src/HOL/Real/Real.ML

-(*  Title       : Real.ML
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
-    Description : The reals
+(*  Title:      HOL/Real/Real.ML
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1998  University of Cambridge
+
+Type "real" is a linear order
 *)
 
-(*** Proving that realrel is an equivalence relation ***)
-
-Goal "[| (x1::preal) + y2 = x2 + y1; x2 + y3 = x3 + y2 |] \
-\            ==> x1 + y3 = x3 + y1";        
-by (res_inst_tac [("C","y2")] preal_add_right_cancel 1);
-by (rotate_tac 1 1 THEN dtac sym 1);
-by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1);
-by (rtac (preal_add_left_commute RS subst) 1);
-by (res_inst_tac [("x1","x1")] (preal_add_assoc RS subst) 1);
-by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1);
-qed "preal_trans_lemma";
-
-(** Natural deduction for realrel **)
-
-Goalw [realrel_def]
-    "(((x1,y1),(x2,y2)): realrel) = (x1 + y2 = x2 + y1)";
-by (Blast_tac 1);
-qed "realrel_iff";
-
-Goalw [realrel_def]
-    "[| x1 + y2 = x2 + y1 |] ==> ((x1,y1),(x2,y2)): realrel";
-by (Blast_tac  1);
-qed "realrelI";
-
-Goalw [realrel_def]
-  "p: realrel --> (EX x1 y1 x2 y2. \
-\                  p = ((x1,y1),(x2,y2)) & x1 + y2 = x2 + y1)";
-by (Blast_tac 1);
-qed "realrelE_lemma";
-
-val [major,minor] = goal thy
-  "[| p: realrel;  \
-\     !!x1 y1 x2 y2. [| p = ((x1,y1),(x2,y2));  x1+y2 = x2+y1 \
-\                    |] ==> Q |] ==> Q";
-by (cut_facts_tac [major RS (realrelE_lemma RS mp)] 1);
-by (REPEAT (eresolve_tac [asm_rl,exE,conjE,minor] 1));
-qed "realrelE";
-
-AddSIs [realrelI];
-AddSEs [realrelE];
-
-Goal "(x,x): realrel";
-by (stac surjective_pairing 1 THEN rtac (refl RS realrelI) 1);
-qed "realrel_refl";
-
-Goalw [equiv_def, refl_def, sym_def, trans_def]
-    "equiv {x::(preal*preal).True} realrel";
-by (fast_tac (claset() addSIs [realrel_refl] 
-                      addSEs [sym,preal_trans_lemma]) 1);
-qed "equiv_realrel";
-
-val equiv_realrel_iff =
-    [TrueI, TrueI] MRS 
-    ([CollectI, CollectI] MRS 
-    (equiv_realrel RS eq_equiv_class_iff));
-
-Goalw  [real_def,realrel_def,quotient_def] "realrel^^{(x,y)}:real";
-by (Blast_tac 1);
-qed "realrel_in_real";
-
-Goal "inj_on Abs_real real";
-by (rtac inj_on_inverseI 1);
-by (etac Abs_real_inverse 1);
-qed "inj_on_Abs_real";
-
-Addsimps [equiv_realrel_iff,inj_on_Abs_real RS inj_on_iff,
-          realrel_iff, realrel_in_real, Abs_real_inverse];
-
-Addsimps [equiv_realrel RS eq_equiv_class_iff];
-val eq_realrelD = equiv_realrel RSN (2,eq_equiv_class);
-
-Goal "inj(Rep_real)";
-by (rtac inj_inverseI 1);
-by (rtac Rep_real_inverse 1);
-qed "inj_Rep_real";
-
-(** real_preal: the injection from preal to real **)
-Goal "inj(real_preal)";
-by (rtac injI 1);
-by (rewtac real_preal_def);
-by (dtac (inj_on_Abs_real RS inj_onD) 1);
-by (REPEAT (rtac realrel_in_real 1));
-by (dtac eq_equiv_class 1);
-by (rtac equiv_realrel 1);
-by (Blast_tac 1);
-by Safe_tac;
-by (Asm_full_simp_tac 1);
-qed "inj_real_preal";
-
-val [prem] = goal thy
-    "(!!x y. z = Abs_real(realrel^^{(x,y)}) ==> P) ==> P";
-by (res_inst_tac [("x1","z")] 
-    (rewrite_rule [real_def] Rep_real RS quotientE) 1);
-by (dres_inst_tac [("f","Abs_real")] arg_cong 1);
-by (res_inst_tac [("p","x")] PairE 1);
-by (rtac prem 1);
-by (asm_full_simp_tac (simpset() addsimps [Rep_real_inverse]) 1);
-qed "eq_Abs_real";
-
-(**** real_minus: additive inverse on real ****)
-
-Goalw [congruent_def]
-  "congruent realrel (%p. split (%x y. realrel^^{(y,x)}) p)";
-by Safe_tac;
-by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1);
-qed "real_minus_congruent";
-
-(*Resolve th against the corresponding facts for real_minus*)
-val real_minus_ize = RSLIST [equiv_realrel, real_minus_congruent];
-
-Goalw [real_minus_def]
-      "%~ (Abs_real(realrel^^{(x,y)})) = Abs_real(realrel ^^ {(y,x)})";
-by (res_inst_tac [("f","Abs_real")] arg_cong 1);
-by (simp_tac (simpset() addsimps 
-   [realrel_in_real RS Abs_real_inverse,real_minus_ize UN_equiv_class]) 1);
-qed "real_minus";
-
-Goal "%~ (%~ z) = z";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (asm_simp_tac (simpset() addsimps [real_minus]) 1);
-qed "real_minus_minus";
-
-Addsimps [real_minus_minus];
-
-Goal "inj(real_minus)";
-by (rtac injI 1);
-by (dres_inst_tac [("f","real_minus")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [real_minus_minus]) 1);
-qed "inj_real_minus";
-
-Goalw [real_zero_def] "%~0r = 0r";
-by (simp_tac (simpset() addsimps [real_minus]) 1);
-qed "real_minus_zero";
-
-Addsimps [real_minus_zero];
-
-Goal "(%~x = 0r) = (x = 0r)"; 
-by (res_inst_tac [("z","x")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_zero_def,
-    real_minus] @ preal_add_ac));
-qed "real_minus_zero_iff";
-
-Addsimps [real_minus_zero_iff];
-
-Goal "(%~x ~= 0r) = (x ~= 0r)"; 
-by Auto_tac;
-qed "real_minus_not_zero_iff";
-
-(*** Congruence property for addition ***)
-Goalw [congruent2_def]
-    "congruent2 realrel (%p1 p2.                  \
-\         split (%x1 y1. split (%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)";
-by Safe_tac;
-by (asm_simp_tac (simpset() addsimps [preal_add_assoc]) 1);
-by (res_inst_tac [("z1.1","x1a")] (preal_add_left_commute RS ssubst) 1);
-by (asm_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1);
-by (asm_simp_tac (simpset() addsimps preal_add_ac) 1);
-qed "real_add_congruent2";
-
-(*Resolve th against the corresponding facts for real_add*)
-val real_add_ize = RSLIST [equiv_realrel, real_add_congruent2];
-
-Goalw [real_add_def]
-  "Abs_real(realrel^^{(x1,y1)}) + Abs_real(realrel^^{(x2,y2)}) = \
-\  Abs_real(realrel^^{(x1+x2, y1+y2)})";
-by (asm_simp_tac
-    (simpset() addsimps [real_add_ize UN_equiv_class2]) 1);
-qed "real_add";
-
-Goal "(z::real) + w = w + z";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (res_inst_tac [("z","w")] eq_Abs_real 1);
-by (asm_simp_tac (simpset() addsimps preal_add_ac @ [real_add]) 1);
-qed "real_add_commute";
-
-Goal "((z1::real) + z2) + z3 = z1 + (z2 + z3)";
-by (res_inst_tac [("z","z1")] eq_Abs_real 1);
-by (res_inst_tac [("z","z2")] eq_Abs_real 1);
-by (res_inst_tac [("z","z3")] eq_Abs_real 1);
