author | paulson |
Thu, 06 May 2004 12:43:00 +0200 | |
changeset 14705 | 14b2c22a7e40 |
parent 14691 | e1eedc8cad37 |
child 14738 | 83f1a514dcb4 |
permissions | -rw-r--r-- |
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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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Conversion to Isar and new proofs by Lawrence C Paulson, 2004 |
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*) |
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header{*Construction of Hyperreals Using Ultrafilters*} |
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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)} \<in> 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, zero, one, plus, times, minus, inverse}" .. |
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defs (overloaded) |
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hypreal_zero_def: |
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"0 == Abs_hypreal(hyprel``{%n. 0})" |
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hypreal_one_def: |
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"1 == Abs_hypreal(hyprel``{%n. 1})" |
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hypreal_minus_def: |
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"- P == Abs_hypreal(\<Union>X \<in> Rep_hypreal(P). hyprel``{%n. - (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. r})" |
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omega :: hypreal -- {*an infinite number @{text "= [<1,2,3,...>]"} *} |
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"omega == Abs_hypreal(hyprel``{%n. real (Suc n)})" |
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epsilon :: hypreal -- {*an infinitesimal number @{text "= [<1,1/2,1/3,...>]"} *} |
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"epsilon == Abs_hypreal(hyprel``{%n. 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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syntax (HTML output) |
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omega :: hypreal ("\<omega>") |
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epsilon :: hypreal ("\<epsilon>") |
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defs (overloaded) |
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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. 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. X n * Y n})" |
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hypreal_le_def: |
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"P \<le> (Q::hypreal) == \<exists>X Y. X \<in> Rep_hypreal(P) & |
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Y \<in> Rep_hypreal(Q) & |
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{n. X n \<le> Y n} \<in> FreeUltrafilterNat" |
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hypreal_less_def: "(x < (y::hypreal)) == (x \<le> y & x \<noteq> y)" |
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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 \<in> 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 \<in> 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 \<in> 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 \<in> FreeUltrafilterNat; Y \<in> FreeUltrafilterNat |] |
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==> X Int Y \<in> FreeUltrafilterNat" |
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apply (insert 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 \<in> FreeUltrafilterNat; X \<subseteq> Y |] |
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==> Y \<in> FreeUltrafilterNat" |
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apply (insert 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 \<in> FreeUltrafilterNat ==> -X \<notin> FreeUltrafilterNat" |
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proof |
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assume "X \<in> \<U>" and "- X \<in> \<U>" |
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hence "X Int - X \<in> \<U>" by (rule FreeUltrafilterNat_Int) |
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thus False by force |
