src/HOL/Hyperreal/HyperDef.thy
author schirmer
Mon Jan 26 10:34:02 2004 +0100 (2004-01-26)
changeset 14361 ad2f5da643b4
parent 14348 744c868ee0b7
child 14365 3d4df8c166ae
permissions -rw-r--r--
* Support for raw latex output in control symbols: \<^raw...>
* Symbols may only start with one backslash: \<...>. \\<...> is no longer
accepted by the scanner.
- Adapted some Isar-theories to fit to this policy
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(*  Title       : HOL/Real/Hyperreal/HyperDef.thy
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    ID          : $Id$
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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    Description : Construction of hyperreals using ultrafilters
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*)
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theory HyperDef = Filter + Real
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files ("fuf.ML"):  (*Warning: file fuf.ML refers to the name Hyperdef!*)
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constdefs
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  FreeUltrafilterNat   :: "nat set set"    ("\<U>")
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    "FreeUltrafilterNat == (SOME U. U \<in> FreeUltrafilter (UNIV:: nat set))"
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  hyprel :: "((nat=>real)*(nat=>real)) set"
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    "hyprel == {p. \<exists>X Y. p = ((X::nat=>real),Y) &
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                   {n::nat. X(n) = Y(n)}: FreeUltrafilterNat}"
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typedef hypreal = "UNIV//hyprel" 
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    by (auto simp add: quotient_def) 
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instance hypreal :: ord ..
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instance hypreal :: zero ..
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instance hypreal :: one ..
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instance hypreal :: plus ..
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instance hypreal :: times ..
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instance hypreal :: minus ..
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instance hypreal :: inverse ..
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defs (overloaded)
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  hypreal_zero_def:
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  "0 == Abs_hypreal(hyprel``{%n::nat. (0::real)})"
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  hypreal_one_def:
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  "1 == Abs_hypreal(hyprel``{%n::nat. (1::real)})"
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  hypreal_minus_def:
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  "- P == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). hyprel``{%n::nat. - (X n)})"
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  hypreal_diff_def:
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  "x - y == x + -(y::hypreal)"
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  hypreal_inverse_def:
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  "inverse P == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P).
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                    hyprel``{%n. if X n = 0 then 0 else inverse (X n)})"
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  hypreal_divide_def:
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  "P / Q::hypreal == P * inverse Q"
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constdefs
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  hypreal_of_real  :: "real => hypreal"
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  "hypreal_of_real r         == Abs_hypreal(hyprel``{%n::nat. r})"
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  omega   :: hypreal   (*an infinite number = [<1,2,3,...>] *)
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  "omega == Abs_hypreal(hyprel``{%n::nat. real (Suc n)})"
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  epsilon :: hypreal   (*an infinitesimal number = [<1,1/2,1/3,...>] *)
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  "epsilon == Abs_hypreal(hyprel``{%n::nat. inverse (real (Suc n))})"
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syntax (xsymbols)
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  omega   :: hypreal   ("\<omega>")
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  epsilon :: hypreal   ("\<epsilon>")
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defs
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  hypreal_add_def:
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  "P + Q == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). \<Union>Y \<in> Rep_hypreal(Q).
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                hyprel``{%n::nat. X n + Y n})"
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  hypreal_mult_def:
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  "P * Q == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). \<Union>Y \<in> Rep_hypreal(Q).
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                hyprel``{%n::nat. X n * Y n})"
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  hypreal_less_def:
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  "P < (Q::hypreal) == \<exists>X Y. X \<in> Rep_hypreal(P) &
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                               Y \<in> Rep_hypreal(Q) &
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                               {n::nat. X n < Y n} \<in> FreeUltrafilterNat"
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  hypreal_le_def:
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  "P <= (Q::hypreal) == ~(Q < P)"
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  hrabs_def:  "abs (r::hypreal) == (if 0 \<le> r then r else -r)"
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subsection{*The Set of Naturals is not Finite*}
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(*** based on James' proof that the set of naturals is not finite ***)
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lemma finite_exhausts [rule_format]:
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     "finite (A::nat set) --> (\<exists>n. \<forall>m. Suc (n + m) \<notin> A)"
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apply (rule impI)
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apply (erule_tac F = A in finite_induct)
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apply (blast, erule exE)
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apply (rule_tac x = "n + x" in exI)
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apply (rule allI, erule_tac x = "x + m" in allE)
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apply (auto simp add: add_ac)
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done
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lemma finite_not_covers [rule_format (no_asm)]:
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     "finite (A :: nat set) --> (\<exists>n. n \<notin>A)"
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by (rule impI, drule finite_exhausts, blast)
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lemma not_finite_nat: "~ finite(UNIV:: nat set)"
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by (fast dest!: finite_exhausts)
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subsection{*Existence of Free Ultrafilter over the Naturals*}
