src/HOL/Integ/IntArith.ML

Numerals and simprocs for types real and hypreal. The abstract
constants 0, 1 and binary numerals work harmoniously.

(*  Title:      HOL/Integ/IntArith.ML
    ID:         $Id$
    Authors:    Larry Paulson and Tobias Nipkow
*)


Goal "x - - y = x + (y::int)";
by (Simp_tac 1); 
qed "int_diff_minus_eq";
Addsimps [int_diff_minus_eq];

Goal "abs(abs(x::int)) = abs(x)";
by(arith_tac 1);
qed "abs_abs";
Addsimps [abs_abs];

Goal "abs(-(x::int)) = abs(x)";
by(arith_tac 1);
qed "abs_minus";
Addsimps [abs_minus];

Goal "abs(x+y) <= abs(x) + abs(y::int)";
by(arith_tac 1);
qed "triangle_ineq";


(*** Intermediate value theorems ***)

Goal "(ALL i<n::nat. abs(f(i+1) - f i) <= 1) --> \
\     f 0 <= k --> k <= f n --> (EX i <= n. f i = (k::int))";
by(induct_tac "n" 1);
 by(Asm_simp_tac 1);
by(strip_tac 1);
by(etac impE 1);
 by(Asm_full_simp_tac 1);
by(eres_inst_tac [("x","n")] allE 1);
by(Asm_full_simp_tac 1);
by(case_tac "k = f(n+1)" 1);
 by(Force_tac 1);
by(etac impE 1);
 by(asm_full_simp_tac (simpset() addsimps [zabs_def] 
                                 addsplits [split_if_asm]) 1);
 by(arith_tac 1);
by(blast_tac (claset() addIs [le_SucI]) 1);
val lemma = result();

bind_thm("nat0_intermed_int_val", ObjectLogic.rulify_no_asm lemma);

Goal "[| !i. m <= i & i < n --> abs(f(i + 1::nat) - f i) <= 1; m < n; \
\        f m <= k; k <= f n |] ==> ? i. m <= i & i <= n & f i = (k::int)";
by(cut_inst_tac [("n","n-m"),("f", "%i. f(i+m)"),("k","k")]lemma 1);
by(Asm_full_simp_tac 1);
by(etac impE 1);
 by(strip_tac 1);
 by(eres_inst_tac [("x","i+m")] allE 1);
 by(arith_tac 1);
by(etac exE 1);
by(res_inst_tac [("x","i+m")] exI 1);
by(arith_tac 1);
qed "nat_intermed_int_val";


(*** Some convenient biconditionals for products of signs ***)

Goal "[| (0::int) < i; 0 < j |] ==> 0 < i*j";
by (dtac zmult_zless_mono1 1);
by Auto_tac; 
qed "zmult_pos";

Goal "[| i < (0::int); j < 0 |] ==> 0 < i*j";
by (dtac zmult_zless_mono1_neg 1);
by Auto_tac; 
qed "zmult_neg";

Goal "[| (0::int) < i; j < 0 |] ==> i*j < 0";
by (dtac zmult_zless_mono1_neg 1);
by Auto_tac; 
qed "zmult_pos_neg";

Goal "((0::int) < x*y) = (0 < x & 0 < y | x < 0 & y < 0)";
by (auto_tac (claset(), 
              simpset() addsimps [order_le_less, linorder_not_less,
	                          zmult_pos, zmult_neg]));
by (ALLGOALS (rtac ccontr)); 
by (auto_tac (claset(), 
	      simpset() addsimps [order_le_less, linorder_not_less]));
by (ALLGOALS (etac rev_mp)); 
by (ALLGOALS (dtac zmult_pos_neg THEN' assume_tac));
by (auto_tac (claset() addDs [order_less_not_sym], 
              simpset() addsimps [zmult_commute]));  
qed "int_0_less_mult_iff";

Goal "((0::int) <= x*y) = (0 <= x & 0 <= y | x <= 0 & y <= 0)";
by (auto_tac (claset(), 
              simpset() addsimps [order_le_less, linorder_not_less,  
                                  int_0_less_mult_iff]));
qed "int_0_le_mult_iff";