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qed |
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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 \<in> 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 \<in> FreeUltrafilterNat) = (-X \<notin> FreeUltrafilterNat)" |
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by (auto simp add: FreeUltrafilterNat_Compl_iff1 [symmetric]) |
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lemma cofinite_mem_FreeUltrafilterNat: "finite (-X) ==> X \<in> FreeUltrafilterNat" |
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apply (drule FreeUltrafilterNat_finite) |
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apply (simp add: FreeUltrafilterNat_Compl_iff2 [symmetric]) |
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done |
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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 \<in> FreeUltrafilterNat |
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==> X \<in> FreeUltrafilterNat | Y \<in> FreeUltrafilterNat" |
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by (auto, ultra) |
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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} \<in> FreeUltrafilterNat)" |
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by (simp add: hyprel_def) |
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lemma hyprel_refl: "(x,x) \<in> hyprel" |
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by (simp add: hyprel_def) |
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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 (simp add: hyprel_def, 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 (simp add: 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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by (simp add: hyprel_def) |
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lemma hypreal_empty_not_mem [simp]: "{} \<notin> hypreal" |
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apply (simp add: 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 (insert Rep_hypreal [of x], 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 (simp add: hypreal_of_real_def split: split_if_asm) |
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done |
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lemma eq_Abs_hypreal: |
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"(!!x. 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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theorem hypreal_cases [case_names Abs_hypreal, cases type: hypreal]: |
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"(!!x. z = Abs_hypreal(hyprel``{x}) ==> P) ==> P" |
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by (rule eq_Abs_hypreal [of z], blast) |
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subsection{*Hyperreal Addition*} |
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lemma hypreal_add_congruent2: |
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"congruent2 hyprel hyprel (%X Y. hyprel``{%n. X n + Y n})" |
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by (simp add: congruent2_def, auto, ultra) |
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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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by (simp add: hypreal_add_def |
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UN_equiv_class2 [OF equiv_hyprel equiv_hyprel hypreal_add_congruent2]) |
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lemma hypreal_add_commute: "(z::hypreal) + w = w + z" |
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apply (cases z, cases w) |
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apply (simp add: add_ac hypreal_add) |
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done |
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lemma hypreal_add_assoc: "((z1::hypreal) + z2) + z3 = z1 + (z2 + z3)" |
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apply (cases z1, cases z2, cases z3) |
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apply (simp add: hypreal_add real_add_assoc) |
330 |
done |