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text{*Also, proof of various properties of @{term FreeUltrafilterNat}: 
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an arbitrary free ultrafilter*}
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lemma FreeUltrafilterNat_Ex: "\<exists>U. U: FreeUltrafilter (UNIV::nat set)"
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by (rule not_finite_nat [THEN FreeUltrafilter_Ex])
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lemma FreeUltrafilterNat_mem [simp]: 
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     "FreeUltrafilterNat: FreeUltrafilter(UNIV:: nat set)"
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apply (unfold FreeUltrafilterNat_def)
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apply (rule FreeUltrafilterNat_Ex [THEN exE])
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apply (rule someI2, assumption+)
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done
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lemma FreeUltrafilterNat_finite: "finite x ==> x \<notin> FreeUltrafilterNat"
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apply (unfold FreeUltrafilterNat_def)
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apply (rule FreeUltrafilterNat_Ex [THEN exE])
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apply (rule someI2, assumption)
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apply (blast dest: mem_FreeUltrafiltersetD1)
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done
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lemma FreeUltrafilterNat_not_finite: "x: FreeUltrafilterNat ==> ~ finite x"
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by (blast dest: FreeUltrafilterNat_finite)
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lemma FreeUltrafilterNat_empty [simp]: "{} \<notin> FreeUltrafilterNat"
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apply (unfold FreeUltrafilterNat_def)
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apply (rule FreeUltrafilterNat_Ex [THEN exE])
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apply (rule someI2, assumption)
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apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter 
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                   Filter_empty_not_mem)
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done
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lemma FreeUltrafilterNat_Int:
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     "[| X: FreeUltrafilterNat;  Y: FreeUltrafilterNat |]   
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      ==> X Int Y \<in> FreeUltrafilterNat"
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apply (cut_tac FreeUltrafilterNat_mem)
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apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter mem_FiltersetD1)
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done
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lemma FreeUltrafilterNat_subset:
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     "[| X: FreeUltrafilterNat;  X <= Y |]  
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      ==> Y \<in> FreeUltrafilterNat"
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apply (cut_tac FreeUltrafilterNat_mem)
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apply (blast dest: FreeUltrafilter_Ultrafilter Ultrafilter_Filter mem_FiltersetD2)
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done
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lemma FreeUltrafilterNat_Compl:
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     "X: FreeUltrafilterNat ==> -X \<notin> FreeUltrafilterNat"
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apply safe
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apply (drule FreeUltrafilterNat_Int, assumption, auto)
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done
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lemma FreeUltrafilterNat_Compl_mem:
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     "X\<notin> FreeUltrafilterNat ==> -X \<in> FreeUltrafilterNat"
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apply (cut_tac FreeUltrafilterNat_mem [THEN FreeUltrafilter_iff [THEN iffD1]])
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apply (safe, drule_tac x = X in bspec)
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apply (auto simp add: UNIV_diff_Compl)
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done
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lemma FreeUltrafilterNat_Compl_iff1:
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     "(X \<notin> FreeUltrafilterNat) = (-X: FreeUltrafilterNat)"
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by (blast dest: FreeUltrafilterNat_Compl FreeUltrafilterNat_Compl_mem)
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lemma FreeUltrafilterNat_Compl_iff2:
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     "(X: FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)"
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by (auto simp add: FreeUltrafilterNat_Compl_iff1 [symmetric])
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lemma FreeUltrafilterNat_UNIV [simp]: "(UNIV::nat set) \<in> FreeUltrafilterNat"
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by (rule FreeUltrafilterNat_mem [THEN FreeUltrafilter_Ultrafilter, THEN Ultrafilter_Filter, THEN mem_FiltersetD4])
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lemma FreeUltrafilterNat_Nat_set [simp]: "UNIV \<in> FreeUltrafilterNat"
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by auto
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lemma FreeUltrafilterNat_Nat_set_refl [intro]:
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     "{n. P(n) = P(n)} \<in> FreeUltrafilterNat"
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by simp
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lemma FreeUltrafilterNat_P: "{n::nat. P} \<in> FreeUltrafilterNat ==> P"
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by (rule ccontr, simp)
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lemma FreeUltrafilterNat_Ex_P: "{n. P(n)} \<in> FreeUltrafilterNat ==> \<exists>n. P(n)"
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by (rule ccontr, simp)
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lemma FreeUltrafilterNat_all: "\<forall>n. P(n) ==> {n. P(n)} \<in> FreeUltrafilterNat"
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by (auto intro: FreeUltrafilterNat_Nat_set)
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text{*Define and use Ultrafilter tactics*}
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use "fuf.ML"
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method_setup fuf = {*
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    Method.ctxt_args (fn ctxt =>
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        Method.METHOD (fn facts =>
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            fuf_tac (Classical.get_local_claset ctxt,
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                     Simplifier.get_local_simpset ctxt) 1)) *}
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    "free ultrafilter tactic"
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method_setup ultra = {*
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    Method.ctxt_args (fn ctxt =>
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        Method.METHOD (fn facts =>
