src/HOL/Arith.ML
author nipkow
Wed Oct 28 11:25:38 1998 +0100 (1998-10-28)
changeset 5771 7c2c8cf20221
parent 5758 27a2b36efd95
child 5983 79e301a6a51b
permissions -rw-r--r--
added nat_diff_split and a few lemmas in Trancl.
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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   1998  University of Cambridge
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Proofs about elementary arithmetic: addition, multiplication, etc.
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Some from the Hoare example from Norbert Galm
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*)
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(*** Basic rewrite rules for the arithmetic operators ***)
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(** Difference **)
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qed_goal "diff_0_eq_0" thy
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    "0 - n = 0"
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 (fn _ => [induct_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_goal "diff_Suc_Suc" thy
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    "Suc(m) - Suc(n) = m - n"
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 (fn _ =>
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  [Simp_tac 1, induct_tac "n" 1, ALLGOALS Asm_simp_tac]);
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Addsimps [diff_0_eq_0, diff_Suc_Suc];
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(* Could be (and is, below) generalized in various ways;
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   However, none of the generalizations are currently in the simpset,
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   and I dread to think what happens if I put them in *)
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Goal "0 < n ==> Suc(n-1) = n";
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by (asm_simp_tac (simpset() addsplits [nat.split]) 1);
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qed "Suc_pred";
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Addsimps [Suc_pred];
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Delsimps [diff_Suc];
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(**** Inductive properties of the operators ****)
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(*** Addition ***)
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qed_goal "add_0_right" thy "m + 0 = m"
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 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
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qed_goal "add_Suc_right" thy "m + Suc(n) = Suc(m+n)"
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 (fn _ => [induct_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" thy "(m + n) + k = m + ((n + k)::nat)"
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 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
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(*Commutative law for addition*)  
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qed_goal "add_commute" thy "m + n = n + (m::nat)"
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 (fn _ =>  [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
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qed_goal "add_left_commute" 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 "(k + m = k + n) = (m=(n::nat))";
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by (induct_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 "(m + k = n + k) = (m=(n::nat))";
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by (induct_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 "(k + m <= k + n) = (m<=(n::nat))";
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by (induct_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 "(k + m < k + n) = (m<(n::nat))";
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by (induct_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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(** Reasoning about m+0=0, etc. **)
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Goal "(m+n = 0) = (m=0 & n=0)";
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by (exhaust_tac "m" 1);
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by (Auto_tac);
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qed "add_is_0";
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AddIffs [add_is_0];
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Goal "(0 = m+n) = (m=0 & n=0)";
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by (exhaust_tac "m" 1);
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by (Auto_tac);
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qed "zero_is_add";