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lemma hypreal_add_zero_left: "(0::hypreal) + z = z" |
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by (cases z, simp add: hypreal_zero_def hypreal_add) |
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instance hypreal :: plus_ac0 |
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by intro_classes |
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(assumption | |
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rule hypreal_add_commute hypreal_add_assoc hypreal_add_zero_left)+ |
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lemma hypreal_add_zero_right [simp]: "z + (0::hypreal) = z" |
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by (simp add: hypreal_add_zero_left hypreal_add_commute) |
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subsection{*Additive inverse on @{typ hypreal}*} |
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lemma hypreal_minus_congruent: |
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"congruent hyprel (%X. hyprel``{%n. - (X n)})" |
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by (force simp add: congruent_def) |
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lemma hypreal_minus: |
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351 |
"- (Abs_hypreal(hyprel``{%n. X n})) = Abs_hypreal(hyprel `` {%n. -(X n)})" |
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by (simp add: hypreal_minus_def Abs_hypreal_inject |
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hyprel_in_hypreal [THEN Abs_hypreal_inverse] |
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UN_equiv_class [OF equiv_hyprel hypreal_minus_congruent]) |
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lemma hypreal_diff: |
357 |
"Abs_hypreal(hyprel``{%n. X n}) - Abs_hypreal(hyprel``{%n. Y n}) = |
|
14299 | 358 |
Abs_hypreal(hyprel``{%n. X n - Y n})" |
14705 | 359 |
by (simp add: hypreal_diff_def hypreal_minus hypreal_add) |
14299 | 360 |
|
14301 | 361 |
lemma hypreal_add_minus [simp]: "z + -z = (0::hypreal)" |
14705 | 362 |
by (cases z, simp add: hypreal_zero_def hypreal_minus hypreal_add) |
14299 | 363 |
|
14331 | 364 |
lemma hypreal_add_minus_left: "-z + z = (0::hypreal)" |
14301 | 365 |
by (simp add: hypreal_add_commute hypreal_add_minus) |
14299 | 366 |
|
14329 | 367 |
|
368 |
subsection{*Hyperreal Multiplication*} |
|
14299 | 369 |
|
370 |
lemma hypreal_mult_congruent2: |
|
14658 | 371 |
"congruent2 hyprel hyprel (%X Y. hyprel``{%n. X n * Y n})" |
372 |
by (simp add: congruent2_def, auto, ultra) |
|
14299 | 373 |
|
374 |
lemma hypreal_mult: |
|
375 |
"Abs_hypreal(hyprel``{%n. X n}) * Abs_hypreal(hyprel``{%n. Y n}) = |
|
376 |
Abs_hypreal(hyprel``{%n. X n * Y n})" |
|
14658 | 377 |
by (simp add: hypreal_mult_def |
378 |
UN_equiv_class2 [OF equiv_hyprel equiv_hyprel hypreal_mult_congruent2]) |
|
14299 | 379 |
|
380 |
lemma hypreal_mult_commute: "(z::hypreal) * w = w * z" |
|
14705 | 381 |
by (cases z, cases w, simp add: hypreal_mult mult_ac) |
14299 | 382 |
|
383 |
lemma hypreal_mult_assoc: "((z1::hypreal) * z2) * z3 = z1 * (z2 * z3)" |
|
14705 | 384 |
by (cases z1, cases z2, cases z3, simp add: hypreal_mult mult_assoc) |
14299 | 385 |
|
14331 | 386 |
lemma hypreal_mult_1: "(1::hypreal) * z = z" |
14705 | 387 |
by (cases z, simp add: hypreal_one_def hypreal_mult) |
14301 | 388 |
|
14329 | 389 |
lemma hypreal_add_mult_distrib: |
390 |
"((z1::hypreal) + z2) * w = (z1 * w) + (z2 * w)" |
|
14705 | 391 |
by (cases z1, cases z2, cases w, simp add: hypreal_mult hypreal_add left_distrib) |
14299 | 392 |
|
14331 | 393 |
text{*one and zero are distinct*} |
14299 | 394 |
lemma hypreal_zero_not_eq_one: "0 \<noteq> (1::hypreal)" |
14468 | 395 |
by (simp add: hypreal_zero_def hypreal_one_def) |
14299 | 396 |
|
397 |
||
14329 | 398 |
subsection{*Multiplicative Inverse on @{typ hypreal} *} |
14299 | 399 |
|
400 |
lemma hypreal_inverse_congruent: |
|
401 |
"congruent hyprel (%X. hyprel``{%n. if X n = 0 then 0 else inverse(X n)})" |
|
14705 | 402 |
by (auto simp add: congruent_def, ultra) |
14299 | 403 |
|
404 |
lemma hypreal_inverse: |
|
405 |
"inverse (Abs_hypreal(hyprel``{%n. X n})) = |
