src/HOL/Real/RealOrd.ML

deleted obsolete lemma_not_leI2

(*  Title:       HOL/Real/Real.ML
    ID:          $Id$
    Author:      Lawrence C. Paulson
                 Jacques D. Fleuriot
    Copyright:   1998  University of Cambridge
    Description: Type "real" is a linear order
*)

Goal "(0r < x) = (? 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 ==> ? 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 ==> ? y. x = real_of_preal y";
by (blast_tac (claset() addDs [real_le_imp_less_or_eq,
			       real_gt_preal_preal_Ex]) 1);
qed "real_ge_preal_preal_Ex";

Goal "y <= 0r ==> !x. y < real_of_preal x";
by (auto_tac (claset() addEs [real_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 "~ 0r < y ==> !x. y < real_of_preal x";
by (blast_tac (claset() addSIs [real_less_all_preal,real_leI]) 1);
qed "real_less_all_real2";

Goal "((x::real) < y) = (-y < -x)";
by (rtac (real_less_sum_gt_0_iff RS subst) 1);
by (res_inst_tac [("W1","x")] (real_less_sum_gt_0_iff RS subst) 1);
by (simp_tac (simpset() addsimps [real_add_commute]) 1);
qed "real_less_swap_iff";

Goal "[| R + L = S; 0r < 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 "? T. 0r < 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) = (? T. 0r < 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 "(0r < x) = (-x < x)";
by Safe_tac;
by (rtac ccontr 2 THEN forward_tac 
    [real_leI RS real_le_imp_less_or_eq] 2);
by (Step_tac 2);
by (dtac (real_minus_zero_less_iff RS iffD2) 2);
by (blast_tac (claset() addIs [real_less_trans]) 2);
by (auto_tac (claset(),
	      simpset() addsimps 
 	        [real_gt_zero_preal_Ex,real_of_preal_minus_less_self]));
qed "real_gt_zero_iff";

Goal "(x < 0r) = (x < -x)";
by (rtac (real_minus_zero_less_iff RS subst) 1);
by (stac real_gt_zero_iff 1);
by (Full_simp_tac 1);
qed "real_lt_zero_iff";

Goalw [real_le_def] "(0r <= x) = (-x <= x)";
by (auto_tac (claset(), simpset() addsimps [real_lt_zero_iff RS sym]));
qed "real_ge_zero_iff";

Goalw [real_le_def] "(x <= 0r) = (x <= -x)";
by (auto_tac (claset(), simpset() addsimps [real_gt_zero_iff RS sym]));
qed "real_le_zero_iff";

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 preal_le_less_trans 1 THEN assume_tac 1);
by (etac preal_less_irrefl 1);
qed "real_of_preal_le_iff";

Goal "[| 0r < x; 0r < y |] ==> 0r < 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 < 0r; y < 0r |] ==> 0r < 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 "[| 0r <= x; 0r <= y |] ==> 0r <= x * y";
by (REPEAT(dtac real_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [real_mult_order, real_less_imp_le],
	      simpset()));
qed "real_le_mult_order";

Goal "[| 0r < x; 0r <= y |] ==> 0r <= x * y";
by (dtac real_le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [real_mult_order,
			      real_less_imp_le],simpset()));
qed "real_less_le_mult_order";

Goal "[| x <= 0r; y <= 0r |] ==> 0r <= x * y";
by (rtac real_less_or_eq_imp_le 1);
by (dtac real_le_imp_less_or_eq 1 THEN etac disjE 1);
by Auto_tac;
by (dtac real_le_imp_less_or_eq 1);
by (auto_tac (claset() addDs [real_mult_less_zero1],simpset()));
qed "real_mult_le_zero1";

Goal "[| 0r <= x; y < 0r |] ==> x * y <= 0r";
by (rtac real_less_or_eq_imp_le 1);
by (dtac real_le_imp_less_or_eq 1 THEN etac disjE 1);
by Auto_tac;
by (dtac (real_minus_zero_less_iff RS iffD2) 1);
by (rtac (real_minus_zero_less_iff RS subst) 1);
by (blast_tac (claset() addDs [real_mult_order] 
	                addIs [real_minus_mult_eq2 RS ssubst]) 1);
qed "real_mult_le_zero";

