(* Title: HOL/Real/Real.ML
ID: $Id$
Author: Jacques D. Fleuriot and Lawrence C. Paulson
Copyright: 1998 University of Cambridge
Description: Type "real" is a linear order
*)
(**** The simproc abel_cancel ****)
(*** Two lemmas needed for the simprocs ***)
(*Deletion of other terms in the formula, seeking the -x at the front of z*)
Goal "((x::real) + (y + z) = y + u) = ((x + z) = u)";
by (stac real_add_left_commute 1);
by (rtac real_add_left_cancel 1);
qed "real_add_cancel_21";
(*A further rule to deal with the case that
everything gets cancelled on the right.*)
Goal "((x::real) + (y + z) = y) = (x = -z)";
by (stac real_add_left_commute 1);
by (res_inst_tac [("t", "y")] (real_add_zero_right RS subst) 1
THEN stac real_add_left_cancel 1);
by (simp_tac (simpset() addsimps [real_eq_diff_eq RS sym]) 1);
qed "real_add_cancel_end";
structure Real_Cancel_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 zero = Const ("0", T)
val restrict_to_left = restrict_to_left
val add_cancel_21 = real_add_cancel_21
val add_cancel_end = real_add_cancel_end
val add_left_cancel = real_add_left_cancel
val add_assoc = real_add_assoc
val add_commute = real_add_commute
val add_left_commute = real_add_left_commute
val add_0 = real_add_zero_left
val add_0_right = real_add_zero_right
val eq_diff_eq = real_eq_diff_eq
val eqI_rules = [real_less_eqI, real_eq_eqI, real_le_eqI]
fun dest_eqI th =
#1 (HOLogic.dest_bin "op =" HOLogic.boolT
(HOLogic.dest_Trueprop (concl_of th)))
val diff_def = real_diff_def
val minus_add_distrib = real_minus_add_distrib
val minus_minus = real_minus_minus
val minus_0 = real_minus_zero
val add_inverses = [real_add_minus, real_add_minus_left]
val cancel_simps = [real_add_minus_cancel, real_minus_add_cancel]
end;
structure Real_Cancel = Abel_Cancel (Real_Cancel_Data);
Addsimprocs [Real_Cancel.sum_conv, Real_Cancel.rel_conv];
Goal "- (z - y) = y - (z::real)";
by (Simp_tac 1);
qed "real_minus_diff_eq";
Addsimps [real_minus_diff_eq];
(**** Theorems about the ordering ****)
Goal "(0 < x) = (EX y. x = real_of_preal y)";
by (auto_tac (claset(), simpset() addsimps [real_of_preal_zero_less]));
by (cut_inst_tac [("x","x")] real_of_preal_trichotomy 1);
by (blast_tac (claset() addSEs [real_less_irrefl,
real_of_preal_not_minus_gt_zero RS notE]) 1);
qed "real_gt_zero_preal_Ex";
Goal "real_of_preal z < x ==> EX y. x = real_of_preal y";
by (blast_tac (claset() addSDs [real_of_preal_zero_less RS real_less_trans]
addIs [real_gt_zero_preal_Ex RS iffD1]) 1);
qed "real_gt_preal_preal_Ex";
Goal "real_of_preal z <= x ==> EX y. x = real_of_preal y";
by (blast_tac (claset() addDs [order_le_imp_less_or_eq,
real_gt_preal_preal_Ex]) 1);
qed "real_ge_preal_preal_Ex";
Goal "y <= 0 ==> ALL x. y < real_of_preal x";
by (auto_tac (claset() addEs [order_le_imp_less_or_eq RS disjE]
addIs [real_of_preal_zero_less RSN(2,real_less_trans)],
simpset() addsimps [real_of_preal_zero_less]));
qed "real_less_all_preal";
Goal "~ 0 < y ==> ALL x. y < real_of_preal x";
by (blast_tac (claset() addSIs [real_less_all_preal,real_leI]) 1);
qed "real_less_all_real2";
Goal "[| R + L = S; (0::real) < L |] ==> R < S";