-by (asm_simp_tac (simpset() addsimps [real_add, preal_add_assoc]) 1);
-qed "real_add_assoc";
-
-(*For AC rewriting*)
-Goal "(x::real)+(y+z)=y+(x+z)";
-by (rtac (real_add_commute RS trans) 1);
-by (rtac (real_add_assoc RS trans) 1);
-by (rtac (real_add_commute RS arg_cong) 1);
-qed "real_add_left_commute";
-
-(* real addition is an AC operator *)
-val real_add_ac = [real_add_assoc,real_add_commute,real_add_left_commute];
-
-Goalw [real_preal_def,real_zero_def] "0r + z = z";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add] @ preal_add_ac) 1);
-qed "real_add_zero_left";
-
-Goal "z + 0r = z";
-by (simp_tac (simpset() addsimps [real_add_zero_left,real_add_commute]) 1);
-qed "real_add_zero_right";
-
-Goalw [real_zero_def] "z + %~z = 0r";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (asm_full_simp_tac (simpset() addsimps [real_minus,
-        real_add, preal_add_commute]) 1);
-qed "real_add_minus";
-
-Goal "%~z + z = 0r";
-by (simp_tac (simpset() addsimps 
-    [real_add_commute,real_add_minus]) 1);
-qed "real_add_minus_left";
-
-Goal "? y. (x::real) + y = 0r";
-by (blast_tac (claset() addIs [real_add_minus]) 1);
-qed "real_minus_ex";
-
-Goal "?! y. (x::real) + y = 0r";
-by (auto_tac (claset() addIs [real_add_minus],simpset()));
-by (dres_inst_tac [("f","%x. ya+x")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute,
-    real_add_zero_right,real_add_zero_left]) 1);
-qed "real_minus_ex1";
-
-Goal "?! y. y + (x::real) = 0r";
-by (auto_tac (claset() addIs [real_add_minus_left],simpset()));
-by (dres_inst_tac [("f","%x. x+ya")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_commute,
-    real_add_zero_right,real_add_zero_left]) 1);
-qed "real_minus_left_ex1";
-
-Goal "x + y = 0r ==> x = %~y";
-by (cut_inst_tac [("z","y")] real_add_minus_left 1);
-by (res_inst_tac [("x1","y")] (real_minus_left_ex1 RS ex1E) 1);
-by (Blast_tac 1);
-qed "real_add_minus_eq_minus";
-
-Goal "? y. x = %~y";
-by (cut_inst_tac [("x","x")] real_minus_ex 1);
-by (etac exE 1 THEN dtac real_add_minus_eq_minus 1);
-by (Blast_tac 1);
-qed "real_as_add_inverse_ex";
-
-(* real_minus_add_distrib *)
-Goal "%~(x + y) = %~x + %~y";
-by (res_inst_tac [("z","x")] eq_Abs_real 1);
-by (res_inst_tac [("z","y")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_minus,real_add]));
-qed "real_minus_add_eq";
-
-val real_minus_add_distrib = real_minus_add_eq;
-
-Goal "((x::real) + y = x + z) = (y = z)";
-by (Step_tac 1);
-by (dres_inst_tac [("f","%t.%~x + t")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_minus_left,
-                 real_add_assoc RS sym,real_add_zero_left]) 1);
-qed "real_add_left_cancel";
-
-Goal "(y + (x::real)= z + x) = (y = z)";
-by (simp_tac (simpset() addsimps [real_add_commute,real_add_left_cancel]) 1);
-qed "real_add_right_cancel";
-
-(*** Congruence property for multiplication ***)
-Goal "!!(x1::preal). [| x1 + y2 = x2 + y1 |] ==> \
-\         x * x1 + y * y1 + (x * y2 + x2 * y) = \
-\         x * x2 + y * y2 + (x * y1 + x1 * y)";
-by (asm_full_simp_tac (simpset() addsimps [preal_add_left_commute,
-    preal_add_assoc RS sym,preal_add_mult_distrib2 RS sym]) 1);
-by (rtac (preal_mult_commute RS subst) 1);
-by (res_inst_tac [("y1","x2")] (preal_mult_commute RS subst) 1);
-by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc,
-    preal_add_mult_distrib2 RS sym]) 1);
-by (asm_full_simp_tac (simpset() addsimps [preal_add_commute]) 1);
-qed "real_mult_congruent2_lemma";
-
-Goal 
-    "congruent2 realrel (%p1 p2.                  \
-\         split (%x1 y1. split (%x2 y2. realrel^^{(x1*x2 + y1*y2, x1*y2+x2*y1)}) p2) p1)";
-by (rtac (equiv_realrel RS congruent2_commuteI) 1);
-by Safe_tac;
-by (rewtac split_def);
-by (asm_simp_tac (simpset() addsimps [preal_mult_commute,preal_add_commute]) 1);
-by (auto_tac (claset(),simpset() addsimps [real_mult_congruent2_lemma]));
-qed "real_mult_congruent2";
-
-(*Resolve th against the corresponding facts for real_mult*)
-val real_mult_ize = RSLIST [equiv_realrel, real_mult_congruent2];
-
-Goalw [real_mult_def]
-   "Abs_real((realrel^^{(x1,y1)})) * Abs_real((realrel^^{(x2,y2)})) =   \
-\   Abs_real(realrel ^^ {(x1*x2+y1*y2,x1*y2+x2*y1)})";
-by (simp_tac (simpset() addsimps [real_mult_ize UN_equiv_class2]) 1);
-qed "real_mult";
-
-Goal "(z::real) * w = w * z";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (res_inst_tac [("z","w")] eq_Abs_real 1);
-by (asm_simp_tac
-    (simpset() addsimps [real_mult] @ preal_add_ac @ preal_mult_ac) 1);
-qed "real_mult_commute";
-
-Goal "((z1::real) * z2) * z3 = z1 * (z2 * z3)";
-by (res_inst_tac [("z","z1")] eq_Abs_real 1);
-by (res_inst_tac [("z","z2")] eq_Abs_real 1);
-by (res_inst_tac [("z","z3")] eq_Abs_real 1);
-by (asm_simp_tac (simpset() addsimps [preal_add_mult_distrib2,real_mult] @ 
-                                     preal_add_ac @ preal_mult_ac) 1);
-qed "real_mult_assoc";
-
-qed_goal "real_mult_left_commute" thy
-    "(z1::real) * (z2 * z3) = z2 * (z1 * z3)"
- (fn _ => [rtac (real_mult_commute RS trans) 1, rtac (real_mult_assoc RS trans) 1,
-           rtac (real_mult_commute RS arg_cong) 1]);
-
-(* real multiplication is an AC operator *)
-val real_mult_ac = [real_mult_assoc, real_mult_commute, real_mult_left_commute];
-
-Goalw [real_one_def,pnat_one_def] "1r * z = z";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (asm_full_simp_tac (simpset() addsimps [real_mult,
-    preal_add_mult_distrib2,preal_mult_1_right] 
-    @ preal_mult_ac @ preal_add_ac) 1);
-qed "real_mult_1";
-
-Goal "z * 1r = z";
-by (simp_tac (simpset() addsimps [real_mult_commute,
-    real_mult_1]) 1);
-qed "real_mult_1_right";
-
-Goalw [real_zero_def,pnat_one_def] "0r * z = 0r";
-by (res_inst_tac [("z","z")] eq_Abs_real 1);
-by (asm_full_simp_tac (simpset() addsimps [real_mult,
-    preal_add_mult_distrib2,preal_mult_1_right] 
-    @ preal_mult_ac @ preal_add_ac) 1);
-qed "real_mult_0";
-
-Goal "z * 0r = 0r";
-by (simp_tac (simpset() addsimps [real_mult_commute,
-    real_mult_0]) 1);
-qed "real_mult_0_right";
-
-Addsimps [real_mult_0_right,real_mult_0];
-
-Goal "%~(x * y) = %~x * y";
-by (res_inst_tac [("z","x")] eq_Abs_real 1);
