src/HOL/Hyperreal/HyperDef.thy

tidied

(*  Title       : HOL/Real/Hyperreal/HyperDef.thy
    ID          : $Id$
    Author      : Jacques D. Fleuriot
    Copyright   : 1998  University of Cambridge
    Description : Construction of hyperreals using ultrafilters
*)

theory HyperDef = Filter + Real
files ("fuf.ML"):  (*Warning: file fuf.ML refers to the name Hyperdef!*)


constdefs

  FreeUltrafilterNat   :: "nat set set"    ("\\<U>")
    "FreeUltrafilterNat == (SOME U. U \<in> FreeUltrafilter (UNIV:: nat set))"

  hyprel :: "((nat=>real)*(nat=>real)) set"
    "hyprel == {p. \<exists>X Y. p = ((X::nat=>real),Y) &
                   {n::nat. X(n) = Y(n)}: FreeUltrafilterNat}"

typedef hypreal = "UNIV//hyprel" 
    by (auto simp add: quotient_def) 

instance hypreal :: ord ..
instance hypreal :: zero ..
instance hypreal :: one ..
instance hypreal :: plus ..
instance hypreal :: times ..
instance hypreal :: minus ..
instance hypreal :: inverse ..


defs (overloaded)

  hypreal_zero_def:
  "0 == Abs_hypreal(hyprel``{%n::nat. (0::real)})"

  hypreal_one_def:
  "1 == Abs_hypreal(hyprel``{%n::nat. (1::real)})"

  hypreal_minus_def:
  "- P == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). hyprel``{%n::nat. - (X n)})"

  hypreal_diff_def:
  "x - y == x + -(y::hypreal)"

  hypreal_inverse_def:
  "inverse P == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P).
                    hyprel``{%n. if X n = 0 then 0 else inverse (X n)})"

  hypreal_divide_def:
  "P / Q::hypreal == P * inverse Q"

constdefs

  hypreal_of_real  :: "real => hypreal"
  "hypreal_of_real r         == Abs_hypreal(hyprel``{%n::nat. r})"

  omega   :: hypreal   (*an infinite number = [<1,2,3,...>] *)
  "omega == Abs_hypreal(hyprel``{%n::nat. real (Suc n)})"

  epsilon :: hypreal   (*an infinitesimal number = [<1,1/2,1/3,...>] *)
  "epsilon == Abs_hypreal(hyprel``{%n::nat. inverse (real (Suc n))})"

syntax (xsymbols)
  omega   :: hypreal   ("\<omega>")
  epsilon :: hypreal   ("\<epsilon>")


defs

  hypreal_add_def:
  "P + Q == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). \<Union>Y \<in> Rep_hypreal(Q).
                hyprel``{%n::nat. X n + Y n})"

  hypreal_mult_def:
  "P * Q == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). \<Union>Y \<in> Rep_hypreal(Q).
                hyprel``{%n::nat. X n * Y n})"

  hypreal_less_def:
  "P < (Q::hypreal) == \<exists>X Y. X \<in> Rep_hypreal(P) &
                               Y \<in> Rep_hypreal(Q) &
                               {n::nat. X n < Y n} \<in> FreeUltrafilterNat"
  hypreal_le_def:
  "P <= (Q::hypreal) == ~(Q < P)"

(*------------------------------------------------------------------------
             Proof that the set of naturals is not finite
 ------------------------------------------------------------------------*)

(*** based on James' proof that the set of naturals is not finite ***)
lemma finite_exhausts [rule_format (no_asm)]: "finite (A::nat set) --> (\<exists>n. \<forall>m. Suc (n + m) \<notin> A)"
apply (rule impI)
apply (erule_tac F = A in finite_induct)
apply (blast, erule exE)
apply (rule_tac x = "n + x" in exI)
apply (rule allI, erule_tac x = "x + m" in allE)
apply (auto simp add: add_ac)
done

lemma finite_not_covers [rule_format (no_asm)]: "finite (A :: nat set) --> (\<exists>n. n \<notin>A)"
by (rule impI, drule finite_exhausts, blast)

lemma not_finite_nat: "~ finite(UNIV:: nat set)"
by (fast dest!: finite_exhausts)

(*------------------------------------------------------------------------
   Existence of free ultrafilter over the naturals and proof of various 
   properties of the FreeUltrafilterNat- an arbitrary free ultrafilter
 ------------------------------------------------------------------------*)

lemma FreeUltrafilterNat_Ex: "\<exists>U. U: FreeUltrafilter (UNIV::nat set)"
by (rule not_finite_nat [THEN FreeUltrafilter_Ex])

lemma FreeUltrafilterNat_mem [simp]: 
     "FreeUltrafilterNat: FreeUltrafilter(UNIV:: nat set)"
apply (unfold FreeUltrafilterNat_def)
apply (rule FreeUltrafilterNat_Ex [THEN exE])
apply (rule someI2, assumption+)
done

lemma FreeUltrafilterNat_finite: "finite x ==> x \<notin> FreeUltrafilterNat"
apply (unfold FreeUltrafilterNat_def)
apply (rule FreeUltrafilterNat_Ex [THEN exE])
apply (rule someI2, assumption)
apply (blast dest: mem_FreeUltrafiltersetD1)
done

lemma FreeUltrafilterNat_not_finite: "x: FreeUltrafilterNat ==> ~ finite x"
by (blast dest: FreeUltrafilterNat_finite)

lemma FreeUltrafilterNat_empty [simp]: "{} \<notin> FreeUltrafilterNat"
apply (unfold FreeUltrafilterNat_def)
apply (rule FreeUltrafilterNat_Ex [THEN exE])
apply (rule someI2, assumption)
apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter 
                   Filter_empty_not_mem)
done

lemma FreeUltrafilterNat_Int: "[| X: FreeUltrafilterNat;  Y: FreeUltrafilterNat |]   
      ==> X Int Y \<in> FreeUltrafilterNat"
apply (cut_tac FreeUltrafilterNat_mem)
apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter mem_FiltersetD1)
done

lemma FreeUltrafilterNat_subset: "[| X: FreeUltrafilterNat;  X <= Y |]  
      ==> Y \<in> FreeUltrafilterNat"
apply (cut_tac FreeUltrafilterNat_mem)
apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter mem_FiltersetD2)
done

lemma FreeUltrafilterNat_Compl: "X: FreeUltrafilterNat ==> -X \<notin> FreeUltrafilterNat"
apply safe
apply (drule FreeUltrafilterNat_Int, assumption, auto)
done

lemma FreeUltrafilterNat_Compl_mem: "X\<notin> FreeUltrafilterNat ==> -X \<in> FreeUltrafilterNat"
apply (cut_tac FreeUltrafilterNat_mem [THEN FreeUltrafilter_iff [THEN iffD1]])
apply (safe, drule_tac x = X in bspec)
apply (auto simp add: UNIV_diff_Compl)
done

lemma FreeUltrafilterNat_Compl_iff1: "(X \<notin> FreeUltrafilterNat) = (-X: FreeUltrafilterNat)"
by (blast dest: FreeUltrafilterNat_Compl FreeUltrafilterNat_Compl_mem)

lemma FreeUltrafilterNat_Compl_iff2: "(X: FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)"
by (auto simp add: FreeUltrafilterNat_Compl_iff1 [symmetric])

lemma FreeUltrafilterNat_UNIV [simp]: "(UNIV::nat set) \<in> FreeUltrafilterNat"
by (rule FreeUltrafilterNat_mem [THEN FreeUltrafilter_Ultrafilter, THEN Ultrafilter_Filter, THEN mem_FiltersetD4])

lemma FreeUltrafilterNat_Nat_set [simp]: "UNIV \<in> FreeUltrafilterNat"
by auto

lemma FreeUltrafilterNat_Nat_set_refl [intro]: "{n. P(n) = P(n)} \<in> FreeUltrafilterNat"
by simp

lemma FreeUltrafilterNat_P: "{n::nat. P} \<in> FreeUltrafilterNat ==> P"
by (rule ccontr, simp)

lemma FreeUltrafilterNat_Ex_P: "{n. P(n)} \<in> FreeUltrafilterNat ==> \<exists>n. P(n)"
by (rule ccontr, simp)

lemma FreeUltrafilterNat_all: "\<forall>n. P(n) ==> {n. P(n)} \<in> FreeUltrafilterNat"
by (auto intro: FreeUltrafilterNat_Nat_set)