by (rtac (real_less_sum_gt_0_iff RS iffD1) 1);
by (auto_tac (claset(), simpset() addsimps real_add_ac));
qed "real_lemma_add_positive_imp_less";
Goal "EX T::real. 0 < T & R + T = S ==> R < S";
by (blast_tac (claset() addIs [real_lemma_add_positive_imp_less]) 1);
qed "real_ex_add_positive_left_less";
(*Alternative definition for real_less. NOT for rewriting*)
Goal "(R < S) = (EX T::real. 0 < T & R + T = S)";
by (blast_tac (claset() addSIs [real_less_add_positive_left_Ex,
real_ex_add_positive_left_less]) 1);
qed "real_less_iff_add";
Goal "(real_of_preal m1 <= real_of_preal m2) = (m1 <= m2)";
by (auto_tac (claset() addSIs [preal_leI],
simpset() addsimps [real_less_le_iff RS sym]));
by (dtac order_le_less_trans 1 THEN assume_tac 1);
by (etac preal_less_irrefl 1);
qed "real_of_preal_le_iff";
Goal "[| 0 < x; 0 < y |] ==> (0::real) < x * y";
by (auto_tac (claset(), simpset() addsimps [real_gt_zero_preal_Ex]));
by (res_inst_tac [("x","y*ya")] exI 1);
by (full_simp_tac (simpset() addsimps [real_of_preal_mult]) 1);
qed "real_mult_order";
Goal "[| x < 0; y < 0 |] ==> (0::real) < x * y";
by (REPEAT(dtac (real_minus_zero_less_iff RS iffD2) 1));
by (dtac real_mult_order 1 THEN assume_tac 1);
by (Asm_full_simp_tac 1);
qed "real_mult_less_zero1";
Goal "[| 0 < x; y < 0 |] ==> x*y < (0::real)";
by (dtac (real_minus_zero_less_iff RS iffD2) 1);
by (dtac real_mult_order 1 THEN assume_tac 1);
by (rtac (real_minus_zero_less_iff RS iffD1) 1);
by (Asm_full_simp_tac 1);
qed "real_mult_less_zero";
Goalw [real_one_def] "0 < (1::real)";
by (auto_tac (claset() addIs [real_gt_zero_preal_Ex RS iffD2],
simpset() addsimps [real_of_preal_def]));
qed "real_zero_less_one";
(*** Monotonicity results ***)
Goal "(v+z < w+z) = (v < (w::real))";
by (Simp_tac 1);
qed "real_add_right_cancel_less";
Goal "(z+v < z+w) = (v < (w::real))";
by (Simp_tac 1);
qed "real_add_left_cancel_less";
Addsimps [real_add_right_cancel_less, real_add_left_cancel_less];
Goal "(v+z <= w+z) = (v <= (w::real))";
by (Simp_tac 1);
qed "real_add_right_cancel_le";
Goal "(z+v <= z+w) = (v <= (w::real))";
by (Simp_tac 1);
qed "real_add_left_cancel_le";
Addsimps [real_add_right_cancel_le, real_add_left_cancel_le];
(*"v<=w ==> v+z <= w+z"*)
bind_thm ("real_add_less_mono1", real_add_right_cancel_less RS iffD2);
(*"v<=w ==> v+z <= w+z"*)
bind_thm ("real_add_le_mono1", real_add_right_cancel_le RS iffD2);
Goal "!!z z'::real. [| w'<w; z'<=z |] ==> w' + z' < w + z";
by (etac (real_add_less_mono1 RS order_less_le_trans) 1);
by (Simp_tac 1);
qed "real_add_less_le_mono";
Goal "!!z z'::real. [| w'<=w; z'<z |] ==> w' + z' < w + z";
by (etac (real_add_le_mono1 RS order_le_less_trans) 1);
by (Simp_tac 1);
qed "real_add_le_less_mono";
Goal "!!(A::real). A < B ==> C + A < C + B";
by (Simp_tac 1);
qed "real_add_less_mono2";
Goal "!!(A::real). A + C < B + C ==> A < B";
by (Full_simp_tac 1);
qed "real_less_add_right_cancel";
Goal "!!(A::real). C + A < C + B ==> A < B";
by (Full_simp_tac 1);
qed "real_less_add_left_cancel";
Goal "!!(A::real). A + C <= B + C ==> A <= B";
by (Full_simp_tac 1);
qed "real_le_add_right_cancel";
Goal "!!(A::real). C + A <= C + B ==> A <= B";
by (Full_simp_tac 1);
