src/HOL/Hyperreal/NSA.thy
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(*  Title       : NSA.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998  University of Cambridge
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Converted to Isar and polished by lcp
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*)
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header{*Infinite Numbers, Infinitesimals, Infinitely Close Relation*}
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theory NSA = HyperArith + RComplete:
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constdefs
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  Infinitesimal  :: "hypreal set"
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   "Infinitesimal == {x. \<forall>r \<in> Reals. 0 < r --> abs x < r}"
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  HFinite :: "hypreal set"
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   "HFinite == {x. \<exists>r \<in> Reals. abs x < r}"
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  HInfinite :: "hypreal set"
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   "HInfinite == {x. \<forall>r \<in> Reals. r < abs x}"
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  (* standard part map *)
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  st        :: "hypreal => hypreal"
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   "st           == (%x. @r. x \<in> HFinite & r \<in> Reals & r @= x)"
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  monad     :: "hypreal => hypreal set"
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   "monad x      == {y. x @= y}"
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  galaxy    :: "hypreal => hypreal set"
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   "galaxy x     == {y. (x + -y) \<in> HFinite}"
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  (* infinitely close *)
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  approx :: "[hypreal, hypreal] => bool"    (infixl "@=" 50)
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   "x @= y       == (x + -y) \<in> Infinitesimal"
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defs (overloaded)
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   (*standard real numbers as a subset of the hyperreals*)
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   SReal_def:      "Reals == {x. \<exists>r. x = hypreal_of_real r}"
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syntax (xsymbols)
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    approx :: "[hypreal, hypreal] => bool"    (infixl "\<approx>" 50)
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subsection{*Closure Laws for  Standard Reals*}
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lemma SReal_add [simp]:
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     "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x + y \<in> Reals"
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apply (auto simp add: SReal_def)
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apply (rule_tac x = "r + ra" in exI, simp)
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done
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lemma SReal_mult: "[| (x::hypreal) \<in> Reals; y \<in> Reals |] ==> x * y \<in> Reals"
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apply (simp add: SReal_def, safe)
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apply (rule_tac x = "r * ra" in exI)
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apply (simp (no_asm) add: hypreal_of_real_mult)
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done
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lemma SReal_inverse: "(x::hypreal) \<in> Reals ==> inverse x \<in> Reals"
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apply (simp add: SReal_def)
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apply (blast intro: hypreal_of_real_inverse [symmetric])
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done
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lemma SReal_divide: "[| (x::hypreal) \<in> Reals;  y \<in> Reals |] ==> x/y \<in> Reals"
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apply (simp (no_asm_simp) add: SReal_mult SReal_inverse hypreal_divide_def)
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done
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lemma SReal_minus: "(x::hypreal) \<in> Reals ==> -x \<in> Reals"
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apply (simp add: SReal_def)
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apply (blast intro: hypreal_of_real_minus [symmetric])
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done
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lemma SReal_minus_iff: "(-x \<in> Reals) = ((x::hypreal) \<in> Reals)"
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apply auto
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apply (erule_tac [2] SReal_minus)
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apply (drule SReal_minus, auto)
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done
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declare SReal_minus_iff [simp]
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lemma SReal_add_cancel:
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     "[| (x::hypreal) + y \<in> Reals; y \<in> Reals |] ==> x \<in> Reals"
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apply (drule_tac x = y in SReal_minus)
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apply (drule SReal_add, assumption, auto)
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done
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lemma SReal_hrabs: "(x::hypreal) \<in> Reals ==> abs x \<in> Reals"
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apply (simp add: SReal_def)
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apply (auto simp add: hypreal_of_real_hrabs)
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done
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lemma SReal_hypreal_of_real: "hypreal_of_real x \<in> Reals"
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by (simp add: SReal_def)
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declare SReal_hypreal_of_real [simp]
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lemma SReal_number_of: "(number_of w ::hypreal) \<in> Reals"
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apply (simp only: hypreal_number_of [symmetric])
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apply (rule SReal_hypreal_of_real)
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done
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declare SReal_number_of [simp]
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(** As always with numerals, 0 and 1 are special cases **)
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lemma Reals_0: "(0::hypreal) \<in> Reals"
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apply (subst numeral_0_eq_0 [symmetric])
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apply (rule SReal_number_of)
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done
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declare Reals_0 [simp]
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lemma Reals_1: "(1::hypreal) \<in> Reals"
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apply (subst numeral_1_eq_1 [symmetric])
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apply (rule SReal_number_of)
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done
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declare Reals_1 [simp]
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lemma SReal_divide_number_of: "r \<in> Reals ==> r/(number_of w::hypreal) \<in> Reals"
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apply (unfold hypreal_divide_def)
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apply (blast intro!: SReal_number_of SReal_mult SReal_inverse)
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done
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(* Infinitesimal epsilon not in Reals *)
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lemma SReal_epsilon_not_mem: "epsilon \<notin> Reals"
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apply (simp add: SReal_def)
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apply (auto simp add: hypreal_of_real_not_eq_epsilon [THEN not_sym])
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done
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lemma SReal_omega_not_mem: "omega \<notin> Reals"
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apply (simp add: SReal_def)
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apply (auto simp add: hypreal_of_real_not_eq_omega [THEN not_sym])
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done
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lemma SReal_UNIV_real: "{x. hypreal_of_real x \<in> Reals} = (UNIV::real set)"
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by (simp add: SReal_def)
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lemma SReal_iff: "(x \<in> Reals) = (\<exists>y. x = hypreal_of_real y)"
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by (simp add: SReal_def)
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lemma hypreal_of_real_image: "hypreal_of_real `(UNIV::real set) = Reals"
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by (auto simp add: SReal_def)
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lemma inv_hypreal_of_real_image: "inv hypreal_of_real ` Reals = UNIV"
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apply (auto simp add: SReal_def)
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apply (rule inj_hypreal_of_real [THEN inv_f_f, THEN subst], blast)
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done
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lemma SReal_hypreal_of_real_image:
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      "[| \<exists>x. x: P; P \<subseteq> Reals |] ==> \<exists>Q. P = hypreal_of_real ` Q"
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apply (simp add: SReal_def, blast)
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done
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lemma SReal_dense:
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     "[| (x::hypreal) \<in> Reals; y \<in> Reals;  x<y |] ==> \<exists>r \<in> Reals. x<r & r<y"
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apply (auto simp add: SReal_iff)
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apply (drule real_dense, safe)
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apply (rule_tac x = "hypreal_of_real r" in bexI, auto)
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done
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(*------------------------------------------------------------------
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                   Completeness of Reals
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 ------------------------------------------------------------------*)
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lemma SReal_sup_lemma:
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     "P \<subseteq> Reals ==> ((\<exists>x \<in> P. y < x) =
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      (\<exists>X. hypreal_of_real X \<in> P & y < hypreal_of_real X))"
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by (blast dest!: SReal_iff [THEN iffD1])
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lemma SReal_sup_lemma2:
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     "[| P \<subseteq> Reals; \<exists>x. x \<in> P; \<exists>y \<in> Reals. \<forall>x \<in> P. x < y |]
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      ==> (\<exists>X. X \<in> {w. hypreal_of_real w \<in> P}) &
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          (\<exists>Y. \<forall>X \<in> {w. hypreal_of_real w \<in> P}. X < Y)"
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apply (rule conjI)
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apply (fast dest!: SReal_iff [THEN iffD1])
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apply (auto, frule subsetD, assumption)
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apply (drule SReal_iff [THEN iffD1])
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apply (auto, rule_tac x = ya in exI, auto)
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done
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(*------------------------------------------------------
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    lifting of ub and property of lub
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 -------------------------------------------------------*)
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lemma hypreal_of_real_isUb_iff:
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      "(isUb (Reals) (hypreal_of_real ` Q) (hypreal_of_real Y)) =
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       (isUb (UNIV :: real set) Q Y)"
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apply (simp add: isUb_def setle_def)
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done
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lemma hypreal_of_real_isLub1:
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     "isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)
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      ==> isLub (UNIV :: real set) Q Y"
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apply (simp add: isLub_def leastP_def)
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apply (auto intro: hypreal_of_real_isUb_iff [THEN iffD2]
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            simp add: hypreal_of_real_isUb_iff setge_def)
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done
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lemma hypreal_of_real_isLub2:
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      "isLub (UNIV :: real set) Q Y
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       ==> isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)"
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apply (simp add: isLub_def leastP_def)
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apply (auto simp add: hypreal_of_real_isUb_iff setge_def)
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apply (frule_tac x2 = x in isUbD2a [THEN SReal_iff [THEN iffD1], THEN exE])
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 prefer 2 apply assumption
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apply (drule_tac x = xa in spec)
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apply (auto simp add: hypreal_of_real_isUb_iff)
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done
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lemma hypreal_of_real_isLub_iff:
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     "(isLub Reals (hypreal_of_real ` Q) (hypreal_of_real Y)) =
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      (isLub (UNIV :: real set) Q Y)"
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by (blast intro: hypreal_of_real_isLub1 hypreal_of_real_isLub2)
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(* lemmas *)
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lemma lemma_isUb_hypreal_of_real:
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     "isUb Reals P Y ==> \<exists>Yo. isUb Reals P (hypreal_of_real Yo)"
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by (auto simp add: SReal_iff isUb_def)
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lemma lemma_isLub_hypreal_of_real:
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     "isLub Reals P Y ==> \<exists>Yo. isLub Reals P (hypreal_of_real Yo)"
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by (auto simp add: isLub_def leastP_def isUb_def SReal_iff)
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lemma lemma_isLub_hypreal_of_real2:
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     "\<exists>Yo. isLub Reals P (hypreal_of_real Yo) ==> \<exists>Y. isLub Reals P Y"
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by (auto simp add: isLub_def leastP_def isUb_def)
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lemma SReal_complete:
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     "[| P \<subseteq> Reals;  \<exists>x. x \<in> P;  \<exists>Y. isUb Reals P Y |]
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      ==> \<exists>t::hypreal. isLub Reals P t"
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apply (frule SReal_hypreal_of_real_image)
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apply (auto, drule lemma_isUb_hypreal_of_real)
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apply (auto intro!: reals_complete lemma_isLub_hypreal_of_real2 simp add: hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff)
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done
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subsection{* Set of Finite Elements is a Subring of the Extended Reals*}
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lemma HFinite_add: "[|x \<in> HFinite; y \<in> HFinite|] ==> (x+y) \<in> HFinite"
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apply (simp add: HFinite_def)
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apply (blast intro!: SReal_add hrabs_add_less)
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done
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lemma HFinite_mult: "[|x \<in> HFinite; y \<in> HFinite|] ==> x*y \<in> HFinite"
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apply (simp add: HFinite_def)
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apply (blast intro!: SReal_mult abs_mult_less)
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done
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lemma HFinite_minus_iff: "(-x \<in> HFinite) = (x \<in> HFinite)"
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by (simp add: HFinite_def)
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lemma SReal_subset_HFinite: "Reals \<subseteq> HFinite"
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apply (auto simp add: SReal_def HFinite_def)
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apply (rule_tac x = "1 + abs (hypreal_of_real r) " in exI)
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apply (auto simp add: hypreal_of_real_hrabs)
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apply (rule_tac x = "1 + abs r" in exI, simp)
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done
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lemma HFinite_hypreal_of_real [simp]: "hypreal_of_real x \<in> HFinite"
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by (auto intro: SReal_subset_HFinite [THEN subsetD])
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lemma HFiniteD: "x \<in> HFinite ==> \<exists>t \<in> Reals. abs x < t"
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by (simp add: HFinite_def)
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lemma HFinite_hrabs_iff: "(abs x \<in> HFinite) = (x \<in> HFinite)"
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by (simp add: HFinite_def)
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declare HFinite_hrabs_iff [iff]
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lemma HFinite_number_of: "number_of w \<in> HFinite"
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by (rule SReal_number_of [THEN SReal_subset_HFinite [THEN subsetD]])
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declare HFinite_number_of [simp]
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(** As always with numerals, 0 and 1 are special cases **)
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lemma HFinite_0: "0 \<in> HFinite"
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apply (subst numeral_0_eq_0 [symmetric])
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apply (rule HFinite_number_of)
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done
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declare HFinite_0 [simp]
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lemma HFinite_1: "1 \<in> HFinite"
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apply (subst numeral_1_eq_1 [symmetric])
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apply (rule HFinite_number_of)
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done
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declare HFinite_1 [simp]
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lemma HFinite_bounded: "[|x \<in> HFinite; y \<le> x; 0 \<le> y |] ==> y \<in> HFinite"
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apply (case_tac "x \<le> 0")
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apply (drule_tac y = x in order_trans)
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apply (drule_tac [2] hypreal_le_anti_sym)
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apply (auto simp add: linorder_not_le)
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apply (auto intro: order_le_less_trans simp add: abs_if HFinite_def)
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done
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subsection{* Set of Infinitesimals is a Subring of the Hyperreals*}
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lemma InfinitesimalD:
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      "x \<in> Infinitesimal ==> \<forall>r \<in> Reals. 0 < r --> abs x < r"
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by (simp add: Infinitesimal_def)
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lemma Infinitesimal_zero: "0 \<in> Infinitesimal"
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by (simp add: Infinitesimal_def)
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declare Infinitesimal_zero [iff]
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lemma hypreal_sum_of_halves: "x/(2::hypreal) + x/(2::hypreal) = x"
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by auto
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lemma hypreal_half_gt_zero: "0 < r ==> 0 < r/(2::hypreal)"
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by auto
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lemma Infinitesimal_add:
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     "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> (x+y) \<in> Infinitesimal"
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apply (auto simp add: Infinitesimal_def)
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   313
apply (rule hypreal_sum_of_halves [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   314
apply (drule hypreal_half_gt_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   315
apply (blast intro: hrabs_add_less hrabs_add_less SReal_divide_number_of)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   316
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   317
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   318
lemma Infinitesimal_minus_iff: "(-x:Infinitesimal) = (x:Infinitesimal)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   319
by (simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   320
declare Infinitesimal_minus_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   321
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   322
lemma Infinitesimal_diff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   323
     "[| x \<in> Infinitesimal;  y \<in> Infinitesimal |] ==> x-y \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   324
by (simp add: hypreal_diff_def Infinitesimal_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   325
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   326
lemma Infinitesimal_mult:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   327
     "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> (x * y) \<in> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   328
apply (auto simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   329
apply (case_tac "y=0")
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   330
apply (cut_tac [2] a = "abs x" and b = 1 and c = "abs y" and d = r in mult_strict_mono, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   331
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   332
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   333
lemma Infinitesimal_HFinite_mult:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   334
     "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> (x * y) \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   335
apply (auto dest!: HFiniteD simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   336
apply (frule hrabs_less_gt_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   337
apply (drule_tac x = "r/t" in bspec)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   338
apply (blast intro: SReal_divide)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   339
apply (simp add: zero_less_divide_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   340
apply (case_tac "x=0 | y=0")
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   341
apply (cut_tac [2] a = "abs x" and b = "r/t" and c = "abs y" in mult_strict_mono)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   342
apply (auto simp add: zero_less_divide_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   343
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   344
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   345
lemma Infinitesimal_HFinite_mult2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   346
     "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> (y * x) \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   347
by (auto dest: Infinitesimal_HFinite_mult simp add: hypreal_mult_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   348
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   349
(*** rather long proof ***)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   350
lemma HInfinite_inverse_Infinitesimal:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   351
     "x \<in> HInfinite ==> inverse x: Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   352
apply (auto simp add: HInfinite_def Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   353
apply (erule_tac x = "inverse r" in ballE)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   354
apply (frule_tac a1 = r and z = "abs x" in positive_imp_inverse_positive [THEN order_less_trans], assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   355
apply (drule inverse_inverse_eq [symmetric, THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   356
apply (rule inverse_less_iff_less [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   357
apply (auto simp add: SReal_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   358
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   359
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   360
lemma HInfinite_mult: "[|x \<in> HInfinite;y \<in> HInfinite|] ==> (x*y) \<in> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   361
apply (simp add: HInfinite_def, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   362
apply (erule_tac x = 1 in ballE)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   363
apply (erule_tac x = r in ballE)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   364
apply (case_tac "y=0")
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   365
apply (cut_tac [2] c = 1 and d = "abs x" and a = r and b = "abs y" in mult_strict_mono)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   366
