# HG changeset patch
# User haftmann
# Date 1273237552 -7200
# Node ID 7f1da69cacb3a1bfab8112fd2d7b0d2c8bfb3495
# Parent  912080b2c4495dc7bf99db1618b249b2e42600cd
split of semiring normalization from Groebner theory; moved field_comp_conv to Numeral_Simproces

diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Groebner_Basis.thy
--- a/src/HOL/Groebner_Basis.thy	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Groebner_Basis.thy	Fri May 07 15:05:52 2010 +0200
@@ -2,341 +2,14 @@
     Author:     Amine Chaieb, TU Muenchen
 *)
 
-header {* Semiring normalization and Groebner Bases *}
+header {* Groebner bases *}
 
 theory Groebner_Basis
-imports Numeral_Simprocs Nat_Transfer
+imports Semiring_Normalization
 uses
-  "Tools/Groebner_Basis/normalizer.ML"
   ("Tools/Groebner_Basis/groebner.ML")
 begin
 
-subsection {* Semiring normalization *}
-
-setup Normalizer.setup
-
-locale normalizing_semiring =
-  fixes add mul pwr r0 r1
-  assumes add_a:"(add x (add y z) = add (add x y) z)"
-    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
-    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
-    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
-    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
-    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
-begin
-
-lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
-proof (induct p)
-  case 0
-  then show ?case by (auto simp add: pwr_0 mul_1)
-next
-  case Suc
-  from this [symmetric] show ?case
-    by (auto simp add: pwr_Suc mul_1 mul_a)
-qed
-
-lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
-proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
-  fix q x y
-  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
-  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
-    by (simp add: mul_a)
-  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
-  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
-  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
-    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
-qed
-
-lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
-proof (induct p arbitrary: q)
-  case 0
-  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
-next
-  case Suc
-  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
-qed
-
-lemma semiring_ops:
-  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
-    and "TERM r0" and "TERM r1" .
-
-lemma semiring_rules:
-  "add (mul a m) (mul b m) = mul (add a b) m"
-  "add (mul a m) m = mul (add a r1) m"
-  "add m (mul a m) = mul (add a r1) m"
-  "add m m = mul (add r1 r1) m"
-  "add r0 a = a"
-  "add a r0 = a"
-  "mul a b = mul b a"
-  "mul (add a b) c = add (mul a c) (mul b c)"
-  "mul r0 a = r0"
-  "mul a r0 = r0"
-  "mul r1 a = a"
-  "mul a r1 = a"
-  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
-  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
-  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
-  "mul (mul lx ly) rx = mul (mul lx rx) ly"
-  "mul (mul lx ly) rx = mul lx (mul ly rx)"
-  "mul lx (mul rx ry) = mul (mul lx rx) ry"
-  "mul lx (mul rx ry) = mul rx (mul lx ry)"
-  "add (add a b) (add c d) = add (add a c) (add b d)"
-  "add (add a b) c = add a (add b c)"
-  "add a (add c d) = add c (add a d)"
-  "add (add a b) c = add (add a c) b"
-  "add a c = add c a"
-  "add a (add c d) = add (add a c) d"
-  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
-  "mul x (pwr x q) = pwr x (Suc q)"
-  "mul (pwr x q) x = pwr x (Suc q)"
-  "mul x x = pwr x 2"
-  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
-  "pwr (pwr x p) q = pwr x (p * q)"
-  "pwr x 0 = r1"
-  "pwr x 1 = x"
-  "mul x (add y z) = add (mul x y) (mul x z)"
-  "pwr x (Suc q) = mul x (pwr x q)"
-  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
-  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
-proof -
-  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
-next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
-next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
-next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
-next show "add r0 a = a" using add_0 by simp
-next show "add a r0 = a" using add_0 add_c by simp
-next show "mul a b = mul b a" using mul_c by simp
-next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
-next show "mul r0 a = r0" using mul_0 by simp
-next show "mul a r0 = r0" using mul_0 mul_c by simp
-next show "mul r1 a = a" using mul_1 by simp
-next show "mul a r1 = a" using mul_1 mul_c by simp
-next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
-    using mul_c mul_a by simp
-next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
-    using mul_a by simp
-next
-  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
-  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
-  finally
-  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
-    using mul_c by simp
-next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
-next
-  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
-next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
-next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
-next show "add (add a b) (add c d) = add (add a c) (add b d)"
-    using add_c add_a by simp
-next show "add (add a b) c = add a (add b c)" using add_a by simp
-next show "add a (add c d) = add c (add a d)"
-    apply (simp add: add_a) by (simp only: add_c)
-next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
-next show "add a c = add c a" by (rule add_c)
-next show "add a (add c d) = add (add a c) d" using add_a by simp
-next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
-next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
-next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
-next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
-next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
-next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
-next show "pwr x 0 = r1" using pwr_0 .
-next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
-next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
-next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
-next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
-next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
-    by (simp add: nat_number' pwr_Suc mul_pwr)
-qed
-
-
