
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1); |
|
1660 | 328 |
by (ALLGOALS (Asm_simp_tac)); |
923 | 329 |
qed "Suc_diff_n"; |
330 |
||
1398 | 331 |
goal Arith.thy "Suc(m)-n = (if m<n then 0 else Suc(m-n))"; |
1552 | 332 |
by (simp_tac (!simpset addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n] |
923 | 333 |
setloop (split_tac [expand_if])) 1); |
334 |
qed "if_Suc_diff_n"; |
|
335 |
||
336 |
goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)"; |
|
337 |
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1); |
|
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1713
diff
changeset
|
338 |
by (ALLGOALS (strip_tac THEN' Simp_tac THEN' TRY o Fast_tac)); |
923 | 339 |
qed "zero_induct_lemma"; |
340 |
||
341 |
val prems = goal Arith.thy "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)"; |
|
342 |
by (rtac (diff_self_eq_0 RS subst) 1); |
|
343 |
by (rtac (zero_induct_lemma RS mp RS mp) 1); |
|
344 |
by (REPEAT (ares_tac ([impI,allI]@prems) 1)); |
|
345 |
qed "zero_induct"; |
|
346 |
||
347 |
(*13 July 1992: loaded in 105.7s*) |
|
348 |
||
1618 | 349 |
|
350 |
(*** Further facts about mod (mainly for mutilated checkerboard ***) |
|
351 |
||
352 |
goal Arith.thy |
|
353 |
"!!m n. 0<n ==> \ |
|
354 |
\ Suc(m) mod n = (if Suc(m mod n) = n then 0 else Suc(m mod n))"; |
|
355 |
by (res_inst_tac [("n","m")] less_induct 1); |
|
356 |
by (excluded_middle_tac "Suc(na)<n" 1); |
|
357 |
(* case Suc(na) < n *) |
|
358 |
by (forward_tac [lessI RS less_trans] 2); |
|
359 |
by (asm_simp_tac (!simpset addsimps [mod_less, less_not_refl2 RS not_sym]) 2); |
|
360 |
(* case n <= Suc(na) *) |
|
361 |
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, mod_geq]) 1); |
|
362 |
by (etac (le_imp_less_or_eq RS disjE) 1); |
|
363 |
by (asm_simp_tac (!simpset addsimps [Suc_diff_n]) 1); |
|
364 |
by (asm_full_simp_tac (!simpset addsimps [not_less_eq RS sym, |
|
365 |
diff_less, mod_geq]) 1); |
|
366 |
by (asm_simp_tac (!simpset addsimps [mod_less]) 1); |
|
367 |
qed "mod_Suc"; |
|
368 |
||
369 |
goal Arith.thy "!!m n. 0<n ==> m mod n < n"; |
|
370 |
by (res_inst_tac [("n","m")] less_induct 1); |
|
371 |
by (excluded_middle_tac "na<n" 1); |
|
372 |
(*case na<n*) |
|
373 |
by (asm_simp_tac (!simpset addsimps [mod_less]) 2); |
|
374 |
(*case n le na*) |
|
375 |
by (asm_full_simp_tac (!simpset addsimps [mod_geq, diff_less]) 1); |
|
376 |
qed "mod_less_divisor"; |
|
377 |
||
378 |
||
1626 | 379 |
(** Evens and Odds **) |
380 |
||
1909 | 381 |
(*With less_zeroE, causes case analysis on b<2*) |
382 |
AddSEs [less_SucE]; |
|
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1713
diff
changeset
|
383 |
|
1626 | 384 |
goal thy "!!k b. b<2 ==> k mod 2 = b | k mod 2 = (if b=1 then 0 else 1)"; |
385 |
by (subgoal_tac "k mod 2 < 2" 1); |
|
386 |
by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2); |
|
387 |
by (asm_simp_tac (!simpset setloop split_tac [expand_if]) 1); |
|
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1713
diff
changeset
|
388 |
by (Fast_tac 1); |
1626 | 389 |
qed "mod2_cases"; |
390 |
||
391 |
goal thy "Suc(Suc(m)) mod 2 = m mod 2"; |
|
392 |
by (subgoal_tac "m mod 2 < 2" 1); |
|
393 |
by (asm_simp_tac (!simpset addsimps [mod_less_divisor]) 2); |
|
1909 | 394 |
by (Step_tac 1); |
1626 | 395 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [mod_Suc]))); |
396 |
qed "mod2_Suc_Suc"; |
|
397 |
Addsimps [mod2_Suc_Suc]; |
|
398 |
||
399 |
goal thy "(m+m) mod 2 = 0"; |
|
400 |
by (nat_ind_tac "m" 1); |
|
401 |
by (simp_tac (!simpset addsimps [mod_less]) 1); |
|
402 |
by (asm_simp_tac (!simpset addsimps [mod2_Suc_Suc, add_Suc_right]) 1); |
|
403 |
qed "mod2_add_self"; |
|
