
diff -r b2267a9e9e28 -r 74e0cc2018d9 src/HOL/NatSimprocs.thy
--- a/src/HOL/NatSimprocs.thy	Thu Jun 21 15:42:12 2007 +0200
+++ b/src/HOL/NatSimprocs.thy	Thu Jun 21 15:42:13 2007 +0200
@@ -1,12 +1,13 @@
 (*  Title:      HOL/NatSimprocs.thy
     ID:         $Id$
-    Copyright   2003 TU Muenchen
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Author:     Amine Chaieb, TU Muenchen
 *)
 
 header {*Simprocs for the Naturals*}
 
 theory NatSimprocs
-imports Groebner_Basis
+imports Groebner_Basis Dense_Linear_Order
 uses
   "~~/src/Provers/Arith/cancel_numeral_factor.ML"
   "~~/src/Provers/Arith/extract_common_term.ML"
@@ -36,12 +37,12 @@
 text{*No longer required as a simprule because of the @{text inverse_fold}
    simproc*}
 lemma Suc_diff_number_of:
-     "neg (number_of (uminus v)::int) ==>  
+     "neg (number_of (uminus v)::int) ==>
       Suc m - (number_of v) = m - (number_of (Numeral.pred v))"
 apply (subst Suc_diff_eq_diff_pred)
 apply simp
 apply (simp del: nat_numeral_1_eq_1)
-apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric] 
+apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric]
                         neg_number_of_pred_iff_0)
 done
 
@@ -52,25 +53,25 @@
 subsection{*For @{term nat_case} and @{term nat_rec}*}
 
 lemma nat_case_number_of [simp]:
-     "nat_case a f (number_of v) =  
-        (let pv = number_of (Numeral.pred v) in  
+     "nat_case a f (number_of v) =
+        (let pv = number_of (Numeral.pred v) in
          if neg pv then a else f (nat pv))"
 by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0)
 
 lemma nat_case_add_eq_if [simp]:
-     "nat_case a f ((number_of v) + n) =  
-       (let pv = number_of (Numeral.pred v) in  
+     "nat_case a f ((number_of v) + n) =
+       (let pv = number_of (Numeral.pred v) in
          if neg pv then nat_case a f n else f (nat pv + n))"
 apply (subst add_eq_if)
 apply (simp split add: nat.split
             del: nat_numeral_1_eq_1
-	    add: numeral_1_eq_Suc_0 [symmetric] Let_def 
+            add: numeral_1_eq_Suc_0 [symmetric] Let_def
                  neg_imp_number_of_eq_0 neg_number_of_pred_iff_0)
 done
 
 lemma nat_rec_number_of [simp]:
-     "nat_rec a f (number_of v) =  
-        (let pv = number_of (Numeral.pred v) in  
+     "nat_rec a f (number_of v) =
+        (let pv = number_of (Numeral.pred v) in
          if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))"
 apply (case_tac " (number_of v) ::nat")
 apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0)
@@ -78,9 +79,9 @@
 done
 
 lemma nat_rec_add_eq_if [simp]:
-     "nat_rec a f (number_of v + n) =  
-        (let pv = number_of (Numeral.pred v) in  
-         if neg pv then nat_rec a f n  
+     "nat_rec a f (number_of v + n) =
+        (let pv = number_of (Numeral.pred v) in
+         if neg pv then nat_rec a f n
                    else f (nat pv + n) (nat_rec a f (nat pv + n)))"
 apply (subst add_eq_if)
 apply (simp split add: nat.split
@@ -123,7 +124,7 @@
 
 lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)"
 apply (subgoal_tac "m mod 2 < 2")
-apply (force simp del: mod_less_divisor, simp) 
+apply (force simp del: mod_less_divisor, simp)
 done
 
 subsubsection{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*}
@@ -273,35 +274,35 @@
 
 subsubsection{*To Simplify Inequalities Where One Side is the Constant 1*}
 
-lemma less_minus_iff_1 [simp]: 
-  fixes b::"'b::{ordered_idom,number_ring}" 
+lemma less_minus_iff_1 [simp]:
+  fixes b::"'b::{ordered_idom,number_ring}"
   shows "(1 < - b) = (b < -1)"
 by auto
 
-lemma le_minus_iff_1 [simp]: 
-  fixes b::"'b::{ordered_idom,number_ring}" 
+lemma le_minus_iff_1 [simp]:
+  fixes b::"'b::{ordered_idom,number_ring}"
   shows "(1 \<le> - b) = (b \<le> -1)"
 by auto
 
