src/HOL/Arith.ML
author nipkow
Mon Jan 04 15:07:47 1999 +0100 (1999-01-04)
changeset 6055 fdf4638bf726
parent 5983 79e301a6a51b
child 6059 aa00e235ea27
permissions -rw-r--r--
Version 1.0 of linear nat arithmetic.
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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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by (etac le_SucI 1);
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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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by(blast_tac (claset() addDs [Suc_lessD]) 1);
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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)";
paulson@3339
   310
by (induct_tac "m" 1);
paulson@3339
   311
by (induct_tac "n" 2);
paulson@3293
   312
by (ALLGOALS Asm_simp_tac);
paulson@3293
   313
qed "mult_is_0";
paulson@3293
   314
Addsimps [mult_is_0];
paulson@3293
   315
paulson@5429
   316
Goal "m <= m*(m::nat)";
paulson@4158
   317
by (induct_tac "m" 1);
paulson@4158
   318
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
paulson@4158
   319
by (etac (le_add2 RSN (2,le_trans)) 1);
paulson@4158
   320
qed "le_square";
paulson@4158
   321
paulson@3234
   322
paulson@3234
   323
(*** Difference ***)
paulson@3234
   324
paulson@3234
   325
paulson@4732
   326
qed_goal "diff_self_eq_0" thy "m - m = 0"
paulson@3339
   327
 (fn _ => [induct_tac "m" 1, ALLGOALS Asm_simp_tac]);
paulson@3234
   328
Addsimps [diff_self_eq_0];
paulson@3234
   329
paulson@3234
   330
(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
wenzelm@5069
   331
Goal "~ m<n --> n+(m-n) = (m::nat)";
paulson@3234
   332
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   333
by (ALLGOALS Asm_simp_tac);
paulson@3381
   334
qed_spec_mp "add_diff_inverse";
paulson@3381
   335
paulson@5143
   336
Goal "n<=m ==> n+(m-n) = (m::nat)";
wenzelm@4089
   337
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
paulson@3381
   338
qed "le_add_diff_inverse";
paulson@3234
   339
paulson@5143
   340
Goal "n<=m ==> (m-n)+n = (m::nat)";
wenzelm@4089
   341
by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
paulson@3381
   342
qed "le_add_diff_inverse2";
paulson@3381
   343
paulson@3381
   344
Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
paulson@3234
   345
paulson@3234
   346
paulson@3234
   347
(*** More results about difference ***)
paulson@3234
   348
paulson@5414
   349
Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
paulson@5316
   350
by (etac rev_mp 1);
paulson@3352
   351
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   352
by (ALLGOALS Asm_simp_tac);
paulson@5414
   353
qed "Suc_diff_le";
paulson@3352
   354
paulson@5429
   355
Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
paulson@5429
   356
by (res_inst_tac [("m","n"),("n","l")] diff_induct 1);
paulson@5429
   357
by (ALLGOALS Asm_simp_tac);
paulson@5429
   358
qed_spec_mp "Suc_diff_add_le";
paulson@5429
   359
wenzelm@5069
   360
Goal "m - n < Suc(m)";
paulson@3234
   361
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3234
   362
by (etac less_SucE 3);
wenzelm@4089
   363
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
paulson@3234
   364
qed "diff_less_Suc";
paulson@3234
   365
paulson@5429
   366
Goal "m - n <= (m::nat)";
paulson@3234
   367
by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
paulson@3234
   368
by (ALLGOALS Asm_simp_tac);
paulson@3234
   369
qed "diff_le_self";
paulson@3903
   370
Addsimps [diff_le_self];
paulson@3234
   371
paulson@4732
   372
(* j<k ==> j-n < k *)
paulson@4732
   373
bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
paulson@4732
   374
wenzelm@5069
   375
Goal "!!i::nat. i-j-k = i - (j+k)";
paulson@3352
   376
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@3352
   377
by (ALLGOALS Asm_simp_tac);
paulson@3352
   378
qed "diff_diff_left";
paulson@3352
   379
wenzelm@5069
   380
Goal "(Suc m - n) - Suc k = m - n - k";
wenzelm@4423
   381
by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
paulson@4736
   382
qed "Suc_diff_diff";
paulson@4736
   383
Addsimps [Suc_diff_diff];
nipkow@4360
   384
paulson@5143
   385
Goal "0<n ==> n - Suc i < n";
berghofe@5183
   386
by (exhaust_tac "n" 1);
paulson@4732
   387
by Safe_tac;
paulson@5497
   388
by (asm_simp_tac (simpset() addsimps le_simps) 1);
paulson@4732
   389
qed "diff_Suc_less";
paulson@4732
   390
Addsimps [diff_Suc_less];
paulson@4732
   391
paulson@5329
   392
Goal "i<n ==> n - Suc i < n - i";
paulson@5329
   393
by (exhaust_tac "n" 1);
paulson@5497
   394
by (auto_tac (claset(),
paulson@5537
   395
	      simpset() addsimps [Suc_diff_le]@le_simps));
paulson@5329
   396
qed "diff_Suc_less_diff";
paulson@5329
   397
wenzelm@3396
   398
(*This and the next few suggested by Florian Kammueller*)
wenzelm@5069
   399
Goal "!!i::nat. i-j-k = i-k-j";
wenzelm@4089
   400
by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
paulson@3352
   401
qed "diff_commute";
paulson@3352
   402
paulson@5429
   403
Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
paulson@3352
   404
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@3352
   405
by (ALLGOALS Asm_simp_tac);
paulson@5414
   406
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
paulson@3352
   407
qed_spec_mp "diff_diff_right";
paulson@3352
   408
paulson@5429
   409
Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
paulson@3352
   410
by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
paulson@3352
   411
by (ALLGOALS Asm_simp_tac);
paulson@3352
   412
qed_spec_mp "diff_add_assoc";
paulson@3352
   413
paulson@5429
   414
Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
paulson@4732
   415
by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
paulson@4732
   416
qed_spec_mp "diff_add_assoc2";
paulson@4732
   417
paulson@5429
   418
Goal "(n+m) - n = (m::nat)";
paulson@3339
   419
by (induct_tac "n" 1);
paulson@3234
   420
by (ALLGOALS Asm_simp_tac);
paulson@3234
   421
qed "diff_add_inverse";
paulson@3234
   422
Addsimps [diff_add_inverse];
