src/HOL/Arith.ML
author paulson
Thu Sep 23 13:06:31 1999 +0200 (1999-09-23)
changeset 7584 5be4bb8e4e3f
parent 7582 2650c9c2ab7f
child 7622 dcb93b295683
permissions -rw-r--r--
tidied; added lemma restrict_to_left
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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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Goal "0 - n = 0";
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by (induct_tac "n" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_0_eq_0";
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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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Goal "Suc(m) - Suc(n) = m - n";
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by (Simp_tac 1);
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by (induct_tac "n" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "diff_Suc_Suc";
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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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Goal "m + 0 = m";
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by (induct_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_0_right";
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Goal "m + Suc(n) = Suc(m+n)";
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by (induct_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_Suc_right";
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Addsimps [add_0_right,add_Suc_right];
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(*Associative law for addition*)
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Goal "(m + n) + k = m + ((n + k)::nat)";
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by (induct_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_assoc";
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(*Commutative law for addition*)  
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Goal "m + n = n + (m::nat)";
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by (induct_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "add_commute";
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Goal "x+(y+z)=y+((x+z)::nat)";
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by (rtac (add_commute RS trans) 1);
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by (rtac (add_assoc RS trans) 1);
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by (rtac (add_commute RS arg_cong) 1);
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qed "add_left_commute";
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(*Addition is an AC-operator*)
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bind_thms ("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, Suc_n_not_n]
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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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Goal "m * 0 = 0";
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by (induct_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "mult_0_right";
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(*right successor law for multiplication*)
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Goal  "m * Suc(n) = m + (m * n)";
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by (induct_tac "m" 1);
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by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
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qed "mult_Suc_right";
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paulson@3293
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Addsimps [mult_0_right, mult_Suc_right];
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wenzelm@5069
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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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Goal "m * n = n * (m::nat)";
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by (induct_tac "m" 1);
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by (ALLGOALS Asm_simp_tac);
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qed "mult_commute";
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(*addition distributes over multiplication*)
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Goal "(m + n)*k = (m*k) + ((n*k)::nat)";
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by (induct_tac "m" 1);
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by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
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qed "add_mult_distrib";
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Goal "k*(m + n) = (k*m) + ((k*n)::nat)";
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by (induct_tac "m" 1);
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by (ALLGOALS(asm_simp_tac (simpset() addsimps add_ac)));
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qed "add_mult_distrib2";
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(*Associative law for multiplication*)
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Goal "(m * n) * k = m * ((n * k)::nat)";
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by (induct_tac "m" 1);
paulson@7007
   321
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_mult_distrib])));
paulson@7007
   322
qed "mult_assoc";
paulson@3234
   323
paulson@7007
   324
Goal "x*(y*z) = y*((x*z)::nat)";
paulson@7007
   325
by (rtac trans 1);
paulson@7007
   326
by (rtac mult_commute 1);
paulson@7007
   327
by (rtac trans 1);
paulson@7007
   328
by (rtac mult_assoc 1);
paulson@7007
   329
by (rtac (mult_commute RS arg_cong) 1);
paulson@7007
   330
qed "mult_left_commute";
paulson@3234
   331
wenzelm@7428
   332
bind_thms ("mult_ac", [mult_assoc,mult_commute,mult_left_commute]);
paulson@3234
   333
wenzelm@5069
   334
Goal "(m*n = 0) = (m=0 | n=0)";
paulson@3339
   335
by (induct_tac "m" 1);
paulson@3339
   336
by (induct_tac "n" 2);
paulson@3293
   337
by (ALLGOALS Asm_simp_tac);
paulson@3293
   338
qed "mult_is_0";
paulson@3293
   339
Addsimps [mult_is_0];
paulson@3293
   340
paulson@5429
   341
Goal "m <= m*(m::nat)";
paulson@4158
   342
by (induct_tac "m" 1);
paulson@4158
   343
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_assoc RS sym])));
paulson@4158
   344
by (etac (le_add2 RSN (2,le_trans)) 1);
paulson@4158
   345
qed "le_square";
paulson@4158
   346
paulson@3234
   347
paulson@3234
   348
(*** Difference ***)
paulson@3234
   349
paulson@7007
   350
Goal "m - m = 0";
paulson@7007
   351
by (induct_tac "m" 1);
paulson@7007
   352
by (ALLGOALS Asm_simp_tac);
paulson@7007
   353
qed "diff_self_eq_0";
paulson@3234
   354
paulson@3234
   355
Addsimps [diff_self_eq_0];
paulson@3234
   356
paulson@3234
   357
(*Addition is the inverse of subtraction: if n<=m then n+(m-n) = m. *)
wenzelm@5069
   358
Goal "~ m<n --> n+(m-n) = (m::nat)";
paulson@3234
   359
