src/HOL/Integ/Bin.ML

Subject reduction and strong normalization of simply-typed lambda terms.

(*  Title:      HOL/Integ/Bin.ML
    ID:         $Id$
    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
                David Spelt, University of Twente 
                Tobias Nipkow, Technical University Munich
    Copyright   1994  University of Cambridge
    Copyright   1996  University of Twente
    Copyright   1999  TU Munich

Arithmetic on binary integers.
*)

(** extra rules for bin_succ, bin_pred, bin_add, bin_mult **)

Goal "NCons Pls False = Pls";
by (Simp_tac 1);
qed "NCons_Pls_0";

Goal "NCons Pls True = Pls BIT True";
by (Simp_tac 1);
qed "NCons_Pls_1";

Goal "NCons Min False = Min BIT False";
by (Simp_tac 1);
qed "NCons_Min_0";

Goal "NCons Min True = Min";
by (Simp_tac 1);
qed "NCons_Min_1";

Goal "bin_succ(w BIT True) = (bin_succ w) BIT False";
by (Simp_tac 1);
qed "bin_succ_1";

Goal "bin_succ(w BIT False) =  NCons w True";
by (Simp_tac 1);
qed "bin_succ_0";

Goal "bin_pred(w BIT True) = NCons w False";
by (Simp_tac 1);
qed "bin_pred_1";

Goal "bin_pred(w BIT False) = (bin_pred w) BIT True";
by (Simp_tac 1);
qed "bin_pred_0";

Goal "bin_minus(w BIT True) = bin_pred (NCons (bin_minus w) False)";
by (Simp_tac 1);
qed "bin_minus_1";

Goal "bin_minus(w BIT False) = (bin_minus w) BIT False";
by (Simp_tac 1);
qed "bin_minus_0";


(*** bin_add: binary addition ***)

Goal "bin_add (v BIT True) (w BIT True) = \
\    NCons (bin_add v (bin_succ w)) False";
by (Simp_tac 1);
qed "bin_add_BIT_11";

Goal "bin_add (v BIT True) (w BIT False) = NCons (bin_add v w) True";
by (Simp_tac 1);
qed "bin_add_BIT_10";

Goal "bin_add (v BIT False) (w BIT y) = NCons (bin_add v w) y";
by Auto_tac;
qed "bin_add_BIT_0";

Goal "bin_add w Pls = w";
by (induct_tac "w" 1);
by Auto_tac;
qed "bin_add_Pls_right";

Goal "bin_add w Min = bin_pred w";
by (induct_tac "w" 1);
by Auto_tac;
qed "bin_add_Min_right";

Goal "bin_add (v BIT x) (w BIT y) = \
\    NCons(bin_add v (if x & y then (bin_succ w) else w)) (x~= y)";
by (Simp_tac 1);
qed "bin_add_BIT_BIT";


(*** bin_mult: binary multiplication ***)

Goal "bin_mult (v BIT True) w = bin_add (NCons (bin_mult v w) False) w";
by (Simp_tac 1);
qed "bin_mult_1";

Goal "bin_mult (v BIT False) w = NCons (bin_mult v w) False";
by (Simp_tac 1);
qed "bin_mult_0";


(**** The carry/borrow functions, bin_succ and bin_pred ****)


(**** number_of ****)

Goal "number_of(NCons w b) = (number_of(w BIT b)::int)";
by (induct_tac "w" 1);
by (ALLGOALS Asm_simp_tac);
qed "number_of_NCons";

Addsimps [number_of_NCons];

Goal "number_of(bin_succ w) = int 1 + number_of w";
by (induct_tac "w" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps zadd_ac)));
qed "number_of_succ";

Goal "number_of(bin_pred w) = - (int 1) + number_of w";
by (induct_tac "w" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps zadd_ac)));
qed "number_of_pred";

Goal "number_of(bin_minus w) = (- (number_of w)::int)";
by (induct_tac "w" 1);
by (Simp_tac 1);
by (Simp_tac 1);
by (asm_simp_tac (simpset()
		  delsimps [bin_pred_Pls, bin_pred_Min, bin_pred_BIT]
		  addsimps [number_of_succ,number_of_pred,
			    zadd_assoc]) 1);
qed "number_of_minus";

 
bind_thms ("bin_add_simps", [bin_add_BIT_BIT, number_of_succ, number_of_pred]);