-lemmas normalizing_semiring_axioms' =
-  normalizing_semiring_axioms [normalizer
-    semiring ops: semiring_ops
-    semiring rules: semiring_rules]
-
-end
-
-sublocale comm_semiring_1
-  < normalizing!: normalizing_semiring plus times power zero one
-proof
-qed (simp_all add: algebra_simps)
-
-declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
-
-locale normalizing_ring = normalizing_semiring +
-  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
-    and neg :: "'a \<Rightarrow> 'a"
-  assumes neg_mul: "neg x = mul (neg r1) x"
-    and sub_add: "sub x y = add x (neg y)"
-begin
-
-lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
-
-lemmas ring_rules = neg_mul sub_add
-
-lemmas normalizing_ring_axioms' =
-  normalizing_ring_axioms [normalizer
-    semiring ops: semiring_ops
-    semiring rules: semiring_rules
-    ring ops: ring_ops
-    ring rules: ring_rules]
-
-end
-
-sublocale comm_ring_1
-  < normalizing!: normalizing_ring plus times power zero one minus uminus
-proof
-qed (simp_all add: diff_minus)
-
-declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
-
-locale normalizing_field = normalizing_ring +
-  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
-    and inverse:: "'a \<Rightarrow> 'a"
-  assumes divide_inverse: "divide x y = mul x (inverse y)"
-     and inverse_divide: "inverse x = divide r1 x"
-begin
-
-lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
-
-lemmas field_rules = divide_inverse inverse_divide
-
-lemmas normalizing_field_axioms' =
-  normalizing_field_axioms [normalizer
-    semiring ops: semiring_ops
-    semiring rules: semiring_rules
-    ring ops: ring_ops
-    ring rules: ring_rules
-    field ops: field_ops
-    field rules: field_rules]
-
-end
-
-locale normalizing_semiring_cancel = normalizing_semiring +
-  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
-  and add_mul_solve: "add (mul w y) (mul x z) =
-    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
-begin
-
-lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
-proof-
-  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
-  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
-    using add_mul_solve by blast
-  finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
-    by simp
-qed
-
-lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
-  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
-proof(clarify)
-  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
-    and eq: "add b (mul r c) = add b (mul r d)"
-  hence "mul r c = mul r d" using cnd add_cancel by simp
-  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
-    using mul_0 add_cancel by simp
-  thus "False" using add_mul_solve nz cnd by simp
-qed
-
-lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
-proof-
-  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
-  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
-qed
-
-declare normalizing_semiring_axioms' [normalizer del]
-
-lemmas normalizing_semiring_cancel_axioms' =
-  normalizing_semiring_cancel_axioms [normalizer
-    semiring ops: semiring_ops
-    semiring rules: semiring_rules
-    idom rules: noteq_reduce add_scale_eq_noteq]
-
-end
-
-locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
-  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
-begin
-
-declare normalizing_ring_axioms' [normalizer del]
-
-lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
-  semiring ops: semiring_ops
-  semiring rules: semiring_rules
-  ring ops: ring_ops
-  ring rules: ring_rules
-  idom rules: noteq_reduce add_scale_eq_noteq
-  ideal rules: subr0_iff add_r0_iff]
-
-end
-
-sublocale idom
-  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
-proof
-  fix w x y z
-  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
-  proof
-    assume "w * y + x * z = w * z + x * y"
-    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
-    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
-    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
-    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
-    then show "w = x \<or> y = z" by auto
-  qed (auto simp add: add_ac)
-qed (simp_all add: algebra_simps)
-
-declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
-
-interpretation normalizing_nat!: normalizing_semiring_cancel
-  "op +" "op *" "op ^" "0::nat" "1"
-proof (unfold_locales, simp add: algebra_simps)
-  fix w x y z ::"nat"
-  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
-    hence "y < z \<or> y > z" by arith
-    moreover {
-      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
-      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
-      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
-      hence "x*k = w*k" by simp
-      hence "w = x" using kp by simp }
-    moreover {
-      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
-      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
-      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
-      hence "w*k = x*k" by simp
-      hence "w = x" using kp by simp }
-    ultimately have "w=x" by blast }
-  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
-qed
-
-declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
-
-locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
-begin
-
-declare normalizing_field_axioms' [normalizer del]
-
-lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
-  semiring ops: semiring_ops
-  semiring rules: semiring_rules
-  ring ops: ring_ops
-  ring rules: ring_rules
-  field ops: field_ops
-  field rules: field_rules
-  idom rules: noteq_reduce add_scale_eq_noteq
-  ideal rules: subr0_iff add_r0_iff]
-
-end
-
-sublocale field 
-  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
-proof
-qed (simp_all add: divide_inverse)
-
-declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
- 
-
 subsection {* Groebner Bases *}
 
 lemmas bool_simps = simp_thms(1-34)
@@ -367,6 +40,11 @@
 
 setup Algebra_Simplification.setup
 
+use "Tools/Groebner_Basis/groebner.ML"
+
+method_setup algebra = Groebner.algebra_method
+  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
+
 declare dvd_def[algebra]
 declare dvd_eq_mod_eq_0[symmetric, algebra]
 declare mod_div_trivial[algebra]
@@ -395,9 +73,4 @@
 declare zmod_eq_dvd_iff[algebra]
 declare nat_mod_eq_iff[algebra]
 
