Revised version with Abelian group simprocs

--- a/src/HOL/Real/PNat.ML	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/PNat.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -518,7 +518,7 @@
 \     |] ==> f(i) <= (f(j)::pnat)";
 by (auto_tac (claset() addSDs [inj_Rep_pnat RS injD],
              simpset() addsimps [pnat_le_iff_Rep_pnat_le,
-                                     le_eq_less_or_eq]));
+				 order_le_less]));
 qed "pnat_less_mono_imp_le_mono";
 
 Goal "!!i j k::pnat. i<=j ==> i + k <= j + k";

--- a/src/HOL/Real/PReal.ML	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/PReal.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -157,28 +157,6 @@
       (*** theorems for ordering ***)
 (* prove introduction and elimination rules for preal_less *)
 
-Goalw [preal_less_def]
-      "R1 < (R2::preal) = (Rep_preal(R1) < Rep_preal(R2))";
-by (Fast_tac 1);
-qed "preal_less_iff";
-
-Goalw [preal_less_def]
-      "!! (R1::preal). R1 < R2 ==> (Rep_preal(R1) < Rep_preal(R2))";
-by (Fast_tac  1);
-qed "preal_lessI";
-
-Goalw [preal_less_def]
-      "R1 < (R2::preal) --> (Rep_preal(R1) < Rep_preal(R2))";
-by (Fast_tac  1);
-qed "preal_lessE_lemma";
-
-Goal "!!P. [| R1 < (R2::preal); \
-\             (Rep_preal(R1) < Rep_preal(R2)) ==> P |] \
-\          ==> P";
-by (dtac (preal_lessE_lemma RS mp) 1);
-by Auto_tac;
-qed "preal_lessE";
-
 (* A positive fraction not in a positive real is an upper bound *)
 (* Gleason p. 122 - Remark (1)                                  *)
 
@@ -806,10 +784,6 @@
 by Auto_tac;
 qed "preal_less_or_eq_imp_le";
 
-Goal "(x <= (y::preal)) = (x < y | x=y)";
-by (REPEAT(ares_tac [iffI, preal_less_or_eq_imp_le, preal_le_imp_less_or_eq] 1));
-qed "preal_le_eq_less_or_eq";
-
 Goalw [preal_le_def] "w <= (w::preal)";
 by (Simp_tac 1);
 qed "preal_le_refl";
@@ -1037,13 +1011,6 @@
     simpset() addsimps [preal_add_commute]));
 qed "preal_add_le_mono1";
  
-Goal "!!k l::preal. [|i<=j;  k<=l |] ==> i + k <= j + l";
-by (etac (preal_add_le_mono1 RS preal_le_trans) 1);
-by (simp_tac (simpset() addsimps [preal_add_commute]) 1);
-(*j moves to the end because it is free while k, l are bound*)
-by (etac preal_add_le_mono1 1);
-qed "preal_add_le_mono";
-
 Goal "!!(A::preal). A + C < B + C ==> A < B";
 by (cut_facts_tac [preal_linear] 1);
 by (auto_tac (claset() addEs [preal_less_irrefl],simpset()));

--- a/src/HOL/Real/RComplete.ML	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/RComplete.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -31,12 +31,12 @@
 \                 EX u. isUb (UNIV::real set) S u \
 \              |] ==> EX t. isLub (UNIV::real set) S t";
 by (res_inst_tac [("x","%#psup({w. %#w : S})")] exI 1);
-by (auto_tac (claset(),simpset() addsimps [isLub_def,leastP_def,isUb_def]));
+by (auto_tac (claset(), simpset() addsimps [isLub_def,leastP_def,isUb_def]));
 by (auto_tac (claset() addSIs [setleI,setgeI] 
-    addSDs [real_gt_zero_preal_Ex RS iffD1],simpset()));
+	               addSDs [real_gt_zero_preal_Ex RS iffD1],simpset()));
 by (forw_inst_tac [("x","y")] bspec 1 THEN assume_tac 1);
 by (dtac (real_gt_zero_preal_Ex RS iffD1) 1);
-by (auto_tac (claset(),simpset() addsimps [real_preal_le_iff]));
+by (auto_tac (claset(), simpset() addsimps [real_preal_le_iff]));
 by (rtac preal_psup_leI2a 1);
 by (forw_inst_tac [("y","%#ya")] setleD 1 THEN assume_tac 1);
 by (forward_tac  [real_ge_preal_preal_Ex] 1);
@@ -46,73 +46,48 @@
 by (forw_inst_tac [("x","x")] bspec 1 THEN assume_tac 1);
 by (forward_tac [isUbD2] 1);
 by (dtac (real_gt_zero_preal_Ex RS iffD1) 1);
-by (auto_tac (claset() addSDs [isUbD,
-    real_ge_preal_preal_Ex],simpset() addsimps [real_preal_le_iff]));
-by (blast_tac (claset() addSDs [setleD] addSIs 
-    [psup_le_ub1] addIs [real_preal_le_iff RS iffD1]) 1);
+by (auto_tac (claset() addSDs [isUbD, real_ge_preal_preal_Ex],
+	      simpset() addsimps [real_preal_le_iff]));
+by (blast_tac (claset() addSDs [setleD] addSIs [psup_le_ub1] 
+	                addIs [real_preal_le_iff RS iffD1]) 1);
 qed "posreals_complete";
 
 
 (*-------------------------------
     Lemmas
  -------------------------------*)
-Goal "! y : {z. ? x: P. z = x + %~xa + 1r} Int {x. 0r < x}. 0r < y";
+Goal "! y : {z. ? x: P. z = x + -xa + 1r} Int {x. 0r < x}. 0r < y";
 by Auto_tac;
 qed "real_sup_lemma3";
  
