(* Title: HOL/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
(simpset_of Int.thy 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
(simpset_of Int.thy 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
(simpset_of Int.thy 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
(simpset_of Int.thy 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_of RealDef.thy
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
(simpset_of Int.thy 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
(simpset_of Int.thy 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 thy th =
asm_full_simplify real_numeral_ss (change_theory thy th);
(*Now insert some identities previously stated for 0 and 1r*)
(** RealDef & Real **)
Addsimps (map (rename_numerals thy)
[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 thy) [real_mult_is_0, real_0_is_mult]);
bind_thm ("real_0_less_mult_iff",
rename_numerals thy real_zero_less_mult_iff);
bind_thm ("real_0_le_mult_iff",
rename_numerals thy real_zero_le_mult_iff);
bind_thm ("real_mult_less_0_iff",
rename_numerals thy real_mult_less_zero_iff);
bind_thm ("real_mult_le_0_iff",
rename_numerals thy 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
(simpset_of Int.thy 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 thy)
[real_add_zero_left, real_add_zero_right];
val mult_1s = map (rename_numerals thy)
[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 (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;
(*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 RealInt.thy) (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 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 "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 thy = RealBin.thy
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];
(*---------------------------------------------------------------------------*)
(* Linear arithmetic *)
(*---------------------------------------------------------------------------*)
(* Instantiation of the generic linear arithmetic package for type real. *)
(* Author: Tobias Nipkow, TU Muenchen
Copyright 1999 TU Muenchen
*)
let
(* reduce contradictory <= to False *)
val simps = [order_less_irrefl, zero_eq_numeral_0, one_eq_numeral_1,
add_real_number_of, minus_real_number_of, diff_real_number_of,
mult_real_number_of, eq_real_number_of, less_real_number_of,
le_real_number_of_eq_not_less, real_diff_def,
real_minus_add_distrib, real_minus_minus];
val add_rules =
map (rename_numerals thy)
[real_add_zero_left, real_add_zero_right,
real_add_minus, real_add_minus_left,
real_mult_0, real_mult_0_right,
real_mult_1, real_mult_1_right,
real_mult_minus_1, real_mult_minus_1_right];
val simprocs = [Real_Times_Assoc.conv, Real_Numeral_Simprocs.combine_numerals]@
Real_Numeral_Simprocs.cancel_numerals;
val mono_ss = simpset() addsimps
[real_add_le_mono,real_add_less_mono,
real_add_less_le_mono,real_add_le_less_mono];
val add_mono_thms =
map (fn s => prove_goal thy s
(fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 1]))
["(i <= j) & (k <= l) ==> i + k <= j + (l::real)",
"(i = j) & (k <= l) ==> i + k <= j + (l::real)",
"(i <= j) & (k = l) ==> i + k <= j + (l::real)",
"(i = j) & (k = l) ==> i + k = j + (l::real)",
"(i < j) & (k = l) ==> i + k < j + (l::real)",
"(i = j) & (k < l) ==> i + k < j + (l::real)",
"(i < j) & (k <= l) ==> i + k < j + (l::real)",
"(i <= j) & (k < l) ==> i + k < j + (l::real)",
"(i < j) & (k < l) ==> i + k < j + (l::real)"];
in
LA_Data_Ref.add_mono_thms := !LA_Data_Ref.add_mono_thms @ add_mono_thms;
LA_Data_Ref.ss_ref := !LA_Data_Ref.ss_ref addsimps simps@add_rules
addsimprocs simprocs
addcongs [if_weak_cong];
LA_Data_Ref.discrete := !LA_Data_Ref.discrete @ [("RealDef.real",false)]
(*We don't change LA_Data_Ref.lessD because the real ordering is dense!*)
end;
let
val real_arith_simproc_pats =
map (fn s => Thm.read_cterm (Theory.sign_of RealDef.thy) (s, HOLogic.boolT))
["(m::real) < n","(m::real) <= n", "(m::real) = n"];
val fast_real_arith_simproc = mk_simproc
"fast_real_arith" real_arith_simproc_pats Fast_Arith.lin_arith_prover;
in
Addsimprocs [fast_real_arith_simproc]
end;
(* Some test data [omitting examples thet assume the ordering to be discrete!]
Goal "!!a::real. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
by (fast_arith_tac 1);
Goal "!!a::real. [| a <= b; b+b <= c |] ==> a+a <= c";
by (fast_arith_tac 1);
Goal "!!a::real. [| a+b <= i+j; a<=b; i<=j |] ==> a+a <= j+j";
by (fast_arith_tac 1);
Goal "!!a::real. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
by (arith_tac 1);
Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a <= l";
by (fast_arith_tac 1);
Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a+a+a+a <= l+l+l+l";
by (fast_arith_tac 1);
Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a+a+a+a+a <= l+l+l+l+i";
by (fast_arith_tac 1);
Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> a+a+a+a+a+a <= l+l+l+l+i+l";
by (fast_arith_tac 1);
Goal "!!a::real. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
\ ==> #6*a <= #5*l+i";
by (fast_arith_tac 1);
*)
(*---------------------------------------------------------------------------*)
(* End of linear arithmetic *)
(*---------------------------------------------------------------------------*)
(*useful??*)
Goal "(z = z + w) = (w = (#0::real))";
by Auto_tac;
qed "real_add_left_cancel0";
Addsimps [real_add_left_cancel0];