apply (auto simp add: mult_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   367
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   368
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   369
lemma HInfinite_add_ge_zero:
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   370
      "[|x \<in> HInfinite; 0 \<le> y; 0 \<le> x|] ==> (x + y): HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   371
by (auto intro!: hypreal_add_zero_less_le_mono 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   372
       simp add: abs_if hypreal_add_commute hypreal_le_add_order HInfinite_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   373
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   374
lemma HInfinite_add_ge_zero2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   375
     "[|x \<in> HInfinite; 0 \<le> y; 0 \<le> x|] ==> (y + x): HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   376
by (auto intro!: HInfinite_add_ge_zero simp add: hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   377
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   378
lemma HInfinite_add_gt_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   379
     "[|x \<in> HInfinite; 0 < y; 0 < x|] ==> (x + y): HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   380
by (blast intro: HInfinite_add_ge_zero order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   381
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   382
lemma HInfinite_minus_iff: "(-x \<in> HInfinite) = (x \<in> HInfinite)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   383
by (simp add: HInfinite_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   384
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   385
lemma HInfinite_add_le_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   386
     "[|x \<in> HInfinite; y \<le> 0; x \<le> 0|] ==> (x + y): HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   387
apply (drule HInfinite_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   388
apply (rule HInfinite_minus_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   389
apply (auto intro: HInfinite_add_ge_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   390
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   391
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   392
lemma HInfinite_add_lt_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   393
     "[|x \<in> HInfinite; y < 0; x < 0|] ==> (x + y): HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   394
by (blast intro: HInfinite_add_le_zero order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   395
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   396
lemma HFinite_sum_squares:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   397
     "[|a: HFinite; b: HFinite; c: HFinite|]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   398
      ==> a*a + b*b + c*c \<in> HFinite"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   399
by (auto intro: HFinite_mult HFinite_add)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   400
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   401
lemma not_Infinitesimal_not_zero: "x \<notin> Infinitesimal ==> x \<noteq> 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   402
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   403
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   404
lemma not_Infinitesimal_not_zero2: "x \<in> HFinite - Infinitesimal ==> x \<noteq> 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   405
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   406
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   407
lemma Infinitesimal_hrabs_iff: "(abs x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   408
by (auto simp add: hrabs_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   409
declare Infinitesimal_hrabs_iff [iff]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   410
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   411
lemma HFinite_diff_Infinitesimal_hrabs:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   412
     "x \<in> HFinite - Infinitesimal ==> abs x \<in> HFinite - Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   413
by blast
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   414
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   415
lemma hrabs_less_Infinitesimal:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   416
      "[| e \<in> Infinitesimal; abs x < e |] ==> x \<in> Infinitesimal"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   417
by (auto simp add: Infinitesimal_def abs_less_iff)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   418
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   419
lemma hrabs_le_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   420
     "[| e \<in> Infinitesimal; abs x \<le> e |] ==> x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   421
by (blast dest: order_le_imp_less_or_eq intro: hrabs_less_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   422
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   423
lemma Infinitesimal_interval:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   424
      "[| e \<in> Infinitesimal; e' \<in> Infinitesimal; e' < x ; x < e |] 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   425
       ==> x \<in> Infinitesimal"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   426
by (auto simp add: Infinitesimal_def abs_less_iff)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   427
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   428
lemma Infinitesimal_interval2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   429
     "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   430
         e' \<le> x ; x \<le> e |] ==> x \<in> Infinitesimal"
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   431
by (auto intro: Infinitesimal_interval simp add: order_le_less)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   432
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   433
lemma not_Infinitesimal_mult:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   434
     "[| x \<notin> Infinitesimal;  y \<notin> Infinitesimal|] ==> (x*y) \<notin>Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   435
apply (unfold Infinitesimal_def, clarify)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   436
apply (simp add: linorder_not_less)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   437
apply (erule_tac x = "r*ra" in ballE)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   438
prefer 2 apply (fast intro: SReal_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   439
apply (auto simp add: zero_less_mult_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   440
apply (cut_tac c = ra and d = "abs y" and a = r and b = "abs x" in mult_mono, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   441
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   442
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   443
lemma Infinitesimal_mult_disj:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   444
     "x*y \<in> Infinitesimal ==> x \<in> Infinitesimal | y \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   445
apply (rule ccontr)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   446
apply (drule de_Morgan_disj [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   447
apply (fast dest: not_Infinitesimal_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   448
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   449
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   450
lemma HFinite_Infinitesimal_not_zero: "x \<in> HFinite-Infinitesimal ==> x \<noteq> 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   451
by blast
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   452
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   453
lemma HFinite_Infinitesimal_diff_mult:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   454
     "[| x \<in> HFinite - Infinitesimal;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   455
                   y \<in> HFinite - Infinitesimal
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   456
                |] ==> (x*y) \<in> HFinite - Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   457
apply clarify
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   458
apply (blast dest: HFinite_mult not_Infinitesimal_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   459
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   460
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   461
lemma Infinitesimal_subset_HFinite:
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   462
      "Infinitesimal \<subseteq> HFinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   463
apply (simp add: Infinitesimal_def HFinite_def, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   464
apply (rule_tac x = 1 in bexI, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   465
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   466
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   467
lemma Infinitesimal_hypreal_of_real_mult:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   468
     "x \<in> Infinitesimal ==> x * hypreal_of_real r \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   469
by (erule HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   470
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   471
lemma Infinitesimal_hypreal_of_real_mult2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   472
     "x \<in> Infinitesimal ==> hypreal_of_real r * x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   473
by (erule HFinite_hypreal_of_real [THEN [2] Infinitesimal_HFinite_mult2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   474
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   475
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   476
subsection{*The Infinitely Close Relation*}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   477
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   478
lemma mem_infmal_iff: "(x \<in> Infinitesimal) = (x @= 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   479
by (simp add: Infinitesimal_def approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   480
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   481
lemma approx_minus_iff: " (x @= y) = (x + -y @= 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   482
by (simp add: approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   483
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   484
lemma approx_minus_iff2: " (x @= y) = (-y + x @= 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   485
by (simp add: approx_def hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   486
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   487
lemma approx_refl: "x @= x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   488
by (simp add: approx_def Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   489
declare approx_refl [iff]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   490
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   491
lemma approx_sym: "x @= y ==> y @= x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   492
apply (simp add: approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   493
apply (rule hypreal_minus_distrib1 [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   494
apply (erule Infinitesimal_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   495
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   496
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   497
lemma approx_trans: "[| x @= y; y @= z |] ==> x @= z"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   498
apply (simp add: approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   499
apply (drule Infinitesimal_add, assumption, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   500
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   501
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   502
lemma approx_trans2: "[| r @= x; s @= x |] ==> r @= s"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   503
by (blast intro: approx_sym approx_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   504
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   505
lemma approx_trans3: "[| x @= r; x @= s|] ==> r @= s"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   506
by (blast intro: approx_sym approx_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   507
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   508
lemma number_of_approx_reorient: "(number_of w @= x) = (x @= number_of w)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   509
by (blast intro: approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   510
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   511
lemma zero_approx_reorient: "(0 @= x) = (x @= 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   512
by (blast intro: approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   513
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   514
lemma one_approx_reorient: "(1 @= x) = (x @= 1)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   515
by (blast intro: approx_sym)
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
   516
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
   517
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   518
ML
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   519
{*
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   520
val SReal_add = thm "SReal_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   521
val SReal_mult = thm "SReal_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   522
val SReal_inverse = thm "SReal_inverse";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   523
val SReal_divide = thm "SReal_divide";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   524
val SReal_minus = thm "SReal_minus";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   525
val SReal_minus_iff = thm "SReal_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   526
val SReal_add_cancel = thm "SReal_add_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   527
val SReal_hrabs = thm "SReal_hrabs";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   528
val SReal_hypreal_of_real = thm "SReal_hypreal_of_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   529
val SReal_number_of = thm "SReal_number_of";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   530
val Reals_0 = thm "Reals_0";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   531
val Reals_1 = thm "Reals_1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   532
val SReal_divide_number_of = thm "SReal_divide_number_of";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   533
val SReal_epsilon_not_mem = thm "SReal_epsilon_not_mem";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   534
val SReal_omega_not_mem = thm "SReal_omega_not_mem";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   535
val SReal_UNIV_real = thm "SReal_UNIV_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   536
val SReal_iff = thm "SReal_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   537
val hypreal_of_real_image = thm "hypreal_of_real_image";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   538
val inv_hypreal_of_real_image = thm "inv_hypreal_of_real_image";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   539
val SReal_hypreal_of_real_image = thm "SReal_hypreal_of_real_image";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   540
val SReal_dense = thm "SReal_dense";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   541
val SReal_sup_lemma = thm "SReal_sup_lemma";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   542
val SReal_sup_lemma2 = thm "SReal_sup_lemma2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   543
val hypreal_of_real_isUb_iff = thm "hypreal_of_real_isUb_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   544
val hypreal_of_real_isLub1 = thm "hypreal_of_real_isLub1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   545
val hypreal_of_real_isLub2 = thm "hypreal_of_real_isLub2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   546
val hypreal_of_real_isLub_iff = thm "hypreal_of_real_isLub_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   547
val lemma_isUb_hypreal_of_real = thm "lemma_isUb_hypreal_of_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   548
val lemma_isLub_hypreal_of_real = thm "lemma_isLub_hypreal_of_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   549
val lemma_isLub_hypreal_of_real2 = thm "lemma_isLub_hypreal_of_real2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   550
val SReal_complete = thm "SReal_complete";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   551
val HFinite_add = thm "HFinite_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   552
val HFinite_mult = thm "HFinite_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   553
val HFinite_minus_iff = thm "HFinite_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   554
val SReal_subset_HFinite = thm "SReal_subset_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   555
val HFinite_hypreal_of_real = thm "HFinite_hypreal_of_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   556
val HFiniteD = thm "HFiniteD";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   557
val HFinite_hrabs_iff = thm "HFinite_hrabs_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   558
val HFinite_number_of = thm "HFinite_number_of";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   559
val HFinite_0 = thm "HFinite_0";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   560
val HFinite_1 = thm "HFinite_1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   561
val HFinite_bounded = thm "HFinite_bounded";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   562
val InfinitesimalD = thm "InfinitesimalD";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   563
val Infinitesimal_zero = thm "Infinitesimal_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   564
val hypreal_sum_of_halves = thm "hypreal_sum_of_halves";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   565
val hypreal_half_gt_zero = thm "hypreal_half_gt_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   566
val Infinitesimal_add = thm "Infinitesimal_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   567
val Infinitesimal_minus_iff = thm "Infinitesimal_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   568
val Infinitesimal_diff = thm "Infinitesimal_diff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   569
val Infinitesimal_mult = thm "Infinitesimal_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   570
val Infinitesimal_HFinite_mult = thm "Infinitesimal_HFinite_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   571
val Infinitesimal_HFinite_mult2 = thm "Infinitesimal_HFinite_mult2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   572
val HInfinite_inverse_Infinitesimal = thm "HInfinite_inverse_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   573
val HInfinite_mult = thm "HInfinite_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   574
val HInfinite_add_ge_zero = thm "HInfinite_add_ge_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   575
val HInfinite_add_ge_zero2 = thm "HInfinite_add_ge_zero2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   576
val HInfinite_add_gt_zero = thm "HInfinite_add_gt_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   577
val HInfinite_minus_iff = thm "HInfinite_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   578
val HInfinite_add_le_zero = thm "HInfinite_add_le_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   579
val HInfinite_add_lt_zero = thm "HInfinite_add_lt_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   580
val HFinite_sum_squares = thm "HFinite_sum_squares";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   581
val not_Infinitesimal_not_zero = thm "not_Infinitesimal_not_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   582
val not_Infinitesimal_not_zero2 = thm "not_Infinitesimal_not_zero2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   583
val Infinitesimal_hrabs_iff = thm "Infinitesimal_hrabs_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   584
val HFinite_diff_Infinitesimal_hrabs = thm "HFinite_diff_Infinitesimal_hrabs";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   585
val hrabs_less_Infinitesimal = thm "hrabs_less_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   586
val hrabs_le_Infinitesimal = thm "hrabs_le_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   587
val Infinitesimal_interval = thm "Infinitesimal_interval";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   588
val Infinitesimal_interval2 = thm "Infinitesimal_interval2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   589
val not_Infinitesimal_mult = thm "not_Infinitesimal_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   590
val Infinitesimal_mult_disj = thm "Infinitesimal_mult_disj";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   591
val HFinite_Infinitesimal_not_zero = thm "HFinite_Infinitesimal_not_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   592
val HFinite_Infinitesimal_diff_mult = thm "HFinite_Infinitesimal_diff_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   593
val Infinitesimal_subset_HFinite = thm "Infinitesimal_subset_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   594
val Infinitesimal_hypreal_of_real_mult = thm "Infinitesimal_hypreal_of_real_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   595
val Infinitesimal_hypreal_of_real_mult2 = thm "Infinitesimal_hypreal_of_real_mult2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   596
val mem_infmal_iff = thm "mem_infmal_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   597
val approx_minus_iff = thm "approx_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   598
val approx_minus_iff2 = thm "approx_minus_iff2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   599
val approx_refl = thm "approx_refl";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   600
val approx_sym = thm "approx_sym";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   601
val approx_trans = thm "approx_trans";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   602
val approx_trans2 = thm "approx_trans2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   603
val approx_trans3 = thm "approx_trans3";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   604
val number_of_approx_reorient = thm "number_of_approx_reorient";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   605
val zero_approx_reorient = thm "zero_approx_reorient";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   606
val one_approx_reorient = thm "one_approx_reorient";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   607
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   608
(*** re-orientation, following HOL/Integ/Bin.ML
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   609
     We re-orient x @=y where x is 0, 1 or a numeral, unless y is as well!
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   610
 ***)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   611
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   612
(*reorientation simprules using ==, for the following simproc*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   613
val meta_zero_approx_reorient = zero_approx_reorient RS eq_reflection;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   614
val meta_one_approx_reorient = one_approx_reorient RS eq_reflection;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   615
val meta_number_of_approx_reorient = number_of_approx_reorient RS eq_reflection
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   616
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   617
(*reorientation simplification procedure: reorients (polymorphic)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   618
  0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   619
fun reorient_proc sg _ (_ $ t $ u) =
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   620
  case u of
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   621
      Const("0", _) => None
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   622
    | Const("1", _) => None
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   623
    | Const("Numeral.number_of", _) $ _ => None
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   624
    | _ => Some (case t of
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   625
                Const("0", _) => meta_zero_approx_reorient
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   626
              | Const("1", _) => meta_one_approx_reorient
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   627
              | Const("Numeral.number_of", _) $ _ =>
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   628
                                 meta_number_of_approx_reorient);
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   629
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   630
val approx_reorient_simproc =
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   631
  Bin_Simprocs.prep_simproc
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   632
    ("reorient_simproc", ["0@=x", "1@=x", "number_of w @= x"], reorient_proc);
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   633
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   634
Addsimprocs [approx_reorient_simproc];
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   635
*}
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   636
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   637
lemma Infinitesimal_approx_minus: "(x-y \<in> Infinitesimal) = (x @= y)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   638
by (auto simp add: hypreal_diff_def approx_minus_iff [symmetric] mem_infmal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   639
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   640
lemma approx_monad_iff: "(x @= y) = (monad(x)=monad(y))"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   641
apply (simp add: monad_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   642
apply (auto dest: approx_sym elim!: approx_trans equalityCE)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   643
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   644
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   645
lemma Infinitesimal_approx:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   646
     "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> x @= y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   647
apply (simp add: mem_infmal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   648
apply (blast intro: approx_trans approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   649
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   650
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   651
lemma approx_add: "[| a @= b; c @= d |] ==> a+c @= b+d"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   652
proof (unfold approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   653
  assume inf: "a + - b \<in> Infinitesimal" "c + - d \<in> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   654
  have "a + c + - (b + d) = (a + - b) + (c + - d)" by arith
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   655
  also have "... \<in> Infinitesimal" using inf by (rule Infinitesimal_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   656
  finally show "a + c + - (b + d) \<in> Infinitesimal" .
