(*  Title:      HOL/Hyperreal/HyperBin.ML

    ID:         $Id$

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   2000  University of Cambridge

Binary arithmetic for the hypreals (integer literals only).

*)

(** hypreal_of_real (coercion from int to real) **)

Goal "hypreal_of_real (number_of w) = number_of w";

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

qed "hypreal_number_of";

Addsimps [hypreal_number_of];

Goalw [hypreal_number_of_def] "(0::hypreal) = Numeral0";

by (simp_tac (simpset() addsimps [hypreal_of_real_zero RS sym]) 1);

qed "zero_eq_numeral_0";

Goalw [hypreal_number_of_def] "1hr = Numeral1";

by (simp_tac (simpset() addsimps [hypreal_of_real_one RS sym]) 1);

qed "one_eq_numeral_1";

(** Addition **)

Goal "(number_of v :: hypreal) + number_of v' = number_of (bin_add v v')";

by (simp_tac

    (HOL_ss addsimps [hypreal_number_of_def, 

                      hypreal_of_real_add RS sym, add_real_number_of]) 1);

qed "add_hypreal_number_of";

Addsimps [add_hypreal_number_of];

(** Subtraction **)

Goalw [hypreal_number_of_def]

     "- (number_of w :: hypreal) = number_of (bin_minus w)";

by (simp_tac

    (HOL_ss addsimps [minus_real_number_of, hypreal_of_real_minus RS sym]) 1);

qed "minus_hypreal_number_of";

Addsimps [minus_hypreal_number_of];

Goalw [hypreal_number_of_def, hypreal_diff_def]

     "(number_of v :: hypreal) - number_of w = \

\     number_of (bin_add v (bin_minus w))";

by (Simp_tac 1); 

qed "diff_hypreal_number_of";

Addsimps [diff_hypreal_number_of];

(** Multiplication **)

Goal "(number_of v :: hypreal) * number_of v' = number_of (bin_mult v v')";

by (simp_tac

    (HOL_ss addsimps [hypreal_number_of_def, 

	              hypreal_of_real_mult RS sym, mult_real_number_of]) 1);

qed "mult_hypreal_number_of";

Addsimps [mult_hypreal_number_of];

Goal "(# 2::hypreal) = Numeral1 + Numeral1";

by (Simp_tac 1);

val lemma = result();

(*For specialist use: NOT as default simprules*)

Goal "# 2 * z = (z+z::hypreal)";

by (simp_tac (simpset ()

	      addsimps [lemma, hypreal_add_mult_distrib,

			one_eq_numeral_1 RS sym]) 1);

qed "hypreal_mult_2";

Goal "z * # 2 = (z+z::hypreal)";

by (stac hypreal_mult_commute 1 THEN rtac hypreal_mult_2 1);

qed "hypreal_mult_2_right";

(*** Comparisons ***)

(** Equals (=) **)

Goal "((number_of v :: hypreal) = number_of v') = \

\     iszero (number_of (bin_add v (bin_minus v')))";

by (simp_tac

    (HOL_ss addsimps [hypreal_number_of_def, 

	              hypreal_of_real_eq_iff, eq_real_number_of]) 1);

qed "eq_hypreal_number_of";

Addsimps [eq_hypreal_number_of];

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

(*"neg" is used in rewrite rules for binary comparisons*)

Goal "((number_of v :: hypreal) < number_of v') = \

\     neg (number_of (bin_add v (bin_minus v')))";

by (simp_tac

    (HOL_ss addsimps [hypreal_number_of_def, hypreal_of_real_less_iff, 

		      less_real_number_of]) 1);

qed "less_hypreal_number_of";

Addsimps [less_hypreal_number_of];

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

Goal "(number_of x <= (number_of y::hypreal)) = \

\     (~ number_of y < (number_of x::hypreal))";

by (rtac (linorder_not_less RS sym) 1);

qed "le_hypreal_number_of_eq_not_less"; 

Addsimps [le_hypreal_number_of_eq_not_less];

(*** New versions of existing theorems involving 0, 1hr ***)

Goal "- Numeral1 = (# -1::hypreal)";

by (Simp_tac 1);

qed "minus_numeral_one";

(*Maps 0 to Numeral0 and 1hr to Numeral1 and -1hr to # -1*)

val hypreal_numeral_ss = 

    real_numeral_ss addsimps [zero_eq_numeral_0, one_eq_numeral_1, 

		              minus_numeral_one];

fun rename_numerals th = 

    asm_full_simplify hypreal_numeral_ss (Thm.transfer (the_context ()) th);