-use "Tools/Groebner_Basis/groebner.ML"
-
-method_setup algebra = Groebner.algebra_method
-  "solve polynomial equations over (semi)rings and ideal membership problems using Groebner bases"
-
 end
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/IsaMakefile	Fri May 07 15:05:52 2010 +0200
@@ -271,6 +271,7 @@
   Random.thy \
   Random_Sequence.thy \
   Recdef.thy \
+  Semiring_Normalization.thy \
   SetInterval.thy \
   Sledgehammer.thy \
   String.thy \
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML
--- a/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Library/Sum_Of_Squares/sum_of_squares.ML	Fri May 07 15:05:52 2010 +0200
@@ -1222,7 +1222,7 @@
    in
   (let val th = tryfind trivial_axiom (keq @ klep @ kltp)
    in
-    (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Normalizer.field_comp_conv) th, RealArith.Trivial)
+    (fconv_rule (arg_conv (arg1_conv real_poly_conv) then_conv Numeral_Simprocs.field_comp_conv) th, RealArith.Trivial)
    end)
    handle Failure _ =>
      (let val proof =
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Library/normarith.ML
--- a/src/HOL/Library/normarith.ML	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Library/normarith.ML	Fri May 07 15:05:52 2010 +0200
@@ -168,7 +168,7 @@
   val real_poly_conv = 
     Normalizer.semiring_normalize_wrapper ctxt
      (the (Normalizer.match ctxt @{cterm "(0::real) + 1"}))
- in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Normalizer.field_comp_conv then_conv real_poly_conv)))
+ in fconv_rule (arg_conv ((rewr_conv @{thm ge_iff_diff_ge_0}) then_conv arg_conv (Numeral_Simprocs.field_comp_conv then_conv real_poly_conv)))
 end;
 
  fun absc cv ct = case term_of ct of 
@@ -190,8 +190,8 @@
  val apply_pth5 = rewr_conv @{thm pth_5};
  val apply_pth6 = rewr_conv @{thm pth_6};
  val apply_pth7 = rewrs_conv @{thms pth_7};
- val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Normalizer.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
- val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Normalizer.field_comp_conv);
+ val apply_pth8 = rewr_conv @{thm pth_8} then_conv arg1_conv Numeral_Simprocs.field_comp_conv then_conv (try_conv (rewr_conv (mk_meta_eq @{thm scaleR_zero_left})));
+ val apply_pth9 = rewrs_conv @{thms pth_9} then_conv arg1_conv (arg1_conv Numeral_Simprocs.field_comp_conv);
  val apply_ptha = rewr_conv @{thm pth_a};
  val apply_pthb = rewrs_conv @{thms pth_b};
  val apply_pthc = rewrs_conv @{thms pth_c};
@@ -204,7 +204,7 @@
  | _ => error "headvector: non-canonical term"
 
 fun vector_cmul_conv ct =
-   ((apply_pth5 then_conv arg1_conv Normalizer.field_comp_conv) else_conv
+   ((apply_pth5 then_conv arg1_conv Numeral_Simprocs.field_comp_conv) else_conv
     (apply_pth6 then_conv binop_conv vector_cmul_conv)) ct
 
 fun vector_add_conv ct = apply_pth7 ct 
@@ -396,7 +396,7 @@
   fun init_conv ctxt = 
    Simplifier.rewrite (Simplifier.context ctxt 
      (HOL_basic_ss addsimps ([(*@{thm vec_0}, @{thm vec_1},*) @{thm dist_norm}, @{thm diff_0_right}, @{thm right_minus}, @{thm diff_self}, @{thm norm_zero}] @ @{thms arithmetic_simps} @ @{thms norm_pths})))
-   then_conv Normalizer.field_comp_conv 
+   then_conv Numeral_Simprocs.field_comp_conv 
    then_conv nnf_conv
 
  fun pure ctxt = fst o RealArith.gen_prover_real_arith ctxt (real_vector_prover ctxt);
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Library/positivstellensatz.ML
--- a/src/HOL/Library/positivstellensatz.ML	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Library/positivstellensatz.ML	Fri May 07 15:05:52 2010 +0200
@@ -751,7 +751,7 @@
       (the (Normalizer.match ctxt @{cterm "(0::real) + 1"})) 
      simple_cterm_ord
 in gen_real_arith ctxt
-   (cterm_of_rat, Normalizer.field_comp_conv, Normalizer.field_comp_conv, Normalizer.field_comp_conv,
+   (cterm_of_rat, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv, Numeral_Simprocs.field_comp_conv,
     main,neg,add,mul, prover)
 end;
 
