--- a/src/HOL/Hyperreal/HyperDef.thy Tue Dec 16 23:24:17 2003 +0100
+++ b/src/HOL/Hyperreal/HyperDef.thy Wed Dec 17 16:23:52 2003 +0100
@@ -5,52 +5,55 @@
Description : Construction of hyperreals using ultrafilters
*)
-HyperDef = Filter + Real +
-
-consts
-
- FreeUltrafilterNat :: nat set set ("\\<U>")
-
-defs
-
- FreeUltrafilterNat_def
- "FreeUltrafilterNat == (@U. U : FreeUltrafilter (UNIV:: nat set))"
+theory HyperDef = Filter + Real
+files ("fuf.ML"): (*Warning: file fuf.ML refers to the name Hyperdef!*)
constdefs
- hyprel :: "((nat=>real)*(nat=>real)) set"
- "hyprel == {p. ? X Y. p = ((X::nat=>real),Y) &
+
+ 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" (quotient_def)
+typedef hypreal = "UNIV//hyprel"
+ by (auto simp add: quotient_def)
-instance
- hypreal :: {ord, zero, one, plus, times, minus, inverse}
+instance hypreal :: ord ..
+instance hypreal :: zero ..
+instance hypreal :: one ..
+instance hypreal :: plus ..
+instance hypreal :: times ..
+instance hypreal :: minus ..
+instance hypreal :: inverse ..
-defs
- hypreal_zero_def
+defs (overloaded)
+
+ hypreal_zero_def:
"0 == Abs_hypreal(hyprel``{%n::nat. (0::real)})"
- hypreal_one_def
+ hypreal_one_def:
"1 == Abs_hypreal(hyprel``{%n::nat. (1::real)})"
- hypreal_minus_def
- "- P == Abs_hypreal(UN X: Rep_hypreal(P). hyprel``{%n::nat. - (X n)})"
+ hypreal_minus_def:
+ "- P == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). hyprel``{%n::nat. - (X n)})"
- hypreal_diff_def
+ hypreal_diff_def:
"x - y == x + -(y::hypreal)"
- hypreal_inverse_def
- "inverse P == Abs_hypreal(UN X: Rep_hypreal(P).
+ 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
+ hypreal_divide_def:
"P / Q::hypreal == P * inverse Q"
constdefs
- hypreal_of_real :: real => hypreal
+ hypreal_of_real :: "real => hypreal"
"hypreal_of_real r == Abs_hypreal(hyprel``{%n::nat. r})"
omega :: hypreal (*an infinite number = [<1,2,3,...>] *)
@@ -60,25 +63,1442 @@
"epsilon == Abs_hypreal(hyprel``{%n::nat. inverse (real (Suc n))})"
syntax (xsymbols)
- omega :: hypreal ("\\<omega>")
- epsilon :: hypreal ("\\<epsilon>")
+ omega :: hypreal ("\<omega>")
+ epsilon :: hypreal ("\<epsilon>")
defs
- hypreal_add_def
- "P + Q == Abs_hypreal(UN X:Rep_hypreal(P). UN Y:Rep_hypreal(Q).
+ 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(UN X:Rep_hypreal(P). UN Y:Rep_hypreal(Q).