(*-------------------------------------------------------
     Define and use Ultrafilter tactics
 -------------------------------------------------------*)
use "fuf.ML"

method_setup fuf = {*
    Method.ctxt_args (fn ctxt =>
        Method.METHOD (fn facts =>
            fuf_tac (Classical.get_local_claset ctxt,
                     Simplifier.get_local_simpset ctxt) 1)) *}
    "free ultrafilter tactic"

method_setup ultra = {*
    Method.ctxt_args (fn ctxt =>
        Method.METHOD (fn facts =>
            ultra_tac (Classical.get_local_claset ctxt,
                       Simplifier.get_local_simpset ctxt) 1)) *}
    "ultrafilter tactic"


(*-------------------------------------------------------
  Now prove one further property of our free ultrafilter
 -------------------------------------------------------*)
lemma FreeUltrafilterNat_Un: "X Un Y: FreeUltrafilterNat  
      ==> X: FreeUltrafilterNat | Y: FreeUltrafilterNat"
apply auto
apply ultra
done

(*-------------------------------------------------------
   Properties of hyprel
 -------------------------------------------------------*)

(** Proving that hyprel is an equivalence relation **)
(** Natural deduction for hyprel **)

lemma hyprel_iff: "((X,Y): hyprel) = ({n. X n = Y n}: FreeUltrafilterNat)"
by (unfold hyprel_def, fast)

lemma hyprel_refl: "(x,x): hyprel"
apply (unfold hyprel_def)
apply (auto simp add: FreeUltrafilterNat_Nat_set)
done

lemma hyprel_sym [rule_format (no_asm)]: "(x,y): hyprel --> (y,x):hyprel"
by (simp add: hyprel_def eq_commute)

lemma hyprel_trans: 
      "[|(x,y): hyprel; (y,z):hyprel|] ==> (x,z):hyprel"
apply (unfold hyprel_def, auto, ultra)
done

lemma equiv_hyprel: "equiv UNIV hyprel"
apply (simp add: equiv_def refl_def sym_def trans_def hyprel_refl)
apply (blast intro: hyprel_sym hyprel_trans) 
done

(* (hyprel `` {x} = hyprel `` {y}) = ((x,y) \<in> hyprel) *)
lemmas equiv_hyprel_iff =
    eq_equiv_class_iff [OF equiv_hyprel UNIV_I UNIV_I, simp] 

lemma hyprel_in_hypreal [simp]: "hyprel``{x}:hypreal"
by (unfold hypreal_def hyprel_def quotient_def, blast)

lemma inj_on_Abs_hypreal: "inj_on Abs_hypreal hypreal"
apply (rule inj_on_inverseI)
apply (erule Abs_hypreal_inverse)
done

declare inj_on_Abs_hypreal [THEN inj_on_iff, simp] 
        Abs_hypreal_inverse [simp]

declare equiv_hyprel [THEN eq_equiv_class_iff, simp]

declare hyprel_iff [iff]

lemmas eq_hyprelD = eq_equiv_class [OF _ equiv_hyprel]

lemma inj_Rep_hypreal: "inj(Rep_hypreal)"
apply (rule inj_on_inverseI)
apply (rule Rep_hypreal_inverse)
done

lemma lemma_hyprel_refl [simp]: "x \<in> hyprel `` {x}"
apply (unfold hyprel_def, safe)
apply (auto intro!: FreeUltrafilterNat_Nat_set)
done

lemma hypreal_empty_not_mem [simp]: "{} \<notin> hypreal"
apply (unfold hypreal_def)
apply (auto elim!: quotientE equalityCE)
done

lemma Rep_hypreal_nonempty [simp]: "Rep_hypreal x \<noteq> {}"
by (cut_tac x = x in Rep_hypreal, auto)


(*------------------------------------------------------------------------
   hypreal_of_real: the injection from real to hypreal
 ------------------------------------------------------------------------*)

lemma inj_hypreal_of_real: "inj(hypreal_of_real)"
apply (rule inj_onI)
apply (unfold hypreal_of_real_def)
apply (drule inj_on_Abs_hypreal [THEN inj_onD])
apply (rule hyprel_in_hypreal)+
apply (drule eq_equiv_class)
apply (rule equiv_hyprel)
apply (simp_all add: split: split_if_asm) 
done

lemma eq_Abs_hypreal:
    "(!!x y. z = Abs_hypreal(hyprel``{x}) ==> P) ==> P"
apply (rule_tac x1=z in Rep_hypreal [unfolded hypreal_def, THEN quotientE])
apply (drule_tac f = Abs_hypreal in arg_cong)
apply (force simp add: Rep_hypreal_inverse)
done

(**** hypreal_minus: additive inverse on hypreal ****)

lemma hypreal_minus_congruent: 
  "congruent hyprel (%X. hyprel``{%n. - (X n)})"
by (force simp add: congruent_def)

lemma hypreal_minus: 
   "- (Abs_hypreal(hyprel``{%n. X n})) = Abs_hypreal(hyprel `` {%n. -(X n)})"
apply (unfold hypreal_minus_def)
apply (rule_tac f = Abs_hypreal in arg_cong)
apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
               UN_equiv_class [OF equiv_hyprel hypreal_minus_congruent])
done

lemma hypreal_minus_minus [simp]: "- (- z) = (z::hypreal)"
apply (rule_tac z = z in eq_Abs_hypreal)
apply (simp add: hypreal_minus)
done

lemma inj_hypreal_minus: "inj(%r::hypreal. -r)"
apply (rule inj_onI)
apply (drule_tac f = uminus in arg_cong)
apply (simp add: hypreal_minus_minus)
done

lemma hypreal_minus_zero [simp]: "- 0 = (0::hypreal)"
apply (unfold hypreal_zero_def)
apply (simp add: hypreal_minus)
done

lemma hypreal_minus_zero_iff [simp]: "(-x = 0) = (x = (0::hypreal))"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (auto simp add: hypreal_zero_def hypreal_minus)
done


(**** hyperreal addition: hypreal_add  ****)

lemma hypreal_add_congruent2: 
    "congruent2 hyprel (%X Y. hyprel``{%n. X n + Y n})"
apply (unfold congruent2_def, auto, ultra)
done

lemma hypreal_add: 
  "Abs_hypreal(hyprel``{%n. X n}) + Abs_hypreal(hyprel``{%n. Y n}) =  
   Abs_hypreal(hyprel``{%n. X n + Y n})"
apply (unfold hypreal_add_def)
apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_add_congruent2])
done

lemma hypreal_diff: "Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) =  
      Abs_hypreal(hyprel``{%n. X n - Y n})"
apply (simp add: hypreal_diff_def hypreal_minus hypreal_add)
done

lemma hypreal_add_commute: "(z::hypreal) + w = w + z"
apply (rule_tac z = z in eq_Abs_hypreal)
apply (rule_tac z = w in eq_Abs_hypreal)
apply (simp add: real_add_ac hypreal_add)
done

lemma hypreal_add_assoc: "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)"
apply (rule_tac z = z1 in eq_Abs_hypreal)
apply (rule_tac z = z2 in eq_Abs_hypreal)
apply (rule_tac z = z3 in eq_Abs_hypreal)
apply (simp add: hypreal_add real_add_assoc)
done

(*For AC rewriting*)
lemma hypreal_add_left_commute: "(x::hypreal)+(y+z)=y+(x+z)"
  apply (rule mk_left_commute [of "op +"])
  apply (rule hypreal_add_assoc)
  apply (rule hypreal_add_commute)
  done