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Semiring_Normalization.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Semiring_Normalization.thy	Fri May 07 15:05:52 2010 +0200
@@ -0,0 +1,336 @@
+(*  Title:      HOL/Semiring_Normalization.thy
+    Author:     Amine Chaieb, TU Muenchen
+*)
+
+header {* Semiring normalization *}
+
+theory Semiring_Normalization
+imports Numeral_Simprocs Nat_Transfer
+uses
+  "Tools/Groebner_Basis/normalizer.ML"
+begin
+
+setup Normalizer.setup
+
+locale normalizing_semiring =
+  fixes add mul pwr r0 r1
+  assumes add_a:"(add x (add y z) = add (add x y) z)"
+    and add_c: "add x y = add y x" and add_0:"add r0 x = x"
+    and mul_a:"mul x (mul y z) = mul (mul x y) z" and mul_c:"mul x y = mul y x"
+    and mul_1:"mul r1 x = x" and  mul_0:"mul r0 x = r0"
+    and mul_d:"mul x (add y z) = add (mul x y) (mul x z)"
+    and pwr_0:"pwr x 0 = r1" and pwr_Suc:"pwr x (Suc n) = mul x (pwr x n)"
+begin
+
+lemma mul_pwr:"mul (pwr x p) (pwr x q) = pwr x (p + q)"
+proof (induct p)
+  case 0
+  then show ?case by (auto simp add: pwr_0 mul_1)
+next
+  case Suc
+  from this [symmetric] show ?case
+    by (auto simp add: pwr_Suc mul_1 mul_a)
+qed
+
+lemma pwr_mul: "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
+proof (induct q arbitrary: x y, auto simp add:pwr_0 pwr_Suc mul_1)
+  fix q x y
+  assume "\<And>x y. pwr (mul x y) q = mul (pwr x q) (pwr y q)"
+  have "mul (mul x y) (mul (pwr x q) (pwr y q)) = mul x (mul y (mul (pwr x q) (pwr y q)))"
+    by (simp add: mul_a)
+  also have "\<dots> = (mul (mul y (mul (pwr y q) (pwr x q))) x)" by (simp add: mul_c)
+  also have "\<dots> = (mul (mul y (pwr y q)) (mul (pwr x q) x))" by (simp add: mul_a)
+  finally show "mul (mul x y) (mul (pwr x q) (pwr y q)) =
+    mul (mul x (pwr x q)) (mul y (pwr y q))" by (simp add: mul_c)
+qed
+
+lemma pwr_pwr: "pwr (pwr x p) q = pwr x (p * q)"
+proof (induct p arbitrary: q)
+  case 0
+  show ?case using pwr_Suc mul_1 pwr_0 by (induct q) auto
+next
+  case Suc
+  thus ?case by (auto simp add: mul_pwr [symmetric] pwr_mul pwr_Suc)
+qed
+
+lemma semiring_ops:
+  shows "TERM (add x y)" and "TERM (mul x y)" and "TERM (pwr x n)"
+    and "TERM r0" and "TERM r1" .
+
+lemma semiring_rules:
+  "add (mul a m) (mul b m) = mul (add a b) m"
+  "add (mul a m) m = mul (add a r1) m"
+  "add m (mul a m) = mul (add a r1) m"
+  "add m m = mul (add r1 r1) m"
+  "add r0 a = a"
+  "add a r0 = a"
+  "mul a b = mul b a"
+  "mul (add a b) c = add (mul a c) (mul b c)"
+  "mul r0 a = r0"
+  "mul a r0 = r0"
+  "mul r1 a = a"
+  "mul a r1 = a"
+  "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
+  "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
+  "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
+  "mul (mul lx ly) rx = mul (mul lx rx) ly"
+  "mul (mul lx ly) rx = mul lx (mul ly rx)"
+  "mul lx (mul rx ry) = mul (mul lx rx) ry"
+  "mul lx (mul rx ry) = mul rx (mul lx ry)"
+  "add (add a b) (add c d) = add (add a c) (add b d)"
+  "add (add a b) c = add a (add b c)"
+  "add a (add c d) = add c (add a d)"
+  "add (add a b) c = add (add a c) b"
+  "add a c = add c a"
+  "add a (add c d) = add (add a c) d"
+  "mul (pwr x p) (pwr x q) = pwr x (p + q)"
+  "mul x (pwr x q) = pwr x (Suc q)"
+  "mul (pwr x q) x = pwr x (Suc q)"
+  "mul x x = pwr x 2"
+  "pwr (mul x y) q = mul (pwr x q) (pwr y q)"
+  "pwr (pwr x p) q = pwr x (p * q)"
+  "pwr x 0 = r1"
+  "pwr x 1 = x"
+  "mul x (add y z) = add (mul x y) (mul x z)"
+  "pwr x (Suc q) = mul x (pwr x q)"
+  "pwr x (2*n) = mul (pwr x n) (pwr x n)"
+  "pwr x (Suc (2*n)) = mul x (mul (pwr x n) (pwr x n))"
+proof -
+  show "add (mul a m) (mul b m) = mul (add a b) m" using mul_d mul_c by simp
+next show"add (mul a m) m = mul (add a r1) m" using mul_d mul_c mul_1 by simp
+next show "add m (mul a m) = mul (add a r1) m" using mul_c mul_d mul_1 add_c by simp
+next show "add m m = mul (add r1 r1) m" using mul_c mul_d mul_1 by simp
+next show "add r0 a = a" using add_0 by simp
+next show "add a r0 = a" using add_0 add_c by simp
+next show "mul a b = mul b a" using mul_c by simp
+next show "mul (add a b) c = add (mul a c) (mul b c)" using mul_c mul_d by simp
+next show "mul r0 a = r0" using mul_0 by simp
+next show "mul a r0 = r0" using mul_0 mul_c by simp
+next show "mul r1 a = a" using mul_1 by simp
+next show "mul a r1 = a" using mul_1 mul_c by simp
+next show "mul (mul lx ly) (mul rx ry) = mul (mul lx rx) (mul ly ry)"
+    using mul_c mul_a by simp
+next show "mul (mul lx ly) (mul rx ry) = mul lx (mul ly (mul rx ry))"
+    using mul_a by simp
+next