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AddIffs [zero_is_add];
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Goal "(m+n=1) = (m=1 & n=0 | m=0 & n=1)";
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by(exhaust_tac "m" 1);
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by(Auto_tac);
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qed "add_is_1";
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Goal "(1=m+n) = (m=1 & n=0 | m=0 & n=1)";
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by(exhaust_tac "m" 1);
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by(Auto_tac);
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qed "one_is_add";
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Goal "(0<m+n) = (0<m | 0<n)";
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by (simp_tac (simpset() delsimps [neq0_conv] addsimps [neq0_conv RS sym]) 1);
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qed "add_gr_0";
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AddIffs [add_gr_0];
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(* FIXME: really needed?? *)
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Goal "((m+n)-1 = 0) = (m=0 & n-1 = 0 | m-1 = 0 & n=0)";
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by (exhaust_tac "m" 1);
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by (ALLGOALS (fast_tac (claset() addss (simpset()))));
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qed "pred_add_is_0";
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Addsimps [pred_add_is_0];
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(* Could be generalized, eg to "k<n ==> m+(n-(Suc k)) = (m+n)-(Suc k)" *)
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Goal "0<n ==> m + (n-1) = (m+n)-1";
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by (exhaust_tac "m" 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps [diff_Suc]
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                                      addsplits [nat.split])));
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qed "add_pred";
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Addsimps [add_pred];
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Goal "m + n = m ==> n = 0";
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by (dtac (add_0_right RS ssubst) 1);
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by (asm_full_simp_tac (simpset() addsimps [add_assoc]
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                                 delsimps [add_0_right]) 1);
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qed "add_eq_self_zero";
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(**** Additional theorems about "less than" ****)
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(*Deleted less_natE; instead use less_eq_Suc_add RS exE*)
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Goal "m<n --> (? k. n=Suc(m+k))";
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by (induct_tac "n" 1);
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by (ALLGOALS (simp_tac (simpset() addsimps [order_le_less])));
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by (blast_tac (claset() addSEs [less_SucE] 
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                        addSIs [add_0_right RS sym, add_Suc_right RS sym]) 1);
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qed_spec_mp "less_eq_Suc_add";
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Goal "n <= ((m + n)::nat)";
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by (induct_tac "m" 1);
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by (ALLGOALS Simp_tac);
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qed "le_add2";
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Goal "n <= ((n + m)::nat)";
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by (simp_tac (simpset() addsimps add_ac) 1);
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by (rtac le_add2 1);
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qed "le_add1";
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bind_thm ("less_add_Suc1", (lessI RS (le_add1 RS le_less_trans)));
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bind_thm ("less_add_Suc2", (lessI RS (le_add2 RS le_less_trans)));
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Goal "(m<n) = (? k. n=Suc(m+k))";
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by (blast_tac (claset() addSIs [less_add_Suc1, less_eq_Suc_add]) 1);
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qed "less_iff_Suc_add";
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(*"i <= j ==> i <= j+m"*)
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bind_thm ("trans_le_add1", le_add1 RSN (2,le_trans));
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(*"i <= j ==> i <= m+j"*)