(*Now insert some identities previously stated for 0 and 1hr*)

(** HyperDef **)

Addsimps (map rename_numerals

	  [hypreal_minus_zero, hypreal_minus_zero_iff,

	   hypreal_add_zero_left, hypreal_add_zero_right, 

	   hypreal_diff_zero, hypreal_diff_zero_right,

	   hypreal_mult_0_right, hypreal_mult_0, 

           hypreal_mult_1_right, hypreal_mult_1,

	   hypreal_inverse_1, hypreal_minus_zero_less_iff]);

bind_thm ("hypreal_0_less_mult_iff", 

	  rename_numerals hypreal_zero_less_mult_iff);

bind_thm ("hypreal_0_le_mult_iff", 

	  rename_numerals hypreal_zero_le_mult_iff);

bind_thm ("hypreal_mult_less_0_iff", 

	  rename_numerals hypreal_mult_less_zero_iff);

bind_thm ("hypreal_mult_le_0_iff", 

	  rename_numerals hypreal_mult_le_zero_iff);

bind_thm ("hypreal_inverse_less_0", rename_numerals hypreal_inverse_less_zero);

bind_thm ("hypreal_inverse_gt_0", rename_numerals hypreal_inverse_gt_zero);

Addsimps [zero_eq_numeral_0,one_eq_numeral_1];

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

Goal "number_of v + (number_of w + z) = (number_of(bin_add v w) + z::hypreal)";

by Auto_tac; 

qed "hypreal_add_number_of_left";

Goal "number_of v *(number_of w * z) = (number_of(bin_mult v w) * z::hypreal)";

by (simp_tac (simpset() addsimps [hypreal_mult_assoc RS sym]) 1);

qed "hypreal_mult_number_of_left";

Goalw [hypreal_diff_def]

    "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::hypreal)";

by (rtac hypreal_add_number_of_left 1);

qed "hypreal_add_number_of_diff1";

Goal "number_of v + (c - number_of w) = \

\     number_of (bin_add v (bin_minus w)) + (c::hypreal)";

by (stac (diff_hypreal_number_of RS sym) 1);

by Auto_tac;

qed "hypreal_add_number_of_diff2";

Addsimps [hypreal_add_number_of_left, hypreal_mult_number_of_left,

	  hypreal_add_number_of_diff1, hypreal_add_number_of_diff2]; 

(**** Simprocs for numeric literals ****)

(** Combining of literal coefficients in sums of products **)

Goal "(x < y) = (x-y < (Numeral0::hypreal))";

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

qed "hypreal_less_iff_diff_less_0";

Goal "(x = y) = (x-y = (Numeral0::hypreal))";

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

qed "hypreal_eq_iff_diff_eq_0";

Goal "(x <= y) = (x-y <= (Numeral0::hypreal))";

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

qed "hypreal_le_iff_diff_le_0";

(** For combine_numerals **)

Goal "i*u + (j*u + k) = (i+j)*u + (k::hypreal)";

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

qed "left_hypreal_add_mult_distrib";

(** For cancel_numerals **)

val rel_iff_rel_0_rls = 

    map (inst "y" "?u+?v")

      [hypreal_less_iff_diff_less_0, hypreal_eq_iff_diff_eq_0, 

       hypreal_le_iff_diff_le_0] @

    map (inst "y" "n")

      [hypreal_less_iff_diff_less_0, hypreal_eq_iff_diff_eq_0, 

       hypreal_le_iff_diff_le_0];

Goal "!!i::hypreal. (i*u + m = j*u + n) = ((i-j)*u + m = n)";

by (asm_simp_tac

    (simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@

	                 hypreal_add_ac@rel_iff_rel_0_rls) 1);

qed "hypreal_eq_add_iff1";

Goal "!!i::hypreal. (i*u + m = j*u + n) = (m = (j-i)*u + n)";

by (asm_simp_tac

    (simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@

                         hypreal_add_ac@rel_iff_rel_0_rls) 1);

qed "hypreal_eq_add_iff2";

Goal "!!i::hypreal. (i*u + m < j*u + n) = ((i-j)*u + m < n)";

by (asm_simp_tac

    (simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@

                         hypreal_add_ac@rel_iff_rel_0_rls) 1);

qed "hypreal_less_add_iff1";

Goal "!!i::hypreal. (i*u + m < j*u + n) = (m < (j-i)*u + n)";

by (asm_simp_tac

    (simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@

                         hypreal_add_ac@rel_iff_rel_0_rls) 1);

qed "hypreal_less_add_iff2";