+  have "mul (mul lx ly) (mul rx ry) = mul (mul rx ry) (mul lx ly)" by (rule mul_c)
+  also have "\<dots> = mul rx (mul ry (mul lx ly))" using mul_a by simp
+  finally
+  show "mul (mul lx ly) (mul rx ry) = mul rx (mul (mul lx ly) ry)"
+    using mul_c by simp
+next show "mul (mul lx ly) rx = mul (mul lx rx) ly" using mul_c mul_a by simp
+next
+  show "mul (mul lx ly) rx = mul lx (mul ly rx)" by (simp add: mul_a)
+next show "mul lx (mul rx ry) = mul (mul lx rx) ry" by (simp add: mul_a )
+next show "mul lx (mul rx ry) = mul rx (mul lx ry)" by (simp add: mul_a,simp add: mul_c)
+next show "add (add a b) (add c d) = add (add a c) (add b d)"
+    using add_c add_a by simp
+next show "add (add a b) c = add a (add b c)" using add_a by simp
+next show "add a (add c d) = add c (add a d)"
+    apply (simp add: add_a) by (simp only: add_c)
+next show "add (add a b) c = add (add a c) b" using add_a add_c by simp
+next show "add a c = add c a" by (rule add_c)
+next show "add a (add c d) = add (add a c) d" using add_a by simp
+next show "mul (pwr x p) (pwr x q) = pwr x (p + q)" by (rule mul_pwr)
+next show "mul x (pwr x q) = pwr x (Suc q)" using pwr_Suc by simp
+next show "mul (pwr x q) x = pwr x (Suc q)" using pwr_Suc mul_c by simp
+next show "mul x x = pwr x 2" by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
+next show "pwr (mul x y) q = mul (pwr x q) (pwr y q)" by (rule pwr_mul)
+next show "pwr (pwr x p) q = pwr x (p * q)" by (rule pwr_pwr)
+next show "pwr x 0 = r1" using pwr_0 .
+next show "pwr x 1 = x" unfolding One_nat_def by (simp add: nat_number' pwr_Suc pwr_0 mul_1 mul_c)
+next show "mul x (add y z) = add (mul x y) (mul x z)" using mul_d by simp
+next show "pwr x (Suc q) = mul x (pwr x q)" using pwr_Suc by simp
+next show "pwr x (2 * n) = mul (pwr x n) (pwr x n)" by (simp add: nat_number' mul_pwr)
+next show "pwr x (Suc (2 * n)) = mul x (mul (pwr x n) (pwr x n))"
+    by (simp add: nat_number' pwr_Suc mul_pwr)
+qed
+
+
+lemmas normalizing_semiring_axioms' =
+  normalizing_semiring_axioms [normalizer
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules]
+
+end
+
+sublocale comm_semiring_1
+  < normalizing!: normalizing_semiring plus times power zero one
+proof
+qed (simp_all add: algebra_simps)
+
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_semiring_axioms'} *}
+
+locale normalizing_ring = normalizing_semiring +
+  fixes sub :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+    and neg :: "'a \<Rightarrow> 'a"
+  assumes neg_mul: "neg x = mul (neg r1) x"
+    and sub_add: "sub x y = add x (neg y)"
+begin
+
+lemma ring_ops: shows "TERM (sub x y)" and "TERM (neg x)" .
+
+lemmas ring_rules = neg_mul sub_add
+
+lemmas normalizing_ring_axioms' =
+  normalizing_ring_axioms [normalizer
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules
+    ring ops: ring_ops
+    ring rules: ring_rules]
+
+end
+
+sublocale comm_ring_1
+  < normalizing!: normalizing_ring plus times power zero one minus uminus
+proof
+qed (simp_all add: diff_minus)
+
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_axioms'} *}
+
+locale normalizing_field = normalizing_ring +
+  fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
+    and inverse:: "'a \<Rightarrow> 'a"
+  assumes divide_inverse: "divide x y = mul x (inverse y)"
+     and inverse_divide: "inverse x = divide r1 x"
+begin
+
+lemma field_ops: shows "TERM (divide x y)" and "TERM (inverse x)" .
+
+lemmas field_rules = divide_inverse inverse_divide
+
+lemmas normalizing_field_axioms' =
+  normalizing_field_axioms [normalizer
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules
+    ring ops: ring_ops
+    ring rules: ring_rules
+    field ops: field_ops
+    field rules: field_rules]
+
+end
+
+locale normalizing_semiring_cancel = normalizing_semiring +
+  assumes add_cancel: "add (x::'a) y = add x z \<longleftrightarrow> y = z"
+  and add_mul_solve: "add (mul w y) (mul x z) =
+    add (mul w z) (mul x y) \<longleftrightarrow> w = x \<or> y = z"
+begin
+
+lemma noteq_reduce: "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
+proof-
+  have "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> \<not> (a = b \<or> c = d)" by simp
+  also have "\<dots> \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
+    using add_mul_solve by blast
+  finally show "a \<noteq> b \<and> c \<noteq> d \<longleftrightarrow> add (mul a c) (mul b d) \<noteq> add (mul a d) (mul b c)"
+    by simp
+qed
+
+lemma add_scale_eq_noteq: "\<lbrakk>r \<noteq> r0 ; (a = b) \<and> ~(c = d)\<rbrakk>
+  \<Longrightarrow> add a (mul r c) \<noteq> add b (mul r d)"