Goal "[| 0r < x; y < 0r |] ==> x*y < 0r";
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 (simpset() addsimps [real_minus_mult_eq2]) 1);
qed "real_mult_less_zero";

Goalw [real_one_def] "0r < 1r";
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 real_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 real_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 "[| 0r < x; 0r < y |] ==> 0r < x + y";
by (etac real_less_trans 1);
by (dtac real_add_less_mono2 1);
by (Full_simp_tac 1);
qed "real_add_order";

Goal "[| 0r <= x; 0r <= y |] ==> 0r <= x + y";
by (REPEAT(dtac real_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [real_add_order, real_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 real_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 real_le_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 + (-1r)")] 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 "0r < 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 "((r::real) <= s) = (-s <= -r)";
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 RS sym];
          
Goal "0r <= 1r";
by (rtac (real_zero_less_one RS real_less_imp_le) 1);
qed "real_zero_le_one";
Addsimps [real_zero_le_one];

Goal "0r <= x*x";
by (res_inst_tac [("R2.0","0r"),("R1.0","x")] real_linear_less2 1);
by (auto_tac (claset() addIs [real_mult_order,
			      real_mult_less_zero1,real_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 = 1r";
by (full_simp_tac (simpset() addsimps [pnat_one_iff RS sym,real_of_preal_def]) 1);
by (fold_tac [real_one_def]);
by (rtac refl 1);
qed "real_of_posnat_one";

Goalw [real_of_posnat_def] "real_of_posnat 1 = 1r + 1r";
by (full_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) + 1r";
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 + 1r";
by (res_inst_tac [("x1","1r")] (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 + 1r";
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_posnat_def] "0r < real_of_posnat n";
by (rtac (real_gt_zero_preal_Ex RS iffD2) 1);
by (Blast_tac 1);
qed "real_of_posnat_less_zero";

Goal "real_of_posnat n ~= 0r";
by (rtac (real_of_posnat_less_zero RS real_not_refl2 RS not_sym) 1);
qed "real_of_posnat_not_eq_zero";
Addsimps[real_of_posnat_not_eq_zero];

Goal "1r <= real_of_posnat n";
by (simp_tac (simpset() addsimps [real_of_posnat_one RS sym]) 1);
by (induct_tac "n" 1);
by (auto_tac (claset(),
	      simpset () addsimps [real_of_posnat_Suc,real_of_posnat_one,
			   real_of_posnat_less_zero, real_less_imp_le]));
qed "real_of_posnat_less_one";
Addsimps [real_of_posnat_less_one];

Goal "rinv(real_of_posnat n) ~= 0r";
by (rtac ((real_of_posnat_less_zero RS 
    real_not_refl2 RS not_sym) RS rinv_not_zero) 1);
qed "real_of_posnat_rinv_not_zero";
Addsimps [real_of_posnat_rinv_not_zero];

Goal "rinv(real_of_posnat x) = rinv(real_of_posnat y) ==> x = y";
by (rtac (inj_real_of_posnat RS injD) 1);
by (res_inst_tac [("n2","x")] 
    (real_of_posnat_rinv_not_zero RS real_mult_left_cancel RS iffD1) 1);
by (full_simp_tac (simpset() addsimps [(real_of_posnat_less_zero RS 
    real_not_refl2 RS not_sym) RS real_mult_inv_left]) 1);
by (asm_full_simp_tac (simpset() addsimps [(real_of_posnat_less_zero RS 
    real_not_refl2 RS not_sym)]) 1);
qed "real_of_posnat_rinv_inj";

Goal "0r < x ==> 0r < rinv x";
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 rinv_not_zero) 1);
by (EVERY1[dtac real_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() addsimps [real_minus_mult_eq1 RS sym]));
qed "real_rinv_gt_zero";

Goal "x < 0r ==> rinv x < 0r";
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 (dtac (real_minus_rinv RS sym) 1);
by (auto_tac (claset() addIs [real_rinv_gt_zero], simpset()));
qed "real_rinv_less_zero";