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   657
qed
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   658
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   659
lemma approx_minus: "a @= b ==> -a @= -b"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   660
apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   661
apply (drule approx_minus_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   662
apply (simp (no_asm) add: hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   663
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   664
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   665
lemma approx_minus2: "-a @= -b ==> a @= b"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   666
by (auto dest: approx_minus)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   667
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   668
lemma approx_minus_cancel: "(-a @= -b) = (a @= b)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   669
by (blast intro: approx_minus approx_minus2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   670
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   671
declare approx_minus_cancel [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   672
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   673
lemma approx_add_minus: "[| a @= b; c @= d |] ==> a + -c @= b + -d"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   674
by (blast intro!: approx_add approx_minus)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   675
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   676
lemma approx_mult1: "[| a @= b; c: HFinite|] ==> a*c @= b*c"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   677
by (simp add: approx_def Infinitesimal_HFinite_mult minus_mult_left 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   678
              left_distrib [symmetric] 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   679
         del: minus_mult_left [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   680
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   681
lemma approx_mult2: "[|a @= b; c: HFinite|] ==> c*a @= c*b"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   682
apply (simp (no_asm_simp) add: approx_mult1 hypreal_mult_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   683
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   684
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   685
lemma approx_mult_subst: "[|u @= v*x; x @= y; v \<in> HFinite|] ==> u @= v*y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   686
by (blast intro: approx_mult2 approx_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   687
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   688
lemma approx_mult_subst2: "[| u @= x*v; x @= y; v \<in> HFinite |] ==> u @= y*v"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   689
by (blast intro: approx_mult1 approx_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   690
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   691
lemma approx_mult_subst_SReal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   692
     "[| u @= x*hypreal_of_real v; x @= y |] ==> u @= y*hypreal_of_real v"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   693
by (auto intro: approx_mult_subst2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   694
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   695
lemma approx_eq_imp: "a = b ==> a @= b"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   696
by (simp add: approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   697
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   698
lemma Infinitesimal_minus_approx: "x \<in> Infinitesimal ==> -x @= x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   699
by (blast intro: Infinitesimal_minus_iff [THEN iffD2] 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   700
                    mem_infmal_iff [THEN iffD1] approx_trans2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   701
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   702
lemma bex_Infinitesimal_iff: "(\<exists>y \<in> Infinitesimal. x + -z = y) = (x @= z)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   703
by (simp add: approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   704
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   705
lemma bex_Infinitesimal_iff2: "(\<exists>y \<in> Infinitesimal. x = z + y) = (x @= z)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   706
by (force simp add: bex_Infinitesimal_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   707
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   708
lemma Infinitesimal_add_approx: "[| y \<in> Infinitesimal; x + y = z |] ==> x @= z"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   709
apply (rule bex_Infinitesimal_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   710
apply (drule Infinitesimal_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   711
apply (auto simp add: minus_add_distrib hypreal_add_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   712
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   713
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   714
lemma Infinitesimal_add_approx_self: "y \<in> Infinitesimal ==> x @= x + y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   715
apply (rule bex_Infinitesimal_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   716
apply (drule Infinitesimal_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   717
apply (auto simp add: minus_add_distrib hypreal_add_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   718
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   719
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   720
lemma Infinitesimal_add_approx_self2: "y \<in> Infinitesimal ==> x @= y + x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   721
by (auto dest: Infinitesimal_add_approx_self simp add: hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   722
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   723
lemma Infinitesimal_add_minus_approx_self: "y \<in> Infinitesimal ==> x @= x + -y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   724
by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   725
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   726
lemma Infinitesimal_add_cancel: "[| y \<in> Infinitesimal; x+y @= z|] ==> x @= z"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   727
apply (drule_tac x = x in Infinitesimal_add_approx_self [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   728
apply (erule approx_trans3 [THEN approx_sym], assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   729
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   730
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   731
lemma Infinitesimal_add_right_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   732
     "[| y \<in> Infinitesimal; x @= z + y|] ==> x @= z"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   733
apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   734
apply (erule approx_trans3 [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   735
apply (simp add: hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   736
apply (erule approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   737
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   738
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   739
lemma approx_add_left_cancel: "d + b  @= d + c ==> b @= c"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   740
apply (drule approx_minus_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   741
apply (simp add: minus_add_distrib approx_minus_iff [symmetric] add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   742
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   743
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   744
lemma approx_add_right_cancel: "b + d @= c + d ==> b @= c"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   745
apply (rule approx_add_left_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   746
apply (simp add: hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   747
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   748
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   749
lemma approx_add_mono1: "b @= c ==> d + b @= d + c"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   750
apply (rule approx_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   751
apply (simp add: minus_add_distrib approx_minus_iff [symmetric] add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   752
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   753
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   754
lemma approx_add_mono2: "b @= c ==> b + a @= c + a"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   755
apply (simp (no_asm_simp) add: hypreal_add_commute approx_add_mono1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   756
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   757
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   758
lemma approx_add_left_iff: "(a + b @= a + c) = (b @= c)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   759
by (fast elim: approx_add_left_cancel approx_add_mono1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   760
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   761
declare approx_add_left_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   762
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   763
lemma approx_add_right_iff: "(b + a @= c + a) = (b @= c)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   764
apply (simp (no_asm) add: hypreal_add_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   765
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   766
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   767
declare approx_add_right_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   768
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   769
lemma approx_HFinite: "[| x \<in> HFinite; x @= y |] ==> y \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   770
apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   771
apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   772
apply (drule HFinite_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   773
apply (auto simp add: hypreal_add_assoc)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   774
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   775
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   776
lemma approx_hypreal_of_real_HFinite: "x @= hypreal_of_real D ==> x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   777
by (rule approx_sym [THEN [2] approx_HFinite], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   778
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   779
lemma approx_mult_HFinite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   780
     "[|a @= b; c @= d; b: HFinite; d: HFinite|] ==> a*c @= b*d"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   781
apply (rule approx_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   782
apply (rule_tac [2] approx_mult2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   783
apply (rule approx_mult1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   784
prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   785
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   786
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   787
lemma approx_mult_hypreal_of_real:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   788
     "[|a @= hypreal_of_real b; c @= hypreal_of_real d |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   789
      ==> a*c @= hypreal_of_real b*hypreal_of_real d"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   790
apply (blast intro!: approx_mult_HFinite approx_hypreal_of_real_HFinite HFinite_hypreal_of_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   791
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   792
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   793
lemma approx_SReal_mult_cancel_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   794
     "[| a \<in> Reals; a \<noteq> 0; a*x @= 0 |] ==> x @= 0"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   795
apply (drule SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   796
apply (auto dest: approx_mult2 simp add: hypreal_mult_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   797
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   798
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   799
(* REM comments: newly added *)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   800
lemma approx_mult_SReal1: "[| a \<in> Reals; x @= 0 |] ==> x*a @= 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   801
by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   802
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   803
lemma approx_mult_SReal2: "[| a \<in> Reals; x @= 0 |] ==> a*x @= 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   804
by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   805
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   806
lemma approx_mult_SReal_zero_cancel_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   807
     "[|a \<in> Reals; a \<noteq> 0 |] ==> (a*x @= 0) = (x @= 0)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   808
by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   809
declare approx_mult_SReal_zero_cancel_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   810
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   811
lemma approx_SReal_mult_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   812
     "[| a \<in> Reals; a \<noteq> 0; a* w @= a*z |] ==> w @= z"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   813
apply (drule SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   814
apply (auto dest: approx_mult2 simp add: hypreal_mult_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   815
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   816
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   817
lemma approx_SReal_mult_cancel_iff1:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   818
     "[| a \<in> Reals; a \<noteq> 0|] ==> (a* w @= a*z) = (w @= z)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   819
by (auto intro!: approx_mult2 SReal_subset_HFinite [THEN subsetD] intro: approx_SReal_mult_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   820
declare approx_SReal_mult_cancel_iff1 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   821
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   822
lemma approx_le_bound: "[| z \<le> f; f @= g; g \<le> z |] ==> f @= z"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   823
apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   824
apply (rule_tac x = "g+y-z" in bexI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   825
apply (simp (no_asm))
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   826
apply (rule Infinitesimal_interval2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   827
apply (rule_tac [2] Infinitesimal_zero, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   828
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   829
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   830
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   831
subsection{* Zero is the Only Infinitesimal that is Also a Real*}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   832
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   833
lemma Infinitesimal_less_SReal:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   834
     "[| x \<in> Reals; y \<in> Infinitesimal; 0 < x |] ==> y < x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   835
apply (simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   836
apply (rule abs_ge_self [THEN order_le_less_trans], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   837
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   838
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   839
lemma Infinitesimal_less_SReal2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   840
     "y \<in> Infinitesimal ==> \<forall>r \<in> Reals. 0 < r --> y < r"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   841
by (blast intro: Infinitesimal_less_SReal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   842
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   843
lemma SReal_not_Infinitesimal:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   844
     "[| 0 < y;  y \<in> Reals|] ==> y \<notin> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   845
apply (simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   846
apply (auto simp add: hrabs_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   847
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   848
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   849
lemma SReal_minus_not_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   850
     "[| y < 0;  y \<in> Reals |] ==> y \<notin> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   851
apply (subst Infinitesimal_minus_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   852
apply (rule SReal_not_Infinitesimal, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   853
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   854
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   855
lemma SReal_Int_Infinitesimal_zero: "Reals Int Infinitesimal = {0}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   856
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   857
apply (cut_tac x = x and y = 0 in linorder_less_linear)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   858
apply (blast dest: SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   859
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   860
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   861
lemma SReal_Infinitesimal_zero: "[| x \<in> Reals; x \<in> Infinitesimal|] ==> x = 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   862
by (cut_tac SReal_Int_Infinitesimal_zero, blast)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   863
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   864
lemma SReal_HFinite_diff_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   865
     "[| x \<in> Reals; x \<noteq> 0 |] ==> x \<in> HFinite - Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   866
by (auto dest: SReal_Infinitesimal_zero SReal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   867
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   868
lemma hypreal_of_real_HFinite_diff_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   869
     "hypreal_of_real x \<noteq> 0 ==> hypreal_of_real x \<in> HFinite - Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   870
by (rule SReal_HFinite_diff_Infinitesimal, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   871
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   872
lemma hypreal_of_real_Infinitesimal_iff_0:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   873
     "(hypreal_of_real x \<in> Infinitesimal) = (x=0)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   874
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   875
apply (rule ccontr)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   876
apply (rule hypreal_of_real_HFinite_diff_Infinitesimal [THEN DiffD2], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   877
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   878
declare hypreal_of_real_Infinitesimal_iff_0 [iff]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   879
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   880
lemma number_of_not_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   881
     "number_of w \<noteq> (0::hypreal) ==> number_of w \<notin> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   882
by (fast dest: SReal_number_of [THEN SReal_Infinitesimal_zero])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   883
declare number_of_not_Infinitesimal [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   884
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   885
(*again: 1 is a special case, but not 0 this time*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   886
lemma one_not_Infinitesimal: "1 \<notin> Infinitesimal"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
   887
apply (subst numeral_1_eq_1 [symmetric])
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   888
apply (rule number_of_not_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   889
apply (simp (no_asm))
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   890
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   891
declare one_not_Infinitesimal [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   892
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   893
lemma approx_SReal_not_zero: "[| y \<in> Reals; x @= y; y\<noteq> 0 |] ==> x \<noteq> 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   894
apply (cut_tac x = 0 and y = y in linorder_less_linear, simp)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   895
apply (blast dest: approx_sym [THEN mem_infmal_iff [THEN iffD2]] SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   896
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   897
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   898
lemma HFinite_diff_Infinitesimal_approx:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   899
     "[| x @= y; y \<in> HFinite - Infinitesimal |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   900
      ==> x \<in> HFinite - Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   901
apply (auto intro: approx_sym [THEN [2] approx_HFinite]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   902
            simp add: mem_infmal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   903
apply (drule approx_trans3, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   904
apply (blast dest: approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   905
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   906
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   907
(*The premise y\<noteq>0 is essential; otherwise x/y =0 and we lose the
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   908
  HFinite premise.*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   909
lemma Infinitesimal_ratio:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   910
     "[| y \<noteq> 0;  y \<in> Infinitesimal;  x/y \<in> HFinite |] ==> x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   911
apply (drule Infinitesimal_HFinite_mult2, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   912
apply (simp add: hypreal_divide_def hypreal_mult_assoc)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   913
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   914
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   915
lemma Infinitesimal_SReal_divide: 
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   916
  "[| x \<in> Infinitesimal; y \<in> Reals |] ==> x/y \<in> Infinitesimal"
14430
5cb24165a2e1 new material from Avigad, and simplified treatment of division by 0
paulson
parents: 14420
diff changeset
   917
apply (simp add: divide_inverse)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   918
apply (auto intro!: Infinitesimal_HFinite_mult 
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   919
            dest!: SReal_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   920
done
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   921
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   922
(*------------------------------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   923
       Standard Part Theorem: Every finite x: R* is infinitely
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   924
       close to a unique real number (i.e a member of Reals)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   925
 ------------------------------------------------------------------*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   926
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   927
subsection{* Uniqueness: Two Infinitely Close Reals are Equal*}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   928
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   929
lemma SReal_approx_iff: "[|x \<in> Reals; y \<in> Reals|] ==> (x @= y) = (x = y)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   930
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   931
apply (simp add: approx_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   932
apply (drule_tac x = y in SReal_minus)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   933
apply (drule SReal_add, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   934
apply (drule SReal_Infinitesimal_zero, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   935
apply (drule sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   936
apply (simp add: hypreal_eq_minus_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   937
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   938
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   939
lemma number_of_approx_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   940
     "(number_of v @= number_of w) = (number_of v = (number_of w :: hypreal))"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   941
by (auto simp add: SReal_approx_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   942
declare number_of_approx_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   943
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   944
(*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   945
lemma [simp]: "(0 @= number_of w) = ((number_of w :: hypreal) = 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   946
              "(number_of w @= 0) = ((number_of w :: hypreal) = 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   947
              "(1 @= number_of w) = ((number_of w :: hypreal) = 1)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   948
              "(number_of w @= 1) = ((number_of w :: hypreal) = 1)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   949
              "~ (0 @= 1)" "~ (1 @= 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   950
by (auto simp only: SReal_number_of SReal_approx_iff Reals_0 Reals_1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   951
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   952
lemma hypreal_of_real_approx_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   953
     "(hypreal_of_real k @= hypreal_of_real m) = (k = m)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   954
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   955
apply (rule inj_hypreal_of_real [THEN injD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   956
apply (rule SReal_approx_iff [THEN iffD1], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   957
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   958
declare hypreal_of_real_approx_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   959
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   960
lemma hypreal_of_real_approx_number_of_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   961
     "(hypreal_of_real k @= number_of w) = (k = number_of w)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   962
by (subst hypreal_of_real_approx_iff [symmetric], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   963
declare hypreal_of_real_approx_number_of_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   964
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   965
(*And also for 0 and 1.*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   966
(*And also for 0 @= #nn and 1 @= #nn, #nn @= 0 and #nn @= 1.*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   967
lemma [simp]: "(hypreal_of_real k @= 0) = (k = 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   968
              "(hypreal_of_real k @= 1) = (k = 1)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   969
  by (simp_all add:  hypreal_of_real_approx_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   970
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   971
lemma approx_unique_real:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   972
     "[| r \<in> Reals; s \<in> Reals; r @= x; s @= x|] ==> r = s"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   973
by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   974
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   975
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   976
subsection{* Existence of Unique Real Infinitely Close*}
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   977
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   978
(* lemma about lubs *)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   979
lemma hypreal_isLub_unique:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   980
     "[| isLub R S x; isLub R S y |] ==> x = (y::hypreal)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   981
apply (frule isLub_isUb)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   982
apply (frule_tac x = y in isLub_isUb)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   983
apply (blast intro!: hypreal_le_anti_sym dest!: isLub_le_isUb)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   984
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   985
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   986
lemma lemma_st_part_ub:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
   987
     "x \<in> HFinite ==> \<exists>u. isUb Reals {s. s \<in> Reals & s < x} u"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   988
apply (drule HFiniteD, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   989
apply (rule exI, rule isUbI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   990
apply (auto intro: setleI isUbI simp add: abs_less_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   991
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   992
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   993
lemma lemma_st_part_nonempty: "x \<in> HFinite ==> \<exists>y. y \<in> {s. s \<in> Reals & s < x}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   994
apply (drule HFiniteD, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   995
apply (drule SReal_minus)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   996
apply (rule_tac x = "-t" in exI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   997
apply (auto simp add: abs_less_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   998
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
   999
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1000
lemma lemma_st_part_subset: "{s. s \<in> Reals & s < x} \<subseteq> Reals"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1001
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1002
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1003
lemma lemma_st_part_lub:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1004
     "x \<in> HFinite ==> \<exists>t. isLub Reals {s. s \<in> Reals & s < x} t"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1005