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            ultra_tac (Classical.get_local_claset ctxt,
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                       Simplifier.get_local_simpset ctxt) 1)) *}
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    "ultrafilter tactic"
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text{*One further property of our free ultrafilter*}
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lemma FreeUltrafilterNat_Un:
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     "X Un Y: FreeUltrafilterNat  
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      ==> X: FreeUltrafilterNat | Y: FreeUltrafilterNat"
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apply auto
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apply ultra
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done
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subsection{*Properties of @{term hyprel}*}
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text{*Proving that @{term hyprel} is an equivalence relation*}
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lemma hyprel_iff: "((X,Y): hyprel) = ({n. X n = Y n}: FreeUltrafilterNat)"
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by (unfold hyprel_def, fast)
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lemma hyprel_refl: "(x,x): hyprel"
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apply (unfold hyprel_def)
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apply (auto simp add: FreeUltrafilterNat_Nat_set)
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done
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lemma hyprel_sym [rule_format (no_asm)]: "(x,y): hyprel --> (y,x):hyprel"
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by (simp add: hyprel_def eq_commute)
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lemma hyprel_trans: 
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      "[|(x,y): hyprel; (y,z):hyprel|] ==> (x,z):hyprel"
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apply (unfold hyprel_def, auto, ultra)
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done
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lemma equiv_hyprel: "equiv UNIV hyprel"
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apply (simp add: equiv_def refl_def sym_def trans_def hyprel_refl)
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apply (blast intro: hyprel_sym hyprel_trans) 
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done
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(* (hyprel `` {x} = hyprel `` {y}) = ((x,y) \<in> hyprel) *)
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lemmas equiv_hyprel_iff =
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    eq_equiv_class_iff [OF equiv_hyprel UNIV_I UNIV_I, simp] 
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lemma hyprel_in_hypreal [simp]: "hyprel``{x}:hypreal"
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by (unfold hypreal_def hyprel_def quotient_def, blast)
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lemma inj_on_Abs_hypreal: "inj_on Abs_hypreal hypreal"
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apply (rule inj_on_inverseI)
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apply (erule Abs_hypreal_inverse)
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done
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declare inj_on_Abs_hypreal [THEN inj_on_iff, simp] 
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        Abs_hypreal_inverse [simp]
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declare equiv_hyprel [THEN eq_equiv_class_iff, simp]
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declare hyprel_iff [iff]
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lemmas eq_hyprelD = eq_equiv_class [OF _ equiv_hyprel]
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lemma inj_Rep_hypreal: "inj(Rep_hypreal)"
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apply (rule inj_on_inverseI)
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apply (rule Rep_hypreal_inverse)
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done
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lemma lemma_hyprel_refl [simp]: "x \<in> hyprel `` {x}"
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apply (unfold hyprel_def, safe)
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apply (auto intro!: FreeUltrafilterNat_Nat_set)
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done
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lemma hypreal_empty_not_mem [simp]: "{} \<notin> hypreal"
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apply (unfold hypreal_def)
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apply (auto elim!: quotientE equalityCE)
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done
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lemma Rep_hypreal_nonempty [simp]: "Rep_hypreal x \<noteq> {}"
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by (cut_tac x = x in Rep_hypreal, auto)
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subsection{*@{term hypreal_of_real}: 
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            the Injection from @{typ real} to @{typ hypreal}*}
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lemma inj_hypreal_of_real: "inj(hypreal_of_real)"
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apply (rule inj_onI)
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apply (unfold hypreal_of_real_def)
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apply (drule inj_on_Abs_hypreal [THEN inj_onD])
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apply (rule hyprel_in_hypreal)+
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apply (drule eq_equiv_class)
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apply (rule equiv_hyprel)
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apply (simp_all add: split: split_if_asm) 
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done
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lemma eq_Abs_hypreal:
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    "(!!x y. z = Abs_hypreal(hyprel``{x}) ==> P) ==> P"
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apply (rule_tac x1=z in Rep_hypreal [unfolded hypreal_def, THEN quotientE])
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apply (drule_tac f = Abs_hypreal in arg_cong)
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apply (force simp add: Rep_hypreal_inverse)
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done
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subsection{*Hyperreal Addition*}
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lemma hypreal_add_congruent2: 
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    "congruent2 hyprel (%X Y. hyprel``{%n. X n + Y n})"
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apply (unfold congruent2_def, auto, ultra)
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done
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lemma hypreal_add: 
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  "Abs_hypreal(hyprel``{%n. X n}) + Abs_hypreal(hyprel``{%n. Y n}) =  
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   Abs_hypreal(hyprel``{%n. X n + Y n})"
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apply (unfold hypreal_add_def)
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apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_add_congruent2])
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done