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bind_thm ("trans_le_add2", le_add2 RSN (2,le_trans));
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(*"i < j ==> i < j+m"*)
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bind_thm ("trans_less_add1", le_add1 RSN (2,less_le_trans));
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(*"i < j ==> i < m+j"*)
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bind_thm ("trans_less_add2", le_add2 RSN (2,less_le_trans));
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Goal "i+j < (k::nat) --> i<k";
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by (induct_tac "j" 1);
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by (ALLGOALS Asm_simp_tac);
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qed_spec_mp "add_lessD1";
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Goal "~ (i+j < (i::nat))";
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by (rtac notI 1);
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by (etac (add_lessD1 RS less_irrefl) 1);
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qed "not_add_less1";
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Goal "~ (j+i < (i::nat))";
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by (simp_tac (simpset() addsimps [add_commute, not_add_less1]) 1);
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qed "not_add_less2";
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AddIffs [not_add_less1, not_add_less2];
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Goal "m+k<=n --> m<=(n::nat)";
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by (induct_tac "k" 1);
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by (ALLGOALS (asm_simp_tac (simpset() addsimps le_simps)));
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qed_spec_mp "add_leD1";
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Goal "m+k<=n ==> k<=(n::nat)";
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by (full_simp_tac (simpset() addsimps [add_commute]) 1);
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by (etac add_leD1 1);
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qed_spec_mp "add_leD2";
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Goal "m+k<=n ==> m<=n & k<=(n::nat)";
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by (blast_tac (claset() addDs [add_leD1, add_leD2]) 1);
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bind_thm ("add_leE", result() RS conjE);
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(*needs !!k for add_ac to work*)
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Goal "!!k:: nat. [| k<l;  m+l = k+n |] ==> m<n";
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by (force_tac (claset(),
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	      simpset() delsimps [add_Suc_right]
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	                addsimps [less_iff_Suc_add,
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				  add_Suc_right RS sym] @ add_ac) 1);
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qed "less_add_eq_less";
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(*** Monotonicity of Addition ***)
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(*strict, in 1st argument*)
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Goal "i < j ==> i + k < j + (k::nat)";
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by (induct_tac "k" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_less_mono1";
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(*strict, in both arguments*)
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Goal "[|i < j; k < l|] ==> i + k < j + (l::nat)";
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by (rtac (add_less_mono1 RS less_trans) 1);
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by (REPEAT (assume_tac 1));
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by (induct_tac "j" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_less_mono";
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(*A [clumsy] way of lifting < monotonicity to <= monotonicity *)
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val [lt_mono,le] = Goal
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     "[| !!i j::nat. i<j ==> f(i) < f(j);       \
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\        i <= j                                 \
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\     |] ==> f(i) <= (f(j)::nat)";
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by (cut_facts_tac [le] 1);
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by (asm_full_simp_tac (simpset() addsimps [order_le_less]) 1);
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by (blast_tac (claset() addSIs [lt_mono]) 1);