(* hypreal addition is an AC operator *)
lemmas hypreal_add_ac =
       hypreal_add_assoc hypreal_add_commute hypreal_add_left_commute

lemma hypreal_add_zero_left [simp]: "(0::hypreal) + z = z"
apply (unfold hypreal_zero_def)
apply (rule_tac z = z in eq_Abs_hypreal)
apply (simp add: hypreal_add)
done

lemma hypreal_add_zero_right [simp]: "z + (0::hypreal) = z"
by (simp add: hypreal_add_zero_left hypreal_add_commute)

lemma hypreal_add_minus [simp]: "z + -z = (0::hypreal)"
apply (unfold hypreal_zero_def)
apply (rule_tac z = z in eq_Abs_hypreal)
apply (simp add: hypreal_minus hypreal_add)
done

lemma hypreal_add_minus_left [simp]: "-z + z = (0::hypreal)"
by (simp add: hypreal_add_commute hypreal_add_minus)

lemma hypreal_minus_ex: "\<exists>y. (x::hypreal) + y = 0"
by (fast intro: hypreal_add_minus)

lemma hypreal_minus_ex1: "EX! y. (x::hypreal) + y = 0"
apply (auto intro: hypreal_add_minus)
apply (drule_tac f = "%x. ya+x" in arg_cong)
apply (simp add: hypreal_add_assoc [symmetric])
apply (simp add: hypreal_add_commute)
done

lemma hypreal_minus_left_ex1: "EX! y. y + (x::hypreal) = 0"
apply (auto intro: hypreal_add_minus_left)
apply (drule_tac f = "%x. x+ya" in arg_cong)
apply (simp add: hypreal_add_assoc)
apply (simp add: hypreal_add_commute)
done

lemma hypreal_add_minus_eq_minus: "x + y = (0::hypreal) ==> x = -y"
apply (cut_tac z = y in hypreal_add_minus_left)
apply (rule_tac x1 = y in hypreal_minus_left_ex1 [THEN ex1E], blast)
done

lemma hypreal_as_add_inverse_ex: "\<exists>y::hypreal. x = -y"
apply (cut_tac x = x in hypreal_minus_ex)
apply (erule exE, drule hypreal_add_minus_eq_minus, fast)
done

lemma hypreal_minus_add_distrib [simp]: "-(x + (y::hypreal)) = -x + -y"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (rule_tac z = y in eq_Abs_hypreal)
apply (auto simp add: hypreal_minus hypreal_add real_minus_add_distrib)
done

lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
by (simp add: hypreal_add_commute)

lemma hypreal_add_left_cancel: "((x::hypreal) + y = x + z) = (y = z)"
apply safe
apply (drule_tac f = "%t.-x + t" in arg_cong)
apply (simp add: hypreal_add_assoc [symmetric])
done

lemma hypreal_add_right_cancel: "(y + (x::hypreal)= z + x) = (y = z)"
by (simp add: hypreal_add_commute hypreal_add_left_cancel)

lemma hypreal_add_minus_cancelA [simp]: "z + ((- z) + w) = (w::hypreal)"
by (simp add: hypreal_add_assoc [symmetric])

lemma hypreal_minus_add_cancelA [simp]: "(-z) + (z + w) = (w::hypreal)"
by (simp add: hypreal_add_assoc [symmetric])

(**** hyperreal multiplication: hypreal_mult  ****)

lemma hypreal_mult_congruent2: 
    "congruent2 hyprel (%X Y. hyprel``{%n. X n * Y n})"
apply (unfold congruent2_def, auto, ultra)
done

lemma hypreal_mult: 
  "Abs_hypreal(hyprel``{%n. X n}) * Abs_hypreal(hyprel``{%n. Y n}) =  
   Abs_hypreal(hyprel``{%n. X n * Y n})"
apply (unfold hypreal_mult_def)
apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_mult_congruent2])
done

lemma hypreal_mult_commute: "(z::hypreal) * w = w * z"
apply (rule_tac z = z in eq_Abs_hypreal)
apply (rule_tac z = w in eq_Abs_hypreal)
apply (simp add: hypreal_mult real_mult_ac)
done

lemma hypreal_mult_assoc: "((z1::hypreal) * z2) * z3 = z1 * (z2 * z3)"
apply (rule_tac z = z1 in eq_Abs_hypreal)
apply (rule_tac z = z2 in eq_Abs_hypreal)
apply (rule_tac z = z3 in eq_Abs_hypreal)
apply (simp add: hypreal_mult real_mult_assoc)
done

lemma hypreal_mult_left_commute: "(z1::hypreal) * (z2 * z3) = z2 * (z1 * z3)"
  apply (rule mk_left_commute [of "op *"])
  apply (rule hypreal_mult_assoc)
  apply (rule hypreal_mult_commute)
  done

(* hypreal multiplication is an AC operator *)
lemmas hypreal_mult_ac =
       hypreal_mult_assoc hypreal_mult_commute hypreal_mult_left_commute

lemma hypreal_mult_1 [simp]: "(1::hypreal) * z = z"
apply (unfold hypreal_one_def)
apply (rule_tac z = z in eq_Abs_hypreal)
apply (simp add: hypreal_mult)
done

lemma hypreal_mult_1_right [simp]: "z * (1::hypreal) = z"
by (simp add: hypreal_mult_commute hypreal_mult_1)

lemma hypreal_mult_0 [simp]: "0 * z = (0::hypreal)"
apply (unfold hypreal_zero_def)
apply (rule_tac z = z in eq_Abs_hypreal)
apply (simp add: hypreal_mult)
done

lemma hypreal_mult_0_right [simp]: "z * 0 = (0::hypreal)"
by (simp add: hypreal_mult_commute)

lemma hypreal_minus_mult_eq1: "-(x * y) = -x * (y::hypreal)"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (rule_tac z = y in eq_Abs_hypreal)
apply (auto simp add: hypreal_minus hypreal_mult real_mult_ac real_add_ac)
done

lemma hypreal_minus_mult_eq2: "-(x * y) = (x::hypreal) * -y"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (rule_tac z = y in eq_Abs_hypreal)
apply (auto simp add: hypreal_minus hypreal_mult real_mult_ac real_add_ac)
done

(*Pull negations out*)
declare hypreal_minus_mult_eq2 [symmetric, simp] 
        hypreal_minus_mult_eq1 [symmetric, simp]

lemma hypreal_mult_minus_1 [simp]: "(- (1::hypreal)) * z = -z"
by simp

lemma hypreal_mult_minus_1_right [simp]: "z * (- (1::hypreal)) = -z"
by (subst hypreal_mult_commute, simp)

lemma hypreal_minus_mult_commute: "(-x) * y = (x::hypreal) * -y"
by auto

(*-----------------------------------------------------------------------------
    A few more theorems
 ----------------------------------------------------------------------------*)
lemma hypreal_add_assoc_cong: "(z::hypreal) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
by (simp add: hypreal_add_assoc [symmetric])

lemma hypreal_add_mult_distrib: "((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)"
apply (rule_tac z = z1 in eq_Abs_hypreal)
apply (rule_tac z = z2 in eq_Abs_hypreal)
apply (rule_tac z = w in eq_Abs_hypreal)
apply (simp add: hypreal_mult hypreal_add real_add_mult_distrib)
done

lemma hypreal_add_mult_distrib2: "(w::hypreal) * (z1 + z2) = (w * z1) + (w * z2)"
by (simp add: hypreal_mult_commute [of w] hypreal_add_mult_distrib)


lemma hypreal_diff_mult_distrib: "((z1::hypreal) - z2) * w = (z1 * w) - (z2 * w)"

apply (unfold hypreal_diff_def)
apply (simp add: hypreal_add_mult_distrib)
done

lemma hypreal_diff_mult_distrib2: "(w::hypreal) * (z1 - z2) = (w * z1) - (w * z2)"
by (simp add: hypreal_mult_commute [of w] hypreal_diff_mult_distrib)