qed "real_le_add_left_cancel";
Goal "[| 0 < x; 0 < y |] ==> (0::real) < x + y";
by (etac order_less_trans 1);
by (dtac real_add_less_mono2 1);
by (Full_simp_tac 1);
qed "real_add_order";
Goal "[| 0 <= x; 0 <= y |] ==> (0::real) <= x + y";
by (REPEAT(dtac order_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [real_add_order, order_less_imp_le],
simpset()));
qed "real_le_add_order";
Goal "[| R1 < S1; R2 < S2 |] ==> R1 + R2 < S1 + (S2::real)";
by (dtac real_add_less_mono1 1);
by (etac order_less_trans 1);
by (etac real_add_less_mono2 1);
qed "real_add_less_mono";
Goal "!!(q1::real). q1 <= q2 ==> x + q1 <= x + q2";
by (Simp_tac 1);
qed "real_add_left_le_mono1";
Goal "[|i<=j; k<=l |] ==> i + k <= j + (l::real)";
by (dtac real_add_le_mono1 1);
by (etac order_trans 1);
by (Simp_tac 1);
qed "real_add_le_mono";
Goal "EX (x::real). x < y";
by (rtac (real_add_zero_right RS subst) 1);
by (res_inst_tac [("x","y + (- (1::real))")] exI 1);
by (auto_tac (claset() addSIs [real_add_less_mono2],
simpset() addsimps [real_minus_zero_less_iff2, real_zero_less_one]));
qed "real_less_Ex";
Goal "(0::real) < r ==> u + (-r) < u";
by (res_inst_tac [("C","r")] real_less_add_right_cancel 1);
by (simp_tac (simpset() addsimps [real_add_assoc]) 1);
qed "real_add_minus_positive_less_self";
Goal "(-s <= -r) = ((r::real) <= s)";
by (rtac sym 1);
by (Step_tac 1);
by (dres_inst_tac [("x","-s")] real_add_left_le_mono1 1);
by (dres_inst_tac [("x","r")] real_add_left_le_mono1 2);
by Auto_tac;
by (dres_inst_tac [("z","-r")] real_add_le_mono1 1);
by (dres_inst_tac [("z","s")] real_add_le_mono1 2);
by (auto_tac (claset(), simpset() addsimps [real_add_assoc]));
qed "real_le_minus_iff";
Addsimps [real_le_minus_iff];
Goal "(0::real) <= x*x";
by (res_inst_tac [("R2.0","0"),("R1.0","x")] real_linear_less2 1);
by (auto_tac (claset() addIs [real_mult_order,
real_mult_less_zero1,order_less_imp_le],
simpset()));
qed "real_le_square";
Addsimps [real_le_square];
(*----------------------------------------------------------------------------
An embedding of the naturals in the reals
----------------------------------------------------------------------------*)
Goalw [real_of_posnat_def] "real_of_posnat 0 = (1::real)";
by (simp_tac (simpset() addsimps [pnat_one_iff RS sym,real_of_preal_def,
symmetric real_one_def]) 1);
qed "real_of_posnat_one";
Goalw [real_of_posnat_def] "real_of_posnat (Suc 0) = (1::real) + (1::real)";
by (simp_tac (simpset() addsimps [real_of_preal_def,real_one_def,
pnat_two_eq,real_add,prat_of_pnat_add RS sym,
preal_of_prat_add RS sym] @ pnat_add_ac) 1);
qed "real_of_posnat_two";
Goalw [real_of_posnat_def]
"real_of_posnat n1 + real_of_posnat n2 = real_of_posnat (n1 + n2) + (1::real)";
by (full_simp_tac (simpset() addsimps [real_of_posnat_one RS sym,
real_of_posnat_def,real_of_preal_add RS sym,preal_of_prat_add RS sym,
prat_of_pnat_add RS sym,pnat_of_nat_add]) 1);
qed "real_of_posnat_add";
Goal "real_of_posnat (n + 1) = real_of_posnat n + (1::real)";
by (res_inst_tac [("x1","(1::real)")] (real_add_right_cancel RS iffD1) 1);
by (rtac (real_of_posnat_add RS subst) 1);