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   360
by (ALLGOALS Asm_simp_tac);
paulson@3381
   361
qed_spec_mp "add_diff_inverse";
paulson@3381
   362
paulson@5143
   363
Goal "n<=m ==> n+(m-n) = (m::nat)";
wenzelm@4089
   364
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, not_less_iff_le]) 1);
paulson@3381
   365
qed "le_add_diff_inverse";
paulson@3234
   366
paulson@5143
   367
Goal "n<=m ==> (m-n)+n = (m::nat)";
wenzelm@4089
   368
by (asm_simp_tac (simpset() addsimps [le_add_diff_inverse, add_commute]) 1);
paulson@3381
   369
qed "le_add_diff_inverse2";
paulson@3381
   370
paulson@3381
   371
Addsimps  [le_add_diff_inverse, le_add_diff_inverse2];
paulson@3234
   372
paulson@3234
   373
paulson@3234
   374
(*** More results about difference ***)
paulson@3234
   375
paulson@5414
   376
Goal "n <= m ==> Suc(m)-n = Suc(m-n)";
paulson@5316
   377
by (etac rev_mp 1);
paulson@3352
   378
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   379
by (ALLGOALS Asm_simp_tac);
paulson@5414
   380
qed "Suc_diff_le";
paulson@3352
   381
wenzelm@5069
   382
Goal "m - n < Suc(m)";
paulson@3234
   383
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3234
   384
by (etac less_SucE 3);
wenzelm@4089
   385
by (ALLGOALS (asm_simp_tac (simpset() addsimps [less_Suc_eq])));
paulson@3234
   386
qed "diff_less_Suc";
paulson@3234
   387
paulson@5429
   388
Goal "m - n <= (m::nat)";
paulson@3234
   389
by (res_inst_tac [("m","m"), ("n","n")] diff_induct 1);
nipkow@6075
   390
by (ALLGOALS (asm_simp_tac (simpset() addsimps [le_SucI])));
paulson@3234
   391
qed "diff_le_self";
paulson@3903
   392
Addsimps [diff_le_self];
paulson@3234
   393
paulson@4732
   394
(* j<k ==> j-n < k *)
paulson@4732
   395
bind_thm ("less_imp_diff_less", diff_le_self RS le_less_trans);
paulson@4732
   396
wenzelm@5069
   397
Goal "!!i::nat. i-j-k = i - (j+k)";
paulson@3352
   398
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@3352
   399
by (ALLGOALS Asm_simp_tac);
paulson@3352
   400
qed "diff_diff_left";
paulson@3352
   401
wenzelm@5069
   402
Goal "(Suc m - n) - Suc k = m - n - k";
wenzelm@4423
   403
by (simp_tac (simpset() addsimps [diff_diff_left]) 1);
paulson@4736
   404
qed "Suc_diff_diff";
paulson@4736
   405
Addsimps [Suc_diff_diff];
nipkow@4360
   406
paulson@5143
   407
Goal "0<n ==> n - Suc i < n";
berghofe@5183
   408
by (exhaust_tac "n" 1);
paulson@4732
   409
by Safe_tac;
paulson@5497
   410
by (asm_simp_tac (simpset() addsimps le_simps) 1);
paulson@4732
   411
qed "diff_Suc_less";
paulson@4732
   412
Addsimps [diff_Suc_less];
paulson@4732
   413
wenzelm@3396
   414
(*This and the next few suggested by Florian Kammueller*)
wenzelm@5069
   415
Goal "!!i::nat. i-j-k = i-k-j";
wenzelm@4089
   416
by (simp_tac (simpset() addsimps [diff_diff_left, add_commute]) 1);
paulson@3352
   417
qed "diff_commute";
paulson@3352
   418
paulson@5429
   419
Goal "k<=j --> j<=i --> i - (j - k) = i - j + (k::nat)";
paulson@3352
   420
by (res_inst_tac [("m","i"),("n","j")] diff_induct 1);
paulson@3352
   421
by (ALLGOALS Asm_simp_tac);
paulson@5414
   422
by (asm_simp_tac (simpset() addsimps [Suc_diff_le, le_Suc_eq]) 1);
paulson@3352
   423
qed_spec_mp "diff_diff_right";
paulson@3352
   424
paulson@5429
   425
Goal "k <= (j::nat) --> (i + j) - k = i + (j - k)";
paulson@3352
   426
by (res_inst_tac [("m","j"),("n","k")] diff_induct 1);
paulson@3352
   427
by (ALLGOALS Asm_simp_tac);
paulson@3352
   428
qed_spec_mp "diff_add_assoc";
paulson@3352
   429
paulson@5429
   430
Goal "k <= (j::nat) --> (j + i) - k = i + (j - k)";
paulson@4732
   431
by (asm_simp_tac (simpset() addsimps [add_commute, diff_add_assoc]) 1);
paulson@4732
   432
qed_spec_mp "diff_add_assoc2";
paulson@4732
   433
paulson@5429
   434
Goal "(n+m) - n = (m::nat)";
paulson@3339
   435
by (induct_tac "n" 1);
paulson@3234
   436
by (ALLGOALS Asm_simp_tac);
paulson@3234
   437
qed "diff_add_inverse";
paulson@3234
   438
Addsimps [diff_add_inverse];
paulson@3234
   439
paulson@5429
   440
Goal "(m+n) - n = (m::nat)";
wenzelm@4089
   441
by (simp_tac (simpset() addsimps [diff_add_assoc]) 1);
paulson@3234
   442
qed "diff_add_inverse2";
paulson@3234
   443
Addsimps [diff_add_inverse2];
paulson@3234
   444
paulson@5429
   445
Goal "i <= (j::nat) ==> (j-i=k) = (j=k+i)";
paulson@3724
   446
by Safe_tac;
paulson@3381
   447
by (ALLGOALS Asm_simp_tac);
paulson@3366
   448
qed "le_imp_diff_is_add";
paulson@3366
   449
paulson@5356
   450
Goal "(m-n = 0) = (m <= n)";
paulson@3234
   451
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@5497
   452
by (ALLGOALS Asm_simp_tac);
paulson@5356
   453
qed "diff_is_0_eq";
paulson@7059
   454
paulson@7059
   455
Goal "(0 = m-n) = (m <= n)";
paulson@7059
   456
by (stac (diff_is_0_eq RS sym) 1);
paulson@7059
   457
by (rtac eq_sym_conv 1);
paulson@7059
   458
qed "zero_is_diff_eq";
paulson@7059
   459
Addsimps [diff_is_0_eq, zero_is_diff_eq];
paulson@3234
   460
paulson@5333
   461
Goal "(0<n-m) = (m<n)";
paulson@3234
   462
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3352
   463
by (ALLGOALS Asm_simp_tac);
paulson@5333
   464
qed "zero_less_diff";
paulson@5333
   465
Addsimps [zero_less_diff];
paulson@3234
   466
paulson@5333
   467
Goal "i < j  ==> ? k. 0<k & i+k = j";
paulson@5078
   468
by (res_inst_tac [("x","j - i")] exI 1);
paulson@5333
   469
by (asm_simp_tac (simpset() addsimps [add_diff_inverse, less_not_sym]) 1);
paulson@5078
   470
qed "less_imp_add_positive";
paulson@5078
   471
wenzelm@5069
   472
Goal "P(k) --> (!n. P(Suc(n))--> P(n)) --> P(k-i)";
paulson@3234
   473
by (res_inst_tac [("m","k"),("n","i")] diff_induct 1);
paulson@3718
   474
by (ALLGOALS (Clarify_tac THEN' Simp_tac THEN' TRY o Blast_tac));
paulson@3234
   475
qed "zero_induct_lemma";
paulson@3234
   476
paulson@5316
   477
val prems = Goal "[| P(k);  !!n. P(Suc(n)) ==> P(n) |] ==> P(0)";
paulson@3234
   478
by (rtac (diff_self_eq_0 RS subst) 1);
paulson@3234
   479
by (rtac (zero_induct_lemma RS mp RS mp) 1);
paulson@3234
   480
by (REPEAT (ares_tac ([impI,allI]@prems) 1));
paulson@3234
   481
qed "zero_induct";
paulson@3234
   482
paulson@5429
   483
Goal "(k+m) - (k+n) = m - (n::nat)";
paulson@3339
   484
by (induct_tac "k" 1);
paulson@3234
   485
by (ALLGOALS Asm_simp_tac);
paulson@3234
   486
qed "diff_cancel";
paulson@3234
   487
Addsimps [diff_cancel];
paulson@3234
   488
paulson@5429
   489
Goal "(m+k) - (n+k) = m - (n::nat)";
paulson@3234
   490
val add_commute_k = read_instantiate [("n","k")] add_commute;
paulson@5537
   491
by (asm_simp_tac (simpset() addsimps [add_commute_k]) 1);
paulson@3234
   492
qed "diff_cancel2";
paulson@3234
   493
Addsimps [diff_cancel2];
paulson@3234
   494
paulson@5429
   495
Goal "n - (n+m) = 0";
paulson@3339
   496
by (induct_tac "n" 1);
paulson@3234
   497
by (ALLGOALS Asm_simp_tac);
paulson@3234
   498
qed "diff_add_0";
paulson@3234
   499
Addsimps [diff_add_0];
paulson@3234
   500
paulson@5409
   501
paulson@3234
   502
