src/HOL/Hyperreal/HyperDef.thy
author paulson
Wed Jan 28 17:01:01 2004 +0100 (2004-01-28)
changeset 14369 c50188fe6366
parent 14365 3d4df8c166ae
child 14370 b0064703967b
permissions -rw-r--r--
tidying up arithmetic for the hyperreals
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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 \<le> (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 \<subseteq> 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) \<in> 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) \<in> 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) \<in> hyprel --> (y,x) \<in> 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) \<in> hyprel; (y,z) \<in> hyprel|] ==> (x,z) \<in> hyprel"
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by (unfold hyprel_def, auto, ultra)
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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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apply (simp add: add_ac hypreal_add)
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done
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paulson@14329
   331
lemma hypreal_add_assoc: "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)"
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   332
apply (rule_tac z = z1 in eq_Abs_hypreal)
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   333
apply (rule_tac z = z2 in eq_Abs_hypreal)
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   334
apply (rule_tac z = z3 in eq_Abs_hypreal)
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   335
apply (simp add: hypreal_add real_add_assoc)
paulson@14329
   336
done
paulson@14329
   337
paulson@14331
   338
lemma hypreal_add_zero_left: "(0::hypreal) + z = z"
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   339
apply (unfold hypreal_zero_def)
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   340
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14329
   341
apply (simp add: hypreal_add)
paulson@14329
   342
done
paulson@14329
   343
paulson@14329
   344
instance hypreal :: plus_ac0
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   345
  by (intro_classes,
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   346
      (assumption | 
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   347
       rule hypreal_add_commute hypreal_add_assoc hypreal_add_zero_left)+)
paulson@14329
   348
paulson@14329
   349
lemma hypreal_add_zero_right [simp]: "z + (0::hypreal) = z"
paulson@14329
   350
by (simp add: hypreal_add_zero_left hypreal_add_commute)
paulson@14329
   351
paulson@14329
   352
paulson@14329
   353
subsection{*Additive inverse on @{typ hypreal}*}
paulson@14299
   354
paulson@14299
   355
lemma hypreal_minus_congruent: 
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   356
  "congruent hyprel (%X. hyprel``{%n. - (X n)})"
paulson@14299
   357
by (force simp add: congruent_def)
paulson@14299
   358
paulson@14299
   359
lemma hypreal_minus: 
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   360
   "- (Abs_hypreal(hyprel``{%n. X n})) = Abs_hypreal(hyprel `` {%n. -(X n)})"
paulson@14299
   361
apply (unfold hypreal_minus_def)
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   362
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14301
   363
apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
paulson@14299
   364
               UN_equiv_class [OF equiv_hyprel hypreal_minus_congruent])
paulson@14299
   365
done
paulson@14299
   366
paulson@14329
   367
lemma hypreal_diff:
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   368
     "Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) =  
paulson@14299
   369
      Abs_hypreal(hyprel``{%n. X n - Y n})"
paulson@14301
   370
apply (simp add: hypreal_diff_def hypreal_minus hypreal_add)
paulson@14299
   371
done
paulson@14299
   372
paulson@14301
   373
lemma hypreal_add_minus [simp]: "z + -z = (0::hypreal)"
paulson@14299
   374
apply (unfold hypreal_zero_def)
paulson@14301
   375
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14299
   376
apply (simp add: hypreal_minus hypreal_add)
paulson@14299
   377
done
paulson@14299
   378
paulson@14331
   379
lemma hypreal_add_minus_left: "-z + z = (0::hypreal)"
paulson@14301
   380
by (simp add: hypreal_add_commute hypreal_add_minus)
paulson@14299
   381
paulson@14329
   382
paulson@14329
   383
subsection{*Hyperreal Multiplication*}
paulson@14299
   384
paulson@14299
   385
lemma hypreal_mult_congruent2: 
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   386
    "congruent2 hyprel (%X Y. hyprel``{%n. X n * Y n})"
paulson@14301
   387
apply (unfold congruent2_def, auto, ultra)
paulson@14299
   388
done
paulson@14299
   389
paulson@14299
   390
lemma hypreal_mult: 
paulson@14299
   391
  "Abs_hypreal(hyprel``{%n. X n}) * Abs_hypreal(hyprel``{%n. Y n}) =  
paulson@14299
   392
   Abs_hypreal(hyprel``{%n. X n * Y n})"
paulson@14299
   393
apply (unfold hypreal_mult_def)
paulson@14299
   394
apply (simp add: UN_equiv_class2 [OF equiv_hyprel hypreal_mult_congruent2])
paulson@14299
   395
done
paulson@14299
   396
paulson@14299
   397
lemma hypreal_mult_commute: "(z::hypreal) * w = w * z"
paulson@14301
   398
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14301
   399
apply (rule_tac z = w in eq_Abs_hypreal)
paulson@14331
   400
apply (simp add: hypreal_mult mult_ac)
paulson@14299
   401
done
paulson@14299
   402
paulson@14299
   403
lemma hypreal_mult_assoc: "((z1::hypreal) * z2) * z3 = z1 * (z2 * z3)"
paulson@14301
   404
apply (rule_tac z = z1 in eq_Abs_hypreal)
paulson@14301
   405
apply (rule_tac z = z2 in eq_Abs_hypreal)
paulson@14301
   406
apply (rule_tac z = z3 in eq_Abs_hypreal)
paulson@14331
   407
apply (simp add: hypreal_mult mult_assoc)
paulson@14299
   408
done
paulson@14299
   409
paulson@14331
   410
lemma hypreal_mult_1: "(1::hypreal) * z = z"
paulson@14299
   411
apply (unfold hypreal_one_def)
paulson@14301
   412
apply (rule_tac z = z in eq_Abs_hypreal)
paulson@14299
   413