by (full_simp_tac (simpset() addsimps [real_of_posnat_two,real_add_assoc]) 1);
qed "real_of_posnat_add_one";
Goal "real_of_posnat (Suc n) = real_of_posnat n + (1::real)";
by (stac (real_of_posnat_add_one RS sym) 1);
by (Simp_tac 1);
qed "real_of_posnat_Suc";
Goal "inj(real_of_posnat)";
by (rtac injI 1);
by (rewtac real_of_posnat_def);
by (dtac (inj_real_of_preal RS injD) 1);
by (dtac (inj_preal_of_prat RS injD) 1);
by (dtac (inj_prat_of_pnat RS injD) 1);
by (etac (inj_pnat_of_nat RS injD) 1);
qed "inj_real_of_posnat";
Goalw [real_of_nat_def] "real (0::nat) = 0";
by (simp_tac (simpset() addsimps [real_of_posnat_one]) 1);
qed "real_of_nat_zero";
Goalw [real_of_nat_def] "real (Suc 0) = (1::real)";
by (simp_tac (simpset() addsimps [real_of_posnat_two, real_add_assoc]) 1);
qed "real_of_nat_one";
Addsimps [real_of_nat_zero, real_of_nat_one];
Goalw [real_of_nat_def]
"real (m + n) = real (m::nat) + real n";
by (simp_tac (simpset() addsimps
[real_of_posnat_add,real_add_assoc RS sym]) 1);
qed "real_of_nat_add";
Addsimps [real_of_nat_add];
(*Not for addsimps: often the LHS is used to represent a positive natural*)
Goalw [real_of_nat_def] "real (Suc n) = real n + (1::real)";
by (simp_tac (simpset() addsimps [real_of_posnat_Suc] @ real_add_ac) 1);
qed "real_of_nat_Suc";
Goalw [real_of_nat_def, real_of_posnat_def]
"(real (n::nat) < real m) = (n < m)";
by Auto_tac;
qed "real_of_nat_less_iff";
AddIffs [real_of_nat_less_iff];
Goal "(real (n::nat) <= real m) = (n <= m)";
by (simp_tac (simpset() addsimps [linorder_not_less RS sym]) 1);
qed "real_of_nat_le_iff";
AddIffs [real_of_nat_le_iff];
Goal "inj (real :: nat => real)";
by (rtac injI 1);
by (auto_tac (claset() addSIs [inj_real_of_posnat RS injD],
simpset() addsimps [real_of_nat_def,real_add_right_cancel]));
qed "inj_real_of_nat";
Goal "0 <= real (n::nat)";
by (induct_tac "n" 1);
by (auto_tac (claset(),
simpset () addsimps [real_of_nat_Suc]));
by (dtac real_add_le_less_mono 1);
by (rtac real_zero_less_one 1);
by (asm_full_simp_tac (simpset() addsimps [order_less_imp_le]) 1);
qed "real_of_nat_ge_zero";
AddIffs [real_of_nat_ge_zero];
Goal "real (m * n) = real (m::nat) * real n";
by (induct_tac "m" 1);
by (auto_tac (claset(),
simpset() addsimps [real_of_nat_Suc,
real_add_mult_distrib, real_add_commute]));
qed "real_of_nat_mult";
Addsimps [real_of_nat_mult];
Goal "(real (n::nat) = real m) = (n = m)";
by (auto_tac (claset() addDs [inj_real_of_nat RS injD], simpset()));
qed "real_of_nat_inject";
AddIffs [real_of_nat_inject];
Goal "n <= m --> real (m - n) = real (m::nat) - real n";
by (induct_tac "m" 1);
by (auto_tac (claset(),
simpset() addsimps [real_diff_def, Suc_diff_le, le_Suc_eq,
real_of_nat_Suc, real_of_nat_zero] @ real_add_ac));
qed_spec_mp "real_of_nat_diff";
Goal "(real (n::nat) = 0) = (n = 0)";
by (auto_tac ((claset() addIs [inj_real_of_nat RS injD], simpset()) delIffs [real_of_nat_inject]));
qed "real_of_nat_zero_iff";
Goal "neg z ==> real (nat z) = 0";
by (asm_simp_tac (simpset() addsimps [neg_nat, real_of_nat_zero]) 1);
qed "real_of_nat_neg_int";
Addsimps [real_of_nat_neg_int];
(*----------------------------------------------------------------------------
inverse, etc.