-(* lemmas re-arranging the terms *)
-Goal "(S <= Y + %~X + Z) = (S + X + %~Z <= Y)";
-by (Step_tac 1);
-by (dres_inst_tac [("x","%~Z")] real_add_le_mono1 1);
-by (dres_inst_tac [("x","Z")] real_add_le_mono1 2);
-by (auto_tac (claset(),simpset() addsimps [real_add_assoc,
-    real_add_minus,real_add_zero_right,real_add_minus_left]));
-by (dres_inst_tac [("x","X")] real_add_le_mono1 1);
-by (dres_inst_tac [("x","%~X")] real_add_le_mono1 2);
-by (auto_tac (claset(),simpset() addsimps [real_add_assoc,
-    real_add_minus,real_add_zero_right,real_add_minus_left]));
-by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
-qed "lemma_le_swap";
-
-Goal "(xa <= S + X + %~Z) = (xa + %~X + Z <= S)";
-by (Step_tac 1);
-by (dres_inst_tac [("x","Z")] real_add_le_mono1 1);
-by (dres_inst_tac [("x","%~Z")] real_add_le_mono1 2);
-by (auto_tac (claset(),simpset() addsimps [real_add_assoc,
-    real_add_minus,real_add_zero_right,real_add_minus_left]));
-by (dres_inst_tac [("x","%~X")] real_add_le_mono1 1);
-by (dres_inst_tac [("x","X")] real_add_le_mono1 2);
-by (auto_tac (claset(),simpset() addsimps [real_add_assoc,
-    real_add_minus,real_add_zero_right,real_add_minus_left]));
-by (auto_tac (claset(),simpset() addsimps [real_add_commute]));
+Goal "(xa <= S + X + -Z) = (xa + -X + Z <= (S::real))";
+by (simp_tac (simpset() addsimps [real_diff_def, real_diff_le_eq RS sym] @
+	                         real_add_ac) 1);
 qed "lemma_le_swap2";
 
-Goal "[| 0r < x + %~X + 1r; x < xa |] ==> 0r < xa + %~X + 1r";
+Goal "[| 0r < x + -X + 1r; x < xa |] ==> 0r < xa + -X + 1r";
 by (dtac real_add_less_mono 1);
 by (assume_tac 1);
-by (dres_inst_tac [("C","%~x"),("A","0r + x")] real_add_less_mono2 1);
+by (dres_inst_tac [("C","-x"),("A","0r + x")] real_add_less_mono2 1);
 by (asm_full_simp_tac (simpset() addsimps [real_add_zero_right,
     real_add_assoc RS sym,real_add_minus_left,real_add_zero_left]) 1);
 by (asm_full_simp_tac (simpset() addsimps real_add_ac) 1);
 qed "lemma_real_complete1";
 
-Goal "!!x. [| x + %~X + 1r <= S; xa < x |] ==> xa + %~X + 1r <= S";
+Goal "!!x. [| x + -X + 1r <= S; xa < x |] ==> xa + -X + 1r <= S";
 by (dtac real_less_imp_le 1);
 by (dtac real_add_le_mono 1);
 by (assume_tac 1);
 by (asm_full_simp_tac (simpset() addsimps real_add_ac) 1);
-by (dres_inst_tac [("x","%~x"),("q2.0","x + S")] real_add_left_le_mono1 1);
-by (asm_full_simp_tac (simpset() addsimps [real_add_assoc RS sym,
-        real_add_minus_left,real_add_zero_left]) 1);
 qed "lemma_real_complete2";
 
-Goal "[| x + %~X + 1r <= S; xa < x |] ==> xa <= S + X + %~1r"; (**)
+Goal "[| x + -X + 1r <= S; xa < x |] ==> xa <= S + X + -1r"; (**)
 by (rtac (lemma_le_swap2 RS iffD2) 1);
 by (etac lemma_real_complete2 1);
 by (assume_tac 1);
 qed "lemma_real_complete2a";
 