404 |
Addsimps [mod2_add_self]; |
|
405 |
||
1909 | 406 |
Delrules [less_SucE]; |
407 |
||
1626 | 408 |
|
923 | 409 |
(**** Additional theorems about "less than" ****) |
410 |
||
1909 | 411 |
goal Arith.thy "? k::nat. n = n+k"; |
412 |
by (res_inst_tac [("x","0")] exI 1); |
|
413 |
by (Simp_tac 1); |
|
414 |
val lemma = result(); |
|
415 |
||
923 | 416 |
goal Arith.thy "!!m. m<n --> (? k. n=Suc(m+k))"; |
417 |
by (nat_ind_tac "n" 1); |
|
1909 | 418 |
by (ALLGOALS (simp_tac (!simpset addsimps [less_Suc_eq]))); |
419 |
by (step_tac (!claset addSIs [lemma]) 1); |
|
923 | 420 |
by (res_inst_tac [("x","Suc(k)")] exI 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
421 |
by (Simp_tac 1); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1475
diff
changeset
|
422 |
qed_spec_mp "less_eq_Suc_add"; |
923 | 423 |
|
424 |
goal Arith.thy "n <= ((m + n)::nat)"; |
|
425 |
by (nat_ind_tac "m" 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
426 |
by (ALLGOALS Simp_tac); |
923 | 427 |
by (etac le_trans 1); |
428 |
by (rtac (lessI RS less_imp_le) 1); |
|
429 |
qed "le_add2"; |
|
430 |
||
431 |
goal Arith.thy "n <= ((n + m)::nat)"; |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
432 |
by (simp_tac (!simpset addsimps add_ac) 1); |
923 | 433 |
by (rtac le_add2 1); |
434 |
qed "le_add1"; |
|
435 |
||
436 |
bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans))); |
|
437 |
bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans))); |
|
438 |
||
439 |
(*"i <= j ==> i <= j+m"*) |
|
440 |
bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans)); |
|
441 |
||
442 |
(*"i <= j ==> i <= m+j"*) |
|
443 |
bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans)); |
|
444 |
||
445 |
(*"i < j ==> i < j+m"*) |
|
446 |
bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans)); |
|
447 |
||
448 |
(*"i < j ==> i < m+j"*) |
|
449 |
bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans)); |
|
450 |
||
1152 | 451 |
goal Arith.thy "!!i. i+j < (k::nat) ==> i<k"; |
1552 | 452 |
by (etac rev_mp 1); |
453 |
by (nat_ind_tac "j" 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
454 |
by (ALLGOALS Asm_simp_tac); |
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1713
diff
changeset
|
455 |
by (fast_tac (!claset addDs [Suc_lessD]) 1); |
1152 | 456 |
qed "add_lessD1"; |
457 |
||
923 | 458 |
goal Arith.thy "!!k::nat. m <= n ==> m <= n+k"; |
1552 | 459 |
by (etac le_trans 1); |
460 |
by (rtac le_add1 1); |
|
923 | 461 |
qed "le_imp_add_le"; |
462 |
||
463 |
goal Arith.thy "!!k::nat. m < n ==> m < n+k"; |
|
1552 | 464 |
by (etac less_le_trans 1); |
465 |
by (rtac le_add1 1); |
|
923 | 466 |
qed "less_imp_add_less"; |
467 |
||
468 |
goal Arith.thy "m+k<=n --> m<=(n::nat)"; |
|
469 |
by (nat_ind_tac "k" 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
470 |
by (ALLGOALS Asm_simp_tac); |
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1713
diff
changeset
|
471 |
by (fast_tac (!claset addDs [Suc_leD]) 1); |
1485
240cc98b94a7
Added qed_spec_mp to avoid renaming of bound vars in 'th RS spec'
nipkow
parents:
1475
diff
changeset
|
472 |
qed_spec_mp "add_leD1"; |
923 | 473 |
|
2498 | 474 |
goal Arith.thy "!!n::nat. m+k<=n ==> k<=n"; |
475 |
by (full_simp_tac (!simpset addsimps [add_commute]) 1); |
|
476 |
by (etac add_leD1 1); |
|
477 |
qed_spec_mp "add_leD2"; |
|
478 |
||
479 |
goal Arith.thy "!!n::nat. m+k<=n ==> m<=n & k<=n"; |
|
480 |
by (fast_tac (!claset addDs [add_leD1, add_leD2]) 1); |
|
481 |
bind_thm ("add_leE", result() RS conjE); |
|
482 |
||
923 | 483 |
goal Arith.thy "!!k l::nat. [| k<l; m+l = k+n |] ==> m<n"; |
1786
8a31d85d27b8
best_tac, deepen_tac and safe_tac now also use default claset.