(*This proof is complicated by the mutual recursion*)
Goal "! w. number_of(bin_add v w) = (number_of v + number_of w::int)";
by (induct_tac "v" 1);
by (simp_tac (simpset() addsimps bin_add_simps) 1);
by (simp_tac (simpset() addsimps bin_add_simps) 1);
by (rtac allI 1);
by (induct_tac "w" 1);
by (ALLGOALS (asm_simp_tac (simpset() addsimps bin_add_simps @ zadd_ac)));
qed_spec_mp "number_of_add";


(*Subtraction*)
Goalw [zdiff_def]
     "number_of v - number_of w = (number_of(bin_add v (bin_minus w))::int)";
by (simp_tac (simpset() addsimps [number_of_add, number_of_minus]) 1);
qed "diff_number_of_eq";

bind_thms ("bin_mult_simps", [zmult_zminus, number_of_minus, number_of_add]);

Goal "number_of(bin_mult v w) = (number_of v * number_of w::int)";
by (induct_tac "v" 1);
by (simp_tac (simpset() addsimps bin_mult_simps) 1);
by (simp_tac (simpset() addsimps bin_mult_simps) 1);
by (asm_simp_tac
    (simpset() addsimps bin_mult_simps @ [zadd_zmult_distrib] @ zadd_ac) 1);
qed "number_of_mult";


(*The correctness of shifting.  But it doesn't seem to give a measurable
  speed-up.*)
Goal "(#2::int) * number_of w = number_of (w BIT False)";
by (induct_tac "w" 1);
by (ALLGOALS (asm_simp_tac
    (simpset() addsimps bin_mult_simps @ [zadd_zmult_distrib] @ zadd_ac)));
qed "double_number_of_BIT";


(** Simplification rules with integer constants **)

Goal "#0 + z = (z::int)";
by (Simp_tac 1);
qed "zadd_0";

Goal "z + #0 = (z::int)";
by (Simp_tac 1);
qed "zadd_0_right";

Addsimps [zadd_0, zadd_0_right];


(** Converting simple cases of (int n) to numerals **)

(*int 0 = #0 *)
bind_thm ("int_0", number_of_Pls RS sym);

Goal "int (Suc n) = #1 + int n";
by (simp_tac (simpset() addsimps [zadd_int]) 1);
qed "int_Suc";

Goal "- (#0) = (#0::int)";
by (Simp_tac 1);
qed "zminus_0";

Addsimps [zminus_0];


Goal "(#0::int) - x = -x";
by (simp_tac (simpset() addsimps [zdiff_def]) 1);
qed "zdiff0";

Goal "x - (#0::int) = x";
by (simp_tac (simpset() addsimps [zdiff_def]) 1);
qed "zdiff0_right";

Goal "x - x = (#0::int)";
by (simp_tac (simpset() addsimps [zdiff_def]) 1);
qed "zdiff_self";

Addsimps [zdiff0, zdiff0_right, zdiff_self];


(** Special simplification, for constants only **)

(*Distributive laws for literals*)
Addsimps (map (inst "w" "number_of ?v")
	  [zadd_zmult_distrib, zadd_zmult_distrib2,
	   zdiff_zmult_distrib, zdiff_zmult_distrib2]);

Addsimps (map (inst "x" "number_of ?v") 
	  [zless_zminus, zle_zminus, equation_zminus]);
Addsimps (map (inst "y" "number_of ?v") 
	  [zminus_zless, zminus_zle, zminus_equation]);

(*Moving negation out of products*)
Addsimps [zmult_zminus, zmult_zminus_right];

(** Special-case simplification for small constants **)

Goal "#0 * z = (#0::int)";
by (Simp_tac 1);
qed "zmult_0";

Goal "z * #0 = (#0::int)";
by (Simp_tac 1);
qed "zmult_0_right";

Goal "#1 * z = (z::int)";
by (Simp_tac 1);
qed "zmult_1";

Goal "z * #1 = (z::int)";
by (Simp_tac 1);
qed "zmult_1_right";

Goal "#-1 * z = -(z::int)";
by (simp_tac (simpset() addsimps zcompare_rls@[zmult_zminus]) 1);
qed "zmult_minus1";

Goal "z * #-1 = -(z::int)";
by (simp_tac (simpset() addsimps zcompare_rls@[zmult_zminus_right]) 1);
qed "zmult_minus1_right";