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lemma hypreal_add_commute: "(z::hypreal) + w = w + z"
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apply (rule_tac z = z in eq_Abs_hypreal)
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apply (rule_tac z = w in eq_Abs_hypreal)
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   329
apply (simp add: add_ac hypreal_add)
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   330
done
paulson@14329
   331
paulson@14329
   332
lemma hypreal_add_assoc: "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)"
paulson@14329
   333
apply (rule_tac z = z1 in eq_Abs_hypreal)
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   334
apply (rule_tac z = z2 in eq_Abs_hypreal)
paulson@14329
   335
apply (rule_tac z = z3 in eq_Abs_hypreal)
paulson@14329
   336
apply (simp add: hypreal_add real_add_assoc)
paulson@14329
   337
done
paulson@14329
   338
paulson@14331
   339
lemma hypreal_add_zero_left: "(0::hypreal) + z = z"
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   340
apply (unfold hypreal_zero_def)
paulson@14329
   341
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14329
   342
apply (simp add: hypreal_add)
paulson@14329
   343
done
paulson@14329
   344
paulson@14329
   345
instance hypreal :: plus_ac0
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   346
  by (intro_classes,
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   347
      (assumption | 
paulson@14329
   348
       rule hypreal_add_commute hypreal_add_assoc hypreal_add_zero_left)+)
paulson@14329
   349
paulson@14329
   350
lemma hypreal_add_zero_right [simp]: "z + (0::hypreal) = z"
paulson@14329
   351
by (simp add: hypreal_add_zero_left hypreal_add_commute)
paulson@14329
   352
paulson@14329
   353
paulson@14329
   354
subsection{*Additive inverse on @{typ hypreal}*}
paulson@14299
   355
paulson@14299
   356
lemma hypreal_minus_congruent: 
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   357
  "congruent hyprel (%X. hyprel``{%n. - (X n)})"
paulson@14299
   358
by (force simp add: congruent_def)
paulson@14299
   359
paulson@14299
   360
lemma hypreal_minus: 
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   361
   "- (Abs_hypreal(hyprel``{%n. X n})) = Abs_hypreal(hyprel `` {%n. -(X n)})"
paulson@14299
   362
apply (unfold hypreal_minus_def)
paulson@14301
   363
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14301
   364
apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
paulson@14299
   365
               UN_equiv_class [OF equiv_hyprel hypreal_minus_congruent])
paulson@14299
   366
done
paulson@14299
   367
paulson@14329
   368
lemma hypreal_diff:
paulson@14329
   369
     "Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) =  
paulson@14299
   370
      Abs_hypreal(hyprel``{%n. X n - Y n})"
paulson@14301
   371
apply (simp add: hypreal_diff_def hypreal_minus hypreal_add)
paulson@14299
   372
done
paulson@14299
   373
paulson@14301
   374
lemma hypreal_add_minus [simp]: "z + -z = (0::hypreal)"
paulson@14299
   375
apply (unfold hypreal_zero_def)
paulson@14301
   376
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14299
   377
apply (simp add: hypreal_minus hypreal_add)
paulson@14299
   378
done
paulson@14299
   379
paulson@14331
   380
lemma hypreal_add_minus_left: "-z + z = (0::hypreal)"
paulson@14301
   381
by (simp add: hypreal_add_commute hypreal_add_minus)
paulson@14299
   382
paulson@14329
   383
paulson@14329
   384
subsection{*Hyperreal Multiplication*}
paulson@14299
   385
paulson@14299
   386
lemma hypreal_mult_congruent2: 
paulson@14299
   387
    "congruent2 hyprel (%X Y. hyprel``{%n. X n * Y n})"
paulson@14301
   388
apply (unfold congruent2_def, auto, ultra)
paulson@14299
   389
done
paulson@14299
   390
paulson@14299
   391
lemma hypreal_mult: 
paulson@14299
   392
  "Abs_hypreal(hyprel``{%n. X n}) * Abs_hypreal(hyprel``{%n. Y n}) =  
paulson@14299
   393
   Abs_hypreal(hyprel``{%n. X n * Y n})"
paulson@14299
   394
apply (unfold hypreal_mult_def)
paulson@14299
   395
apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_mult_congruent2])
paulson@14299
   396
done
paulson@14299
   397
paulson@14299
   398
lemma hypreal_mult_commute: "(z::hypreal) * w = w * z"
paulson@14301
   399
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14301
   400
apply (rule_tac z = w in eq_Abs_hypreal)
paulson@14331
   401
apply (simp add: hypreal_mult mult_ac)
paulson@14299
   402
done
paulson@14299
   403
paulson@14299
   404
lemma hypreal_mult_assoc: "((z1::hypreal) * z2) * z3 = z1 * (z2 * z3)"
paulson@14301
   405
apply (rule_tac z = z1 in eq_Abs_hypreal)
paulson@14301
   406
apply (rule_tac z = z2 in eq_Abs_hypreal)
paulson@14301
   407
apply (rule_tac z = z3 in eq_Abs_hypreal)
paulson@14331
   408
apply (simp add: hypreal_mult mult_assoc)
paulson@14299
   409
done
paulson@14299
   410
paulson@14331
   411
lemma hypreal_mult_1: "(1::hypreal) * z = z"
paulson@14299
   412
apply (unfold hypreal_one_def)
paulson@14301
   413
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14299
   414
apply (simp add: hypreal_mult)
paulson@14299
   415
done
paulson@14301
   416
paulson@14329
   417
lemma hypreal_add_mult_distrib:
paulson@14329
   418
     "((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14301
   419
apply (rule_tac z = z1 in eq_Abs_hypreal)
paulson@14301
   420
apply (rule_tac z = z2 in eq_Abs_hypreal)
paulson@14301
   421
apply (rule_tac z = w in eq_Abs_hypreal)
paulson@14334
   422
apply (simp add: hypreal_mult hypreal_add left_distrib)
paulson@14299
   423
done
paulson@14299
   424
paulson@14331
   425
text{*one and zero are distinct*}
paulson@14299
   426
lemma hypreal_zero_not_eq_one: "0 \<noteq> (1::hypreal)"
paulson@14299
   427
apply (unfold hypreal_zero_def hypreal_one_def)
paulson@14299
   428
apply (auto simp add: real_zero_not_eq_one)
paulson@14299
   429
done
paulson@14299
   430
paulson@14299
   431
paulson@14329
   432
subsection{*Multiplicative Inverse on @{typ hypreal} *}
paulson@14299
   433
paulson@14299
   434
lemma hypreal_inverse_congruent: 
paulson@14299
   435
  "congruent hyprel (%X. hyprel``{%n. if X n = 0 then 0 else inverse(X n)})"
paulson@14299
   436
apply (unfold congruent_def)
paulson@14301
   437
apply (auto, ultra)
paulson@14299
   438
done
paulson@14299
   439
paulson@14299
   440
lemma hypreal_inverse: 
paulson@14299
   441
      "inverse (Abs_hypreal(hyprel``{%n. X n})) =  
paulson@14299
   442
       Abs_hypreal(hyprel `` {%n. if X n = 0 then 0 else inverse(X n)})"
paulson@14299
   443
apply (unfold hypreal_inverse_def)
paulson@14301
   444
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14301
   445
apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
paulson@14299
   446
           UN_equiv_class [OF equiv_hyprel hypreal_inverse_congruent])
paulson@14299
   447
done
paulson@14299
   448
paulson@14331
   449
lemma hypreal_mult_inverse: 
paulson@14299
   450
     "x \<noteq> 0 ==> x*inverse(x) = (1::hypreal)"
paulson@14299
   451
apply (unfold hypreal_one_def hypreal_zero_def)
paulson@14301
   452
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14299
   453
apply (simp add: hypreal_inverse hypreal_mult)
paulson@14299
   454
apply (drule FreeUltrafilterNat_Compl_mem)
paulson@14334
   455
apply (blast intro!: right_inverse FreeUltrafilterNat_subset)
paulson@14299
   456
done
paulson@14299
   457
paulson@14331
   458
lemma hypreal_mult_inverse_left:
paulson@14329
   459
     "x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)"
paulson@14301
   460
by (simp add: hypreal_mult_inverse hypreal_mult_commute)
paulson@14299
   461
paulson@14331
   462
instance hypreal :: field
paulson@14331
   463
proof
paulson@14331
   464
  fix x y z :: hypreal