berghofe
parents:
1767
diff
changeset
|
484 |
by (safe_tac (!claset addSDs [less_eq_Suc_add])); |
923 | 485 |
by (asm_full_simp_tac |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
486 |
(!simpset delsimps [add_Suc_right] |
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
487 |
addsimps ([add_Suc_right RS sym, add_left_cancel] @add_ac)) 1); |
1552 | 488 |
by (etac subst 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
489 |
by (simp_tac (!simpset addsimps [less_add_Suc1]) 1); |
923 | 490 |
qed "less_add_eq_less"; |
491 |
||
492 |
||
1713 | 493 |
(*** Monotonicity of Addition ***) |
923 | 494 |
|
495 |
(*strict, in 1st argument*) |
|
496 |
goal Arith.thy "!!i j k::nat. i < j ==> i + k < j + k"; |
|
497 |
by (nat_ind_tac "k" 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
498 |
by (ALLGOALS Asm_simp_tac); |
923 | 499 |
qed "add_less_mono1"; |
500 |
||
501 |
(*strict, in both arguments*) |
|
502 |
goal Arith.thy "!!i j k::nat. [|i < j; k < l|] ==> i + k < j + l"; |
|
503 |
by (rtac (add_less_mono1 RS less_trans) 1); |
|
1198 | 504 |
by (REPEAT (assume_tac 1)); |
923 | 505 |
by (nat_ind_tac "j" 1); |
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
506 |
by (ALLGOALS Asm_simp_tac); |
923 | 507 |
qed "add_less_mono"; |
508 |
||
509 |
(*A [clumsy] way of lifting < monotonicity to <= monotonicity *) |
|
510 |
val [lt_mono,le] = goal Arith.thy |
|
1465 | 511 |
"[| !!i j::nat. i<j ==> f(i) < f(j); \ |
512 |
\ i <= j \ |
|
923 | 513 |
\ |] ==> f(i) <= (f(j)::nat)"; |
514 |
by (cut_facts_tac [le] 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
515 |
by (asm_full_simp_tac (!simpset addsimps [le_eq_less_or_eq]) 1); |
1760
6f41a494f3b1
Replaced fast_tac by Fast_tac (which uses default claset)
berghofe
parents:
1713
diff
changeset
|
516 |
by (fast_tac (!claset addSIs [lt_mono]) 1); |
923 | 517 |
qed "less_mono_imp_le_mono"; |
518 |
||
519 |
(*non-strict, in 1st argument*) |
|
520 |
goal Arith.thy "!!i j k::nat. i<=j ==> i + k <= j + k"; |
|
521 |
by (res_inst_tac [("f", "%j.j+k")] less_mono_imp_le_mono 1); |
|
1552 | 522 |
by (etac add_less_mono1 1); |
923 | 523 |
by (assume_tac 1); |
524 |
qed "add_le_mono1"; |
|
525 |
||
526 |
(*non-strict, in both arguments*) |
|
527 |
goal Arith.thy "!!k l::nat. [|i<=j; k<=l |] ==> i + k <= j + l"; |
|
528 |
by (etac (add_le_mono1 RS le_trans) 1); |
|
1264
3eb91524b938
added local simpsets; removed IOA from 'make test'
clasohm
parents:
1198
diff
changeset
|
529 |
by (simp_tac (!simpset addsimps [add_commute]) 1); |
923 | 530 |
(*j moves to the end because it is free while k, l are bound*) |
1552 | 531 |
by (etac add_le_mono1 1); |
923 | 532 |
qed "add_le_mono"; |
1713 | 533 |
|
534 |
(*** Monotonicity of Multiplication ***) |
|
535 |
||
536 |
goal Arith.thy "!!i::nat. i<=j ==> i*k<=j*k"; |
|
537 |
by (nat_ind_tac "k" 1); |
|
538 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_le_mono]))); |
|
539 |
qed "mult_le_mono1"; |
|
540 |
||
541 |
(*<=monotonicity, BOTH arguments*) |
|
542 |