by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty lemma_st_part_subset)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1006
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1007
lemma lemma_hypreal_le_left_cancel: "((t::hypreal) + r \<le> t) = (r \<le> 0)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1008
apply safe
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1009
apply (drule_tac c = "-t" in add_left_mono)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1010
apply (drule_tac [2] c = t in add_left_mono)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1011
apply (auto simp add: hypreal_add_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1012
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1013
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1014
lemma lemma_st_part_le1:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1015
     "[| x \<in> HFinite;  isLub Reals {s. s \<in> Reals & s < x} t;
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1016
         r \<in> Reals;  0 < r |] ==> x \<le> t + r"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1017
apply (frule isLubD1a)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1018
apply (rule ccontr, drule linorder_not_le [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1019
apply (drule_tac x = t in SReal_add, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1020
apply (drule_tac y = "t + r" in isLubD1 [THEN setleD], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1021
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1022
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1023
lemma hypreal_setle_less_trans:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1024
     "!!x::hypreal. [| S *<= x; x < y |] ==> S *<= y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1025
apply (simp add: setle_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1026
apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1027
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1028
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1029
lemma hypreal_gt_isUb:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1030
     "!!x::hypreal. [| isUb R S x; x < y; y \<in> R |] ==> isUb R S y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1031
apply (simp add: isUb_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1032
apply (blast intro: hypreal_setle_less_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1033
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1034
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1035
lemma lemma_st_part_gt_ub:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1036
     "[| x \<in> HFinite; x < y; y \<in> Reals |]
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1037
      ==> isUb Reals {s. s \<in> Reals & s < x} y"
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1038
by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1039
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1040
lemma lemma_minus_le_zero: "t \<le> t + -r ==> r \<le> (0::hypreal)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1041
apply (drule_tac c = "-t" in add_left_mono)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1042
apply (auto simp add: hypreal_add_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1043
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1044
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1045
lemma lemma_st_part_le2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1046
     "[| x \<in> HFinite;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1047
         isLub Reals {s. s \<in> Reals & s < x} t;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1048
         r \<in> Reals; 0 < r |]
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1049
      ==> t + -r \<le> x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1050
apply (frule isLubD1a)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1051
apply (rule ccontr, drule linorder_not_le [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1052
apply (drule SReal_minus, drule_tac x = t in SReal_add, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1053
apply (drule lemma_st_part_gt_ub, assumption+)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1054
apply (drule isLub_le_isUb, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1055
apply (drule lemma_minus_le_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1056
apply (auto dest: order_less_le_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1057
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1058
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1059
lemma lemma_hypreal_le_swap: "((x::hypreal) \<le> t + r) = (x + -t \<le> r)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1060
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1061
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1062
lemma lemma_st_part1a:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1063
     "[| x \<in> HFinite;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1064
         isLub Reals {s. s \<in> Reals & s < x} t;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1065
         r \<in> Reals; 0 < r |]
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1066
      ==> x + -t \<le> r"
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1067
by (blast intro!: lemma_hypreal_le_swap [THEN iffD1] lemma_st_part_le1)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1068
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1069
lemma lemma_hypreal_le_swap2: "(t + -r \<le> x) = (-(x + -t) \<le> (r::hypreal))"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1070
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1071
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1072
lemma lemma_st_part2a:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1073
     "[| x \<in> HFinite;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1074
         isLub Reals {s. s \<in> Reals & s < x} t;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1075
         r \<in> Reals;  0 < r |]
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1076
      ==> -(x + -t) \<le> r"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1077
apply (blast intro!: lemma_hypreal_le_swap2 [THEN iffD1] lemma_st_part_le2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1078
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1079
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1080
lemma lemma_SReal_ub:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1081
     "(x::hypreal) \<in> Reals ==> isUb Reals {s. s \<in> Reals & s < x} x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1082
by (auto intro: isUbI setleI order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1083
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1084
lemma lemma_SReal_lub:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1085
     "(x::hypreal) \<in> Reals ==> isLub Reals {s. s \<in> Reals & s < x} x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1086
apply (auto intro!: isLubI2 lemma_SReal_ub setgeI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1087
apply (frule isUbD2a)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1088
apply (rule_tac x = x and y = y in linorder_cases)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1089
apply (auto intro!: order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1090
apply (drule SReal_dense, assumption, assumption, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1091
apply (drule_tac y = r in isUbD)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1092
apply (auto dest: order_less_le_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1093
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1094
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1095
lemma lemma_st_part_not_eq1:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1096
     "[| x \<in> HFinite;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1097
         isLub Reals {s. s \<in> Reals & s < x} t;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1098
         r \<in> Reals; 0 < r |]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1099
      ==> x + -t \<noteq> r"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1100
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1101
apply (frule isLubD1a [THEN SReal_minus])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1102
apply (drule SReal_add_cancel, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1103
apply (drule_tac x = x in lemma_SReal_lub)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1104
apply (drule hypreal_isLub_unique, assumption, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1105
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1106
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1107
lemma lemma_st_part_not_eq2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1108
     "[| x \<in> HFinite;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1109
         isLub Reals {s. s \<in> Reals & s < x} t;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1110
         r \<in> Reals; 0 < r |]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1111
      ==> -(x + -t) \<noteq> r"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1112
apply (auto simp add: minus_add_distrib)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1113
apply (frule isLubD1a)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1114
apply (drule SReal_add_cancel, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1115
apply (drule_tac x = "-x" in SReal_minus, simp)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1116
apply (drule_tac x = x in lemma_SReal_lub)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1117
apply (drule hypreal_isLub_unique, assumption, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1118
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1119
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1120
lemma lemma_st_part_major:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1121
     "[| x \<in> HFinite;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1122
         isLub Reals {s. s \<in> Reals & s < x} t;
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1123
         r \<in> Reals; 0 < r |]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1124
      ==> abs (x + -t) < r"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1125
apply (frule lemma_st_part1a)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1126
apply (frule_tac [4] lemma_st_part2a, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1127
apply (drule order_le_imp_less_or_eq)+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1128
apply (auto dest: lemma_st_part_not_eq1 lemma_st_part_not_eq2 simp add: abs_less_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1129
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1130
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1131
lemma lemma_st_part_major2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1132
     "[| x \<in> HFinite; isLub Reals {s. s \<in> Reals & s < x} t |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1133
      ==> \<forall>r \<in> Reals. 0 < r --> abs (x + -t) < r"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1134
by (blast dest!: lemma_st_part_major)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1135
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1136
(*----------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1137
  Existence of real and Standard Part Theorem
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1138
 ----------------------------------------------*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1139
lemma lemma_st_part_Ex:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1140
     "x \<in> HFinite ==> \<exists>t \<in> Reals. \<forall>r \<in> Reals. 0 < r --> abs (x + -t) < r"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1141
apply (frule lemma_st_part_lub, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1142
apply (frule isLubD1a)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1143
apply (blast dest: lemma_st_part_major2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1144
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1145
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1146
lemma st_part_Ex:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1147
     "x \<in> HFinite ==> \<exists>t \<in> Reals. x @= t"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1148
apply (simp add: approx_def Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1149
apply (drule lemma_st_part_Ex, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1150
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1151
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1152
(*--------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1153
  Unique real infinitely close
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1154
 -------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1155
lemma st_part_Ex1: "x \<in> HFinite ==> EX! t. t \<in> Reals & x @= t"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1156
apply (drule st_part_Ex, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1157
apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1158
apply (auto intro!: approx_unique_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1159
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1160
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1161
subsection{* Finite, Infinite and Infinitesimal*}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1162
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1163
lemma HFinite_Int_HInfinite_empty: "HFinite Int HInfinite = {}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1164
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1165
apply (simp add: HFinite_def HInfinite_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1166
apply (auto dest: order_less_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1167
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1168
declare HFinite_Int_HInfinite_empty [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1169
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1170
lemma HFinite_not_HInfinite: 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1171
  assumes x: "x \<in> HFinite" shows "x \<notin> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1172
proof
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1173
  assume x': "x \<in> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1174
  with x have "x \<in> HFinite \<inter> HInfinite" by blast
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1175
  thus False by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1176
qed
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1177
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1178
lemma not_HFinite_HInfinite: "x\<notin> HFinite ==> x \<in> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1179
apply (simp add: HInfinite_def HFinite_def, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1180
apply (drule_tac x = "r + 1" in bspec)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1181
apply (auto simp add: SReal_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1182
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1183
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1184
lemma HInfinite_HFinite_disj: "x \<in> HInfinite | x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1185
by (blast intro: not_HFinite_HInfinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1186
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1187
lemma HInfinite_HFinite_iff: "(x \<in> HInfinite) = (x \<notin> HFinite)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1188
by (blast dest: HFinite_not_HInfinite not_HFinite_HInfinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1189
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1190
lemma HFinite_HInfinite_iff: "(x \<in> HFinite) = (x \<notin> HInfinite)"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1191
by (simp add: HInfinite_HFinite_iff)
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1192
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1193
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1194
lemma HInfinite_diff_HFinite_Infinitesimal_disj:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1195
     "x \<notin> Infinitesimal ==> x \<in> HInfinite | x \<in> HFinite - Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1196
by (fast intro: not_HFinite_HInfinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1197
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1198
lemma HFinite_inverse:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1199
     "[| x \<in> HFinite; x \<notin> Infinitesimal |] ==> inverse x \<in> HFinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1200
apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1201
apply (auto dest!: HInfinite_inverse_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1202
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1203
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1204
lemma HFinite_inverse2: "x \<in> HFinite - Infinitesimal ==> inverse x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1205
by (blast intro: HFinite_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1206
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1207
(* stronger statement possible in fact *)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1208
lemma Infinitesimal_inverse_HFinite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1209
     "x \<notin> Infinitesimal ==> inverse(x) \<in> HFinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1210
apply (drule HInfinite_diff_HFinite_Infinitesimal_disj)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1211
apply (blast intro: HFinite_inverse HInfinite_inverse_Infinitesimal Infinitesimal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1212
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1213
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1214
lemma HFinite_not_Infinitesimal_inverse:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1215
     "x \<in> HFinite - Infinitesimal ==> inverse x \<in> HFinite - Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1216
apply (auto intro: Infinitesimal_inverse_HFinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1217
apply (drule Infinitesimal_HFinite_mult2, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1218
apply (simp add: not_Infinitesimal_not_zero hypreal_mult_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1219
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1220
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1221
lemma approx_inverse:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1222
     "[| x @= y; y \<in>  HFinite - Infinitesimal |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1223
      ==> inverse x @= inverse y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1224
apply (frule HFinite_diff_Infinitesimal_approx, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1225
apply (frule not_Infinitesimal_not_zero2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1226
apply (frule_tac x = x in not_Infinitesimal_not_zero2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1227
apply (drule HFinite_inverse2)+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1228
apply (drule approx_mult2, assumption, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1229
apply (drule_tac c = "inverse x" in approx_mult1, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1230
apply (auto intro: approx_sym simp add: hypreal_mult_assoc)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1231
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1232
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1233
(*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1234
lemmas hypreal_of_real_approx_inverse =  hypreal_of_real_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1235
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1236
lemma inverse_add_Infinitesimal_approx:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1237
     "[| x \<in> HFinite - Infinitesimal;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1238
         h \<in> Infinitesimal |] ==> inverse(x + h) @= inverse x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1239
apply (auto intro: approx_inverse approx_sym Infinitesimal_add_approx_self)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1240
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1241
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1242
lemma inverse_add_Infinitesimal_approx2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1243
     "[| x \<in> HFinite - Infinitesimal;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1244
         h \<in> Infinitesimal |] ==> inverse(h + x) @= inverse x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1245
apply (rule hypreal_add_commute [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1246
apply (blast intro: inverse_add_Infinitesimal_approx)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1247
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1248
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1249
lemma inverse_add_Infinitesimal_approx_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1250
     "[| x \<in> HFinite - Infinitesimal;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1251
         h \<in> Infinitesimal |] ==> inverse(x + h) + -inverse x @= h"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1252
apply (rule approx_trans2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1253
apply (auto intro: inverse_add_Infinitesimal_approx simp add: mem_infmal_iff approx_minus_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1254
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1255
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1256
lemma Infinitesimal_square_iff: "(x \<in> Infinitesimal) = (x*x \<in> Infinitesimal)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1257
apply (auto intro: Infinitesimal_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1258
apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1259
apply (frule not_Infinitesimal_not_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1260
apply (auto dest: Infinitesimal_HFinite_mult simp add: hypreal_mult_assoc)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1261
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1262
declare Infinitesimal_square_iff [symmetric, simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1263
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1264
lemma HFinite_square_iff: "(x*x \<in> HFinite) = (x \<in> HFinite)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1265
apply (auto intro: HFinite_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1266
apply (auto dest: HInfinite_mult simp add: HFinite_HInfinite_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1267
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1268
declare HFinite_square_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1269
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1270
lemma HInfinite_square_iff: "(x*x \<in> HInfinite) = (x \<in> HInfinite)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1271
by (auto simp add: HInfinite_HFinite_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1272
declare HInfinite_square_iff [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1273
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1274
lemma approx_HFinite_mult_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1275
     "[| a: HFinite-Infinitesimal; a* w @= a*z |] ==> w @= z"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1276
apply safe
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1277
apply (frule HFinite_inverse, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1278
apply (drule not_Infinitesimal_not_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1279
apply (auto dest: approx_mult2 simp add: hypreal_mult_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1280
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1281
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1282
lemma approx_HFinite_mult_cancel_iff1:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1283
     "a: HFinite-Infinitesimal ==> (a * w @= a * z) = (w @= z)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1284
by (auto intro: approx_mult2 approx_HFinite_mult_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1285
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1286
lemma HInfinite_HFinite_add_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1287
     "[| x + y \<in> HInfinite; y \<in> HFinite |] ==> x \<in> HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1288
apply (rule ccontr)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1289
apply (drule HFinite_HInfinite_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1290
apply (auto dest: HFinite_add simp add: HInfinite_HFinite_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1291
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1292
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1293
lemma HInfinite_HFinite_add:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1294
     "[| x \<in> HInfinite; y \<in> HFinite |] ==> x + y \<in> HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1295
apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1296
apply (auto simp add: hypreal_add_assoc HFinite_minus_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1297
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1298
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1299
lemma HInfinite_ge_HInfinite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1300
     "[| x \<in> HInfinite; x \<le> y; 0 \<le> x |] ==> y \<in> HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1301
by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1302
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1303
lemma Infinitesimal_inverse_HInfinite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1304
     "[| x \<in> Infinitesimal; x \<noteq> 0 |] ==> inverse x \<in> HInfinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1305
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1306
apply (auto dest: Infinitesimal_HFinite_mult2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1307
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1308
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1309
lemma HInfinite_HFinite_not_Infinitesimal_mult:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1310
     "[| x \<in> HInfinite; y \<in> HFinite - Infinitesimal |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1311
      ==> x * y \<in> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1312
apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1313
apply (frule HFinite_Infinitesimal_not_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1314
apply (drule HFinite_not_Infinitesimal_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1315
apply (safe, drule HFinite_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1316
apply (auto simp add: hypreal_mult_assoc HFinite_HInfinite_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1317
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1318
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1319
lemma HInfinite_HFinite_not_Infinitesimal_mult2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1320
     "[| x \<in> HInfinite; y \<in> HFinite - Infinitesimal |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1321
      ==> y * x \<in> HInfinite"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1322
by (auto simp add: hypreal_mult_commute HInfinite_HFinite_not_Infinitesimal_mult)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1323
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1324
lemma HInfinite_gt_SReal: "[| x \<in> HInfinite; 0 < x; y \<in> Reals |] ==> y < x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1325
by (auto dest!: bspec simp add: HInfinite_def hrabs_def order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1326
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1327
lemma HInfinite_gt_zero_gt_one: "[| x \<in> HInfinite; 0 < x |] ==> 1 < x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1328
by (auto intro: HInfinite_gt_SReal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1329
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1330
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1331
lemma not_HInfinite_one: "1 \<notin> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1332
apply (simp (no_asm) add: HInfinite_HFinite_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1333
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1334
declare not_HInfinite_one [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1335
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1336
lemma approx_hrabs_disj: "abs x @= x | abs x @= -x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1337
by (cut_tac x = x in hrabs_disj, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1338
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1339
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1340
subsection{*Theorems about Monads*}
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1341
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1342
lemma monad_hrabs_Un_subset: "monad (abs x) \<le> monad(x) Un monad(-x)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1343
by (rule_tac x1 = x in hrabs_disj [THEN disjE], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1344
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1345
lemma Infinitesimal_monad_eq: "e \<in> Infinitesimal ==> monad (x+e) = monad x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1346
by (fast intro!: Infinitesimal_add_approx_self [THEN approx_sym] approx_monad_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1347
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1348
lemma mem_monad_iff: "(u \<in> monad x) = (-u \<in> monad (-x))"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1349
by (simp add: monad_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1350
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1351
lemma Infinitesimal_monad_zero_iff: "(x \<in> Infinitesimal) = (x \<in> monad 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1352
by (auto intro: approx_sym simp add: monad_def mem_infmal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1353
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1354
lemma monad_zero_minus_iff: "(x \<in> monad 0) = (-x \<in> monad 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1355
apply (simp (no_asm) add: Infinitesimal_monad_zero_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1356
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1357
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1358
lemma monad_zero_hrabs_iff: "(x \<in> monad 0) = (abs x \<in> monad 0)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1359
apply (rule_tac x1 = x in hrabs_disj [THEN disjE])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1360