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qed "less_mono_imp_le_mono";
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(*non-strict, in 1st argument*)
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Goal "i<=j ==> i + k <= j + (k::nat)";
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by (res_inst_tac [("f", "%j. j+k")] less_mono_imp_le_mono 1);
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by (etac add_less_mono1 1);
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by (assume_tac 1);
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qed "add_le_mono1";
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(*non-strict, in both arguments*)
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Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::nat)";
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by (etac (add_le_mono1 RS le_trans) 1);
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by (simp_tac (simpset() addsimps [add_commute]) 1);
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qed "add_le_mono";
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(*** Multiplication ***)
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(*right annihilation in product*)
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qed_goal "mult_0_right" thy "m * 0 = 0"
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 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
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(*right successor law for multiplication*)
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qed_goal "mult_Suc_right" thy  "m * Suc(n) = m + (m * n)"
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 (fn _ => [induct_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 "1 * n = n";
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by (Asm_simp_tac 1);
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qed "mult_1";
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Goal "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" thy "m * n = n * (m::nat)"
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 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
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(*addition distributes over multiplication*)
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qed_goal "add_mult_distrib" thy "(m + n)*k = (m*k) + ((n*k)::nat)"
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 (fn _ => [induct_tac "m" 1,
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           ALLGOALS(asm_simp_tac (simpset() addsimps add_ac))]);
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qed_goal "add_mult_distrib2" thy "k*(m + n) = (k*m) + ((k*n)::nat)"
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 (fn _ => [induct_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" thy "(m * n) * k = m * ((n * k)::nat)"
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  (fn _ => [induct_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" 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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Goal "(m*n = 0) = (m=0 | n=0)";
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by (induct_tac "m" 1);
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by (induct_tac "n" 2);
paulson@3293
   310
by (ALLGOALS Asm_simp_tac);
paulson@3293
   311
qed "mult_is_0";
paulson@3293
   312
Addsimps [mult_is_0];
paulson@3293
   313
paulson@5429
   314
Goal "m <= m*(m::nat)";
paulson@4158
   315
by (induct_tac "m" 1);
paulson@4158
   316
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
paulson@4158
   317
by (etac (le_add2 RSN (2,le_trans)) 1);
paulson@4158
   318
qed "le_square";
paulson@4158
   319
paulson@3234
   320
paulson@3234
   321
(*** Difference ***)
paulson@3234
   322
paulson@3234
   323
paulson@4732
   324
qed_goal "diff_self_eq_0" thy "m - m = 0"
paulson@3339
   325
 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
paulson@3234
   326
Addsimps [diff_self_eq_0];
paulson@3234
   327
paulson@3234
   328
(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
wenzelm@5069
   329
Goal "~ m<n --> n+(m-n) = (m::nat)";
paulson@3234
   330
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   331
by (ALLGOALS Asm_simp_tac);
paulson@3381
   332
qed_spec_mp "add_diff_inverse";
paulson@3381
   333
paulson@5143
   334
Goal "n<=m ==> n+(m-n) = (m::nat)";
wenzelm@4089
   335
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
paulson@3381
   336
qed "le_add_diff_inverse";
paulson@3234
   337
paulson@5143
   338
Goal "n<=m ==> (m-n)+n = (m::nat)";
wenzelm@4089
   339
by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
paulson@3381
   340
qed "le_add_diff_inverse2";
paulson@3381
   341
paulson@3381
   342
Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