+ 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) == EX X Y. X: Rep_hypreal(P) &
- Y: Rep_hypreal(Q) &
- {n::nat. X n < Y n} : FreeUltrafilterNat"
- hypreal_le_def
+ 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)"
+apply (rule impI , drule finite_exhausts)
+apply blast
+done
+
+lemma not_finite_nat: "~ finite(UNIV:: nat set)"
+apply (fast dest!: finite_exhausts)
+done
+
+(*------------------------------------------------------------------------
+ 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)"
+apply (rule not_finite_nat [THEN FreeUltrafilter_Ex])
+done
+
+lemma FreeUltrafilterNat_mem:
+ "FreeUltrafilterNat: FreeUltrafilter(UNIV:: nat set)"
+apply (unfold FreeUltrafilterNat_def)
+apply (rule FreeUltrafilterNat_Ex [THEN exE])
+apply (rule someI2)
+apply assumption+
+done
+declare FreeUltrafilterNat_mem [simp]
+
+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"
+apply (blast dest: FreeUltrafilterNat_finite)
+done
+
+lemma FreeUltrafilterNat_empty: "{} \<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
+declare FreeUltrafilterNat_empty [simp]
+
+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)
+apply 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)"
+apply (blast dest: FreeUltrafilterNat_Compl FreeUltrafilterNat_Compl_mem)
+done
+
+lemma FreeUltrafilterNat_Compl_iff2: "(X: FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)"
+apply (auto simp add: FreeUltrafilterNat_Compl_iff1 [symmetric])
+done
+
+lemma FreeUltrafilterNat_UNIV: "(UNIV::nat set) \<in> FreeUltrafilterNat"
+apply (rule FreeUltrafilterNat_mem [THEN FreeUltrafilter_Ultrafilter, THEN Ultrafilter_Filter, THEN mem_FiltersetD4])
+done
+declare FreeUltrafilterNat_UNIV [simp]
+
+lemma FreeUltrafilterNat_Nat_set: "UNIV \<in> FreeUltrafilterNat"
+apply auto
+done
+declare FreeUltrafilterNat_Nat_set [simp]
+
+lemma FreeUltrafilterNat_Nat_set_refl: "{n. P(n) = P(n)} \<in> FreeUltrafilterNat"
+apply (simp (no_asm))
+done
+declare FreeUltrafilterNat_Nat_set_refl [intro]
+
+lemma FreeUltrafilterNat_P: "{n::nat. P} \<in> FreeUltrafilterNat ==> P"
+apply (rule ccontr)
+apply simp
+done
+
+lemma FreeUltrafilterNat_Ex_P: "{n. P(n)} \<in> FreeUltrafilterNat ==> \<exists>n. P(n)"
+apply (rule ccontr)
+apply simp
+done
+
+lemma FreeUltrafilterNat_all: "\<forall>n. P(n) ==> {n. P(n)} \<in> FreeUltrafilterNat"
+apply (auto intro: FreeUltrafilterNat_Nat_set)
+done
+
+(*-------------------------------------------------------
+ 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)"
+apply (unfold hyprel_def)
+apply fast
+done
+
+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"
+apply (simp add: hyprel_def eq_commute)
+done
+
+lemma hyprel_trans:
+ "[|(x,y): hyprel; (y,z):hyprel|] ==> (x,z):hyprel"
+apply (unfold hyprel_def)
+apply auto
+apply (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: "hyprel``{x}:hypreal"
+apply (unfold hypreal_def hyprel_def quotient_def)
+apply blast
+done
+
+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]
+ hyprel_in_hypreal [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: "x \<in> hyprel `` {x}"
+apply (unfold hyprel_def)
+apply (safe)
+apply (auto intro!: FreeUltrafilterNat_Nat_set)
+done
+
+declare lemma_hyprel_refl [simp]
+
+lemma hypreal_empty_not_mem: "{} \<notin> hypreal"
+apply (unfold hypreal_def)
+apply (auto elim!: quotientE equalityCE)
+done
+
+declare hypreal_empty_not_mem [simp]
+
+lemma Rep_hypreal_nonempty: "Rep_hypreal x \<noteq> {}"
+apply (cut_tac x = "x" in Rep_hypreal)
+apply auto
+done
+
+declare Rep_hypreal_nonempty [simp]
+
+(*------------------------------------------------------------------------
+ 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 (no_asm) add: hyprel_in_hypreal [THEN Abs_hypreal_inverse]
+ UN_equiv_class [OF equiv_hyprel hypreal_minus_congruent])
+done
+
+lemma hypreal_minus_minus: "- (- z) = (z::hypreal)"
+apply (rule_tac z = "z" in eq_Abs_hypreal)
+apply (simp (no_asm_simp) add: hypreal_minus)
+done
+
+declare hypreal_minus_minus [simp]
+
+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: "- 0 = (0::hypreal)"