paulson@3234
   423
paulson@5429
   424
Goal "(m+n) - n = (m::nat)";
wenzelm@4089
   425
by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
paulson@3234
   426
qed "diff_add_inverse2";
paulson@3234
   427
Addsimps [diff_add_inverse2];
paulson@3234
   428
paulson@5429
   429
Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
paulson@3724
   430
by Safe_tac;
paulson@3381
   431
by (ALLGOALS Asm_simp_tac);
paulson@3366
   432
qed "le_imp_diff_is_add";
paulson@3366
   433
paulson@5356
   434
Goal "(m-n = 0) = (m <= n)";
paulson@3234
   435
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@5497
   436
by (ALLGOALS Asm_simp_tac);
paulson@5356
   437
qed "diff_is_0_eq";
paulson@5356
   438
Addsimps [diff_is_0_eq RS iffD2];
paulson@3234
   439
paulson@5333
   440
Goal "(0<n-m) = (m<n)";
paulson@3234
   441
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   442
by (ALLGOALS Asm_simp_tac);
paulson@5333
   443
qed "zero_less_diff";
paulson@5333
   444
Addsimps [zero_less_diff];
paulson@3234
   445
paulson@5333
   446
Goal "i < j  ==> ? k. 0<k & i+k = j";
paulson@5078
   447
by (res_inst_tac [("x","j - i")] exI 1);
paulson@5333
   448
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
paulson@5078
   449
qed "less_imp_add_positive";
paulson@5078
   450
wenzelm@5069
   451
Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
paulson@5414
   452
by (simp_tac (simpset() addsimps [leI, Suc_le_eq, Suc_diff_le]) 1);
paulson@5414
   453
qed "if_Suc_diff_le";
paulson@3234
   454
wenzelm@5069
   455
Goal "Suc(m)-n <= Suc(m-n)";
paulson@5414
   456
by (simp_tac (simpset() addsimps [if_Suc_diff_le]) 1);
paulson@4672
   457
qed "diff_Suc_le_Suc_diff";
paulson@4672
   458
wenzelm@5069
   459
Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
paulson@3234
   460
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
paulson@3718
   461
by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
paulson@3234
   462
qed "zero_induct_lemma";
paulson@3234
   463
paulson@5316
   464
val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
paulson@3234
   465
by (rtac (diff_self_eq_0 RS subst) 1);
paulson@3234
   466
by (rtac (zero_induct_lemma RS mp RS mp) 1);
paulson@3234
   467
by (REPEAT (ares_tac ([impI,allI]@prems) 1));
paulson@3234
   468
qed "zero_induct";
paulson@3234
   469
paulson@5429
   470
Goal "(k+m) - (k+n) = m - (n::nat)";
paulson@3339
   471
by (induct_tac "k" 1);
paulson@3234
   472
by (ALLGOALS Asm_simp_tac);
paulson@3234
   473
qed "diff_cancel";
paulson@3234
   474
Addsimps [diff_cancel];
paulson@3234
   475
paulson@5429
   476
Goal "(m+k) - (n+k) = m - (n::nat)";
paulson@3234
   477
val add_commute_k = read_instantiate [("n","k")] add_commute;
paulson@5537
   478
by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1);
paulson@3234
   479
qed "diff_cancel2";
paulson@3234
   480
Addsimps [diff_cancel2];
paulson@3234
   481
paulson@5414
   482
(*From Clemens Ballarin, proof by lcp*)
paulson@5429
   483
Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
paulson@5414
   484
by (REPEAT (etac rev_mp 1));
paulson@5414
   485
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@5414
   486
by (ALLGOALS Asm_simp_tac);
paulson@5414
   487
(*a confluence problem*)
paulson@5414
   488
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
paulson@3234
   489
qed "diff_right_cancel";
paulson@3234
   490
paulson@5429
   491
Goal "n - (n+m) = 0";
paulson@3339
   492
by (induct_tac "n" 1);
paulson@3234
   493
by (ALLGOALS Asm_simp_tac);
paulson@3234
   494
qed "diff_add_0";
paulson@3234
   495
Addsimps [diff_add_0];
paulson@3234
   496
paulson@5409
   497
paulson@3234
   498
(** Difference distributes over multiplication **)
paulson@3234
   499
wenzelm@5069
   500
Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
paulson@3234
   501
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3234
   502
by (ALLGOALS Asm_simp_tac);
paulson@3234
   503
qed "diff_mult_distrib" ;
paulson@3234
   504
wenzelm@5069
   505
Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
paulson@3234
   506
val mult_commute_k = read_instantiate [("m","k")] mult_commute;
wenzelm@4089
   507
by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
paulson@3234
   508
qed "diff_mult_distrib2" ;
paulson@3234
   509
(*NOT added as rewrites, since sometimes they are used from right-to-left*)
paulson@3234
   510
paulson@3234
   511
paulson@1713
   512
(*** Monotonicity of Multiplication ***)
paulson@1713
   513
paulson@5429
   514
Goal "i <= (j::nat) ==> i*k<=j*k";
paulson@3339
   515
by (induct_tac "k" 1);
wenzelm@4089
   516
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
paulson@1713
   517
qed "mult_le_mono1";
paulson@1713
   518
paulson@1713
   519
(*<=monotonicity, BOTH arguments*)
paulson@5429
   520
Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
paulson@2007
   521
by (etac (mult_le_mono1 RS le_trans) 1);
paulson@1713
   522
by (rtac le_trans 1);
paulson@2007
   523
by (stac mult_commute 2);
paulson@2007
   524
by (etac mult_le_mono1 2);
wenzelm@4089
   525
by (simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@1713
   526
qed "mult_le_mono";
paulson@1713
   527
paulson@1713
   528
(*strict, in 1st argument; proof is by induction on k>0*)
paulson@5429
   529
Goal "[| i<j; 0<k |] ==> k*i < k*j";
paulson@5078
   530
by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
paulson@1713
   531
by (Asm_simp_tac 1);
paulson@3339
   532
by (induct_tac "x" 1);
wenzelm@4089
   533
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
paulson@1713
   534
qed "mult_less_mono2";
paulson@1713
   535
paulson@5429
   536
Goal "[| i<j; 0<k |] ==> i*k < j*k";
paulson@3457
   537
by (dtac mult_less_mono2 1);
wenzelm@4089
   538
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
paulson@3234
   539
qed "mult_less_mono1";
paulson@3234
   540
wenzelm@5069
   541
Goal "(0 < m*n) = (0<m & 0<n)";
paulson@3339
   542
by (induct_tac "m" 1);
paulson@3339
   543
by (induct_tac "n" 2);
paulson@1713
   544
by (ALLGOALS Asm_simp_tac);
paulson@1713
   545