Goal "0r < rinv(real_of_posnat n)";
by (rtac (real_of_posnat_less_zero RS real_rinv_gt_zero) 1);
qed "real_of_posnat_rinv_gt_zero";
Addsimps [real_of_posnat_rinv_gt_zero];

Goal "x+x = x*(1r+1r)";
by (simp_tac (simpset() addsimps 
              [real_add_mult_distrib2]) 1);
qed "real_add_self";

Goal "x < x + 1r";
by (rtac (real_less_sum_gt_0_iff RS iffD1) 1);
by (full_simp_tac (simpset() addsimps [real_zero_less_one,
				real_add_assoc, real_add_left_commute]) 1);
qed "real_self_less_add_one";

Goal "1r < 1r + 1r";
by (rtac real_self_less_add_one 1);
qed "real_one_less_two";

Goal "0r < 1r + 1r";
by (rtac ([real_zero_less_one,
	   real_one_less_two] MRS real_less_trans) 1);
qed "real_zero_less_two";

Goal "1r + 1r ~= 0r";
by (rtac (real_zero_less_two RS real_not_refl2 RS not_sym) 1);
qed "real_two_not_zero";

Addsimps [real_two_not_zero];

Goal "x*rinv(1r + 1r) + x*rinv(1r + 1r) = x";
by (stac real_add_self 1);
by (full_simp_tac (simpset() addsimps [real_mult_assoc]) 1);
qed "real_sum_of_halves";

Goal "[| 0r<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_minus_mult_eq2 RS sym, real_mult_commute ]) 1);
qed "real_mult_less_mono1";

Goal "[| 0r<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 "[| 0r<z; x*z<y*z |] ==> x<y";
by (forw_inst_tac [("x","x*z")] (real_rinv_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 "[| 0r<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 "0r < 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 "0r < 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];

Goal "[| 0r<=z; x<y |] ==> x*z<=y*z";
by (EVERY1 [rtac real_less_or_eq_imp_le, dtac real_le_imp_less_or_eq]);
by (auto_tac (claset() addIs [real_mult_less_mono1],simpset()));
qed "real_mult_le_less_mono1";

Goal "[| 0r<=z; x<y |] ==> z*x<=z*y";
by (asm_simp_tac (simpset() addsimps [real_mult_commute,real_mult_le_less_mono1]) 1);
qed "real_mult_le_less_mono2";

Goal "[| 0r<=z; x<=y |] ==> z*x<=z*y";
by (dres_inst_tac [("x","x")] real_le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [real_mult_le_less_mono2], simpset()));
qed "real_mult_le_le_mono1";

Goal "[| 0r < r1; r1 <r2; 0r < x; x < y|] ==> r1 * x < r2 * y";
by (dres_inst_tac [("x","x")] real_mult_less_mono2 1);
by (dres_inst_tac [("R1.0","0r")] real_less_trans 2);
by (dres_inst_tac [("x","r1")] real_mult_less_mono1 3);
by Auto_tac;
by (blast_tac (claset() addIs [real_less_trans]) 1);
qed "real_mult_less_mono";

Goal "[| 0r < r1; r1 <r2; 0r < y|] ==> 0r < r2 * y";
by (dres_inst_tac [("R1.0","0r")] real_less_trans 1);
by (assume_tac 1);
by (blast_tac (claset() addIs [real_mult_order]) 1);
qed "real_mult_order_trans";

Goal "[| 0r < r1; r1 <r2; 0r <= x; x < y|] ==> r1 * x < r2 * y";
by (auto_tac (claset() addSDs [real_le_imp_less_or_eq] 
	               addIs [real_mult_less_mono,real_mult_order_trans],
              simpset()));
qed "real_mult_less_mono3";

Goal "[| 0r <= r1; r1 <r2; 0r <= x; x < y|] ==> r1 * x < r2 * y";
by (auto_tac (claset() addSDs [real_le_imp_less_or_eq] 
	               addIs [real_mult_less_mono,real_mult_order_trans,
			      real_mult_order],
	      simpset()));
by (dres_inst_tac [("R2.0","x")] real_less_trans 1);
by (assume_tac 1);
by (blast_tac (claset() addIs [real_mult_order]) 1);
qed "real_mult_less_mono4";