+apply (unfold hypreal_zero_def)
+apply (simp (no_asm) add: hypreal_minus)
+done
+declare hypreal_minus_zero [simp]
+
+lemma hypreal_minus_zero_iff: "(-x = 0) = (x = (0::hypreal))"
+apply (rule_tac z = "x" in eq_Abs_hypreal)
+apply (auto simp add: hypreal_zero_def hypreal_minus)
+done
+declare hypreal_minus_zero_iff [simp]
+
+
+(**** hyperreal addition: hypreal_add ****)
+
+lemma hypreal_add_congruent2:
+ "congruent2 hyprel (%X Y. hyprel``{%n. X n + Y n})"
+apply (unfold congruent2_def)
+apply (auto );
+apply 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 (no_asm) 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 (no_asm_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 (no_asm_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: "(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: "z + (0::hypreal) = z"
+apply (simp (no_asm) add: hypreal_add_zero_left hypreal_add_commute)
+done
+
+lemma hypreal_add_minus: "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: "-z + z = (0::hypreal)"
+apply (simp (no_asm) add: hypreal_add_commute hypreal_add_minus)
+done
+
+declare hypreal_add_minus [simp] hypreal_add_minus_left [simp]
+ hypreal_add_zero_left [simp] hypreal_add_zero_right [simp]
+
+lemma hypreal_minus_ex: "\<exists>y. (x::hypreal) + y = 0"
+apply (fast intro: hypreal_add_minus)
+done
+
+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])
+apply 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)
+apply fast
+done
+
+lemma hypreal_minus_add_distrib: "-(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
+declare hypreal_minus_add_distrib [simp]
+
+lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
+apply (simp (no_asm) add: hypreal_add_commute)
+done
+
+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)"
+apply (simp (no_asm) add: hypreal_add_commute hypreal_add_left_cancel)
+done
+
+lemma hypreal_add_minus_cancelA: "z + ((- z) + w) = (w::hypreal)"
+apply (simp (no_asm) add: hypreal_add_assoc [symmetric])
+done
+
+lemma hypreal_minus_add_cancelA: "(-z) + (z + w) = (w::hypreal)"
+apply (simp (no_asm) add: hypreal_add_assoc [symmetric])
+done
+
+declare hypreal_add_minus_cancelA [simp] hypreal_minus_add_cancelA [simp]
+
+(**** hyperreal multiplication: hypreal_mult ****)
+
+lemma hypreal_mult_congruent2:
+ "congruent2 hyprel (%X Y. hyprel``{%n. X n * Y n})"
+apply (unfold congruent2_def)
+apply auto
+apply (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 (no_asm_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 (no_asm_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: "(1::hypreal) * z = z"
+apply (unfold hypreal_one_def)
+apply (rule_tac z = "z" in eq_Abs_hypreal)
+apply (simp add: hypreal_mult)
+done
+declare hypreal_mult_1 [simp]
+
+lemma hypreal_mult_1_right: "z * (1::hypreal) = z"
+apply (simp (no_asm) add: hypreal_mult_commute hypreal_mult_1)
+done
+declare hypreal_mult_1_right [simp]
+
+lemma hypreal_mult_0: "0 * z = (0::hypreal)"
+apply (unfold hypreal_zero_def)
+apply (rule_tac z = "z" in eq_Abs_hypreal)
+apply (simp add: hypreal_mult)
+done
+declare hypreal_mult_0 [simp]
+
+lemma hypreal_mult_0_right: "z * 0 = (0::hypreal)"
+apply (simp (no_asm) add: hypreal_mult_commute)
+done
+declare hypreal_mult_0_right [simp]
+
+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: "(- (1::hypreal)) * z = -z"
+apply (simp (no_asm))
+done
+declare hypreal_mult_minus_1 [simp]
+
+lemma hypreal_mult_minus_1_right: "z * (- (1::hypreal)) = -z"
+apply (subst hypreal_mult_commute)
+apply (simp (no_asm))
+done
+declare hypreal_mult_minus_1_right [simp]
+
+lemma hypreal_minus_mult_commute: "(-x) * y = (x::hypreal) * -y"
+apply auto
+done
+
+(*-----------------------------------------------------------------------------
+ A few more theorems
+ ----------------------------------------------------------------------------*)
+lemma hypreal_add_assoc_cong: "(z::hypreal) + v = z' + v' ==> z + (v + w) = z' + (v' + w)"
+apply (simp (no_asm_simp) add: hypreal_add_assoc [symmetric])
+done
+