-Goal "[| x + %~X + 1r <= S; xa <= x |] ==> xa <= S + X + %~1r";
+Goal "[| x + -X + 1r <= S; xa <= x |] ==> xa <= S + X + -1r";
 by (rotate_tac 1 1);
 by (etac (real_le_imp_less_or_eq RS disjE) 1);
 by (blast_tac (claset() addIs [lemma_real_complete2a]) 1);
@@ -126,20 +101,22 @@
 \                             EX Y. isUb (UNIV::real set) S Y \
 \                          |] ==> EX t. isLub (UNIV :: real set) S t";
 by (Step_tac 1);
-by (subgoal_tac "? u. u: {z. ? x: S. z = x + %~X + 1r} \
+by (subgoal_tac "? u. u: {z. ? x: S. z = x + -X + 1r} \
 \                Int {x. 0r < x}" 1);
-by (subgoal_tac "isUb (UNIV::real set) ({z. ? x: S. z = x + %~X + 1r} \
-\                Int {x. 0r < x})  (Y + %~X + 1r)" 1); 
+by (subgoal_tac "isUb (UNIV::real set) ({z. ? x: S. z = x + -X + 1r} \
+\                Int {x. 0r < x})  (Y + -X + 1r)" 1); 
 by (cut_inst_tac [("P","S"),("xa","X")] real_sup_lemma3 1);
 by (EVERY1[forward_tac [exI RSN (3,posreals_complete)], Blast_tac, Blast_tac, Step_tac]);
-by (res_inst_tac [("x","t + X + %~1r")] exI 1);
+by (res_inst_tac [("x","t + X + -1r")] exI 1);
 by (rtac isLubI2 1);
 by (rtac setgeI 2 THEN Step_tac 2);
-by (subgoal_tac "isUb (UNIV:: real set) ({z. ? x: S. z = x + %~X + 1r} \
-\                Int {x. 0r < x})  (y + %~X + 1r)" 2); 
-by (dres_inst_tac [("y","(y + %~ X + 1r)")] isLub_le_isUb 2 
+by (subgoal_tac "isUb (UNIV:: real set) ({z. ? x: S. z = x + -X + 1r} \
+\                Int {x. 0r < x})  (y + -X + 1r)" 2); 
+by (dres_inst_tac [("y","(y + - X + 1r)")] isLub_le_isUb 2 
       THEN assume_tac 2);
-by (etac (lemma_le_swap RS subst) 2);
+by (full_simp_tac
+    (simpset() addsimps [real_diff_def, real_diff_le_eq RS sym] @
+                        real_add_ac) 2);
 by (rtac (setleI RS isUbI) 1);
 by (Step_tac 1);
 by (res_inst_tac [("R1.0","x"),("R2.0","y")] real_linear_less2 1);
@@ -154,27 +131,20 @@
 by (rtac lemma_real_complete2b 1);
 by (etac real_less_imp_le 2);
 by (blast_tac (claset() addSIs [isLubD2]) 1 THEN Step_tac 1);
-by (blast_tac (claset() addDs [isUbD] addSIs [(setleI RS isUbI)]
-    addIs [real_add_le_mono1,real_add_assoc RS ssubst]) 1);
-by (blast_tac (claset() addDs [isUbD] addSIs [(setleI RS isUbI)]
-    addIs [real_add_le_mono1,real_add_assoc RS ssubst]) 1);
-by (auto_tac (claset(),simpset() addsimps [real_add_assoc RS sym,
-     real_add_minus,real_add_zero_left,real_zero_less_one]));
+by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
+by (blast_tac (claset() addDs [isUbD] addSIs [setleI RS isUbI]
+                        addIs [real_add_le_mono1]) 1);
+by (blast_tac (claset() addDs [isUbD] addSIs [setleI RS isUbI]
+                        addIs [real_add_le_mono1]) 1);
+by (auto_tac (claset(),
+	      simpset() addsimps [real_add_assoc RS sym,
+				  real_zero_less_one]));
 qed "reals_complete";
 
 (*----------------------------------------------------------------
         Related property: Archimedean property of reals
  ----------------------------------------------------------------*)
 
-Goal "(ALL m. x*%%#m + x <= t) = (ALL m. x*%%#m <= t + %~x)";
-by Auto_tac;
-by (ALLGOALS(dres_inst_tac [("x","m")] spec));
-by (dres_inst_tac [("x","%~x")] real_add_le_mono1 1);
-by (dres_inst_tac [("x","x")] real_add_le_mono1 2);
-by (auto_tac (claset(),simpset() addsimps [real_add_assoc,
-      real_add_minus,real_add_minus_left,real_add_zero_right]));
-qed "lemma_arch";
-
 Goal "0r < x ==> EX n. rinv(%%#n) < x";
 by (stac real_nat_rinv_Ex_iff 1);
 by (EVERY1[rtac ccontr, Asm_full_simp_tac]);
@@ -187,15 +157,15 @@
 by (asm_full_simp_tac (simpset() addsimps 
    [real_nat_Suc,real_add_mult_distrib2]) 1);
 by (blast_tac (claset() addIs [isLubD2]) 2);
-by (asm_full_simp_tac (simpset() addsimps [lemma_arch]) 1);
-by (subgoal_tac "isUb (UNIV::real set) {z. EX n. z = x*%%#n} (t + %~x)" 1);
+by (asm_full_simp_tac
+    (simpset() addsimps [real_le_diff_eq RS sym, real_diff_def]) 1);
+by (subgoal_tac "isUb (UNIV::real set) {z. EX n. z = x*%%#n} (t + -x)" 1);
 by (blast_tac (claset() addSIs [isUbI,setleI]) 2);
-by (dres_inst_tac [("y","t+%~x")] isLub_le_isUb 1);
-by (dres_inst_tac [("x","%~t")] real_add_left_le_mono1 2);
+by (dres_inst_tac [("y","t+-x")] isLub_le_isUb 1);
+by (dres_inst_tac [("x","-t")] real_add_left_le_mono1 2);
 by (auto_tac (claset() addDs [real_le_less_trans,
-    (real_minus_zero_less_iff2 RS iffD2)], simpset() 
-    addsimps [real_less_not_refl,real_add_assoc RS sym,
-    real_add_minus_left,real_add_zero_left]));
+			      (real_minus_zero_less_iff2 RS iffD2)], 
+	      simpset() addsimps [real_less_not_refl,real_add_assoc RS sym]));
 qed "reals_Archimedean";
 
 Goal "EX n. (x::real) < %%#n";
@@ -203,15 +173,17 @@
 by (res_inst_tac [("x","0")] exI 1);
 by (res_inst_tac [("x","0")] exI 2);
 by (auto_tac (claset() addEs [real_less_trans],
-    simpset() addsimps [real_nat_one,real_zero_less_one]));
+	      simpset() addsimps [real_nat_one,real_zero_less_one]));
 by (forward_tac [(real_rinv_gt_zero RS reals_Archimedean)] 1);
 by (Step_tac 1 THEN res_inst_tac [("x","n")] exI 1);
 by (forw_inst_tac [("y","rinv x")] real_mult_less_mono1 1);
 by (auto_tac (claset(),simpset() addsimps [real_not_refl2 RS not_sym]));
 by (dres_inst_tac [("n1","n"),("y","1r")] 
      (real_nat_less_zero RS real_mult_less_mono2)  1);
-by (auto_tac (claset(),simpset() addsimps [real_nat_less_zero,
-    real_not_refl2 RS not_sym,real_mult_assoc RS sym]));
+by (auto_tac (claset(),
+	      simpset() addsimps [real_nat_less_zero,
+				  real_not_refl2 RS not_sym,
+				  real_mult_assoc RS sym]));
 qed "reals_Archimedean2";
 