----------------------------------------------------------------------------*)
Goal "0 < x ==> 0 < inverse (x::real)";
by (EVERY1[rtac ccontr, dtac real_leI]);
by (forward_tac [real_minus_zero_less_iff2 RS iffD2] 1);
by (forward_tac [real_not_refl2 RS not_sym] 1);
by (dtac (real_not_refl2 RS not_sym RS real_inverse_not_zero) 1);
by (EVERY1[dtac order_le_imp_less_or_eq, Step_tac]);
by (dtac real_mult_less_zero1 1 THEN assume_tac 1);
by (auto_tac (claset() addIs [real_zero_less_one RS real_less_asym],
simpset()));
qed "real_inverse_gt_zero";
Goal "x < 0 ==> inverse (x::real) < 0";
by (ftac real_not_refl2 1);
by (dtac (real_minus_zero_less_iff RS iffD2) 1);
by (rtac (real_minus_zero_less_iff RS iffD1) 1);
by (stac (real_minus_inverse RS sym) 1);
by (auto_tac (claset() addIs [real_inverse_gt_zero], simpset()));
qed "real_inverse_less_zero";
Goal "[| (0::real) < z; x < y |] ==> x*z < y*z";
by (rotate_tac 1 1);
by (dtac real_less_sum_gt_zero 1);
by (rtac real_sum_gt_zero_less 1);
by (dtac real_mult_order 1 THEN assume_tac 1);
by (asm_full_simp_tac
(simpset() addsimps [real_add_mult_distrib2, real_mult_commute ]) 1);
qed "real_mult_less_mono1";
Goal "[| (0::real) < z; x < y |] ==> z * x < z * y";
by (asm_simp_tac
(simpset() addsimps [real_mult_commute,real_mult_less_mono1]) 1);
qed "real_mult_less_mono2";
Goal "[| (0::real) < z; x * z < y * z |] ==> x < y";
by (forw_inst_tac [("x","x*z")]
(real_inverse_gt_zero RS real_mult_less_mono1) 1);
by (auto_tac (claset(),
simpset() addsimps [real_mult_assoc,real_not_refl2 RS not_sym]));
qed "real_mult_less_cancel1";
Goal "[| (0::real) < z; z*x < z*y |] ==> x < y";
by (etac real_mult_less_cancel1 1);
by (asm_full_simp_tac (simpset() addsimps [real_mult_commute]) 1);
qed "real_mult_less_cancel2";
Goal "(0::real) < z ==> (x*z < y*z) = (x < y)";
by (blast_tac
(claset() addIs [real_mult_less_mono1, real_mult_less_cancel1]) 1);
qed "real_mult_less_iff1";
Goal "(0::real) < z ==> (z*x < z*y) = (x < y)";
by (blast_tac
(claset() addIs [real_mult_less_mono2, real_mult_less_cancel2]) 1);
qed "real_mult_less_iff2";
Addsimps [real_mult_less_iff1,real_mult_less_iff2];
(* 05/00 *)
Goalw [real_le_def] "(0::real) < z ==> (x*z <= y*z) = (x <= y)";
by (Auto_tac);
qed "real_mult_le_cancel_iff1";
Goalw [real_le_def] "(0::real) < z ==> (z*x <= z*y) = (x <= y)";
by (Auto_tac);
qed "real_mult_le_cancel_iff2";
Addsimps [real_mult_le_cancel_iff1,real_mult_le_cancel_iff2];
Goal "[| (0::real) <= z; x < y |] ==> x*z <= y*z";
by (EVERY1 [rtac real_less_or_eq_imp_le, dtac order_le_imp_less_or_eq]);
by (auto_tac (claset() addIs [real_mult_less_mono1],simpset()));
qed "real_mult_le_less_mono1";
Goal "[| u<v; x<y; (0::real) < v; 0 < x |] ==> u*x < v* y";
by (etac (real_mult_less_mono1 RS order_less_trans) 1);
by (assume_tac 1);
by (etac real_mult_less_mono2 1);
by (assume_tac 1);
qed "real_mult_less_mono";
(*Variant of the theorem above; sometimes it's stronger*)
Goal "[| x < y; r1 < r2; (0::real) <= r1; 0 <= x|] ==> r1 * x < r2 * y";
by (subgoal_tac "0<r2" 1);
by (blast_tac (claset() addIs [order_le_less_trans]) 2);
by (case_tac "x=0" 1);
by (auto_tac (claset() addSDs [order_le_imp_less_or_eq]
addIs [real_mult_less_mono, real_mult_order],
simpset()));
qed "real_mult_less_mono'";
Goal "(1::real) <= x ==> 0 < x";
by (rtac ccontr 1 THEN dtac real_leI 1);
by (dtac order_trans 1 THEN assume_tac 1);
by (auto_tac (claset() addDs [real_zero_less_one RSN (2,order_le_less_trans)],
simpset()));
qed "real_gt_zero";
Goal "[| (1::real) < r; (1::real) <= x |] ==> x <= r * x";
by (dtac (real_gt_zero RS order_less_imp_le) 1);
by (auto_tac (claset() addSDs [real_mult_le_less_mono1],