Addsimps [zmult_0, zmult_0_right, 
	  zmult_1, zmult_1_right,
	  zmult_minus1, zmult_minus1_right];

(*Negation of a coefficient*)
Goal "- (number_of w) * z = number_of(bin_minus w) * (z::int)";
by (simp_tac (simpset_of Int.thy addsimps [number_of_minus, zmult_zminus]) 1);
qed "zminus_number_of_zmult";

Addsimps [zminus_number_of_zmult];


(** Inequality reasoning **)

Goal "(m*n = (#0::int)) = (m = #0 | n = #0)";
by (stac (int_0 RS sym) 1 THEN rtac zmult_eq_int0_iff 1);
qed "zmult_eq_0_iff";

Goal "((#0::int) = m*n) = (#0 = m | #0 = n)";
by (stac eq_commute 1 THEN stac zmult_eq_0_iff 1);
by Auto_tac;
qed "zmult_0_eq_iff";

Addsimps [zmult_eq_0_iff, zmult_0_eq_iff];

Goal "(w < z + (#1::int)) = (w<z | w=z)";
by (simp_tac (simpset() addsimps [zless_add_int_Suc_eq]) 1);
qed "zless_add1_eq";

Goal "(w + (#1::int) <= z) = (w<z)";
by (simp_tac (simpset() addsimps [add_int_Suc_zle_eq]) 1);
qed "add1_zle_eq";

Goal "((#1::int) + w <= z) = (w<z)";
by (stac zadd_commute 1);
by (rtac add1_zle_eq 1);
qed "add1_left_zle_eq";

Goal "neg x = (x < #0)";
by (simp_tac (simpset() addsimps [neg_eq_less_int0]) 1); 
qed "neg_eq_less_0"; 

Goal "(~neg x) = (#0 <= x)";
by (simp_tac (simpset() addsimps [not_neg_eq_ge_int0]) 1); 
qed "not_neg_eq_ge_0"; 

Goal "#0 <= int m";
by (Simp_tac 1);
qed "zero_zle_int";
AddIffs [zero_zle_int];


(** Needed because (int 0) rewrites to #0.
    Can these be generalized without evaluating large numbers?**)

Goal "~ (int k < #0)";
by (Simp_tac 1);
qed "int_less_0_conv";

Goal "(int k <= #0) = (k=0)";
by (Simp_tac 1);
qed "int_le_0_conv";

Goal "(int k = #0) = (k=0)";
by (Simp_tac 1);
qed "int_eq_0_conv";

Goal "(#0 = int k) = (k=0)";
by Auto_tac;
qed "int_eq_0_conv'";

Addsimps [int_less_0_conv, int_le_0_conv, int_eq_0_conv, int_eq_0_conv'];


(** Simplification rules for comparison of binary numbers (Norbert Voelker) **)

(** Equals (=) **)

Goalw [iszero_def]
  "((number_of x::int) = number_of y) = \
\  iszero (number_of (bin_add x (bin_minus y)))"; 
by (simp_tac (simpset() addsimps
              (zcompare_rls @ [number_of_add, number_of_minus])) 1); 
qed "eq_number_of_eq"; 

Goalw [iszero_def] "iszero ((number_of Pls)::int)"; 
by (Simp_tac 1); 
qed "iszero_number_of_Pls"; 

Goalw [iszero_def] "~ iszero ((number_of Min)::int)"; 
by (Simp_tac 1);
qed "nonzero_number_of_Min"; 

Goalw [iszero_def]
     "iszero (number_of (w BIT x)) = (~x & iszero (number_of w::int))"; 
by (Simp_tac 1);
by (int_case_tac "number_of w" 1); 
by (ALLGOALS (asm_simp_tac 
	      (simpset() addsimps zcompare_rls @ 
				  [zminus_zadd_distrib RS sym, 
				   zadd_int]))); 
qed "iszero_number_of_BIT"; 

Goal "iszero (number_of (w BIT False)) = iszero (number_of w::int)"; 
by (simp_tac (HOL_ss addsimps [iszero_number_of_BIT]) 1); 
qed "iszero_number_of_0"; 

Goal "~ iszero (number_of (w BIT True)::int)"; 
by (simp_tac (HOL_ss addsimps [iszero_number_of_BIT]) 1); 
qed "iszero_number_of_1"; 