 

--- a/src/HOL/Real/ROOT.ML	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/ROOT.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -11,5 +11,7 @@
 writeln"Root file for HOL/Real";
 
 set proof_timing;
+time_use_thy "RealDef";
+use          "simproc";
 time_use_thy "RealAbs";
 time_use_thy "RComplete";

--- a/src/HOL/Real/Real.ML	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/Real.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -1,872 +1,12 @@
-(*  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)";
@@ -896,78 +36,7 @@
 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);
@@ -975,42 +44,42 @@
 
 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";
+				real_ex_add_positive_left_less]) 1);
+qed "real_less_iff_add";
 
-Goal "(0r < x) = (%~x < x)";
+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";
 
@@ -1035,8 +104,8 @@
 
 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";
@@ -1125,93 +194,89 @@
 by (Blast_tac 1);
 qed "posreal_complete";
 
-(*------------------------------------------------------------------
+
+
+(*** 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 "!!(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";
+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
  ---------------------------------------------------------------------------------*)
@@ -1267,13 +332,9 @@
 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";
@@ -1318,8 +379,7 @@
 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";
@@ -1328,7 +388,7 @@
 
 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";
@@ -1358,7 +418,8 @@
 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";
 
@@ -1390,7 +451,7 @@
 
 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)";
@@ -1402,7 +463,7 @@
 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);
@@ -1419,7 +480,8 @@
                        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";
 
@@ -1435,17 +497,20 @@
     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";
 
@@ -1458,3 +523,5 @@
 qed "real_ubD";
 
 *)
+
+

--- a/src/HOL/Real/Real.thy	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/Real.thy	Thu Oct 01 18:18:01 1998 +0200
@@ -1,61 +1,14 @@
-(*  Title       : Real.thy
-    Author      : Jacques D. Fleuriot
-    Copyright   : 1998  University of Cambridge
-    Description : The reals
-*) 
-
-Real = PReal +
-
-constdefs
-    realrel   ::  "((preal * preal) * (preal * preal)) set"
-    "realrel  ==  {p. ? x1 y1 x2 y2. p=((x1::preal,y1),(x2,y2)) & x1+y2 = x2+y1}" 
-
-typedef real = "{x::(preal*preal).True}/realrel"          (Equiv.quotient_def)
-
-
-instance
-   real  :: {ord,plus,times}
-
-consts 
-
-  "0r"       :: real               ("0r")   
-  "1r"       :: real               ("1r")  
-
-defs
-
-  real_zero_def      "0r == Abs_real(realrel^^{(@#($#1p),@#($#1p))})"
-  real_one_def       "1r == Abs_real(realrel^^{(@#($#1p) + @#($#1p),@#($#1p))})"
-
-constdefs
+(*  Title:      Real/Real.thy
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1998  University of Cambridge
 
-  real_preal :: preal => real              ("%#_" [80] 80)
-  "%# m     == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
-
-  real_minus :: real => real               ("%~ _" [80] 80) 
-  "%~ R     ==  Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
-
-  rinv       :: real => real
-  "rinv(R)   == (@S. R ~= 0r & S*R = 1r)"
-
-  real_nat :: nat => real                  ("%%# _" [80] 80) 
-  "%%# n      == %#(@#($#(*# n)))"
-
-defs
+Type "real" is a linear order
+*)
 
-  real_add_def  
-  "P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
-                split(%x1 y1. split(%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"
-  
-  real_mult_def  
-  "P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
-                split(%x1 y1. split(%x2 y2. realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
+Real = RealDef +
 
-  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)" 
-
-  real_le_def
-  "P <= (Q::real) == ~(Q < P)"
+instance real :: order (real_le_refl,real_le_trans,real_le_anti_sym,real_less_le)
+instance real :: linorder (real_le_linear)
 
 end

--- a/src/HOL/Real/RealAbs.ML	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/RealAbs.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -4,13 +4,11 @@
     Description : Absolute value function for the reals
 *) 
 
-open RealAbs;
-
 (*----------------------------------------------------------------------------
        Properties of the absolute value function over the reals
        (adapted version of previously proved theorems about abs)
  ----------------------------------------------------------------------------*)
-Goalw [rabs_def] "rabs r = (if 0r<=r then r else %~r)";
+Goalw [rabs_def] "rabs r = (if 0r<=r then r else -r)";
 by Auto_tac;
 qed "rabs_iff";
 
@@ -20,7 +18,7 @@
 
 Addsimps [rabs_zero];
 
-Goalw [rabs_def] "rabs 0r = %~0r";
+Goalw [rabs_def] "rabs 0r = -0r";
 by (stac real_minus_zero 1);
 by (rtac if_cancel 1);
 qed "rabs_minus_zero";
@@ -33,19 +31,19 @@
 by (simp_tac (simpset() addsimps [(prem RS real_less_imp_le),rabs_eqI1]) 1);
 qed "rabs_eqI2";
 
-val [prem] = goalw thy [rabs_def,real_le_def] "x<0r ==> rabs x = %~x";
+val [prem] = goalw thy [rabs_def,real_le_def] "x<0r ==> rabs x = -x";
 by (simp_tac (simpset() addsimps [prem,if_not_P]) 1);
 qed "rabs_minus_eqI2";
 
