(*  Title:      HOL/Integ/int_arith2.ML

     ID:         $Id$

     Authors:    Larry Paulson and Tobias Nipkow

 *)

 Goal "(w <= z - (1::int)) = (w<(z::int))";

 by (arith_tac 1);

 qed "zle_diff1_eq";

 Addsimps [zle_diff1_eq];

 Goal "(w < z + 1) = (w<=(z::int))";

 by (arith_tac 1);

 qed "zle_add1_eq_le";

 Addsimps [zle_add1_eq_le];

 Goal "(z = z + w) = (w = (0::int))";

 by (arith_tac 1);

 qed "zadd_left_cancel0";

 Addsimps [zadd_left_cancel0];

 (* nat *)

 Goal "0 <= z ==> int (nat z) = z"; 

 by (asm_full_simp_tac

     (simpset() addsimps [neg_eq_less_0, zle_def, not_neg_nat]) 1); 

 qed "nat_0_le"; 

 Goal "z <= 0 ==> nat z = 0"; 

 by (case_tac "z = 0" 1);

 by (asm_simp_tac (simpset() addsimps [nat_le_int0]) 1); 

 by (asm_full_simp_tac 

     (simpset() addsimps [neg_eq_less_0, neg_nat, linorder_neq_iff]) 1);

 qed "nat_le_0"; 

 Addsimps [nat_0_le, nat_le_0];

 val [major,minor] = Goal "[| 0 <= z;  !!m. z = int m ==> P |] ==> P"; 

 by (rtac (major RS nat_0_le RS sym RS minor) 1);

 qed "nonneg_eq_int"; 

 Goal "(nat w = m) = (if 0 <= w then w = int m else m=0)";

 by Auto_tac;

 qed "nat_eq_iff";

 Goal "(m = nat w) = (if 0 <= w then w = int m else m=0)";

 by Auto_tac;

 qed "nat_eq_iff2";

 Goal "0 <= w ==> (nat w < m) = (w < int m)";

 by (rtac iffI 1);

 by (asm_full_simp_tac 

     (simpset() delsimps [zless_int] addsimps [zless_int RS sym]) 2);

 by (etac (nat_0_le RS subst) 1);

 by (Simp_tac 1);

 qed "nat_less_iff";

 Goal "(int m = z) = (m = nat z & 0 <= z)";

 by (auto_tac (claset(), simpset() addsimps [nat_eq_iff2]));  

 qed "int_eq_iff";

 (*Users don't want to see (int 0), int(Suc 0) or w + - z*)

 Addsimps (map symmetric [Zero_int_def, One_int_def, zdiff_def]);

 Goal "nat 0 = 0";

 by (simp_tac (simpset() addsimps [nat_eq_iff]) 1);

 qed "nat_0";

 Goal "nat 1 = Suc 0";

 by (stac nat_eq_iff 1);

 by (Simp_tac 1);

 qed "nat_1";

 Goal "nat 2 = Suc (Suc 0)";

 by (stac nat_eq_iff 1);

 by (Simp_tac 1);

 qed "nat_2";

 Goal "0 <= w ==> (nat w < nat z) = (w<z)";

 by (case_tac "neg z" 1);

 by (auto_tac (claset(), simpset() addsimps [nat_less_iff]));

 by (auto_tac (claset() addIs [zless_trans], 

 	      simpset() addsimps [neg_eq_less_0, zle_def]));

 qed "nat_less_eq_zless";

 Goal "0 < w | 0 <= z ==> (nat w <= nat z) = (w<=z)";

 by (auto_tac (claset(), 

 	      simpset() addsimps [linorder_not_less RS sym, 

 				  zless_nat_conj]));

 qed "nat_le_eq_zle";

 (*** abs: absolute value, as an integer ****)

 (* Simpler: use zabs_def as rewrite rule;

    but arith_tac is not parameterized by such simp rules

 *)

 Goalw [zabs_def]

  "P(abs(i::int)) = ((0 <= i --> P i) & (i < 0 --> P(-i)))";

 by(Simp_tac 1);

 qed "zabs_split";

 Goal "0 <= abs (z::int)";

 by (simp_tac (simpset() addsimps [zabs_def]) 1); 

 qed "zero_le_zabs";

 AddIffs [zero_le_zabs];

 (*Not sure why this simprule is required!*)

 Addsimps [inst "z" "number_of ?v" int_eq_iff];

 (*continued in IntArith.ML ...*)