paulson@3234
   343
paulson@3234
   344
paulson@3234
   345
(*** More results about difference ***)
paulson@3234
   346
paulson@5414
   347
Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
paulson@5316
   348
by (etac rev_mp 1);
paulson@3352
   349
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   350
by (ALLGOALS Asm_simp_tac);
paulson@5414
   351
qed "Suc_diff_le";
paulson@3352
   352
paulson@5429
   353
Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
paulson@5429
   354
by (res_inst_tac [("m","n"),("n","l")] diff_induct 1);
paulson@5429
   355
by (ALLGOALS Asm_simp_tac);
paulson@5429
   356
qed_spec_mp "Suc_diff_add_le";
paulson@5429
   357
wenzelm@5069
   358
Goal "m - n < Suc(m)";
paulson@3234
   359
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3234
   360
by (etac less_SucE 3);
wenzelm@4089
   361
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
paulson@3234
   362
qed "diff_less_Suc";
paulson@3234
   363
paulson@5429
   364
Goal "m - n <= (m::nat)";
paulson@3234
   365
by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
paulson@3234
   366
by (ALLGOALS Asm_simp_tac);
paulson@3234
   367
qed "diff_le_self";
paulson@3903
   368
Addsimps [diff_le_self];
paulson@3234
   369
paulson@4732
   370
(* j<k ==> j-n < k *)
paulson@4732
   371
bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
paulson@4732
   372
wenzelm@5069
   373
Goal "!!i::nat. i-j-k = i - (j+k)";
paulson@3352
   374
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@3352
   375
by (ALLGOALS Asm_simp_tac);
paulson@3352
   376
qed "diff_diff_left";
paulson@3352
   377
wenzelm@5069
   378
Goal "(Suc m - n) - Suc k = m - n - k";
wenzelm@4423
   379
by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
paulson@4736
   380
qed "Suc_diff_diff";
paulson@4736
   381
Addsimps [Suc_diff_diff];
nipkow@4360
   382
paulson@5143
   383
Goal "0<n ==> n - Suc i < n";
berghofe@5183
   384
by (exhaust_tac "n" 1);
paulson@4732
   385
by Safe_tac;
paulson@5497
   386
by (asm_simp_tac (simpset() addsimps le_simps) 1);
paulson@4732
   387
qed "diff_Suc_less";
paulson@4732
   388
Addsimps [diff_Suc_less];
paulson@4732
   389
paulson@5329
   390
Goal "i<n ==> n - Suc i < n - i";
paulson@5329
   391
by (exhaust_tac "n" 1);
paulson@5497
   392
by (auto_tac (claset(),
paulson@5537
   393
	      simpset() addsimps [Suc_diff_le]@le_simps));
paulson@5329
   394
qed "diff_Suc_less_diff";
paulson@5329
   395
wenzelm@3396
   396
(*This and the next few suggested by Florian Kammueller*)
wenzelm@5069
   397
Goal "!!i::nat. i-j-k = i-k-j";
wenzelm@4089
   398
by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
paulson@3352
   399
qed "diff_commute";
paulson@3352
   400
paulson@5429
   401
Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
paulson@3352
   402
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@3352
   403
by (ALLGOALS Asm_simp_tac);
paulson@5414
   404
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
paulson@3352
   405
qed_spec_mp "diff_diff_right";
paulson@3352
   406
paulson@5429
   407
Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
paulson@3352
   408
by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
paulson@3352
   409
by (ALLGOALS Asm_simp_tac);
paulson@3352
   410
qed_spec_mp "diff_add_assoc";
paulson@3352
   411
paulson@5429
   412
Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
paulson@4732
   413
by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
paulson@4732
   414
qed_spec_mp "diff_add_assoc2";
paulson@4732
   415
paulson@5429
   416
Goal "(n+m) - n = (m::nat)";
paulson@3339
   417
by (induct_tac "n" 1);
paulson@3234
   418
by (ALLGOALS Asm_simp_tac);
paulson@3234
   419
qed "diff_add_inverse";
paulson@3234
   420
Addsimps [diff_add_inverse];
paulson@3234
   421
paulson@5429
   422
Goal "(m+n) - n = (m::nat)";
wenzelm@4089
   423
by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
paulson@3234
   424
qed "diff_add_inverse2";
paulson@3234
   425
Addsimps [diff_add_inverse2];
paulson@3234
   426
paulson@5429
   427
Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
paulson@3724
   428
by Safe_tac;
paulson@3381
   429
by (ALLGOALS Asm_simp_tac);
paulson@3366
   430
qed "le_imp_diff_is_add";
paulson@3366
   431
paulson@5356
   432
Goal "(m-n = 0) = (m <= n)";
paulson@3234
   433
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@5497
   434
by (ALLGOALS Asm_simp_tac);
paulson@5356
   435
qed "diff_is_0_eq";
paulson@5356
   436
Addsimps [diff_is_0_eq RS iffD2];
paulson@3234
   437
paulson@5333
   438
Goal "(0<n-m) = (m<n)";
paulson@3234
   439
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   440
by (ALLGOALS Asm_simp_tac);
paulson@5333
   441
qed "zero_less_diff";
paulson@5333
   442
Addsimps [zero_less_diff];
paulson@3234
   443
paulson@5333
   444
Goal "i < j  ==> ? k. 0<k & i+k = j";