-lemma equation_minus_iff_1 [simp]: 
-  fixes b::"'b::number_ring" 
+lemma equation_minus_iff_1 [simp]:
+  fixes b::"'b::number_ring"
   shows "(1 = - b) = (b = -1)"
-by (subst equation_minus_iff, auto) 
+by (subst equation_minus_iff, auto)
 
-lemma minus_less_iff_1 [simp]: 
-  fixes a::"'b::{ordered_idom,number_ring}" 
+lemma minus_less_iff_1 [simp]:
+  fixes a::"'b::{ordered_idom,number_ring}"
   shows "(- a < 1) = (-1 < a)"
 by auto
 
-lemma minus_le_iff_1 [simp]: 
-  fixes a::"'b::{ordered_idom,number_ring}" 
+lemma minus_le_iff_1 [simp]:
+  fixes a::"'b::{ordered_idom,number_ring}"
   shows "(- a \<le> 1) = (-1 \<le> a)"
 by auto
 
-lemma minus_equation_iff_1 [simp]: 
-  fixes a::"'b::number_ring" 
+lemma minus_equation_iff_1 [simp]:
+  fixes a::"'b::number_ring"
   shows "(- a = 1) = (a = -1)"
-by (subst minus_equation_iff, auto) 
+by (subst minus_equation_iff, auto)
 
 
 subsubsection {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *}
@@ -366,7 +367,7 @@
 subsubsection{*Division By @{text "-1"}*}
 
 lemma divide_minus1 [simp]:
-     "x/-1 = -(x::'a::{field,division_by_zero,number_ring})" 
+     "x/-1 = -(x::'a::{field,division_by_zero,number_ring})"
 by simp
 
 lemma minus1_divide [simp]:
@@ -392,27 +393,27 @@
 val minus1_divide = @{thm minus1_divide};
 *}
 
+
 section{* Installing Groebner Bases for Fields *}
 
-
-interpretation class_fieldgb: 
+interpretation class_fieldgb:
   fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse)
 
 lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp
-lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0" 
+lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0"
   by simp
-lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)" 
+lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)"
   by simp
-lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y" 
+lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   by simp
-lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y" 
+lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z  = (x*z) / y"
   by simp
 
 lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp
 
-lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y" 
+lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y"
   by (simp add: add_divide_distrib)
-lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y" 
+lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y"
   by (simp add: add_divide_distrib)
 
 declaration{*
@@ -421,28 +422,28 @@
  val zT = ctyp_of_term zr
  val geq = @{cpat "op ="}
  val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd
- val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} 
+ val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"}
  val add_frac_num = mk_meta_eq @{thm "add_frac_num"}
  val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
 
  fun prove_nz ctxt =
-  let val ss = local_simpset_of ctxt 
-  in fn T => fn t => 
-    let 
-      val z = instantiate_cterm ([(zT,T)],[]) zr 
+  let val ss = local_simpset_of ctxt
+  in fn T => fn t =>
+    let
+      val z = instantiate_cterm ([(zT,T)],[]) zr
       val eq = instantiate_cterm ([(eqT,T)],[]) geq
-      val th = Simplifier.rewrite (ss addsimps simp_thms) 
-           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} 
+      val th = Simplifier.rewrite (ss addsimps simp_thms)
+           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
                   (Thm.capply (Thm.capply eq t) z)))
     in equal_elim (symmetric th) TrueI
     end
   end
 
- fun proc ctxt phi ss ct = 
-  let 
-    val ((x,y),(w,z)) = 
+ fun proc ctxt phi ss ct =
+  let
+    val ((x,y),(w,z)) =
          (Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct
-    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] 
+    val _ = map (HOLogic.dest_number o term_of) [x,y,z,w]
     val T = ctyp_of_term x
     val [y_nz, z_nz] = map (prove_nz ctxt T) [y, z]
     val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
@@ -450,18 +451,18 @@
   end
   handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
 
- fun proc2 ctxt phi ss ct = 
-  let 
+ fun proc2 ctxt phi ss ct =
+  let
     val (l,r) = Thm.dest_binop ct
     val T = ctyp_of_term l
   in (case (term_of l, term_of r) of
-      (Const(@{const_name "HOL.divide"},_)$_$_, _) => 
+      (Const(@{const_name "HOL.divide"},_)$_$_, _) =>
         let val (x,y) = Thm.dest_binop l val z = r
             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
             val ynz = prove_nz ctxt T y
         in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
         end
-     | (_, Const (@{const_name "HOL.divide"},_)$_$_) => 
+     | (_, Const (@{const_name "HOL.divide"},_)$_$_) =>
         let val (x,y) = Thm.dest_binop r val z = l
             val _ = map (HOLogic.dest_number o term_of) [x,y,z]
             val ynz = prove_nz ctxt T y
@@ -478,43 +479,43 @@
 