qed "zero_less_mult_iff";
nipkow@4356
   546
Addsimps [zero_less_mult_iff];
paulson@1713
   547
wenzelm@5069
   548
Goal "(m*n = 1) = (m=1 & n=1)";
paulson@3339
   549
by (induct_tac "m" 1);
paulson@1795
   550
by (Simp_tac 1);
paulson@3339
   551
by (induct_tac "n" 1);
paulson@1795
   552
by (Simp_tac 1);
wenzelm@4089
   553
by (fast_tac (claset() addss simpset()) 1);
paulson@1795
   554
qed "mult_eq_1_iff";
nipkow@4356
   555
Addsimps [mult_eq_1_iff];
paulson@1795
   556
paulson@5143
   557
Goal "0<k ==> (m*k < n*k) = (m<n)";
wenzelm@4089
   558
by (safe_tac (claset() addSIs [mult_less_mono1]));
paulson@3234
   559
by (cut_facts_tac [less_linear] 1);
paulson@4389
   560
by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
paulson@3234
   561
qed "mult_less_cancel2";
paulson@3234
   562
paulson@5143
   563
Goal "0<k ==> (k*m < k*n) = (m<n)";
paulson@3457
   564
by (dtac mult_less_cancel2 1);
wenzelm@4089
   565
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@3234
   566
qed "mult_less_cancel1";
paulson@3234
   567
Addsimps [mult_less_cancel1, mult_less_cancel2];
paulson@3234
   568
wenzelm@5069
   569
Goal "(Suc k * m < Suc k * n) = (m < n)";
wenzelm@4423
   570
by (rtac mult_less_cancel1 1);
wenzelm@4297
   571
by (Simp_tac 1);
wenzelm@4297
   572
qed "Suc_mult_less_cancel1";
wenzelm@4297
   573
wenzelm@5069
   574
Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
wenzelm@4297
   575
by (simp_tac (simpset_of HOL.thy) 1);
wenzelm@4423
   576
by (rtac Suc_mult_less_cancel1 1);
wenzelm@4297
   577
qed "Suc_mult_le_cancel1";
wenzelm@4297
   578
paulson@5143
   579
Goal "0<k ==> (m*k = n*k) = (m=n)";
paulson@3234
   580
by (cut_facts_tac [less_linear] 1);
paulson@3724
   581
by Safe_tac;
paulson@3457
   582
by (assume_tac 2);
paulson@3234
   583
by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
paulson@3234
   584
by (ALLGOALS Asm_full_simp_tac);
paulson@3234
   585
qed "mult_cancel2";
paulson@3234
   586
paulson@5143
   587
Goal "0<k ==> (k*m = k*n) = (m=n)";
paulson@3457
   588
by (dtac mult_cancel2 1);
wenzelm@4089
   589
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@3234
   590
qed "mult_cancel1";
paulson@3234
   591
Addsimps [mult_cancel1, mult_cancel2];
paulson@3234
   592
wenzelm@5069
   593
Goal "(Suc k * m = Suc k * n) = (m = n)";
wenzelm@4423
   594
by (rtac mult_cancel1 1);
wenzelm@4297
   595
by (Simp_tac 1);
wenzelm@4297
   596
qed "Suc_mult_cancel1";
wenzelm@4297
   597
paulson@3234
   598
paulson@1795
   599
(** Lemma for gcd **)
paulson@1795
   600
paulson@5143
   601
Goal "m = m*n ==> n=1 | m=0";
paulson@1795
   602
by (dtac sym 1);
paulson@1795
   603
by (rtac disjCI 1);
paulson@1795
   604
by (rtac nat_less_cases 1 THEN assume_tac 2);
wenzelm@4089
   605
by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
nipkow@4356
   606
by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
paulson@1795
   607
qed "mult_eq_self_implies_10";
paulson@1795
   608
paulson@1795
   609
nipkow@5983
   610
nipkow@5983
   611
nipkow@5983
   612
(*---------------------------------------------------------------------------*)
nipkow@5983
   613
(* Various arithmetic proof procedures                                       *)
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   614
(*---------------------------------------------------------------------------*)
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   615
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   616
(*---------------------------------------------------------------------------*)
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   617
(* 1. Cancellation of common terms                                           *)
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   618
(*---------------------------------------------------------------------------*)
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   619
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   620
(*  Title:      HOL/arith_data.ML
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   621
    ID:         $Id$
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   622
    Author:     Markus Wenzel and Stefan Berghofer, TU Muenchen
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   623
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   624
Setup various arithmetic proof procedures.
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   625
*)
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   626
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   627
signature ARITH_DATA =
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   628
sig
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   629
  val nat_cancel_sums_add: simproc list
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   630
  val nat_cancel_sums: simproc list
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   631
  val nat_cancel_factor: simproc list
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   632
  val nat_cancel: simproc list
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   633
end;
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   634
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   635
structure ArithData: ARITH_DATA =
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   636
struct
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   637
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   638
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   639
(** abstract syntax of structure nat: 0, Suc, + **)
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   640
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   641
(* mk_sum, mk_norm_sum *)
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   642
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   643
val one = HOLogic.mk_nat 1;
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   644
val mk_plus = HOLogic.mk_binop "op +";
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   645
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   646
fun mk_sum [] = HOLogic.zero