|
406 |
Abs_hypreal(hyprel `` {%n. if X n = 0 then 0 else inverse(X n)})" |
|
14705 | 407 |
by (simp add: hypreal_inverse_def Abs_hypreal_inject |
408 |
hyprel_in_hypreal [THEN Abs_hypreal_inverse] |
|
409 |
UN_equiv_class [OF equiv_hyprel hypreal_inverse_congruent]) |
|
14299 | 410 |
|
14331 | 411 |
lemma hypreal_mult_inverse: |
14299 | 412 |
"x \<noteq> 0 ==> x*inverse(x) = (1::hypreal)" |
14468 | 413 |
apply (cases x) |
14705 | 414 |
apply (simp add: hypreal_one_def hypreal_zero_def hypreal_inverse hypreal_mult) |
14299 | 415 |
apply (drule FreeUltrafilterNat_Compl_mem) |
14334 | 416 |
apply (blast intro!: right_inverse FreeUltrafilterNat_subset) |
14299 | 417 |
done |
418 |
||
14331 | 419 |
lemma hypreal_mult_inverse_left: |
14329 | 420 |
"x \<noteq> 0 ==> inverse(x)*x = (1::hypreal)" |
14301 | 421 |
by (simp add: hypreal_mult_inverse hypreal_mult_commute) |
14299 | 422 |
|
14331 | 423 |
instance hypreal :: field |
424 |
proof |
|
425 |
fix x y z :: hypreal |
|
426 |
show "(x + y) + z = x + (y + z)" by (rule hypreal_add_assoc) |
|
427 |
show "x + y = y + x" by (rule hypreal_add_commute) |
|
428 |
show "0 + x = x" by simp |
|
429 |
show "- x + x = 0" by (simp add: hypreal_add_minus_left) |
|
430 |
show "x - y = x + (-y)" by (simp add: hypreal_diff_def) |
|
431 |
show "(x * y) * z = x * (y * z)" by (rule hypreal_mult_assoc) |
|
432 |
show "x * y = y * x" by (rule hypreal_mult_commute) |
|
433 |
show "1 * x = x" by (simp add: hypreal_mult_1) |
|
434 |
show "(x + y) * z = x * z + y * z" by (simp add: hypreal_add_mult_distrib) |
|
435 |
show "0 \<noteq> (1::hypreal)" by (rule hypreal_zero_not_eq_one) |
|
436 |
show "x \<noteq> 0 ==> inverse x * x = 1" by (simp add: hypreal_mult_inverse_left) |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
437 |
show "x / y = x * inverse y" by (simp add: hypreal_divide_def) |
14331 | 438 |
qed |
439 |
||
440 |
||
441 |
instance hypreal :: division_by_zero |
|
442 |
proof |
|
14430
5cb24165a2e1
new material from Avigad, and simplified treatment of division by 0
paulson
parents:
14421
diff
changeset
|
443 |
show "inverse 0 = (0::hypreal)" |
14421
ee97b6463cb4
new Ring_and_Field hierarchy, eliminating redundant axioms
paulson
parents:
14387
diff
changeset
|
444 |
by (simp add: hypreal_inverse hypreal_zero_def) |
14331 | 445 |
qed |
446 |
||
14329 | 447 |
|
448 |
subsection{*Properties of The @{text "\<le>"} Relation*} |
|
14299 | 449 |
|
450 |
lemma hypreal_le: |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14361
diff
changeset
|
451 |
"(Abs_hypreal(hyprel``{%n. X n}) \<le> Abs_hypreal(hyprel``{%n. Y n})) = |
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14361
diff
changeset
|
452 |
({n. X n \<le> Y n} \<in> FreeUltrafilterNat)" |
14468 | 453 |
apply (simp add: hypreal_le_def) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
454 |
apply (auto intro!: lemma_hyprel_refl, ultra) |
14299 | 455 |
done |
456 |
||
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14361
diff
changeset
|
457 |
lemma hypreal_le_refl: "w \<le> (w::hypreal)" |
14705 | 458 |
by (cases w, simp add: hypreal_le) |
14299 | 459 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14361
diff
changeset
|
460 |
lemma hypreal_le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::hypreal)" |
14705 | 461 |
by (cases i, cases j, cases k, simp add: hypreal_le, ultra) |
14299 | 462 |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14361
diff
changeset
|
463 |
lemma hypreal_le_anti_sym: "[| z \<le> w; w \<le> z |] ==> z = (w::hypreal)" |
14705 | 464 |
by (cases z, cases w, simp add: hypreal_le, ultra) |
14299 | 465 |
|
466 |
(* Axiom 'order_less_le' of class 'order': *) |
|
14365
3d4df8c166ae
replacing HOL/Real/PRat, PNat by the rational number development
paulson
parents:
14361
diff
changeset
|
467 |
lemma hypreal_less_le: "((w::hypreal) < z) = (w \<le> z & w \<noteq> z)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
468 |
by (simp add: hypreal_less_def) |
14299 | 469 |
|
14329 | 470 |
instance hypreal :: order |
14691 | 471 |
by intro_classes |
472 |
(assumption | |
|
473 |
rule hypreal_le_refl hypreal_le_trans hypreal_le_anti_sym hypreal_less_le)+ |
|
14370 | 474 |