(** Difference distributes over multiplication **)
paulson@3234
   503
wenzelm@5069
   504
Goal "!!m::nat. (m - n) * k = (m * k) - (n * k)";
paulson@3234
   505
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
paulson@3234
   506
by (ALLGOALS Asm_simp_tac);
paulson@3234
   507
qed "diff_mult_distrib" ;
paulson@3234
   508
wenzelm@5069
   509
Goal "!!m::nat. k * (m - n) = (k * m) - (k * n)";
paulson@3234
   510
val mult_commute_k = read_instantiate [("m","k")] mult_commute;
wenzelm@4089
   511
by (simp_tac (simpset() addsimps [diff_mult_distrib, mult_commute_k]) 1);
paulson@3234
   512
qed "diff_mult_distrib2" ;
paulson@3234
   513
(*NOT added as rewrites, since sometimes they are used from right-to-left*)
paulson@3234
   514
paulson@3234
   515
paulson@1713
   516
(*** Monotonicity of Multiplication ***)
paulson@1713
   517
paulson@5429
   518
Goal "i <= (j::nat) ==> i*k<=j*k";
paulson@3339
   519
by (induct_tac "k" 1);
wenzelm@4089
   520
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_le_mono])));
paulson@1713
   521
qed "mult_le_mono1";
paulson@1713
   522
paulson@6987
   523
Goal "i <= (j::nat) ==> k*i <= k*j";
paulson@6987
   524
by (dtac mult_le_mono1 1);
paulson@6987
   525
by (asm_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@6987
   526
qed "mult_le_mono2";
paulson@6987
   527
paulson@6987
   528
(* <= monotonicity, BOTH arguments*)
paulson@5429
   529
Goal "[| i <= (j::nat); k <= l |] ==> i*k <= j*l";
paulson@2007
   530
by (etac (mult_le_mono1 RS le_trans) 1);
paulson@6987
   531
by (etac mult_le_mono2 1);
paulson@1713
   532
qed "mult_le_mono";
paulson@1713
   533
paulson@1713
   534
(*strict, in 1st argument; proof is by induction on k>0*)
paulson@5429
   535
Goal "[| i<j; 0<k |] ==> k*i < k*j";
paulson@5078
   536
by (eres_inst_tac [("m1","0")] (less_eq_Suc_add RS exE) 1);
paulson@1713
   537
by (Asm_simp_tac 1);
paulson@3339
   538
by (induct_tac "x" 1);
wenzelm@4089
   539
by (ALLGOALS (asm_simp_tac (simpset() addsimps [add_less_mono])));
paulson@1713
   540
qed "mult_less_mono2";
paulson@1713
   541
paulson@5429
   542
Goal "[| i<j; 0<k |] ==> i*k < j*k";
paulson@3457
   543
by (dtac mult_less_mono2 1);
wenzelm@4089
   544
by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [mult_commute])));
paulson@3234
   545
qed "mult_less_mono1";
paulson@3234
   546
wenzelm@5069
   547
Goal "(0 < m*n) = (0<m & 0<n)";
paulson@3339
   548
by (induct_tac "m" 1);
paulson@3339
   549
by (induct_tac "n" 2);
paulson@1713
   550
by (ALLGOALS Asm_simp_tac);
paulson@1713
   551
qed "zero_less_mult_iff";
nipkow@4356
   552
Addsimps [zero_less_mult_iff];
paulson@1713
   553
wenzelm@5069
   554
Goal "(m*n = 1) = (m=1 & n=1)";
paulson@3339
   555
by (induct_tac "m" 1);
paulson@1795
   556
by (Simp_tac 1);
paulson@3339
   557
by (induct_tac "n" 1);
paulson@1795
   558
by (Simp_tac 1);
wenzelm@4089
   559
by (fast_tac (claset() addss simpset()) 1);
paulson@1795
   560
qed "mult_eq_1_iff";
nipkow@4356
   561
Addsimps [mult_eq_1_iff];
paulson@1795
   562
paulson@5143
   563
Goal "0<k ==> (m*k < n*k) = (m<n)";
wenzelm@4089
   564
by (safe_tac (claset() addSIs [mult_less_mono1]));
paulson@3234
   565
by (cut_facts_tac [less_linear] 1);
paulson@4389
   566
by (blast_tac (claset() addIs [mult_less_mono1] addEs [less_asym]) 1);
paulson@3234
   567
qed "mult_less_cancel2";
paulson@3234
   568
paulson@5143
   569
Goal "0<k ==> (k*m < k*n) = (m<n)";
paulson@3457
   570
by (dtac mult_less_cancel2 1);
wenzelm@4089
   571
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@3234
   572
qed "mult_less_cancel1";
paulson@3234
   573
Addsimps [mult_less_cancel1, mult_less_cancel2];
paulson@3234
   574
paulson@6864
   575
Goal "0<k ==> (m*k <= n*k) = (m<=n)";
paulson@6864
   576
by (asm_full_simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
paulson@6864
   577
qed "mult_le_cancel2";
paulson@6864
   578
paulson@6864
   579
Goal "0<k ==> (k*m <= k*n) = (m<=n)";
paulson@6864
   580
by (asm_full_simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
paulson@6864
   581
qed "mult_le_cancel1";
paulson@6864
   582
Addsimps [mult_le_cancel1, mult_le_cancel2];
paulson@6864
   583
wenzelm@5069
   584
Goal "(Suc k * m < Suc k * n) = (m < n)";
wenzelm@4423
   585
by (rtac mult_less_cancel1 1);
wenzelm@4297
   586
by (Simp_tac 1);
wenzelm@4297
   587
qed "Suc_mult_less_cancel1";
wenzelm@4297
   588
wenzelm@5069
   589
Goalw [le_def] "(Suc k * m <= Suc k * n) = (m <= n)";
wenzelm@4297
   590
by (simp_tac (simpset_of HOL.thy) 1);
wenzelm@4423
   591
by (rtac Suc_mult_less_cancel1 1);
wenzelm@4297
   592
qed "Suc_mult_le_cancel1";
wenzelm@4297
   593
paulson@5143
   594
Goal "0<k ==> (m*k = n*k) = (m=n)";
paulson@3234
   595
by (cut_facts_tac [less_linear] 1);
paulson@3724
   596
by Safe_tac;
paulson@3457
   597
by (assume_tac 2);
paulson@3234
   598
by (ALLGOALS (dtac mult_less_mono1 THEN' assume_tac));
paulson@3234
   599
by (ALLGOALS Asm_full_simp_tac);
paulson@3234
   600
qed "mult_cancel2";
paulson@3234
   601
paulson@5143
   602
Goal "0<k ==> (k*m = k*n) = (m=n)";
paulson@3457
   603
by (dtac mult_cancel2 1);
wenzelm@4089
   604
by (asm_full_simp_tac (simpset() addsimps [mult_commute]) 1);
paulson@3234
   605
qed "mult_cancel1";
paulson@3234
   606
Addsimps [mult_cancel1, mult_cancel2];
paulson@3234
   607
wenzelm@5069
   608
Goal "(Suc k * m = Suc k * n) = (m = n)";
wenzelm@4423
   609
by (rtac mult_cancel1 1);
wenzelm@4297
   610
by (Simp_tac 1);
wenzelm@4297
   611
qed "Suc_mult_cancel1";
wenzelm@4297
   612
paulson@3234
   613
paulson@1795
   614
(** Lemma for gcd **)
paulson@1795
   615
paulson@5143
   616
Goal "m = m*n ==> n=1 | m=0";
paulson@1795
   617
by (dtac sym 1);
paulson@1795
   618
by (rtac disjCI 1);
paulson@1795
   619
by (rtac nat_less_cases 1 THEN assume_tac 2);
wenzelm@4089
   620
by (fast_tac (claset() addSEs [less_SucE] addss simpset()) 1);
nipkow@4356
   621
by (best_tac (claset() addDs [mult_less_mono2] addss simpset()) 1);
paulson@1795
   622
qed "mult_eq_self_implies_10";
paulson@1795
   623
paulson@1795
   624
nipkow@5983
   625
nipkow@5983
   626
nipkow@5983
   627
(*---------------------------------------------------------------------------*)
nipkow@5983
   628
(* Various arithmetic proof procedures                                       *)
nipkow@5983
   629
(*---------------------------------------------------------------------------*)
nipkow@5983
   630
nipkow@5983
   631
(*---------------------------------------------------------------------------*)
nipkow@5983
   632
(* 1. Cancellation of common terms                                           *)
nipkow@5983
   633
(*---------------------------------------------------------------------------*)
nipkow@5983
   634
nipkow@5983
   635
(*  Title:      HOL/arith_data.ML
nipkow@5983
   636
    ID:         $Id$
nipkow@5983
   637
    Author:     Markus Wenzel and Stefan Berghofer, TU Muenchen
nipkow@5983
   638
nipkow@5983
   639
Setup various arithmetic proof procedures.