paulson@5078
   445
by (res_inst_tac [("x","j - i")] exI 1);
paulson@5333
   446
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
paulson@5078
   447
qed "less_imp_add_positive";
paulson@5078
   448
wenzelm@5069
   449
Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
paulson@5414
   450
by (simp_tac (simpset() addsimps [leI, Suc_le_eq, Suc_diff_le]) 1);
paulson@5414
   451
qed "if_Suc_diff_le";
paulson@3234
   452
wenzelm@5069
   453
Goal "Suc(m)-n <= Suc(m-n)";
paulson@5414
   454
by (simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
paulson@4672
   455
qed "diff_Suc_le_Suc_diff";
paulson@4672
   456
wenzelm@5069
   457
Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
paulson@3234
   458
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
paulson@3718
   459
by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
paulson@3234
   460
qed "zero_induct_lemma";
paulson@3234
   461
paulson@5316
   462
val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
paulson@3234
   463
by (rtac (diff_self_eq_0 RS subst) 1);
paulson@3234
   464
by (rtac (zero_induct_lemma RS mp RS mp) 1);
paulson@3234
   465
by (REPEAT (ares_tac ([impI,allI]@prems) 1));
paulson@3234
   466
qed "zero_induct";
paulson@3234
   467
paulson@5429
   468
Goal "(k+m) - (k+n) = m - (n::nat)";
paulson@3339
   469
by (induct_tac "k" 1);
paulson@3234
   470
by (ALLGOALS Asm_simp_tac);
paulson@3234
   471
qed "diff_cancel";
paulson@3234
   472
Addsimps [diff_cancel];
paulson@3234
   473
paulson@5429
   474
Goal "(m+k) - (n+k) = m - (n::nat)";
paulson@3234
   475
val add_commute_k = read_instantiate [("n","k")] add_commute;
paulson@5537
   476
by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1);
paulson@3234
   477
qed "diff_cancel2";
paulson@3234
   478
Addsimps [diff_cancel2];
paulson@3234
   479
paulson@5414
   480
(*From Clemens Ballarin, proof by lcp*)
paulson@5429
   481
Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
paulson@5414
   482
by (REPEAT (etac rev_mp 1));
paulson@5414
   483
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@5414
   484
by (ALLGOALS Asm_simp_tac);
paulson@5414
   485
(*a confluence problem*)
paulson@5414
   486
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
paulson@3234
   487
qed "diff_right_cancel";
paulson@3234
   488
paulson@5429
   489
Goal "n - (n+m) = 0";
paulson@3339
   490
by (induct_tac "n" 1);
paulson@3234
   491
by (ALLGOALS Asm_simp_tac);
paulson@3234
   492
qed "diff_add_0";
paulson@3234
   493
Addsimps [diff_add_0];
paulson@3234
   494
paulson@5409
   495
paulson@3234
   496
(** Difference distributes over multiplication **)
paulson@3234
   497
wenzelm@5069
   498
Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
paulson@3234
   499
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3234
   500
by (ALLGOALS Asm_simp_tac);
paulson@3234
   501
qed "diff_mult_distrib" ;
paulson@3234
   502
wenzelm@5069
   503
Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
paulson@3234
   504
val mult_commute_k = read_instantiate [("m","k")] mult_commute;
wenzelm@4089
   505
by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
paulson@3234
   506
qed "diff_mult_distrib2" ;
paulson@3234
   507
(*NOT added as rewrites, since sometimes they are used from right-to-left*)
paulson@3234
   508
paulson@3234
   509
paulson@1713
   510
(*** Monotonicity of Multiplication ***)
paulson@1713
   511
paulson@5429
   512
Goal "i <= (j::nat) ==> i*k<=j*k";
paulson@3339
   513
by (induct_tac "k" 1);
wenzelm@4089
   514
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
paulson@1713
   515
qed "mult_le_mono1";
paulson@1713
   516
paulson@1713
   517
(*<=monotonicity, BOTH arguments*)
paulson@5429
   518
Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
paulson@2007
   519
by (etac (mult_le_mono1 RS le_trans) 1);
paulson@1713
   520
by (rtac le_trans 1);
paulson@2007
   521
by (stac mult_commute 2);
paulson@2007
   522
by (etac mult_le_mono1 2);
wenzelm@4089
   523
by (simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@1713
   524
qed "mult_le_mono";
paulson@1713
   525
paulson@1713
   526
(*strict, in 1st argument; proof is by induction on k>0*)
paulson@5429
   527
Goal "[| i<j; 0<k |] ==> k*i < k*j";
paulson@5078
   528
by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
paulson@1713
   529
by (Asm_simp_tac 1);
paulson@3339
   530
by (induct_tac "x" 1);
wenzelm@4089
   531
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
paulson@1713
   532
qed "mult_less_mono2";
paulson@1713
   533
paulson@5429
   534
Goal "[| i<j; 0<k |] ==> i*k < j*k";
paulson@3457
   535
by (dtac mult_less_mono2 1);
wenzelm@4089
   536
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
paulson@3234
   537
qed "mult_less_mono1";
paulson@3234
   538
wenzelm@5069
   539
Goal "(0 < m*n) = (0<m & 0<n)";