  fun proc3 phi ss ct =
   (case term_of ct of
-    Const(@{const_name "Orderings.less"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => 
-      let 
+    Const(@{const_name "Orderings.less"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
+      let
         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
         val _ = map is_number [a,b,c]
         val T = ctyp_of_term c
         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"}
       in SOME (mk_meta_eq th) end
-  | Const(@{const_name "Orderings.less_eq"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => 
-      let 
+  | Const(@{const_name "Orderings.less_eq"},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
+      let
         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
         val _ = map is_number [a,b,c]
         val T = ctyp_of_term c
         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"}
       in SOME (mk_meta_eq th) end
-  | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => 
-      let 
+  | Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ =>
+      let
         val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop
         val _ = map is_number [a,b,c]
         val T = ctyp_of_term c
         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"}
       in SOME (mk_meta_eq th) end
-  | Const(@{const_name "Orderings.less"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => 
-    let 
+  | Const(@{const_name "Orderings.less"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
+    let
       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
         val _ = map is_number [a,b,c]
         val T = ctyp_of_term c
         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"}
       in SOME (mk_meta_eq th) end
-  | Const(@{const_name "Orderings.less_eq"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => 
-    let 
+  | Const(@{const_name "Orderings.less_eq"},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
+    let
       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
         val _ = map is_number [a,b,c]
         val T = ctyp_of_term c
         val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"}
       in SOME (mk_meta_eq th) end
-  | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => 
-    let 
+  | Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) =>
+    let
       val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop
         val _ = map is_number [a,b,c]
         val T = ctyp_of_term c
@@ -523,49 +524,49 @@
   | _ => NONE)
   handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
 
-fun add_frac_frac_simproc ctxt = 
-       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], 
+fun add_frac_frac_simproc ctxt =
+       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
                      name = "add_frac_frac_simproc",
                      proc = proc ctxt, identifier = []}
 
-fun add_frac_num_simproc ctxt = 
-       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], 
+fun add_frac_num_simproc ctxt =
+       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
                      name = "add_frac_num_simproc",
                      proc = proc2 ctxt, identifier = []}
 
-val ord_frac_simproc = 
-  make_simproc 
-    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, 
-             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"}, 
-             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, 
+val ord_frac_simproc =
+  make_simproc
+    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
+             @{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"},
+             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
              @{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"},
              @{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"},
              @{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}],
              name = "ord_frac_simproc", proc = proc3, identifier = []}
 
-val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", 
+val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of",
                "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]
 
-val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", 
-                 "add_Suc", "add_number_of_left", "mult_number_of_left", 
+val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0",
+                 "add_Suc", "add_number_of_left", "mult_number_of_left",
                  "Suc_eq_add_numeral_1"])@
                  (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"])
-                 @ arith_simps@ nat_arith @ rel_simps 
-val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 
-           @{thm "divide_Numeral1"}, 
+                 @ arith_simps@ nat_arith @ rel_simps
+val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
+           @{thm "divide_Numeral1"},
            @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"},
            @{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"},
-           @{thm "mult_num_frac"}, @{thm "mult_frac_num"}, 
-           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, 
+           @{thm "mult_num_frac"}, @{thm "mult_frac_num"},
+           @{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"},
            @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"},
-           @{thm "diff_def"}, @{thm "minus_divide_left"}, 
+           @{thm "diff_def"}, @{thm "minus_divide_left"},
            @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym]
 
 local
 open Conv
 in
-fun comp_conv ctxt = (Simplifier.rewrite  
-(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"} 
+fun comp_conv ctxt = (Simplifier.rewrite
+(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"}
               addsimps ths addsimps comp_arith addsimps simp_thms
               addsimprocs field_cancel_numeral_factors
                addsimprocs [add_frac_frac_simproc ctxt, add_frac_num_simproc ctxt,
@@ -575,9 +576,9 @@
   [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
 end
 
-fun numeral_is_const ct = 
-  case term_of ct of 
-   Const (@{const_name "HOL.divide"},_) $ a $ b => 
+fun numeral_is_const ct =
+  case term_of ct of
+   Const (@{const_name "HOL.divide"},_) $ a $ b =>
      numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct)
  | Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct)
  | t => can HOLogic.dest_number t
@@ -587,16 +588,16 @@
     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
 
-fun mk_const phi cT x = 
+fun mk_const phi cT x =
  let val (a, b) = Rat.quotient_of_rat x
  in if b = 1 then Normalizer.mk_cnumber cT a
-    else Thm.capply 
-         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) 
+    else Thm.capply
+         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
                      (Normalizer.mk_cnumber cT a))
          (Normalizer.mk_cnumber cT b)
   end
 