nipkow@5983
   640
*)
nipkow@5983
   641
nipkow@5983
   642
signature ARITH_DATA =
nipkow@5983
   643
sig
nipkow@6055
   644
  val nat_cancel_sums_add: simproc list
nipkow@5983
   645
  val nat_cancel_sums: simproc list
nipkow@5983
   646
  val nat_cancel_factor: simproc list
nipkow@5983
   647
  val nat_cancel: simproc list
nipkow@5983
   648
end;
nipkow@5983
   649
nipkow@5983
   650
structure ArithData: ARITH_DATA =
nipkow@5983
   651
struct
nipkow@5983
   652
nipkow@5983
   653
nipkow@5983
   654
(** abstract syntax of structure nat: 0, Suc, + **)
nipkow@5983
   655
nipkow@5983
   656
(* mk_sum, mk_norm_sum *)
nipkow@5983
   657
nipkow@5983
   658
val one = HOLogic.mk_nat 1;
nipkow@5983
   659
val mk_plus = HOLogic.mk_binop "op +";
nipkow@5983
   660
nipkow@5983
   661
fun mk_sum [] = HOLogic.zero
nipkow@5983
   662
  | mk_sum [t] = t
nipkow@5983
   663
  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);
nipkow@5983
   664
nipkow@5983
   665
(*normal form of sums: Suc (... (Suc (a + (b + ...))))*)
nipkow@5983
   666
fun mk_norm_sum ts =
nipkow@5983
   667
  let val (ones, sums) = partition (equal one) ts in
nipkow@5983
   668
    funpow (length ones) HOLogic.mk_Suc (mk_sum sums)
nipkow@5983
   669
  end;
nipkow@5983
   670
nipkow@5983
   671
nipkow@5983
   672
(* dest_sum *)
nipkow@5983
   673
nipkow@5983
   674
val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT;
nipkow@5983
   675
nipkow@5983
   676
fun dest_sum tm =
nipkow@5983
   677
  if HOLogic.is_zero tm then []
nipkow@5983
   678
  else
nipkow@5983
   679
    (case try HOLogic.dest_Suc tm of
nipkow@5983
   680
      Some t => one :: dest_sum t
nipkow@5983
   681
    | None =>
nipkow@5983
   682
        (case try dest_plus tm of
nipkow@5983
   683
          Some (t, u) => dest_sum t @ dest_sum u
nipkow@5983
   684
        | None => [tm]));
nipkow@5983
   685
nipkow@5983
   686
nipkow@5983
   687
(** generic proof tools **)
nipkow@5983
   688
nipkow@5983
   689
(* prove conversions *)
nipkow@5983
   690
nipkow@5983
   691
val mk_eqv = HOLogic.mk_Trueprop o HOLogic.mk_eq;
nipkow@5983
   692
nipkow@5983
   693
fun prove_conv expand_tac norm_tac sg (t, u) =
nipkow@5983
   694
  mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv (t, u)))
nipkow@5983
   695
    (K [expand_tac, norm_tac]))
nipkow@5983
   696
  handle ERROR => error ("The error(s) above occurred while trying to prove " ^
nipkow@5983
   697
    (string_of_cterm (cterm_of sg (mk_eqv (t, u)))));
nipkow@5983
   698
nipkow@5983
   699
val subst_equals = prove_goal HOL.thy "[| t = s; u = t |] ==> u = s"
nipkow@5983
   700
  (fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]);
nipkow@5983
   701
nipkow@5983
   702
nipkow@5983
   703
(* rewriting *)
nipkow@5983
   704
nipkow@5983
   705
fun simp_all rules = ALLGOALS (simp_tac (HOL_ss addsimps rules));
nipkow@5983
   706
nipkow@5983
   707
val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right];
nipkow@5983
   708
val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right];
nipkow@5983
   709
nipkow@5983
   710
nipkow@5983
   711
nipkow@5983
   712
(** cancel common summands **)
nipkow@5983
   713
nipkow@5983
   714
structure Sum =
nipkow@5983
   715
struct
nipkow@5983
   716
  val mk_sum = mk_norm_sum;
nipkow@5983
   717
  val dest_sum = dest_sum;
nipkow@5983
   718
  val prove_conv = prove_conv;
nipkow@5983
   719
  val norm_tac = simp_all add_rules THEN simp_all add_ac;
nipkow@5983
   720
end;
nipkow@5983
   721
nipkow@5983
   722
fun gen_uncancel_tac rule ct =
nipkow@5983
   723
  rtac (instantiate' [] [None, Some ct] (rule RS subst_equals)) 1;
nipkow@5983
   724
nipkow@5983
   725
nipkow@5983
   726
(* nat eq *)
nipkow@5983
   727
nipkow@5983
   728
structure EqCancelSums = CancelSumsFun
nipkow@5983
   729
(struct
nipkow@5983
   730
  open Sum;
nipkow@5983
   731
  val mk_bal = HOLogic.mk_eq;
nipkow@5983
   732
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
nipkow@5983
   733
  val uncancel_tac = gen_uncancel_tac add_left_cancel;
nipkow@5983
   734
end);
nipkow@5983
   735
nipkow@5983
   736
nipkow@5983
   737
(* nat less *)
nipkow@5983
   738
nipkow@5983
   739
structure LessCancelSums = CancelSumsFun
nipkow@5983
   740
(struct
nipkow@5983
   741
  open Sum;
nipkow@5983
   742
  val mk_bal = HOLogic.mk_binrel "op <";
nipkow@5983
   743
  val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;
nipkow@5983
   744
  val uncancel_tac = gen_uncancel_tac add_left_cancel_less;
nipkow@5983
   745
end);
nipkow@5983
   746
nipkow@5983
   747
nipkow@5983
   748
(* nat le *)
nipkow@5983
   749
nipkow@5983
   750
structure LeCancelSums = CancelSumsFun
nipkow@5983
   751
(struct
nipkow@5983
   752
  open Sum;
nipkow@5983
   753
  val mk_bal = HOLogic.mk_binrel "op <=";
nipkow@5983
   754
  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;
nipkow@5983
   755
  val uncancel_tac = gen_uncancel_tac add_left_cancel_le;
nipkow@5983
   756
end);
