--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/arith.ML Thu Sep 16 12:21:07 1993 +0200
@@ -0,0 +1,229 @@
+(* Title: HOL/arith.ML
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+For HOL/arith.thy.
+
+Proofs about elementary arithmetic: addition, multiplication, etc.
+Tests definitions and simplifier.
+*)
+
+open Arith;
+
+(*** Basic rewrite and congruence rules for the arithmetic operators ***)
+
+val [add_0,add_Suc] = nat_recs add_def;
+val [mult_0,mult_Suc] = nat_recs mult_def;
+
+(** Difference **)
+
+val diff_0 = diff_def RS def_nat_rec_0;
+
+val diff_0_eq_0 = prove_goalw Arith.thy [diff_def] "0 - n = 0"
+ (fn _ => [nat_ind_tac "n" 1, ALLGOALS(asm_simp_tac nat_ss)]);
+
+(*Must simplify BEFORE the induction!! (Else we get a critical pair)
+ Suc(m) - Suc(n) rewrites to pred(Suc(m) - n) *)
+val diff_Suc_Suc = prove_goalw Arith.thy [diff_def] "Suc(m) - Suc(n) = m - n"
+ (fn _ =>
+ [simp_tac nat_ss 1, nat_ind_tac "n" 1, ALLGOALS(asm_simp_tac nat_ss)]);
+
+(*** Simplification over add, mult, diff ***)
+
+val arith_simps = [add_0, add_Suc,
+ mult_0, mult_Suc,
+ diff_0, diff_0_eq_0, diff_Suc_Suc];
+
+val arith_ss = nat_ss addsimps arith_simps;
+
+(**** Inductive properties of the operators ****)
+
+(*** Addition ***)
+
+val add_0_right = prove_goal Arith.thy "m + 0 = m"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS(asm_simp_tac arith_ss)]);
+
+val add_Suc_right = prove_goal Arith.thy "m + Suc(n) = Suc(m+n)"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS(asm_simp_tac arith_ss)]);
+
+val arith_ss = arith_ss addsimps [add_0_right,add_Suc_right];
+
+(*Associative law for addition*)
+val add_assoc = prove_goal Arith.thy "(m + n) + k = m + (n + k)::nat"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS(asm_simp_tac arith_ss)]);
+
+(*Commutative law for addition. Must simplify after first induction!
+ Orientation of rewrites is delicate*)
+val add_commute = prove_goal Arith.thy "m + n = n + m::nat"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS(asm_simp_tac arith_ss)]);
+
+(*** Multiplication ***)
+
+(*right annihilation in product*)
+val mult_0_right = prove_goal Arith.thy "m * 0 = 0"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS(asm_simp_tac arith_ss)]);
+
+(*right Sucessor law for multiplication*)
+val mult_Suc_right = prove_goal Arith.thy
+ "m * Suc(n) = m + (m * n)"
+ (fn _ =>
+ [nat_ind_tac "m" 1,
+ ALLGOALS(asm_simp_tac(arith_ss addsimps [add_assoc RS sym])),
+ (*The final goal requires the commutative law for addition*)
+ stac add_commute 1, rtac refl 1]);
+
+val arith_ss = arith_ss addsimps [mult_0_right,mult_Suc_right];
+
+(*Commutative law for multiplication*)
+val mult_commute = prove_goal Arith.thy "m * n = n * m::nat"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS (asm_simp_tac arith_ss)]);
+
+(*addition distributes over multiplication*)
+val add_mult_distrib = prove_goal Arith.thy "(m + n)*k = (m*k) + (n*k)::nat"
+ (fn _ => [nat_ind_tac "m" 1,
+ ALLGOALS(asm_simp_tac(arith_ss addsimps [add_assoc]))]);
+
+(*Associative law for multiplication*)
+val mult_assoc = prove_goal Arith.thy "(m * n) * k = m * (n * k)::nat"
+ (fn _ => [nat_ind_tac "m" 1,
+ ALLGOALS(asm_simp_tac(arith_ss addsimps [add_mult_distrib]))]);
+
+
+(*** Difference ***)
+
+val diff_self_eq_0 = prove_goal Arith.thy "m - m = 0"
+ (fn _ => [nat_ind_tac "m" 1, ALLGOALS(asm_simp_tac arith_ss)]);
+
+(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
+val [prem] = goal Arith.thy "[| ~ m<n |] ==> n + (m-n) = m::nat";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS(asm_simp_tac arith_ss));
+val add_diff_inverse = result();
+
+
+(*** Remainder ***)
+
+goal Arith.thy "m - n < Suc(m)";
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (etac less_SucE 3);
+by (ALLGOALS(asm_simp_tac arith_ss));
+val diff_less_Suc = result();
+
+(*In ordinary notation: if 0<n and n<=m then m-n < m *)
+goal Arith.thy "!!m. [| 0<n; ~ m<n |] ==> m - n < m";