(*** one and zero are distinct ***)
lemma hypreal_zero_not_eq_one: "0 \<noteq> (1::hypreal)"
apply (unfold hypreal_zero_def hypreal_one_def)
apply (auto simp add: real_zero_not_eq_one)
done


(**** multiplicative inverse on hypreal ****)

lemma hypreal_inverse_congruent: 
  "congruent hyprel (%X. hyprel``{%n. if X n = 0 then 0 else inverse(X n)})"
apply (unfold congruent_def)
apply (auto, ultra)
done

lemma hypreal_inverse: 
      "inverse (Abs_hypreal(hyprel``{%n. X n})) =  
       Abs_hypreal(hyprel `` {%n. if X n = 0 then 0 else inverse(X n)})"
apply (unfold hypreal_inverse_def)
apply (rule_tac f = Abs_hypreal in arg_cong)
apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
           UN_equiv_class [OF equiv_hyprel hypreal_inverse_congruent])
done

lemma HYPREAL_INVERSE_ZERO: "inverse 0 = (0::hypreal)"
by (simp add: hypreal_inverse hypreal_zero_def)

lemma HYPREAL_DIVISION_BY_ZERO: "a / (0::hypreal) = 0"
by (simp add: hypreal_divide_def HYPREAL_INVERSE_ZERO)

lemma hypreal_inverse_inverse [simp]: "inverse (inverse (z::hypreal)) = z"
apply (case_tac "z=0", simp add: HYPREAL_INVERSE_ZERO)
apply (rule_tac z = z in eq_Abs_hypreal)
apply (simp add: hypreal_inverse hypreal_zero_def)
done

lemma hypreal_inverse_1 [simp]: "inverse((1::hypreal)) = (1::hypreal)"
apply (unfold hypreal_one_def)
apply (simp add: hypreal_inverse real_zero_not_eq_one [THEN not_sym])
done


(*** existence of inverse ***)

lemma hypreal_mult_inverse [simp]: 
     "x \<noteq> 0 ==> x*inverse(x) = (1::hypreal)"
apply (unfold hypreal_one_def hypreal_zero_def)
apply (rule_tac z = x in eq_Abs_hypreal)
apply (simp add: hypreal_inverse hypreal_mult)
apply (drule FreeUltrafilterNat_Compl_mem)
apply (blast intro!: real_mult_inv_right FreeUltrafilterNat_subset)
done

lemma hypreal_mult_inverse_left [simp]: "x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)"
by (simp add: hypreal_mult_inverse hypreal_mult_commute)

lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
apply auto
apply (drule_tac f = "%x. x*inverse c" in arg_cong)
apply (simp add: hypreal_mult_inverse hypreal_mult_ac)
done
    
lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
apply safe
apply (drule_tac f = "%x. x*inverse c" in arg_cong)
apply (simp add: hypreal_mult_inverse hypreal_mult_ac)
done

lemma hypreal_inverse_not_zero: "x \<noteq> 0 ==> inverse (x::hypreal) \<noteq> 0"
apply (unfold hypreal_zero_def)
apply (rule_tac z = x in eq_Abs_hypreal)
apply (simp add: hypreal_inverse hypreal_mult)
done


lemma hypreal_mult_not_0: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::hypreal)"
apply safe
apply (drule_tac f = "%z. inverse x*z" in arg_cong)
apply (simp add: hypreal_mult_assoc [symmetric])
done

lemma hypreal_mult_zero_disj: "x*y = (0::hypreal) ==> x = 0 | y = 0"
by (auto intro: ccontr dest: hypreal_mult_not_0)

lemma hypreal_minus_inverse: "inverse(-x) = -inverse(x::hypreal)"
apply (case_tac "x=0", simp add: HYPREAL_INVERSE_ZERO)
apply (rule hypreal_mult_right_cancel [of "-x", THEN iffD1], simp) 
apply (subst hypreal_mult_inverse_left, auto)
done

lemma hypreal_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::hypreal)"
apply (case_tac "x=0", simp add: HYPREAL_INVERSE_ZERO)
apply (case_tac "y=0", simp add: HYPREAL_INVERSE_ZERO)
apply (frule_tac y = y in hypreal_mult_not_0, assumption)
apply (rule_tac c1 = x in hypreal_mult_left_cancel [THEN iffD1])
apply (auto simp add: hypreal_mult_assoc [symmetric])
apply (rule_tac c1 = y in hypreal_mult_left_cancel [THEN iffD1])
apply (auto simp add: hypreal_mult_left_commute)
apply (simp add: hypreal_mult_assoc [symmetric])
done

(*------------------------------------------------------------------
                   Theorems for ordering 
 ------------------------------------------------------------------*)

(* prove introduction and elimination rules for hypreal_less *)

lemma hypreal_less_iff: 
 "(P < (Q::hypreal)) = (\<exists>X Y. X \<in> Rep_hypreal(P) &  
                              Y \<in> Rep_hypreal(Q) &  
                              {n. X n < Y n} \<in> FreeUltrafilterNat)"

apply (unfold hypreal_less_def, fast)
done

lemma hypreal_lessI: 
 "[| {n. X n < Y n} \<in> FreeUltrafilterNat;  
          X \<in> Rep_hypreal(P);  
          Y \<in> Rep_hypreal(Q) |] ==> P < (Q::hypreal)"
apply (unfold hypreal_less_def, fast)
done


lemma hypreal_lessE: 
     "!! R1. [| R1 < (R2::hypreal);  
          !!X Y. {n. X n < Y n} \<in> FreeUltrafilterNat ==> P;  
          !!X. X \<in> Rep_hypreal(R1) ==> P;   
          !!Y. Y \<in> Rep_hypreal(R2) ==> P |]  
      ==> P"

apply (unfold hypreal_less_def, auto)
done

lemma hypreal_lessD: 
 "R1 < (R2::hypreal) ==> (\<exists>X Y. {n. X n < Y n} \<in> FreeUltrafilterNat &  
                                   X \<in> Rep_hypreal(R1) &  
                                   Y \<in> Rep_hypreal(R2))"
apply (unfold hypreal_less_def, fast)
done

lemma hypreal_less_not_refl: "~ (R::hypreal) < R"
apply (rule_tac z = R in eq_Abs_hypreal)
apply (auto simp add: hypreal_less_def, ultra)
done

(*** y < y ==> P ***)
lemmas hypreal_less_irrefl = hypreal_less_not_refl [THEN notE, standard]
declare hypreal_less_irrefl [elim!]

lemma hypreal_not_refl2: "!!(x::hypreal). x < y ==> x \<noteq> y"
by (auto simp add: hypreal_less_not_refl)

lemma hypreal_less_trans: "!!(R1::hypreal). [| R1 < R2; R2 < R3 |] ==> R1 < R3"
apply (rule_tac z = R1 in eq_Abs_hypreal)
apply (rule_tac z = R2 in eq_Abs_hypreal)
apply (rule_tac z = R3 in eq_Abs_hypreal)
apply (auto intro!: exI simp add: hypreal_less_def, ultra)
done

lemma hypreal_less_asym: "!! (R1::hypreal). [| R1 < R2; R2 < R1 |] ==> P"
apply (drule hypreal_less_trans, assumption)
apply (simp add: hypreal_less_not_refl)
done

(*-------------------------------------------------------
  TODO: The following theorem should have been proved 
  first and then used througout the proofs as it probably 
  makes many of them more straightforward. 
 -------------------------------------------------------*)
lemma hypreal_less: 
      "(Abs_hypreal(hyprel``{%n. X n}) <  
            Abs_hypreal(hyprel``{%n. Y n})) =  
       ({n. X n < Y n} \<in> FreeUltrafilterNat)"
apply (unfold hypreal_less_def)
apply (auto intro!: lemma_hyprel_refl, ultra)
done