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   647
  | mk_sum [t] = t
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   648
  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
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   649
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   650
(*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
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   651
fun mk_norm_sum ts =
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   652
  let val (ones, sums) = partition (equal one) ts in
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   653
    funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
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   654
  end;
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   655
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   656
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   657
(* dest_sum *)
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   658
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   659
val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT;
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   660
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   661
fun dest_sum tm =
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   662
  if HOLogic.is_zero tm then []
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   663
  else
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   664
    (case try HOLogic.dest_Suc tm of
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   665
      Some t => one :: dest_sum t
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   666
    | None =>
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   667
        (case try dest_plus tm of
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   668
          Some (t, u) => dest_sum t @ dest_sum u
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   669
        | None => [tm]));
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   670
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   671
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   672
(** generic proof tools **)
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   673
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   674
(* prove conversions *)
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   675
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   676
val mk_eqv = HOLogic.mk_Trueprop o HOLogic.mk_eq;
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   677
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   678
fun prove_conv expand_tac norm_tac sg (t, u) =
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   679
  mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv (t, u)))
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   680
    (K [expand_tac, norm_tac]))
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   681
  handle ERROR => error ("The error(s) above occurred while trying to prove " ^
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   682
    (string_of_cterm (cterm_of sg (mk_eqv (t, u)))));
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   683
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   684
val subst_equals = prove_goal HOL.thy "[| t = s; u = t |] ==> u = s"
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   685
  (fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]);
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   686
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   687
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   688
(* rewriting *)
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   689
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   690
fun simp_all rules = ALLGOALS (simp_tac (HOL_ss addsimps rules));
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   691
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   692
val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right];
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   693
val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right];
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   694
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   695
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   696
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   697
(** cancel common summands **)
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   698
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   699
structure Sum =
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   700
struct
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   701
  val mk_sum = mk_norm_sum;
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   702
  val dest_sum = dest_sum;
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   703
  val prove_conv = prove_conv;
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   704
  val norm_tac = simp_all add_rules THEN simp_all add_ac;
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   705
end;
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   706
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   707
fun gen_uncancel_tac rule ct =
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   708
  rtac (instantiate' [] [None, Some ct] (rule RS subst_equals)) 1;
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   709
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   710
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   711
(* nat eq *)
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   712
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   713
structure EqCancelSums = CancelSumsFun
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   714
(struct
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   715
  open Sum;
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   716
  val mk_bal = HOLogic.mk_eq;
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   717
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