+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 (no_asm_simp) add: hypreal_mult hypreal_add real_add_mult_distrib)
+done
+
+lemma hypreal_add_mult_distrib2: "(w::hypreal) * (z1 + z2) = (w * z1) + (w * z2)"
+apply (simp add: hypreal_mult_commute [of w] hypreal_add_mult_distrib)
+done
+
+
+lemma hypreal_diff_mult_distrib: "((z1::hypreal) - z2) * w = (z1 * w) - (z2 * w)"
+
+apply (unfold hypreal_diff_def)
+apply (simp (no_asm) add: hypreal_add_mult_distrib)
+done
+
+lemma hypreal_diff_mult_distrib2: "(w::hypreal) * (z1 - z2) = (w * z1) - (w * z2)"
+apply (simp add: hypreal_mult_commute [of w] hypreal_diff_mult_distrib)
+done
+
+(*** 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 (no_asm) 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)"
+apply (simp (no_asm) add: hypreal_inverse hypreal_zero_def)
+done
+
+lemma HYPREAL_DIVISION_BY_ZERO: "a / (0::hypreal) = 0"
+apply (simp (no_asm) add: hypreal_divide_def HYPREAL_INVERSE_ZERO)
+done
+
+lemma hypreal_inverse_inverse: "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
+declare hypreal_inverse_inverse [simp]
+
+lemma hypreal_inverse_1: "inverse((1::hypreal)) = (1::hypreal)"
+apply (unfold hypreal_one_def)
+apply (simp (no_asm_use) add: hypreal_inverse real_zero_not_eq_one [THEN not_sym])
+done
+declare hypreal_inverse_1 [simp]
+
+
+(*** existence of inverse ***)
+
+lemma hypreal_mult_inverse:
+ "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: "x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)"
+apply (simp (no_asm_simp) add: hypreal_mult_inverse hypreal_mult_commute)
+done
+
+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
+
+declare hypreal_mult_inverse [simp] hypreal_mult_inverse_left [simp]
+
+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"
+apply (auto intro: ccontr dest: hypreal_mult_not_0)
+done
+
+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])
+apply (simp add: );
+apply (subst hypreal_mult_inverse_left)
+apply 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 (no_asm_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)
+apply 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)
+apply 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)
+apply 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)
+apply 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)
+apply (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"
+apply (auto simp add: hypreal_less_not_refl)
+done
+
+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)
+apply 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)
+apply (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)
+apply (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)
+apply auto
+apply (drule_tac x = "x" in spec)
+apply (drule_tac x = "xa" in spec)
+apply auto
+apply (ultra)
+done
+
+lemma hypreal_trichotomyE:
+ "[| (0::hypreal) < x ==> P;
+ x = 0 ==> P;
+ x < 0 ==> P |] ==> P"
+apply (insert hypreal_trichotomy [of x])
+apply (blast intro: elim:);
+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])
+apply 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])
+apply auto
+done
+
+(* 07/00 *)
+lemma hypreal_diff_zero: "(0::hypreal) - x = -x"
+apply (simp (no_asm) add: hypreal_diff_def)
+done
+
+lemma hypreal_diff_zero_right: "x - (0::hypreal) = x"
+apply (simp (no_asm) add: hypreal_diff_def)
+done
+
+lemma hypreal_diff_self: "x - x = (0::hypreal)"
+apply (simp (no_asm) add: hypreal_diff_def)
+done
+
+declare hypreal_diff_zero [simp] hypreal_diff_zero_right [simp] hypreal_diff_self [simp]
+
+lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
+apply (auto simp add: hypreal_add_assoc)
+done
+
+lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
+apply (auto dest: hypreal_eq_minus_iff [THEN iffD2])
+done
+
+
+(*** 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)
+apply auto
+done
+
+lemma hypreal_neq_iff: "((w::hypreal) \<noteq> z) = (w<z | z<w)"
+apply (cut_tac hypreal_linear)
+apply blast
+done
+
+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)
+apply 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)