|
475 |
||
476 |
(* Axiom 'linorder_linear' of class 'linorder': *) |
|
477 |
lemma hypreal_le_linear: "(z::hypreal) \<le> w | w \<le> z" |
|
14468 | 478 |
apply (cases z, cases w) |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
479 |
apply (auto simp add: hypreal_le, ultra) |
14370 | 480 |
done |
14329 | 481 |
|
482 |
instance hypreal :: linorder |
|
14691 | 483 |
by intro_classes (rule hypreal_le_linear) |
14329 | 484 |
|
14370 | 485 |
lemma hypreal_not_refl2: "!!(x::hypreal). x < y ==> x \<noteq> y" |
486 |
by (auto simp add: order_less_irrefl) |
|
14329 | 487 |
|
14370 | 488 |
lemma hypreal_add_left_mono: "x \<le> y ==> z + x \<le> z + (y::hypreal)" |
14468 | 489 |
apply (cases x, cases y, cases z) |
14370 | 490 |
apply (auto simp add: hypreal_le hypreal_add) |
14329 | 491 |
done |
492 |
||
493 |
lemma hypreal_mult_less_mono2: "[| (0::hypreal)<z; x<y |] ==> z*x<z*y" |
|
14468 | 494 |
apply (cases x, cases y, cases z) |
14370 | 495 |
apply (auto simp add: hypreal_zero_def hypreal_le hypreal_mult |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
496 |
linorder_not_le [symmetric], ultra) |
14329 | 497 |
done |
498 |
||
14370 | 499 |
|
14329 | 500 |
subsection{*The Hyperreals Form an Ordered Field*} |
501 |
||
502 |
instance hypreal :: ordered_field |
|
503 |
proof |
|
504 |
fix x y z :: hypreal |
|
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
505 |
show "x \<le> y ==> z + x \<le> z + y" |
14370 | 506 |
by (rule hypreal_add_left_mono) |
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
507 |
show "x < y ==> 0 < z ==> z * x < z * y" |
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset
|
508 |
by (simp add: hypreal_mult_less_mono2) |
14329 | 509 |
show "\<bar>x\<bar> = (if x < 0 then -x else x)" |
510 |
by (auto dest: order_le_less_trans simp add: hrabs_def linorder_not_le) |
|
511 |
qed |
|
512 |
||
14331 | 513 |
lemma hypreal_eq_minus_iff: "((x::hypreal) = y) = (x + - y = 0)" |
514 |
apply auto |
|
515 |
apply (rule Ring_and_Field.add_right_cancel [of _ "-y", THEN iffD1], auto) |
|
516 |
done |
|
517 |
||
14329 | 518 |
lemma hypreal_mult_left_cancel: "(c::hypreal) \<noteq> 0 ==> (c*a=c*b) = (a=b)" |
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
519 |
by auto |
14329 | 520 |
|
521 |
lemma hypreal_mult_right_cancel: "(c::hypreal) \<noteq> 0 ==> (a*c=b*c) = (a=b)" |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
522 |
by auto |
14329 | 523 |
|
524 |
||
14371
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14370
diff
changeset
|
525 |
subsection{*The Embedding @{term hypreal_of_real} Preserves Field and |
c78c7da09519
Conversion of HyperNat to Isar format and its declaration as a semiring
paulson
parents:
14370
diff
changeset
|
526 |
Order Properties*} |
14329 | 527 |
|
14301 | 528 |
lemma hypreal_of_real_add [simp]: |
14369 | 529 |
"hypreal_of_real (w + z) = hypreal_of_real w + hypreal_of_real z" |
14705 | 530 |
by (simp add: hypreal_of_real_def, simp add: hypreal_add left_distrib) |
14299 | 531 |
|
14301 | 532 |
lemma hypreal_of_real_mult [simp]: |
14369 | 533 |
"hypreal_of_real (w * z) = hypreal_of_real w * hypreal_of_real z" |
14705 | 534 |
by (simp add: hypreal_of_real_def, simp add: hypreal_mult right_distrib) |
14299 | 535 |
|
14301 | 536 |
lemma hypreal_of_real_one [simp]: "hypreal_of_real 1 = (1::hypreal)" |
14468 | 537 |
by (simp add: hypreal_of_real_def hypreal_one_def) |
14299 | 538 |
|
14301 | 539 |
lemma hypreal_of_real_zero [simp]: "hypreal_of_real 0 = 0" |
14468 | 540 |
by (simp add: hypreal_of_real_def hypreal_zero_def) |
14299 | 541 |
|
14370 | 542 |
lemma hypreal_of_real_le_iff [simp]: |
543 |
"(hypreal_of_real w \<le> hypreal_of_real z) = (w \<le> z)" |
|
14468 | 544 |
apply (simp add: hypreal_le_def hypreal_of_real_def, auto) |
14369 | 545 |
apply (rule_tac [2] x = "%n. w" in exI, safe) |
546 |
apply (rule_tac [3] x = "%n. z" in exI, auto) |
|
547 |
apply (rule FreeUltrafilterNat_P, ultra) |
|
548 |
done |
|
549 |
||
14370 | 550 |
lemma hypreal_of_real_less_iff [simp]: |
551 |
"(hypreal_of_real w < hypreal_of_real z) = (w < z)" |
|
552 |
by (simp add: linorder_not_le [symmetric]) |
|
14369 | 553 |
|
554 |
lemma hypreal_of_real_eq_iff [simp]: |