(*----------------------------------------------------------------------------
		 Trichotomy: the hyperreals are linearly ordered
  ---------------------------------------------------------------------------*)

lemma lemma_hyprel_0_mem: "\<exists>x. x: hyprel `` {%n. 0}"

apply (unfold hyprel_def)
apply (rule_tac x = "%n. 0" in exI, safe)
apply (auto intro!: FreeUltrafilterNat_Nat_set)
done

lemma hypreal_trichotomy: "0 <  x | x = 0 | x < (0::hypreal)"
apply (unfold hypreal_zero_def)
apply (rule_tac z = x in eq_Abs_hypreal)
apply (auto simp add: hypreal_less_def)
apply (cut_tac lemma_hyprel_0_mem, erule exE)
apply (drule_tac x = xa in spec)
apply (drule_tac x = x in spec)
apply (cut_tac x = x in lemma_hyprel_refl, auto)
apply (drule_tac x = x in spec)
apply (drule_tac x = xa in spec, auto, ultra)
done

lemma hypreal_trichotomyE:
     "[| (0::hypreal) < x ==> P;  
         x = 0 ==> P;  
         x < 0 ==> P |] ==> P"
apply (insert hypreal_trichotomy [of x], blast) 
done

(*----------------------------------------------------------------------------
            More properties of <
 ----------------------------------------------------------------------------*)

lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (rule_tac z = y in eq_Abs_hypreal)
apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
done

lemma hypreal_less_minus_iff2: "((x::hypreal) < y) = (x + -y < 0)"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (rule_tac z = y in eq_Abs_hypreal)
apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
done

lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)"
apply auto
apply (rule_tac x1 = "-y" in hypreal_add_right_cancel [THEN iffD1], auto)
done

lemma hypreal_eq_minus_iff2: "((x::hypreal) = y) = (0 = y + - x)"
apply auto
apply (rule_tac x1 = "-x" in hypreal_add_right_cancel [THEN iffD1], auto)
done

(* 07/00 *)
lemma hypreal_diff_zero [simp]: "(0::hypreal) - x = -x"
by (simp add: hypreal_diff_def)

lemma hypreal_diff_zero_right [simp]: "x - (0::hypreal) = x"
by (simp add: hypreal_diff_def)

lemma hypreal_diff_self [simp]: "x - x = (0::hypreal)"
by (simp add: hypreal_diff_def)

lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
by (auto simp add: hypreal_add_assoc)

lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])


(*** linearity ***)

lemma hypreal_linear: "(x::hypreal) < y | x = y | y < x"
apply (subst hypreal_eq_minus_iff2)
apply (rule_tac x1 = x in hypreal_less_minus_iff [THEN ssubst])
apply (rule_tac x1 = y in hypreal_less_minus_iff2 [THEN ssubst])
apply (rule hypreal_trichotomyE, auto)
done

lemma hypreal_neq_iff: "((w::hypreal) \<noteq> z) = (w<z | z<w)"
by (cut_tac hypreal_linear, blast)

lemma hypreal_linear_less2: "!!(x::hypreal). [| x < y ==> P;  x = y ==> P;  
           y < x ==> P |] ==> P"
apply (cut_tac x = x and y = y in hypreal_linear, auto)
done

(*------------------------------------------------------------------------------
                            Properties of <=
 ------------------------------------------------------------------------------*)
(*------ hypreal le iff reals le a.e ------*)

lemma hypreal_le: 
      "(Abs_hypreal(hyprel``{%n. X n}) <=  
            Abs_hypreal(hyprel``{%n. Y n})) =  
       ({n. X n <= Y n} \<in> FreeUltrafilterNat)"
apply (unfold hypreal_le_def real_le_def)
apply (auto simp add: hypreal_less)
apply (ultra+)
done

(*---------------------------------------------------------*)
(*---------------------------------------------------------*)
lemma hypreal_leI: 
     "~(w < z) ==> z <= (w::hypreal)"
apply (unfold hypreal_le_def, assumption)
done

lemma hypreal_leD: 
      "z<=w ==> ~(w<(z::hypreal))"
apply (unfold hypreal_le_def, assumption)
done

lemma hypreal_less_le_iff: "(~(w < z)) = (z <= (w::hypreal))"
by (fast intro!: hypreal_leI hypreal_leD)

lemma not_hypreal_leE: "~ z <= w ==> w<(z::hypreal)"
by (unfold hypreal_le_def, fast)

lemma hypreal_le_imp_less_or_eq: "!!(x::hypreal). x <= y ==> x < y | x = y"
apply (unfold hypreal_le_def)
apply (cut_tac hypreal_linear)
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
done

lemma hypreal_less_or_eq_imp_le: "z<w | z=w ==> z <=(w::hypreal)"
apply (unfold hypreal_le_def)
apply (cut_tac hypreal_linear)
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
done

lemma hypreal_le_eq_less_or_eq: "(x <= (y::hypreal)) = (x < y | x=y)"
by (blast intro!: hypreal_less_or_eq_imp_le dest: hypreal_le_imp_less_or_eq) 

lemmas hypreal_le_less = hypreal_le_eq_less_or_eq

lemma hypreal_le_refl: "w <= (w::hypreal)"
by (simp add: hypreal_le_eq_less_or_eq)

(* Axiom 'linorder_linear' of class 'linorder': *)
lemma hypreal_le_linear: "(z::hypreal) <= w | w <= z"
apply (simp add: hypreal_le_less)
apply (cut_tac hypreal_linear, blast)
done

lemma hypreal_le_trans: "[| i <= j; j <= k |] ==> i <= (k::hypreal)"
apply (drule hypreal_le_imp_less_or_eq) 
apply (drule hypreal_le_imp_less_or_eq) 
apply (rule hypreal_less_or_eq_imp_le) 
apply (blast intro: hypreal_less_trans) 
done

lemma hypreal_le_anti_sym: "[| z <= w; w <= z |] ==> z = (w::hypreal)"
apply (drule hypreal_le_imp_less_or_eq) 
apply (drule hypreal_le_imp_less_or_eq) 
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
done

lemma not_less_not_eq_hypreal_less: "[| ~ y < x; y \<noteq> x |] ==> x < (y::hypreal)"
apply (rule not_hypreal_leE)
apply (fast dest: hypreal_le_imp_less_or_eq)
done

(* Axiom 'order_less_le' of class 'order': *)
lemma hypreal_less_le: "((w::hypreal) < z) = (w <= z & w \<noteq> z)"
apply (simp add: hypreal_le_def hypreal_neq_iff)
apply (blast intro: hypreal_less_asym)
done

lemma hypreal_minus_zero_less_iff [simp]: "(0 < -R) = (R < (0::hypreal))"
apply (rule_tac z = R in eq_Abs_hypreal)
apply (auto simp add: hypreal_zero_def hypreal_less hypreal_minus)
done

lemma hypreal_minus_zero_less_iff2 [simp]: "(-R < 0) = ((0::hypreal) < R)"
apply (rule_tac z = R in eq_Abs_hypreal)
apply (auto simp add: hypreal_zero_def hypreal_less hypreal_minus)
done

lemma hypreal_minus_zero_le_iff [simp]: "((0::hypreal) <= -r) = (r <= 0)"
apply (unfold hypreal_le_def)
apply (simp add: hypreal_minus_zero_less_iff2)
done

lemma hypreal_minus_zero_le_iff2 [simp]: "(-r <= (0::hypreal)) = (0 <= r)"
apply (unfold hypreal_le_def)
apply (simp add: hypreal_minus_zero_less_iff2)
done