nipkow@5983
   757
nipkow@5983
   758
nipkow@5983
   759
(* nat diff *)
nipkow@5983
   760
nipkow@5983
   761
structure DiffCancelSums = CancelSumsFun
nipkow@5983
   762
(struct
nipkow@5983
   763
  open Sum;
nipkow@5983
   764
  val mk_bal = HOLogic.mk_binop "op -";
nipkow@5983
   765
  val dest_bal = HOLogic.dest_bin "op -" HOLogic.natT;
nipkow@5983
   766
  val uncancel_tac = gen_uncancel_tac diff_cancel;
nipkow@5983
   767
end);
nipkow@5983
   768
nipkow@5983
   769
nipkow@5983
   770
nipkow@5983
   771
(** cancel common factor **)
nipkow@5983
   772
nipkow@5983
   773
structure Factor =
nipkow@5983
   774
struct
nipkow@5983
   775
  val mk_sum = mk_norm_sum;
nipkow@5983
   776
  val dest_sum = dest_sum;
nipkow@5983
   777
  val prove_conv = prove_conv;
nipkow@5983
   778
  val norm_tac = simp_all (add_rules @ mult_rules) THEN simp_all add_ac;
nipkow@5983
   779
end;
nipkow@5983
   780
wenzelm@6394
   781
fun mk_cnat n = cterm_of (Theory.sign_of Nat.thy) (HOLogic.mk_nat n);
nipkow@5983
   782
nipkow@5983
   783
fun gen_multiply_tac rule k =
nipkow@5983
   784
  if k > 0 then
nipkow@5983
   785
    rtac (instantiate' [] [None, Some (mk_cnat (k - 1))] (rule RS subst_equals)) 1
nipkow@5983
   786
  else no_tac;
nipkow@5983
   787
nipkow@5983
   788
nipkow@5983
   789
(* nat eq *)
nipkow@5983
   790
nipkow@5983
   791
structure EqCancelFactor = CancelFactorFun
nipkow@5983
   792
(struct
nipkow@5983
   793
  open Factor;
nipkow@5983
   794
  val mk_bal = HOLogic.mk_eq;
nipkow@5983
   795
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT;
nipkow@5983
   796
  val multiply_tac = gen_multiply_tac Suc_mult_cancel1;
nipkow@5983
   797
end);
nipkow@5983
   798
nipkow@5983
   799
nipkow@5983
   800
(* nat less *)
nipkow@5983
   801
nipkow@5983
   802
structure LessCancelFactor = CancelFactorFun
nipkow@5983
   803
(struct
nipkow@5983
   804
  open Factor;
nipkow@5983
   805
  val mk_bal = HOLogic.mk_binrel "op <";
nipkow@5983
   806
  val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT;
nipkow@5983
   807
  val multiply_tac = gen_multiply_tac Suc_mult_less_cancel1;
nipkow@5983
   808
end);
nipkow@5983
   809
nipkow@5983
   810
nipkow@5983
   811
(* nat le *)
nipkow@5983
   812
nipkow@5983
   813
structure LeCancelFactor = CancelFactorFun
nipkow@5983
   814
(struct
nipkow@5983
   815
  open Factor;
nipkow@5983
   816
  val mk_bal = HOLogic.mk_binrel "op <=";
nipkow@5983
   817
  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT;
nipkow@5983
   818
  val multiply_tac = gen_multiply_tac Suc_mult_le_cancel1;
nipkow@5983
   819
end);
nipkow@5983
   820
nipkow@5983
   821
nipkow@5983
   822
nipkow@5983
   823
(** prepare nat_cancel simprocs **)
nipkow@5983
   824
wenzelm@6394
   825
fun prep_pat s = Thm.read_cterm (Theory.sign_of Arith.thy) (s, HOLogic.termTVar);
nipkow@5983
   826
val prep_pats = map prep_pat;
nipkow@5983
   827
nipkow@5983
   828
fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
nipkow@5983
   829
nipkow@5983
   830
val eq_pats = prep_pats ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"];
nipkow@5983
   831
val less_pats = prep_pats ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"];
nipkow@5983
   832
val le_pats = prep_pats ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"];
nipkow@5983
   833
val diff_pats = prep_pats ["((l::nat) + m) - n", "(l::nat) - (m + n)", "Suc m - n", "m - Suc n"];
nipkow@5983
   834
nipkow@6055
   835
val nat_cancel_sums_add = map prep_simproc
nipkow@5983
   836
  [("nateq_cancel_sums", eq_pats, EqCancelSums.proc),
nipkow@5983
   837
   ("natless_cancel_sums", less_pats, LessCancelSums.proc),
nipkow@6055
   838
   ("natle_cancel_sums", le_pats, LeCancelSums.proc)];
nipkow@6055
   839
nipkow@6055
   840
val nat_cancel_sums = nat_cancel_sums_add @
nipkow@6055
   841
  [prep_simproc("natdiff_cancel_sums", diff_pats, DiffCancelSums.proc)];
nipkow@5983
   842
nipkow@5983
   843
val nat_cancel_factor = map prep_simproc
nipkow@5983
   844
  [("nateq_cancel_factor", eq_pats, EqCancelFactor.proc),
nipkow@5983
   845
   ("natless_cancel_factor", less_pats, LessCancelFactor.proc),
nipkow@5983
   846
   ("natle_cancel_factor", le_pats, LeCancelFactor.proc)];
nipkow@5983
   847
nipkow@5983
   848
val nat_cancel = nat_cancel_factor @ nat_cancel_sums;
nipkow@5983
   849
nipkow@5983
   850
nipkow@5983
   851
end;
nipkow@5983
   852
nipkow@5983
   853
open ArithData;
nipkow@5983
   854
nipkow@5983
   855
Addsimprocs nat_cancel;
nipkow@5983
   856
nipkow@5983
   857
(*---------------------------------------------------------------------------*)
nipkow@5983
   858
(* 2. Linear arithmetic                                                      *)
nipkow@5983
   859
(*---------------------------------------------------------------------------*)
nipkow@5983
   860
nipkow@6101
   861
(* Parameters data for general linear arithmetic functor *)
nipkow@6101
   862
nipkow@6101
   863