+proof(clarify)
+  assume nz: "r\<noteq> r0" and cnd: "c\<noteq>d"
+    and eq: "add b (mul r c) = add b (mul r d)"
+  hence "mul r c = mul r d" using cnd add_cancel by simp
+  hence "add (mul r0 d) (mul r c) = add (mul r0 c) (mul r d)"
+    using mul_0 add_cancel by simp
+  thus "False" using add_mul_solve nz cnd by simp
+qed
+
+lemma add_r0_iff: " x = add x a \<longleftrightarrow> a = r0"
+proof-
+  have "a = r0 \<longleftrightarrow> add x a = add x r0" by (simp add: add_cancel)
+  thus "x = add x a \<longleftrightarrow> a = r0" by (auto simp add: add_c add_0)
+qed
+
+declare normalizing_semiring_axioms' [normalizer del]
+
+lemmas normalizing_semiring_cancel_axioms' =
+  normalizing_semiring_cancel_axioms [normalizer
+    semiring ops: semiring_ops
+    semiring rules: semiring_rules
+    idom rules: noteq_reduce add_scale_eq_noteq]
+
+end
+
+locale normalizing_ring_cancel = normalizing_semiring_cancel + normalizing_ring + 
+  assumes subr0_iff: "sub x y = r0 \<longleftrightarrow> x = y"
+begin
+
+declare normalizing_ring_axioms' [normalizer del]
+
+lemmas normalizing_ring_cancel_axioms' = normalizing_ring_cancel_axioms [normalizer
+  semiring ops: semiring_ops
+  semiring rules: semiring_rules
+  ring ops: ring_ops
+  ring rules: ring_rules
+  idom rules: noteq_reduce add_scale_eq_noteq
+  ideal rules: subr0_iff add_r0_iff]
+
+end
+
+sublocale idom
+  < normalizing!: normalizing_ring_cancel plus times power zero one minus uminus
+proof
+  fix w x y z
+  show "w * y + x * z = w * z + x * y \<longleftrightarrow> w = x \<or> y = z"
+  proof
+    assume "w * y + x * z = w * z + x * y"
+    then have "w * y + x * z - w * z - x * y = 0" by (simp add: algebra_simps)
+    then have "w * (y - z) - x * (y - z) = 0" by (simp add: algebra_simps)
+    then have "(y - z) * (w - x) = 0" by (simp add: algebra_simps)
+    then have "y - z = 0 \<or> w - x = 0" by (rule divisors_zero)
+    then show "w = x \<or> y = z" by auto
+  qed (auto simp add: add_ac)
+qed (simp_all add: algebra_simps)
+
+declaration {* Normalizer.semiring_funs @{thm normalizing.normalizing_ring_cancel_axioms'} *}
+
+interpretation normalizing_nat!: normalizing_semiring_cancel
+  "op +" "op *" "op ^" "0::nat" "1"
+proof (unfold_locales, simp add: algebra_simps)
+  fix w x y z ::"nat"
+  { assume p: "w * y + x * z = w * z + x * y" and ynz: "y \<noteq> z"
+    hence "y < z \<or> y > z" by arith
+    moreover {
+      assume lt:"y <z" hence "\<exists>k. z = y + k \<and> k > 0" by (rule_tac x="z - y" in exI, auto)
+      then obtain k where kp: "k>0" and yz:"z = y + k" by blast
+      from p have "(w * y + x *y) + x*k = (w * y + x*y) + w*k" by (simp add: yz algebra_simps)
+      hence "x*k = w*k" by simp
+      hence "w = x" using kp by simp }
+    moreover {
+      assume lt: "y >z" hence "\<exists>k. y = z + k \<and> k>0" by (rule_tac x="y - z" in exI, auto)
+      then obtain k where kp: "k>0" and yz:"y = z + k" by blast
+      from p have "(w * z + x *z) + w*k = (w * z + x*z) + x*k" by (simp add: yz algebra_simps)
+      hence "w*k = x*k" by simp
+      hence "w = x" using kp by simp }
+    ultimately have "w=x" by blast }
+  thus "(w * y + x * z = w * z + x * y) = (w = x \<or> y = z)" by auto
+qed
+
+declaration {* Normalizer.semiring_funs @{thm normalizing_nat.normalizing_semiring_cancel_axioms'} *}
+
+locale normalizing_field_cancel = normalizing_ring_cancel + normalizing_field
+begin
+
+declare normalizing_field_axioms' [normalizer del]
+
+lemmas normalizing_field_cancel_axioms' = normalizing_field_cancel_axioms [normalizer
+  semiring ops: semiring_ops
+  semiring rules: semiring_rules
+  ring ops: ring_ops
+  ring rules: ring_rules
+  field ops: field_ops
+  field rules: field_rules
+  idom rules: noteq_reduce add_scale_eq_noteq
+  ideal rules: subr0_iff add_r0_iff]
+
+end
+
+sublocale field 
+  < normalizing!: normalizing_field_cancel plus times power zero one minus uminus divide inverse
+proof
+qed (simp_all add: divide_inverse)
+
+declaration {* Normalizer.field_funs @{thm normalizing.normalizing_field_cancel_axioms'} *}
+
+end
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Tools/Groebner_Basis/normalizer.ML
--- a/src/HOL/Tools/Groebner_Basis/normalizer.ML	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Tools/Groebner_Basis/normalizer.ML	Fri May 07 15:05:52 2010 +0200
@@ -31,7 +31,6 @@
   val semiring_normalizers_ord_wrapper:  Proof.context -> entry ->
     (cterm -> cterm -> bool) ->
       {add: conv, mul: conv, neg: conv, main: conv, pow: conv, sub: conv}
-  val field_comp_conv: conv
 