-in 
+in
  NormalizerData.funs @{thm class_fieldgb.axioms}
    {is_const = K numeral_is_const,
     dest_const = K dest_const,
@@ -606,4 +607,334 @@
 
 *}
 
+
+subsection {* Instantiation of quantifier elimination in dense linear order
+    (Ferrante and Rackoff algorithm) to linear arithmetic over ordered fields.*}
+
+lemma neg_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
+proof-
+  assume H: "c < 0"
+  have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_eq_simps)
+  also have "\<dots> = (0 < x)" by simp
+  finally show  "(c*x < 0) == (x > 0)" by simp
+qed
+
+lemma pos_prod_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
+proof-
+  assume H: "c > 0"
+  hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_eq_simps)
+  also have "\<dots> = (0 > x)" by simp
+  finally show  "(c*x < 0) == (x < 0)" by simp
+qed
+
+lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
+proof-
+  assume H: "c < 0"
+  have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] ring_eq_simps)
+  also have "\<dots> = ((- 1/c)*t < x)" by simp
+  finally show  "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
+proof-
+  assume H: "c > 0"
+  have "c*x + t< 0 = (c*x < -t)"  by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] ring_eq_simps)
+  also have "\<dots> = ((- 1/c)*t > x)" by simp
+  finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
+qed
+
+lemma sum_lt:"((x::'a::pordered_ab_group_add) + t < 0) == (x < - t)"
+  using less_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma neg_prod_le:"(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
+proof-
+  assume H: "c < 0"
+  have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_eq_simps)
+  also have "\<dots> = (0 <= x)" by simp
+  finally show  "(c*x <= 0) == (x >= 0)" by simp
+qed
+
+lemma pos_prod_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
+proof-
+  assume H: "c > 0"
+  hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_eq_simps)
+  also have "\<dots> = (0 >= x)" by simp
+  finally show  "(c*x <= 0) == (x <= 0)" by simp
+qed
+
+lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>ordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
+proof-
+  assume H: "c < 0"
+  have "c*x + t <= 0 = (c*x <= -t)"  by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] ring_eq_simps)
+  also have "\<dots> = ((- 1/c)*t <= x)" by simp
+  finally show  "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
+qed
+
+lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>ordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
+proof-
+  assume H: "c > 0"
+  have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] ring_eq_simps)
+  also have "\<dots> = ((- 1/c)*t >= x)" by simp
+  finally show  "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
+qed
+
+lemma sum_le:"((x::'a::pordered_ab_group_add) + t <= 0) == (x <= - t)"
+  using le_diff_eq[where a= x and b=t and c=0] by simp
+
+lemma nz_prod_eq:"(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
+lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
+proof-
+  assume H: "c \<noteq> 0"
+  have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
+  also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] ring_eq_simps)
+  finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
+qed
+lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"
+  using eq_diff_eq[where a= x and b=t and c=0] by simp
+
+
+interpretation class_ordered_field_dense_linear_order: dense_linear_order
+ ["op <=" "op <"
+   "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,number_ring}) + y)"]
+proof (unfold_locales,
+  simp_all only: ordered_field_no_ub ordered_field_no_lb,
+    auto simp add: linorder_not_le)
+  fix x y::'a assume lt: "x < y"
+  from  less_half_sum[OF lt] show "x < (x + y) /2" by simp
+next
+  fix x y::'a assume lt: "x < y"
+  from  gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
+qed
+
+declaration{*
+let
+fun earlier [] x y = false
+        | earlier (h::t) x y =
+    if h aconvc y then false else if h aconvc x then true else earlier t x y;
+
+fun dest_frac ct = case term_of ct of
+   Const (@{const_name "HOL.divide"},_) $ a $ b=>
+    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
+ | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
+
+fun mk_frac phi cT x =
+ let val (a, b) = Rat.quotient_of_rat x
+ in if b = 1 then Normalizer.mk_cnumber cT a
+    else Thm.capply
+         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
+                     (Normalizer.mk_cnumber cT a))
+         (Normalizer.mk_cnumber cT b)
+ end
+
+fun whatis x ct = case term_of ct of
+  Const(@{const_name "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>
+     if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
+     else ("Nox",[])
+| Const(@{const_name "HOL.plus"}, _)$y$_ =>
+     if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
+     else ("Nox",[])
+| Const(@{const_name "HOL.times"}, _)$_$y =>
+     if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
+     else ("Nox",[])
+| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
+
+fun xnormalize_conv ctxt [] ct = reflexive ct
+| xnormalize_conv ctxt (vs as (x::_)) ct =
+   case term_of ct of
+   Const(@{const_name "Orderings.less"},_)$_$Const(@{const_name "HOL.zero"},_) =>
+    (case whatis x (Thm.dest_arg1 ct) of
+    ("c*x+t",[c,t]) =>
+       let
+        val cr = dest_frac c