apply (simp add: hypreal_mult)
paulson@14299
   414
done
paulson@14301
   415
paulson@14329
   416
lemma hypreal_add_mult_distrib:
paulson@14329
   417
     "((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)"
paulson@14301
   418
apply (rule_tac z = z1 in eq_Abs_hypreal)
paulson@14301
   419
apply (rule_tac z = z2 in eq_Abs_hypreal)
paulson@14301
   420
apply (rule_tac z = w in eq_Abs_hypreal)
paulson@14334
   421
apply (simp add: hypreal_mult hypreal_add left_distrib)
paulson@14299
   422
done
paulson@14299
   423
paulson@14331
   424
text{*one and zero are distinct*}
paulson@14299
   425
lemma hypreal_zero_not_eq_one: "0 \<noteq> (1::hypreal)"
paulson@14299
   426
apply (unfold hypreal_zero_def hypreal_one_def)
paulson@14299
   427
apply (auto simp add: real_zero_not_eq_one)
paulson@14299
   428
done
paulson@14299
   429
paulson@14299
   430
paulson@14329
   431
subsection{*Multiplicative Inverse on @{typ hypreal} *}
paulson@14299
   432
paulson@14299
   433
lemma hypreal_inverse_congruent: 
paulson@14299
   434
  "congruent hyprel (%X. hyprel``{%n. if X n = 0 then 0 else inverse(X n)})"
paulson@14299
   435
apply (unfold congruent_def)
paulson@14301
   436
apply (auto, ultra)
paulson@14299
   437
done
paulson@14299
   438
paulson@14299
   439
lemma hypreal_inverse: 
paulson@14299
   440
      "inverse (Abs_hypreal(hyprel``{%n. X n})) =  
paulson@14299
   441
       Abs_hypreal(hyprel `` {%n. if X n = 0 then 0 else inverse(X n)})"
paulson@14299
   442
apply (unfold hypreal_inverse_def)
paulson@14301
   443
apply (rule_tac f = Abs_hypreal in arg_cong)
paulson@14301
   444
apply (simp add: hyprel_in_hypreal [THEN Abs_hypreal_inverse] 
paulson@14299
   445
           UN_equiv_class [OF equiv_hyprel hypreal_inverse_congruent])
paulson@14299
   446
done
paulson@14299
   447
paulson@14331
   448
lemma hypreal_mult_inverse: 
paulson@14299
   449
     "x \<noteq> 0 ==> x*inverse(x) = (1::hypreal)"
paulson@14299
   450
apply (unfold hypreal_one_def hypreal_zero_def)
paulson@14301
   451
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14299
   452
apply (simp add: hypreal_inverse hypreal_mult)
paulson@14299
   453
apply (drule FreeUltrafilterNat_Compl_mem)
paulson@14334
   454
apply (blast intro!: right_inverse FreeUltrafilterNat_subset)
paulson@14299
   455
done
paulson@14299
   456
paulson@14331
   457
lemma hypreal_mult_inverse_left:
paulson@14329
   458
     "x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)"
paulson@14301
   459
by (simp add: hypreal_mult_inverse hypreal_mult_commute)
paulson@14299
   460
paulson@14331
   461
instance hypreal :: field
paulson@14331
   462
proof
paulson@14331
   463
  fix x y z :: hypreal
paulson@14331
   464
  show "(x + y) + z = x + (y + z)" by (rule hypreal_add_assoc)
paulson@14331
   465
  show "x + y = y + x" by (rule hypreal_add_commute)
paulson@14331
   466
  show "0 + x = x" by simp
paulson@14331
   467
  show "- x + x = 0" by (simp add: hypreal_add_minus_left)
paulson@14331
   468
  show "x - y = x + (-y)" by (simp add: hypreal_diff_def)
paulson@14331
   469
  show "(x * y) * z = x * (y * z)" by (rule hypreal_mult_assoc)
paulson@14331
   470
  show "x * y = y * x" by (rule hypreal_mult_commute)
paulson@14331
   471
  show "1 * x = x" by (simp add: hypreal_mult_1)
paulson@14331
   472
  show "(x + y) * z = x * z + y * z" by (simp add: hypreal_add_mult_distrib)
paulson@14331
   473
  show "0 \<noteq> (1::hypreal)" by (rule hypreal_zero_not_eq_one)
paulson@14331
   474
  show "x \<noteq> 0 ==> inverse x * x = 1" by (simp add: hypreal_mult_inverse_left)
paulson@14331
   475
  show "y \<noteq> 0 ==> x / y = x * inverse y" by (simp add: hypreal_divide_def)
paulson@14341
   476
  assume eq: "z+x = z+y" 
paulson@14341
   477
    hence "(-z + z) + x = (-z + z) + y" by (simp only: eq hypreal_add_assoc)
paulson@14341
   478
    thus "x = y" by (simp add: hypreal_add_minus_left)
paulson@14331
   479
qed
paulson@14331
   480
paulson@14331
   481
paulson@14331
   482
lemma HYPREAL_INVERSE_ZERO: "inverse 0 = (0::hypreal)"
paulson@14331
   483
by (simp add: hypreal_inverse hypreal_zero_def)
paulson@14331
   484
paulson@14331
   485
lemma HYPREAL_DIVISION_BY_ZERO: "a / (0::hypreal) = 0"
paulson@14331
   486
by (simp add: hypreal_divide_def HYPREAL_INVERSE_ZERO 
paulson@14331
   487
              hypreal_mult_commute [of a])
paulson@14331
   488
paulson@14331
   489
instance hypreal :: division_by_zero
paulson@14331
   490
proof
paulson@14331
   491
  fix x :: hypreal
paulson@14331
   492
  show "inverse 0 = (0::hypreal)" by (rule HYPREAL_INVERSE_ZERO)
paulson@14331
   493
  show "x/0 = 0" by (rule HYPREAL_DIVISION_BY_ZERO) 
paulson@14331
   494
qed
paulson@14331
   495
paulson@14329
   496
paulson@14329
   497
subsection{*Theorems for Ordering*}
paulson@14329
   498
paulson@14329
   499
text{*TODO: define @{text "\<le>"} as the primitive concept and quickly 
paulson@14329
   500
establish membership in class @{text linorder}. Then proofs could be
paulson@14329
   501
simplified, since properties of @{text "<"} would be generic.*}
paulson@14299
   502
paulson@14329
   503
text{*TODO: The following theorem should be used througout the proofs
paulson@14329
   504
  as it probably makes many of them more straightforward.*}
paulson@14329
   505
lemma hypreal_less: 
paulson@14329
   506
      "(Abs_hypreal(hyprel``{%n. X n}) < Abs_hypreal(hyprel``{%n. Y n})) =  
paulson@14329
   507
       ({n. X n < Y n} \<in> FreeUltrafilterNat)"
paulson@14329
   508
apply (unfold hypreal_less_def)
paulson@14329
   509
apply (auto intro!: lemma_hyprel_refl, ultra)
paulson@14299
   510
done
paulson@14299
   511
paulson@14299
   512
lemma hypreal_less_not_refl: "~ (R::hypreal) < R"
paulson@14301
   513
apply (rule_tac z = R in eq_Abs_hypreal)
paulson@14301
   514
apply (auto simp add: hypreal_less_def, ultra)
paulson@14299
   515
done
paulson@14299
   516
paulson@14299
   517
lemmas hypreal_less_irrefl = hypreal_less_not_refl [THEN notE, standard]
paulson@14299
   518
declare hypreal_less_irrefl [elim!]