-Goal "x<=0r ==> rabs x = %~x";
+Goal "x<=0r ==> rabs x = -x";
 by (dtac real_le_imp_less_or_eq 1);
 by (blast_tac (HOL_cs addIs [rabs_minus_zero,rabs_minus_eqI2]) 1);
 qed "rabs_minus_eqI1";
 
 Goalw [rabs_def,real_le_def] "0r<= rabs x";
-by (full_simp_tac (simpset()  setloop (split_tac [expand_if])) 1);
+by (Full_simp_tac 1);
 by (blast_tac (claset() addDs [real_minus_zero_less_iff RS iffD2,
-    real_less_asym]) 1);
+			       real_less_asym]) 1);
 qed "rabs_ge_zero";
 
 Goal "rabs(rabs x)=rabs x";
@@ -54,29 +52,27 @@
 qed "rabs_idempotent";
 
 Goalw [rabs_def] "(x=0r) = (rabs x = 0r)";
-by (full_simp_tac (simpset() setloop (split_tac [expand_if])) 1);
+by (Full_simp_tac 1);
 qed "rabs_zero_iff";
 
 Goal  "(x ~= 0r) = (rabs x ~= 0r)";
-by (full_simp_tac (simpset() addsimps [rabs_zero_iff RS sym] 
-    setloop (split_tac [expand_if])) 1);
+by (full_simp_tac (simpset() addsimps [rabs_zero_iff RS sym]) 1);
 qed "rabs_not_zero_iff";
 
 Goalw [rabs_def] "x<=rabs x";
-by (full_simp_tac (simpset() addsimps [real_le_refl] setloop (split_tac [expand_if])) 1);
+by (Full_simp_tac 1);
 by (auto_tac (claset() addDs [not_real_leE RS real_less_imp_le],
-    simpset() addsimps [real_le_zero_iff]));
+	      simpset() addsimps [real_le_zero_iff]));
 qed "rabs_ge_self";
 
-Goalw [rabs_def] "%~x<=rabs x";
-by (full_simp_tac (simpset() addsimps [real_le_refl,
-    real_ge_zero_iff] setloop (split_tac [expand_if])) 1);
+Goalw [rabs_def] "-x<=rabs x";
+by (full_simp_tac (simpset() addsimps [real_ge_zero_iff]) 1);
 qed "rabs_ge_minus_self";
 
 (* case splits nightmare *)
 Goalw [rabs_def] "rabs(x*y) = (rabs x)*(rabs y)";
 by (auto_tac (claset(),simpset() addsimps [real_minus_mult_eq1,
-   real_minus_mult_commute,real_minus_mult_eq2] setloop (split_tac [expand_if])));
+   real_minus_mult_commute,real_minus_mult_eq2]));
 by (blast_tac (claset() addDs [real_le_mult_order]) 1);
 by (auto_tac (claset() addSDs [not_real_leE],simpset()));
 by (EVERY1[dtac real_mult_le_zero, assume_tac, dtac real_le_anti_sym]);
@@ -88,7 +84,7 @@
 
 Goalw [rabs_def] "x~= 0r ==> rabs(rinv(x)) = rinv(rabs(x))";
 by (auto_tac (claset(),simpset() addsimps [real_minus_rinv] 
-    setloop (split_tac [expand_if])));
+   ));
 by (ALLGOALS(dtac not_real_leE));
 by (etac real_less_asym 1);
 by (blast_tac (claset() addDs [real_le_imp_less_or_eq,
@@ -108,28 +104,30 @@
 
 Goal "rabs(x+y) <= rabs x + rabs y";
 by (EVERY1 [res_inst_tac [("Q1","0r<=x+y")] (expand_if RS ssubst), rtac conjI]);
-by (asm_simp_tac (simpset() addsimps [rabs_eqI1,real_add_le_mono,rabs_ge_self]) 1);
-by (asm_simp_tac (simpset() addsimps [not_real_leE,rabs_minus_eqI2,real_add_le_mono,
-                                     rabs_ge_minus_self,real_minus_add_eq]) 1);
+by (asm_simp_tac
+    (simpset() addsimps [rabs_eqI1,real_add_le_mono,rabs_ge_self]) 1);
+by (asm_simp_tac 
+    (simpset() addsimps [not_real_leE,rabs_minus_eqI2,real_add_le_mono,
+			 rabs_ge_minus_self]) 1);
 qed "rabs_triangle_ineq";
 
 Goal "rabs(w + x + y + z) <= rabs(w) + rabs(x) + rabs(y) + rabs(z)";
 by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
 by (blast_tac (claset() addSIs [(rabs_triangle_ineq RS real_le_trans),
-                real_add_left_le_mono1,real_le_refl]) 1);
+				real_add_left_le_mono1]) 1);
 qed "rabs_triangle_ineq_four";
 
-Goalw [rabs_def] "rabs(%~x)=rabs(x)";
+Goalw [rabs_def] "rabs(-x)=rabs(x)";
 by (auto_tac (claset() addSDs [not_real_leE,real_less_asym] addIs [real_le_anti_sym],
-   simpset() addsimps [real_ge_zero_iff] setloop (split_tac [expand_if])));
+   simpset() addsimps [real_ge_zero_iff]));
 qed "rabs_minus_cancel";
 
-Goal "rabs(x + %~y) <= rabs x + rabs y";
+Goal "rabs(x + -y) <= rabs x + rabs y";
 by (res_inst_tac [("x1","y")] (rabs_minus_cancel RS subst) 1);
 by (rtac rabs_triangle_ineq 1);
 qed "rabs_triangle_minus_ineq";
 