paulson@3339
   540
by (induct_tac "m" 1);
paulson@3339
   541
by (induct_tac "n" 2);
paulson@1713
   542
by (ALLGOALS Asm_simp_tac);
paulson@1713
   543
qed "zero_less_mult_iff";
nipkow@4356
   544
Addsimps [zero_less_mult_iff];
paulson@1713
   545
wenzelm@5069
   546
Goal "(m*n = 1) = (m=1 & n=1)";
paulson@3339
   547
by (induct_tac "m" 1);
paulson@1795
   548
by (Simp_tac 1);
paulson@3339
   549
by (induct_tac "n" 1);
paulson@1795
   550
by (Simp_tac 1);
wenzelm@4089
   551
by (fast_tac (claset() addss simpset()) 1);
paulson@1795
   552
qed "mult_eq_1_iff";
nipkow@4356
   553
Addsimps [mult_eq_1_iff];
paulson@1795
   554
paulson@5143
   555
Goal "0<k ==> (m*k < n*k) = (m<n)";
wenzelm@4089
   556
by (safe_tac (claset() addSIs [mult_less_mono1]));
paulson@3234
   557
by (cut_facts_tac [less_linear] 1);
paulson@4389
   558
by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
paulson@3234
   559
qed "mult_less_cancel2";
paulson@3234
   560
paulson@5143
   561
Goal "0<k ==> (k*m < k*n) = (m<n)";
paulson@3457
   562
by (dtac mult_less_cancel2 1);
wenzelm@4089
   563
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@3234
   564
qed "mult_less_cancel1";
paulson@3234
   565
Addsimps [mult_less_cancel1, mult_less_cancel2];
paulson@3234
   566
wenzelm@5069
   567
Goal "(Suc k * m < Suc k * n) = (m < n)";
wenzelm@4423
   568
by (rtac mult_less_cancel1 1);
wenzelm@4297
   569
by (Simp_tac 1);
wenzelm@4297
   570
qed "Suc_mult_less_cancel1";
wenzelm@4297
   571
wenzelm@5069
   572
Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
wenzelm@4297
   573
by (simp_tac (simpset_of HOL.thy) 1);
wenzelm@4423
   574
by (rtac Suc_mult_less_cancel1 1);
wenzelm@4297
   575
qed "Suc_mult_le_cancel1";
wenzelm@4297
   576
paulson@5143
   577
Goal "0<k ==> (m*k = n*k) = (m=n)";
paulson@3234
   578
by (cut_facts_tac [less_linear] 1);
paulson@3724
   579
by Safe_tac;
paulson@3457
   580
by (assume_tac 2);
paulson@3234
   581
by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
paulson@3234
   582
by (ALLGOALS Asm_full_simp_tac);
paulson@3234
   583
qed "mult_cancel2";
paulson@3234
   584
paulson@5143
   585
Goal "0<k ==> (k*m = k*n) = (m=n)";
paulson@3457
   586
by (dtac mult_cancel2 1);
wenzelm@4089
   587
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@3234
   588
qed "mult_cancel1";
paulson@3234
   589
Addsimps [mult_cancel1, mult_cancel2];
paulson@3234
   590
wenzelm@5069
   591
Goal "(Suc k * m = Suc k * n) = (m = n)";
wenzelm@4423
   592
by (rtac mult_cancel1 1);
wenzelm@4297
   593
by (Simp_tac 1);
wenzelm@4297
   594
qed "Suc_mult_cancel1";
wenzelm@4297
   595
paulson@3234
   596
paulson@1795
   597
(** Lemma for gcd **)
paulson@1795
   598
paulson@5143
   599
Goal "m = m*n ==> n=1 | m=0";
paulson@1795
   600
by (dtac sym 1);
paulson@1795
   601
by (rtac disjCI 1);
paulson@1795
   602
by (rtac nat_less_cases 1 THEN assume_tac 2);
wenzelm@4089
   603
by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
nipkow@4356
   604
by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
paulson@1795
   605
qed "mult_eq_self_implies_10";
paulson@1795
   606
paulson@1795
   607
paulson@4736
   608
(*** Subtraction laws -- mostly from Clemens Ballarin ***)
paulson@3234
   609
paulson@5429
   610
Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
paulson@3234
   611
by (subgoal_tac "c+(a-c) < c+(b-c)" 1);
paulson@3381
   612
by (Full_simp_tac 1);
paulson@3234
   613
by (subgoal_tac "c <= b" 1);
wenzelm@4089
   614
by (blast_tac (claset() addIs [less_imp_le, le_trans]) 2);
paulson@3381
   615
by (Asm_simp_tac 1);
paulson@3234
   616
qed "diff_less_mono";
paulson@3234
   617
paulson@5429
   618
Goal "a+b < (c::nat) ==> a < c-b";
paulson@3457
   619
by (dtac diff_less_mono 1);
paulson@3457
   620
by (rtac le_add2 1);
paulson@3234
   621
by (Asm_full_simp_tac 1);
paulson@3234
   622
qed "add_less_imp_less_diff";
paulson@3234
   623
nipkow@5427
   624
Goal "(i < j-k) = (i+k < (j::nat))";
paulson@5497
   625
by (rtac iffI 1);
paulson@5497
   626
 by (case_tac "k <= j" 1);
paulson@5497
   627
  by (dtac le_add_diff_inverse2 1);
paulson@5497
   628
  by (dres_inst_tac [("k","k")] add_less_mono1 1);
paulson@5497
   629
  by (Asm_full_simp_tac 1);
paulson@5497
   630
 by (rotate_tac 1 1);
paulson@5497
   631
 by (asm_full_simp_tac (simpset() addSolver cut_trans_tac) 1);
paulson@5497
   632
by (etac add_less_imp_less_diff 1);
nipkow@5427
   633
qed "less_diff_conv";
nipkow@5427
   634
paulson@5497
   635
Goal "(j-k <= (i::nat)) = (j <= i+k)";
paulson@5497
   636
by (simp_tac (simpset() addsimps [less_diff_conv, le_def]) 1);
paulson@5485
   637
qed "le_diff_conv";
paulson@5485
   638
paulson@5497
   639
Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))";
paulson@5497
   640
by (asm_full_simp_tac
paulson@5497
   641
    (simpset() delsimps [less_Suc_eq_le]