paulson@14299
   519
paulson@14299
   520
lemma hypreal_not_refl2: "!!(x::hypreal). x < y ==> x \<noteq> y"
paulson@14301
   521
by (auto simp add: hypreal_less_not_refl)
paulson@14299
   522
paulson@14299
   523
lemma hypreal_less_trans: "!!(R1::hypreal). [| R1 < R2; R2 < R3 |] ==> R1 < R3"
paulson@14301
   524
apply (rule_tac z = R1 in eq_Abs_hypreal)
paulson@14301
   525
apply (rule_tac z = R2 in eq_Abs_hypreal)
paulson@14301
   526
apply (rule_tac z = R3 in eq_Abs_hypreal)
paulson@14301
   527
apply (auto intro!: exI simp add: hypreal_less_def, ultra)
paulson@14299
   528
done
paulson@14299
   529
paulson@14299
   530
lemma hypreal_less_asym: "!! (R1::hypreal). [| R1 < R2; R2 < R1 |] ==> P"
paulson@14301
   531
apply (drule hypreal_less_trans, assumption)
paulson@14299
   532
apply (simp add: hypreal_less_not_refl)
paulson@14299
   533
done
paulson@14299
   534
paulson@14299
   535
paulson@14329
   536
subsection{*Trichotomy: the hyperreals are Linearly Ordered*}
paulson@14299
   537
paulson@14365
   538
lemma lemma_hyprel_0_mem: "\<exists>x. x \<in> hyprel `` {%n. 0}"
paulson@14299
   539
apply (unfold hyprel_def)
paulson@14301
   540
apply (rule_tac x = "%n. 0" in exI, safe)
paulson@14299
   541
apply (auto intro!: FreeUltrafilterNat_Nat_set)
paulson@14299
   542
done
paulson@14299
   543
paulson@14299
   544
lemma hypreal_trichotomy: "0 <  x | x = 0 | x < (0::hypreal)"
paulson@14299
   545
apply (unfold hypreal_zero_def)
paulson@14301
   546
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14299
   547
apply (auto simp add: hypreal_less_def)
paulson@14301
   548
apply (cut_tac lemma_hyprel_0_mem, erule exE)
paulson@14301
   549
apply (drule_tac x = xa in spec)
paulson@14301
   550
apply (drule_tac x = x in spec)
paulson@14301
   551
apply (cut_tac x = x in lemma_hyprel_refl, auto)
paulson@14301
   552
apply (drule_tac x = x in spec)
paulson@14301
   553
apply (drule_tac x = xa in spec, auto, ultra)
paulson@14299
   554
done
paulson@14299
   555
paulson@14299
   556
lemma hypreal_trichotomyE:
paulson@14299
   557
     "[| (0::hypreal) < x ==> P;  
paulson@14299
   558
         x = 0 ==> P;  
paulson@14299
   559
         x < 0 ==> P |] ==> P"
paulson@14301
   560
apply (insert hypreal_trichotomy [of x], blast) 
paulson@14299
   561
done
paulson@14299
   562
paulson@14299
   563
lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
paulson@14301
   564
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14301
   565
apply (rule_tac z = y in eq_Abs_hypreal)
paulson@14299
   566
apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
paulson@14299
   567
done
paulson@14299
   568
paulson@14299
   569
lemma hypreal_less_minus_iff2: "((x::hypreal) < y) = (x + -y < 0)"
paulson@14301
   570
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14301
   571
apply (rule_tac z = y in eq_Abs_hypreal)
paulson@14299
   572
apply (auto simp add: hypreal_add hypreal_zero_def hypreal_minus hypreal_less)
paulson@14299
   573
done
paulson@14299
   574
paulson@14299
   575
lemma hypreal_eq_minus_iff2: "((x::hypreal) = y) = (0 = y + - x)"
paulson@14299
   576
apply auto
paulson@14331
   577
apply (rule Ring_and_Field.add_right_cancel [of _ "-x", THEN iffD1], auto)
paulson@14299
   578
done
paulson@14299
   579
paulson@14299
   580
lemma hypreal_linear: "(x::hypreal) < y | x = y | y < x"
paulson@14299
   581
apply (subst hypreal_eq_minus_iff2)
paulson@14301
   582
apply (rule_tac x1 = x in hypreal_less_minus_iff [THEN ssubst])
paulson@14301
   583
apply (rule_tac x1 = y in hypreal_less_minus_iff2 [THEN ssubst])
paulson@14301
   584
apply (rule hypreal_trichotomyE, auto)
paulson@14299
   585
done
paulson@14299
   586
paulson@14299
   587
lemma hypreal_neq_iff: "((w::hypreal) \<noteq> z) = (w<z | z<w)"
paulson@14301
   588
by (cut_tac hypreal_linear, blast)
paulson@14299
   589
paulson@14329
   590
paulson@14329
   591
subsection{*Properties of The @{text "\<le>"} Relation*}
paulson@14299
   592
paulson@14299
   593
lemma hypreal_le: 
paulson@14365
   594
      "(Abs_hypreal(hyprel``{%n. X n}) \<le> Abs_hypreal(hyprel``{%n. Y n})) =  
paulson@14365
   595
       ({n. X n \<le> Y n} \<in> FreeUltrafilterNat)"
paulson@14365
   596
apply (auto simp add: hypreal_less hypreal_le_def linorder_not_less[symmetric])
paulson@14299
   597
apply (ultra+)
paulson@14299
   598
done
paulson@14299
   599
paulson@14365
   600
lemma hypreal_le_imp_less_or_eq: "!!(x::hypreal). x \<le> y ==> x < y | x = y"
paulson@14299
   601
apply (unfold hypreal_le_def)
paulson@14299
   602
apply (cut_tac hypreal_linear)
paulson@14299
   603
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