+        val clt = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
+             (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+      in rth end
+    | ("x+t",[t]) =>
+       let
+        val T = ctyp_of_term x
+        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+       in  rth end
+    | ("c*x",[c]) =>
+       let
+        val cr = dest_frac c
+        val clt = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+             (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
+        val rth = th
+      in rth end
+    | _ => reflexive ct)
+
+
+|  Const(@{const_name "Orderings.less_eq"},_)$_$Const(@{const_name "HOL.zero"},_) =>
+   (case whatis x (Thm.dest_arg1 ct) of
+    ("c*x+t",[c,t]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
+             (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+      in rth end
+    | ("x+t",[t]) =>
+       let
+        val T = ctyp_of_term x
+        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+       in  rth end
+    | ("c*x",[c]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
+        val cz = Thm.dest_arg ct
+        val neg = cr </ Rat.zero
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+               (Thm.capply @{cterm "Trueprop"}
+                  (if neg then Thm.capply (Thm.capply clt c) cz
+                    else Thm.capply (Thm.capply clt cz) c))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
+             (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
+        val rth = th
+      in rth end
+    | _ => reflexive ct)
+
+|  Const("op =",_)$_$Const(@{const_name "HOL.zero"},_) =>
+   (case whatis x (Thm.dest_arg1 ct) of
+    ("c*x+t",[c,t]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val ceq = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+            (Thm.capply @{cterm "Trueprop"}
+             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val th = implies_elim
+                 (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+      in rth end
+    | ("x+t",[t]) =>
+       let
+        val T = ctyp_of_term x
+        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
+        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
+              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
+       in  rth end
+    | ("c*x",[c]) =>
+       let
+        val T = ctyp_of_term x
+        val cr = dest_frac c
+        val ceq = Thm.dest_fun2 ct
+        val cz = Thm.dest_arg ct
+        val cthp = Simplifier.rewrite (local_simpset_of ctxt)
+            (Thm.capply @{cterm "Trueprop"}
+             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
+        val cth = equal_elim (symmetric cthp) TrueI
+        val rth = implies_elim
+                 (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
+      in rth end
+    | _ => reflexive ct);
+
+local
+  val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
+  val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
+  val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
+in
+fun field_isolate_conv phi ctxt vs ct = case term_of ct of
+  Const(@{const_name "Orderings.less"},_)$a$b =>
+   let val (ca,cb) = Thm.dest_binop ct
+       val T = ctyp_of_term ca
+       val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
+       val nth = Conv.fconv_rule
+         (Conv.arg_conv (Conv.arg1_conv
+              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+   in rth end
+| Const(@{const_name "Orderings.less_eq"},_)$a$b =>
+   let val (ca,cb) = Thm.dest_binop ct
+       val T = ctyp_of_term ca
+       val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
+       val nth = Conv.fconv_rule
+         (Conv.arg_conv (Conv.arg1_conv
+              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+   in rth end
+
+| Const("op =",_)$a$b =>
+   let val (ca,cb) = Thm.dest_binop ct
+       val T = ctyp_of_term ca
+       val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
+       val nth = Conv.fconv_rule
+         (Conv.arg_conv (Conv.arg1_conv
+              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
+       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
+   in rth end
+| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
+| _ => reflexive ct
+end;
+
+fun classfield_whatis phi =
+ let
+  fun h x t =
+   case term_of t of
+     Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
+                            else Ferrante_Rackoff_Data.Nox
+   | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
+                            else Ferrante_Rackoff_Data.Nox
+   | Const(@{const_name "Orderings.less"},_)$y$z =>
+       if term_of x aconv y then Ferrante_Rackoff_Data.Lt
+        else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
+        else Ferrante_Rackoff_Data.Nox
+   | Const (@{const_name "Orderings.less_eq"},_)$y$z =>
+         if term_of x aconv y then Ferrante_Rackoff_Data.Le
+         else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
+         else Ferrante_Rackoff_Data.Nox
+   | _ => Ferrante_Rackoff_Data.Nox
+ in h end;
+fun class_field_ss phi =
+   HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
+   addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
+
+in
+Ferrante_Rackoff_Data.funs @{thm "class_ordered_field_dense_linear_order.ferrack_axiom"}
+  {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
 end
+*}
+
+end