(*----------------------------------------------------------
  hypreal_of_real preserves field and order properties
 -----------------------------------------------------------*)
lemma hypreal_of_real_add [simp]: 
     "hypreal_of_real (z1 + z2) = hypreal_of_real z1 + hypreal_of_real z2"
apply (unfold hypreal_of_real_def)
apply (simp add: hypreal_add hypreal_add_mult_distrib)
done

lemma hypreal_of_real_mult [simp]: 
     "hypreal_of_real (z1 * z2) = hypreal_of_real z1 * hypreal_of_real z2"
apply (unfold hypreal_of_real_def)
apply (simp add: hypreal_mult hypreal_add_mult_distrib2)
done

lemma hypreal_of_real_less_iff [simp]: 
     "(hypreal_of_real z1 <  hypreal_of_real z2) = (z1 < z2)"
apply (unfold hypreal_less_def hypreal_of_real_def, auto)
apply (rule_tac [2] x = "%n. z1" in exI, safe)
apply (rule_tac [3] x = "%n. z2" in exI, auto)
apply (rule FreeUltrafilterNat_P, ultra)
done

lemma hypreal_of_real_le_iff [simp]: 
     "(hypreal_of_real z1 <= hypreal_of_real z2) = (z1 <= z2)"
apply (unfold hypreal_le_def real_le_def, auto)
done

lemma hypreal_of_real_eq_iff [simp]: "(hypreal_of_real z1 = hypreal_of_real z2) = (z1 = z2)"
by (force intro: order_antisym hypreal_of_real_le_iff [THEN iffD1])

lemma hypreal_of_real_minus [simp]: "hypreal_of_real (-r) = - hypreal_of_real  r"
apply (unfold hypreal_of_real_def)
apply (auto simp add: hypreal_minus)
done

lemma hypreal_of_real_one [simp]: "hypreal_of_real 1 = (1::hypreal)"
by (unfold hypreal_of_real_def hypreal_one_def, simp)

lemma hypreal_of_real_zero [simp]: "hypreal_of_real 0 = 0"
by (unfold hypreal_of_real_def hypreal_zero_def, simp)

lemma hypreal_of_real_zero_iff: "(hypreal_of_real r = 0) = (r = 0)"
by (auto intro: FreeUltrafilterNat_P simp add: hypreal_of_real_def hypreal_zero_def FreeUltrafilterNat_Nat_set)

lemma hypreal_of_real_inverse [simp]: "hypreal_of_real (inverse r) = inverse (hypreal_of_real r)"
apply (case_tac "r=0")
apply (simp add: DIVISION_BY_ZERO INVERSE_ZERO HYPREAL_INVERSE_ZERO)
apply (rule_tac c1 = "hypreal_of_real r" in hypreal_mult_left_cancel [THEN iffD1])
apply (auto simp add: hypreal_of_real_zero_iff hypreal_of_real_mult [symmetric])
done

lemma hypreal_of_real_divide [simp]: "hypreal_of_real (z1 / z2) = hypreal_of_real z1 / hypreal_of_real z2"
by (simp add: hypreal_divide_def real_divide_def)


(*** Division lemmas ***)

lemma hypreal_zero_divide: "(0::hypreal)/x = 0"
by (simp add: hypreal_divide_def)

lemma hypreal_divide_one: "x/(1::hypreal) = x"
by (simp add: hypreal_divide_def)
declare hypreal_zero_divide [simp] hypreal_divide_one [simp]

lemma hypreal_times_divide1_eq [simp]: "(x::hypreal) * (y/z) = (x*y)/z"
by (simp add: hypreal_divide_def hypreal_mult_assoc)

lemma hypreal_times_divide2_eq [simp]: "(y/z) * (x::hypreal) = (y*x)/z"
by (simp add: hypreal_divide_def hypreal_mult_ac)


lemma hypreal_divide_divide1_eq [simp]: "(x::hypreal) / (y/z) = (x*z)/y"
by (simp add: hypreal_divide_def hypreal_inverse_distrib hypreal_mult_ac)

lemma hypreal_divide_divide2_eq [simp]: "((x::hypreal) / y) / z = x/(y*z)"
by (simp add: hypreal_divide_def hypreal_inverse_distrib hypreal_mult_assoc)


(** As with multiplication, pull minus signs OUT of the / operator **)

lemma hypreal_minus_divide_eq [simp]: "(-x) / (y::hypreal) = - (x/y)"
by (simp add: hypreal_divide_def)

lemma hypreal_divide_minus_eq [simp]: "(x / -(y::hypreal)) = - (x/y)"
by (simp add: hypreal_divide_def hypreal_minus_inverse)

lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
by (simp add: hypreal_divide_def hypreal_add_mult_distrib)

lemma hypreal_inverse_add: "[|(x::hypreal) \<noteq> 0;  y \<noteq> 0 |]   
      ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)"
apply (simp add: hypreal_inverse_distrib hypreal_add_mult_distrib hypreal_mult_assoc [symmetric])
apply (subst hypreal_mult_assoc)
apply (rule hypreal_mult_left_commute [THEN subst])
apply (simp add: hypreal_add_commute)
done

lemma hypreal_self_eq_minus_self_zero: "x = -x ==> x = (0::hypreal)"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (auto simp add: hypreal_minus hypreal_zero_def, ultra)
done

lemma hypreal_add_self_zero_cancel [simp]: "(x + x = 0) = (x = (0::hypreal))"
apply (rule_tac z = x in eq_Abs_hypreal)
apply (auto simp add: hypreal_add hypreal_zero_def)
done

lemma hypreal_add_self_zero_cancel2 [simp]: "(x + x + y = y) = (x = (0::hypreal))"
apply auto
apply (drule hypreal_eq_minus_iff [THEN iffD1])
apply (auto simp add: hypreal_add_assoc hypreal_self_eq_minus_self_zero)
done

lemma hypreal_add_self_zero_cancel2a [simp]: "(x + (x + y) = y) = (x = (0::hypreal))"
by (simp add: hypreal_add_assoc [symmetric])

lemma hypreal_minus_eq_swap: "(b = -a) = (-b = (a::hypreal))"
by auto

lemma hypreal_minus_eq_cancel [simp]: "(-b = -a) = (b = (a::hypreal))"
by (simp add: hypreal_minus_eq_swap)

lemma hypreal_less_eq_diff: "(x<y) = (x-y < (0::hypreal))"
apply (unfold hypreal_diff_def)
apply (rule hypreal_less_minus_iff2)
done

(*** Subtraction laws ***)

lemma hypreal_add_diff_eq: "x + (y - z) = (x + y) - (z::hypreal)"
by (simp add: hypreal_diff_def hypreal_add_ac)

lemma hypreal_diff_add_eq: "(x - y) + z = (x + z) - (y::hypreal)"
by (simp add: hypreal_diff_def hypreal_add_ac)

lemma hypreal_diff_diff_eq: "(x - y) - z = x - (y + (z::hypreal))"
by (simp add: hypreal_diff_def hypreal_add_ac)

lemma hypreal_diff_diff_eq2: "x - (y - z) = (x + z) - (y::hypreal)"
by (simp add: hypreal_diff_def hypreal_add_ac)

lemma hypreal_diff_less_eq: "(x-y < z) = (x < z + (y::hypreal))"
apply (subst hypreal_less_eq_diff)
apply (rule_tac y1 = z in hypreal_less_eq_diff [THEN ssubst])
apply (simp add: hypreal_diff_def hypreal_add_ac)
done

lemma hypreal_less_diff_eq: "(x < z-y) = (x + (y::hypreal) < z)"
apply (subst hypreal_less_eq_diff)
apply (rule_tac y1 = "z-y" in hypreal_less_eq_diff [THEN ssubst])
apply (simp add: hypreal_diff_def hypreal_add_ac)
done

lemma hypreal_diff_le_eq: "(x-y <= z) = (x <= z + (y::hypreal))"
apply (unfold hypreal_le_def)
apply (simp add: hypreal_less_diff_eq)
done

lemma hypreal_le_diff_eq: "(x <= z-y) = (x + (y::hypreal) <= z)"
apply (unfold hypreal_le_def)
apply (simp add: hypreal_diff_less_eq)
done

lemma hypreal_diff_eq_eq: "(x-y = z) = (x = z + (y::hypreal))"
apply (unfold hypreal_diff_def)
apply (auto simp add: hypreal_add_assoc)
done

lemma hypreal_eq_diff_eq: "(x = z-y) = (x + (y::hypreal) = z)"
apply (unfold hypreal_diff_def)
apply (auto simp add: hypreal_add_assoc)
done