-by (res_inst_tac [("z","y")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_minus,real_mult] 
-    @ preal_mult_ac @ preal_add_ac));
-qed "real_minus_mult_eq1";
-
-Goal "%~(x * y) = x * %~y";
-by (res_inst_tac [("z","x")] eq_Abs_real 1);
-by (res_inst_tac [("z","y")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_minus,real_mult] 
-    @ preal_mult_ac @ preal_add_ac));
-qed "real_minus_mult_eq2";
-
-Goal "%~x*%~y = x*y";
-by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym,
-    real_minus_mult_eq1 RS sym]) 1);
-qed "real_minus_mult_cancel";
-
-Addsimps [real_minus_mult_cancel];
-
-Goal "%~x*y = x*%~y";
-by (full_simp_tac (simpset() addsimps [real_minus_mult_eq2 RS sym,
-    real_minus_mult_eq1 RS sym]) 1);
-qed "real_minus_mult_commute";
-
-(*-----------------------------------------------------------------------------
-
- -----------------------------------------------------------------------------*)
-
-(** Lemmas **)
-
-qed_goal "real_add_assoc_cong" thy
-    "!!z. (z::real) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
- (fn _ => [(asm_simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1)]);
-
-qed_goal "real_add_assoc_swap" thy "(z::real) + (v + w) = v + (z + w)"
- (fn _ => [(REPEAT (ares_tac [real_add_commute RS real_add_assoc_cong] 1))]);
-
-Goal "((z1::real) + z2) * w = (z1 * w) + (z2 * w)";
-by (res_inst_tac [("z","z1")] eq_Abs_real 1);
-by (res_inst_tac [("z","z2")] eq_Abs_real 1);
-by (res_inst_tac [("z","w")] eq_Abs_real 1);
-by (asm_simp_tac 
-    (simpset() addsimps [preal_add_mult_distrib2, real_add, real_mult] @ 
-                        preal_add_ac @ preal_mult_ac) 1);
-qed "real_add_mult_distrib";
-
-val real_mult_commute'= read_instantiate [("z","w")] real_mult_commute;
-
-Goal "(w::real) * (z1 + z2) = (w * z1) + (w * z2)";
-by (simp_tac (simpset() addsimps [real_mult_commute',real_add_mult_distrib]) 1);
-qed "real_add_mult_distrib2";
-
-val real_mult_simps = [real_mult_1, real_mult_1_right];
-Addsimps real_mult_simps;
-
-(*** one and zero are distinct ***)
-Goalw [real_zero_def,real_one_def] "0r ~= 1r";
-by (auto_tac (claset(),simpset() addsimps 
-   [preal_self_less_add_left RS preal_not_refl2]));
-qed "real_zero_not_eq_one";
-
-(*** existence of inverse ***)
-(** lemma -- alternative definition for 0r **)
-Goalw [real_zero_def] "0r = Abs_real (realrel ^^ {(x, x)})";
-by (auto_tac (claset(),simpset() addsimps [preal_add_commute]));
-qed "real_zero_iff";
-
-Goalw [real_zero_def,real_one_def] 
-          "!!(x::real). x ~= 0r ==> ? y. x*y = 1r";
-by (res_inst_tac [("z","x")] eq_Abs_real 1);
-by (cut_inst_tac [("r1.0","xa"),("r2.0","y")] preal_linear 1);
-by (auto_tac (claset() addSDs [preal_less_add_left_Ex],
-           simpset() addsimps [real_zero_iff RS sym]));
-by (res_inst_tac [("x","Abs_real (realrel ^^ {(@#$#1p,pinv(D)+@#$#1p)})")] exI 1);
-by (res_inst_tac [("x","Abs_real (realrel ^^ {(pinv(D)+@#$#1p,@#$#1p)})")] exI 2);
-by (auto_tac (claset(),simpset() addsimps [real_mult,
-    pnat_one_def,preal_mult_1_right,preal_add_mult_distrib2,
-    preal_add_mult_distrib,preal_mult_1,preal_mult_inv_right] 
-    @ preal_add_ac @ preal_mult_ac));
-qed "real_mult_inv_right_ex";
-
-Goal "!!(x::real). x ~= 0r ==> ? y. y*x = 1r";
-by (asm_simp_tac (simpset() addsimps [real_mult_commute,
-    real_mult_inv_right_ex]) 1);
-qed "real_mult_inv_left_ex";
-
-Goalw [rinv_def] "!!(x::real). x ~= 0r ==> rinv(x)*x = 1r";
-by (forward_tac [real_mult_inv_left_ex] 1);
-by (Step_tac 1);
-by (rtac selectI2 1);
-by Auto_tac;
-qed "real_mult_inv_left";
-
-Goal "!!(x::real). x ~= 0r ==> x*rinv(x) = 1r";
-by (auto_tac (claset() addIs [real_mult_commute RS subst],
-              simpset() addsimps [real_mult_inv_left]));
-qed "real_mult_inv_right";
-
-Goal "(c::real) ~= 0r ==> (c*a=c*b) = (a=b)";
-by Auto_tac;
-by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac)  1);
-qed "real_mult_left_cancel";
-    
-Goal "(c::real) ~= 0r ==> (a*c=b*c) = (a=b)";
-by (Step_tac 1);
-by (dres_inst_tac [("f","%x. x*rinv c")] arg_cong 1);
-by (asm_full_simp_tac (simpset() addsimps [real_mult_inv_right] @ real_mult_ac)  1);
-qed "real_mult_right_cancel";
-
-Goalw [rinv_def] "x ~= 0r ==> rinv(x) ~= 0r";
-by (forward_tac [real_mult_inv_left_ex] 1);
-by (etac exE 1);
-by (rtac selectI2 1);
-by (auto_tac (claset(),simpset() addsimps [real_mult_0,
-    real_zero_not_eq_one]));
-qed "rinv_not_zero";
-
-Addsimps [real_mult_inv_left,real_mult_inv_right];
-
-Goal "x ~= 0r ==> rinv(rinv x) = x";
-by (res_inst_tac [("c1","rinv x")] (real_mult_right_cancel RS iffD1) 1);
-by (etac rinv_not_zero 1);
-by (auto_tac (claset() addDs [rinv_not_zero],simpset()));
-qed "real_rinv_rinv";
-
-Goalw [rinv_def] "rinv(1r) = 1r";
-by (cut_facts_tac [real_zero_not_eq_one RS 
-       not_sym RS real_mult_inv_left_ex] 1);
-by (etac exE 1);
-by (rtac selectI2 1);
-by (auto_tac (claset(),simpset() addsimps 
-    [real_zero_not_eq_one RS not_sym]));
-qed "real_rinv_1";
-
-Goal "x ~= 0r ==> rinv(%~x) = %~rinv(x)";
-by (res_inst_tac [("c1","%~x")] (real_mult_right_cancel RS iffD1) 1);
-by Auto_tac;
-qed "real_minus_rinv";
-
-      (*** theorems for ordering ***)
-(* prove introduction and elimination rules for real_less *)
-
-Goalw [real_less_def]
- "P < (Q::real) = (EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & \
-\                                  (x1,y1::preal):Rep_real(P) & \
-\                                  (x2,y2):Rep_real(Q))";
-by (Blast_tac 1);
-qed "real_less_iff";
-
-Goalw [real_less_def]
- "[| x1 + y2 < x2 + y1; (x1,y1::preal):Rep_real(P); \
-\         (x2,y2):Rep_real(Q) |] ==> P < (Q::real)";
-by (Blast_tac 1);
-qed "real_lessI";
-
-Goalw [real_less_def]
- "!!P. [| R1 < (R2::real); \
-\         !!x1 x2 y1 y2. x1 + y2 < x2 + y1 ==> P; \
-\         !!x1 y1. (x1,y1::preal):Rep_real(R1) ==> P; \ 
-\         !!x2 y2. (x2,y2::preal):Rep_real(R2) ==> P |] \
-\     ==> P";
-by Auto_tac;
-qed "real_lessE";
-
-Goalw [real_less_def]
- "R1 < (R2::real) ==> (EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & \
-\                                  (x1,y1::preal):Rep_real(R1) & \
-\                                  (x2,y2):Rep_real(R2))";
-by (Blast_tac 1);
-qed "real_lessD";