Goal "!!i::hypreal. (i*u + m <= j*u + n) = ((i-j)*u + m <= n)";

by (asm_simp_tac

    (simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@

                         hypreal_add_ac@rel_iff_rel_0_rls) 1);

qed "hypreal_le_add_iff1";

Goal "!!i::hypreal. (i*u + m <= j*u + n) = (m <= (j-i)*u + n)";

by (asm_simp_tac

    (simpset() addsimps [hypreal_diff_def, hypreal_add_mult_distrib]@

                        hypreal_add_ac@rel_iff_rel_0_rls) 1);

qed "hypreal_le_add_iff2";

Goal "(z::hypreal) * # -1 = -z";

by (stac (minus_numeral_one RS sym) 1);

by (stac (hypreal_minus_mult_eq2 RS sym) 1); 

by Auto_tac;  

qed "hypreal_mult_minus_1_right";

Addsimps [hypreal_mult_minus_1_right];

Goal "# -1 * (z::hypreal) = -z";

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

qed "hypreal_mult_minus_1";

Addsimps [hypreal_mult_minus_1];

structure Hyperreal_Numeral_Simprocs =

struct

(*Utilities*)

val hyprealT = Type("HyperDef.hypreal",[]);

fun mk_numeral n = HOLogic.number_of_const hyprealT $ HOLogic.mk_bin n;

val dest_numeral = Real_Numeral_Simprocs.dest_numeral;

val find_first_numeral = Real_Numeral_Simprocs.find_first_numeral;

val zero = mk_numeral 0;

val mk_plus = HOLogic.mk_binop "op +";

val uminus_const = Const ("uminus", hyprealT --> hyprealT);

(*Thus mk_sum[t] yields t+Numeral0; longer sums don't have a trailing zero*)

fun mk_sum []        = zero

  | mk_sum [t,u]     = mk_plus (t, u)

  | mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

(*this version ALWAYS includes a trailing zero*)

fun long_mk_sum []        = zero

  | long_mk_sum (t :: ts) = mk_plus (t, mk_sum ts);

val dest_plus = HOLogic.dest_bin "op +" hyprealT;

(*decompose additions AND subtractions as a sum*)

fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =

        dest_summing (pos, t, dest_summing (pos, u, ts))

  | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =

        dest_summing (pos, t, dest_summing (not pos, u, ts))

  | dest_summing (pos, t, ts) =

	if pos then t::ts else uminus_const$t :: ts;

fun dest_sum t = dest_summing (true, t, []);

val mk_diff = HOLogic.mk_binop "op -";

val dest_diff = HOLogic.dest_bin "op -" hyprealT;

val one = mk_numeral 1;

val mk_times = HOLogic.mk_binop "op *";

fun mk_prod [] = one

  | mk_prod [t] = t

  | mk_prod (t :: ts) = if t = one then mk_prod ts

                        else mk_times (t, mk_prod ts);

val dest_times = HOLogic.dest_bin "op *" hyprealT;

fun dest_prod t =

      let val (t,u) = dest_times t 

      in  dest_prod t @ dest_prod u  end

      handle TERM _ => [t];

(*DON'T do the obvious simplifications; that would create special cases*) 

fun mk_coeff (k, ts) = mk_times (mk_numeral k, ts);

(*Express t as a product of (possibly) a numeral with other sorted terms*)

fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t

  | dest_coeff sign t =

    let val ts = sort Term.term_ord (dest_prod t)

	val (n, ts') = find_first_numeral [] ts

                          handle TERM _ => (1, ts)

    in (sign*n, mk_prod ts') end;

(*Find first coefficient-term THAT MATCHES u*)

fun find_first_coeff past u [] = raise TERM("find_first_coeff", []) 

  | find_first_coeff past u (t::terms) =

	let val (n,u') = dest_coeff 1 t

	in  if u aconv u' then (n, rev past @ terms)

			  else find_first_coeff (t::past) u terms

	end

	handle TERM _ => find_first_coeff (t::past) u terms;

(*Simplify Numeral1*n and n*Numeral1 to n*)

val add_0s = map rename_numerals

                 [hypreal_add_zero_left, hypreal_add_zero_right];

val mult_plus_1s = map rename_numerals

                      [hypreal_mult_1, hypreal_mult_1_right];

val mult_minus_1s = map rename_numerals

                      [hypreal_mult_minus_1, hypreal_mult_minus_1_right];

val mult_1s = mult_plus_1s @ mult_minus_1s;