paulson@14299
   604
done
paulson@14299
   605
paulson@14365
   606
lemma hypreal_less_or_eq_imp_le: "z<w | z=w ==> z \<le>(w::hypreal)"
paulson@14299
   607
apply (unfold hypreal_le_def)
paulson@14299
   608
apply (cut_tac hypreal_linear)
paulson@14299
   609
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
paulson@14299
   610
done
paulson@14299
   611
paulson@14365
   612
lemma hypreal_le_eq_less_or_eq: "(x \<le> (y::hypreal)) = (x < y | x=y)"
paulson@14299
   613
by (blast intro!: hypreal_less_or_eq_imp_le dest: hypreal_le_imp_less_or_eq) 
paulson@14299
   614
paulson@14299
   615
lemmas hypreal_le_less = hypreal_le_eq_less_or_eq
paulson@14299
   616
paulson@14365
   617
lemma hypreal_le_refl: "w \<le> (w::hypreal)"
paulson@14301
   618
by (simp add: hypreal_le_eq_less_or_eq)
paulson@14299
   619
paulson@14299
   620
(* Axiom 'linorder_linear' of class 'linorder': *)
paulson@14365
   621
lemma hypreal_le_linear: "(z::hypreal) \<le> w | w \<le> z"
paulson@14301
   622
apply (simp add: hypreal_le_less)
paulson@14301
   623
apply (cut_tac hypreal_linear, blast)
paulson@14299
   624
done
paulson@14299
   625
paulson@14365
   626
lemma hypreal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::hypreal)"
paulson@14299
   627
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   628
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   629
apply (rule hypreal_less_or_eq_imp_le) 
paulson@14299
   630
apply (blast intro: hypreal_less_trans) 
paulson@14299
   631
done
paulson@14299
   632
paulson@14365
   633
lemma hypreal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::hypreal)"
paulson@14299
   634
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   635
apply (drule hypreal_le_imp_less_or_eq) 
paulson@14299
   636
apply (fast elim: hypreal_less_irrefl hypreal_less_asym)
paulson@14299
   637
done
paulson@14299
   638
paulson@14299
   639
(* Axiom 'order_less_le' of class 'order': *)
paulson@14365
   640
lemma hypreal_less_le: "((w::hypreal) < z) = (w \<le> z & w \<noteq> z)"
paulson@14301
   641
apply (simp add: hypreal_le_def hypreal_neq_iff)
paulson@14299
   642
apply (blast intro: hypreal_less_asym)
paulson@14299
   643
done
paulson@14299
   644
paulson@14329
   645
instance hypreal :: order
paulson@14329
   646
  by (intro_classes,
paulson@14329
   647
      (assumption | 
paulson@14329
   648
       rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym
paulson@14329
   649
            hypreal_less_le)+)
paulson@14329
   650
paulson@14329
   651
instance hypreal :: linorder 
paulson@14329
   652
  by (intro_classes, rule hypreal_le_linear)
paulson@14329
   653
paulson@14329
   654
paulson@14329
   655
lemma hypreal_add_less_mono1: "(A::hypreal) < B ==> A + C < B + C"
paulson@14329
   656
apply (rule_tac z = A in eq_Abs_hypreal)
paulson@14329
   657
apply (rule_tac z = B in eq_Abs_hypreal)
paulson@14329
   658
apply (rule_tac z = C in eq_Abs_hypreal)
paulson@14329
   659
apply (auto intro!: exI simp add: hypreal_less_def hypreal_add, ultra)
paulson@14329
   660
done
paulson@14329
   661
paulson@14329
   662
lemma hypreal_mult_order: "[| 0 < x; 0 < y |] ==> (0::hypreal) < x * y"
paulson@14329
   663
apply (unfold hypreal_zero_def)
paulson@14329
   664
apply (rule_tac z = x in eq_Abs_hypreal)
paulson@14329
   665
apply (rule_tac z = y in eq_Abs_hypreal)
paulson@14329
   666
apply (auto intro!: exI simp add: hypreal_less_def hypreal_mult, ultra)
paulson@14329
   667
apply (auto intro: real_mult_order)
paulson@14329
   668
done
paulson@14329
   669
paulson@14329
   670
lemma hypreal_add_left_le_mono1: "(q1::hypreal) \<le> q2  ==> x + q1 \<le> x + q2"
paulson@14329
   671
apply (drule order_le_imp_less_or_eq)
paulson@14329
   672
apply (auto intro: order_less_imp_le hypreal_add_less_mono1 simp add: hypreal_add_commute)
paulson@14329
   673
done
paulson@14329
   674
paulson@14329
   675
lemma hypreal_mult_less_mono1: "[| (0::hypreal) < z; x < y |] ==> x*z < y*z"
paulson@14329
   676
apply (rotate_tac 1)
paulson@14329
   677
apply (drule hypreal_less_minus_iff [THEN iffD1])
paulson@14329
   678
apply (rule hypreal_less_minus_iff [THEN iffD2])
paulson@14329
   679
apply (drule hypreal_mult_order, assumption)
paulson@14331
   680
apply (simp add: right_distrib hypreal_mult_commute)
paulson@14329
   681
done
paulson@14329
   682
paulson@14329
   683
lemma hypreal_mult_less_mono2: "[| (0::hypreal)<z; x<y |] ==> z*x<z*y"
paulson@14329
   684
apply (simp (no_asm_simp) add: hypreal_mult_commute hypreal_mult_less_mono1)
paulson@14329
   685
done
paulson@14329
   686
paulson@14329
   687
subsection{*The Hyperreals Form an Ordered Field*}
paulson@14329
   688
