src/HOL/Real/RealBin.ML

fixed deps;

(*  Title:      HOL/Real/RealBin.ML
    ID:         $Id$
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1999  University of Cambridge

Binary arithmetic for the reals (integer literals only).
*)

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

Goal "real_of_int (number_of w) = number_of w";
by (simp_tac (simpset() addsimps [real_number_of_def]) 1);
qed "real_number_of";
Addsimps [real_number_of];

Goalw [real_number_of_def] "(0::real) = #0";
by (simp_tac (simpset() addsimps [real_of_int_zero RS sym]) 1);
qed "zero_eq_numeral_0";

Goalw [real_number_of_def] "1r = #1";
by (simp_tac (simpset() addsimps [real_of_int_one RS sym]) 1);
qed "one_eq_numeral_1";

(** Addition **)

Goal "(number_of v :: real) + number_of v' = number_of (bin_add v v')";
by (simp_tac
    (HOL_ss addsimps [real_number_of_def, 
				  real_of_int_add, number_of_add]) 1);
qed "add_real_number_of";

Addsimps [add_real_number_of];


(** Subtraction **)

Goalw [real_number_of_def] "- (number_of w :: real) = number_of (bin_minus w)";
by (simp_tac
    (HOL_ss addsimps [number_of_minus, real_of_int_minus]) 1);
qed "minus_real_number_of";

Goalw [real_number_of_def]
   "(number_of v :: real) - number_of w = number_of (bin_add v (bin_minus w))";
by (simp_tac
    (HOL_ss addsimps [diff_number_of_eq, real_of_int_diff]) 1);
qed "diff_real_number_of";

Addsimps [minus_real_number_of, diff_real_number_of];


(** Multiplication **)

Goal "(number_of v :: real) * number_of v' = number_of (bin_mult v v')";
by (simp_tac
    (HOL_ss addsimps [real_number_of_def, 
				  real_of_int_mult, number_of_mult]) 1);
qed "mult_real_number_of";

Addsimps [mult_real_number_of];

Goal "(#2::real) = #1 + #1";
by (Simp_tac 1);
val lemma = result();

(*For specialist use: NOT as default simprules*)
Goal "#2 * z = (z+z::real)";
by (simp_tac (simpset ()
	      addsimps [lemma, real_add_mult_distrib,
			one_eq_numeral_1 RS sym]) 1);
qed "real_mult_2";

Goal "z * #2 = (z+z::real)";
by (stac real_mult_commute 1 THEN rtac real_mult_2 1);
qed "real_mult_2_right";


(*** Comparisons ***)

(** Equals (=) **)

Goal "((number_of v :: real) = number_of v') = \
\     iszero (number_of (bin_add v (bin_minus v')))";
by (simp_tac
    (HOL_ss addsimps [real_number_of_def, 
				  real_of_int_eq_iff, eq_number_of_eq]) 1);
qed "eq_real_number_of";

Addsimps [eq_real_number_of];

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

(*"neg" is used in rewrite rules for binary comparisons*)
Goal "((number_of v :: real) < number_of v') = \
\     neg (number_of (bin_add v (bin_minus v')))";
by (simp_tac
    (HOL_ss addsimps [real_number_of_def, real_of_int_less_iff, 
				  less_number_of_eq_neg]) 1);
qed "less_real_number_of";

Addsimps [less_real_number_of];


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

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

Addsimps [le_real_number_of_eq_not_less];

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

Goal "- #1 = (#-1::real)";
by (Simp_tac 1);
qed "minus_numeral_one";


(*Maps 0 to #0 and 1r to #1 and -1r to #-1*)
val real_numeral_ss = 
    HOL_ss addsimps [zero_eq_numeral_0, one_eq_numeral_1, 
		     minus_numeral_one];

fun rename_numerals th = 
    asm_full_simplify real_numeral_ss (Thm.transfer (the_context ()) th);


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

(** RealDef & Real **)