-Goal "rabs (x + y + (%~l + %~m)) <= rabs(x + %~l) + rabs(y + %~m)";
+Goal "rabs (x + y + (-l + -m)) <= rabs(x + -l) + rabs(y + -m)";
 by (full_simp_tac (simpset() addsimps [real_add_assoc]) 1);
 by (res_inst_tac [("x1","y")] (real_add_left_commute RS ssubst) 1);
 by (rtac (real_add_assoc RS subst) 1);
@@ -142,7 +140,7 @@
 by (REPEAT (ares_tac [real_add_less_mono] 1));
 qed "rabs_add_less";
 
-Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+ %~y) < r+s";
+Goal "[| rabs x < r; rabs y < s |] ==> rabs(x+ -y) < r+s";
 by (rotate_tac 1 1);
 by (dtac (rabs_minus_cancel RS ssubst) 1);
 by (asm_simp_tac (simpset() addsimps [rabs_add_less]) 1);
@@ -176,8 +174,7 @@
 			     real_le_less_trans]) 1);
 qed "rabs_mult_less";
 
-Goal "[| rabs x < r; rabs y < s |] \
-\          ==> rabs(x)*rabs(y)<r*s";
+Goal "[| rabs x < r; rabs y < s |] ==> rabs(x)*rabs(y)<r*s";
 by (auto_tac (claset() addIs [rabs_mult_less],
               simpset() addsimps [rabs_mult RS sym]));
 qed "rabs_mult_less2";
@@ -186,13 +183,13 @@
 by (cut_inst_tac [("x1","y")] (rabs_ge_zero RS real_le_imp_less_or_eq) 1);
 by (EVERY1[etac disjE,rtac real_less_imp_le]);
 by (dres_inst_tac [("W","1r")]  real_less_sum_gt_zero 1);
-by (forw_inst_tac [("y","rabs x + %~1r")] real_mult_order 1);
+by (forw_inst_tac [("y","rabs x + -1r")] real_mult_order 1);
 by (assume_tac 1);
 by (rtac real_sum_gt_zero_less 1);
 by (asm_full_simp_tac (simpset() addsimps [real_add_mult_distrib2,
-    rabs_mult, real_mult_commute,real_minus_mult_eq1 RS sym]) 1);
+    real_mult_commute, rabs_mult]) 1);
 by (dtac sym 1);
-by (asm_full_simp_tac (simpset() addsimps [real_le_refl,rabs_mult]) 1);
+by (asm_full_simp_tac (simpset() addsimps [rabs_mult]) 1);
 qed "rabs_mult_le";
 
 Goal "[| 1r < rabs x; r < rabs y|] ==> r < rabs(x*y)";
@@ -205,27 +202,27 @@
 
 Goalw [rabs_def] "rabs 1r = 1r";
 by (auto_tac (claset() addSDs [not_real_leE RS real_less_asym],
-   simpset() addsimps [real_zero_less_one] setloop (split_tac [expand_if])));
+   simpset() addsimps [real_zero_less_one]));
 qed "rabs_one";
 
 Goal "[| 0r < x ; x < r |] ==> rabs x < r";
 by (asm_simp_tac (simpset() addsimps [rabs_eqI2]) 1);
 qed "rabs_lessI";
 
-Goal "rabs x =x | rabs x = %~x";
+Goal "rabs x =x | rabs x = -x";
 by (cut_inst_tac [("R1.0","0r"),("R2.0","x")] real_linear 1);
 by (blast_tac (claset() addIs [rabs_eqI2,rabs_minus_eqI2,
                             rabs_zero,rabs_minus_zero]) 1);
 qed "rabs_disj";
 
-Goal "rabs x = y ==> x = y | %~x = y";
+Goal "rabs x = y ==> x = y | -x = y";
 by (dtac sym 1);
 by (hyp_subst_tac 1);
 by (res_inst_tac [("x1","x")] (rabs_disj RS disjE) 1);
 by (REPEAT(Asm_simp_tac 1));
 qed "rabs_eq_disj";
 
-Goal "(rabs x < r) = (%~r<x & x<r)";
+Goal "(rabs x < r) = (-r<x & x<r)";
 by Safe_tac;
 by (rtac (real_less_swap_iff RS iffD2) 1);
 by (asm_simp_tac (simpset() addsimps [(rabs_ge_minus_self 

--- a/src/HOL/Real/RealAbs.thy	Tue Sep 29 18:13:05 1998 +0200
+++ b/src/HOL/Real/RealAbs.thy	Thu Oct 01 18:18:01 1998 +0200
@@ -8,6 +8,6 @@
 
 constdefs
    rabs   :: real => real
-   "rabs r      == if 0r<=r then r else %~r" 
+   "rabs r      == if 0r<=r then r else -r" 
 