Goal "[| 0r < r1; r1 <= r2; 0r <= x; x <= y |] ==> r1 * x <= r2 * y";
by (rtac real_less_or_eq_imp_le 1);
by (REPEAT(dtac real_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [real_mult_less_mono,
			      real_mult_order_trans,real_mult_order],
	      simpset()));
qed "real_mult_le_mono";

Goal "[| 0r < r1; r1 < r2; 0r <= x; x <= y |] ==> r1 * x <= r2 * y";
by (rtac real_less_or_eq_imp_le 1);
by (REPEAT(dtac real_le_imp_less_or_eq 1));
by (auto_tac (claset() addIs [real_mult_less_mono, real_mult_order_trans,
			      real_mult_order],
	      simpset()));
qed "real_mult_le_mono2";

Goal "[| 0r <= r1; r1 < r2; 0r <= x; x <= y |] ==> r1 * x <= r2 * y";
by (dtac real_le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [real_mult_le_mono2],simpset()));
by (dtac real_le_trans 1 THEN assume_tac 1);
by (auto_tac (claset() addIs [real_less_le_mult_order], simpset()));
qed "real_mult_le_mono3";

Goal "[| 0r <= r1; r1 <= r2; 0r <= x; x <= y |] ==> r1 * x <= r2 * y";
by (dres_inst_tac [("x","r1")] real_le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [real_mult_le_mono3, real_mult_le_le_mono1],
	      simpset()));
qed "real_mult_le_mono4";

Goal "1r <= x ==> 0r < x";
by (rtac ccontr 1 THEN dtac real_leI 1);
by (dtac real_le_trans 1 THEN assume_tac 1);
by (auto_tac (claset() addDs [real_zero_less_one RSN (2,real_le_less_trans)],
	      simpset() addsimps [real_less_not_refl]));
qed "real_gt_zero";

Goal "[| 1r < r; 1r <= x |]  ==> x <= r * x";
by (dtac (real_gt_zero RS real_less_imp_le) 1);
by (auto_tac (claset() addSDs [real_mult_le_less_mono1],
    simpset()));
qed "real_mult_self_le";

Goal "[| 1r <= r; 1r <= x |]  ==> x <= r * x";
by (dtac real_le_imp_less_or_eq 1);
by (auto_tac (claset() addIs [real_mult_self_le],
	      simpset() addsimps [real_le_refl]));
qed "real_mult_self_le2";

Goal "x < y ==> x < (x + y)*rinv(1r + 1r)";
by (dres_inst_tac [("C","x")] real_add_less_mono2 1);
by (dtac (real_add_self RS subst) 1);
by (dtac (real_zero_less_two RS real_rinv_gt_zero RS 
          real_mult_less_mono1) 1);
by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1);
qed "real_less_half_sum";

Goal "x < y ==> (x + y)*rinv(1r + 1r) < y";
by (dtac real_add_less_mono1 1);
by (dtac (real_add_self RS subst) 1);
by (dtac (real_zero_less_two RS real_rinv_gt_zero RS 
          real_mult_less_mono1) 1);
by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1);
qed "real_gt_half_sum";

Goal "x < y ==> EX r::real. x < r & r < y";
by (blast_tac (claset() addSIs [real_less_half_sum,
				real_gt_half_sum]) 1);
qed "real_dense";

Goal "(EX n. rinv(real_of_posnat n) < r) = (EX n. 1r < r * real_of_posnat n)";
by (Step_tac 1);
by (dres_inst_tac [("n1","n")] (real_of_posnat_less_zero 
				RS real_mult_less_mono1) 1);
by (dres_inst_tac [("n2","n")] (real_of_posnat_less_zero RS 
				real_rinv_gt_zero RS real_mult_less_mono1) 2);
by (auto_tac (claset(),
	      simpset() addsimps [(real_of_posnat_less_zero RS 
				   real_not_refl2 RS not_sym),
				  real_mult_assoc]));
qed "real_of_posnat_rinv_Ex_iff";