paulson@14331
   465
  show "(x + y) + z = x + (y + z)" by (rule hypreal_add_assoc)
paulson@14331
   466
  show "x + y = y + x" by (rule hypreal_add_commute)
paulson@14331
   467
  show "0 + x = x" by simp
paulson@14331
   468
  show "- x + x = 0" by (simp add: hypreal_add_minus_left)
paulson@14331
   469
  show "x - y = x + (-y)" by (simp add: hypreal_diff_def)
paulson@14331
   470
  show "(x * y) * z = x * (y * z)" by (rule hypreal_mult_assoc)
paulson@14331
   471
  show "x * y = y * x" by (rule hypreal_mult_commute)
paulson@14331
   472
  show "1 * x = x" by (simp add: hypreal_mult_1)
paulson@14331
   473
  show "(x + y) * z = x * z + y * z" by (simp add: hypreal_add_mult_distrib)
paulson@14331
   474
  show "0 \<noteq> (1::hypreal)" by (rule hypreal_zero_not_eq_one)
paulson@14331
   475
  show "x \<noteq> 0 ==> inverse x * x = 1" by (simp add: hypreal_mult_inverse_left)
paulson@14331
   476
  show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: hypreal_divide_def)
paulson@14341
   477
  assume eq: "z+x = z+y" 
paulson@14341
   478
    hence "(-z + z) + x = (-z + z) + y" by (simp only: eq hypreal_add_assoc)
paulson@14341
   479
    thus "x = y" by (simp add: hypreal_add_minus_left)
paulson@14331
   480
qed
paulson@14331
   481
paulson@14331
   482
paulson@14331
   483
lemma HYPREAL_INVERSE_ZERO: "inverse 0 = (0::hypreal)"
paulson@14331
   484
by (simp add: hypreal_inverse hypreal_zero_def)
paulson@14331
   485
paulson@14331
   486
lemma HYPREAL_DIVISION_BY_ZERO: "a / (0::hypreal) = 0"
paulson@14331
   487
by (simp add: hypreal_divide_def HYPREAL_INVERSE_ZERO 
paulson@14331
   488
              hypreal_mult_commute [of a])
paulson@14331
   489
paulson@14331
   490
instance hypreal :: division_by_zero
paulson@14331
   491
proof
paulson@14331
   492
  fix x :: hypreal
paulson@14331
   493
  show "inverse 0 = (0::hypreal)" by (rule HYPREAL_INVERSE_ZERO)
paulson@14331
   494
  show "x/0 = 0" by (rule HYPREAL_DIVISION_BY_ZERO) 
paulson@14331
   495
qed
paulson@14331
   496
paulson@14329
   497
paulson@14329
   498
subsection{*Theorems for Ordering*}
paulson@14329
   499
paulson@14329
   500
text{*TODO: define @{text "\<le>"} as the primitive concept and quickly 
paulson@14329
   501
establish membership in class @{text linorder}. Then proofs could be
paulson@14329
   502
simplified, since properties of @{text "<"} would be generic.*}
paulson@14299
   503
paulson@14329
   504
text{*TODO: The following theorem should be used througout the proofs
paulson@14329
   505
  as it probably makes many of them more straightforward.*}
paulson@14329
   506
lemma hypreal_less: 
paulson@14329
   507
      "(Abs_hypreal(hyprel``{%n. X n}) < Abs_hypreal(hyprel``{%n. Y n})) =  
paulson@14329
   508
       ({n. X n < Y n} \<in> FreeUltrafilterNat)"
paulson@14329
   509
apply (unfold hypreal_less_def)
paulson@14329
   510
apply (auto intro!: lemma_hyprel_refl, ultra)
paulson@14299
   511
done
paulson@14299
   512
paulson@14299
   513
lemma hypreal_less_not_refl: "~ (R::hypreal) < R"
paulson@14301
   514
apply (rule_tac z = R in eq_Abs_hypreal)
paulson@14301
   515
apply (auto simp add: hypreal_less_def, ultra)
paulson@14299
   516
done
paulson@14299
   517
paulson@14299
   518
lemmas hypreal_less_irrefl = hypreal_less_not_refl [THEN notE, standard]
paulson@14299
   519
declare hypreal_less_irrefl [elim!]
paulson@14299
   520
paulson@14299
   521
lemma hypreal_not_refl2: "!!(x::hypreal). x < y ==> x \<noteq> y"
paulson@14301
   522
by (auto simp add: hypreal_less_not_refl)
paulson@14299
   523
paulson@14299
   524
lemma hypreal_less_trans: "!!(R1::hypreal). [| R1 < R2; R2 < R3 |] ==> R1 < R3"
paulson@14301
   525
apply (rule_tac z = R1 in eq_Abs_hypreal)
paulson@14301
   526
apply (rule_tac z = R2 in eq_Abs_hypreal)
paulson@14301
   527
apply (rule_tac z = R3 in eq_Abs_hypreal)
paulson@14301
   528
apply (auto intro!: exI simp add: hypreal_less_def, ultra)
paulson@14299
   529
done
paulson@14299
   530
paulson@14299
   531
lemma hypreal_less_asym: "!! (R1::hypreal). [| R1 < R2; R2 < R1 |] ==> P"
paulson@14301
   532
apply (drule hypreal_less_trans, assumption)
paulson@14299
   533
apply (simp add: hypreal_less_not_refl)
paulson@14299
   534
done
paulson@14299
   535
paulson@14299
   536
paulson@14329
   537
subsection{*Trichotomy: the hyperreals are Linearly Ordered*}
paulson@14299
   538
paulson@14299
   539
lemma lemma_hyprel_0_mem: "\<exists>x. x: hyprel `` {%n. 0}"
paulson@14299
   540
apply (unfold hyprel_def)
paulson@14301
   541
apply (rule_tac x = "%n. 0" in exI, safe)
paulson@14299
   542
apply (auto intro!: FreeUltrafilterNat_Nat_set)
paulson@14299
   543
done
paulson@14299
   544
paulson@14299
   545
lemma hypreal_trichotomy: "0 <  x | x = 0 | x < (0::hypreal)"
paulson@14299
   546
apply (unfold hypreal_zero_def)
paulson@14301
   547
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14299
   548
apply (auto simp add: hypreal_less_def)
paulson@14301
   549
apply (cut_tac lemma_hyprel_0_mem, erule exE)
paulson@14301
   550
apply (drule_tac x = xa in spec)
paulson@14301
   551
apply (drule_tac x = x in spec)
paulson@14301
   552
apply (cut_tac x = x in lemma_hyprel_refl, auto)
paulson@14301
   553
apply (drule_tac x = x in spec)
paulson@14301
   554
apply (drule_tac x = xa in spec, auto, ultra)
paulson@14299
   555
done
paulson@14299
   556
paulson@14299
   557
lemma hypreal_trichotomyE:
paulson@14299
   558
     "[| (0::hypreal) < x ==> P;  
paulson@14299
   559
         x = 0 ==> P;  
paulson@14299
   560
         x < 0 ==> P |] ==> P"
paulson@14301
   561
apply (insert hypreal_trichotomy [of x], blast) 
paulson@14299
   562
done
paulson@14299
   563
paulson@14299
   564
lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
paulson@14301
   565
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14301
   566
apply (rule_tac z = y in eq_Abs_hypreal)
paulson@14299
   567
apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
paulson@14299
   568
done
paulson@14299
   569
paulson@14299
   570
lemma hypreal_less_minus_iff2: "((x::hypreal) < y) = (x + -y < 0)"
paulson@14301
   571
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14301
   572
apply (rule_tac z = y in eq_Abs_hypreal)
paulson@14299
   573
apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
paulson@14299
   574
done
paulson@14299
   575
paulson@14299
   576
lemma hypreal_eq_minus_iff2: "((x::hypreal) = y) = (0 = y + - x)"
paulson@14299
   577
apply auto
paulson@14331
   578
apply (rule Ring_and_Field.add_right_cancel [of _ "-x", THEN iffD1], auto)
paulson@14299
   579
done
paulson@14299
   580
paulson@14299
   581
lemma hypreal_linear: "(x::hypreal) < y | x = y | y < x"
paulson@14299
   582
apply (subst hypreal_eq_minus_iff2)
paulson@14301
   583
apply (rule_tac x1 = x in hypreal_less_minus_iff [THEN ssubst])
paulson@14301
   584
apply (rule_tac x1 = y in hypreal_less_minus_iff2 [THEN ssubst])
paulson@14301
   585
apply (rule hypreal_trichotomyE, auto)
paulson@14299
   586
done
paulson@14299
   587
paulson@14299
   588
lemma hypreal_neq_iff: "((w::hypreal) \<noteq> z) = (w<z | z<w)"
paulson@14301
   589
by (cut_tac hypreal_linear, blast)
paulson@14299
   590
paulson@14299
   591
lemma hypreal_linear_less2: "!!(x::hypreal). [| x < y ==> P;  x = y ==> P;  
paulson@14299
   592
           y < x ==> P |] ==> P"
paulson@14301
   593
apply (cut_tac x = x and y = y in hypreal_linear, auto)
paulson@14299
   594
done
paulson@14299
   595
paulson@14329
   596
paulson@14329
   597
subsection{*Properties of The @{text "\<le>"} Relation*}
paulson@14299
   598
paulson@14299
   599
lemma hypreal_le: 
paulson@14299
   600
      "(Abs_hypreal(hyprel``{%n. X n}) <=  
paulson@14299