(*To perform binary arithmetic*)

val bin_simps =

    [add_hypreal_number_of, hypreal_add_number_of_left, 

     minus_hypreal_number_of, diff_hypreal_number_of, mult_hypreal_number_of, 

     hypreal_mult_number_of_left] @ bin_arith_simps @ bin_rel_simps;

(*To evaluate binary negations of coefficients*)

val hypreal_minus_simps = NCons_simps @

                   [minus_hypreal_number_of, 

		    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,

		    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];

(*To let us treat subtraction as addition*)

val diff_simps = [hypreal_diff_def, hypreal_minus_add_distrib, 

                  hypreal_minus_minus];

(*push the unary minus down: - x * y = x * - y *)

val hypreal_minus_mult_eq_1_to_2 = 

    [hypreal_minus_mult_eq1 RS sym, hypreal_minus_mult_eq2] MRS trans 

    |> standard;

(*to extract again any uncancelled minuses*)

val hypreal_minus_from_mult_simps = 

    [hypreal_minus_minus, hypreal_minus_mult_eq1 RS sym, 

     hypreal_minus_mult_eq2 RS sym];

(*combine unary minus with numeric literals, however nested within a product*)

val hypreal_mult_minus_simps =

    [hypreal_mult_assoc, hypreal_minus_mult_eq1, hypreal_minus_mult_eq_1_to_2];

(*Apply the given rewrite (if present) just once*)

fun trans_tac None      = all_tac

  | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));

fun prove_conv name tacs sg (hyps: thm list) (t,u) =

  if t aconv u then None

  else

  let val ct = cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u)))

  in Some

     (prove_goalw_cterm [] ct (K tacs)

      handle ERROR => error 

	  ("The error(s) above occurred while trying to prove " ^

	   string_of_cterm ct ^ "\nInternal failure of simproc " ^ name))

  end;

(*version without the hyps argument*)

fun prove_conv_nohyps name tacs sg = prove_conv name tacs sg [];

(*Final simplification: cancel + and *  *)

val simplify_meta_eq = 

    Int_Numeral_Simprocs.simplify_meta_eq

         [hypreal_add_zero_left, hypreal_add_zero_right,

 	  hypreal_mult_0, hypreal_mult_0_right, hypreal_mult_1, 

          hypreal_mult_1_right];

val prep_simproc = Real_Numeral_Simprocs.prep_simproc;

val prep_pats = map Real_Numeral_Simprocs.prep_pat;

structure CancelNumeralsCommon =

  struct

  val mk_sum    	= mk_sum

  val dest_sum		= dest_sum

  val mk_coeff		= mk_coeff

  val dest_coeff	= dest_coeff 1

  val find_first_coeff	= find_first_coeff []

  val trans_tac         = trans_tac

  val norm_tac = 

     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@

                                         hypreal_minus_simps@hypreal_add_ac))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hypreal_mult_minus_simps))

     THEN ALLGOALS

              (simp_tac (HOL_ss addsimps hypreal_minus_from_mult_simps@

                                         hypreal_add_ac@hypreal_mult_ac))

  val numeral_simp_tac	= ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))

  val simplify_meta_eq  = simplify_meta_eq

  end;

structure EqCancelNumerals = CancelNumeralsFun

 (open CancelNumeralsCommon

  val prove_conv = prove_conv "hyprealeq_cancel_numerals"

  val mk_bal   = HOLogic.mk_eq

  val dest_bal = HOLogic.dest_bin "op =" hyprealT

  val bal_add1 = hypreal_eq_add_iff1 RS trans

  val bal_add2 = hypreal_eq_add_iff2 RS trans

);

structure LessCancelNumerals = CancelNumeralsFun

 (open CancelNumeralsCommon

  val prove_conv = prove_conv "hyprealless_cancel_numerals"

  val mk_bal   = HOLogic.mk_binrel "op <"

  val dest_bal = HOLogic.dest_bin "op <" hyprealT

  val bal_add1 = hypreal_less_add_iff1 RS trans

  val bal_add2 = hypreal_less_add_iff2 RS trans

);

structure LeCancelNumerals = CancelNumeralsFun

 (open CancelNumeralsCommon

  val prove_conv = prove_conv "hyprealle_cancel_numerals"

  val mk_bal   = HOLogic.mk_binrel "op <="

  val dest_bal = HOLogic.dest_bin "op <=" hyprealT

  val bal_add1 = hypreal_le_add_iff1 RS trans

  val bal_add2 = hypreal_le_add_iff2 RS trans

);