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   718
  val uncancel_tac = gen_uncancel_tac add_left_cancel;
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   719
end);
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   720
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   721
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   722
(* nat less *)
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   723
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   724
structure LessCancelSums = CancelSumsFun
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   725
(struct
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   726
  open Sum;
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   727
  val mk_bal = HOLogic.mk_binrel "op <";
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   728
  val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;
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   729
  val uncancel_tac = gen_uncancel_tac add_left_cancel_less;
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   730
end);
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   731
nipkow@5983
   732
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   733
(* nat le *)
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   734
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   735
structure LeCancelSums = CancelSumsFun
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   736
(struct
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   737
  open Sum;
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   738
  val mk_bal = HOLogic.mk_binrel "op <=";
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   739
  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;
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   740
  val uncancel_tac = gen_uncancel_tac add_left_cancel_le;
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   741
end);
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   742
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   743
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   744
(* nat diff *)
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   745
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   746
structure DiffCancelSums = CancelSumsFun
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   747
(struct
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   748
  open Sum;
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   749
  val mk_bal = HOLogic.mk_binop "op -";
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   750
  val dest_bal = HOLogic.dest_bin "op -" HOLogic.natT;
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   751
  val uncancel_tac = gen_uncancel_tac diff_cancel;
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   752
end);
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   753
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   754
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   755
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   756
(** cancel common factor **)
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   757
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   758
structure Factor =
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   759
struct
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   760
  val mk_sum = mk_norm_sum;
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   761
  val dest_sum = dest_sum;
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   762
  val prove_conv = prove_conv;
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   763
  val norm_tac = simp_all (add_rules @ mult_rules) THEN simp_all add_ac;
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   764
end;
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   765
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   766
fun mk_cnat n = cterm_of (sign_of Nat.thy) (HOLogic.mk_nat n);
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   767
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   768
fun gen_multiply_tac rule k =
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   769
  if k > 0 then
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   770
    rtac (instantiate' [] [None, Some (mk_cnat (k - 1))] (rule RS subst_equals)) 1
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   771
  else no_tac;
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   772
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   773
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   774
(* nat eq *)
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   775
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   776
structure EqCancelFactor = CancelFactorFun
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   777
(struct
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   778
  open Factor;
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   779
  val mk_bal = HOLogic.mk_eq;
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   780
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
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   781
  val multiply_tac = gen_multiply_tac Suc_mult_cancel1;
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   782
end);
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   783
nipkow@5983
   784
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   785
(* nat less *)
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   786
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   787
structure LessCancelFactor = CancelFactorFun
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   788
(struct
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   789
  open Factor;
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   790
  val mk_bal = HOLogic.mk_binrel "op <";
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   791
  val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;
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   792