(** For the cancellation simproc.
    The idea is to cancel like terms on opposite sides by subtraction **)

lemma hypreal_less_eqI: "(x::hypreal) - y = x' - y' ==> (x<y) = (x'<y')"
apply (subst hypreal_less_eq_diff)
apply (rule_tac y1 = y in hypreal_less_eq_diff [THEN ssubst], simp)
done

lemma hypreal_le_eqI: "(x::hypreal) - y = x' - y' ==> (y<=x) = (y'<=x')"
apply (drule hypreal_less_eqI)
apply (simp add: hypreal_le_def)
done

lemma hypreal_eq_eqI: "(x::hypreal) - y = x' - y' ==> (x=y) = (x'=y')"
apply safe
apply (simp_all add: hypreal_eq_diff_eq hypreal_diff_eq_eq)
done

lemma hypreal_zero_num: "0 = Abs_hypreal (hyprel `` {%n. 0})"
by (simp add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric])

lemma hypreal_one_num: "1 = Abs_hypreal (hyprel `` {%n. 1})"
by (simp add: hypreal_one_def [THEN meta_eq_to_obj_eq, symmetric])

lemma hypreal_omega_gt_zero [simp]: "0 < omega"
apply (unfold omega_def)
apply (auto simp add: hypreal_less hypreal_zero_num)
done

ML
{*
val hypreal_zero_def = thm "hypreal_zero_def";
val hypreal_one_def = thm "hypreal_one_def";
val hypreal_minus_def = thm "hypreal_minus_def";
val hypreal_diff_def = thm "hypreal_diff_def";
val hypreal_inverse_def = thm "hypreal_inverse_def";
val hypreal_divide_def = thm "hypreal_divide_def";
val hypreal_of_real_def = thm "hypreal_of_real_def";
val omega_def = thm "omega_def";
val epsilon_def = thm "epsilon_def";
val hypreal_add_def = thm "hypreal_add_def";
val hypreal_mult_def = thm "hypreal_mult_def";
val hypreal_less_def = thm "hypreal_less_def";
val hypreal_le_def = thm "hypreal_le_def";