paulson@14329
   689
instance hypreal :: ordered_field
paulson@14329
   690
proof
paulson@14329
   691
  fix x y z :: hypreal
paulson@14348
   692
  show "0 < (1::hypreal)" 
paulson@14348
   693
    by (unfold hypreal_one_def hypreal_zero_def hypreal_less_def, force)
paulson@14348
   694
  show "x \<le> y ==> z + x \<le> z + y" 
paulson@14348
   695
    by (rule hypreal_add_left_le_mono1)
paulson@14348
   696
  show "x < y ==> 0 < z ==> z * x < z * y" 
paulson@14348
   697
    by (simp add: hypreal_mult_less_mono2)
paulson@14329
   698
  show "\<bar>x\<bar> = (if x < 0 then -x else x)"
paulson@14329
   699
    by (auto dest: order_le_less_trans simp add: hrabs_def linorder_not_le)
paulson@14329
   700
qed
paulson@14329
   701
paulson@14331
   702
lemma hypreal_mult_1_right: "z * (1::hypreal) = z"
paulson@14331
   703
  by (rule Ring_and_Field.mult_1_right)
paulson@14331
   704
paulson@14331
   705
lemma hypreal_mult_minus_1 [simp]: "(- (1::hypreal)) * z = -z"
paulson@14331
   706
by (simp)
paulson@14331
   707
paulson@14331
   708
lemma hypreal_mult_minus_1_right [simp]: "z * (- (1::hypreal)) = -z"
paulson@14331
   709
by (subst hypreal_mult_commute, simp)
paulson@14329
   710
paulson@14329
   711
(*Used ONCE: in NSA.ML*)
paulson@14329
   712
lemma hypreal_minus_distrib1: "-(y + -(x::hypreal)) = x + -y"
paulson@14329
   713
by (simp add: hypreal_add_commute)
paulson@14329
   714
paulson@14329
   715
(*Used ONCE: in Lim.ML*)
paulson@14329
   716
lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
paulson@14329
   717
by (auto simp add: hypreal_add_assoc)
paulson@14329
   718
paulson@14331
   719
lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)"
paulson@14331
   720
apply auto
paulson@14331
   721
apply (rule Ring_and_Field.add_right_cancel [of _ "-y", THEN iffD1], auto)
paulson@14331
   722
done
paulson@14331
   723
paulson@14331
   724
(*Used 3 TIMES: in Lim.ML*)
paulson@14329
   725
lemma hypreal_not_eq_minus_iff: "(x \<noteq> a) = (x + -a \<noteq> (0::hypreal))"
paulson@14329
   726
by (auto dest: hypreal_eq_minus_iff [THEN iffD2])
paulson@14329
   727
paulson@14329
   728
lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)"
paulson@14329
   729
apply auto
paulson@14329
   730
done
paulson@14329
   731
    
paulson@14329
   732
lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)"
paulson@14329
   733
apply auto
paulson@14329
   734
done
paulson@14329
   735
paulson@14329
   736
lemma hypreal_inverse_not_zero: "x \<noteq> 0 ==> inverse (x::hypreal) \<noteq> 0"
paulson@14329
   737
  by (rule Ring_and_Field.nonzero_imp_inverse_nonzero)
paulson@14329
   738
paulson@14329
   739
lemma hypreal_mult_not_0: "[| x \<noteq> 0; y \<noteq> 0 |] ==> x * y \<noteq> (0::hypreal)"
paulson@14329
   740
by simp
paulson@14329
   741
paulson@14329
   742
lemma hypreal_minus_inverse: "inverse(-x) = -inverse(x::hypreal)"
paulson@14329
   743
  by (rule Ring_and_Field.inverse_minus_eq)
paulson@14329
   744
paulson@14329
   745
lemma hypreal_inverse_distrib: "inverse(x*y) = inverse(x)*inverse(y::hypreal)"
paulson@14329
   746
  by (rule Ring_and_Field.inverse_mult_distrib)
paulson@14329
   747
paulson@14329
   748
paulson@14329
   749
subsection{* Division lemmas *}
paulson@14329
   750
paulson@14329
   751
lemma hypreal_divide_one: "x/(1::hypreal) = x"
paulson@14329
   752
by (simp add: hypreal_divide_def)
paulson@14329
   753
paulson@14329
   754
paulson@14329
   755
(** As with multiplication, pull minus signs OUT of the / operator **)
paulson@14329
   756
paulson@14329
   757
lemma hypreal_add_divide_distrib: "(x+y)/(z::hypreal) = x/z + y/z"
paulson@14329
   758
  by (rule Ring_and_Field.add_divide_distrib)
paulson@14329
   759
paulson@14329
   760
lemma hypreal_inverse_add:
paulson@14329
   761
     "[|(x::hypreal) \<noteq> 0;  y \<noteq> 0 |]   
paulson@14329
   762
      ==> inverse(x) + inverse(y) = (x + y)*inverse(x*y)"
paulson@14329
   763
by (simp add: Ring_and_Field.inverse_add mult_assoc)
paulson@14329
   764
paulson@14329
   765
paulson@14329
   766
subsection{*@{term hypreal_of_real} Preserves Field and Order Properties*}
paulson@14329
   767
paulson@14301
   768
lemma hypreal_of_real_add [simp]: 
paulson@14369
   769
     "hypreal_of_real (w + z) = hypreal_of_real w + hypreal_of_real z"
paulson@14299
   770
apply (unfold hypreal_of_real_def)
paulson@14331
   771
apply (simp add: hypreal_add left_distrib)
paulson@14299
   772
done
paulson@14299
   773
paulson@14301
   774
lemma hypreal_of_real_mult [simp]: 
paulson@14369
   775
     "hypreal_of_real (w * z) = hypreal_of_real w * hypreal_of_real z"
paulson@14299
   776
apply (unfold hypreal_of_real_def)