structure LA_Logic: LIN_ARITH_LOGIC =
nipkow@5983
   864
struct
nipkow@5983
   865
val ccontr = ccontr;
nipkow@5983
   866
val conjI = conjI;
nipkow@6101
   867
val neqE = linorder_neqE;
nipkow@5983
   868
val notI = notI;
nipkow@5983
   869
val sym = sym;
nipkow@6109
   870
val not_lessD = linorder_not_less RS iffD1;
nipkow@6128
   871
val not_leD = linorder_not_le RS iffD1;
nipkow@5983
   872
nipkow@6128
   873
wenzelm@6968
   874
fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI);
nipkow@6128
   875
nipkow@6073
   876
val mk_Trueprop = HOLogic.mk_Trueprop;
nipkow@6073
   877
nipkow@6079
   878
fun neg_prop(TP$(Const("Not",_)$t)) = TP$t
nipkow@6079
   879
  | neg_prop(TP$t) = TP $ (Const("Not",HOLogic.boolT-->HOLogic.boolT)$t);
nipkow@6073
   880
nipkow@6101
   881
fun is_False thm =
nipkow@6101
   882
  let val _ $ t = #prop(rep_thm thm)
nipkow@6101
   883
  in t = Const("False",HOLogic.boolT) end;
nipkow@6101
   884
nipkow@6128
   885
fun is_nat(t) = fastype_of1 t = HOLogic.natT;
nipkow@6128
   886
nipkow@6128
   887
fun mk_nat_thm sg t =
nipkow@6128
   888
  let val ct = cterm_of sg t  and cn = cterm_of sg (Var(("n",0),HOLogic.natT))
nipkow@6128
   889
  in instantiate ([],[(cn,ct)]) le0 end;
nipkow@6128
   890
nipkow@6101
   891
end;
nipkow@6101
   892
nipkow@7582
   893
signature LIN_ARITH_DATA2 =
nipkow@7582
   894
sig
nipkow@7582
   895
  include LIN_ARITH_DATA
nipkow@7582
   896
  val discrete: (string * bool)list ref
nipkow@7582
   897
end;
nipkow@5983
   898
nipkow@7582
   899
structure LA_Data_Ref: LIN_ARITH_DATA2 =
nipkow@7582
   900
struct
nipkow@7582
   901
  val add_mono_thms = ref ([]:thm list);
nipkow@7582
   902
  val lessD = ref ([]:thm list);
nipkow@7582
   903
  val ss_ref = ref HOL_basic_ss;
nipkow@7582
   904
  val discrete = ref ([]:(string*bool)list);
nipkow@5983
   905
nipkow@7548
   906
(* Decomposition of terms *)
nipkow@7548
   907
nipkow@7548
   908
fun nT (Type("fun",[N,_])) = N = HOLogic.natT
nipkow@7548
   909
  | nT _ = false;
nipkow@7548
   910
nipkow@7548
   911
fun add_atom(t,m,(p,i)) = (case assoc(p,t) of None => ((t,m)::p,i)
nipkow@7548
   912
                           | Some n => (overwrite(p,(t,n+m:int)), i));
nipkow@7548
   913
nipkow@7548
   914
(* Turn term into list of summand * multiplicity plus a constant *)
nipkow@7548
   915
fun poly(Const("op +",_) $ s $ t, m, pi) = poly(s,m,poly(t,m,pi))
nipkow@7548
   916
  | poly(all as Const("op -",T) $ s $ t, m, pi) =
nipkow@7548
   917
      if nT T then add_atom(all,m,pi)
nipkow@7548
   918
      else poly(s,m,poly(t,~1*m,pi))
nipkow@7548
   919
  | poly(Const("uminus",_) $ t, m, pi) = poly(t,~1*m,pi)
nipkow@7548
   920
  | poly(Const("0",_), _, pi) = pi
nipkow@7548
   921
  | poly(Const("Suc",_)$t, m, (p,i)) = poly(t, m, (p,i+m))
nipkow@7548
   922
  | poly(all as Const("op *",_) $ (Const("Numeral.number_of",_)$c) $ t, m, pi)=
nipkow@7548
   923
      (poly(t,m*HOLogic.dest_binum c,pi)
nipkow@7548
   924
       handle TERM _ => add_atom(all,m,pi))
nipkow@7548
   925
  | poly(all as Const("op *",_) $ t $ (Const("Numeral.number_of",_)$c), m, pi)=
nipkow@7548
   926
      (poly(t,m*HOLogic.dest_binum c,pi)
nipkow@7548
   927
       handle TERM _ => add_atom(all,m,pi))
nipkow@7548
   928
  | poly(all as Const("Numeral.number_of",_)$t,m,(p,i)) =
nipkow@7548
   929
     ((p,i + m*HOLogic.dest_binum t)
nipkow@7548
   930
      handle TERM _ => add_atom(all,m,(p,i)))
nipkow@7548
   931
  | poly x  = add_atom x;
nipkow@7548
   932
nipkow@7548
   933
fun decomp2(rel,lhs,rhs) =
nipkow@7548
   934
  let val (p,i) = poly(lhs,1,([],0)) and (q,j) = poly(rhs,1,([],0))
nipkow@7548
   935
  in case rel of
nipkow@7548
   936
       "op <"  => Some(p,i,"<",q,j)
nipkow@7548
   937
     | "op <=" => Some(p,i,"<=",q,j)
nipkow@7548
   938
     | "op ="  => Some(p,i,"=",q,j)
nipkow@7548
   939
     | _       => None
nipkow@7548
   940
  end;
nipkow@7548
   941
nipkow@7548
   942
fun negate(Some(x,i,rel,y,j,d)) = Some(x,i,"~"^rel,y,j,d)
nipkow@7548
   943
  | negate None = None;
nipkow@7548
   944
nipkow@7582
   945
fun decomp1 (T,xxx) =
nipkow@7548
   946
  (case T of
nipkow@7548
   947
     Type("fun",[Type(D,[]),_]) =>
nipkow@7582
   948
       (case assoc(!discrete,D) of
nipkow@7548
   949
          None => None
nipkow@7548
   950
        | Some d => (case decomp2 xxx of
nipkow@7548
   951
                       None => None
nipkow@7548
   952
                     | Some(p,i,rel,q,j) => Some(p,i,rel,q,j,d)))
nipkow@7548
   953
   | _ => None);
nipkow@7548
   954
nipkow@7582
   955
fun decomp (_$(Const(rel,T)$lhs$rhs)) = decomp1 (T,(rel,lhs,rhs))
nipkow@7582
   956
  | decomp (_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) =
nipkow@7582
   957
      negate(decomp1 (T,(rel,lhs,rhs)))
nipkow@7582
   958
  | decomp _ = None
nipkow@6128
   959
end;
nipkow@6055
   960
nipkow@7582
   961
let
nipkow@7582
   962
nipkow@7582
   963
(* reduce contradictory <= to False.