val cancel_numerals = 

  map prep_simproc

   [("hyprealeq_cancel_numerals",

     prep_pats ["(l::hypreal) + m = n", "(l::hypreal) = m + n", 

		"(l::hypreal) - m = n", "(l::hypreal) = m - n", 

		"(l::hypreal) * m = n", "(l::hypreal) = m * n"], 

     EqCancelNumerals.proc),

    ("hyprealless_cancel_numerals", 

     prep_pats ["(l::hypreal) + m < n", "(l::hypreal) < m + n", 

		"(l::hypreal) - m < n", "(l::hypreal) < m - n", 

		"(l::hypreal) * m < n", "(l::hypreal) < m * n"], 

     LessCancelNumerals.proc),

    ("hyprealle_cancel_numerals", 

     prep_pats ["(l::hypreal) + m <= n", "(l::hypreal) <= m + n", 

		"(l::hypreal) - m <= n", "(l::hypreal) <= m - n", 

		"(l::hypreal) * m <= n", "(l::hypreal) <= m * n"], 

     LeCancelNumerals.proc)];

structure CombineNumeralsData =

  struct

  val add		= op + : int*int -> int 

  val mk_sum    	= long_mk_sum    (*to work for e.g. # 2*x + # 3*x *)

  val dest_sum		= dest_sum

  val mk_coeff		= mk_coeff

  val dest_coeff	= dest_coeff 1

  val left_distrib	= left_hypreal_add_mult_distrib RS trans

  val prove_conv	= prove_conv_nohyps "hypreal_combine_numerals"

  val trans_tac          = trans_tac

  val norm_tac = 

     ALLGOALS (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@

                                         hypreal_minus_simps@hypreal_add_ac))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps bin_simps@hypreal_mult_minus_simps))

     THEN ALLGOALS (simp_tac (HOL_ss addsimps hypreal_minus_from_mult_simps@

                                              hypreal_add_ac@hypreal_mult_ac))

  val numeral_simp_tac	= ALLGOALS 

                    (simp_tac (HOL_ss addsimps add_0s@bin_simps))

  val simplify_meta_eq  = simplify_meta_eq

  end;

structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);

val combine_numerals = 

    prep_simproc ("hypreal_combine_numerals",

		  prep_pats ["(i::hypreal) + j", "(i::hypreal) - j"],

		  CombineNumerals.proc);

(** Declarations for ExtractCommonTerm **)

(*this version ALWAYS includes a trailing one*)

fun long_mk_prod []        = one

  | long_mk_prod (t :: ts) = mk_times (t, mk_prod ts);

(*Find first term that matches u*)

fun find_first past u []         = raise TERM("find_first", []) 

  | find_first past u (t::terms) =

	if u aconv t then (rev past @ terms)

        else find_first (t::past) u terms

	handle TERM _ => find_first (t::past) u terms;

(*Final simplification: cancel + and *  *)

fun cancel_simplify_meta_eq cancel_th th = 

    Int_Numeral_Simprocs.simplify_meta_eq 

        [hypreal_mult_1, hypreal_mult_1_right] 

        (([th, cancel_th]) MRS trans);

end;

Addsimprocs Hyperreal_Numeral_Simprocs.cancel_numerals;

Addsimprocs [Hyperreal_Numeral_Simprocs.combine_numerals];

(*The Abel_Cancel simprocs are now obsolete*)

Delsimprocs [Hyperreal_Cancel.sum_conv, Hyperreal_Cancel.rel_conv];

(*examples:

print_depth 22;

set timing;

set trace_simp;

fun test s = (Goal s, by (Simp_tac 1)); 

test "l + # 2 + # 2 + # 2 + (l + # 2) + (oo + # 2) = (uu::hypreal)";

test "# 2*u = (u::hypreal)";

test "(i + j + # 12 + (k::hypreal)) - # 15 = y";

test "(i + j + # 12 + (k::hypreal)) - # 5 = y";

test "y - b < (b::hypreal)";

test "y - (# 3*b + c) < (b::hypreal) - # 2*c";

test "(# 2*x - (u*v) + y) - v*# 3*u = (w::hypreal)";

test "(# 2*x*u*v + (u*v)*# 4 + y) - v*u*# 4 = (w::hypreal)";

test "(# 2*x*u*v + (u*v)*# 4 + y) - v*u = (w::hypreal)";

test "u*v - (x*u*v + (u*v)*# 4 + y) = (w::hypreal)";

test "(i + j + # 12 + (k::hypreal)) = u + # 15 + y";

test "(i + j*# 2 + # 12 + (k::hypreal)) = j + # 5 + y";

test "# 2*y + # 3*z + # 6*w + # 2*y + # 3*z + # 2*u = # 2*y' + # 3*z' + # 6*w' + # 2*y' + # 3*z' + u + (vv::hypreal)";

test "a + -(b+c) + b = (d::hypreal)";

test "a + -(b+c) - b = (d::hypreal)";