+apply assumption
+done
+
+lemma hypreal_leD:
+ "z<=w ==> ~(w<(z::hypreal))"
+apply (unfold hypreal_le_def)
+apply assumption
+done
+
+lemma hypreal_less_le_iff: "(~(w < z)) = (z <= (w::hypreal))"
+apply (fast intro!: hypreal_leI hypreal_leD)
+done
+
+lemma not_hypreal_leE: "~ z <= w ==> w<(z::hypreal)"
+apply (unfold hypreal_le_def)
+apply fast
+done
+
+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)"
+apply (simp (no_asm) add: hypreal_le_eq_less_or_eq)
+done
+
+(* Axiom 'linorder_linear' of class 'linorder': *)
+lemma hypreal_le_linear: "(z::hypreal) <= w | w <= z"
+apply (simp (no_asm) add: hypreal_le_less)
+apply (cut_tac hypreal_linear)
+apply 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 (no_asm) add: hypreal_le_def hypreal_neq_iff)
+apply (blast intro: hypreal_less_asym)
+done
+
+lemma hypreal_minus_zero_less_iff: "(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
+declare hypreal_minus_zero_less_iff [simp]
+
+lemma hypreal_minus_zero_less_iff2: "(-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
+declare hypreal_minus_zero_less_iff2 [simp]
+
+lemma hypreal_minus_zero_le_iff: "((0::hypreal) <= -r) = (r <= 0)"
+apply (unfold hypreal_le_def)
+apply (simp (no_asm) add: hypreal_minus_zero_less_iff2)
+done
+declare hypreal_minus_zero_le_iff [simp]
+
+lemma hypreal_minus_zero_le_iff2: "(-r <= (0::hypreal)) = (0 <= r)"
+apply (unfold hypreal_le_def)
+apply (simp (no_asm) add: hypreal_minus_zero_less_iff2)
+done
+declare hypreal_minus_zero_le_iff2 [simp]
+
+(*----------------------------------------------------------
+ hypreal_of_real preserves field and order properties
+ -----------------------------------------------------------*)
+lemma hypreal_of_real_add:
+ "hypreal_of_real (z1 + z2) = hypreal_of_real z1 + hypreal_of_real z2"
+apply (unfold hypreal_of_real_def)
+apply (simp (no_asm) add: hypreal_add hypreal_add_mult_distrib)
+done
+declare hypreal_of_real_add [simp]
+
+lemma hypreal_of_real_mult:
+ "hypreal_of_real (z1 * z2) = hypreal_of_real z1 * hypreal_of_real z2"
+apply (unfold hypreal_of_real_def)
+apply (simp (no_asm) add: hypreal_mult hypreal_add_mult_distrib2)
+done
+declare hypreal_of_real_mult [simp]
+
+lemma hypreal_of_real_less_iff:
+ "(hypreal_of_real z1 < hypreal_of_real z2) = (z1 < z2)"
+apply (unfold hypreal_less_def hypreal_of_real_def)
+apply auto
+apply (rule_tac [2] x = "%n. z1" in exI)
+apply (safe)
+apply (rule_tac [3] x = "%n. z2" in exI)
+apply auto
+apply (rule FreeUltrafilterNat_P)
+apply (ultra)
+done
+declare hypreal_of_real_less_iff [simp]
+
+lemma hypreal_of_real_le_iff:
+ "(hypreal_of_real z1 <= hypreal_of_real z2) = (z1 <= z2)"
+apply (unfold hypreal_le_def real_le_def)
+apply auto
+done
+declare hypreal_of_real_le_iff [simp]
+
+lemma hypreal_of_real_eq_iff: "(hypreal_of_real z1 = hypreal_of_real z2) = (z1 = z2)"
+apply (force intro: order_antisym hypreal_of_real_le_iff [THEN iffD1])
+done
+declare hypreal_of_real_eq_iff [simp]
+
+lemma hypreal_of_real_minus: "hypreal_of_real (-r) = - hypreal_of_real r"
+apply (unfold hypreal_of_real_def)
+apply (auto simp add: hypreal_minus)
+done
+declare hypreal_of_real_minus [simp]
+
+lemma hypreal_of_real_one: "hypreal_of_real 1 = (1::hypreal)"
+apply (unfold hypreal_of_real_def hypreal_one_def)
+apply (simp (no_asm))
+done
+declare hypreal_of_real_one [simp]
+
+lemma hypreal_of_real_zero: "hypreal_of_real 0 = 0"
+apply (unfold hypreal_of_real_def hypreal_zero_def)
+apply (simp (no_asm))
+done
+declare hypreal_of_real_zero [simp]
+
+lemma hypreal_of_real_zero_iff: "(hypreal_of_real r = 0) = (r = 0)"
+apply (auto intro: FreeUltrafilterNat_P simp add: hypreal_of_real_def hypreal_zero_def FreeUltrafilterNat_Nat_set)
+done
+
+lemma hypreal_of_real_inverse: "hypreal_of_real (inverse r) = inverse (hypreal_of_real r)"
+apply (case_tac "r=0")
+apply (simp (no_asm_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
+declare hypreal_of_real_inverse [simp]
+
+lemma hypreal_of_real_divide: "hypreal_of_real (z1 / z2) = hypreal_of_real z1 / hypreal_of_real z2"