|
555 |
"(hypreal_of_real w = hypreal_of_real z) = (w = z)" |
|
556 |
by (force intro: order_antisym hypreal_of_real_le_iff [THEN iffD1]) |
|
557 |
||
558 |
text{*As above, for 0*} |
|
559 |
||
560 |
declare hypreal_of_real_less_iff [of 0, simplified, simp] |
|
561 |
declare hypreal_of_real_le_iff [of 0, simplified, simp] |
|
562 |
declare hypreal_of_real_eq_iff [of 0, simplified, simp] |
|
563 |
||
564 |
declare hypreal_of_real_less_iff [of _ 0, simplified, simp] |
|
565 |
declare hypreal_of_real_le_iff [of _ 0, simplified, simp] |
|
566 |
declare hypreal_of_real_eq_iff [of _ 0, simplified, simp] |
|
567 |
||
568 |
text{*As above, for 1*} |
|
569 |
||
570 |
declare hypreal_of_real_less_iff [of 1, simplified, simp] |
|
571 |
declare hypreal_of_real_le_iff [of 1, simplified, simp] |
|
572 |
declare hypreal_of_real_eq_iff [of 1, simplified, simp] |
|
573 |
||
574 |
declare hypreal_of_real_less_iff [of _ 1, simplified, simp] |
|
575 |
declare hypreal_of_real_le_iff [of _ 1, simplified, simp] |
|
576 |
declare hypreal_of_real_eq_iff [of _ 1, simplified, simp] |
|
577 |
||
578 |
lemma hypreal_of_real_minus [simp]: |
|
579 |
"hypreal_of_real (-r) = - hypreal_of_real r" |
|
14370 | 580 |
by (auto simp add: hypreal_of_real_def hypreal_minus) |
14299 | 581 |
|
14329 | 582 |
lemma hypreal_of_real_inverse [simp]: |
583 |
"hypreal_of_real (inverse r) = inverse (hypreal_of_real r)" |
|
14370 | 584 |
apply (case_tac "r=0", simp) |
14299 | 585 |
apply (rule_tac c1 = "hypreal_of_real r" in hypreal_mult_left_cancel [THEN iffD1]) |
14369 | 586 |
apply (auto simp add: hypreal_of_real_mult [symmetric]) |
14299 | 587 |
done |
588 |
||
14329 | 589 |
lemma hypreal_of_real_divide [simp]: |
14369 | 590 |
"hypreal_of_real (w / z) = hypreal_of_real w / hypreal_of_real z" |
14301 | 591 |
by (simp add: hypreal_divide_def real_divide_def) |
14299 | 592 |
|
593 |
||
14329 | 594 |
subsection{*Misc Others*} |
14299 | 595 |
|
14370 | 596 |
lemma hypreal_less: |
597 |
"(Abs_hypreal(hyprel``{%n. X n}) < Abs_hypreal(hyprel``{%n. Y n})) = |
|
598 |
({n. X n < Y n} \<in> FreeUltrafilterNat)" |
|
14705 | 599 |
by (auto simp add: hypreal_le linorder_not_le [symmetric], ultra+) |
14370 | 600 |
|
14299 | 601 |
lemma hypreal_zero_num: "0 = Abs_hypreal (hyprel `` {%n. 0})" |
14301 | 602 |
by (simp add: hypreal_zero_def [THEN meta_eq_to_obj_eq, symmetric]) |
14299 | 603 |
|
604 |
lemma hypreal_one_num: "1 = Abs_hypreal (hyprel `` {%n. 1})" |
|
14301 | 605 |
by (simp add: hypreal_one_def [THEN meta_eq_to_obj_eq, symmetric]) |
14299 | 606 |
|
14301 | 607 |
lemma hypreal_omega_gt_zero [simp]: "0 < omega" |
14705 | 608 |
by (auto simp add: omega_def hypreal_less hypreal_zero_num) |
14299 | 609 |
|
14329 | 610 |
lemma hypreal_hrabs: |
611 |
"abs (Abs_hypreal (hyprel `` {X})) = |
|
612 |
Abs_hypreal(hyprel `` {%n. abs (X n)})" |
|
613 |
apply (auto simp add: hrabs_def hypreal_zero_def hypreal_le hypreal_minus) |
|
614 |
apply (ultra, arith)+ |
|
615 |
done |
|
616 |
||
14370 | 617 |
|
618 |
||
619 |
lemma hypreal_add_zero_less_le_mono: "[|r < x; (0::hypreal) \<le> y|] ==> r < x+y" |
|
620 |
by (auto dest: add_less_le_mono) |
|
621 |
||
622 |
text{*The precondition could be weakened to @{term "0\<le>x"}*} |
|
623 |
lemma hypreal_mult_less_mono: |
|
624 |
"[| u<v; x<y; (0::hypreal) < v; 0 < x |] ==> u*x < v* y" |
|
625 |
by (simp add: Ring_and_Field.mult_strict_mono order_less_imp_le) |
|
626 |
||
627 |
||
628 |
subsection{*Existence of Infinite Hyperreal Number*} |
|
629 |
||
630 |
lemma Rep_hypreal_omega: "Rep_hypreal(omega) \<in> hypreal" |
|
14468 | 631 |
by (simp add: omega_def) |
14370 | 632 |
|
633 |
text{*Existence of infinite number not corresponding to any real number. |
|
634 |
Use assumption that member @{term FreeUltrafilterNat} is not finite.*} |
|
635 |
||
636 |
||
637 |
text{*A few lemmas first*} |
|
638 |
||
639 |
lemma lemma_omega_empty_singleton_disj: "{n::nat. x = real n} = {} | |
|
640 |
(\<exists>y. {n::nat. x = real n} = {y})" |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
641 |
by force |
14370 | 642 |
|
643 |
lemma lemma_finite_omega_set: "finite {n::nat. x = real n}" |
|
644 |