-
-(* real_less is a strong order i.e nonreflexive and transitive *)
-(*** lemmas ***)
-Goal "!!(x::preal). [| x = y; x1 = y1 |] ==> x + y1 = x1 + y";
-by (asm_simp_tac (simpset() addsimps [preal_add_commute]) 1);
-qed "preal_lemma_eq_rev_sum";
-
-Goal "!!(b::preal). x + (b + y) = x1 + (b + y1) ==> x + y = x1 + y1";
-by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1);
-qed "preal_add_left_commute_cancel";
-
-Goal 
-     "!!(x::preal). [| x + y2a = x2a + y; \
-\                      x + y2b = x2b + y |] \
-\                   ==> x2a + y2b = x2b + y2a";
-by (dtac preal_lemma_eq_rev_sum 1);
-by (assume_tac 1);
-by (thin_tac "x + y2b = x2b + y" 1);
-by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1);
-by (dtac preal_add_left_commute_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1);
-qed "preal_lemma_for_not_refl";
-
-Goal "~ (R::real) < R";
-by (res_inst_tac [("z","R")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_less_def]));
-by (dtac preal_lemma_for_not_refl 1);
-by (assume_tac 1 THEN rotate_tac 2 1);
-by (auto_tac (claset(),simpset() addsimps [preal_less_not_refl]));
-qed "real_less_not_refl";
-
-(*** y < y ==> P ***)
-bind_thm("real_less_irrefl",real_less_not_refl RS notE);
-
-Goal "!!(x::real). x < y ==> x ~= y";
-by (auto_tac (claset(),simpset() addsimps [real_less_not_refl]));
-qed "real_not_refl2";
-
-(* lemma re-arranging and eliminating terms *)
-Goal "!! (a::preal). [| a + b = c + d; \
-\            x2b + d + (c + y2e) < a + y2b + (x2e + b) |] \
-\         ==> x2b + y2e < x2e + y2b";
-by (asm_full_simp_tac (simpset() addsimps preal_add_ac) 1);
-by (res_inst_tac [("C","c+d")] preal_add_left_less_cancel 1);
-by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1);
-qed "preal_lemma_trans";
-
-(** heavy re-writing involved*)
-Goal "!!(R1::real). [| R1 < R2; R2 < R3 |] ==> R1 < R3";
-by (res_inst_tac [("z","R1")] eq_Abs_real 1);
-by (res_inst_tac [("z","R2")] eq_Abs_real 1);
-by (res_inst_tac [("z","R3")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_less_def]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (dtac preal_lemma_for_not_refl 1 THEN assume_tac 1);
-by (blast_tac (claset() addDs [preal_add_less_mono] 
-    addIs [preal_lemma_trans]) 1);
-qed "real_less_trans";
-
-Goal "!! (R1::real). [| R1 < R2; R2 < R1 |] ==> P";
-by (dtac real_less_trans 1 THEN assume_tac 1);
-by (asm_full_simp_tac (simpset() addsimps [real_less_not_refl]) 1);
-qed "real_less_asym";
-
-(****)(****)(****)(****)(****)(****)(****)(****)(****)(****)
-    (****** Map and more real_less ******)
-(*** mapping from preal into real ***)
-Goalw [real_preal_def] 
-            "%#((z1::preal) + z2) = %#z1 + %#z2";
-by (asm_simp_tac (simpset() addsimps [real_add,
-       preal_add_mult_distrib,preal_mult_1] addsimps preal_add_ac) 1);
-qed "real_preal_add";
-
-Goalw [real_preal_def] 
-            "%#((z1::preal) * z2) = %#z1* %#z2";
-by (full_simp_tac (simpset() addsimps [real_mult,
-        preal_add_mult_distrib2,preal_mult_1,
-        preal_mult_1_right,pnat_one_def] 
-        @ preal_add_ac @ preal_mult_ac) 1);
-qed "real_preal_mult";
-
-Goalw [real_preal_def]
-      "!!(x::preal). y < x ==> ? m. Abs_real (realrel ^^ {(x,y)}) = %#m";
-by (auto_tac (claset() addSDs [preal_less_add_left_Ex],
-    simpset() addsimps preal_add_ac));
-qed "real_preal_ExI";
-
-Goalw [real_preal_def]
-      "!!(x::preal). ? m. Abs_real (realrel ^^ {(x,y)}) = %#m ==> y < x";
-by (auto_tac (claset(),simpset() addsimps 
-    [preal_add_commute,preal_add_assoc]));
-by (asm_full_simp_tac (simpset() addsimps 
-    [preal_add_assoc RS sym,preal_self_less_add_left]) 1);
-qed "real_preal_ExD";
-
-Goal "(? m. Abs_real (realrel ^^ {(x,y)}) = %#m) = (y < x)";
-by (blast_tac (claset() addSIs [real_preal_ExI,real_preal_ExD]) 1);
-qed "real_preal_iff";
-
-(*** Gleason prop 9-4.4 p 127 ***)
-Goalw [real_preal_def,real_zero_def] 
-      "? m. (x::real) = %#m | x = 0r | x = %~(%#m)";
-by (res_inst_tac [("z","x")] eq_Abs_real 1);
-by (auto_tac (claset(),simpset() addsimps [real_minus] @ preal_add_ac));
-by (cut_inst_tac [("r1.0","x"),("r2.0","y")] preal_linear 1);
-by (auto_tac (claset() addSDs [preal_less_add_left_Ex],
-    simpset() addsimps [preal_add_assoc RS sym]));
-by (auto_tac (claset(),simpset() addsimps [preal_add_commute]));
-qed "real_preal_trichotomy";
-
-Goal "!!P. [| !!m. x = %#m ==> P; \
-\             x = 0r ==> P; \
-\             !!m. x = %~(%#m) ==> P |] ==> P";
-by (cut_inst_tac [("x","x")] real_preal_trichotomy 1);
-by Auto_tac;
-qed "real_preal_trichotomyE";
-
-Goalw [real_preal_def] "%#m1 < %#m2 ==> m1 < m2";
-by (auto_tac (claset(),simpset() addsimps [real_less_def] @ preal_add_ac));
-by (auto_tac (claset(),simpset() addsimps [preal_add_assoc RS sym]));
-by (auto_tac (claset(),simpset() addsimps preal_add_ac));
-qed "real_preal_lessD";
-
-Goal "m1 < m2 ==> %#m1 < %#m2";
-by (dtac preal_less_add_left_Ex 1);
-by (auto_tac (claset(),simpset() addsimps [real_preal_add,
-    real_preal_def,real_less_def]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (simp_tac (simpset() addsimps [preal_self_less_add_left] 
-    delsimps [preal_add_less_iff2]) 1);
-qed "real_preal_lessI";
-
-Goal "(%#m1 < %#m2) = (m1 < m2)";
-by (blast_tac (claset() addIs [real_preal_lessI,real_preal_lessD]) 1);
-qed "real_preal_less_iff1";
-
-Addsimps [real_preal_less_iff1];
-
-Goal "%~ %#m < %#m";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_minus]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (full_simp_tac (simpset() addsimps preal_add_ac) 1);
-by (full_simp_tac (simpset() addsimps [preal_self_less_add_right,
-    preal_add_assoc RS sym]) 1);
-qed "real_preal_minus_less_self";
-
-Goalw [real_zero_def] "%~ %#m < 0r";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_minus]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (full_simp_tac (simpset() addsimps 
-  [preal_self_less_add_right] @ preal_add_ac) 1);
-qed "real_preal_minus_less_zero";
-
-Goal "~ 0r < %~ %#m";
-by (cut_facts_tac [real_preal_minus_less_zero] 1);
-by (fast_tac (claset() addDs [real_less_trans] 
-                        addEs [real_less_irrefl]) 1);