(*negative numerals*)

test "(i + j + # -2 + (k::hypreal)) - (u + # 5 + y) = zz";

test "(i + j + # -3 + (k::hypreal)) < u + # 5 + y";

test "(i + j + # 3 + (k::hypreal)) < u + # -6 + y";

test "(i + j + # -12 + (k::hypreal)) - # 15 = y";

test "(i + j + # 12 + (k::hypreal)) - # -15 = y";

test "(i + j + # -12 + (k::hypreal)) - # -15 = y";

*)

(** Constant folding for hypreal plus and times **)

(*We do not need

    structure Hyperreal_Plus_Assoc = Assoc_Fold (Hyperreal_Plus_Assoc_Data);

  because combine_numerals does the same thing*)

structure Hyperreal_Times_Assoc_Data : ASSOC_FOLD_DATA =

struct

  val ss		= HOL_ss

  val eq_reflection	= eq_reflection

  val sg_ref    = Sign.self_ref (Theory.sign_of (the_context ()))

  val T	     = Hyperreal_Numeral_Simprocs.hyprealT

  val plus   = Const ("op *", [T,T] ---> T)

  val add_ac = hypreal_mult_ac

end;

structure Hyperreal_Times_Assoc = Assoc_Fold (Hyperreal_Times_Assoc_Data);

Addsimprocs [Hyperreal_Times_Assoc.conv];

Addsimps [rename_numerals hypreal_of_real_zero_iff];

(*Simplification of  x-y < 0, etc.*)

AddIffs [hypreal_less_iff_diff_less_0 RS sym];

AddIffs [hypreal_eq_iff_diff_eq_0 RS sym];

AddIffs [hypreal_le_iff_diff_le_0 RS sym];

(** number_of related to hypreal_of_real **)

Goal "(number_of w < hypreal_of_real z) = (number_of w < z)";

by (stac (hypreal_of_real_less_iff RS sym) 1); 

by (Simp_tac 1); 

qed "number_of_less_hypreal_of_real_iff";

Addsimps [number_of_less_hypreal_of_real_iff];

Goal "(number_of w <= hypreal_of_real z) = (number_of w <= z)";

by (stac (hypreal_of_real_le_iff RS sym) 1); 

by (Simp_tac 1); 

qed "number_of_le_hypreal_of_real_iff";

Addsimps [number_of_le_hypreal_of_real_iff];

Goal "(hypreal_of_real z < number_of w) = (z < number_of w)";

by (stac (hypreal_of_real_less_iff RS sym) 1); 

by (Simp_tac 1); 

qed "hypreal_of_real_less_number_of_iff";

Addsimps [hypreal_of_real_less_number_of_iff];

Goal "(hypreal_of_real z <= number_of w) = (z <= number_of w)";

by (stac (hypreal_of_real_le_iff RS sym) 1); 

by (Simp_tac 1); 

qed "hypreal_of_real_le_number_of_iff";

Addsimps [hypreal_of_real_le_number_of_iff];

(** <= monotonicity results: needed for arithmetic **)

Goal "[| i <= j;  (0::hypreal) <= k |] ==> i*k <= j*k";

by (auto_tac (claset(), 

              simpset() addsimps [order_le_less, hypreal_mult_less_mono1]));  

qed "hypreal_mult_le_mono1";

Goal "[| i <= j;  (0::hypreal) <= k |] ==> k*i <= k*j";

by (dtac hypreal_mult_le_mono1 1);

by (ALLGOALS (asm_full_simp_tac (simpset() addsimps [hypreal_mult_commute])));

qed "hypreal_mult_le_mono2";

Goal "[| u <= v;  x <= y;  0 <= v;  (0::hypreal) <= x |] ==> u * x <= v * y";

by (etac (hypreal_mult_le_mono1 RS order_trans) 1);

by (assume_tac 1);

by (etac hypreal_mult_le_mono2 1);

by (assume_tac 1);

qed "hypreal_mult_le_mono";