+by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
+by (fast_tac HOL_cs 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS(asm_simp_tac(arith_ss addsimps [diff_less_Suc])));
+val div_termination = result();
+
+val wf_less_trans = wf_pred_nat RS wf_trancl RSN (2, def_wfrec RS trans);
+
+goalw Nat.thy [less_def] "<m,n> : pred_nat^+ = (m<n)";
+by (rtac refl 1);
+val less_eq = result();
+
+(*
+val div_simps = [div_termination, cut_apply, less_eq];
+val div_ss = nat_ss addsimps div_simps;
+val div_ss = nat_ss addsimps div_simps;
+*)
+
+goal Arith.thy "!!m. m<n ==> m mod n = m";
+by (rtac (mod_def RS wf_less_trans) 1);
+by(asm_simp_tac HOL_ss 1);
+val mod_less = result();
+
+goal Arith.thy "!!m. [| 0<n; ~m<n |] ==> m mod n = (m-n) mod n";
+by (rtac (mod_def RS wf_less_trans) 1);
+by(asm_simp_tac (nat_ss addsimps [div_termination, cut_apply, less_eq]) 1);
+val mod_geq = result();
+
+
+(*** Quotient ***)
+
+goal Arith.thy "!!m. m<n ==> m div n = 0";
+by (rtac (div_def RS wf_less_trans) 1);
+by(asm_simp_tac nat_ss 1);
+val div_less = result();
+
+goal Arith.thy "!!M. [| 0<n; ~m<n |] ==> m div n = Suc((m-n) div n)";
+by (rtac (div_def RS wf_less_trans) 1);
+by(asm_simp_tac (nat_ss addsimps [div_termination, cut_apply, less_eq]) 1);
+val div_geq = result();
+
+(*Main Result about quotient and remainder.*)
+goal Arith.thy "!!m. 0<n ==> (m div n)*n + m mod n = m";
+by (res_inst_tac [("n","m")] less_induct 1);
+by (rename_tac "k" 1); (*Variable name used in line below*)
+by (res_inst_tac [("Q","k<n")] (excluded_middle RS disjE) 1);
+by (ALLGOALS (asm_simp_tac(arith_ss addsimps
+ [mod_less, mod_geq, div_less, div_geq,
+ add_assoc, add_diff_inverse, div_termination])));
+val mod_div_equality = result();
+
+
+(*** More results about difference ***)
+
+val [prem] = goal Arith.thy "m < Suc(n) ==> m-n = 0";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS (asm_simp_tac arith_ss));
+val less_imp_diff_is_0 = result();
+
+val prems = goal Arith.thy "m-n = 0 --> n-m = 0 --> m=n";
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (REPEAT(simp_tac arith_ss 1 THEN TRY(atac 1)));
+val diffs0_imp_equal_lemma = result();
+
+(* [| m-n = 0; n-m = 0 |] ==> m=n *)
+val diffs0_imp_equal = standard (diffs0_imp_equal_lemma RS mp RS mp);
+
+val [prem] = goal Arith.thy "m<n ==> 0<n-m";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS(asm_simp_tac arith_ss));
+val less_imp_diff_positive = result();
+
+val [prem] = goal Arith.thy "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
+by (rtac (prem RS rev_mp) 1);
+by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
+by (ALLGOALS(asm_simp_tac arith_ss));
+val Suc_diff_n = result();
+
+goal Arith.thy "Suc(m)-n = if(m<n, 0, Suc(m-n))";
+by(simp_tac (nat_ss addsimps [less_imp_diff_is_0, not_less_eq, Suc_diff_n]
+ setloop (split_tac [expand_if])) 1);
+val if_Suc_diff_n = result();
+
+goal Arith.thy "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
+by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
+by (ALLGOALS (strip_tac THEN' simp_tac arith_ss THEN' fast_tac HOL_cs));
+val zero_induct_lemma = result();
+
+val prems = goal Arith.thy "[| P(k); !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
+by (rtac (diff_self_eq_0 RS subst) 1);
+by (rtac (zero_induct_lemma RS mp RS mp) 1);
+by (REPEAT (ares_tac ([impI,allI]@prems) 1));
+val zero_induct = result();
+
+(*13 July 1992: loaded in 105.7s*)
+
+(**** Additional theorems about "less than" ****)
+
+goal Arith.thy "n <= (m + n)::nat";
+by (nat_ind_tac "m" 1);
+by (ALLGOALS(simp_tac (arith_ss addsimps [le_refl])));
+by (etac le_trans 1);
+by (rtac (lessI RS less_imp_le) 1);
+val le_add2 = result();
+
+goal Arith.thy "m <= (m + n)::nat";
+by (stac add_commute 1);
+by (rtac le_add2 1);
+val le_add1 = result();
+
+val less_add_Suc1 = standard (lessI RS (le_add1 RS le_less_trans));
+val less_add_Suc2 = standard (lessI RS (le_add2 RS le_less_trans));