paulson@14331
   777
apply (simp add: hypreal_mult right_distrib)
paulson@14299
   778
done
paulson@14299
   779
paulson@14301
   780
lemma hypreal_of_real_one [simp]: "hypreal_of_real 1 = (1::hypreal)"
paulson@14301
   781
by (unfold hypreal_of_real_def hypreal_one_def, simp)
paulson@14299
   782
paulson@14301
   783
lemma hypreal_of_real_zero [simp]: "hypreal_of_real 0 = 0"
paulson@14301
   784
by (unfold hypreal_of_real_def hypreal_zero_def, simp)
paulson@14299
   785
paulson@14369
   786
lemma hypreal_of_real_less_iff [simp]: 
paulson@14369
   787
     "(hypreal_of_real w <  hypreal_of_real z) = (w < z)"
paulson@14369
   788
apply (unfold hypreal_less_def hypreal_of_real_def, auto)
paulson@14369
   789
apply (rule_tac [2] x = "%n. w" in exI, safe)
paulson@14369
   790
apply (rule_tac [3] x = "%n. z" in exI, auto)
paulson@14369
   791
apply (rule FreeUltrafilterNat_P, ultra)
paulson@14369
   792
done
paulson@14369
   793
paulson@14369
   794
lemma hypreal_of_real_le_iff [simp]: 
paulson@14369
   795
     "(hypreal_of_real w \<le> hypreal_of_real z) = (w \<le> z)"
paulson@14369
   796
by (force simp add: hypreal_less hypreal_le_def linorder_not_less[symmetric])
paulson@14369
   797
paulson@14369
   798
lemma hypreal_of_real_eq_iff [simp]:
paulson@14369
   799
     "(hypreal_of_real w = hypreal_of_real z) = (w = z)"
paulson@14369
   800
by (force intro: order_antisym hypreal_of_real_le_iff [THEN iffD1])
paulson@14369
   801
paulson@14369
   802
text{*As above, for 0*}
paulson@14369
   803
paulson@14369
   804
declare hypreal_of_real_less_iff [of 0, simplified, simp]
paulson@14369
   805
declare hypreal_of_real_le_iff   [of 0, simplified, simp]
paulson@14369
   806
declare hypreal_of_real_eq_iff   [of 0, simplified, simp]
paulson@14369
   807
paulson@14369
   808
declare hypreal_of_real_less_iff [of _ 0, simplified, simp]
paulson@14369
   809
declare hypreal_of_real_le_iff   [of _ 0, simplified, simp]
paulson@14369
   810
declare hypreal_of_real_eq_iff   [of _ 0, simplified, simp]
paulson@14369
   811
paulson@14369
   812
text{*As above, for 1*}
paulson@14369
   813
paulson@14369
   814
declare hypreal_of_real_less_iff [of 1, simplified, simp]
paulson@14369
   815
declare hypreal_of_real_le_iff   [of 1, simplified, simp]
paulson@14369
   816
declare hypreal_of_real_eq_iff   [of 1, simplified, simp]
paulson@14369
   817
paulson@14369
   818
declare hypreal_of_real_less_iff [of _ 1, simplified, simp]
paulson@14369
   819
declare hypreal_of_real_le_iff   [of _ 1, simplified, simp]
paulson@14369
   820
declare hypreal_of_real_eq_iff   [of _ 1, simplified, simp]
paulson@14369
   821
paulson@14369
   822
lemma hypreal_of_real_minus [simp]:
paulson@14369
   823
     "hypreal_of_real (-r) = - hypreal_of_real  r"
paulson@14369
   824
apply (unfold hypreal_of_real_def)
paulson@14369
   825
apply (auto simp add: hypreal_minus)
paulson@14369
   826
done
paulson@14299
   827
paulson@14329
   828
lemma hypreal_of_real_inverse [simp]:
paulson@14329
   829
     "hypreal_of_real (inverse r) = inverse (hypreal_of_real r)"
paulson@14299
   830
apply (case_tac "r=0")
paulson@14301
   831
apply (simp add: DIVISION_BY_ZERO INVERSE_ZERO HYPREAL_INVERSE_ZERO)
paulson@14299
   832
apply (rule_tac c1 = "hypreal_of_real r" in hypreal_mult_left_cancel [THEN iffD1])
paulson@14369
   833
apply (auto simp add: hypreal_of_real_mult [symmetric])
paulson@14299
   834
done
paulson@14299
   835
paulson@14329
   836
lemma hypreal_of_real_divide [simp]:
paulson@14369
   837
     "hypreal_of_real (w / z) = hypreal_of_real w / hypreal_of_real z"
paulson@14301
   838
by (simp add: hypreal_divide_def real_divide_def)
paulson@14299
   839
paulson@14299
   840
paulson@14329
   841
subsection{*Misc Others*}
paulson@14299
   842
paulson@14299
   843
lemma hypreal_zero_num: "0 = Abs_hypreal (hyprel `` {%n. 0})"
paulson@14301
   844
by (simp add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric])
paulson@14299
   845
paulson@14299
   846
lemma hypreal_one_num: "1 = Abs_hypreal (hyprel `` {%n. 1})"
paulson@14301
   847
by (simp add: hypreal_one_def [THEN meta_eq_to_obj_eq, symmetric])
paulson@14299
   848
paulson@14301
   849
lemma hypreal_omega_gt_zero [simp]: "0 < omega"
paulson@14299
   850
apply (unfold omega_def)
paulson@14299
   851
apply (auto simp add: hypreal_less hypreal_zero_num)
paulson@14299
   852
done
paulson@14299
   853
paulson@14329
   854
paulson@14329
   855
lemma hypreal_hrabs:
paulson@14329
   856
     "abs (Abs_hypreal (hyprel `` {X})) = 
paulson@14329
   857
      Abs_hypreal(hyprel `` {%n. abs (X n)})"
paulson@14329
   858
apply (auto simp add: hrabs_def hypreal_zero_def hypreal_le hypreal_minus)