  val multiply_tac = gen_multiply_tac Suc_mult_less_cancel1;
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   793
end);
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   794
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   795
nipkow@5983
   796
(* nat le *)
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   797
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   798
structure LeCancelFactor = CancelFactorFun
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   799
(struct
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   800
  open Factor;
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   801
  val mk_bal = HOLogic.mk_binrel "op <=";
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   802
  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;
nipkow@5983
   803
  val multiply_tac = gen_multiply_tac Suc_mult_le_cancel1;
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   804
end);
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   805
nipkow@5983
   806
nipkow@5983
   807
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   808
(** prepare nat_cancel simprocs **)
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   809
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   810
fun prep_pat s = Thm.read_cterm (sign_of Arith.thy) (s, HOLogic.termTVar);
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   811
val prep_pats = map prep_pat;
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   812
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   813
fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
nipkow@5983
   814
nipkow@5983
   815
val eq_pats = prep_pats ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"];
nipkow@5983
   816
val less_pats = prep_pats ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"];
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   817
val le_pats = prep_pats ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"];
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   818
val diff_pats = prep_pats ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"];
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   819
nipkow@6055
   820
val nat_cancel_sums_add = map prep_simproc
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   821
  [("nateq_cancel_sums", eq_pats, EqCancelSums.proc),
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   822
   ("natless_cancel_sums", less_pats, LessCancelSums.proc),
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   823
   ("natle_cancel_sums", le_pats, LeCancelSums.proc)];
nipkow@6055
   824
nipkow@6055
   825
val nat_cancel_sums = nat_cancel_sums_add @
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   826
  [prep_simproc("natdiff_cancel_sums", diff_pats, DiffCancelSums.proc)];
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   827
nipkow@5983
   828
val nat_cancel_factor = map prep_simproc
nipkow@5983
   829
  [("nateq_cancel_factor", eq_pats, EqCancelFactor.proc),
nipkow@5983
   830
   ("natless_cancel_factor", less_pats, LessCancelFactor.proc),
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   831
   ("natle_cancel_factor", le_pats, LeCancelFactor.proc)];
nipkow@5983
   832
nipkow@5983
   833
val nat_cancel = nat_cancel_factor @ nat_cancel_sums;
nipkow@5983
   834
nipkow@5983
   835
nipkow@5983
   836
end;
nipkow@5983
   837
nipkow@5983
   838
open ArithData;
nipkow@5983
   839
nipkow@5983
   840
Addsimprocs nat_cancel;
nipkow@5983
   841
nipkow@5983
   842
(*---------------------------------------------------------------------------*)
nipkow@5983
   843
(* 2. Linear arithmetic                                                      *)
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   844
(*---------------------------------------------------------------------------*)
nipkow@5983
   845
nipkow@5983
   846
(* Parameter data for general linear arithmetic functor *)
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   847
structure Nat_LA_Data: LIN_ARITH_DATA =
nipkow@5983
   848
struct
nipkow@5983
   849
val ccontr = ccontr;
nipkow@5983
   850
val conjI = conjI;
nipkow@5983
   851
val lessD = Suc_leI;
nipkow@5983
   852
val nat_neqE = nat_neqE;
nipkow@5983
   853
val notI = notI;
nipkow@5983
   854
val not_leD = not_leE RS Suc_leI;
nipkow@5983
   855
val not_lessD = leI;
nipkow@5983
   856
val sym = sym;
nipkow@5983
   857
nipkow@5983
   858
val nat = Type("nat",[]);
nipkow@6055
   859
val boolT = Type("bool",[]);
nipkow@5983
   860
nipkow@6055
   861
fun nnb T = T = ([nat,nat] ---> boolT);
nipkow@5983
   862
nipkow@5983
   863
(* Turn term into list of summand * multiplicity plus a constant *)
nipkow@5983
   864
fun poly(Const("Suc",_)$t, (p,i)) = poly(t, (p,i+1))
nipkow@5983
   865
  | poly(Const("op +",Type("fun",[Type("nat",[]),_])) $ s $ t, pi) =
nipkow@5983
   866
      poly(s,poly(t,pi))
nipkow@5983
   867
  | poly(t,(p,i)) =
nipkow@5983
   868
      if t = Const("0",nat) then (p,i)
nipkow@5983
   869
      else (case assoc(p,t) of None => ((t,1)::p,i)
nipkow@5983
   870
            | Some m => (overwrite(p,(t,m+1)), i))
nipkow@5983
   871
nipkow@5983
   872
fun decomp2(rel,T,lhs,rhs) =
nipkow@5983
   873
  if not(nnb T) then None else
nipkow@5983
   874
  let val (p,i) = poly(lhs,([],0)) and (q,j) = poly(rhs,([],0))
nipkow@5983
   875
  in case rel of
nipkow@5983
   876
       "op <"  => Some(p,i,"<",q,j)
nipkow@5983
   877
     | "op <=" => Some(p,i,"<=",q,j)
nipkow@5983
   878
     | "op ="  => Some(p,i,"=",q,j)
nipkow@5983
   879
     | _       => None
nipkow@5983
   880
  end;
nipkow@5983
   881
nipkow@5983
   882
fun negate(Some(x,i,rel,y,j)) = Some(x,i,"~"^rel,y,j)
nipkow@5983
   883
  | negate None = None;
nipkow@5983
   884
nipkow@5983
   885
fun decomp(_$(Const(rel,T)$lhs$rhs)) = decomp2(rel,T,lhs,rhs)
nipkow@5983
   886
  | decomp(_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
nipkow@5983
   887
      negate(decomp2(rel,T,lhs,rhs))
nipkow@5983
   888
  | decomp _ = None
nipkow@6055
   889
nipkow@5983
   890
(* reduce contradictory <= to False.