Goal "(x*y < (0::int)) = (0 < x & y < 0 | x < 0 & 0 < y)";
by (auto_tac (claset(), 
              simpset() addsimps [int_0_le_mult_iff, 
                                  linorder_not_le RS sym]));
by (auto_tac (claset() addDs [order_less_not_sym],  
              simpset() addsimps [linorder_not_le]));
qed "zmult_less_0_iff";

Goal "(x*y <= (0::int)) = (0 <= x & y <= 0 | x <= 0 & 0 <= y)";
by (auto_tac (claset() addDs [order_less_not_sym], 
              simpset() addsimps [int_0_less_mult_iff, 
                                  linorder_not_less RS sym]));
qed "zmult_le_0_iff";

Goal "abs (x * y) = abs x * abs (y::int)";
by (simp_tac (simpset () delsimps [thm "number_of_reorient"] addsplits [zabs_split] 
                         addsplits [zabs_split] 
                         addsimps [zmult_less_0_iff, zle_def]) 1);
qed "abs_mult";

Goal "(abs x = 0) = (x = (0::int))";
by (simp_tac (simpset () addsplits [zabs_split]) 1);
qed "abs_eq_0";
AddIffs [abs_eq_0];

Goal "(0 < abs x) = (x ~= (0::int))";
by (simp_tac (simpset () addsplits [zabs_split]) 1);
by (arith_tac 1);
qed "zero_less_abs_iff";
AddIffs [zero_less_abs_iff];

Goal "0 <= x * (x::int)";
by (subgoal_tac "(- x) * x <= 0" 1);
 by (Asm_full_simp_tac 1);
by (simp_tac (HOL_basic_ss addsimps [zmult_le_0_iff]) 1);
by Auto_tac;
qed "square_nonzero";
Addsimps [square_nonzero];
AddIs [square_nonzero];


(*** Products and 1, by T. M. Rasmussen ***)

Goal "(m = m*(n::int)) = (n = 1 | m = 0)";
by Auto_tac;
by (subgoal_tac "m*1 = m*n" 1);
by (dtac (zmult_cancel1 RS iffD1) 1); 
by Auto_tac;
qed "zmult_eq_self_iff";

Goal "[| 1 < m; 1 < n |] ==> 1 < m*(n::int)";
by (res_inst_tac [("y","1*n")] order_less_trans 1);
by (rtac zmult_zless_mono1 2);
by (ALLGOALS Asm_simp_tac);
qed "zless_1_zmult";

Goal "[| 0 < n; n ~= 1 |] ==> 1 < (n::int)";
by (arith_tac 1);
val lemma = result();

Goal "0 < (m::int) ==> (m * n = 1) = (m = 1 & n = 1)";
by Auto_tac;
by (case_tac "m=1" 1);
by (case_tac "n=1" 2);
by (case_tac "m=1" 4);
by (case_tac "n=1" 5);
by Auto_tac;
by distinct_subgoals_tac;
by (subgoal_tac "1<m*n" 1);
by (Asm_full_simp_tac 1);
by (rtac zless_1_zmult 1);
by (ALLGOALS (rtac lemma));
by Auto_tac;  
by (subgoal_tac "0<m*n" 1);
by (Asm_simp_tac 2);
by (dtac (int_0_less_mult_iff RS iffD1) 1);
by Auto_tac;  
qed "pos_zmult_eq_1_iff";

Goal "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))";
by (case_tac "0<m" 1);
by (asm_simp_tac (simpset() addsimps [pos_zmult_eq_1_iff]) 1);
by (case_tac "m=0" 1);
by (asm_simp_tac (simpset () delsimps [thm "number_of_reorient"]) 1);
by (subgoal_tac "0 < -m" 1);
by (arith_tac 2);
by (dres_inst_tac [("n","-n")] pos_zmult_eq_1_iff 1); 
by Auto_tac;  
qed "zmult_eq_1_iff";