-qed "real_preal_not_minus_gt_zero";
-
-Goalw [real_zero_def] " 0r < %#m";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_minus]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (full_simp_tac (simpset() addsimps 
-  [preal_self_less_add_right] @ preal_add_ac) 1);
-qed "real_preal_zero_less";
-
-Goal "~ %#m < 0r";
-by (cut_facts_tac [real_preal_zero_less] 1);
-by (blast_tac (claset() addDs [real_less_trans] 
-               addEs [real_less_irrefl]) 1);
-qed "real_preal_not_less_zero";
-
-Goal "0r < %~ %~ %#m";
-by (simp_tac (simpset() addsimps 
-    [real_preal_zero_less]) 1);
-qed "real_minus_minus_zero_less";
-
-(* another lemma *)
-Goalw [real_zero_def] " 0r < %#m + %#m1";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_add]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (full_simp_tac (simpset() addsimps preal_add_ac) 1);
-by (full_simp_tac (simpset() addsimps [preal_self_less_add_right,
-    preal_add_assoc RS sym]) 1);
-qed "real_preal_sum_zero_less";
-
-Goal "%~ %#m < %#m1";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_minus]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (full_simp_tac (simpset() addsimps preal_add_ac) 1);
-by (full_simp_tac (simpset() addsimps [preal_self_less_add_right,
-    preal_add_assoc RS sym]) 1);
-qed "real_preal_minus_less_all";
-
-Goal "~ %#m < %~ %#m1";
-by (cut_facts_tac [real_preal_minus_less_all] 1);
-by (blast_tac (claset() addDs [real_less_trans] 
-               addEs [real_less_irrefl]) 1);
-qed "real_preal_not_minus_gt_all";
-
-Goal "%~ %#m1 < %~ %#m2 ==> %#m2 < %#m1";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_minus]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (auto_tac (claset(),simpset() addsimps preal_add_ac));
-by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1);
-by (auto_tac (claset(),simpset() addsimps preal_add_ac));
-qed "real_preal_minus_less_rev1";
-
-Goal "%#m1 < %#m2 ==> %~ %#m2 < %~ %#m1";
-by (auto_tac (claset(),simpset() addsimps 
-    [real_preal_def,real_less_def,real_minus]));
-by (REPEAT(rtac exI 1));
-by (EVERY[rtac conjI 1, rtac conjI 2]);
-by (REPEAT(Blast_tac 2));
-by (auto_tac (claset(),simpset() addsimps preal_add_ac));
-by (asm_full_simp_tac (simpset() addsimps [preal_add_assoc RS sym]) 1);
-by (auto_tac (claset(),simpset() addsimps preal_add_ac));
-qed "real_preal_minus_less_rev2";
-
-Goal "(%~ %#m1 < %~ %#m2) = (%#m2 < %#m1)";
-by (blast_tac (claset() addSIs [real_preal_minus_less_rev1,
-               real_preal_minus_less_rev2]) 1);
-qed "real_preal_minus_less_rev_iff";
-
-Addsimps [real_preal_minus_less_rev_iff];
-
-(*** linearity ***)
-Goal "(R1::real) < R2 | R1 = R2 | R2 < R1";
-by (res_inst_tac [("x","R1")]  real_preal_trichotomyE 1);
-by (ALLGOALS(res_inst_tac [("x","R2")]  real_preal_trichotomyE));
-by (auto_tac (claset() addSDs [preal_le_anti_sym],
-              simpset() addsimps [preal_less_le_iff,real_preal_minus_less_zero,
-               real_preal_zero_less,real_preal_minus_less_all]));
-qed "real_linear";
-
-Goal "!!(R1::real). [| R1 < R2 ==> P;  R1 = R2 ==> P; \
-\                      R2 < R1 ==> P |] ==> P";
-by (cut_inst_tac [("R1.0","R1"),("R2.0","R2")] real_linear 1);
-by Auto_tac;
-qed "real_linear_less2";
-
-(*** Properties of <= ***)
-
-Goalw [real_le_def] "~(w < z) ==> z <= (w::real)";
-by (assume_tac 1);
-qed "real_leI";
-
-Goalw [real_le_def] "z<=w ==> ~(w<(z::real))";
-by (assume_tac 1);
-qed "real_leD";
-
-val real_leE = make_elim real_leD;
-
-Goal "(~(w < z)) = (z <= (w::real))";
-by (blast_tac (claset() addSIs [real_leI,real_leD]) 1);
-qed "real_less_le_iff";
-
-Goalw [real_le_def] "~ z <= w ==> w<(z::real)";
-by (Blast_tac 1);
-qed "not_real_leE";
-
-Goalw [real_le_def] "z < w ==> z <= (w::real)";
-by (blast_tac (claset() addEs [real_less_asym]) 1);
-qed "real_less_imp_le";
-
-Goalw [real_le_def] "!!(x::real). x <= y ==> x < y | x = y";
-by (cut_facts_tac [real_linear] 1);
-by (blast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1);
-qed "real_le_imp_less_or_eq";
-
-Goalw [real_le_def] "z<w | z=w ==> z <=(w::real)";
-by (cut_facts_tac [real_linear] 1);
-by (fast_tac (claset() addEs [real_less_irrefl,real_less_asym]) 1);
-qed "real_less_or_eq_imp_le";
-
-Goal "(x <= (y::real)) = (x < y | x=y)";
-by (REPEAT(ares_tac [iffI, real_less_or_eq_imp_le, real_le_imp_less_or_eq] 1));
-qed "real_le_eq_less_or_eq";
-
-Goal "w <= (w::real)";
-by (simp_tac (simpset() addsimps [real_le_eq_less_or_eq]) 1);
-qed "real_le_refl";
-
-val prems = goal Real.thy "!!i. [| i <= j; j < k |] ==> i < (k::real)";
-by (dtac real_le_imp_less_or_eq 1);
-by (blast_tac (claset() addIs [real_less_trans]) 1);
-qed "real_le_less_trans";
-
-Goal "!! (i::real). [| i < j; j <= k |] ==> i < k";
-by (dtac real_le_imp_less_or_eq 1);
-by (blast_tac (claset() addIs [real_less_trans]) 1);
-qed "real_less_le_trans";
-
-Goal "[| i <= j; j <= k |] ==> i <= (k::real)";
-by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq,
-            rtac real_less_or_eq_imp_le, blast_tac (claset() addIs [real_less_trans])]);
-qed "real_le_trans";
-
-Goal "[| z <= w; w <= z |] ==> z = (w::real)";
-by (EVERY1 [dtac real_le_imp_less_or_eq, dtac real_le_imp_less_or_eq,
-            fast_tac (claset() addEs [real_less_irrefl,real_less_asym])]);
-qed "real_le_anti_sym";
-
-Goal "[| ~ y < x; y ~= x |] ==> x < (y::real)";
-by (rtac not_real_leE 1);
-by (blast_tac (claset() addDs [real_le_imp_less_or_eq]) 1);
-qed "not_less_not_eq_real_less";
-
-Goal "(0r < %~R) = (R < 0r)";
-by (res_inst_tac [("x","R")]  real_preal_trichotomyE 1);
-by (auto_tac (claset(),simpset() addsimps [real_preal_not_minus_gt_zero,
-                        real_preal_not_less_zero,real_preal_zero_less,
-                        real_preal_minus_less_zero]));
-qed "real_minus_zero_less_iff";
-
-Addsimps [real_minus_zero_less_iff];
-
-Goal "(%~R < 0r) = (0r < R)";
-by (res_inst_tac [("x","R")]  real_preal_trichotomyE 1);
-by (auto_tac (claset(),simpset() addsimps [real_preal_not_minus_gt_zero,
-                        real_preal_not_less_zero,real_preal_zero_less,
-                        real_preal_minus_less_zero]));
-qed "real_minus_zero_less_iff2";
+
 