nipkow@7582
   964
   Most of the work is done by the cancel tactics.
nipkow@7582
   965
*)
nipkow@7582
   966
val add_rules = [add_0,add_0_right,Zero_not_Suc,Suc_not_Zero,le_0_eq];
nipkow@7582
   967
nipkow@7582
   968
val add_mono_thms = map (fn s => prove_goal Arith.thy s
nipkow@7582
   969
 (fn prems => [cut_facts_tac prems 1,
nipkow@7582
   970
               blast_tac (claset() addIs [add_le_mono]) 1]))
nipkow@7582
   971
["(i <= j) & (k <= l) ==> i + k <= j + (l::nat)",
nipkow@7582
   972
 "(i  = j) & (k <= l) ==> i + k <= j + (l::nat)",
nipkow@7582
   973
 "(i <= j) & (k  = l) ==> i + k <= j + (l::nat)",
nipkow@7582
   974
 "(i  = j) & (k  = l) ==> i + k  = j + (l::nat)"
nipkow@7582
   975
];
nipkow@7582
   976
nipkow@7582
   977
in
nipkow@7582
   978
LA_Data_Ref.add_mono_thms := !LA_Data_Ref.add_mono_thms @ add_mono_thms;
nipkow@7582
   979
LA_Data_Ref.lessD := !LA_Data_Ref.lessD @ [Suc_leI];
nipkow@7582
   980
LA_Data_Ref.ss_ref := !LA_Data_Ref.ss_ref addsimps add_rules
nipkow@7582
   981
                      addsimprocs nat_cancel_sums_add;
nipkow@7582
   982
LA_Data_Ref.discrete := !LA_Data_Ref.discrete @ [("nat",true)]
nipkow@5983
   983
end;
nipkow@5983
   984
nipkow@6128
   985
structure Fast_Arith =
nipkow@6128
   986
  Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref);
nipkow@5983
   987
nipkow@6128
   988
val fast_arith_tac = Fast_Arith.lin_arith_tac;
nipkow@6073
   989
nipkow@7582
   990
let
nipkow@6128
   991
val nat_arith_simproc_pats =
wenzelm@6394
   992
  map (fn s => Thm.read_cterm (Theory.sign_of Arith.thy) (s, HOLogic.boolT))
nipkow@6128
   993
      ["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"];
nipkow@5983
   994
nipkow@7582
   995
val fast_nat_arith_simproc = mk_simproc
nipkow@7582
   996
  "fast_nat_arith" nat_arith_simproc_pats Fast_Arith.lin_arith_prover;
nipkow@7582
   997
in
nipkow@7582
   998
Addsimprocs [fast_nat_arith_simproc]
nipkow@7582
   999
end;
nipkow@6073
  1000
nipkow@6073
  1001
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only
nipkow@6073
  1002
useful to detect inconsistencies among the premises for subgoals which are
nipkow@6073
  1003
*not* themselves (in)equalities, because the latter activate
nipkow@6073
  1004
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
nipkow@6073
  1005
solver all the time rather than add the additional check. *)
nipkow@6073
  1006
nipkow@7570
  1007
simpset_ref () := (simpset() addSolver
nipkow@7570
  1008
   (mk_solver "lin. arith." Fast_Arith.cut_lin_arith_tac));
nipkow@6055
  1009
nipkow@6055
  1010
(* Elimination of `-' on nat due to John Harrison *)
nipkow@6055
  1011
Goal "P(a - b::nat) = (!d. (b = a + d --> P 0) & (a = b + d --> P d))";
paulson@6301
  1012
by (case_tac "a <= b" 1);
paulson@7059
  1013
by Auto_tac;
paulson@6301
  1014
by (eres_inst_tac [("x","b-a")] allE 1);
paulson@7059
  1015
by (asm_simp_tac (simpset() addsimps [diff_is_0_eq RS iffD2]) 1);
nipkow@6055
  1016
qed "nat_diff_split";
nipkow@6055
  1017
nipkow@6055
  1018
(* FIXME: K true should be replaced by a sensible test to speed things up
nipkow@6157
  1019
   in case there are lots of irrelevant terms involved;
nipkow@6157
  1020
   elimination of min/max can be optimized:
nipkow@6157
  1021
   (max m n + k <= r) = (m+k <= r & n+k <= r)
nipkow@6157
  1022
   (l <= min m n + k) = (l <= m+k & l <= n+k)
nipkow@6055
  1023
*)
nipkow@7582
  1024
val arith_tac_split_thms = ref [nat_diff_split,split_min,split_max];
nipkow@7582
  1025
fun arith_tac i =
nipkow@7582
  1026
  refute_tac (K true) (REPEAT o split_tac (!arith_tac_split_thms))
nipkow@7582
  1027
             ((REPEAT_DETERM o etac linorder_neqE) THEN' fast_arith_tac) i;
nipkow@6055
  1028
wenzelm@7131
  1029
wenzelm@7131
  1030
(* proof method setup *)
wenzelm@7131
  1031
wenzelm@7428
  1032
val arith_method =
wenzelm@7428
  1033
  Method.METHOD (fn facts => FIRSTGOAL (Method.insert_tac facts THEN' arith_tac));
wenzelm@7131
  1034
wenzelm@7131
  1035
val arith_setup =
wenzelm@7131
  1036
 [Method.add_methods
wenzelm@7131
  1037
  [("arith", Method.no_args arith_method, "decide linear arithmethic")]];
wenzelm@7131
  1038
nipkow@5983
  1039
(*---------------------------------------------------------------------------*)
nipkow@5983
  1040
(* End of proof procedures. Now go and USE them!                             *)
nipkow@5983
  1041
(*---------------------------------------------------------------------------*)
nipkow@5983
  1042
paulson@4736
  1043
(*** Subtraction laws -- mostly from Clemens Ballarin ***)
paulson@3234
  1044
paulson@5429
  1045
Goal "[| a < (b::nat); c <= a |] ==> a-c < b-c";
paulson@6301
  1046
by (arith_tac 1);
paulson@3234
  1047
qed "diff_less_mono";
paulson@3234
  1048
paulson@5429
  1049
Goal "a+b < (c::nat) ==> a < c-b";
paulson@6301
  1050
by (arith_tac 1);
paulson@3234
  1051
qed "add_less_imp_less_diff";
paulson@3234
  1052
nipkow@5427
  1053
Goal "(i < j-k) = (i+k < (j::nat))";
paulson@6301
  1054
by (arith_tac 1);
nipkow@5427
  1055
qed "less_diff_conv";
nipkow@5427
  1056