Addsimps (map rename_numerals
	  [real_minus_zero, real_minus_zero_iff,
	   real_add_zero_left, real_add_zero_right, 
	   real_diff_0, real_diff_0_right,
	   real_mult_0_right, real_mult_0, real_mult_1_right, real_mult_1,
	   real_mult_minus_1_right, real_mult_minus_1, real_rinv_1,
	   real_minus_zero_less_iff]);

AddIffs (map rename_numerals [real_mult_is_0, real_0_is_mult]);

bind_thm ("real_0_less_mult_iff", 
	  rename_numerals real_zero_less_mult_iff);
bind_thm ("real_0_le_mult_iff", 
	  rename_numerals real_zero_le_mult_iff);
bind_thm ("real_mult_less_0_iff", 
	  rename_numerals real_mult_less_zero_iff);
bind_thm ("real_mult_le_0_iff", 
	  rename_numerals real_mult_le_zero_iff);


(*Perhaps add some theorems that aren't in the default simpset, as
  done in Integ/NatBin.ML*)


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::real)";
by Auto_tac; 
qed "real_add_number_of_left";

Goal "number_of v * (number_of w * z) = (number_of(bin_mult v w) * z::real)";
by (simp_tac (simpset() addsimps [real_mult_assoc RS sym]) 1);
qed "real_mult_number_of_left";

Goalw [real_diff_def]
    "number_of v + (number_of w - c) = number_of(bin_add v w) - (c::real)";
by (rtac real_add_number_of_left 1);
qed "real_add_number_of_diff1";

Goal "number_of v + (c - number_of w) = \
\    number_of (bin_add v (bin_minus w)) + (c::real)";
by (stac (diff_real_number_of RS sym) 1);
by Auto_tac;
qed "real_add_number_of_diff2";

Addsimps [real_add_number_of_left, real_mult_number_of_left,
	  real_add_number_of_diff1, real_add_number_of_diff2]; 


(*"neg" is used in rewrite rules for binary comparisons*)
Goal "real_of_nat (number_of v :: nat) = \
\        (if neg (number_of v) then #0 \
\         else (number_of v :: real))";
by (simp_tac
    (HOL_ss addsimps [nat_number_of_def, real_of_nat_real_of_int,
				  real_of_nat_neg_int, real_number_of,
				  zero_eq_numeral_0]) 1);
qed "real_of_nat_number_of";
Addsimps [real_of_nat_number_of];


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

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

Goal "(x < y) = (x-y < (#0::real))";
by (simp_tac (simpset() addsimps [real_diff_less_eq]) 1);   
qed "real_less_iff_diff_less_0";

Goal "(x = y) = (x-y = (#0::real))";
by (simp_tac (simpset() addsimps [real_diff_eq_eq]) 1);   
qed "real_eq_iff_diff_eq_0";

Goal "(x <= y) = (x-y <= (#0::real))";
by (simp_tac (simpset() addsimps [real_diff_le_eq]) 1);   
qed "real_le_iff_diff_le_0";


(** For combine_numerals **)

Goal "i*u + (j*u + k) = (i+j)*u + (k::real)";
by (asm_simp_tac (simpset() addsimps [real_add_mult_distrib]) 1);
qed "left_real_add_mult_distrib";


(** For cancel_numerals **)

val rel_iff_rel_0_rls = map (inst "y" "?u+?v")
                          [real_less_iff_diff_less_0, real_eq_iff_diff_eq_0, 
			   real_le_iff_diff_le_0] @
		        map (inst "y" "n")
                          [real_less_iff_diff_less_0, real_eq_iff_diff_eq_0, 
			   real_le_iff_diff_le_0];

Goal "!!i::real. (i*u + m = j*u + n) = ((i-j)*u + m = n)";
by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
		                     real_add_ac@rel_iff_rel_0_rls) 1);
qed "real_eq_add_iff1";

Goal "!!i::real. (i*u + m = j*u + n) = (m = (j-i)*u + n)";
by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
                                     real_add_ac@rel_iff_rel_0_rls) 1);
qed "real_eq_add_iff2";