   601
            Abs_hypreal(hyprel``{%n. Y n})) =  
paulson@14299
   602
       ({n. X n <= Y n} \<in> FreeUltrafilterNat)"
paulson@14299
   603
apply (unfold hypreal_le_def real_le_def)
paulson@14299
   604
apply (auto simp add: hypreal_less)
paulson@14299
   605
apply (ultra+)
paulson@14299
   606
done
paulson@14299
   607
paulson@14299
   608
lemma hypreal_le_imp_less_or_eq: "!!(x::hypreal). x <= y ==> x < y | x = y"
paulson@14299
   609
apply (unfold hypreal_le_def)
paulson@14299
   610
apply (cut_tac hypreal_linear)
paulson@14299
   611
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
paulson@14299
   612
done
paulson@14299
   613
paulson@14299
   614
lemma hypreal_less_or_eq_imp_le: "z<w | z=w ==> z <=(w::hypreal)"
paulson@14299
   615
apply (unfold hypreal_le_def)
paulson@14299
   616
apply (cut_tac hypreal_linear)
paulson@14299
   617
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
paulson@14299
   618
done
paulson@14299
   619
paulson@14299
   620
lemma hypreal_le_eq_less_or_eq: "(x <= (y::hypreal)) = (x < y | x=y)"
paulson@14299
   621
by (blast intro!: hypreal_less_or_eq_imp_le dest: hypreal_le_imp_less_or_eq) 
paulson@14299
   622
paulson@14299
   623
lemmas hypreal_le_less = hypreal_le_eq_less_or_eq
paulson@14299
   624
paulson@14299
   625
lemma hypreal_le_refl: "w <= (w::hypreal)"
paulson@14301
   626
by (simp add: hypreal_le_eq_less_or_eq)
paulson@14299
   627
paulson@14299
   628
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14299
   629
lemma hypreal_le_linear: "(z::hypreal) <= w | w <= z"
paulson@14301
   630
apply (simp add: hypreal_le_less)
paulson@14301
   631
apply (cut_tac hypreal_linear, blast)
paulson@14299
   632
done
paulson@14299
   633
paulson@14299
   634
lemma hypreal_le_trans: "[| i <= j; j <= k |] ==> i <= (k::hypreal)"
paulson@14299
   635
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   636
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   637
apply (rule hypreal_less_or_eq_imp_le) 
paulson@14299
   638
apply (blast intro: hypreal_less_trans) 
paulson@14299
   639
done
paulson@14299
   640
paulson@14299
   641
lemma hypreal_le_anti_sym: "[| z <= w; w <= z |] ==> z = (w::hypreal)"
paulson@14299
   642
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   643
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   644
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
paulson@14299
   645
done
paulson@14299
   646
paulson@14299
   647
(* Axiom 'order_less_le' of class 'order': *)
paulson@14299
   648
lemma hypreal_less_le: "((w::hypreal) < z) = (w <= z & w \<noteq> z)"
paulson@14301
   649
apply (simp add: hypreal_le_def hypreal_neq_iff)
paulson@14299
   650
apply (blast intro: hypreal_less_asym)
paulson@14299
   651
done
paulson@14299
   652
paulson@14329
   653
instance hypreal :: order
paulson@14329
   654
  by (intro_classes,
paulson@14329
   655
      (assumption | 
paulson@14329
   656
       rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym
paulson@14329
   657
            hypreal_less_le)+)
paulson@14329
   658
paulson@14329
   659
instance hypreal :: linorder 
paulson@14329
   660
  by (intro_classes, rule hypreal_le_linear)
paulson@14329
   661
paulson@14329
   662
paulson@14329
   663
lemma hypreal_add_less_mono1: "(A::hypreal) < B ==> A + C < B + C"
paulson@14329
   664
apply (rule_tac z = A in eq_Abs_hypreal)
paulson@14329
   665
apply (rule_tac z = B in eq_Abs_hypreal)
paulson@14329
   666
apply (rule_tac z = C in eq_Abs_hypreal)
paulson@14329
   667
apply (auto intro!: exI simp add: hypreal_less_def hypreal_add, ultra)
paulson@14329
   668
done
paulson@14329
   669
paulson@14329
   670
lemma hypreal_mult_order: "[| 0 < x; 0 < y |] ==> (0::hypreal) < x * y"
paulson@14329
   671
apply (unfold hypreal_zero_def)
paulson@14329
   672
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14329
   673
apply (rule_tac z = y in eq_Abs_hypreal)
paulson@14329
   674
apply (auto intro!: exI simp add: hypreal_less_def hypreal_mult, ultra)
paulson@14329
   675
apply (auto intro: real_mult_order)
paulson@14329
   676
done
paulson@14329
   677
paulson@14329
   678
lemma hypreal_add_left_le_mono1: "(q1::hypreal) \<le> q2  ==> x + q1 \<le> x + q2"
paulson@14329
   679
apply (drule order_le_imp_less_or_eq)
paulson@14329
   680
apply (auto intro: order_less_imp_le hypreal_add_less_mono1 simp add: hypreal_add_commute)
paulson@14329
   681
done
paulson@14329
   682
paulson@14329
   683
lemma hypreal_mult_less_mono1: "[| (0::hypreal) < z; x < y |] ==> x*z < y*z"
paulson@14329
   684
apply (rotate_tac 1)
paulson@14329
   685
apply (drule hypreal_less_minus_iff [THEN iffD1])
paulson@14329
   686
apply (rule hypreal_less_minus_iff [THEN iffD2])
paulson@14329
   687
apply (drule hypreal_mult_order, assumption)
paulson@14331
   688
apply (simp add: right_distrib hypreal_mult_commute)
paulson@14329
   689
done
paulson@14329
   690
paulson@14329
   691
lemma hypreal_mult_less_mono2: "[| (0::hypreal)<z; x<y |] ==> z*x<z*y"
paulson@14329
   692
apply (simp (no_asm_simp) add: hypreal_mult_commute hypreal_mult_less_mono1)
paulson@14329
   693
done
paulson@14329
   694
paulson@14329
   695
subsection{*The Hyperreals Form an Ordered Field*}
paulson@14329
   696
paulson@14329
   697
instance hypreal :: ordered_field
paulson@14329
   698
proof
paulson@14329
   699
  fix x y z :: hypreal
paulson@14348
   700
  show "0 < (1::hypreal)" 
paulson@14348
   701
    by (unfold hypreal_one_def hypreal_zero_def hypreal_less_def, force)
paulson@14348
   702
  show "x \<le> y ==> z + x \<le> z + y" 
paulson@14348
   703
    by (rule hypreal_add_left_le_mono1)
paulson@14348
   704
  show "x < y ==> 0 < z ==> z * x < z * y" 
paulson@14348
   705
    by (simp add: hypreal_mult_less_mono2)
paulson@14329
   706
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
paulson@14329
   707
    by (auto dest: order_le_less_trans simp add: hrabs_def linorder_not_le)
paulson@14329
   708
qed
paulson@14329
   709
paulson@14331
   710
lemma hypreal_mult_1_right: "z * (1::hypreal) = z"
paulson@14331
   711
  by (rule Ring_and_Field.mult_1_right)
paulson@14331
   712
paulson@14331
   713
lemma hypreal_mult_minus_1 [simp]: "(- (1::hypreal)) * z = -z"
paulson@14331
   714
by (simp)
paulson@14331
   715
paulson@14331
   716
lemma hypreal_mult_minus_1_right [simp]: "z * (- (1::hypreal)) = -z"
paulson@14331
   717
by (subst hypreal_mult_commute, simp)
paulson@14329
   718
paulson@14329
   719
(*Used ONCE: in NSA.ML*)
paulson@14329
   720
lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
paulson@14329
   721
by (simp add: hypreal_add_commute)
paulson@14329
   722
paulson@14329
   723
(*Used ONCE: in Lim.ML*)
paulson@14329
   724
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
paulson@14329
   725
by (auto simp add: hypreal_add_assoc)
paulson@14329
   726
paulson@14331
   727
lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)"
paulson@14331
   728
apply auto
paulson@14331
   729
apply (rule Ring_and_Field.add_right_cancel [of _ "-y", THEN iffD1], auto)
paulson@14331
   730
done
paulson@14331
   731
paulson@14331
   732
(*Used 3 TIMES: in Lim.ML*)
paulson@14329
   733
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
paulson@14329
   734
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
paulson@14329
   735
paulson@14329
   736
lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14329
   737
apply auto
paulson@14329
   738
done
paulson@14329
   739
    
paulson@14329
   740
lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14329
   741
apply auto
paulson@14329
   742
done
paulson@14329
   743
paulson@14329
   744
lemma hypreal_inverse_not_zero: "x \<noteq> 0 ==> inverse (x::hypreal) \<noteq> 0"