apply (auto simp add: monad_zero_minus_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1361
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1362
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1363
lemma mem_monad_self: "x \<in> monad x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1364
by (simp add: monad_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1365
declare mem_monad_self [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1366
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1367
(*------------------------------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1368
         Proof that x @= y ==> abs x @= abs y
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1369
 ------------------------------------------------------------------*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1370
lemma approx_subset_monad: "x @= y ==> {x,y}\<le>monad x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1371
apply (simp (no_asm))
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1372
apply (simp add: approx_monad_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1373
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1374
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1375
lemma approx_subset_monad2: "x @= y ==> {x,y}\<le>monad y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1376
apply (drule approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1377
apply (fast dest: approx_subset_monad)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1378
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1379
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1380
lemma mem_monad_approx: "u \<in> monad x ==> x @= u"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1381
by (simp add: monad_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1382
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1383
lemma approx_mem_monad: "x @= u ==> u \<in> monad x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1384
by (simp add: monad_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1385
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1386
lemma approx_mem_monad2: "x @= u ==> x \<in> monad u"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1387
apply (simp add: monad_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1388
apply (blast intro!: approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1389
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1390
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1391
lemma approx_mem_monad_zero: "[| x @= y;x \<in> monad 0 |] ==> y \<in> monad 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1392
apply (drule mem_monad_approx)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1393
apply (fast intro: approx_mem_monad approx_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1394
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1395
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1396
lemma Infinitesimal_approx_hrabs:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1397
     "[| x @= y; x \<in> Infinitesimal |] ==> abs x @= abs y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1398
apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1399
apply (blast intro: approx_mem_monad_zero monad_zero_hrabs_iff [THEN iffD1] mem_monad_approx approx_trans3)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1400
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1401
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1402
lemma less_Infinitesimal_less:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1403
     "[| 0 < x;  x \<notin>Infinitesimal;  e :Infinitesimal |] ==> e < x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1404
apply (rule ccontr)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1405
apply (auto intro: Infinitesimal_zero [THEN [2] Infinitesimal_interval] 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1406
            dest!: order_le_imp_less_or_eq simp add: linorder_not_less)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1407
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1408
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1409
lemma Ball_mem_monad_gt_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1410
     "[| 0 < x;  x \<notin> Infinitesimal; u \<in> monad x |] ==> 0 < u"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1411
apply (drule mem_monad_approx [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1412
apply (erule bex_Infinitesimal_iff2 [THEN iffD2, THEN bexE])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1413
apply (drule_tac e = "-xa" in less_Infinitesimal_less, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1414
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1415
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1416
lemma Ball_mem_monad_less_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1417
     "[| x < 0; x \<notin> Infinitesimal; u \<in> monad x |] ==> u < 0"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1418
apply (drule mem_monad_approx [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1419
apply (erule bex_Infinitesimal_iff [THEN iffD2, THEN bexE])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1420
apply (cut_tac x = "-x" and e = xa in less_Infinitesimal_less, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1421
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1422
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1423
lemma lemma_approx_gt_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1424
     "[|0 < x; x \<notin> Infinitesimal; x @= y|] ==> 0 < y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1425
by (blast dest: Ball_mem_monad_gt_zero approx_subset_monad)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1426
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1427
lemma lemma_approx_less_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1428
     "[|x < 0; x \<notin> Infinitesimal; x @= y|] ==> y < 0"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1429
by (blast dest: Ball_mem_monad_less_zero approx_subset_monad)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1430
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1431
lemma approx_hrabs1:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1432
     "[| x @= y; x < 0; x \<notin> Infinitesimal |] ==> abs x @= abs y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1433
apply (frule lemma_approx_less_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1434
apply (assumption+)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1435
apply (simp add: abs_if) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1436
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1437
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1438
lemma approx_hrabs2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1439
     "[| x @= y; 0 < x; x \<notin> Infinitesimal |] ==> abs x @= abs y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1440
apply (frule lemma_approx_gt_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1441
apply (assumption+)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1442
apply (simp add: abs_if) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1443
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1444
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1445
lemma approx_hrabs: "x @= y ==> abs x @= abs y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1446
apply (rule_tac Q = "x \<in> Infinitesimal" in excluded_middle [THEN disjE])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1447
apply (rule_tac x1 = x and y1 = 0 in linorder_less_linear [THEN disjE])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1448
apply (auto intro: approx_hrabs1 approx_hrabs2 Infinitesimal_approx_hrabs)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1449
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1450
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1451
lemma approx_hrabs_zero_cancel: "abs(x) @= 0 ==> x @= 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1452
apply (cut_tac x = x in hrabs_disj)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1453
apply (auto dest: approx_minus)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1454
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1455
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1456
lemma approx_hrabs_add_Infinitesimal: "e \<in> Infinitesimal ==> abs x @= abs(x+e)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1457
by (fast intro: approx_hrabs Infinitesimal_add_approx_self)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1458
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1459
lemma approx_hrabs_add_minus_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1460
     "e \<in> Infinitesimal ==> abs x @= abs(x + -e)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1461
by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1462
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1463
lemma hrabs_add_Infinitesimal_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1464
     "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1465
         abs(x+e) = abs(y+e')|] ==> abs x @= abs y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1466
apply (drule_tac x = x in approx_hrabs_add_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1467
apply (drule_tac x = y in approx_hrabs_add_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1468
apply (auto intro: approx_trans2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1469
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1470
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1471
lemma hrabs_add_minus_Infinitesimal_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1472
     "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1473
         abs(x + -e) = abs(y + -e')|] ==> abs x @= abs y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1474
apply (drule_tac x = x in approx_hrabs_add_minus_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1475
apply (drule_tac x = y in approx_hrabs_add_minus_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1476
apply (auto intro: approx_trans2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1477
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1478
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1479
lemma hypreal_less_minus_iff: "((x::hypreal) < y) = (0 < y + -x)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1480
by arith
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
  1481
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1482
(* interesting slightly counterintuitive theorem: necessary
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1483
   for proving that an open interval is an NS open set
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1484
*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1485
lemma Infinitesimal_add_hypreal_of_real_less:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1486
     "[| x < y;  u \<in> Infinitesimal |]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1487
      ==> hypreal_of_real x + u < hypreal_of_real y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1488
apply (simp add: Infinitesimal_def)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1489
apply (drule_tac x = "hypreal_of_real y + -hypreal_of_real x" in bspec, simp)  
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1490
apply (auto simp add: add_commute abs_less_iff SReal_add SReal_minus)
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1491
apply (simp add: compare_rls) 
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1492
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1493
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1494
lemma Infinitesimal_add_hrabs_hypreal_of_real_less:
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1495
     "[| x \<in> Infinitesimal; abs(hypreal_of_real r) < hypreal_of_real y |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1496
      ==> abs (hypreal_of_real r + x) < hypreal_of_real y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1497
apply (drule_tac x = "hypreal_of_real r" in approx_hrabs_add_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1498
apply (drule approx_sym [THEN bex_Infinitesimal_iff2 [THEN iffD2]])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1499
apply (auto intro!: Infinitesimal_add_hypreal_of_real_less simp add: hypreal_of_real_hrabs)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1500
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1501
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1502
lemma Infinitesimal_add_hrabs_hypreal_of_real_less2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1503
     "[| x \<in> Infinitesimal;  abs(hypreal_of_real r) < hypreal_of_real y |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1504
      ==> abs (x + hypreal_of_real r) < hypreal_of_real y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1505
apply (rule hypreal_add_commute [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1506
apply (erule Infinitesimal_add_hrabs_hypreal_of_real_less, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1507
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1508
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1509
lemma hypreal_of_real_le_add_Infininitesimal_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1510
     "[| u \<in> Infinitesimal; v \<in> Infinitesimal;
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1511
         hypreal_of_real x + u \<le> hypreal_of_real y + v |]
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1512
      ==> hypreal_of_real x \<le> hypreal_of_real y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1513
apply (simp add: linorder_not_less [symmetric], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1514
apply (drule_tac u = "v-u" in Infinitesimal_add_hypreal_of_real_less)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1515
apply (auto simp add: Infinitesimal_diff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1516
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1517
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1518
lemma hypreal_of_real_le_add_Infininitesimal_cancel2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1519
     "[| u \<in> Infinitesimal; v \<in> Infinitesimal;
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1520
         hypreal_of_real x + u \<le> hypreal_of_real y + v |]
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1521
      ==> x \<le> y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1522
apply (blast intro!: hypreal_of_real_le_iff [THEN iffD1] hypreal_of_real_le_add_Infininitesimal_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1523
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1524
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1525
lemma hypreal_of_real_less_Infinitesimal_le_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1526
     "[| hypreal_of_real x < e; e \<in> Infinitesimal |] ==> hypreal_of_real x \<le> 0"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1527
apply (rule linorder_not_less [THEN iffD1], safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1528
apply (drule Infinitesimal_interval)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1529
apply (drule_tac [4] SReal_hypreal_of_real [THEN SReal_Infinitesimal_zero], auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1530
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1531
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1532
(*used once, in Lim/NSDERIV_inverse*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1533
lemma Infinitesimal_add_not_zero:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1534
     "[| h \<in> Infinitesimal; x \<noteq> 0 |] ==> hypreal_of_real x + h \<noteq> 0"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1535
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1536
apply (subgoal_tac "h = - hypreal_of_real x", auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1537
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1538
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1539
lemma Infinitesimal_square_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1540
     "x*x + y*y \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1541
apply (rule Infinitesimal_interval2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1542
apply (rule_tac [3] zero_le_square, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1543
apply (auto simp add: zero_le_square)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1544
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1545
declare Infinitesimal_square_cancel [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1546
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1547
lemma HFinite_square_cancel: "x*x + y*y \<in> HFinite ==> x*x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1548
apply (rule HFinite_bounded, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1549
apply (auto simp add: zero_le_square)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1550
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1551
declare HFinite_square_cancel [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1552
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1553
lemma Infinitesimal_square_cancel2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1554
     "x*x + y*y \<in> Infinitesimal ==> y*y \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1555
apply (rule Infinitesimal_square_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1556
apply (rule hypreal_add_commute [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1557
apply (simp (no_asm))
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1558
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1559
declare Infinitesimal_square_cancel2 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1560
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1561
lemma HFinite_square_cancel2: "x*x + y*y \<in> HFinite ==> y*y \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1562
apply (rule HFinite_square_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1563
apply (rule hypreal_add_commute [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1564
apply (simp (no_asm))
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1565
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1566
declare HFinite_square_cancel2 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1567
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1568
lemma Infinitesimal_sum_square_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1569
     "x*x + y*y + z*z \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1570
apply (rule Infinitesimal_interval2, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1571
apply (rule_tac [2] zero_le_square, simp)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1572
apply (insert zero_le_square [of y]) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1573
apply (insert zero_le_square [of z], simp)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1574
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1575
declare Infinitesimal_sum_square_cancel [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1576
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1577
lemma HFinite_sum_square_cancel: "x*x + y*y + z*z \<in> HFinite ==> x*x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1578
apply (rule HFinite_bounded, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1579
apply (rule_tac [2] zero_le_square)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1580
apply (insert zero_le_square [of y]) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1581
apply (insert zero_le_square [of z], simp)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1582
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1583
declare HFinite_sum_square_cancel [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1584
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1585
lemma Infinitesimal_sum_square_cancel2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1586
     "y*y + x*x + z*z \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1587
apply (rule Infinitesimal_sum_square_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1588
apply (simp add: add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1589
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1590
declare Infinitesimal_sum_square_cancel2 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1591
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1592
lemma HFinite_sum_square_cancel2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1593
     "y*y + x*x + z*z \<in> HFinite ==> x*x \<in> HFinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1594
apply (rule HFinite_sum_square_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1595
apply (simp add: add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1596
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1597
declare HFinite_sum_square_cancel2 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1598
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1599
lemma Infinitesimal_sum_square_cancel3:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1600
     "z*z + y*y + x*x \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1601
apply (rule Infinitesimal_sum_square_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1602
apply (simp add: add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1603
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1604
declare Infinitesimal_sum_square_cancel3 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1605
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1606
lemma HFinite_sum_square_cancel3:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1607
     "z*z + y*y + x*x \<in> HFinite ==> x*x \<in> HFinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1608
apply (rule HFinite_sum_square_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1609
apply (simp add: add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1610
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1611
declare HFinite_sum_square_cancel3 [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1612
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1613
lemma monad_hrabs_less: "[| y \<in> monad x; 0 < hypreal_of_real e |]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1614
      ==> abs (y + -x) < hypreal_of_real e"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1615
apply (drule mem_monad_approx [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1616
apply (drule bex_Infinitesimal_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1617
apply (auto dest!: InfinitesimalD)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1618
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1619
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1620
lemma mem_monad_SReal_HFinite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1621
     "x \<in> monad (hypreal_of_real  a) ==> x \<in> HFinite"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1622
apply (drule mem_monad_approx [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1623
apply (drule bex_Infinitesimal_iff2 [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1624
apply (safe dest!: Infinitesimal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1625
apply (erule SReal_hypreal_of_real [THEN SReal_subset_HFinite [THEN subsetD], THEN HFinite_add])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1626
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1627
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1628
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1629
subsection{* Theorems about Standard Part*}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1630
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1631
lemma st_approx_self: "x \<in> HFinite ==> st x @= x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1632
apply (simp add: st_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1633
apply (frule st_part_Ex, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1634
apply (rule someI2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1635
apply (auto intro: approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1636
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1637
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1638
lemma st_SReal: "x \<in> HFinite ==> st x \<in> Reals"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1639
apply (simp add: st_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1640
apply (frule st_part_Ex, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1641
apply (rule someI2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1642
apply (auto intro: approx_sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1643
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1644
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1645
lemma st_HFinite: "x \<in> HFinite ==> st x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1646
by (erule st_SReal [THEN SReal_subset_HFinite [THEN subsetD]])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1647
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1648
lemma st_SReal_eq: "x \<in> Reals ==> st x = x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1649
apply (simp add: st_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1650
apply (rule some_equality)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1651
apply (fast intro: SReal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1652
apply (blast dest: SReal_approx_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1653
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1654
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1655
(* ???should be added to simpset *)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1656
lemma st_hypreal_of_real: "st (hypreal_of_real x) = hypreal_of_real x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1657
by (rule SReal_hypreal_of_real [THEN st_SReal_eq])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1658
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1659
lemma st_eq_approx: "[| x \<in> HFinite; y \<in> HFinite; st x = st y |] ==> x @= y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1660
by (auto dest!: st_approx_self elim!: approx_trans3)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1661
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1662
lemma approx_st_eq: 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1663
  assumes "x \<in> HFinite" and "y \<in> HFinite" and "x @= y" 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1664
  shows "st x = st y"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1665
proof -
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1666
  have "st x @= x" "st y @= y" "st x \<in> Reals" "st y \<in> Reals"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1667
    by (simp_all add: st_approx_self st_SReal prems) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1668
  with prems show ?thesis 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1669
    by (fast elim: approx_trans approx_trans2 SReal_approx_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1670
qed
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1671
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1672
lemma st_eq_approx_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1673
     "[| x \<in> HFinite; y \<in> HFinite|]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1674
                   ==> (x @= y) = (st x = st y)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1675
by (blast intro: approx_st_eq st_eq_approx)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1676
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1677
lemma st_Infinitesimal_add_SReal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1678
     "[| x \<in> Reals; e \<in> Infinitesimal |] ==> st(x + e) = x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1679
apply (frule st_SReal_eq [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1680
prefer 2 apply assumption
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1681
apply (frule SReal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1682
apply (frule Infinitesimal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1683
apply (drule st_SReal_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1684
apply (rule approx_st_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1685
apply (auto intro: HFinite_add simp add: Infinitesimal_add_approx_self [THEN approx_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1686
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1687
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1688
lemma st_Infinitesimal_add_SReal2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1689
     "[| x \<in> Reals; e \<in> Infinitesimal |] ==> st(e + x) = x"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1690
apply (rule hypreal_add_commute [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1691
apply (blast intro!: st_Infinitesimal_add_SReal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1692
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1693
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1694
lemma HFinite_st_Infinitesimal_add:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1695
     "x \<in> HFinite ==> \<exists>e \<in> Infinitesimal. x = st(x) + e"
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1696
by (blast dest!: st_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1697
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1698
lemma st_add: 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1699
  assumes x: "x \<in> HFinite" and y: "y \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1700
  shows "st (x + y) = st(x) + st(y)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1701
proof -
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1702
  from HFinite_st_Infinitesimal_add [OF x]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1703
  obtain ex where ex: "ex \<in> Infinitesimal" "st x + ex = x" 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1704
    by (blast intro: sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1705
  from HFinite_st_Infinitesimal_add [OF y]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1706
  obtain ey where ey: "ey \<in> Infinitesimal" "st y + ey = y" 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1707
    by (blast intro: sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1708
  have "st (x + y) = st ((st x + ex) + (st y + ey))"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1709
    by (simp add: ex ey) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1710
  also have "... = st ((ex + ey) + (st x + st y))" by (simp add: add_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1711
  also have "... = st x + st y" 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1712
    by (simp add: prems st_SReal SReal_add Infinitesimal_add 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1713
                  st_Infinitesimal_add_SReal2) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1714
  finally show ?thesis .