by (cut_tac x = x in lemma_omega_empty_singleton_disj, auto) |
|
645 |
||
646 |
lemma not_ex_hypreal_of_real_eq_omega: |
|
647 |
"~ (\<exists>x. hypreal_of_real x = omega)" |
|
14468 | 648 |
apply (simp add: omega_def hypreal_of_real_def) |
14370 | 649 |
apply (auto simp add: real_of_nat_Suc diff_eq_eq [symmetric] |
650 |
lemma_finite_omega_set [THEN FreeUltrafilterNat_finite]) |
|
651 |
done |
|
652 |
||
653 |
lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x \<noteq> omega" |
|
14705 | 654 |
by (insert not_ex_hypreal_of_real_eq_omega, auto) |
14370 | 655 |
|
656 |
text{*Existence of infinitesimal number also not corresponding to any |
|
657 |
real number*} |
|
658 |
||
659 |
lemma lemma_epsilon_empty_singleton_disj: |
|
660 |
"{n::nat. x = inverse(real(Suc n))} = {} | |
|
661 |
(\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})" |
|
14387
e96d5c42c4b0
Polymorphic treatment of binary arithmetic using axclasses
paulson
parents:
14378
diff
changeset
|
662 |
by auto |
14370 | 663 |
|
664 |
lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}" |
|
665 |
by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto) |
|
666 |
||
14705 | 667 |
lemma not_ex_hypreal_of_real_eq_epsilon: "~ (\<exists>x. hypreal_of_real x = epsilon)" |
668 |
by (auto simp add: epsilon_def hypreal_of_real_def |
|
669 |
lemma_finite_epsilon_set [THEN FreeUltrafilterNat_finite]) |
|
14370 | 670 |
|
671 |
lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x \<noteq> epsilon" |
|
14705 | 672 |
by (insert not_ex_hypreal_of_real_eq_epsilon, auto) |
14370 | 673 |
|
674 |
lemma hypreal_epsilon_not_zero: "epsilon \<noteq> 0" |
|
14468 | 675 |
by (simp add: epsilon_def hypreal_zero_def) |
14370 | 676 |
|
677 |
lemma hypreal_epsilon_inverse_omega: "epsilon = inverse(omega)" |
|
678 |
by (simp add: hypreal_inverse omega_def epsilon_def) |
|
679 |
||
680 |
||
14299 | 681 |
ML |
682 |
{* |
|
14329 | 683 |
val hrabs_def = thm "hrabs_def"; |
684 |
val hypreal_hrabs = thm "hypreal_hrabs"; |
|
685 |
||
14299 | 686 |
val hypreal_zero_def = thm "hypreal_zero_def"; |
687 |
val hypreal_one_def = thm "hypreal_one_def"; |
|
688 |
val hypreal_minus_def = thm "hypreal_minus_def"; |
|
689 |
val hypreal_diff_def = thm "hypreal_diff_def"; |
|
690 |
val hypreal_inverse_def = thm "hypreal_inverse_def"; |
|
691 |
val hypreal_divide_def = thm "hypreal_divide_def"; |
|
692 |
val hypreal_of_real_def = thm "hypreal_of_real_def"; |
|
693 |
val omega_def = thm "omega_def"; |
|
694 |
val epsilon_def = thm "epsilon_def"; |
|
695 |
val hypreal_add_def = thm "hypreal_add_def"; |
|
696 |
val hypreal_mult_def = thm "hypreal_mult_def"; |
|
697 |
val hypreal_less_def = thm "hypreal_less_def"; |
|
698 |
val hypreal_le_def = thm "hypreal_le_def"; |
|
699 |
||
700 |
val finite_exhausts = thm "finite_exhausts"; |
|
701 |
val finite_not_covers = thm "finite_not_covers"; |
|
702 |
val not_finite_nat = thm "not_finite_nat"; |
|
703 |
val FreeUltrafilterNat_Ex = thm "FreeUltrafilterNat_Ex"; |
|
704 |
val FreeUltrafilterNat_mem = thm "FreeUltrafilterNat_mem"; |
|
705 |
val FreeUltrafilterNat_finite = thm "FreeUltrafilterNat_finite"; |
|
706 |
val FreeUltrafilterNat_not_finite = thm "FreeUltrafilterNat_not_finite"; |
|
707 |
val FreeUltrafilterNat_empty = thm "FreeUltrafilterNat_empty"; |
|
708 |
val FreeUltrafilterNat_Int = thm "FreeUltrafilterNat_Int"; |
|
709 |
val FreeUltrafilterNat_subset = thm "FreeUltrafilterNat_subset"; |
|
710 |
val FreeUltrafilterNat_Compl = thm "FreeUltrafilterNat_Compl"; |
|
711 |
val FreeUltrafilterNat_Compl_mem = thm "FreeUltrafilterNat_Compl_mem"; |
|
712 |
val FreeUltrafilterNat_Compl_iff1 = thm "FreeUltrafilterNat_Compl_iff1"; |
|
713 |
val FreeUltrafilterNat_Compl_iff2 = thm "FreeUltrafilterNat_Compl_iff2"; |
|
714 |
val FreeUltrafilterNat_UNIV = thm "FreeUltrafilterNat_UNIV"; |
|
715 |
val FreeUltrafilterNat_Nat_set = thm "FreeUltrafilterNat_Nat_set"; |
|
716 |
val FreeUltrafilterNat_Nat_set_refl = thm "FreeUltrafilterNat_Nat_set_refl"; |
|
717 |
val FreeUltrafilterNat_P = thm "FreeUltrafilterNat_P"; |
|
718 |
val FreeUltrafilterNat_Ex_P = thm "FreeUltrafilterNat_Ex_P"; |
|
719 |
val FreeUltrafilterNat_all = thm "FreeUltrafilterNat_all"; |
|
720 |