simpset()));
qed "real_mult_self_le";
Goal "[| (1::real) <= r; (1::real) <= x |] ==> x <= r * x";
by (dtac order_le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [real_mult_self_le], simpset()));
qed "real_mult_self_le2";
Goal "[| 0 < r; r < x |] ==> inverse x < inverse (r::real)";
by (ftac order_less_trans 1 THEN assume_tac 1);
by (ftac real_inverse_gt_zero 1);
by (forw_inst_tac [("x","x")] real_inverse_gt_zero 1);
by (forw_inst_tac [("x","r"),("z","inverse r")] real_mult_less_mono1 1);
by (assume_tac 1);
by (asm_full_simp_tac (simpset() addsimps [real_not_refl2 RS
not_sym RS real_mult_inv_right]) 1);
by (ftac real_inverse_gt_zero 1);
by (forw_inst_tac [("x","(1::real)"),("z","inverse x")] real_mult_less_mono2 1);
by (assume_tac 1);
by (asm_full_simp_tac (simpset() addsimps [real_not_refl2 RS
not_sym RS real_mult_inv_left,real_mult_assoc RS sym]) 1);
qed "real_inverse_less_swap";
Goal "(x*y = 0) = (x = 0 | y = (0::real))";
by Auto_tac;
by (blast_tac (claset() addIs [ccontr] addDs [real_mult_not_zero]) 1);
qed "real_mult_is_0";
AddIffs [real_mult_is_0];
Goal "[| x ~= 0; y ~= 0 |] \
\ ==> inverse x + inverse y = (x + y) * inverse (x * (y::real))";
by (asm_full_simp_tac (simpset() addsimps
[real_inverse_distrib,real_add_mult_distrib,
real_mult_assoc RS sym]) 1);
by (stac real_mult_assoc 1);
by (rtac (real_mult_left_commute RS subst) 1);
by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1);
qed "real_inverse_add";
(* 05/00 *)
Goal "(0 <= -R) = (R <= (0::real))";
by (auto_tac (claset() addDs [sym],
simpset() addsimps [real_le_less]));
qed "real_minus_zero_le_iff";
Goal "(-R <= 0) = ((0::real) <= R)";
by (auto_tac (claset(),simpset() addsimps
[real_minus_zero_less_iff2,real_le_less]));
qed "real_minus_zero_le_iff2";
Addsimps [real_minus_zero_le_iff, real_minus_zero_le_iff2];
Goal "x * x + y * y = 0 ==> x = (0::real)";
by (dtac real_add_minus_eq_minus 1);
by (cut_inst_tac [("x","x")] real_le_square 1);
by (Auto_tac THEN dtac real_le_anti_sym 1);
by Auto_tac;
qed "real_sum_squares_cancel";
Goal "x * x + y * y = 0 ==> y = (0::real)";
by (res_inst_tac [("y","x")] real_sum_squares_cancel 1);
by (asm_full_simp_tac (simpset() addsimps [real_add_commute]) 1);
qed "real_sum_squares_cancel2";
(*----------------------------------------------------------------------------
Some convenient biconditionals for products of signs (lcp)
----------------------------------------------------------------------------*)
Goal "((0::real) < x*y) = (0 < x & 0 < y | x < 0 & y < 0)";
by (auto_tac (claset(),
simpset() addsimps [order_le_less, linorder_not_less,
real_mult_order, real_mult_less_zero1]));
by (ALLGOALS (rtac ccontr));
by (auto_tac (claset(), simpset() addsimps [order_le_less, linorder_not_less]));
by (ALLGOALS (etac rev_mp));
by (ALLGOALS (dtac real_mult_less_zero THEN' assume_tac));
by (auto_tac (claset() addDs [order_less_not_sym],
simpset() addsimps [real_mult_commute]));
qed "real_zero_less_mult_iff";
Goal "((0::real) <= x*y) = (0 <= x & 0 <= y | x <= 0 & y <= 0)";
by (auto_tac (claset(),
simpset() addsimps [order_le_less, linorder_not_less,
real_zero_less_mult_iff]));
qed "real_zero_le_mult_iff";
Goal "(x*y < (0::real)) = (0 < x & y < 0 | x < 0 & 0 < y)";
by (auto_tac (claset(),
simpset() addsimps [real_zero_le_mult_iff,
linorder_not_le RS sym]));
by (auto_tac (claset() addDs [order_less_not_sym],
simpset() addsimps [linorder_not_le]));
qed "real_mult_less_zero_iff";
Goal "(x*y <= (0::real)) = (0 <= x & y <= 0 | x <= 0 & 0 <= y)";
by (auto_tac (claset() addDs [order_less_not_sym],
simpset() addsimps [real_zero_less_mult_iff,
linorder_not_less RS sym]));
qed "real_mult_le_zero_iff";