nipkow@5983
   891
   Most of the work is done by the cancel tactics.
nipkow@5983
   892
*)
nipkow@5983
   893
val add_rules = [Zero_not_Suc,Suc_not_Zero,le_0_eq];
nipkow@5983
   894
nipkow@5983
   895
val cancel_sums_ss = HOL_basic_ss addsimps add_rules
nipkow@6055
   896
                                  addsimprocs nat_cancel_sums_add;
nipkow@5983
   897
nipkow@5983
   898
val simp = simplify cancel_sums_ss;
nipkow@5983
   899
nipkow@5983
   900
val add_mono_thms = map (fn s => prove_goal Arith.thy s
nipkow@5983
   901
 (fn prems => [cut_facts_tac prems 1,
nipkow@5983
   902
               blast_tac (claset() addIs [add_le_mono]) 1]))
nipkow@5983
   903
["(i <= j) & (k <= l) ==> i + k <= j + (l::nat)",
nipkow@6055
   904
 "(i  = j) & (k <= l) ==> i + k <= j + (l::nat)",
nipkow@6055
   905
 "(i <= j) & (k  = l) ==> i + k <= j + (l::nat)",
nipkow@6055
   906
 "(i  = j) & (k  = l) ==> i + k  = j + (l::nat)"
nipkow@5983
   907
];
nipkow@5983
   908
nipkow@6055
   909
fun is_False thm =
nipkow@6055
   910
  let val _ $ t = #prop(rep_thm thm)
nipkow@6055
   911
  in t = Const("False",boolT) end;
nipkow@6055
   912
nipkow@6055
   913
fun is_nat(t) = fastype_of1 t = nat;
nipkow@6055
   914
nipkow@6055
   915
fun mk_nat_thm sg t =
nipkow@6055
   916
  let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),nat))
nipkow@6055
   917
  in instantiate ([],[(cn,ct)]) le0 end;
nipkow@6055
   918
nipkow@5983
   919
end;
nipkow@5983
   920
nipkow@5983
   921
structure Fast_Nat_Arith = Fast_Lin_Arith(Nat_LA_Data);
nipkow@5983
   922
nipkow@5983
   923
simpset_ref () := (simpset() addSolver Fast_Nat_Arith.cut_lin_arith_tac);
nipkow@5983
   924
nipkow@6055
   925
val fast_nat_arith_tac = Fast_Nat_Arith.lin_arith_tac;
nipkow@6055
   926
nipkow@6055
   927
(* Elimination of `-' on nat due to John Harrison *)
nipkow@6055
   928
Goal "P(a - b::nat) = (!d. (b = a + d --> P 0) & (a = b + d --> P d))";
nipkow@6055
   929
by(case_tac "a <= b" 1);
nipkow@6055
   930
by(Auto_tac);
nipkow@6055
   931
by(eres_inst_tac [("x","b-a")] allE 1);
nipkow@6055
   932
by(Auto_tac);
nipkow@6055
   933
qed "nat_diff_split";
nipkow@6055
   934
nipkow@6055
   935
(* FIXME: K true should be replaced by a sensible test to speed things up
nipkow@6055
   936
   in case there are lots of irrelevant terms involved
nipkow@6055
   937
*)
nipkow@6055
   938
val nat_arith_tac =
nipkow@6055
   939
  refute_tac (K true) (REPEAT o split_tac[nat_diff_split])
nipkow@6055
   940
             ((REPEAT o etac nat_neqE) THEN' fast_nat_arith_tac);
nipkow@6055
   941
nipkow@5983
   942
(*---------------------------------------------------------------------------*)
nipkow@5983
   943
(* End of proof procedures. Now go and USE them!                             *)
nipkow@5983
   944
(*---------------------------------------------------------------------------*)
nipkow@5983
   945
paulson@4736
   946
(*** Subtraction laws -- mostly from Clemens Ballarin ***)
paulson@3234
   947
paulson@5429
   948
Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
nipkow@6055
   949
by(nat_arith_tac 1);
paulson@3234
   950
qed "diff_less_mono";
paulson@3234
   951
paulson@5429
   952
Goal "a+b < (c::nat) ==> a < c-b";
nipkow@6055
   953
by(nat_arith_tac 1);
paulson@3234
   954
qed "add_less_imp_less_diff";
paulson@3234
   955
nipkow@5427
   956
Goal "(i < j-k) = (i+k < (j::nat))";
nipkow@6055
   957
by(nat_arith_tac 1);
nipkow@5427
   958