 end
\ No newline at end of file

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/RealDef.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -0,0 +1,1042 @@
+(*  Title       : Real/RealDef.ML
+    Author      : Jacques D. Fleuriot
+    Copyright   : 1998  University of Cambridge
+    Description : The reals
+*)
+
+(*** 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::real)";
+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(%r::real. -r)";
+by (rtac injI 1);
+by (dres_inst_tac [("f","uminus")] 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";
+Addsimps [real_add_zero_left];
+
+Goal "z + 0r = z";
+by (simp_tac (simpset() addsimps [real_add_commute]) 1);
+qed "real_add_zero_right";
+Addsimps [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";
+Addsimps [real_add_minus];
+
+Goal "-z + z = 0r";
+by (simp_tac (simpset() addsimps [real_add_commute]) 1);
+qed "real_add_minus_left";
+Addsimps [real_add_minus_left];
+
+
+Goal "z + (- z + w) = (w::real)";
+by (simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1);
+qed "real_add_minus_cancel";
+
+Goal "(-z) + (z + w) = (w::real)";
+by (simp_tac (simpset() addsimps [real_add_assoc RS sym]) 1);
+qed "real_minus_add_cancel";
+
+Addsimps [real_add_minus_cancel, real_minus_add_cancel];
+
+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]) 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]) 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 "-(x + y) = -x + -(y::real)";
+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_distrib";
+
+Addsimps [real_minus_add_distrib];
+
+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_assoc RS sym]) 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";
+
+Goal "0r - x = -x";
+by (simp_tac (simpset() addsimps [real_diff_def]) 1);
+qed "real_diff_0";
+
+Goal "x - 0r = x";
+by (simp_tac (simpset() addsimps [real_diff_def]) 1);
+qed "real_diff_0_right";
+
+Goal "x - x = 0r";
+by (simp_tac (simpset() addsimps [real_diff_def]) 1);
+qed "real_diff_self";
+
+Addsimps [real_diff_0, real_diff_0_right, real_diff_self];
+
+
+(*** 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";
+
+Addsimps [real_mult_1];
+
+Goal "z * 1r = z";
+by (simp_tac (simpset() addsimps [real_mult_commute]) 1);
+qed "real_mult_1_right";
+
+Addsimps [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::real)";
+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::real)";
+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 "- 1r * z = -z";
+by (simp_tac (simpset() addsimps [real_minus_mult_eq1 RS sym]) 1);
+qed "real_mult_minus_1";
+
+Addsimps [real_mult_minus_1];
+
+Goal "z * - 1r = -z";
+by (stac real_mult_commute 1);
+by (Simp_tac 1);
+qed "real_mult_minus_1_right";
+
+Addsimps [real_mult_minus_1_right];
+
+Goal "-x * -y = x * (y::real)";
+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::real)";
+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";
+
+(*** 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 *)
+
+(* 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);
+AddSEs [real_less_irrefl];
+
+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 "!!w::real. (w ~= z) = (w<z | z<w)";
+by (cut_facts_tac [real_linear] 1);
+by (Blast_tac 1);
+qed "real_neq_iff";
+
+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_less";
+
+Goal "w <= (w::real)";
+by (simp_tac (simpset() addsimps [real_le_less]) 1);
+qed "real_le_refl";
+
+AddIffs [real_le_refl];
+
+(* Axiom 'linorder_linear' of class 'linorder': *)
+Goal "(z::real) <= w | w <= z";
+by (simp_tac (simpset() addsimps [real_le_less]) 1);
+by (cut_facts_tac [real_linear] 1);
+by (Blast_tac 1);
+qed "real_le_linear";
+
+Goal "[| 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";
+
+(* Axiom 'order_less_le' of class 'order': *)
+Goal "(w::real) < z = (w <= z & w ~= z)";
+by (simp_tac (simpset() addsimps [real_le_def, real_neq_iff]) 1);
+by (blast_tac (claset() addSEs [real_less_asym]) 1);
+qed "real_less_le";
+
+
+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";
+
+
+(*Alternative definition for real_less*)
+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]));
+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_assoc]));
+qed "real_less_add_positive_left_Ex";
+
+
+
+(** 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::real. T = S + W ==> S = T + -W";
+by (asm_simp_tac (simpset() addsimps 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]));
+by (EVERY1[dtac real_less_add_positive_left_Ex, etac exE, etac conjE]);
+by (Asm_full_simp_tac 1);
+by (dtac real_lemma_change_eq_subj 1);
+by Auto_tac;
+by (dtac real_less_sum_gt_zero 1);
+by (asm_full_simp_tac (simpset() addsimps 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()));
+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";
+
+
+Goalw [real_diff_def] "(x<y) = (x-y < 0r)";
+by (stac (real_minus_zero_less_iff RS sym) 1);
+by (simp_tac (simpset() addsimps [real_add_commute,
+				  real_less_sum_gt_0_iff]) 1);
+qed "real_less_eq_diff";
+
+
+(*** Subtraction laws ***)
+
+Goal "x + (y - z) = (x + y) - (z::real)";
+by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1);
+qed "real_add_diff_eq";
+
+Goal "(x - y) + z = (x + z) - (y::real)";
+by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1);
+qed "real_diff_add_eq";
+
+Goal "(x - y) - z = x - (y + (z::real))";
+by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1);
+qed "real_diff_diff_eq";
+
+Goal "x - (y - z) = (x + z) - (y::real)";
+by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1);
+qed "real_diff_diff_eq2";
+
+Goal "(x-y < z) = (x < z + (y::real))";
+by (stac real_less_eq_diff 1);
+by (res_inst_tac [("y1", "z")] (real_less_eq_diff RS ssubst) 1);
+by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1);
+qed "real_diff_less_eq";
+
+Goal "(x < z-y) = (x + (y::real) < z)";
+by (stac real_less_eq_diff 1);
+by (res_inst_tac [("y1", "z-y")] (real_less_eq_diff RS ssubst) 1);
+by (simp_tac (simpset() addsimps real_diff_def::real_add_ac) 1);
+qed "real_less_diff_eq";
+
+Goalw [real_le_def] "(x-y <= z) = (x <= z + (y::real))";
+by (simp_tac (simpset() addsimps [real_less_diff_eq]) 1);
+qed "real_diff_le_eq";
+
+Goalw [real_le_def] "(x <= z-y) = (x + (y::real) <= z)";
+by (simp_tac (simpset() addsimps [real_diff_less_eq]) 1);
+qed "real_le_diff_eq";
+
+Goalw [real_diff_def] "(x-y = z) = (x = z + (y::real))";
+by (auto_tac (claset(), simpset() addsimps [real_add_assoc]));
+qed "real_diff_eq_eq";
+
+Goalw [real_diff_def] "(x = z-y) = (x + (y::real) = z)";
+by (auto_tac (claset(), simpset() addsimps [real_add_assoc]));
+qed "real_eq_diff_eq";
+
+(*This list of rewrites simplifies (in)equalities by bringing subtractions
+  to the top and then moving negative terms to the other side.  
+  Use with real_add_ac*)
+val real_compare_rls = 
+  [symmetric real_diff_def,
+   real_add_diff_eq, real_diff_add_eq, real_diff_diff_eq, real_diff_diff_eq2, 
+   real_diff_less_eq, real_less_diff_eq, real_diff_le_eq, real_le_diff_eq, 
+   real_diff_eq_eq, real_eq_diff_eq];
+
+
+(** For the cancellation simproc.
+    The idea is to cancel like terms on opposite sides by subtraction **)
+
+Goal "(x::real) - y = x' - y' ==> (x<y) = (x'<y')";
+by (stac real_less_eq_diff 1);
+by (res_inst_tac [("y1", "y")] (real_less_eq_diff RS ssubst) 1);
+by (Asm_simp_tac 1);
+qed "real_less_eqI";
+
+Goal "(x::real) - y = x' - y' ==> (y<=x) = (y'<=x')";
+by (dtac real_less_eqI 1);
+by (asm_simp_tac (simpset() addsimps [real_le_def]) 1);
+qed "real_le_eqI";
+
+Goal "(x::real) - y = x' - y' ==> (x=y) = (x'=y')";
+by Safe_tac;
+by (ALLGOALS
+    (asm_full_simp_tac
+     (simpset() addsimps [real_eq_diff_eq, real_diff_eq_eq])));
+qed "real_eq_eqI";

