(*  Title:      HOL/Integ/IntArith.ML

     ID:         $Id$

     Authors:    Larry Paulson and Tobias Nipkow

 *)

 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";