Goal "!!i::real. (i*u + m < j*u + n) = ((i-j)*u + m < n)";
by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
                                     real_add_ac@rel_iff_rel_0_rls) 1);
qed "real_less_add_iff1";

Goal "!!i::real. (i*u + m < j*u + n) = (m < (j-i)*u + n)";
by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
                                     real_add_ac@rel_iff_rel_0_rls) 1);
qed "real_less_add_iff2";

Goal "!!i::real. (i*u + m <= j*u + n) = ((i-j)*u + m <= n)";
by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]@
                                     real_add_ac@rel_iff_rel_0_rls) 1);
qed "real_le_add_iff1";

Goal "!!i::real. (i*u + m <= j*u + n) = (m <= (j-i)*u + n)";
by (asm_simp_tac (simpset() addsimps [real_diff_def, real_add_mult_distrib]
                                     @real_add_ac@rel_iff_rel_0_rls) 1);
qed "real_le_add_iff2";


structure Real_Numeral_Simprocs =
struct

(*Utilities*)

fun mk_numeral n = HOLogic.number_of_const HOLogic.realT $ 
                   NumeralSyntax.mk_bin n;

(*Decodes a binary real constant*)
fun dest_numeral (Const("Numeral.number_of", _) $ w) = 
     (NumeralSyntax.dest_bin w
      handle Match => raise TERM("Real_Numeral_Simprocs.dest_numeral:1", [w]))
  | dest_numeral t = raise TERM("Real_Numeral_Simprocs.dest_numeral:2", [t]);

fun find_first_numeral past (t::terms) =
	((dest_numeral t, rev past @ terms)
	 handle TERM _ => find_first_numeral (t::past) terms)
  | find_first_numeral past [] = raise TERM("find_first_numeral", []);

val zero = mk_numeral 0;
val mk_plus = HOLogic.mk_binop "op +";

val uminus_const = Const ("uminus", HOLogic.realT --> HOLogic.realT);

(*Thus mk_sum[t] yields t+#0; 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 +" HOLogic.realT;

(*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 -" HOLogic.realT;

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 *" HOLogic.realT;

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 #1*n and n*#1 to n*)
val add_0s = map rename_numerals
                 [real_add_zero_left, real_add_zero_right];
val mult_1s = map rename_numerals
                  [real_mult_1, real_mult_1_right, 
		   real_mult_minus_1, real_mult_minus_1_right];

(*To perform binary arithmetic*)
val bin_simps =
    [add_real_number_of, real_add_number_of_left, minus_real_number_of, 
     diff_real_number_of] @ 
    bin_arith_simps @ bin_rel_simps;

(*To evaluate binary negations of coefficients*)
val real_minus_simps = NCons_simps @
                   [minus_real_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 = [real_diff_def, real_minus_add_distrib, real_minus_minus];

(*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
         [real_add_zero_left, real_add_zero_right,
 	  real_mult_0, real_mult_0_right, real_mult_1, real_mult_1_right];

fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc;
fun prep_pat s = Thm.read_cterm (Theory.sign_of (the_context ())) (s, HOLogic.termT);
val prep_pats = map 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@
                                                real_minus_simps@real_add_ac))
                 THEN ALLGOALS
                    (simp_tac (HOL_ss addsimps [real_minus_mult_eq2]@
                                         bin_simps@real_add_ac@real_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 "realeq_cancel_numerals"
  val mk_bal   = HOLogic.mk_eq
  val dest_bal = HOLogic.dest_bin "op =" HOLogic.realT
  val bal_add1 = real_eq_add_iff1 RS trans
  val bal_add2 = real_eq_add_iff2 RS trans
);

structure LessCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val prove_conv = prove_conv "realless_cancel_numerals"
  val mk_bal   = HOLogic.mk_binrel "op <"
  val dest_bal = HOLogic.dest_bin "op <" HOLogic.realT
  val bal_add1 = real_less_add_iff1 RS trans
  val bal_add2 = real_less_add_iff2 RS trans
);