qed "less_diff_conv";
nipkow@5427
   959
paulson@5497
   960
Goal "(j-k <= (i::nat)) = (j <= i+k)";
nipkow@6055
   961
by(nat_arith_tac 1);
paulson@5485
   962
qed "le_diff_conv";
paulson@5485
   963
paulson@5497
   964
Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))";
nipkow@6055
   965
by(nat_arith_tac 1);
paulson@5497
   966
qed "le_diff_conv2";
paulson@5497
   967
paulson@5143
   968
Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
nipkow@6055
   969
by(nat_arith_tac 1);
paulson@3234
   970
qed "Suc_diff_Suc";
paulson@3234
   971
paulson@5429
   972
Goal "i <= (n::nat) ==> n - (n - i) = i";
nipkow@6055
   973
by(nat_arith_tac 1);
paulson@3234
   974
qed "diff_diff_cancel";
paulson@3381
   975
Addsimps [diff_diff_cancel];
paulson@3234
   976
paulson@5429
   977
Goal "k <= (n::nat) ==> m <= n + m - k";
nipkow@6055
   978
by(nat_arith_tac 1);
paulson@3234
   979
qed "le_add_diff";
paulson@3234
   980
nipkow@6055
   981
Goal "[| 0<k; j<i |] ==> j+k-i < k";
nipkow@6055
   982
by(nat_arith_tac 1);
nipkow@6055
   983
qed "add_diff_less";
paulson@3234
   984
paulson@5356
   985
Goal "m-1 < n ==> m <= n";
nipkow@6055
   986
by(nat_arith_tac 1);
paulson@5356
   987
qed "pred_less_imp_le";
paulson@5356
   988
paulson@5356
   989
Goal "j<=i ==> i - j < Suc i - j";
nipkow@6055
   990
by(nat_arith_tac 1);
paulson@5356
   991
qed "diff_less_Suc_diff";
paulson@5356
   992
paulson@5356
   993
Goal "i - j <= Suc i - j";
nipkow@6055
   994
by(nat_arith_tac 1);
paulson@5356
   995
qed "diff_le_Suc_diff";
paulson@5356
   996
AddIffs [diff_le_Suc_diff];
paulson@5356
   997
paulson@5356
   998
Goal "n - Suc i <= n - i";
nipkow@6055
   999
by(nat_arith_tac 1);
paulson@5356
  1000
qed "diff_Suc_le_diff";
paulson@5356
  1001
AddIffs [diff_Suc_le_diff];
paulson@5356
  1002
paulson@5409
  1003
Goal "0 < n ==> (m <= n-1) = (m<n)";
nipkow@6055
  1004
by(nat_arith_tac 1);
paulson@5409
  1005
qed "le_pred_eq";
paulson@5409
  1006
paulson@5409
  1007
Goal "0 < n ==> (m-1 < n) = (m<=n)";
nipkow@6055
  1008
by(nat_arith_tac 1);
paulson@5409
  1009
qed "less_pred_eq";
paulson@5409
  1010
paulson@5414
  1011
(*In ordinary notation: if 0<n and n<=m then m-n < m *)
paulson@5414
  1012
Goal "[| 0<n; ~ m<n |] ==> m - n < m";
nipkow@6055
  1013
by(nat_arith_tac 1);
paulson@5414
  1014
qed "diff_less";
paulson@5414
  1015
paulson@5414
  1016
Goal "[| 0<n; n<=m |] ==> m - n < m";
nipkow@6055
  1017
by(nat_arith_tac 1);
paulson@5414
  1018
qed "le_diff_less";
paulson@5414
  1019
paulson@5356
  1020
paulson@4732
  1021
nipkow@3484
  1022
(** (Anti)Monotonicity of subtraction -- by Stefan Merz **)
nipkow@3484
  1023
nipkow@3484
  1024
(* Monotonicity of subtraction in first argument *)
nipkow@6055
  1025
Goal "m <= (n::nat) ==> (m-l) <= (n-l)";
nipkow@6055
  1026
by(nat_arith_tac 1);
nipkow@6055
  1027
qed "diff_le_mono";
nipkow@3484
  1028
paulson@5429
  1029
Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
nipkow@6055
  1030
by(nat_arith_tac 1);
nipkow@6055
  1031
qed "diff_le_mono2";
nipkow@5983
  1032
nipkow@5983
  1033
nipkow@5983
  1034
(*This proof requires natdiff_cancel_sums*)
nipkow@6055
  1035
Goal "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)";
nipkow@6055
  1036
by(nat_arith_tac 1);
nipkow@6055
  1037
qed "diff_less_mono2";
nipkow@5983
  1038
nipkow@6055
  1039
Goal "[| m-n = 0; n-m = 0 |] ==>  m=n";
nipkow@6055
  1040
by(nat_arith_tac 1);
nipkow@6055
  1041
qed "diffs0_imp_equal";