val FreeUltrafilterNat_Un = thm "FreeUltrafilterNat_Un"; |
|
721 |
val hyprel_iff = thm "hyprel_iff"; |
|
722 |
val hyprel_in_hypreal = thm "hyprel_in_hypreal"; |
|
723 |
val Abs_hypreal_inverse = thm "Abs_hypreal_inverse"; |
|
724 |
val inj_on_Abs_hypreal = thm "inj_on_Abs_hypreal"; |
|
725 |
val inj_Rep_hypreal = thm "inj_Rep_hypreal"; |
|
726 |
val lemma_hyprel_refl = thm "lemma_hyprel_refl"; |
|
727 |
val hypreal_empty_not_mem = thm "hypreal_empty_not_mem"; |
|
728 |
val Rep_hypreal_nonempty = thm "Rep_hypreal_nonempty"; |
|
729 |
val inj_hypreal_of_real = thm "inj_hypreal_of_real"; |
|
730 |
val eq_Abs_hypreal = thm "eq_Abs_hypreal"; |
|
731 |
val hypreal_minus_congruent = thm "hypreal_minus_congruent"; |
|
732 |
val hypreal_minus = thm "hypreal_minus"; |
|
733 |
val hypreal_add = thm "hypreal_add"; |
|
734 |
val hypreal_diff = thm "hypreal_diff"; |
|
735 |
val hypreal_add_commute = thm "hypreal_add_commute"; |
|
736 |
val hypreal_add_assoc = thm "hypreal_add_assoc"; |
|
737 |
val hypreal_add_zero_left = thm "hypreal_add_zero_left"; |
|
738 |
val hypreal_add_zero_right = thm "hypreal_add_zero_right"; |
|
739 |
val hypreal_add_minus = thm "hypreal_add_minus"; |
|
740 |
val hypreal_add_minus_left = thm "hypreal_add_minus_left"; |
|
741 |
val hypreal_mult = thm "hypreal_mult"; |
|
742 |
val hypreal_mult_commute = thm "hypreal_mult_commute"; |
|
743 |
val hypreal_mult_assoc = thm "hypreal_mult_assoc"; |
|
744 |
val hypreal_mult_1 = thm "hypreal_mult_1"; |
|
745 |
val hypreal_zero_not_eq_one = thm "hypreal_zero_not_eq_one"; |
|
746 |
val hypreal_inverse_congruent = thm "hypreal_inverse_congruent"; |
|
747 |
val hypreal_inverse = thm "hypreal_inverse"; |
|
748 |
val hypreal_mult_inverse = thm "hypreal_mult_inverse"; |
|
749 |
val hypreal_mult_inverse_left = thm "hypreal_mult_inverse_left"; |
|
750 |
val hypreal_mult_left_cancel = thm "hypreal_mult_left_cancel"; |
|
751 |
val hypreal_mult_right_cancel = thm "hypreal_mult_right_cancel"; |
|
752 |
val hypreal_not_refl2 = thm "hypreal_not_refl2"; |
|
753 |
val hypreal_less = thm "hypreal_less"; |
|
754 |
val hypreal_eq_minus_iff = thm "hypreal_eq_minus_iff"; |
|
755 |
val hypreal_le = thm "hypreal_le"; |
|
756 |
val hypreal_le_refl = thm "hypreal_le_refl"; |
|
757 |
val hypreal_le_linear = thm "hypreal_le_linear"; |
|
758 |
val hypreal_le_trans = thm "hypreal_le_trans"; |
|
759 |
val hypreal_le_anti_sym = thm "hypreal_le_anti_sym"; |
|
760 |
val hypreal_less_le = thm "hypreal_less_le"; |
|
761 |
val hypreal_of_real_add = thm "hypreal_of_real_add"; |
|
762 |
val hypreal_of_real_mult = thm "hypreal_of_real_mult"; |
|
763 |
val hypreal_of_real_less_iff = thm "hypreal_of_real_less_iff"; |
|
764 |
val hypreal_of_real_le_iff = thm "hypreal_of_real_le_iff"; |
|
765 |
val hypreal_of_real_eq_iff = thm "hypreal_of_real_eq_iff"; |
|
766 |
val hypreal_of_real_minus = thm "hypreal_of_real_minus"; |
|
767 |
val hypreal_of_real_one = thm "hypreal_of_real_one"; |
|
768 |
val hypreal_of_real_zero = thm "hypreal_of_real_zero"; |
|
769 |
val hypreal_of_real_inverse = thm "hypreal_of_real_inverse"; |
|
770 |
val hypreal_of_real_divide = thm "hypreal_of_real_divide"; |
|
771 |
val hypreal_zero_num = thm "hypreal_zero_num"; |
|
772 |
val hypreal_one_num = thm "hypreal_one_num"; |
|
773 |
val hypreal_omega_gt_zero = thm "hypreal_omega_gt_zero"; |
|
14370 | 774 |
|
775 |
val hypreal_add_zero_less_le_mono = thm"hypreal_add_zero_less_le_mono"; |
|
776 |
val Rep_hypreal_omega = thm"Rep_hypreal_omega"; |
|
777 |
val lemma_omega_empty_singleton_disj = thm"lemma_omega_empty_singleton_disj"; |
|
778 |
val lemma_finite_omega_set = thm"lemma_finite_omega_set"; |
|
779 |
val not_ex_hypreal_of_real_eq_omega = thm"not_ex_hypreal_of_real_eq_omega"; |
|
780 |
val hypreal_of_real_not_eq_omega = thm"hypreal_of_real_not_eq_omega"; |
|
781 |
val not_ex_hypreal_of_real_eq_epsilon = thm"not_ex_hypreal_of_real_eq_epsilon"; |
|
782 |
val hypreal_of_real_not_eq_epsilon = thm"hypreal_of_real_not_eq_epsilon"; |
|
783 |
val hypreal_epsilon_not_zero = thm"hypreal_epsilon_not_zero"; |
|
784 |
val hypreal_epsilon_inverse_omega = thm"hypreal_epsilon_inverse_omega"; |
|
14299 | 785 |
*} |
786 |
||
10751 | 787 |
end |