paulson@14329
   745
  by (rule Ring_and_Field.nonzero_imp_inverse_nonzero)
paulson@14329
   746
paulson@14329
   747
lemma hypreal_mult_not_0: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::hypreal)"
paulson@14329
   748
by simp
paulson@14329
   749
paulson@14329
   750
lemma hypreal_minus_inverse: "inverse(-x) = -inverse(x::hypreal)"
paulson@14329
   751
  by (rule Ring_and_Field.inverse_minus_eq)
paulson@14329
   752
paulson@14329
   753
lemma hypreal_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::hypreal)"
paulson@14329
   754
  by (rule Ring_and_Field.inverse_mult_distrib)
paulson@14329
   755
paulson@14329
   756
paulson@14329
   757
subsection{* Division lemmas *}
paulson@14329
   758
paulson@14329
   759
lemma hypreal_divide_one: "x/(1::hypreal) = x"
paulson@14329
   760
by (simp add: hypreal_divide_def)
paulson@14329
   761
paulson@14329
   762
paulson@14329
   763
(** As with multiplication, pull minus signs OUT of the / operator **)
paulson@14329
   764
paulson@14329
   765
lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
paulson@14329
   766
  by (rule Ring_and_Field.add_divide_distrib)
paulson@14329
   767
paulson@14329
   768
lemma hypreal_inverse_add:
paulson@14329
   769
     "[|(x::hypreal) \<noteq> 0;  y \<noteq> 0 |]   
paulson@14329
   770
      ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)"
paulson@14329
   771
by (simp add: Ring_and_Field.inverse_add mult_assoc)
paulson@14329
   772
paulson@14329
   773
paulson@14329
   774
subsection{*@{term hypreal_of_real} Preserves Field and Order Properties*}
paulson@14329
   775
paulson@14301
   776
lemma hypreal_of_real_add [simp]: 
paulson@14299
   777
     "hypreal_of_real (z1 + z2) = hypreal_of_real z1 + hypreal_of_real z2"
paulson@14299
   778
apply (unfold hypreal_of_real_def)
paulson@14331
   779
apply (simp add: hypreal_add left_distrib)
paulson@14299
   780
done
paulson@14299
   781
paulson@14301
   782
lemma hypreal_of_real_mult [simp]: 
paulson@14299
   783
     "hypreal_of_real (z1 * z2) = hypreal_of_real z1 * hypreal_of_real z2"
paulson@14299
   784
apply (unfold hypreal_of_real_def)
paulson@14331
   785
apply (simp add: hypreal_mult right_distrib)
paulson@14299
   786
done
paulson@14299
   787
paulson@14301
   788
lemma hypreal_of_real_less_iff [simp]: 
paulson@14299
   789
     "(hypreal_of_real z1 <  hypreal_of_real z2) = (z1 < z2)"
paulson@14301
   790
apply (unfold hypreal_less_def hypreal_of_real_def, auto)
paulson@14301
   791
apply (rule_tac [2] x = "%n. z1" in exI, safe)
paulson@14301
   792
apply (rule_tac [3] x = "%n. z2" in exI, auto)
paulson@14301
   793
apply (rule FreeUltrafilterNat_P, ultra)
paulson@14299
   794
done
paulson@14299
   795
paulson@14301
   796
lemma hypreal_of_real_le_iff [simp]: 
paulson@14299
   797
     "(hypreal_of_real z1 <= hypreal_of_real z2) = (z1 <= z2)"
paulson@14301
   798
apply (unfold hypreal_le_def real_le_def, auto)
paulson@14299
   799
done
paulson@14299
   800
paulson@14329
   801
lemma hypreal_of_real_eq_iff [simp]:
paulson@14329
   802
     "(hypreal_of_real z1 = hypreal_of_real z2) = (z1 = z2)"
paulson@14301
   803
by (force intro: order_antisym hypreal_of_real_le_iff [THEN iffD1])
paulson@14299
   804
paulson@14329
   805
lemma hypreal_of_real_minus [simp]:
paulson@14329
   806
     "hypreal_of_real (-r) = - hypreal_of_real  r"
paulson@14299
   807
apply (unfold hypreal_of_real_def)
paulson@14299
   808
apply (auto simp add: hypreal_minus)
paulson@14299
   809
done
paulson@14299
   810
paulson@14301
   811
lemma hypreal_of_real_one [simp]: "hypreal_of_real 1 = (1::hypreal)"
paulson@14301
   812
by (unfold hypreal_of_real_def hypreal_one_def, simp)
paulson@14299
   813
paulson@14301
   814
lemma hypreal_of_real_zero [simp]: "hypreal_of_real 0 = 0"
paulson@14301
   815
by (unfold hypreal_of_real_def hypreal_zero_def, simp)
paulson@14299
   816
paulson@14299
   817
lemma hypreal_of_real_zero_iff: "(hypreal_of_real r = 0) = (r = 0)"
paulson@14301
   818
by (auto intro: FreeUltrafilterNat_P simp add: hypreal_of_real_def hypreal_zero_def FreeUltrafilterNat_Nat_set)
paulson@14299
   819
paulson@14329
   820
lemma hypreal_of_real_inverse [simp]:
paulson@14329
   821
     "hypreal_of_real (inverse r) = inverse (hypreal_of_real r)"
paulson@14299
   822
apply (case_tac "r=0")
paulson@14301
   823
apply (simp add: DIVISION_BY_ZERO INVERSE_ZERO HYPREAL_INVERSE_ZERO)
paulson@14299
   824
apply (rule_tac c1 = "hypreal_of_real r" in hypreal_mult_left_cancel [THEN iffD1])
paulson@14299
   825
apply (auto simp add: hypreal_of_real_zero_iff hypreal_of_real_mult [symmetric])
paulson@14299
   826
done
paulson@14299
   827
paulson@14329
   828
lemma hypreal_of_real_divide [simp]:
paulson@14329
   829
     "hypreal_of_real (z1 / z2) = hypreal_of_real z1 / hypreal_of_real z2"
paulson@14301
   830
by (simp add: hypreal_divide_def real_divide_def)
paulson@14299
   831
paulson@14299
   832
paulson@14329
   833
subsection{*Misc Others*}
paulson@14299
   834
paulson@14299
   835
lemma hypreal_zero_num: "0 = Abs_hypreal (hyprel `` {%n. 0})"
paulson@14301
   836
by (simp add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric])
paulson@14299
   837
paulson@14299
   838
lemma hypreal_one_num: "1 = Abs_hypreal (hyprel `` {%n. 1})"
paulson@14301
   839
by (simp add: hypreal_one_def [THEN meta_eq_to_obj_eq, symmetric])
paulson@14299
   840
paulson@14301
   841
lemma hypreal_omega_gt_zero [simp]: "0 < omega"
paulson@14299
   842
apply (unfold omega_def)
paulson@14299
   843
apply (auto simp add: hypreal_less hypreal_zero_num)
paulson@14299
   844
done
paulson@14299
   845
paulson@14329
   846
paulson@14329
   847
lemma hypreal_hrabs:
paulson@14329
   848
     "abs (Abs_hypreal (hyprel `` {X})) = 
paulson@14329
   849
      Abs_hypreal(hyprel `` {%n. abs (X n)})"
paulson@14329
   850
apply (auto simp add: hrabs_def hypreal_zero_def hypreal_le hypreal_minus)
paulson@14329
   851
apply (ultra, arith)+
paulson@14329
   852
done
paulson@14329
   853
paulson@14299
   854
ML
paulson@14299
   855
{*
paulson@14329
   856
val hrabs_def = thm "hrabs_def";
paulson@14329
   857
val hypreal_hrabs = thm "hypreal_hrabs";
paulson@14329
   858
paulson@14299
   859
val hypreal_zero_def = thm "hypreal_zero_def";
paulson@14299
   860
val hypreal_one_def = thm "hypreal_one_def";
paulson@14299
   861
val hypreal_minus_def = thm "hypreal_minus_def";
paulson@14299
   862
val hypreal_diff_def = thm "hypreal_diff_def";
paulson@14299
   863
val hypreal_inverse_def = thm "hypreal_inverse_def";
paulson@14299
   864
val hypreal_divide_def = thm "hypreal_divide_def";
paulson@14299
   865
val hypreal_of_real_def = thm "hypreal_of_real_def";
paulson@14299
   866
val omega_def = thm "omega_def";
paulson@14299
   867
val epsilon_def = thm "epsilon_def";
paulson@14299
   868
val hypreal_add_def = thm "hypreal_add_def";
paulson@14299
   869
val hypreal_mult_def = thm "hypreal_mult_def";
paulson@14299
   870
val hypreal_less_def = thm "hypreal_less_def";
paulson@14299
   871
val hypreal_le_def = thm "hypreal_le_def";
paulson@14299
   872
paulson@14299
   873
val finite_exhausts = thm "finite_exhausts";
paulson@14299
   874
val finite_not_covers = thm "finite_not_covers";
paulson@14299
   875
val not_finite_nat = thm "not_finite_nat";
paulson@14299
   876
val FreeUltrafilterNat_Ex = thm "FreeUltrafilterNat_Ex";
paulson@14299
   877
val FreeUltrafilterNat_mem = thm "FreeUltrafilterNat_mem";
paulson@14299
   878
val FreeUltrafilterNat_finite = thm "FreeUltrafilterNat_finite";
paulson@14299
   879
val FreeUltrafilterNat_not_finite = thm "FreeUltrafilterNat_not_finite";
paulson@14299
   880
val FreeUltrafilterNat_empty = thm "FreeUltrafilterNat_empty";