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/RealDef.thy	Thu Oct 01 18:18:01 1998 +0200
@@ -0,0 +1,62 @@
+(*  Title       : Real/RealDef.thy
+    Author      : Jacques D. Fleuriot
+    Copyright   : 1998  University of Cambridge
+    Description : The reals
+*) 
+
+RealDef = PReal +
+
+constdefs
+  realrel   ::  "((preal * preal) * (preal * preal)) set"
+  "realrel == {p. ? x1 y1 x2 y2. p = ((x1,y1),(x2,y2)) & x1+y2 = x2+y1}" 
+
+typedef real = "{x::(preal*preal).True}/realrel"          (Equiv.quotient_def)
+
+
+instance
+   real  :: {ord, plus, times, minus}
+
+consts 
+
+  "0r"       :: real               ("0r")   
+  "1r"       :: real               ("1r")  
+
+defs
+
+  real_zero_def  "0r == Abs_real(realrel^^{(@#($#1p),@#($#1p))})"
+  real_one_def   "1r == Abs_real(realrel^^{(@#($#1p) + @#($#1p),@#($#1p))})"
+
+  real_minus_def
+    "- R ==  Abs_real(UN p:Rep_real(R). split (%x y. realrel^^{(y,x)}) p)"
+
+  real_diff_def "x - y == x + -(y::real)"
+
+constdefs
+
+  real_preal :: preal => real              ("%#_" [80] 80)
+  "%# m     == Abs_real(realrel^^{(m+@#($#1p),@#($#1p))})"
+
+  rinv       :: real => real
+  "rinv(R)   == (@S. R ~= 0r & S*R = 1r)"
+
+  real_nat :: nat => real                  ("%%# _" [80] 80) 
+  "%%# n      == %#(@#($#(*# n)))"
+
+defs
+
+  real_add_def  
+  "P + Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
+                split(%x1 y1. split(%x2 y2. realrel^^{(x1+x2, y1+y2)}) p2) p1)"
+  
+  real_mult_def  
+  "P * Q == Abs_real(UN p1:Rep_real(P). UN p2:Rep_real(Q).
+                split(%x1 y1. split(%x2 y2. realrel^^{(x1*x2+y1*y2,x1*y2+x2*y1)}) p2) p1)"
+
+  real_less_def
+  "P < Q == EX x1 y1 x2 y2. x1 + y2 < x2 + y1 & 
+                                   (x1,y1):Rep_real(P) &
+                                   (x2,y2):Rep_real(Q)" 
+  real_le_def
+  "P <= (Q::real) == ~(Q < P)"
+
+end

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Real/simproc.ML	Thu Oct 01 18:18:01 1998 +0200
@@ -0,0 +1,62 @@
+(*  Title:      HOL/Real/simproc.ML
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1998  University of Cambridge
+
+Apply Abel_Cancel to the reals
+*)
+
+(*** Two lemmas needed for the simprocs ***)
+
+(*Deletion of other terms in the formula, seeking the -x at the front of z*)
+val real_add_cancel_21 = prove_goal RealDef.thy
+    "((x::real) + (y + z) = y + u) = ((x + z) = u)"
+  (fn _ => [stac real_add_left_commute 1,
+	    rtac real_add_left_cancel 1]);
+
+(*A further rule to deal with the case that
+  everything gets cancelled on the right.*)
+val real_add_cancel_end = prove_goal RealDef.thy
+    "((x::real) + (y + z) = y) = (x = -z)"
+  (fn _ => [stac real_add_left_commute 1,
+	    res_inst_tac [("t", "y")] (real_add_zero_right RS subst) 1,
+	    stac real_add_left_cancel 1,
+	    simp_tac (simpset() addsimps [real_eq_diff_eq RS sym]) 1]);
+
+
+structure Real_Cancel_Data =
+struct
+  val ss		= HOL_ss
+  val mk_eq		= HOLogic.mk_Trueprop o HOLogic.mk_eq
+  fun mk_meta_eq r	= r RS eq_reflection
+
+  val thy		= RealDef.thy
+  val T			= Type ("RealDef.real", [])
+  val zero		= Const ("RealDef.0r", T)
+  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];
+