   val setup: theory -> theory
 end
@@ -41,156 +40,6 @@
 
 (** some conversion **)
 
-local
- val zr = @{cpat "0"}
- 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_num = mk_meta_eq @{thm "add_frac_num"}
- val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
-
- fun prove_nz ss T t =
-    let
-      val z = instantiate_cterm ([(zT,T)],[]) zr
-      val eq = instantiate_cterm ([(eqT,T)],[]) geq
-      val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
-           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
-                  (Thm.capply (Thm.capply eq t) z)))
-    in equal_elim (symmetric th) TrueI
-    end
-
- fun proc 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 T = ctyp_of_term x
-    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
-    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
-  in SOME (implies_elim (implies_elim th y_nz) z_nz)
-  end
-  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
-
- fun proc2 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 Rings.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 ss T y
-        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
-        end
-     | (_, Const (@{const_name Rings.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 ss T y
-        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
-        end
-     | _ => NONE)
-  end
-  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
-
- fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
-   | is_number t = can HOLogic.dest_number t
-
- val is_number = is_number o term_of
-
- fun proc3 phi ss ct =
-  (case term_of ct of
-    Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
-      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},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
-      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 =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
-      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 Rings.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 Rings.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 Rings.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 "eq_divide_eq"}
-      in SOME (mk_meta_eq th) end
-  | _ => NONE)
-  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
-
-val add_frac_frac_simproc =
-       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
-                     name = "add_frac_frac_simproc",
-                     proc = proc, identifier = []}
-
-val add_frac_num_simproc =
-       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
-                     name = "add_frac_num_simproc",
-                     proc = proc2, identifier = []}
-
-val ord_frac_simproc =
-  make_simproc
-    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
-             @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
-             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
-             @{cpat "?c <= (?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 ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
-           @{thm "divide_Numeral1"},
-           @{thm "divide_zero"}, @{thm "divide_Numeral0"},
-           @{thm "divide_divide_eq_left"}, 
-           @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
-           @{thm "times_divide_times_eq"},
-           @{thm "divide_divide_eq_right"},
-           @{thm "diff_def"}, @{thm "minus_divide_left"},
-           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
-           @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
-           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
-           (@{thm field_divide_inverse} RS sym)]
-
-in
-
-val field_comp_conv = (Simplifier.rewrite
-(HOL_basic_ss addsimps @{thms "semiring_norm"}
-              addsimps ths addsimps @{thms simp_thms}
-              addsimprocs Numeral_Simprocs.field_cancel_numeral_factors
-               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
-                            ord_frac_simproc]
-                addcongs [@{thm "if_weak_cong"}]))
-then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
-  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
-
-end
 