paulson@14299
   881
val FreeUltrafilterNat_Int = thm "FreeUltrafilterNat_Int";
paulson@14299
   882
val FreeUltrafilterNat_subset = thm "FreeUltrafilterNat_subset";
paulson@14299
   883
val FreeUltrafilterNat_Compl = thm "FreeUltrafilterNat_Compl";
paulson@14299
   884
val FreeUltrafilterNat_Compl_mem = thm "FreeUltrafilterNat_Compl_mem";
paulson@14299
   885
val FreeUltrafilterNat_Compl_iff1 = thm "FreeUltrafilterNat_Compl_iff1";
paulson@14299
   886
val FreeUltrafilterNat_Compl_iff2 = thm "FreeUltrafilterNat_Compl_iff2";
paulson@14299
   887
val FreeUltrafilterNat_UNIV = thm "FreeUltrafilterNat_UNIV";
paulson@14299
   888
val FreeUltrafilterNat_Nat_set = thm "FreeUltrafilterNat_Nat_set";
paulson@14299
   889
val FreeUltrafilterNat_Nat_set_refl = thm "FreeUltrafilterNat_Nat_set_refl";
paulson@14299
   890
val FreeUltrafilterNat_P = thm "FreeUltrafilterNat_P";
paulson@14299
   891
val FreeUltrafilterNat_Ex_P = thm "FreeUltrafilterNat_Ex_P";
paulson@14299
   892
val FreeUltrafilterNat_all = thm "FreeUltrafilterNat_all";
paulson@14299
   893
val FreeUltrafilterNat_Un = thm "FreeUltrafilterNat_Un";
paulson@14299
   894
val hyprel_iff = thm "hyprel_iff";
paulson@14299
   895
val hyprel_refl = thm "hyprel_refl";
paulson@14299
   896
val hyprel_sym = thm "hyprel_sym";
paulson@14299
   897
val hyprel_trans = thm "hyprel_trans";
paulson@14299
   898
val equiv_hyprel = thm "equiv_hyprel";
paulson@14299
   899
val hyprel_in_hypreal = thm "hyprel_in_hypreal";
paulson@14299
   900
val Abs_hypreal_inverse = thm "Abs_hypreal_inverse";
paulson@14299
   901
val inj_on_Abs_hypreal = thm "inj_on_Abs_hypreal";
paulson@14299
   902
val inj_Rep_hypreal = thm "inj_Rep_hypreal";
paulson@14299
   903
val lemma_hyprel_refl = thm "lemma_hyprel_refl";
paulson@14299
   904
val hypreal_empty_not_mem = thm "hypreal_empty_not_mem";
paulson@14299
   905
val Rep_hypreal_nonempty = thm "Rep_hypreal_nonempty";
paulson@14299
   906
val inj_hypreal_of_real = thm "inj_hypreal_of_real";
paulson@14299
   907
val eq_Abs_hypreal = thm "eq_Abs_hypreal";
paulson@14299
   908
val hypreal_minus_congruent = thm "hypreal_minus_congruent";
paulson@14299
   909
val hypreal_minus = thm "hypreal_minus";
paulson@14299
   910
val hypreal_add_congruent2 = thm "hypreal_add_congruent2";
paulson@14299
   911
val hypreal_add = thm "hypreal_add";
paulson@14299
   912
val hypreal_diff = thm "hypreal_diff";
paulson@14299
   913
val hypreal_add_commute = thm "hypreal_add_commute";
paulson@14299
   914
val hypreal_add_assoc = thm "hypreal_add_assoc";
paulson@14299
   915
val hypreal_add_zero_left = thm "hypreal_add_zero_left";
paulson@14299
   916
val hypreal_add_zero_right = thm "hypreal_add_zero_right";
paulson@14299
   917
val hypreal_add_minus = thm "hypreal_add_minus";
paulson@14299
   918
val hypreal_add_minus_left = thm "hypreal_add_minus_left";
paulson@14299
   919
val hypreal_minus_distrib1 = thm "hypreal_minus_distrib1";
paulson@14299
   920
val hypreal_mult_congruent2 = thm "hypreal_mult_congruent2";
paulson@14299
   921
val hypreal_mult = thm "hypreal_mult";
paulson@14299
   922
val hypreal_mult_commute = thm "hypreal_mult_commute";
paulson@14299
   923
val hypreal_mult_assoc = thm "hypreal_mult_assoc";
paulson@14299
   924
val hypreal_mult_1 = thm "hypreal_mult_1";
paulson@14299
   925
val hypreal_mult_1_right = thm "hypreal_mult_1_right";
paulson@14299
   926
val hypreal_mult_minus_1 = thm "hypreal_mult_minus_1";
paulson@14299
   927
val hypreal_mult_minus_1_right = thm "hypreal_mult_minus_1_right";
paulson@14299
   928
val hypreal_zero_not_eq_one = thm "hypreal_zero_not_eq_one";
paulson@14299
   929
val hypreal_inverse_congruent = thm "hypreal_inverse_congruent";
paulson@14299
   930
val hypreal_inverse = thm "hypreal_inverse";
paulson@14299
   931
val HYPREAL_INVERSE_ZERO = thm "HYPREAL_INVERSE_ZERO";
paulson@14299
   932
val HYPREAL_DIVISION_BY_ZERO = thm "HYPREAL_DIVISION_BY_ZERO";
paulson@14299
   933
val hypreal_mult_inverse = thm "hypreal_mult_inverse";
paulson@14299
   934
val hypreal_mult_inverse_left = thm "hypreal_mult_inverse_left";
paulson@14299
   935
val hypreal_mult_left_cancel = thm "hypreal_mult_left_cancel";
paulson@14299
   936
val hypreal_mult_right_cancel = thm "hypreal_mult_right_cancel";
paulson@14299
   937
val hypreal_inverse_not_zero = thm "hypreal_inverse_not_zero";
paulson@14299
   938
val hypreal_mult_not_0 = thm "hypreal_mult_not_0";
paulson@14299
   939
val hypreal_minus_inverse = thm "hypreal_minus_inverse";
paulson@14299
   940
val hypreal_inverse_distrib = thm "hypreal_inverse_distrib";
paulson@14299
   941
val hypreal_less_not_refl = thm "hypreal_less_not_refl";
paulson@14331
   942
val hypreal_less_irrefl = thm"hypreal_less_irrefl";
paulson@14299
   943
val hypreal_not_refl2 = thm "hypreal_not_refl2";
paulson@14299
   944
val hypreal_less_trans = thm "hypreal_less_trans";
paulson@14299
   945
val hypreal_less_asym = thm "hypreal_less_asym";
paulson@14299
   946
val hypreal_less = thm "hypreal_less";
paulson@14299
   947
val hypreal_trichotomy = thm "hypreal_trichotomy";
paulson@14299
   948
val hypreal_less_minus_iff = thm "hypreal_less_minus_iff";
paulson@14299
   949
val hypreal_less_minus_iff2 = thm "hypreal_less_minus_iff2";
paulson@14299
   950
val hypreal_eq_minus_iff = thm "hypreal_eq_minus_iff";
paulson@14299
   951
val hypreal_eq_minus_iff2 = thm "hypreal_eq_minus_iff2";
paulson@14299
   952
val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
paulson@14299
   953
val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
paulson@14299
   954
val hypreal_linear = thm "hypreal_linear";
paulson@14299
   955
val hypreal_neq_iff = thm "hypreal_neq_iff";
paulson@14299
   956
val hypreal_linear_less2 = thm "hypreal_linear_less2";
paulson@14299
   957
val hypreal_le = thm "hypreal_le";
paulson@14299
   958
val hypreal_le_imp_less_or_eq = thm "hypreal_le_imp_less_or_eq";
paulson@14299
   959
val hypreal_le_eq_less_or_eq = thm "hypreal_le_eq_less_or_eq";
paulson@14299
   960
val hypreal_le_refl = thm "hypreal_le_refl";
paulson@14299
   961
val hypreal_le_linear = thm "hypreal_le_linear";
paulson@14299
   962
val hypreal_le_trans = thm "hypreal_le_trans";
paulson@14299
   963
val hypreal_le_anti_sym = thm "hypreal_le_anti_sym";
paulson@14299
   964
val hypreal_less_le = thm "hypreal_less_le";
paulson@14299
   965
val hypreal_of_real_add = thm "hypreal_of_real_add";
paulson@14299
   966
val hypreal_of_real_mult = thm "hypreal_of_real_mult";
paulson@14299
   967
val hypreal_of_real_less_iff = thm "hypreal_of_real_less_iff";
paulson@14299
   968
val hypreal_of_real_le_iff = thm "hypreal_of_real_le_iff";
paulson@14299
   969
val hypreal_of_real_eq_iff = thm "hypreal_of_real_eq_iff";
paulson@14299
   970
val hypreal_of_real_minus = thm "hypreal_of_real_minus";
paulson@14299
   971
val hypreal_of_real_one = thm "hypreal_of_real_one";
paulson@14299
   972
val hypreal_of_real_zero = thm "hypreal_of_real_zero";
paulson@14299
   973
val hypreal_of_real_zero_iff = thm "hypreal_of_real_zero_iff";
paulson@14299
   974
val hypreal_of_real_inverse = thm "hypreal_of_real_inverse";
paulson@14299
   975
val hypreal_of_real_divide = thm "hypreal_of_real_divide";
paulson@14299
   976
val hypreal_divide_one = thm "hypreal_divide_one";
paulson@14299
   977
val hypreal_add_divide_distrib = thm "hypreal_add_divide_distrib";
paulson@14299
   978
val hypreal_inverse_add = thm "hypreal_inverse_add";
paulson@14299
   979
val hypreal_zero_num = thm "hypreal_zero_num";
paulson@14299
   980
val hypreal_one_num = thm "hypreal_one_num";
paulson@14299
   981
val hypreal_omega_gt_zero = thm "hypreal_omega_gt_zero";
paulson@14299
   982
*}
paulson@14299
   983
paulson@10751
   984
end