goal Arith.thy "!!i::nat. [| i<=j; k<=l |] ==> i*k<=j*l"; |
|
2007 | 543 |
by (etac (mult_le_mono1 RS le_trans) 1); |
1713 | 544 |
by (rtac le_trans 1); |
2007 | 545 |
by (stac mult_commute 2); |
546 |
by (etac mult_le_mono1 2); |
|
547 |
by (simp_tac (!simpset addsimps [mult_commute]) 1); |
|
1713 | 548 |
qed "mult_le_mono"; |
549 |
||
550 |
(*strict, in 1st argument; proof is by induction on k>0*) |
|
551 |
goal Arith.thy "!!i::nat. [| i<j; 0<k |] ==> k*i < k*j"; |
|
2031 | 552 |
by (etac zero_less_natE 1); |
1713 | 553 |
by (Asm_simp_tac 1); |
554 |
by (nat_ind_tac "x" 1); |
|
555 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [add_less_mono]))); |
|
556 |
qed "mult_less_mono2"; |
|
557 |
||
558 |
goal Arith.thy "(0 < m*n) = (0<m & 0<n)"; |
|
559 |
by (nat_ind_tac "m" 1); |
|
560 |
by (nat_ind_tac "n" 2); |
|
561 |
by (ALLGOALS Asm_simp_tac); |
|
562 |
qed "zero_less_mult_iff"; |
|
563 |
||
1795 | 564 |
goal Arith.thy "(m*n = 1) = (m=1 & n=1)"; |
565 |
by (nat_ind_tac "m" 1); |
|
566 |
by (Simp_tac 1); |
|
567 |
by (nat_ind_tac "n" 1); |
|
568 |
by (Simp_tac 1); |
|
569 |
by (fast_tac (!claset addss !simpset) 1); |
|
570 |
qed "mult_eq_1_iff"; |
|
571 |
||
1713 | 572 |
(*Cancellation law for division*) |
573 |
goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) div (k*n) = m div n"; |
|
574 |
by (res_inst_tac [("n","m")] less_induct 1); |
|
575 |
by (case_tac "na<n" 1); |
|
576 |
by (asm_simp_tac (!simpset addsimps [div_less, zero_less_mult_iff, |
|
2031 | 577 |
mult_less_mono2]) 1); |
1713 | 578 |
by (subgoal_tac "~ k*na < k*n" 1); |
579 |
by (asm_simp_tac |
|
580 |
(!simpset addsimps [zero_less_mult_iff, div_geq, |
|
2031 | 581 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
1713 | 582 |
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, |
2031 | 583 |
le_refl RS mult_le_mono]) 1); |
1713 | 584 |
qed "div_cancel"; |
585 |
||
586 |
goal Arith.thy "!!k. [| 0<n; 0<k |] ==> (k*m) mod (k*n) = k * (m mod n)"; |
|
587 |
by (res_inst_tac [("n","m")] less_induct 1); |
|
588 |
by (case_tac "na<n" 1); |
|
589 |
by (asm_simp_tac (!simpset addsimps [mod_less, zero_less_mult_iff, |
|
2031 | 590 |
mult_less_mono2]) 1); |
1713 | 591 |
by (subgoal_tac "~ k*na < k*n" 1); |
592 |
by (asm_simp_tac |
|
593 |
(!simpset addsimps [zero_less_mult_iff, mod_geq, |
|
2031 | 594 |
diff_mult_distrib2 RS sym, diff_less]) 1); |
1713 | 595 |
by (asm_full_simp_tac (!simpset addsimps [not_less_iff_le, |
2031 | 596 |
le_refl RS mult_le_mono]) 1); |
1713 | 597 |
qed "mult_mod_distrib"; |
598 |
||
599 |
||
1795 | 600 |
(** Lemma for gcd **) |
601 |
||
602 |
goal Arith.thy "!!m n. m = m*n ==> n=1 | m=0"; |
|
603 |
by (dtac sym 1); |
|
604 |
by (rtac disjCI 1); |
|
605 |
by (rtac nat_less_cases 1 THEN assume_tac 2); |
|
1909 | 606 |
by (fast_tac (!claset addSEs [less_SucE] addss !simpset) 1); |
1979 | 607 |
by (best_tac (!claset addDs [mult_less_mono2] |
1795 | 608 |
addss (!simpset addsimps [zero_less_eq RS sym])) 1); |
609 |
qed "mult_eq_self_implies_10"; |
|
610 |
||
611 |