structure LeCancelNumerals = CancelNumeralsFun
 (open CancelNumeralsCommon
  val prove_conv = prove_conv "realle_cancel_numerals"
  val mk_bal   = HOLogic.mk_binrel "op <="
  val dest_bal = HOLogic.dest_bin "op <=" HOLogic.realT
  val bal_add1 = real_le_add_iff1 RS trans
  val bal_add2 = real_le_add_iff2 RS trans
);

val cancel_numerals = 
  map prep_simproc
   [("realeq_cancel_numerals",
     prep_pats ["(l::real) + m = n", "(l::real) = m + n", 
		"(l::real) - m = n", "(l::real) = m - n", 
		"(l::real) * m = n", "(l::real) = m * n"], 
     EqCancelNumerals.proc),
    ("realless_cancel_numerals", 
     prep_pats ["(l::real) + m < n", "(l::real) < m + n", 
		"(l::real) - m < n", "(l::real) < m - n", 
		"(l::real) * m < n", "(l::real) < m * n"], 
     LessCancelNumerals.proc),
    ("realle_cancel_numerals", 
     prep_pats ["(l::real) + m <= n", "(l::real) <= m + n", 
		"(l::real) - m <= n", "(l::real) <= m - n", 
		"(l::real) * m <= n", "(l::real) <= 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_real_add_mult_distrib RS trans
  val prove_conv	= prove_conv_nohyps "real_combine_numerals"
  val trans_tac          = trans_tac
  val norm_tac = ALLGOALS
                   (simp_tac (HOL_ss addsimps add_0s@mult_1s@diff_simps@
                                              real_minus_simps@real_add_ac))
                 THEN ALLGOALS
                    (simp_tac (HOL_ss addsimps [real_minus_mult_eq2]@
                                               bin_simps@real_add_ac@real_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 ("real_combine_numerals",
		  prep_pats ["(i::real) + j", "(i::real) - j"],
		  CombineNumerals.proc);

end;


Addsimprocs Real_Numeral_Simprocs.cancel_numerals;
Addsimprocs [Real_Numeral_Simprocs.combine_numerals];

(*The Abel_Cancel simprocs are now obsolete*)
Delsimprocs [Real_Cancel.sum_conv, Real_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::real)";

test "#2*u = (u::real)";
test "(i + j + #12 + (k::real)) - #15 = y";
test "(i + j + #12 + (k::real)) - #5 = y";

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

test "(#2*x - (u*v) + y) - v*#3*u = (w::real)";
test "(#2*x*u*v + (u*v)*#4 + y) - v*u*#4 = (w::real)";
test "(#2*x*u*v + (u*v)*#4 + y) - v*u = (w::real)";
test "u*v - (x*u*v + (u*v)*#4 + y) = (w::real)";

test "(i + j + #12 + (k::real)) = u + #15 + y";
test "(i + j*#2 + #12 + (k::real)) = 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::real)";

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

(*negative numerals*)
test "(i + j + #-2 + (k::real)) - (u + #5 + y) = zz";
test "(i + j + #-3 + (k::real)) < u + #5 + y";
test "(i + j + #3 + (k::real)) < u + #-6 + y";
test "(i + j + #-12 + (k::real)) - #15 = y";
test "(i + j + #12 + (k::real)) - #-15 = y";
test "(i + j + #-12 + (k::real)) - #-15 = y";
*)


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

(*We do not need
    structure Real_Plus_Assoc = Assoc_Fold (Real_Plus_Assoc_Data);
  because combine_numerals does the same thing*)

structure Real_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	     = HOLogic.realT
  val plus   = Const ("op *", [HOLogic.realT,HOLogic.realT] ---> HOLogic.realT)
  val add_ac = real_mult_ac
end;

structure Real_Times_Assoc = Assoc_Fold (Real_Times_Assoc_Data);

Addsimprocs [Real_Times_Assoc.conv];