Goal "(rinv(real_of_posnat n) < r) = (1r < r * real_of_posnat n)";
by (Step_tac 1);
by (dres_inst_tac [("n1","n")] (real_of_posnat_less_zero 
                       RS real_mult_less_mono1) 1);
by (dres_inst_tac [("n2","n")] (real_of_posnat_less_zero RS 
				real_rinv_gt_zero RS real_mult_less_mono1) 2);
by (auto_tac (claset(), simpset() addsimps [real_mult_assoc]));
qed "real_of_posnat_rinv_iff";

Goal "(rinv(real_of_posnat n) <= r) = (1r <= r * real_of_posnat n)";
by (Step_tac 1);
by (dres_inst_tac [("n2","n")] (real_of_posnat_less_zero RS 
    real_less_imp_le RS real_mult_le_le_mono1) 1);
by (dres_inst_tac [("n3","n")] (real_of_posnat_less_zero RS 
        real_rinv_gt_zero RS real_less_imp_le RS 
        real_mult_le_le_mono1) 2);
by (auto_tac (claset(), simpset() addsimps real_mult_ac));
qed "real_of_posnat_rinv_le_iff";

Goalw [real_of_posnat_def] "(real_of_posnat n < real_of_posnat m) = (n < m)";
by Auto_tac;
qed "real_of_posnat_less_iff";

Addsimps [real_of_posnat_less_iff];

Goal "0r < u  ==> (u < rinv (real_of_posnat n)) = (real_of_posnat n < rinv(u))";
by (Step_tac 1);
by (res_inst_tac [("n2","n")] (real_of_posnat_less_zero RS 
    real_rinv_gt_zero RS real_mult_less_cancel1) 1);
by (res_inst_tac [("x1","u")] ( real_rinv_gt_zero
   RS real_mult_less_cancel1) 2);
by (auto_tac (claset(),
	      simpset() addsimps [real_of_posnat_less_zero, 
    real_not_refl2 RS not_sym]));
by (res_inst_tac [("z","u")] real_mult_less_cancel2 1);
by (res_inst_tac [("n1","n")] (real_of_posnat_less_zero RS 
    real_mult_less_cancel2) 3);
by (auto_tac (claset(),
	      simpset() addsimps [real_of_posnat_less_zero, 
    real_not_refl2 RS not_sym,real_mult_assoc RS sym]));
qed "real_of_posnat_less_rinv_iff";

Goal "0r < u ==> (u = rinv(real_of_posnat n)) = (real_of_posnat n = rinv u)";
by (auto_tac (claset(),
	      simpset() addsimps [real_rinv_rinv,
    real_of_posnat_less_zero,real_not_refl2 RS not_sym]));
qed "real_of_posnat_rinv_eq_iff";

Goal "[| 0r < r; r < x |] ==> rinv x < rinv r";
by (ftac real_less_trans 1 THEN assume_tac 1);
by (ftac real_rinv_gt_zero 1);
by (forw_inst_tac [("x","x")] real_rinv_gt_zero 1);
by (forw_inst_tac [("x","r"),("z","rinv 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_rinv_gt_zero 1);
by (forw_inst_tac [("x","1r"),("z","rinv 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_rinv_less_swap";

Goal "[| 0r < r; 0r < x|] ==> (r < x) = (rinv x < rinv r)";
by (auto_tac (claset() addIs [real_rinv_less_swap],simpset()));
by (res_inst_tac [("t","r")] (real_rinv_rinv RS subst) 1);
by (etac (real_not_refl2 RS not_sym) 1);
by (res_inst_tac [("t","x")] (real_rinv_rinv RS subst) 1);
by (etac (real_not_refl2 RS not_sym) 1);
by (auto_tac (claset() addIs [real_rinv_less_swap],
	      simpset() addsimps [real_rinv_gt_zero]));
qed "real_rinv_less_iff";

Goal "r < r + rinv(real_of_posnat n)";
by (res_inst_tac [("C","-r")] real_less_add_left_cancel 1);
by (full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1);
qed "real_add_rinv_real_of_posnat_less";
Addsimps [real_add_rinv_real_of_posnat_less];