paulson@5497
   642
               addsimps [less_Suc_eq_le RS sym, less_diff_conv,
paulson@5497
   643
			 Suc_diff_le RS sym]) 1);
paulson@5497
   644
qed "le_diff_conv2";
paulson@5497
   645
paulson@5143
   646
Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
paulson@5497
   647
by (asm_full_simp_tac (simpset() addsimps [Suc_diff_le RS sym]) 1);
paulson@3234
   648
qed "Suc_diff_Suc";
paulson@3234
   649
paulson@5429
   650
Goal "i <= (n::nat) ==> n - (n - i) = i";
paulson@3903
   651
by (etac rev_mp 1);
paulson@3903
   652
by (res_inst_tac [("m","n"),("n","i")] diff_induct 1);
wenzelm@4089
   653
by (ALLGOALS (asm_simp_tac  (simpset() addsimps [Suc_diff_le])));
paulson@3234
   654
qed "diff_diff_cancel";
paulson@3381
   655
Addsimps [diff_diff_cancel];
paulson@3234
   656
paulson@5429
   657
Goal "k <= (n::nat) ==> m <= n + m - k";
paulson@3457
   658
by (etac rev_mp 1);
paulson@3234
   659
by (res_inst_tac [("m", "k"), ("n", "n")] diff_induct 1);
paulson@3234
   660
by (Simp_tac 1);
paulson@5497
   661
by (simp_tac (simpset() addsimps [le_add2, less_imp_le]) 1);
paulson@3234
   662
by (Simp_tac 1);
paulson@3234
   663
qed "le_add_diff";
paulson@3234
   664
paulson@5429
   665
Goal "0<k ==> j<i --> j+k-i < k";
paulson@4736
   666
by (res_inst_tac [("m","j"),("n","i")] diff_induct 1);
paulson@4736
   667
by (ALLGOALS Asm_simp_tac);
paulson@4736
   668
qed_spec_mp "add_diff_less";
paulson@4736
   669
paulson@3234
   670
paulson@5356
   671
Goal "m-1 < n ==> m <= n";
paulson@5356
   672
by (exhaust_tac "m" 1);
paulson@5356
   673
by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
paulson@5356
   674
qed "pred_less_imp_le";
paulson@5356
   675
paulson@5356
   676
Goal "j<=i ==> i - j < Suc i - j";
paulson@5356
   677
by (REPEAT (etac rev_mp 1));
paulson@5356
   678
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@5356
   679
by Auto_tac;
paulson@5356
   680
qed "diff_less_Suc_diff";
paulson@5356
   681
paulson@5356
   682
Goal "i - j <= Suc i - j";
paulson@5356
   683
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@5356
   684
by Auto_tac;
paulson@5356
   685
qed "diff_le_Suc_diff";
paulson@5356
   686
AddIffs [diff_le_Suc_diff];
paulson@5356
   687
paulson@5356
   688
Goal "n - Suc i <= n - i";
paulson@5356
   689
by (case_tac "i<n" 1);
paulson@5497
   690
by (dtac diff_Suc_less_diff 1);
paulson@5604
   691
by (auto_tac (claset(), simpset() addsimps [less_imp_le, leI]));
paulson@5356
   692
qed "diff_Suc_le_diff";
paulson@5356
   693
AddIffs [diff_Suc_le_diff];
paulson@5356
   694
paulson@5409
   695
Goal "0 < n ==> (m <= n-1) = (m<n)";
paulson@5409
   696
by (exhaust_tac "n" 1);
paulson@5497
   697
by (auto_tac (claset(), simpset() addsimps le_simps));
paulson@5409
   698
qed "le_pred_eq";
paulson@5409
   699
paulson@5409
   700
Goal "0 < n ==> (m-1 < n) = (m<=n)";
paulson@5409
   701
by (exhaust_tac "m" 1);
paulson@5409
   702
by (auto_tac (claset(), simpset() addsimps [Suc_le_eq]));
paulson@5409
   703
qed "less_pred_eq";
paulson@5409
   704
paulson@5414
   705
(*In ordinary notation: if 0<n and n<=m then m-n < m *)
paulson@5414
   706
Goal "[| 0<n; ~ m<n |] ==> m - n < m";
paulson@5414
   707
by (subgoal_tac "0<n --> ~ m<n --> m - n < m" 1);
paulson@5414
   708
by (Blast_tac 1);
paulson@5414
   709
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@5414
   710
by (ALLGOALS(asm_simp_tac(simpset() addsimps [diff_less_Suc])));
paulson@5414
   711
qed "diff_less";
paulson@5414
   712
paulson@5414
   713
Goal "[| 0<n; n<=m |] ==> m - n < m";
paulson@5414
   714
by (asm_simp_tac (simpset() addsimps [diff_less, not_less_iff_le]) 1);
paulson@5414
   715
qed "le_diff_less";
paulson@5414
   716
paulson@5356
   717
paulson@4732
   718
nipkow@3484
   719
(** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
nipkow@3484
   720
nipkow@3484
   721
(* Monotonicity of subtraction in first argument *)
paulson@5429
   722
Goal "m <= (n::nat) --> (m-l) <= (n-l)";
nipkow@3484
   723
by (induct_tac "n" 1);
nipkow@3484
   724
by (Simp_tac 1);
wenzelm@4089
   725
by (simp_tac (simpset() addsimps [le_Suc_eq]) 1);
paulson@4732
   726
by (blast_tac (claset() addIs [diff_le_Suc_diff, le_trans]) 1);
nipkow@3484
   727
qed_spec_mp "diff_le_mono";
nipkow@3484
   728
paulson@5429
   729
Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
nipkow@3484
   730
by (induct_tac "l" 1);
nipkow@3484
   731
by (Simp_tac 1);
berghofe@5183
   732
by (case_tac "n <= na" 1);
berghofe@5183
   733
by (subgoal_tac "m <= na" 1);
wenzelm@4089
   734
by (asm_simp_tac (simpset() addsimps [Suc_diff_le]) 1);
wenzelm@4089
   735
by (fast_tac (claset() addEs [le_trans]) 1);
nipkow@3484
   736
by (dtac not_leE 1);
paulson@5414
   737
by (asm_simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
nipkow@3484
   738
qed_spec_mp "diff_le_mono2";