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1715
qed
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1716
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1717
lemma st_number_of: "st (number_of w) = number_of w"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1718
by (rule SReal_number_of [THEN st_SReal_eq])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1719
declare st_number_of [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1720
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1721
(*the theorem above for the special cases of zero and one*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1722
lemma [simp]: "st 0 = 0" "st 1 = 1"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1723
by (simp_all add: st_SReal_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1724
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1725
lemma st_minus: assumes "y \<in> HFinite" shows "st(-y) = -st(y)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1726
proof -
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1727
  have "st (- y) + st y = 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1728
   by (simp add: prems st_add [symmetric] HFinite_minus_iff) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1729
  thus ?thesis by arith
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1730
qed
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1731
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1732
lemma st_diff: "[| x \<in> HFinite; y \<in> HFinite |] ==> st (x-y) = st(x) - st(y)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1733
apply (simp add: hypreal_diff_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1734
apply (frule_tac y1 = y in st_minus [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1735
apply (drule_tac x1 = y in HFinite_minus_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1736
apply (simp (no_asm_simp) add: st_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1737
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1738
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1739
(* lemma *)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1740
lemma lemma_st_mult:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1741
     "[| x \<in> HFinite; y \<in> HFinite; e \<in> Infinitesimal; ea \<in> Infinitesimal |]
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1742
      ==> e*y + x*ea + e*ea \<in> Infinitesimal"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1743
apply (frule_tac x = e and y = y in Infinitesimal_HFinite_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1744
apply (frule_tac [2] x = ea and y = x in Infinitesimal_HFinite_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1745
apply (drule_tac [3] Infinitesimal_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1746
apply (auto intro: Infinitesimal_add simp add: add_ac mult_ac)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1747
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1748
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1749
lemma st_mult: "[| x \<in> HFinite; y \<in> HFinite |] ==> st (x * y) = st(x) * st(y)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1750
apply (frule HFinite_st_Infinitesimal_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1751
apply (frule_tac x = y in HFinite_st_Infinitesimal_add, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1752
apply (subgoal_tac "st (x * y) = st ((st x + e) * (st y + ea))")
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1753
apply (drule_tac [2] sym, drule_tac [2] sym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1754
 prefer 2 apply simp 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1755
apply (erule_tac V = "x = st x + e" in thin_rl)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1756
apply (erule_tac V = "y = st y + ea" in thin_rl)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1757
apply (simp add: left_distrib right_distrib)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1758
apply (drule st_SReal)+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1759
apply (simp (no_asm_use) add: hypreal_add_assoc)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1760
apply (rule st_Infinitesimal_add_SReal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1761
apply (blast intro!: SReal_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1762
apply (drule SReal_subset_HFinite [THEN subsetD])+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1763
apply (rule hypreal_add_assoc [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1764
apply (blast intro!: lemma_st_mult)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1765
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1766
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1767
lemma st_Infinitesimal: "x \<in> Infinitesimal ==> st x = 0"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1768
apply (subst numeral_0_eq_0 [symmetric])
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1769
apply (rule st_number_of [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1770
apply (rule approx_st_eq)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1771
apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1772
            simp add: mem_infmal_iff [symmetric])
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1773
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1774
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1775
lemma st_not_Infinitesimal: "st(x) \<noteq> 0 ==> x \<notin> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1776
by (fast intro: st_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1777
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1778
lemma st_inverse:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1779
     "[| x \<in> HFinite; st x \<noteq> 0 |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1780
      ==> st(inverse x) = inverse (st x)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1781
apply (rule_tac c1 = "st x" in hypreal_mult_left_cancel [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1782
apply (auto simp add: st_mult [symmetric] st_not_Infinitesimal HFinite_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1783
apply (subst hypreal_mult_inverse, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1784
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1785
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1786
lemma st_divide:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1787
     "[| x \<in> HFinite; y \<in> HFinite; st y \<noteq> 0 |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1788
      ==> st(x/y) = (st x) / (st y)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1789
apply (auto simp add: hypreal_divide_def st_mult st_not_Infinitesimal HFinite_inverse st_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1790
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1791
declare st_divide [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1792
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1793
lemma st_idempotent: "x \<in> HFinite ==> st(st(x)) = st(x)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1794
by (blast intro: st_HFinite st_approx_self approx_st_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1795
declare st_idempotent [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1796
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1797
lemma Infinitesimal_add_st_less:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1798
     "[| x \<in> HFinite; y \<in> HFinite; u \<in> Infinitesimal; st x < st y |] 
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1799
      ==> st x + u < st y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1800
apply (drule st_SReal)+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1801
apply (auto intro!: Infinitesimal_add_hypreal_of_real_less simp add: SReal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1802
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1803
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1804
lemma Infinitesimal_add_st_le_cancel:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1805
     "[| x \<in> HFinite; y \<in> HFinite;
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1806
         u \<in> Infinitesimal; st x \<le> st y + u
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1807
      |] ==> st x \<le> st y"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1808
apply (simp add: linorder_not_less [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1809
apply (auto dest: Infinitesimal_add_st_less)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1810
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1811
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1812
lemma st_le: "[| x \<in> HFinite; y \<in> HFinite; x \<le> y |] ==> st(x) \<le> st(y)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1813
apply (frule HFinite_st_Infinitesimal_add)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1814
apply (rotate_tac 1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1815
apply (frule HFinite_st_Infinitesimal_add, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1816
apply (rule Infinitesimal_add_st_le_cancel)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1817
apply (rule_tac [3] x = ea and y = e in Infinitesimal_diff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1818
apply (auto simp add: hypreal_add_assoc [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1819
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1820
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1821
lemma st_zero_le: "[| 0 \<le> x;  x \<in> HFinite |] ==> 0 \<le> st x"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1822
apply (subst numeral_0_eq_0 [symmetric])
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1823
apply (rule st_number_of [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1824
apply (rule st_le, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1825
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1826
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1827
lemma st_zero_ge: "[| x \<le> 0;  x \<in> HFinite |] ==> st x \<le> 0"
14387
e96d5c42c4b0 Polymorphic treatment of binary arithmetic using axclasses
paulson
parents: 14378
diff changeset
  1828
apply (subst numeral_0_eq_0 [symmetric])
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1829
apply (rule st_number_of [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1830
apply (rule st_le, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1831
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1832
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1833
lemma st_hrabs: "x \<in> HFinite ==> abs(st x) = st(abs x)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1834
apply (simp add: linorder_not_le st_zero_le abs_if st_minus
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1835
   linorder_not_less)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1836
apply (auto dest!: st_zero_ge [OF order_less_imp_le]) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1837
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1838
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1839
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1840
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1841
subsection{*Alternative Definitions for @{term HFinite} using Free Ultrafilter*}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1842
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1843
lemma FreeUltrafilterNat_Rep_hypreal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1844
     "[| X \<in> Rep_hypreal x; Y \<in> Rep_hypreal x |]
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1845
      ==> {n. X n = Y n} \<in> FreeUltrafilterNat"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1846
by (rule_tac z = x in eq_Abs_hypreal, auto, ultra)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1847
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1848
lemma HFinite_FreeUltrafilterNat:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1849
    "x \<in> HFinite 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1850
     ==> \<exists>X \<in> Rep_hypreal x. \<exists>u. {n. abs (X n) < u} \<in> FreeUltrafilterNat"
14468
6be497cacab5 heavy tidying
paulson
parents: 14430
diff changeset
  1851
apply (cases x)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1852
apply (auto simp add: HFinite_def abs_less_iff minus_less_iff [of x] 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1853
              hypreal_less SReal_iff hypreal_minus hypreal_of_real_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1854
apply (rule_tac x=x in bexI) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1855
apply (rule_tac x=y in exI, auto, ultra)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1856
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1857
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1858
lemma FreeUltrafilterNat_HFinite:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1859
     "\<exists>X \<in> Rep_hypreal x.
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1860
       \<exists>u. {n. abs (X n) < u} \<in> FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1861
       ==>  x \<in> HFinite"
14468
6be497cacab5 heavy tidying
paulson
parents: 14430
diff changeset
  1862
apply (cases x)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1863
apply (auto simp add: HFinite_def abs_less_iff minus_less_iff [of x])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1864
apply (rule_tac x = "hypreal_of_real u" in bexI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1865
apply (auto simp add: hypreal_less SReal_iff hypreal_minus hypreal_of_real_def, ultra+)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1866
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1867
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1868
lemma HFinite_FreeUltrafilterNat_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1869
     "(x \<in> HFinite) = (\<exists>X \<in> Rep_hypreal x.
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1870
           \<exists>u. {n. abs (X n) < u} \<in> FreeUltrafilterNat)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1871
apply (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1872
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1873
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1874
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1875
subsection{*Alternative Definitions for @{term HInfinite} using Free Ultrafilter*}
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1876
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1877
lemma lemma_Compl_eq: "- {n. (u::real) < abs (xa n)} = {n. abs (xa n) \<le> u}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1878
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1879
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1880
lemma lemma_Compl_eq2: "- {n. abs (xa n) < (u::real)} = {n. u \<le> abs (xa n)}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1881
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1882
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1883
lemma lemma_Int_eq1:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1884
     "{n. abs (xa n) \<le> (u::real)} Int {n. u \<le> abs (xa n)}
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1885
          = {n. abs(xa n) = u}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1886
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1887
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1888
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1889
lemma lemma_FreeUltrafilterNat_one:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1890
     "{n. abs (xa n) = u} \<le> {n. abs (xa n) < u + (1::real)}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1891
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1892
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1893
(*-------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1894
  Exclude this type of sets from free
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1895
  ultrafilter for Infinite numbers!
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1896
 -------------------------------------*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1897
lemma FreeUltrafilterNat_const_Finite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1898
     "[| xa: Rep_hypreal x;
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1899
                  {n. abs (xa n) = u} \<in> FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1900
               |] ==> x \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1901
apply (rule FreeUltrafilterNat_HFinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1902
apply (rule_tac x = xa in bexI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1903
apply (rule_tac x = "u + 1" in exI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1904
apply (ultra, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1905
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1906
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1907
lemma HInfinite_FreeUltrafilterNat:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1908
     "x \<in> HInfinite ==> \<exists>X \<in> Rep_hypreal x.
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1909
           \<forall>u. {n. u < abs (X n)} \<in> FreeUltrafilterNat"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1910
apply (frule HInfinite_HFinite_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1911
apply (cut_tac x = x in Rep_hypreal_nonempty)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1912
apply (auto simp del: Rep_hypreal_nonempty simp add: HFinite_FreeUltrafilterNat_iff Bex_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1913
apply (drule spec)+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1914
apply auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1915
apply (drule_tac x = u in spec)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1916
apply (drule FreeUltrafilterNat_Compl_mem)+
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1917
apply (drule FreeUltrafilterNat_Int, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1918
apply (simp add: lemma_Compl_eq lemma_Compl_eq2 lemma_Int_eq1)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1919
apply (auto dest: FreeUltrafilterNat_const_Finite simp
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1920
      add: HInfinite_HFinite_iff [THEN iffD1])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1921
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1922
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1923
lemma lemma_Int_HI:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1924
     "{n. abs (Xa n) < u} Int {n. X n = Xa n} \<subseteq> {n. abs (X n) < (u::real)}"
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1925
by auto
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1926
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1927
lemma lemma_Int_HIa: "{n. u < abs (X n)} Int {n. abs (X n) < (u::real)} = {}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1928
by (auto intro: order_less_asym)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1929
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1930
lemma FreeUltrafilterNat_HInfinite:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1931
     "\<exists>X \<in> Rep_hypreal x. \<forall>u.
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1932
               {n. u < abs (X n)} \<in> FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1933
               ==>  x \<in> HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1934
apply (rule HInfinite_HFinite_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1935
apply (safe, drule HFinite_FreeUltrafilterNat, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1936
apply (drule_tac x = u in spec)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1937
apply (drule FreeUltrafilterNat_Rep_hypreal, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1938
apply (drule_tac Y = "{n. X n = Xa n}" in FreeUltrafilterNat_Int, simp) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1939
apply (drule lemma_Int_HI [THEN [2] FreeUltrafilterNat_subset])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1940
apply (drule_tac Y = "{n. abs (X n) < u}" in FreeUltrafilterNat_Int)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1941
apply (auto simp add: lemma_Int_HIa FreeUltrafilterNat_empty)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1942
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1943
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1944
lemma HInfinite_FreeUltrafilterNat_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1945
     "(x \<in> HInfinite) = (\<exists>X \<in> Rep_hypreal x.
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1946
           \<forall>u. {n. u < abs (X n)} \<in> FreeUltrafilterNat)"
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1947
by (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1948
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1949
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1950
subsection{*Alternative Definitions for @{term Infinitesimal} using Free Ultrafilter*}
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
  1951
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1952
lemma Infinitesimal_FreeUltrafilterNat:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1953
          "x \<in> Infinitesimal ==> \<exists>X \<in> Rep_hypreal x.