 
 (** data **)
@@ -365,7 +214,7 @@
      {is_const = K numeral_is_const,
       dest_const = K dest_const,
       mk_const = mk_const,
-      conv = K (K field_comp_conv)}
+      conv = K (K Numeral_Simprocs.field_comp_conv)}
   end;
 
 
diff -r 912080b2c449 -r 7f1da69cacb3 src/HOL/Tools/numeral_simprocs.ML
--- a/src/HOL/Tools/numeral_simprocs.ML	Fri May 07 10:00:24 2010 +0200
+++ b/src/HOL/Tools/numeral_simprocs.ML	Fri May 07 15:05:52 2010 +0200
@@ -1,7 +1,7 @@
 (* Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   2000  University of Cambridge
 
-Simprocs for the integer numerals.
+Simprocs for the (integer) numerals.
 *)
 
 (*To quote from Provers/Arith/cancel_numeral_factor.ML:
@@ -24,6 +24,7 @@
   val field_combine_numerals: simproc
   val field_cancel_numeral_factors: simproc list
   val num_ss: simpset
+  val field_comp_conv: conv
 end;
 
 structure Numeral_Simprocs : NUMERAL_SIMPROCS =
@@ -602,6 +603,157 @@
       "(l::'a::field_inverse_zero) / (m * n)"],
      K DivideCancelFactor.proc)];
 
+local
+ val zr = @{cpat "0"}
+ 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_num = mk_meta_eq @{thm "add_frac_num"}
+ val add_num_frac = mk_meta_eq @{thm "add_num_frac"}
+
+ fun prove_nz ss T t =
+    let
+      val z = instantiate_cterm ([(zT,T)],[]) zr
+      val eq = instantiate_cterm ([(eqT,T)],[]) geq
+      val th = Simplifier.rewrite (ss addsimps @{thms simp_thms})
+           (Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"}
+                  (Thm.capply (Thm.capply eq t) z)))
+    in equal_elim (symmetric th) TrueI
+    end
+
+ fun proc 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 T = ctyp_of_term x
+    val [y_nz, z_nz] = map (prove_nz ss T) [y, z]
+    val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq
+  in SOME (implies_elim (implies_elim th y_nz) z_nz)
+  end
+  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
+
+ fun proc2 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 Rings.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 ss T y
+        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz)
+        end
+     | (_, Const (@{const_name Rings.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 ss T y
+        in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz)
+        end
+     | _ => NONE)
+  end
+  handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE
+
+ fun is_number (Const(@{const_name Rings.divide},_)$a$b) = is_number a andalso is_number b
+   | is_number t = can HOLogic.dest_number t
+
+ val is_number = is_number o term_of
+
+ fun proc3 phi ss ct =
+  (case term_of ct of
+    Const(@{const_name Orderings.less},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
+      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},_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
+      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 =",_)$(Const(@{const_name Rings.divide},_)$_$_)$_ =>
+      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 Rings.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 Rings.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 Rings.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 "eq_divide_eq"}
+      in SOME (mk_meta_eq th) end
+  | _ => NONE)
+  handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE
+
+val add_frac_frac_simproc =
+       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}],
+                     name = "add_frac_frac_simproc",
+                     proc = proc, identifier = []}
+
+val add_frac_num_simproc =
+       make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}],
+                     name = "add_frac_num_simproc",
+                     proc = proc2, identifier = []}
+
+val ord_frac_simproc =
+  make_simproc
+    {lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"},
+             @{cpat "(?a::(?'a::{field, ord}))/?b <= ?c"},
+             @{cpat "?c < (?a::(?'a::{field, ord}))/?b"},
+             @{cpat "?c <= (?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 ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"},
+           @{thm "divide_Numeral1"},
+           @{thm "divide_zero"}, @{thm "divide_Numeral0"},
+           @{thm "divide_divide_eq_left"}, 
+           @{thm "times_divide_eq_left"}, @{thm "times_divide_eq_right"},
+           @{thm "times_divide_times_eq"},
+           @{thm "divide_divide_eq_right"},
+           @{thm "diff_def"}, @{thm "minus_divide_left"},
+           @{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym,
+           @{thm field_divide_inverse} RS sym, @{thm inverse_divide}, 
+           Conv.fconv_rule (Conv.arg_conv (Conv.arg1_conv (Conv.rewr_conv (mk_meta_eq @{thm mult_commute}))))   
+           (@{thm field_divide_inverse} RS sym)]
+
+in
+
+val field_comp_conv = (Simplifier.rewrite
+(HOL_basic_ss addsimps @{thms "semiring_norm"}
+              addsimps ths addsimps @{thms simp_thms}
+              addsimprocs field_cancel_numeral_factors
+               addsimprocs [add_frac_frac_simproc, add_frac_num_simproc,
+                            ord_frac_simproc]
+                addcongs [@{thm "if_weak_cong"}]))
+then_conv (Simplifier.rewrite (HOL_basic_ss addsimps
+  [@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)}))
+
+end
+
 end;
 
 Addsimprocs Numeral_Simprocs.cancel_numerals;