+apply (simp (no_asm) add: hypreal_divide_def real_divide_def)
+done
+declare hypreal_of_real_divide [simp]
+
+
+(*** Division lemmas ***)
+
+lemma hypreal_zero_divide: "(0::hypreal)/x = 0"
+apply (simp (no_asm) add: hypreal_divide_def)
+done
+
+lemma hypreal_divide_one: "x/(1::hypreal) = x"
+apply (simp (no_asm) add: hypreal_divide_def)
+done
+declare hypreal_zero_divide [simp] hypreal_divide_one [simp]
+
+lemma hypreal_times_divide1_eq: "(x::hypreal) * (y/z) = (x*y)/z"
+apply (simp (no_asm) add: hypreal_divide_def hypreal_mult_assoc)
+done
+
+lemma hypreal_times_divide2_eq: "(y/z) * (x::hypreal) = (y*x)/z"
+apply (simp (no_asm) add: hypreal_divide_def hypreal_mult_ac)
+done
+
+declare hypreal_times_divide1_eq [simp] hypreal_times_divide2_eq [simp]
+
+lemma hypreal_divide_divide1_eq: "(x::hypreal) / (y/z) = (x*z)/y"
+apply (simp (no_asm) add: hypreal_divide_def hypreal_inverse_distrib hypreal_mult_ac)
+done
+
+lemma hypreal_divide_divide2_eq: "((x::hypreal) / y) / z = x/(y*z)"
+apply (simp (no_asm) add: hypreal_divide_def hypreal_inverse_distrib hypreal_mult_assoc)
+done
+
+declare hypreal_divide_divide1_eq [simp] hypreal_divide_divide2_eq [simp]
+
+(** As with multiplication, pull minus signs OUT of the / operator **)
+
+lemma hypreal_minus_divide_eq: "(-x) / (y::hypreal) = - (x/y)"
+apply (simp (no_asm) add: hypreal_divide_def)
+done
+declare hypreal_minus_divide_eq [simp]
+
+lemma hypreal_divide_minus_eq: "(x / -(y::hypreal)) = - (x/y)"
+apply (simp (no_asm) add: hypreal_divide_def hypreal_minus_inverse)
+done
+declare hypreal_divide_minus_eq [simp]
+
+lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
+apply (simp (no_asm) add: hypreal_divide_def hypreal_add_mult_distrib)
+done
+
+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)
+apply (ultra)
+done
+
+lemma hypreal_add_self_zero_cancel: "(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
+declare hypreal_add_self_zero_cancel [simp]
+
+lemma hypreal_add_self_zero_cancel2: "(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
+declare hypreal_add_self_zero_cancel2 [simp]
+
+lemma hypreal_add_self_zero_cancel2a: "(x + (x + y) = y) = (x = (0::hypreal))"
+apply (simp (no_asm) add: hypreal_add_assoc [symmetric])
+done
+declare hypreal_add_self_zero_cancel2a [simp]
+
+lemma hypreal_minus_eq_swap: "(b = -a) = (-b = (a::hypreal))"
+apply auto
+done
+
+lemma hypreal_minus_eq_cancel: "(-b = -a) = (b = (a::hypreal))"
+apply (simp add: hypreal_minus_eq_swap)
+done
+declare hypreal_minus_eq_cancel [simp]
+
+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)"
+apply (simp (no_asm) add: hypreal_diff_def hypreal_add_ac)
+done
+
+lemma hypreal_diff_add_eq: "(x - y) + z = (x + z) - (y::hypreal)"
+apply (simp (no_asm) add: hypreal_diff_def hypreal_add_ac)
+done
+
+lemma hypreal_diff_diff_eq: "(x - y) - z = x - (y + (z::hypreal))"
+apply (simp (no_asm) add: hypreal_diff_def hypreal_add_ac)
+done
+
+lemma hypreal_diff_diff_eq2: "x - (y - z) = (x + z) - (y::hypreal)"
+apply (simp (no_asm) add: hypreal_diff_def hypreal_add_ac)
+done
+
+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 (no_asm) 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 (no_asm) 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 (no_asm) 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 (no_asm) 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])
+apply (simp (no_asm_simp))
+done
+
+lemma hypreal_le_eqI: "(x::hypreal) - y = x' - y' ==> (y<=x) = (y'<=x')"
+apply (drule hypreal_less_eqI)
+apply (simp (no_asm_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})"
+apply (simp (no_asm) add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric])
+done
+
+lemma hypreal_one_num: "1 = Abs_hypreal (hyprel `` {%n. 1})"
+apply (simp (no_asm) add: hypreal_one_def [THEN meta_eq_to_obj_eq, symmetric])
+done
+
+lemma hypreal_omega_gt_zero: "0 < omega"
+apply (unfold omega_def)
+apply (auto simp add: hypreal_less hypreal_zero_num)
+done
+declare hypreal_omega_gt_zero [simp]
+
+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