paulson@14329
   859
apply (ultra, arith)+
paulson@14329
   860
done
paulson@14329
   861
paulson@14299
   862
ML
paulson@14299
   863
{*
paulson@14329
   864
val hrabs_def = thm "hrabs_def";
paulson@14329
   865
val hypreal_hrabs = thm "hypreal_hrabs";
paulson@14329
   866
paulson@14299
   867
val hypreal_zero_def = thm "hypreal_zero_def";
paulson@14299
   868
val hypreal_one_def = thm "hypreal_one_def";
paulson@14299
   869
val hypreal_minus_def = thm "hypreal_minus_def";
paulson@14299
   870
val hypreal_diff_def = thm "hypreal_diff_def";
paulson@14299
   871
val hypreal_inverse_def = thm "hypreal_inverse_def";
paulson@14299
   872
val hypreal_divide_def = thm "hypreal_divide_def";
paulson@14299
   873
val hypreal_of_real_def = thm "hypreal_of_real_def";
paulson@14299
   874
val omega_def = thm "omega_def";
paulson@14299
   875
val epsilon_def = thm "epsilon_def";
paulson@14299
   876
val hypreal_add_def = thm "hypreal_add_def";
paulson@14299
   877
val hypreal_mult_def = thm "hypreal_mult_def";
paulson@14299
   878
val hypreal_less_def = thm "hypreal_less_def";
paulson@14299
   879
val hypreal_le_def = thm "hypreal_le_def";
paulson@14299
   880
paulson@14299
   881
val finite_exhausts = thm "finite_exhausts";
paulson@14299
   882
val finite_not_covers = thm "finite_not_covers";
paulson@14299
   883
val not_finite_nat = thm "not_finite_nat";
paulson@14299
   884
val FreeUltrafilterNat_Ex = thm "FreeUltrafilterNat_Ex";
paulson@14299
   885
val FreeUltrafilterNat_mem = thm "FreeUltrafilterNat_mem";
paulson@14299
   886
val FreeUltrafilterNat_finite = thm "FreeUltrafilterNat_finite";
paulson@14299
   887
val FreeUltrafilterNat_not_finite = thm "FreeUltrafilterNat_not_finite";
paulson@14299
   888
val FreeUltrafilterNat_empty = thm "FreeUltrafilterNat_empty";
paulson@14299
   889
val FreeUltrafilterNat_Int = thm "FreeUltrafilterNat_Int";
paulson@14299
   890
val FreeUltrafilterNat_subset = thm "FreeUltrafilterNat_subset";
paulson@14299
   891
val FreeUltrafilterNat_Compl = thm "FreeUltrafilterNat_Compl";
paulson@14299
   892
val FreeUltrafilterNat_Compl_mem = thm "FreeUltrafilterNat_Compl_mem";
paulson@14299
   893
val FreeUltrafilterNat_Compl_iff1 = thm "FreeUltrafilterNat_Compl_iff1";
paulson@14299
   894
val FreeUltrafilterNat_Compl_iff2 = thm "FreeUltrafilterNat_Compl_iff2";
paulson@14299
   895
val FreeUltrafilterNat_UNIV = thm "FreeUltrafilterNat_UNIV";
paulson@14299
   896
val FreeUltrafilterNat_Nat_set = thm "FreeUltrafilterNat_Nat_set";
paulson@14299
   897
val FreeUltrafilterNat_Nat_set_refl = thm "FreeUltrafilterNat_Nat_set_refl";
paulson@14299
   898
val FreeUltrafilterNat_P = thm "FreeUltrafilterNat_P";
paulson@14299
   899
val FreeUltrafilterNat_Ex_P = thm "FreeUltrafilterNat_Ex_P";
paulson@14299
   900
val FreeUltrafilterNat_all = thm "FreeUltrafilterNat_all";
paulson@14299
   901
val FreeUltrafilterNat_Un = thm "FreeUltrafilterNat_Un";
paulson@14299
   902
val hyprel_iff = thm "hyprel_iff";
paulson@14299
   903
val hyprel_refl = thm "hyprel_refl";
paulson@14299
   904
val hyprel_sym = thm "hyprel_sym";
paulson@14299
   905
val hyprel_trans = thm "hyprel_trans";
paulson@14299
   906
val equiv_hyprel = thm "equiv_hyprel";
paulson@14299
   907
val hyprel_in_hypreal = thm "hyprel_in_hypreal";
paulson@14299
   908
val Abs_hypreal_inverse = thm "Abs_hypreal_inverse";
paulson@14299
   909
val inj_on_Abs_hypreal = thm "inj_on_Abs_hypreal";
paulson@14299
   910
val inj_Rep_hypreal = thm "inj_Rep_hypreal";
paulson@14299
   911
val lemma_hyprel_refl = thm "lemma_hyprel_refl";
paulson@14299
   912
val hypreal_empty_not_mem = thm "hypreal_empty_not_mem";
paulson@14299
   913
val Rep_hypreal_nonempty = thm "Rep_hypreal_nonempty";
paulson@14299
   914
val inj_hypreal_of_real = thm "inj_hypreal_of_real";
paulson@14299
   915
val eq_Abs_hypreal = thm "eq_Abs_hypreal";
paulson@14299
   916
val hypreal_minus_congruent = thm "hypreal_minus_congruent";
paulson@14299
   917
val hypreal_minus = thm "hypreal_minus";
paulson@14299
   918
val hypreal_add_congruent2 = thm "hypreal_add_congruent2";
paulson@14299
   919
val hypreal_add = thm "hypreal_add";
paulson@14299
   920
val hypreal_diff = thm "hypreal_diff";
paulson@14299
   921
val hypreal_add_commute = thm "hypreal_add_commute";
paulson@14299
   922
val hypreal_add_assoc = thm "hypreal_add_assoc";
paulson@14299
   923
val hypreal_add_zero_left = thm "hypreal_add_zero_left";
paulson@14299
   924
val hypreal_add_zero_right = thm "hypreal_add_zero_right";
paulson@14299
   925
val hypreal_add_minus = thm "hypreal_add_minus";
paulson@14299
   926
val hypreal_add_minus_left = thm "hypreal_add_minus_left";