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1954
           \<forall>u. 0 < u --> {n. abs (X n) < u} \<in> FreeUltrafilterNat"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1955
apply (simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1956
apply (auto simp add: abs_less_iff minus_less_iff [of x])
14468
6be497cacab5 heavy tidying
paulson
parents: 14430
diff changeset
  1957
apply (cases x)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1958
apply (auto, rule bexI [OF _ lemma_hyprel_refl], safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1959
apply (drule hypreal_of_real_less_iff [THEN iffD2])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1960
apply (drule_tac x = "hypreal_of_real u" in bspec, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1961
apply (auto simp add: hypreal_less hypreal_minus hypreal_of_real_def, ultra)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1962
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1963
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1964
lemma FreeUltrafilterNat_Infinitesimal:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1965
     "\<exists>X \<in> Rep_hypreal x.
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1966
            \<forall>u. 0 < u --> {n. abs (X n) < u} \<in> FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1967
      ==> x \<in> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1968
apply (simp add: Infinitesimal_def)
14468
6be497cacab5 heavy tidying
paulson
parents: 14430
diff changeset
  1969
apply (cases x)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1970
apply (auto simp add: abs_less_iff abs_interval_iff minus_less_iff [of x])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1971
apply (auto simp add: SReal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1972
apply (drule_tac [!] x=y in spec) 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1973
apply (auto simp add: hypreal_less hypreal_minus hypreal_of_real_def, ultra+)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1974
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1975
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1976
lemma Infinitesimal_FreeUltrafilterNat_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1977
     "(x \<in> Infinitesimal) = (\<exists>X \<in> Rep_hypreal x.
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1978
           \<forall>u. 0 < u --> {n. abs (X n) < u} \<in> FreeUltrafilterNat)"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1979
apply (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1980
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1981
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1982
(*------------------------------------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1983
         Infinitesimals as smaller than 1/n for all n::nat (> 0)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1984
 ------------------------------------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1985
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1986
lemma lemma_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1987
     "(\<forall>r. 0 < r --> x < r) = (\<forall>n. x < inverse(real (Suc n)))"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1988
apply (auto simp add: real_of_nat_Suc_gt_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1989
apply (blast dest!: reals_Archimedean intro: order_less_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1990
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1991
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  1992
lemma of_nat_in_Reals [simp]: "(of_nat n::hypreal) \<in> \<real>"
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  1993
apply (induct n)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1994
apply (simp_all add: SReal_add)
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  1995
done 
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  1996
 
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1997
lemma lemma_Infinitesimal2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  1998
     "(\<forall>r \<in> Reals. 0 < r --> x < r) =
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  1999
      (\<forall>n. x < inverse(hypreal_of_nat (Suc n)))"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2000
apply safe
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2001
apply (drule_tac x = "inverse (hypreal_of_real (real (Suc n))) " in bspec)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2002
apply (simp (no_asm_use) add: SReal_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2003
apply (rule real_of_nat_Suc_gt_zero [THEN positive_imp_inverse_positive, THEN hypreal_of_real_less_iff [THEN iffD2], THEN [2] impE])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2004
prefer 2 apply assumption
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  2005
apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_eq)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2006
apply (auto dest!: reals_Archimedean simp add: SReal_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2007
apply (drule hypreal_of_real_less_iff [THEN iffD2])
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  2008
apply (simp add: real_of_nat_Suc_gt_zero hypreal_of_nat_eq)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2009
apply (blast intro: order_less_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2010
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2011
14378
69c4d5997669 generic of_nat and of_int functions, and generalization of iszero
paulson
parents: 14371
diff changeset
  2012
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2013
lemma Infinitesimal_hypreal_of_nat_iff:
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2014
     "Infinitesimal = {x. \<forall>n. abs x < inverse (hypreal_of_nat (Suc n))}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2015
apply (simp add: Infinitesimal_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2016
apply (auto simp add: lemma_Infinitesimal2)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2017
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2018
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2019
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2020
(*-------------------------------------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2021
       Proof that omega is an infinite number and
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2022
       hence that epsilon is an infinitesimal number.
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2023
 -------------------------------------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2024
lemma Suc_Un_eq: "{n. n < Suc m} = {n. n < m} Un {n. n = m}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2025
by (auto simp add: less_Suc_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2026
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2027
(*-------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2028
  Prove that any segment is finite and
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2029
  hence cannot belong to FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2030
 -------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2031
lemma finite_nat_segment: "finite {n::nat. n < m}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2032
apply (induct_tac "m")
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2033
apply (auto simp add: Suc_Un_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2034
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2035
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2036
lemma finite_real_of_nat_segment: "finite {n::nat. real n < real (m::nat)}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2037
by (auto intro: finite_nat_segment)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2038
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2039
lemma finite_real_of_nat_less_real: "finite {n::nat. real n < u}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2040
apply (cut_tac x = u in reals_Archimedean2, safe)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2041
apply (rule finite_real_of_nat_segment [THEN [2] finite_subset])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2042
apply (auto dest: order_less_trans)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2043
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2044
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2045
lemma lemma_real_le_Un_eq:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2046
     "{n. f n \<le> u} = {n. f n < u} Un {n. u = (f n :: real)}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2047
by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2048
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2049
lemma finite_real_of_nat_le_real: "finite {n::nat. real n \<le> u}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2050
by (auto simp add: lemma_real_le_Un_eq lemma_finite_omega_set finite_real_of_nat_less_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2051
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2052
lemma finite_rabs_real_of_nat_le_real: "finite {n::nat. abs(real n) \<le> u}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2053
apply (simp (no_asm) add: real_of_nat_Suc_gt_zero finite_real_of_nat_le_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2054
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2055
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2056
lemma rabs_real_of_nat_le_real_FreeUltrafilterNat:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2057
     "{n. abs(real n) \<le> u} \<notin> FreeUltrafilterNat"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2058
by (blast intro!: FreeUltrafilterNat_finite finite_rabs_real_of_nat_le_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2059
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2060
lemma FreeUltrafilterNat_nat_gt_real: "{n. u < real n} \<in> FreeUltrafilterNat"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2061
apply (rule ccontr, drule FreeUltrafilterNat_Compl_mem)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2062
apply (subgoal_tac "- {n::nat. u < real n} = {n. real n \<le> u}")
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2063
prefer 2 apply force
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2064
apply (simp add: finite_real_of_nat_le_real [THEN FreeUltrafilterNat_finite])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2065
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2066
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2067
(*--------------------------------------------------------------
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2068
 The complement of {n. abs(real n) \<le> u} =
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2069
 {n. u < abs (real n)} is in FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2070
 by property of (free) ultrafilters
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2071
 --------------------------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2072
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2073
lemma Compl_real_le_eq: "- {n::nat. real n \<le> u} = {n. u < real n}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2074
by (auto dest!: order_le_less_trans simp add: linorder_not_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2075
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2076
(*-----------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2077
       Omega is a member of HInfinite
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2078
 -----------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2079
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2080
lemma hypreal_omega: "hyprel``{%n::nat. real (Suc n)} \<in> hypreal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2081
by auto
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2082
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2083
lemma FreeUltrafilterNat_omega: "{n. u < real n} \<in> FreeUltrafilterNat"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2084
apply (cut_tac u = u in rabs_real_of_nat_le_real_FreeUltrafilterNat)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2085
apply (auto dest: FreeUltrafilterNat_Compl_mem simp add: Compl_real_le_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2086
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2087
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2088
lemma HInfinite_omega: "omega: HInfinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2089
apply (simp add: omega_def)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2090
apply (auto intro!: FreeUltrafilterNat_HInfinite)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2091
apply (rule bexI)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2092
apply (rule_tac [2] lemma_hyprel_refl, auto)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2093
apply (simp (no_asm) add: real_of_nat_Suc diff_less_eq [symmetric] FreeUltrafilterNat_omega)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2094
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2095
declare HInfinite_omega [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2096
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2097
(*-----------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2098
       Epsilon is a member of Infinitesimal
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2099
 -----------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2100
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2101
lemma Infinitesimal_epsilon: "epsilon \<in> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2102
by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega simp add: hypreal_epsilon_inverse_omega)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2103
declare Infinitesimal_epsilon [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2104
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2105
lemma HFinite_epsilon: "epsilon \<in> HFinite"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2106
by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2107
declare HFinite_epsilon [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2108
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2109
lemma epsilon_approx_zero: "epsilon @= 0"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2110
apply (simp (no_asm) add: mem_infmal_iff [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2111
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2112
declare epsilon_approx_zero [simp]
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2113
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2114
(*------------------------------------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2115
  Needed for proof that we define a hyperreal [<X(n)] @= hypreal_of_real a given
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2116
  that \<forall>n. |X n - a| < 1/n. Used in proof of NSLIM => LIM.
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2117
 -----------------------------------------------------------------------*)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2118
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2119
lemma real_of_nat_less_inverse_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2120
     "0 < u  ==> (u < inverse (real(Suc n))) = (real(Suc n) < inverse u)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2121
apply (simp add: inverse_eq_divide)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2122
apply (subst pos_less_divide_eq, assumption)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2123
apply (subst pos_less_divide_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2124
 apply (simp add: real_of_nat_Suc_gt_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2125
apply (simp add: real_mult_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2126
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2127
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2128
lemma finite_inverse_real_of_posnat_gt_real:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2129
     "0 < u ==> finite {n. u < inverse(real(Suc n))}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2130
apply (simp (no_asm_simp) add: real_of_nat_less_inverse_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2131
apply (simp (no_asm_simp) add: real_of_nat_Suc less_diff_eq [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2132
apply (rule finite_real_of_nat_less_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2133
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2134
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2135
lemma lemma_real_le_Un_eq2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2136
     "{n. u \<le> inverse(real(Suc n))} =
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2137
     {n. u < inverse(real(Suc n))} Un {n. u = inverse(real(Suc n))}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2138
apply (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2139
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2140
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2141
lemma real_of_nat_inverse_le_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2142
     "(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2143
apply (simp (no_asm) add: linorder_not_less [symmetric])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2144
apply (simp (no_asm) add: inverse_eq_divide)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2145
apply (subst pos_less_divide_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2146
apply (simp (no_asm) add: real_of_nat_Suc_gt_zero)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2147
apply (simp (no_asm) add: real_mult_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2148
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2149
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2150
lemma real_of_nat_inverse_eq_iff:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2151
     "(u = inverse (real(Suc n))) = (real(Suc n) = inverse u)"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2152
by (auto simp add: inverse_inverse_eq real_of_nat_Suc_gt_zero real_not_refl2 [THEN not_sym])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2153
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2154
lemma lemma_finite_omega_set2: "finite {n::nat. u = inverse(real(Suc n))}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2155
apply (simp (no_asm_simp) add: real_of_nat_inverse_eq_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2156
apply (cut_tac x = "inverse u - 1" in lemma_finite_omega_set)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2157
apply (simp add: real_of_nat_Suc diff_eq_eq [symmetric] eq_commute)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2158
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2159
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2160
lemma finite_inverse_real_of_posnat_ge_real:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2161
     "0 < u ==> finite {n. u \<le> inverse(real(Suc n))}"
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2162
by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_omega_set2 finite_inverse_real_of_posnat_gt_real)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2163
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2164
lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2165
     "0 < u ==> {n. u \<le> inverse(real(Suc n))} \<notin> FreeUltrafilterNat"
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2166
by (blast intro!: FreeUltrafilterNat_finite finite_inverse_real_of_posnat_ge_real)
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2167
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2168
(*--------------------------------------------------------------
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2169
    The complement of  {n. u \<le> inverse(real(Suc n))} =
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2170
    {n. inverse(real(Suc n)) < u} is in FreeUltrafilterNat
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2171
    by property of (free) ultrafilters
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2172
 --------------------------------------------------------------*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2173
lemma Compl_le_inverse_eq:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2174
     "- {n. u \<le> inverse(real(Suc n))} =
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2175
      {n. inverse(real(Suc n)) < u}"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2176
apply (auto dest!: order_le_less_trans simp add: linorder_not_le)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2177
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2178
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2179
lemma FreeUltrafilterNat_inverse_real_of_posnat:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2180
     "0 < u ==>
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2181
      {n. inverse(real(Suc n)) < u} \<in> FreeUltrafilterNat"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2182
apply (cut_tac u = u in inverse_real_of_posnat_ge_real_FreeUltrafilterNat)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2183
apply (auto dest: FreeUltrafilterNat_Compl_mem simp add: Compl_le_inverse_eq)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2184
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2185
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2186
text{* Example where we get a hyperreal from a real sequence
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2187
      for which a particular property holds. The theorem is
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2188
      used in proofs about equivalence of nonstandard and
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2189
      standard neighbourhoods. Also used for equivalence of
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2190
      nonstandard ans standard definitions of pointwise
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2191
      limit.*}
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2192
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2193
(*-----------------------------------------------------
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2194
    |X(n) - x| < 1/n ==> [<X n>] - hypreal_of_real x| \<in> Infinitesimal
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2195
 -----------------------------------------------------*)
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2196
lemma real_seq_to_hypreal_Infinitesimal:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2197
     "\<forall>n. abs(X n + -x) < inverse(real(Suc n))
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2198
     ==> Abs_hypreal(hyprel``{X}) + -hypreal_of_real x \<in> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2199
apply (auto intro!: bexI dest: FreeUltrafilterNat_inverse_real_of_posnat FreeUltrafilterNat_all FreeUltrafilterNat_Int intro: order_less_trans FreeUltrafilterNat_subset simp add: hypreal_minus hypreal_of_real_def hypreal_add Infinitesimal_FreeUltrafilterNat_iff hypreal_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2200
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2201
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2202
lemma real_seq_to_hypreal_approx:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2203
     "\<forall>n. abs(X n + -x) < inverse(real(Suc n))
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2204
      ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2205
apply (subst approx_minus_iff)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2206
apply (rule mem_infmal_iff [THEN subst])
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2207
apply (erule real_seq_to_hypreal_Infinitesimal)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2208
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2209
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2210
lemma real_seq_to_hypreal_approx2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2211
     "\<forall>n. abs(x + -X n) < inverse(real(Suc n))
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2212
               ==> Abs_hypreal(hyprel``{X}) @= hypreal_of_real x"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2213
apply (simp add: abs_minus_add_cancel real_seq_to_hypreal_approx)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2214
done
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2215
14420
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2216
lemma real_seq_to_hypreal_Infinitesimal2:
4e72cd222e0b converted Hyperreal/HTranscendental to Isar script
paulson
parents: 14387
diff changeset
  2217