 (** lemma **)
 Goal "(0r < x) = (? y. x = %#y)";
 by (auto_tac (claset(),simpset() addsimps [real_preal_zero_less]));
 by (cut_inst_tac [("x","x")] real_preal_trichotomy 1);

 
 Goal "~ 0r < y ==> !x. y < %#x";
 by (blast_tac (claset() addSIs [real_less_all_preal,real_leI]) 1);
 qed "real_less_all_real2";
 
-(**** Derive alternative definition for real_less ****)
-(** lemma **)
-Goal "!!(R::real). ? A. S + A = R";
-by (res_inst_tac [("x","%~S + R")] exI 1);
-by (simp_tac (simpset() addsimps [real_add_minus,
-    real_add_zero_right] @ real_add_ac) 1);
-qed "real_lemma_add_left_ex";
-
-Goal "!!(R::real). R < S ==> ? T. R + T = S";
-by (res_inst_tac [("x","R")]  real_preal_trichotomyE 1);
-by (ALLGOALS(res_inst_tac [("x","S")]  real_preal_trichotomyE));
-by (auto_tac (claset() addSDs [preal_le_anti_sym] addSDs [preal_less_add_left_Ex],
-              simpset() addsimps [preal_less_le_iff,real_preal_add,real_minus_add_eq,
-               real_preal_minus_less_zero,real_less_not_refl,real_minus_ex,real_add_assoc,
-               real_preal_zero_less,real_preal_minus_less_all,real_add_minus_left,
-               real_preal_not_less_zero,real_add_zero_left,real_lemma_add_left_ex]));
-qed "real_less_add_left_Ex";
-
-Goal "!!(R::real). R < S ==> ? T. 0r < T & R + T = S";
-by (res_inst_tac [("x","R")]  real_preal_trichotomyE 1);
-by (ALLGOALS(res_inst_tac [("x","S")]  real_preal_trichotomyE));
-by (auto_tac (claset() addSDs [preal_less_add_left_Ex],
-                         simpset() addsimps [real_preal_not_minus_gt_all,
-            real_preal_add, real_preal_not_less_zero,real_less_not_refl,
-    real_preal_not_minus_gt_zero,real_add_zero_left,real_minus_add_eq]));
-by (res_inst_tac [("x","%#D")] exI 1);
-by (res_inst_tac [("x","%#m+%#ma")] exI 2);
-by (res_inst_tac [("x","%#m")] exI 3);
-by (res_inst_tac [("x","%#D")] exI 4);
-by (auto_tac (claset(),simpset() addsimps [real_preal_zero_less,
-    real_preal_sum_zero_less,real_add_minus_left,real_add_assoc,
-                          real_add_minus,real_add_zero_right]));
-by (simp_tac (simpset() addsimps [real_add_assoc RS sym, 
-            real_add_minus_left,real_add_zero_left]) 1);
-qed "real_less_add_positive_left_Ex";
-
-(* lemmas *)
-(** change naff name(s)! **)
-Goal "(W < S) ==> (0r < S + %~W)";
-by (dtac real_less_add_positive_left_Ex 1);
-by (auto_tac (claset(),simpset() addsimps [real_add_minus,
-    real_add_zero_right] @ real_add_ac));
-qed "real_less_sum_gt_zero";
-
-Goal "!!S. T = S + W ==> S = T + %~W";
-by (asm_simp_tac (simpset() addsimps [real_add_minus, real_add_zero_right] 
-		                     @ real_add_ac) 1);
-qed "real_lemma_change_eq_subj";
-
-(* FIXME: long! *)
-Goal "(0r < S + %~W) ==> (W < S)";
-by (rtac ccontr 1);
-by (dtac (real_leI RS real_le_imp_less_or_eq) 1);
-by (auto_tac (claset(),
-    simpset() addsimps [real_less_not_refl,real_add_minus]));
-by (EVERY1[dtac real_less_add_positive_left_Ex, etac exE, etac conjE]);
-by (asm_full_simp_tac (simpset() addsimps [real_add_zero_left]) 1);
-by (dtac real_lemma_change_eq_subj 1);
-by (auto_tac (claset(),simpset() addsimps [real_minus_minus]));
-by (dtac real_less_sum_gt_zero 1);
-by (asm_full_simp_tac (simpset() addsimps [real_minus_add_eq] @ real_add_ac) 1);
-by (EVERY1[rotate_tac 1, dtac (real_add_left_commute RS ssubst)]);
-by (auto_tac (claset() addEs [real_less_asym],
-              simpset() addsimps [real_add_minus,real_add_zero_right]));
-qed "real_sum_gt_zero_less";
-
-Goal "(0r < S + %~W) = (W < S)";
-by (blast_tac (claset() addIs [real_less_sum_gt_zero,
-    real_sum_gt_zero_less]) 1);
-qed "real_less_sum_gt_0_iff";
-
-Goal "((x::real) < y) = (%~y < %~x)";
+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_minus,real_add_zero_right] @ real_add_ac));
+by (auto_tac (claset(), simpset() addsimps real_add_ac));
 qed "real_lemma_add_positive_imp_less";
 