Goal "r <= r + rinv(real_of_posnat n)";
by (rtac real_less_imp_le 1);
by (Simp_tac 1);
qed "real_add_rinv_real_of_posnat_le";
Addsimps [real_add_rinv_real_of_posnat_le];

Goal "r + (-rinv(real_of_posnat n)) < r";
by (res_inst_tac [("C","-r")] real_less_add_left_cancel 1);
by (full_simp_tac (simpset() addsimps [real_add_assoc RS sym,
				       real_minus_zero_less_iff2]) 1);
qed "real_add_minus_rinv_real_of_posnat_less";
Addsimps [real_add_minus_rinv_real_of_posnat_less];

Goal "r + (-rinv(real_of_posnat n)) <= r";
by (rtac real_less_imp_le 1);
by (Simp_tac 1);
qed "real_add_minus_rinv_real_of_posnat_le";
Addsimps [real_add_minus_rinv_real_of_posnat_le];

Goal "0r < r ==> r*(1r + (-rinv(real_of_posnat n))) < r";
by (simp_tac (simpset() addsimps [real_add_mult_distrib2]) 1);
by (res_inst_tac [("C","-r")] real_less_add_left_cancel 1);
by (auto_tac (claset() addIs [real_mult_order],
	      simpset() addsimps [real_add_assoc RS sym,
				  real_minus_mult_eq2 RS sym,
				  real_minus_zero_less_iff2]));
qed "real_mult_less_self";

Goal "0r <= 1r + (-rinv(real_of_posnat n))";
by (res_inst_tac [("C","rinv(real_of_posnat n)")] real_le_add_right_cancel 1);
by (simp_tac (simpset() addsimps [real_add_assoc,
				  real_of_posnat_rinv_le_iff]) 1);
qed "real_add_one_minus_rinv_ge_zero";

Goal "0r < r ==> 0r <= r*(1r + (-rinv(real_of_posnat n)))";
by (dtac (real_add_one_minus_rinv_ge_zero RS real_mult_le_less_mono1) 1);
by Auto_tac;
qed "real_mult_add_one_minus_ge_zero";

Goal "x*y = 0r ==> x = 0r | y = 0r";
by (auto_tac (claset() addIs [ccontr] addDs [real_mult_not_zero],
	      simpset()));
qed "real_mult_zero_disj";
 
Goal "x + x*y = x*(1r + y)";
by (simp_tac (simpset() addsimps [real_add_mult_distrib2]) 1);
qed "real_add_mult_distrib_one";

Goal "[| x ~= 1r; y * x = y |] ==> y = 0r";
by (dtac (sym RS (real_eq_minus_iff RS iffD1)) 1);
by (dtac sym 1);
by (asm_full_simp_tac (simpset() addsimps [real_minus_mult_eq2,
    real_add_mult_distrib_one]) 1);
by (dtac real_mult_zero_disj 1);
by (auto_tac (claset(),
	      simpset() addsimps [real_eq_minus_iff2 RS sym]));
qed "real_mult_eq_self_zero";
Addsimps [real_mult_eq_self_zero];

Goal "[| x ~= 1r; y = y * x |] ==> y = 0r";
by (dtac sym 1);
by (Asm_full_simp_tac 1);
qed "real_mult_eq_self_zero2";
Addsimps [real_mult_eq_self_zero2];

Goal "[| 0r <= x*y; 0r < x |] ==> 0r <= y";
by (ftac real_rinv_gt_zero 1);
by (dres_inst_tac [("x","rinv x")] real_less_le_mult_order 1);
by (dtac (real_not_refl2 RS not_sym RS real_mult_inv_left) 2);
by (auto_tac (claset(),
	      simpset() addsimps [real_mult_assoc RS sym]));
qed "real_mult_ge_zero_cancel";

Goal "[|x ~= 0r; y ~= 0r |] ==> rinv(x) + rinv(y) = (x + y)*rinv(x*y)";
by (asm_full_simp_tac (simpset() addsimps 
		       [real_rinv_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_rinv_add";