     "\<forall>n. abs(X n + -Y n) < inverse(real(Suc n))
14370
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2218
      ==> Abs_hypreal(hyprel``{X}) +
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2219
          -Abs_hypreal(hyprel``{Y}) \<in> Infinitesimal"
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2220
by (auto intro!: bexI
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2221
	 dest: FreeUltrafilterNat_inverse_real_of_posnat 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2222
	       FreeUltrafilterNat_all FreeUltrafilterNat_Int
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2223
	 intro: order_less_trans FreeUltrafilterNat_subset 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2224
	 simp add: Infinitesimal_FreeUltrafilterNat_iff hypreal_minus 
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2225
                   hypreal_add hypreal_inverse)
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2226
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2227
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2228
ML
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2229
{*
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2230
val Infinitesimal_def = thm"Infinitesimal_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2231
val HFinite_def = thm"HFinite_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2232
val HInfinite_def = thm"HInfinite_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2233
val st_def = thm"st_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2234
val monad_def = thm"monad_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2235
val galaxy_def = thm"galaxy_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2236
val approx_def = thm"approx_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2237
val SReal_def = thm"SReal_def";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2238
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2239
val Infinitesimal_approx_minus = thm "Infinitesimal_approx_minus";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2240
val approx_monad_iff = thm "approx_monad_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2241
val Infinitesimal_approx = thm "Infinitesimal_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2242
val approx_add = thm "approx_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2243
val approx_minus = thm "approx_minus";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2244
val approx_minus2 = thm "approx_minus2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2245
val approx_minus_cancel = thm "approx_minus_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2246
val approx_add_minus = thm "approx_add_minus";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2247
val approx_mult1 = thm "approx_mult1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2248
val approx_mult2 = thm "approx_mult2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2249
val approx_mult_subst = thm "approx_mult_subst";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2250
val approx_mult_subst2 = thm "approx_mult_subst2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2251
val approx_mult_subst_SReal = thm "approx_mult_subst_SReal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2252
val approx_eq_imp = thm "approx_eq_imp";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2253
val Infinitesimal_minus_approx = thm "Infinitesimal_minus_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2254
val bex_Infinitesimal_iff = thm "bex_Infinitesimal_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2255
val bex_Infinitesimal_iff2 = thm "bex_Infinitesimal_iff2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2256
val Infinitesimal_add_approx = thm "Infinitesimal_add_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2257
val Infinitesimal_add_approx_self = thm "Infinitesimal_add_approx_self";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2258
val Infinitesimal_add_approx_self2 = thm "Infinitesimal_add_approx_self2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2259
val Infinitesimal_add_minus_approx_self = thm "Infinitesimal_add_minus_approx_self";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2260
val Infinitesimal_add_cancel = thm "Infinitesimal_add_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2261
val Infinitesimal_add_right_cancel = thm "Infinitesimal_add_right_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2262
val approx_add_left_cancel = thm "approx_add_left_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2263
val approx_add_right_cancel = thm "approx_add_right_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2264
val approx_add_mono1 = thm "approx_add_mono1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2265
val approx_add_mono2 = thm "approx_add_mono2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2266
val approx_add_left_iff = thm "approx_add_left_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2267
val approx_add_right_iff = thm "approx_add_right_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2268
val approx_HFinite = thm "approx_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2269
val approx_hypreal_of_real_HFinite = thm "approx_hypreal_of_real_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2270
val approx_mult_HFinite = thm "approx_mult_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2271
val approx_mult_hypreal_of_real = thm "approx_mult_hypreal_of_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2272
val approx_SReal_mult_cancel_zero = thm "approx_SReal_mult_cancel_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2273
val approx_mult_SReal1 = thm "approx_mult_SReal1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2274
val approx_mult_SReal2 = thm "approx_mult_SReal2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2275
val approx_mult_SReal_zero_cancel_iff = thm "approx_mult_SReal_zero_cancel_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2276
val approx_SReal_mult_cancel = thm "approx_SReal_mult_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2277
val approx_SReal_mult_cancel_iff1 = thm "approx_SReal_mult_cancel_iff1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2278
val approx_le_bound = thm "approx_le_bound";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2279
val Infinitesimal_less_SReal = thm "Infinitesimal_less_SReal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2280
val Infinitesimal_less_SReal2 = thm "Infinitesimal_less_SReal2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2281
val SReal_not_Infinitesimal = thm "SReal_not_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2282
val SReal_minus_not_Infinitesimal = thm "SReal_minus_not_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2283
val SReal_Int_Infinitesimal_zero = thm "SReal_Int_Infinitesimal_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2284
val SReal_Infinitesimal_zero = thm "SReal_Infinitesimal_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2285
val SReal_HFinite_diff_Infinitesimal = thm "SReal_HFinite_diff_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2286
val hypreal_of_real_HFinite_diff_Infinitesimal = thm "hypreal_of_real_HFinite_diff_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2287
val hypreal_of_real_Infinitesimal_iff_0 = thm "hypreal_of_real_Infinitesimal_iff_0";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2288
val number_of_not_Infinitesimal = thm "number_of_not_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2289
val one_not_Infinitesimal = thm "one_not_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2290
val approx_SReal_not_zero = thm "approx_SReal_not_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2291
val HFinite_diff_Infinitesimal_approx = thm "HFinite_diff_Infinitesimal_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2292
val Infinitesimal_ratio = thm "Infinitesimal_ratio";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2293
val SReal_approx_iff = thm "SReal_approx_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2294
val number_of_approx_iff = thm "number_of_approx_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2295
val hypreal_of_real_approx_iff = thm "hypreal_of_real_approx_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2296
val hypreal_of_real_approx_number_of_iff = thm "hypreal_of_real_approx_number_of_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2297
val approx_unique_real = thm "approx_unique_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2298
val hypreal_isLub_unique = thm "hypreal_isLub_unique";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2299
val hypreal_setle_less_trans = thm "hypreal_setle_less_trans";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2300
val hypreal_gt_isUb = thm "hypreal_gt_isUb";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2301
val st_part_Ex = thm "st_part_Ex";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2302
val st_part_Ex1 = thm "st_part_Ex1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2303
val HFinite_Int_HInfinite_empty = thm "HFinite_Int_HInfinite_empty";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2304
val HFinite_not_HInfinite = thm "HFinite_not_HInfinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2305
val not_HFinite_HInfinite = thm "not_HFinite_HInfinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2306
val HInfinite_HFinite_disj = thm "HInfinite_HFinite_disj";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2307
val HInfinite_HFinite_iff = thm "HInfinite_HFinite_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2308
val HFinite_HInfinite_iff = thm "HFinite_HInfinite_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2309
val HInfinite_diff_HFinite_Infinitesimal_disj = thm "HInfinite_diff_HFinite_Infinitesimal_disj";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2310
val HFinite_inverse = thm "HFinite_inverse";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2311
val HFinite_inverse2 = thm "HFinite_inverse2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2312
val Infinitesimal_inverse_HFinite = thm "Infinitesimal_inverse_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2313
val HFinite_not_Infinitesimal_inverse = thm "HFinite_not_Infinitesimal_inverse";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2314
val approx_inverse = thm "approx_inverse";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2315
val hypreal_of_real_approx_inverse = thm "hypreal_of_real_approx_inverse";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2316
val inverse_add_Infinitesimal_approx = thm "inverse_add_Infinitesimal_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2317
val inverse_add_Infinitesimal_approx2 = thm "inverse_add_Infinitesimal_approx2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2318
val inverse_add_Infinitesimal_approx_Infinitesimal = thm "inverse_add_Infinitesimal_approx_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2319
val Infinitesimal_square_iff = thm "Infinitesimal_square_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2320
val HFinite_square_iff = thm "HFinite_square_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2321
val HInfinite_square_iff = thm "HInfinite_square_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2322
val approx_HFinite_mult_cancel = thm "approx_HFinite_mult_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2323
val approx_HFinite_mult_cancel_iff1 = thm "approx_HFinite_mult_cancel_iff1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2324
val approx_hrabs_disj = thm "approx_hrabs_disj";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2325
val monad_hrabs_Un_subset = thm "monad_hrabs_Un_subset";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2326
val Infinitesimal_monad_eq = thm "Infinitesimal_monad_eq";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2327
val mem_monad_iff = thm "mem_monad_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2328
val Infinitesimal_monad_zero_iff = thm "Infinitesimal_monad_zero_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2329
val monad_zero_minus_iff = thm "monad_zero_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2330
val monad_zero_hrabs_iff = thm "monad_zero_hrabs_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2331
val mem_monad_self = thm "mem_monad_self";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2332
val approx_subset_monad = thm "approx_subset_monad";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2333
val approx_subset_monad2 = thm "approx_subset_monad2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2334
val mem_monad_approx = thm "mem_monad_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2335
val approx_mem_monad = thm "approx_mem_monad";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2336
val approx_mem_monad2 = thm "approx_mem_monad2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2337
val approx_mem_monad_zero = thm "approx_mem_monad_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2338
val Infinitesimal_approx_hrabs = thm "Infinitesimal_approx_hrabs";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2339
val less_Infinitesimal_less = thm "less_Infinitesimal_less";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2340
val Ball_mem_monad_gt_zero = thm "Ball_mem_monad_gt_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2341
val Ball_mem_monad_less_zero = thm "Ball_mem_monad_less_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2342
val approx_hrabs1 = thm "approx_hrabs1";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2343
val approx_hrabs2 = thm "approx_hrabs2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2344
val approx_hrabs = thm "approx_hrabs";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2345
val approx_hrabs_zero_cancel = thm "approx_hrabs_zero_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2346
val approx_hrabs_add_Infinitesimal = thm "approx_hrabs_add_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2347
val approx_hrabs_add_minus_Infinitesimal = thm "approx_hrabs_add_minus_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2348
val hrabs_add_Infinitesimal_cancel = thm "hrabs_add_Infinitesimal_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2349
val hrabs_add_minus_Infinitesimal_cancel = thm "hrabs_add_minus_Infinitesimal_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2350
val hypreal_less_minus_iff = thm "hypreal_less_minus_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2351
val Infinitesimal_add_hypreal_of_real_less = thm "Infinitesimal_add_hypreal_of_real_less";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2352
val Infinitesimal_add_hrabs_hypreal_of_real_less = thm "Infinitesimal_add_hrabs_hypreal_of_real_less";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2353
val Infinitesimal_add_hrabs_hypreal_of_real_less2 = thm "Infinitesimal_add_hrabs_hypreal_of_real_less2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2354
val hypreal_of_real_le_add_Infininitesimal_cancel2 = thm"hypreal_of_real_le_add_Infininitesimal_cancel2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2355
val hypreal_of_real_less_Infinitesimal_le_zero = thm "hypreal_of_real_less_Infinitesimal_le_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2356
val Infinitesimal_add_not_zero = thm "Infinitesimal_add_not_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2357
val Infinitesimal_square_cancel = thm "Infinitesimal_square_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2358
val HFinite_square_cancel = thm "HFinite_square_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2359
val Infinitesimal_square_cancel2 = thm "Infinitesimal_square_cancel2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2360
val HFinite_square_cancel2 = thm "HFinite_square_cancel2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2361
val Infinitesimal_sum_square_cancel = thm "Infinitesimal_sum_square_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2362
val HFinite_sum_square_cancel = thm "HFinite_sum_square_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2363
val Infinitesimal_sum_square_cancel2 = thm "Infinitesimal_sum_square_cancel2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2364
val HFinite_sum_square_cancel2 = thm "HFinite_sum_square_cancel2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2365
val Infinitesimal_sum_square_cancel3 = thm "Infinitesimal_sum_square_cancel3";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2366
val HFinite_sum_square_cancel3 = thm "HFinite_sum_square_cancel3";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2367
val monad_hrabs_less = thm "monad_hrabs_less";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2368
val mem_monad_SReal_HFinite = thm "mem_monad_SReal_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2369
val st_approx_self = thm "st_approx_self";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2370
val st_SReal = thm "st_SReal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2371
val st_HFinite = thm "st_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2372
val st_SReal_eq = thm "st_SReal_eq";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2373
val st_hypreal_of_real = thm "st_hypreal_of_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2374
val st_eq_approx = thm "st_eq_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2375
val approx_st_eq = thm "approx_st_eq";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2376
val st_eq_approx_iff = thm "st_eq_approx_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2377
val st_Infinitesimal_add_SReal = thm "st_Infinitesimal_add_SReal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2378
val st_Infinitesimal_add_SReal2 = thm "st_Infinitesimal_add_SReal2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2379
val HFinite_st_Infinitesimal_add = thm "HFinite_st_Infinitesimal_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2380
val st_add = thm "st_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2381
val st_number_of = thm "st_number_of";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2382
val st_minus = thm "st_minus";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2383
val st_diff = thm "st_diff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2384
val st_mult = thm "st_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2385
val st_Infinitesimal = thm "st_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2386
val st_not_Infinitesimal = thm "st_not_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2387
val st_inverse = thm "st_inverse";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2388
val st_divide = thm "st_divide";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2389
val st_idempotent = thm "st_idempotent";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2390
val Infinitesimal_add_st_less = thm "Infinitesimal_add_st_less";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2391
val Infinitesimal_add_st_le_cancel = thm "Infinitesimal_add_st_le_cancel";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2392
val st_le = thm "st_le";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2393
val st_zero_le = thm "st_zero_le";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2394
val st_zero_ge = thm "st_zero_ge";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2395
val st_hrabs = thm "st_hrabs";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2396
val FreeUltrafilterNat_HFinite = thm "FreeUltrafilterNat_HFinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2397
val HFinite_FreeUltrafilterNat_iff = thm "HFinite_FreeUltrafilterNat_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2398
val FreeUltrafilterNat_const_Finite = thm "FreeUltrafilterNat_const_Finite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2399
val FreeUltrafilterNat_HInfinite = thm "FreeUltrafilterNat_HInfinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2400
val HInfinite_FreeUltrafilterNat_iff = thm "HInfinite_FreeUltrafilterNat_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2401
val Infinitesimal_FreeUltrafilterNat = thm "Infinitesimal_FreeUltrafilterNat";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2402
val FreeUltrafilterNat_Infinitesimal = thm "FreeUltrafilterNat_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2403
val Infinitesimal_FreeUltrafilterNat_iff = thm "Infinitesimal_FreeUltrafilterNat_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2404
val Infinitesimal_hypreal_of_nat_iff = thm "Infinitesimal_hypreal_of_nat_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2405
val Suc_Un_eq = thm "Suc_Un_eq";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2406
val finite_nat_segment = thm "finite_nat_segment";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2407
val finite_real_of_nat_segment = thm "finite_real_of_nat_segment";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2408
val finite_real_of_nat_less_real = thm "finite_real_of_nat_less_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2409
val finite_real_of_nat_le_real = thm "finite_real_of_nat_le_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2410
val finite_rabs_real_of_nat_le_real = thm "finite_rabs_real_of_nat_le_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2411
val rabs_real_of_nat_le_real_FreeUltrafilterNat = thm "rabs_real_of_nat_le_real_FreeUltrafilterNat";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2412
val FreeUltrafilterNat_nat_gt_real = thm "FreeUltrafilterNat_nat_gt_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2413
val hypreal_omega = thm "hypreal_omega";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2414
val FreeUltrafilterNat_omega = thm "FreeUltrafilterNat_omega";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2415
val HInfinite_omega = thm "HInfinite_omega";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2416
val Infinitesimal_epsilon = thm "Infinitesimal_epsilon";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2417
val HFinite_epsilon = thm "HFinite_epsilon";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2418
val epsilon_approx_zero = thm "epsilon_approx_zero";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2419
val real_of_nat_less_inverse_iff = thm "real_of_nat_less_inverse_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2420
val finite_inverse_real_of_posnat_gt_real = thm "finite_inverse_real_of_posnat_gt_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2421
val real_of_nat_inverse_le_iff = thm "real_of_nat_inverse_le_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2422
val real_of_nat_inverse_eq_iff = thm "real_of_nat_inverse_eq_iff";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2423
val finite_inverse_real_of_posnat_ge_real = thm "finite_inverse_real_of_posnat_ge_real";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2424
val inverse_real_of_posnat_ge_real_FreeUltrafilterNat = thm "inverse_real_of_posnat_ge_real_FreeUltrafilterNat";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2425
val FreeUltrafilterNat_inverse_real_of_posnat = thm "FreeUltrafilterNat_inverse_real_of_posnat";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2426
val real_seq_to_hypreal_Infinitesimal = thm "real_seq_to_hypreal_Infinitesimal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2427
val real_seq_to_hypreal_approx = thm "real_seq_to_hypreal_approx";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2428
val real_seq_to_hypreal_approx2 = thm "real_seq_to_hypreal_approx2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2429
val real_seq_to_hypreal_Infinitesimal2 = thm "real_seq_to_hypreal_Infinitesimal2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2430
val HInfinite_HFinite_add = thm "HInfinite_HFinite_add";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2431
val HInfinite_ge_HInfinite = thm "HInfinite_ge_HInfinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2432
val Infinitesimal_inverse_HInfinite = thm "Infinitesimal_inverse_HInfinite";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2433
val HInfinite_HFinite_not_Infinitesimal_mult = thm "HInfinite_HFinite_not_Infinitesimal_mult";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2434
val HInfinite_HFinite_not_Infinitesimal_mult2 = thm "HInfinite_HFinite_not_Infinitesimal_mult2";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2435
val HInfinite_gt_SReal = thm "HInfinite_gt_SReal";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2436
val HInfinite_gt_zero_gt_one = thm "HInfinite_gt_zero_gt_one";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2437
val not_HInfinite_one = thm "not_HInfinite_one";
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2438
*}
b0064703967b simplifications in the hyperreals
paulson
parents: 12114
diff changeset
  2439
10751
a81ea5d3dd41 separation of HOL-Hyperreal from HOL-Real
paulson
parents:
diff changeset
  2440
end