val finite_exhausts = thm "finite_exhausts";
val finite_not_covers = thm "finite_not_covers";
val not_finite_nat = thm "not_finite_nat";
val FreeUltrafilterNat_Ex = thm "FreeUltrafilterNat_Ex";
val FreeUltrafilterNat_mem = thm "FreeUltrafilterNat_mem";
val FreeUltrafilterNat_finite = thm "FreeUltrafilterNat_finite";
val FreeUltrafilterNat_not_finite = thm "FreeUltrafilterNat_not_finite";
val FreeUltrafilterNat_empty = thm "FreeUltrafilterNat_empty";
val FreeUltrafilterNat_Int = thm "FreeUltrafilterNat_Int";
val FreeUltrafilterNat_subset = thm "FreeUltrafilterNat_subset";
val FreeUltrafilterNat_Compl = thm "FreeUltrafilterNat_Compl";
val FreeUltrafilterNat_Compl_mem = thm "FreeUltrafilterNat_Compl_mem";
val FreeUltrafilterNat_Compl_iff1 = thm "FreeUltrafilterNat_Compl_iff1";
val FreeUltrafilterNat_Compl_iff2 = thm "FreeUltrafilterNat_Compl_iff2";
val FreeUltrafilterNat_UNIV = thm "FreeUltrafilterNat_UNIV";
val FreeUltrafilterNat_Nat_set = thm "FreeUltrafilterNat_Nat_set";
val FreeUltrafilterNat_Nat_set_refl = thm "FreeUltrafilterNat_Nat_set_refl";
val FreeUltrafilterNat_P = thm "FreeUltrafilterNat_P";
val FreeUltrafilterNat_Ex_P = thm "FreeUltrafilterNat_Ex_P";
val FreeUltrafilterNat_all = thm "FreeUltrafilterNat_all";
val FreeUltrafilterNat_Un = thm "FreeUltrafilterNat_Un";
val hyprel_iff = thm "hyprel_iff";
val hyprel_refl = thm "hyprel_refl";
val hyprel_sym = thm "hyprel_sym";
val hyprel_trans = thm "hyprel_trans";
val equiv_hyprel = thm "equiv_hyprel";
val hyprel_in_hypreal = thm "hyprel_in_hypreal";
val Abs_hypreal_inverse = thm "Abs_hypreal_inverse";
val inj_on_Abs_hypreal = thm "inj_on_Abs_hypreal";
val inj_Rep_hypreal = thm "inj_Rep_hypreal";
val lemma_hyprel_refl = thm "lemma_hyprel_refl";
val hypreal_empty_not_mem = thm "hypreal_empty_not_mem";
val Rep_hypreal_nonempty = thm "Rep_hypreal_nonempty";
val inj_hypreal_of_real = thm "inj_hypreal_of_real";
val eq_Abs_hypreal = thm "eq_Abs_hypreal";
val hypreal_minus_congruent = thm "hypreal_minus_congruent";
val hypreal_minus = thm "hypreal_minus";
val hypreal_minus_minus = thm "hypreal_minus_minus";
val inj_hypreal_minus = thm "inj_hypreal_minus";
val hypreal_minus_zero = thm "hypreal_minus_zero";
val hypreal_minus_zero_iff = thm "hypreal_minus_zero_iff";
val hypreal_add_congruent2 = thm "hypreal_add_congruent2";
val hypreal_add = thm "hypreal_add";
val hypreal_diff = thm "hypreal_diff";
val hypreal_add_commute = thm "hypreal_add_commute";
val hypreal_add_assoc = thm "hypreal_add_assoc";
val hypreal_add_left_commute = thm "hypreal_add_left_commute";
val hypreal_add_zero_left = thm "hypreal_add_zero_left";
val hypreal_add_zero_right = thm "hypreal_add_zero_right";
val hypreal_add_minus = thm "hypreal_add_minus";
val hypreal_add_minus_left = thm "hypreal_add_minus_left";
val hypreal_minus_ex = thm "hypreal_minus_ex";
val hypreal_minus_ex1 = thm "hypreal_minus_ex1";
val hypreal_minus_left_ex1 = thm "hypreal_minus_left_ex1";
val hypreal_add_minus_eq_minus = thm "hypreal_add_minus_eq_minus";
val hypreal_as_add_inverse_ex = thm "hypreal_as_add_inverse_ex";
val hypreal_minus_add_distrib = thm "hypreal_minus_add_distrib";
val hypreal_minus_distrib1 = thm "hypreal_minus_distrib1";
val hypreal_add_left_cancel = thm "hypreal_add_left_cancel";
val hypreal_add_right_cancel = thm "hypreal_add_right_cancel";
val hypreal_add_minus_cancelA = thm "hypreal_add_minus_cancelA";
val hypreal_minus_add_cancelA = thm "hypreal_minus_add_cancelA";
val hypreal_mult_congruent2 = thm "hypreal_mult_congruent2";
val hypreal_mult = thm "hypreal_mult";
val hypreal_mult_commute = thm "hypreal_mult_commute";
val hypreal_mult_assoc = thm "hypreal_mult_assoc";
val hypreal_mult_left_commute = thm "hypreal_mult_left_commute";
val hypreal_mult_1 = thm "hypreal_mult_1";
val hypreal_mult_1_right = thm "hypreal_mult_1_right";
val hypreal_mult_0 = thm "hypreal_mult_0";
val hypreal_mult_0_right = thm "hypreal_mult_0_right";
val hypreal_minus_mult_eq1 = thm "hypreal_minus_mult_eq1";
val hypreal_minus_mult_eq2 = thm "hypreal_minus_mult_eq2";
val hypreal_mult_minus_1 = thm "hypreal_mult_minus_1";
val hypreal_mult_minus_1_right = thm "hypreal_mult_minus_1_right";
val hypreal_minus_mult_commute = thm "hypreal_minus_mult_commute";
val hypreal_add_assoc_cong = thm "hypreal_add_assoc_cong";
val hypreal_add_mult_distrib = thm "hypreal_add_mult_distrib";
val hypreal_add_mult_distrib2 = thm "hypreal_add_mult_distrib2";
val hypreal_diff_mult_distrib = thm "hypreal_diff_mult_distrib";
val hypreal_diff_mult_distrib2 = thm "hypreal_diff_mult_distrib2";
val hypreal_zero_not_eq_one = thm "hypreal_zero_not_eq_one";
val hypreal_inverse_congruent = thm "hypreal_inverse_congruent";
val hypreal_inverse = thm "hypreal_inverse";
val HYPREAL_INVERSE_ZERO = thm "HYPREAL_INVERSE_ZERO";
val HYPREAL_DIVISION_BY_ZERO = thm "HYPREAL_DIVISION_BY_ZERO";
val hypreal_inverse_inverse = thm "hypreal_inverse_inverse";
val hypreal_inverse_1 = thm "hypreal_inverse_1";
val hypreal_mult_inverse = thm "hypreal_mult_inverse";
val hypreal_mult_inverse_left = thm "hypreal_mult_inverse_left";
val hypreal_mult_left_cancel = thm "hypreal_mult_left_cancel";
val hypreal_mult_right_cancel = thm "hypreal_mult_right_cancel";
val hypreal_inverse_not_zero = thm "hypreal_inverse_not_zero";
val hypreal_mult_not_0 = thm "hypreal_mult_not_0";
val hypreal_mult_zero_disj = thm "hypreal_mult_zero_disj";
val hypreal_minus_inverse = thm "hypreal_minus_inverse";
val hypreal_inverse_distrib = thm "hypreal_inverse_distrib";
val hypreal_less_iff = thm "hypreal_less_iff";
val hypreal_lessI = thm "hypreal_lessI";
val hypreal_lessE = thm "hypreal_lessE";
val hypreal_lessD = thm "hypreal_lessD";
val hypreal_less_not_refl = thm "hypreal_less_not_refl";
val hypreal_not_refl2 = thm "hypreal_not_refl2";
val hypreal_less_trans = thm "hypreal_less_trans";
val hypreal_less_asym = thm "hypreal_less_asym";
val hypreal_less = thm "hypreal_less";
val hypreal_trichotomy = thm "hypreal_trichotomy";
val hypreal_trichotomyE = thm "hypreal_trichotomyE";
val hypreal_less_minus_iff = thm "hypreal_less_minus_iff";
val hypreal_less_minus_iff2 = thm "hypreal_less_minus_iff2";
val hypreal_eq_minus_iff = thm "hypreal_eq_minus_iff";
val hypreal_eq_minus_iff2 = thm "hypreal_eq_minus_iff2";
val hypreal_diff_zero = thm "hypreal_diff_zero";
val hypreal_diff_zero_right = thm "hypreal_diff_zero_right";
val hypreal_diff_self = thm "hypreal_diff_self";
val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
val hypreal_linear = thm "hypreal_linear";
val hypreal_neq_iff = thm "hypreal_neq_iff";
val hypreal_linear_less2 = thm "hypreal_linear_less2";
val hypreal_le = thm "hypreal_le";
val hypreal_leI = thm "hypreal_leI";
val hypreal_leD = thm "hypreal_leD";
val hypreal_less_le_iff = thm "hypreal_less_le_iff";
val not_hypreal_leE = thm "not_hypreal_leE";
val hypreal_le_imp_less_or_eq = thm "hypreal_le_imp_less_or_eq";
val hypreal_less_or_eq_imp_le = thm "hypreal_less_or_eq_imp_le";
val hypreal_le_eq_less_or_eq = thm "hypreal_le_eq_less_or_eq";
val hypreal_le_refl = thm "hypreal_le_refl";
val hypreal_le_linear = thm "hypreal_le_linear";
val hypreal_le_trans = thm "hypreal_le_trans";
val hypreal_le_anti_sym = thm "hypreal_le_anti_sym";
val not_less_not_eq_hypreal_less = thm "not_less_not_eq_hypreal_less";
val hypreal_less_le = thm "hypreal_less_le";
val hypreal_minus_zero_less_iff = thm "hypreal_minus_zero_less_iff";
val hypreal_minus_zero_less_iff2 = thm "hypreal_minus_zero_less_iff2";
val hypreal_minus_zero_le_iff = thm "hypreal_minus_zero_le_iff";
val hypreal_minus_zero_le_iff2 = thm "hypreal_minus_zero_le_iff2";
val hypreal_of_real_add = thm "hypreal_of_real_add";
val hypreal_of_real_mult = thm "hypreal_of_real_mult";
val hypreal_of_real_less_iff = thm "hypreal_of_real_less_iff";
val hypreal_of_real_le_iff = thm "hypreal_of_real_le_iff";
val hypreal_of_real_eq_iff = thm "hypreal_of_real_eq_iff";
val hypreal_of_real_minus = thm "hypreal_of_real_minus";
val hypreal_of_real_one = thm "hypreal_of_real_one";
val hypreal_of_real_zero = thm "hypreal_of_real_zero";
val hypreal_of_real_zero_iff = thm "hypreal_of_real_zero_iff";
val hypreal_of_real_inverse = thm "hypreal_of_real_inverse";
val hypreal_of_real_divide = thm "hypreal_of_real_divide";
val hypreal_zero_divide = thm "hypreal_zero_divide";
val hypreal_divide_one = thm "hypreal_divide_one";
val hypreal_times_divide1_eq = thm "hypreal_times_divide1_eq";
val hypreal_times_divide2_eq = thm "hypreal_times_divide2_eq";
val hypreal_divide_divide1_eq = thm "hypreal_divide_divide1_eq";
val hypreal_divide_divide2_eq = thm "hypreal_divide_divide2_eq";
val hypreal_minus_divide_eq = thm "hypreal_minus_divide_eq";
val hypreal_divide_minus_eq = thm "hypreal_divide_minus_eq";
val hypreal_add_divide_distrib = thm "hypreal_add_divide_distrib";
val hypreal_inverse_add = thm "hypreal_inverse_add";
val hypreal_self_eq_minus_self_zero = thm "hypreal_self_eq_minus_self_zero";
val hypreal_add_self_zero_cancel = thm "hypreal_add_self_zero_cancel";
val hypreal_add_self_zero_cancel2 = thm "hypreal_add_self_zero_cancel2";
val hypreal_add_self_zero_cancel2a = thm "hypreal_add_self_zero_cancel2a";
val hypreal_minus_eq_swap = thm "hypreal_minus_eq_swap";
val hypreal_minus_eq_cancel = thm "hypreal_minus_eq_cancel";
val hypreal_less_eq_diff = thm "hypreal_less_eq_diff";
val hypreal_add_diff_eq = thm "hypreal_add_diff_eq";
val hypreal_diff_add_eq = thm "hypreal_diff_add_eq";
val hypreal_diff_diff_eq = thm "hypreal_diff_diff_eq";
val hypreal_diff_diff_eq2 = thm "hypreal_diff_diff_eq2";
val hypreal_diff_less_eq = thm "hypreal_diff_less_eq";
val hypreal_less_diff_eq = thm "hypreal_less_diff_eq";
val hypreal_diff_le_eq = thm "hypreal_diff_le_eq";
val hypreal_le_diff_eq = thm "hypreal_le_diff_eq";
val hypreal_diff_eq_eq = thm "hypreal_diff_eq_eq";
val hypreal_eq_diff_eq = thm "hypreal_eq_diff_eq";
val hypreal_less_eqI = thm "hypreal_less_eqI";
val hypreal_le_eqI = thm "hypreal_le_eqI";
val hypreal_eq_eqI = thm "hypreal_eq_eqI";
val hypreal_zero_num = thm "hypreal_zero_num";
val hypreal_one_num = thm "hypreal_one_num";
val hypreal_omega_gt_zero = thm "hypreal_omega_gt_zero";
*}


end