paulson@5497
  1057
Goal "(j-k <= (i::nat)) = (j <= i+k)";
paulson@6301
  1058
by (arith_tac 1);
paulson@5485
  1059
qed "le_diff_conv";
paulson@5485
  1060
paulson@5497
  1061
Goal "k <= j ==> (i <= j-k) = (i+k <= (j::nat))";
paulson@6301
  1062
by (arith_tac 1);
paulson@5497
  1063
qed "le_diff_conv2";
paulson@5497
  1064
paulson@5143
  1065
Goal "Suc i <= n ==> Suc (n - Suc i) = n - i";
paulson@6301
  1066
by (arith_tac 1);
paulson@3234
  1067
qed "Suc_diff_Suc";
paulson@3234
  1068
paulson@5429
  1069
Goal "i <= (n::nat) ==> n - (n - i) = i";
paulson@6301
  1070
by (arith_tac 1);
paulson@3234
  1071
qed "diff_diff_cancel";
paulson@3381
  1072
Addsimps [diff_diff_cancel];
paulson@3234
  1073
paulson@5429
  1074
Goal "k <= (n::nat) ==> m <= n + m - k";
paulson@6301
  1075
by (arith_tac 1);
paulson@3234
  1076
qed "le_add_diff";
paulson@3234
  1077
nipkow@6055
  1078
Goal "[| 0<k; j<i |] ==> j+k-i < k";
paulson@6301
  1079
by (arith_tac 1);
nipkow@6055
  1080
qed "add_diff_less";
paulson@3234
  1081
paulson@5356
  1082
Goal "m-1 < n ==> m <= n";
paulson@6301
  1083
by (arith_tac 1);
paulson@5356
  1084
qed "pred_less_imp_le";
paulson@5356
  1085
paulson@5356
  1086
Goal "j<=i ==> i - j < Suc i - j";
paulson@6301
  1087
by (arith_tac 1);
paulson@5356
  1088
qed "diff_less_Suc_diff";
paulson@5356
  1089
paulson@5356
  1090
Goal "i - j <= Suc i - j";
paulson@6301
  1091
by (arith_tac 1);
paulson@5356
  1092
qed "diff_le_Suc_diff";
paulson@5356
  1093
AddIffs [diff_le_Suc_diff];
paulson@5356
  1094
paulson@5356
  1095
Goal "n - Suc i <= n - i";
paulson@6301
  1096
by (arith_tac 1);
paulson@5356
  1097
qed "diff_Suc_le_diff";
paulson@5356
  1098
AddIffs [diff_Suc_le_diff];
paulson@5356
  1099
paulson@5409
  1100
Goal "0 < n ==> (m <= n-1) = (m<n)";
paulson@6301
  1101
by (arith_tac 1);
paulson@5409
  1102
qed "le_pred_eq";
paulson@5409
  1103
paulson@5409
  1104
Goal "0 < n ==> (m-1 < n) = (m<=n)";
paulson@6301
  1105
by (arith_tac 1);
paulson@5409
  1106
qed "less_pred_eq";
paulson@5409
  1107
paulson@7059
  1108
(*Replaces the previous diff_less and le_diff_less, which had the stronger
paulson@7059
  1109
  second premise n<=m*)
paulson@7059
  1110
Goal "[| 0<n; 0<m |] ==> m - n < m";
paulson@6301
  1111
by (arith_tac 1);
paulson@5414
  1112
qed "diff_less";
paulson@5414
  1113
paulson@4732
  1114
paulson@7128
  1115
(*** Reducting subtraction to addition ***)
paulson@7128
  1116
paulson@7128
  1117
(*Intended for use with linear arithmetic, but useful in its own right*)
paulson@7128
  1118
Goal "P (x-y) = (ALL z. (x<y --> P 0) & (x = y+z --> P z))";
paulson@7128
  1119
by (case_tac "x<y" 1);
paulson@7128
  1120
by (auto_tac (claset(),  simpset() addsimps [diff_is_0_eq RS iffD2]));
paulson@7128
  1121
qed "split_diff";
paulson@7128
  1122
paulson@7128
  1123
val remove_diff_ss = 
paulson@7128
  1124
    simpset()
paulson@7128
  1125
      delsimps ex_simps@all_simps
paulson@7128
  1126
      addsimps [le_diff_conv2, le_diff_conv, le_imp_diff_is_add, 
paulson@7128
  1127
		diff_diff_right] 
paulson@7128
  1128
      addcongs [conj_cong]
paulson@7128
  1129
      addsplits [split_diff];
paulson@7128
  1130
paulson@7128
  1131
Goal "n<=(l::nat) --> Suc l - n + m = Suc (l - n + m)";
paulson@7128
  1132
by (simp_tac remove_diff_ss 1);
paulson@7128
  1133
qed_spec_mp "Suc_diff_add_le";
paulson@7128
  1134
paulson@7128
  1135
Goal "i<n ==> n - Suc i < n - i";
paulson@7128
  1136
by (asm_simp_tac remove_diff_ss 1);
paulson@7128
  1137
qed "diff_Suc_less_diff";
paulson@7128
  1138
paulson@7128
  1139
Goal "Suc(m)-n = (if m<n then 0 else Suc(m-n))";
paulson@7128
  1140
by (simp_tac remove_diff_ss 1);
paulson@7128
  1141
qed "if_Suc_diff_le";
paulson@7128
  1142
paulson@7128
  1143
Goal "Suc(m)-n <= Suc(m-n)";
paulson@7128
  1144
by (simp_tac remove_diff_ss 1);
paulson@7128
  1145
qed "diff_Suc_le_Suc_diff";
paulson@7128
  1146
paulson@7128
  1147
Goal "[| k<=n; n<=m |] ==> (m-k) - (n-k) = m-(n::nat)";
paulson@7128
  1148
by (asm_simp_tac remove_diff_ss 1);
paulson@7128
  1149
qed "diff_right_cancel";
paulson@7128
  1150
paulson@7128
  1151
wenzelm@7108
  1152
(** (Anti)Monotonicity of subtraction -- by Stephan Merz **)
nipkow@3484
  1153
nipkow@3484
  1154
(* Monotonicity of subtraction in first argument *)
nipkow@6055
  1155
Goal "m <= (n::nat) ==> (m-l) <= (n-l)";
paulson@7128
  1156
by (asm_simp_tac remove_diff_ss 1);
nipkow@6055
  1157
qed "diff_le_mono";
nipkow@3484
  1158
paulson@5429
  1159
Goal "m <= (n::nat) ==> (l-n) <= (l-m)";
paulson@7128
  1160
by (asm_simp_tac remove_diff_ss 1);
nipkow@6055
  1161
qed "diff_le_mono2";
nipkow@5983
  1162
nipkow@5983
  1163
(*This proof requires natdiff_cancel_sums*)
nipkow@6055
  1164
Goal "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)";
paulson@7128
  1165
by (asm_simp_tac remove_diff_ss 1);
nipkow@6055
  1166
qed "diff_less_mono2";
nipkow@5983
  1167
nipkow@6055
  1168
Goal "[| m-n = 0; n-m = 0 |] ==>  m=n";
paulson@7128
  1169
by (asm_full_simp_tac remove_diff_ss 1);
nipkow@6055
  1170
qed "diffs0_imp_equal";