(** Less-than (<) **)

Goalw [zless_def,zdiff_def] 
    "(number_of x::int) < number_of y \
\    = neg (number_of (bin_add x (bin_minus y)))";
by (simp_tac (simpset() addsimps bin_mult_simps) 1);
qed "less_number_of_eq_neg"; 

Goal "~ neg (number_of Pls)"; 
by (Simp_tac 1); 
qed "not_neg_number_of_Pls"; 

Goal "neg (number_of Min)"; 
by (Simp_tac 1);
qed "neg_number_of_Min"; 

Goal "neg (number_of (w BIT x)) = neg (number_of w)"; 
by (Asm_simp_tac 1); 
by (int_case_tac "number_of w" 1); 
by (ALLGOALS (asm_simp_tac 
	      (simpset() addsimps [zadd_int, neg_eq_less_int0, 
				   symmetric zdiff_def] @ zcompare_rls))); 
qed "neg_number_of_BIT"; 


(** Less-than-or-equals (<=) **)

Goal "(number_of x <= (number_of y::int)) = \
\     (~ number_of y < (number_of x::int))";
by (rtac (linorder_not_less RS sym) 1);
qed "le_number_of_eq_not_less"; 

(*Delete the original rewrites, with their clumsy conditional expressions*)
Delsimps [bin_succ_BIT, bin_pred_BIT, bin_minus_BIT, 
          NCons_Pls, NCons_Min, bin_add_BIT, bin_mult_BIT];

(*Hide the binary representation of integer constants*)
Delsimps [number_of_Pls, number_of_Min, number_of_BIT];

(*Simplification of arithmetic operations on integer constants.
  Note that bin_pred_Pls, etc. were added to the simpset by primrec.*)

bind_thms ("NCons_simps", [NCons_Pls_0, NCons_Pls_1, NCons_Min_0, NCons_Min_1, NCons_BIT]);

bind_thms ("bin_arith_extra_simps",
    [number_of_add RS sym,
     number_of_minus RS sym,
     number_of_mult RS sym,
     bin_succ_1, bin_succ_0, 
     bin_pred_1, bin_pred_0, 
     bin_minus_1, bin_minus_0,  
     bin_add_Pls_right, bin_add_Min_right,
     bin_add_BIT_0, bin_add_BIT_10, bin_add_BIT_11,
     diff_number_of_eq, 
     bin_mult_1, bin_mult_0] @ NCons_simps);

(*For making a minimal simpset, one must include these default simprules
  of thy.  Also include simp_thms, or at least (~False)=True*)
bind_thms ("bin_arith_simps",
    [bin_pred_Pls, bin_pred_Min,
     bin_succ_Pls, bin_succ_Min,
     bin_add_Pls, bin_add_Min,
     bin_minus_Pls, bin_minus_Min,
     bin_mult_Pls, bin_mult_Min] @ bin_arith_extra_simps);

(*Simplification of relational operations*)
bind_thms ("bin_rel_simps",
    [eq_number_of_eq, iszero_number_of_Pls, nonzero_number_of_Min,
     iszero_number_of_0, iszero_number_of_1,
     less_number_of_eq_neg,
     not_neg_number_of_Pls, neg_number_of_Min, neg_number_of_BIT,
     le_number_of_eq_not_less]);

Addsimps bin_arith_extra_simps;
Addsimps bin_rel_simps;


(** Simplification of arithmetic when nested to the right **)

Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::int)";
by (simp_tac (simpset() addsimps [zadd_assoc RS sym]) 1);
qed "add_number_of_left";

Goal "number_of v * (number_of w * z) = (number_of(bin_mult v w) * z::int)";
by (simp_tac (simpset() addsimps [zmult_assoc RS sym]) 1);
qed "mult_number_of_left";

Goalw [zdiff_def]
    "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::int)";
by (rtac add_number_of_left 1);
qed "add_number_of_diff1";

Goal "number_of v + (c - number_of w) = \
\    number_of (bin_add v (bin_minus w)) + (c::int)";
by (stac (diff_number_of_eq RS sym) 1);
by Auto_tac;
qed "add_number_of_diff2";

Addsimps [add_number_of_left, mult_number_of_left,
	  add_number_of_diff1, add_number_of_diff2];