paulson@14299
   927
val hypreal_minus_distrib1 = thm "hypreal_minus_distrib1";
paulson@14299
   928
val hypreal_mult_congruent2 = thm "hypreal_mult_congruent2";
paulson@14299
   929
val hypreal_mult = thm "hypreal_mult";
paulson@14299
   930
val hypreal_mult_commute = thm "hypreal_mult_commute";
paulson@14299
   931
val hypreal_mult_assoc = thm "hypreal_mult_assoc";
paulson@14299
   932
val hypreal_mult_1 = thm "hypreal_mult_1";
paulson@14299
   933
val hypreal_mult_1_right = thm "hypreal_mult_1_right";
paulson@14299
   934
val hypreal_mult_minus_1 = thm "hypreal_mult_minus_1";
paulson@14299
   935
val hypreal_mult_minus_1_right = thm "hypreal_mult_minus_1_right";
paulson@14299
   936
val hypreal_zero_not_eq_one = thm "hypreal_zero_not_eq_one";
paulson@14299
   937
val hypreal_inverse_congruent = thm "hypreal_inverse_congruent";
paulson@14299
   938
val hypreal_inverse = thm "hypreal_inverse";
paulson@14299
   939
val HYPREAL_INVERSE_ZERO = thm "HYPREAL_INVERSE_ZERO";
paulson@14299
   940
val HYPREAL_DIVISION_BY_ZERO = thm "HYPREAL_DIVISION_BY_ZERO";
paulson@14299
   941
val hypreal_mult_inverse = thm "hypreal_mult_inverse";
paulson@14299
   942
val hypreal_mult_inverse_left = thm "hypreal_mult_inverse_left";
paulson@14299
   943
val hypreal_mult_left_cancel = thm "hypreal_mult_left_cancel";
paulson@14299
   944
val hypreal_mult_right_cancel = thm "hypreal_mult_right_cancel";
paulson@14299
   945
val hypreal_inverse_not_zero = thm "hypreal_inverse_not_zero";
paulson@14299
   946
val hypreal_mult_not_0 = thm "hypreal_mult_not_0";
paulson@14299
   947
val hypreal_minus_inverse = thm "hypreal_minus_inverse";
paulson@14299
   948
val hypreal_inverse_distrib = thm "hypreal_inverse_distrib";
paulson@14299
   949
val hypreal_less_not_refl = thm "hypreal_less_not_refl";
paulson@14331
   950
val hypreal_less_irrefl = thm"hypreal_less_irrefl";
paulson@14299
   951
val hypreal_not_refl2 = thm "hypreal_not_refl2";
paulson@14299
   952
val hypreal_less_trans = thm "hypreal_less_trans";
paulson@14299
   953
val hypreal_less_asym = thm "hypreal_less_asym";
paulson@14299
   954
val hypreal_less = thm "hypreal_less";
paulson@14299
   955
val hypreal_trichotomy = thm "hypreal_trichotomy";
paulson@14299
   956
val hypreal_less_minus_iff = thm "hypreal_less_minus_iff";
paulson@14299
   957
val hypreal_less_minus_iff2 = thm "hypreal_less_minus_iff2";
paulson@14299
   958
val hypreal_eq_minus_iff = thm "hypreal_eq_minus_iff";
paulson@14299
   959
val hypreal_eq_minus_iff2 = thm "hypreal_eq_minus_iff2";
paulson@14299
   960
val hypreal_eq_minus_iff3 = thm "hypreal_eq_minus_iff3";
paulson@14299
   961
val hypreal_not_eq_minus_iff = thm "hypreal_not_eq_minus_iff";
paulson@14299
   962
val hypreal_linear = thm "hypreal_linear";
paulson@14299
   963
val hypreal_le = thm "hypreal_le";
paulson@14299
   964
val hypreal_le_imp_less_or_eq = thm "hypreal_le_imp_less_or_eq";
paulson@14299
   965
val hypreal_le_eq_less_or_eq = thm "hypreal_le_eq_less_or_eq";
paulson@14299
   966
val hypreal_le_refl = thm "hypreal_le_refl";
paulson@14299
   967
val hypreal_le_linear = thm "hypreal_le_linear";
paulson@14299
   968
val hypreal_le_trans = thm "hypreal_le_trans";
paulson@14299
   969
val hypreal_le_anti_sym = thm "hypreal_le_anti_sym";
paulson@14299
   970
val hypreal_less_le = thm "hypreal_less_le";
paulson@14299
   971
val hypreal_of_real_add = thm "hypreal_of_real_add";
paulson@14299
   972
val hypreal_of_real_mult = thm "hypreal_of_real_mult";
paulson@14299
   973
val hypreal_of_real_less_iff = thm "hypreal_of_real_less_iff";
paulson@14299
   974
val hypreal_of_real_le_iff = thm "hypreal_of_real_le_iff";
paulson@14299
   975
val hypreal_of_real_eq_iff = thm "hypreal_of_real_eq_iff";
paulson@14299
   976
val hypreal_of_real_minus = thm "hypreal_of_real_minus";
paulson@14299
   977
val hypreal_of_real_one = thm "hypreal_of_real_one";
paulson@14299
   978
val hypreal_of_real_zero = thm "hypreal_of_real_zero";
paulson@14299
   979
val hypreal_of_real_inverse = thm "hypreal_of_real_inverse";
paulson@14299
   980
val hypreal_of_real_divide = thm "hypreal_of_real_divide";
paulson@14299
   981
val hypreal_divide_one = thm "hypreal_divide_one";
paulson@14299
   982
val hypreal_add_divide_distrib = thm "hypreal_add_divide_distrib";
paulson@14299
   983
val hypreal_inverse_add = thm "hypreal_inverse_add";
paulson@14299
   984
val hypreal_zero_num = thm "hypreal_zero_num";
paulson@14299
   985
val hypreal_one_num = thm "hypreal_one_num";
paulson@14299
   986
val hypreal_omega_gt_zero = thm "hypreal_omega_gt_zero";
paulson@14299
   987
*}
paulson@14299
   988
paulson@10751
   989
end