(*----------------------------------------------------------------------------
     Another embedding of the naturals in the reals (see real_of_posnat)
 ----------------------------------------------------------------------------*)
Goalw [real_of_nat_def] "real_of_nat 0 = 0r";
by (full_simp_tac (simpset() addsimps [real_of_posnat_one]) 1);
qed "real_of_nat_zero";

Goalw [real_of_nat_def] "real_of_nat 1 = 1r";
by (full_simp_tac (simpset() addsimps [real_of_posnat_two,
    real_add_assoc]) 1);
qed "real_of_nat_one";

Goalw [real_of_nat_def]
          "real_of_nat n1 + real_of_nat n2 = real_of_nat (n1 + n2)";
by (simp_tac (simpset() addsimps 
    [real_of_posnat_add,real_add_assoc RS sym]) 1);
qed "real_of_nat_add";

Goalw [real_of_nat_def] "real_of_nat (Suc n) = real_of_nat n + 1r";
by (simp_tac (simpset() addsimps [real_of_posnat_Suc] @ real_add_ac) 1);
qed "real_of_nat_Suc";
    
Goalw [real_of_nat_def] "(n < m) = (real_of_nat n < real_of_nat m)";
by Auto_tac;
qed "real_of_nat_less_iff";

Addsimps [real_of_nat_less_iff RS sym];

Goal "inj real_of_nat";
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";

Goalw [real_of_nat_def] "0r <= real_of_nat n";
by (res_inst_tac [("C","1r")] real_le_add_right_cancel 1);
by (asm_full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
qed "real_of_nat_ge_zero";
Addsimps [real_of_nat_ge_zero];

Goal "real_of_nat n1 * real_of_nat n2 = real_of_nat (n1 * n2)";
by (induct_tac "n1" 1);
by (dtac sym 2);
by (auto_tac (claset(),
	      simpset() addsimps [real_of_nat_zero,
				  real_of_nat_add RS sym,real_of_nat_Suc,
				  real_add_mult_distrib, real_add_commute]));
qed "real_of_nat_mult";

Goal "(real_of_nat n = real_of_nat m) = (n = m)";
by (auto_tac (claset() addDs [inj_real_of_nat RS injD],
              simpset()));
qed "real_of_nat_eq_cancel";

(*------- lemmas -------*)
context NatDef.thy;

Goal "!!m. [| m < Suc n; n <= m |] ==> m = n";
by (auto_tac (claset() addSDs [le_imp_less_or_eq] addIs [less_asym],
	      simpset() addsimps [less_Suc_eq]));
qed "lemma_nat_not_dense";

Goal "!!m. [| m <= Suc n; n < m |] ==> m = Suc n";
by (auto_tac (claset() addSDs [le_imp_less_or_eq] addIs [less_asym],
	      simpset() addsimps [le_Suc_eq]));
qed "lemma_nat_not_dense2";

Goal "!!m. m < Suc n ==> ~ Suc n <= m";
by (blast_tac (claset() addDs [less_le_trans] addIs [less_asym]) 1);
qed "lemma_not_leI";

(*------- lemmas -------*)
context Arith.thy;

Goal "n < Suc(m) ==> Suc(m)-n = Suc(m-n)";
by (dtac rev_mp 1);
by (res_inst_tac [("m","m"),("n","n")] diff_induct 1);
by (ALLGOALS Asm_simp_tac);
qed "Suc_diff_n";


context thy;

(* Goalw  [real_of_nat_def] 
   "real_of_nat (n1 - n2) = real_of_nat n1 + -real_of_nat n2";*)

Goal "n2 < n1 --> real_of_nat (n1 - n2) = real_of_nat n1 + (-real_of_nat n2)";
by (induct_tac "n1" 1);
by (Step_tac 1 THEN dtac leI 1 THEN dtac sym 2);
by (dtac lemma_nat_not_dense 1);
by (auto_tac (claset(),
	      simpset() addsimps [real_of_nat_Suc, real_of_nat_zero] @ 
	                         real_add_ac));
by (asm_full_simp_tac (simpset() addsimps [real_of_nat_one RS sym,
					   real_of_nat_add,Suc_diff_n]) 1);
qed "real_of_nat_minus";