 Goal "!!(R::real). ? 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 ***)
-Goal "!!(R::real). (? T. 0r < T & R + T = S) = (R < S)";
+(*Alternative definition for real_less.  NOT for rewriting*)
+Goal "!!(R::real). (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_iffdef";
-
-Goal "(0r < x) = (%~x < x)";
+				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_preal_minus_less_self]));
+by (auto_tac (claset(),
+	      simpset() addsimps 
+	      [real_gt_zero_preal_Ex,real_preal_minus_less_self]));
 qed "real_gt_zero_iff";
 
-Goal "(x < 0r) = (x < %~x)";
+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)";
+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)";
+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 "(%#m1 <= %#m2) = (m1 <= m2)";
 by (auto_tac (claset() addSIs [preal_leI],

 by (Asm_full_simp_tac 1);
 qed "real_mult_less_zero1";
 
 Goal "!!(x::real). [| 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() addsimps [real_le_refl]));
+by (auto_tac (claset() addIs [real_mult_order, real_less_imp_le],
+	      simpset()));
 qed "real_le_mult_order";
 
 Goal "!!(x::real). [| 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 (Blast_tac 1);
 by (Blast_tac 1);
 by (Blast_tac 1);
 qed "posreal_complete";
 
-(*------------------------------------------------------------------
-
- ------------------------------------------------------------------*)
-
-Goal "!!(A::real). A < B ==> A + C < B + C";
-by (dtac (real_less_iffdef RS iffD2) 1);
-by (rtac (real_less_iffdef RS iffD1) 1);
-by (REPEAT(Step_tac 1));
-by (full_simp_tac (simpset() addsimps real_add_ac) 1);
-qed "real_add_less_mono1";
+
+
+(*** 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_mono";
+
 
 Goal "!!(A::real). A < B ==> C + A < C + B";
-by (auto_tac (claset() addIs [real_add_less_mono1],
-    simpset() addsimps [real_add_commute]));
+by (Simp_tac 1);
 qed "real_add_less_mono2";
 
 Goal "!!(A::real). A + C < B + C ==> A < B";
-by (dres_inst_tac [("C","%~C")] real_add_less_mono1 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc,
-    real_add_minus,real_add_zero_right]) 1);
+by (Full_simp_tac 1);
 qed "real_less_add_right_cancel";
 
 Goal "!!(A::real). C + A < C + B ==> A < B";
-by (dres_inst_tac [("C","%~C")] real_add_less_mono2 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym,
-    real_add_minus_left,real_add_zero_left]) 1);
+by (Full_simp_tac 1);
 qed "real_less_add_left_cancel";
 
 Goal "[| 0r < x; 0r < y |] ==> 0r < x + y";
-by (REPEAT(dtac (real_gt_zero_preal_Ex RS iffD1) 1));
-by (rtac (real_gt_zero_preal_Ex RS iffD2) 1);
-by (Step_tac 1);
-by (res_inst_tac [("x","y + ya")] exI 1);
-by (full_simp_tac (simpset() addsimps [real_preal_add]) 1);
+be real_less_trans 1;
+bd real_add_less_mono2 1;
+by (Full_simp_tac 1);
 qed "real_add_order";
 
 Goal "!!(x::real). [| 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() addsimps [real_add_zero_left,
-    real_add_zero_right,real_le_refl]));
+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_less_iffdef RS iffD2) 1);
-by (dtac (real_less_iffdef RS iffD2) 1);
-by (rtac (real_less_iffdef RS iffD1) 1);
-by Auto_tac;
-by (res_inst_tac [("x","T + Ta")] exI 1);
-by (auto_tac (claset(),simpset() addsimps [real_add_order] @ real_add_ac));
+Goal "[| R1 < S1; R2 < S2 |] ==> R1 + R2 < S1 + (S2::real)";
+bd real_add_less_mono1 1;
+be real_less_trans 1;
+be real_add_less_mono2 1;
 qed "real_add_less_mono";
 
-Goal "!!(x::real). [| 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() addsimps [real_add_zero_left,
-    real_add_zero_right,real_le_refl]));
-qed "real_le_add_order";
-
 Goal "!!(q1::real). q1 <= q2  ==> x + q1 <= x + q2";
-by (dtac real_le_imp_less_or_eq 1);
-by (Step_tac 1);
-by (auto_tac (claset() addSIs [real_le_refl,
-    real_less_imp_le,real_add_less_mono1],
-    simpset() addsimps [real_add_commute]));
+by (Simp_tac 1);
 qed "real_add_left_le_mono1";
 
-Goal "!!(q1::real). q1 <= q2  ==> q1 + x <= q2 + x";
-by (auto_tac (claset() addDs [real_add_left_le_mono1],
-    simpset() addsimps [real_add_commute]));
-qed "real_add_le_mono1";
-
-Goal "!!k l::real. [|i<=j;  k<=l |] ==> i + k <= j + l";
-by (etac (real_add_le_mono1 RS real_le_trans) 1);
-by (simp_tac (simpset() addsimps [real_add_commute]) 1);
-(*j moves to the end because it is free while k, l are bound*)
-by (etac real_add_le_mono1 1);
+Goal "[|i<=j;  k<=l |] ==> i + k <= j + (l::real)";
+bd real_add_le_mono1 1;
+be 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 (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]));
+	  simpset() addsimps [real_minus_zero_less_iff2, real_zero_less_one]));
 qed "real_less_Ex";
+
 (*---------------------------------------------------------------------------------
              An embedding of the naturals in the reals
  ---------------------------------------------------------------------------------*)
 
 Goalw [real_nat_def] "%%#0 = 1r";

 qed "real_nat_less_zero";
 
 Goal "1r <= %%#n";
 by (simp_tac (simpset() addsimps [real_nat_one RS sym]) 1);
 by (induct_tac "n" 1);
-by (auto_tac (claset(),simpset () 
-    addsimps [real_nat_Suc,real_le_refl,real_nat_one]));
-by (res_inst_tac [("t","1r")] (real_add_zero_left RS subst) 1);
-by (rtac real_add_le_mono 1);
-by (auto_tac (claset(),simpset () 
-    addsimps [real_le_refl,real_nat_less_zero,
-    real_less_imp_le,real_add_zero_left]));
+by (auto_tac (claset(),
+	      simpset () addsimps [real_nat_Suc,real_nat_one,
+				   real_nat_less_zero, real_less_imp_le]));
 qed "real_nat_less_one";
 
 Goal "rinv(%%#n) ~= 0r";
 by (rtac ((real_nat_less_zero RS 
     real_not_refl2 RS not_sym) RS rinv_not_zero) 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_minus,real_add_zero_right,
-    real_add_left_commute]) 1);
+				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);
+	   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";

 qed "real_mult_less_mono2";
 
 Goal "!!(x::real). [| 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 
+by (auto_tac (claset(),
+	      simpset() addsimps 
      [real_mult_assoc,real_not_refl2 RS not_sym]));
 qed "real_mult_less_cancel1";
 
 Goal "!!(x::real). [| 0r<z; z*x<z*y |] ==> x<y";
 by (etac real_mult_less_cancel1 1);

 by (asm_simp_tac (simpset() addsimps [real_mult_commute,real_mult_le_less_mono1]) 1);
 qed "real_mult_le_less_mono2";
 
 Goal "!!x y (z::real). [| 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,real_le_refl],simpset()));
+by (auto_tac (claset() addIs [real_mult_le_less_mono2], simpset()));
 qed "real_mult_le_le_mono1";
 
 Goal "!!(x::real). 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);

           real_mult_less_mono1) 1);
 by (asm_full_simp_tac (simpset() addsimps [real_mult_assoc]) 1);
 qed "real_less_half_sum";
 
 Goal "!!(x::real). x < y ==> (x + y)*rinv(1r + 1r) < y";
-by (dres_inst_tac [("C","y")] real_add_less_mono1 1);
+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";

 by (Step_tac 1);
 by (dres_inst_tac [("n1","n")] (real_nat_less_zero 
                        RS real_mult_less_mono1) 1);
 by (dres_inst_tac [("n2","n")] (real_nat_less_zero RS 
         real_rinv_gt_zero RS real_mult_less_mono1) 2);
-by (auto_tac (claset(),simpset() addsimps [(real_nat_less_zero RS 
+by (auto_tac (claset(),
+	      simpset() addsimps [(real_nat_less_zero RS 
     real_not_refl2 RS not_sym),real_mult_assoc]));
 qed "real_nat_rinv_Ex_iff";
 
 Goalw [real_nat_def] "(%%#n < %%#m) = (n < m)";
 by Auto_tac;

 by (Step_tac 1);
 by (res_inst_tac [("n2","n")] (real_nat_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_nat_less_zero, 
+by (auto_tac (claset(),
+	      simpset() addsimps [real_nat_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_nat_less_zero RS 
     real_mult_less_cancel2) 3);
-by (auto_tac (claset(),simpset() addsimps [real_nat_less_zero, 
+by (auto_tac (claset(),
+	      simpset() addsimps [real_nat_less_zero, 
     real_not_refl2 RS not_sym,real_mult_assoc RS sym]));
 qed "real_nat_less_rinv_iff";
 
 Goal "0r < u ==> (u = rinv(%%#n)) = (%%#n = rinv u)";
-by (auto_tac (claset(),simpset() addsimps [real_rinv_rinv,
+by (auto_tac (claset(),
+	      simpset() addsimps [real_rinv_rinv,
     real_nat_less_zero,real_not_refl2 RS not_sym]));
 qed "real_nat_rinv_eq_iff";
 
 (*
 (*------------------------------------------------------------------

 Goalw [real_ub_def] "[| real_ub u S; x : S |] ==> x <= u";
 by Auto_tac;
 qed "real_ubD";
 
 *)
+
+