author wenzelm Tue Nov 01 00:44:24 2016 +0100 (2016-11-01) changeset 64435 c93b0e6131c3 parent 64434 af5235830c16 child 64436 254c9411fc48
misc tuning and modernization;
```     1.1 --- a/src/HOL/Nonstandard_Analysis/Free_Ultrafilter.thy	Mon Oct 31 16:26:36 2016 +0100
1.2 +++ b/src/HOL/Nonstandard_Analysis/Free_Ultrafilter.thy	Tue Nov 01 00:44:24 2016 +0100
1.3 @@ -2,14 +2,15 @@
1.4      Author:     Jacques D. Fleuriot, University of Cambridge
1.5      Author:     Lawrence C Paulson
1.6      Author:     Brian Huffman
1.7 -*)
1.8 +*)
1.9
1.10  section \<open>Filters and Ultrafilters\<close>
1.11
1.12  theory Free_Ultrafilter
1.13 -imports "~~/src/HOL/Library/Infinite_Set"
1.14 +  imports "~~/src/HOL/Library/Infinite_Set"
1.15  begin
1.16
1.17 +
1.18  subsection \<open>Definitions and basic properties\<close>
1.19
1.20  subsubsection \<open>Ultrafilters\<close>
1.21 @@ -43,28 +44,29 @@
1.22
1.23  end
1.24
1.25 +
1.26  subsection \<open>Maximal filter = Ultrafilter\<close>
1.27
1.28  text \<open>
1.29 -   A filter F is an ultrafilter iff it is a maximal filter,
1.30 -   i.e. whenever G is a filter and @{term "F \<subseteq> G"} then @{term "F = G"}
1.31 +   A filter \<open>F\<close> is an ultrafilter iff it is a maximal filter,
1.32 +   i.e. whenever \<open>G\<close> is a filter and @{prop "F \<subseteq> G"} then @{prop "F = G"}
1.33  \<close>
1.34 +
1.35  text \<open>
1.36 -  Lemmas that shows existence of an extension to what was assumed to
1.37 +  Lemma that shows existence of an extension to what was assumed to
1.38    be a maximal filter. Will be used to derive contradiction in proof of
1.39    property of ultrafilter.
1.40  \<close>
1.41
1.42 -lemma extend_filter:
1.43 -  "frequently P F \<Longrightarrow> inf F (principal {x. P x}) \<noteq> bot"
1.44 -  unfolding trivial_limit_def eventually_inf_principal by (simp add: not_eventually)
1.45 +lemma extend_filter: "frequently P F \<Longrightarrow> inf F (principal {x. P x}) \<noteq> bot"
1.46 +  by (simp add: trivial_limit_def eventually_inf_principal not_eventually)
1.47
1.48  lemma max_filter_ultrafilter:
1.49 -  assumes proper: "F \<noteq> bot"
1.50 +  assumes "F \<noteq> bot"
1.51    assumes max: "\<And>G. G \<noteq> bot \<Longrightarrow> G \<le> F \<Longrightarrow> F = G"
1.52    shows "ultrafilter F"
1.53  proof
1.54 -  fix P show "eventually P F \<or> (\<forall>\<^sub>Fx in F. \<not> P x)"
1.55 +  show "eventually P F \<or> (\<forall>\<^sub>Fx in F. \<not> P x)" for P
1.56    proof (rule disjCI)
1.57      assume "\<not> (\<forall>\<^sub>Fx in F. \<not> P x)"
1.58      then have "inf F (principal {x. P x}) \<noteq> bot"
1.59 @@ -84,7 +86,9 @@
1.60    done
1.61
1.62  lemma (in ultrafilter) max_filter:
1.63 -  assumes G: "G \<noteq> bot" and sub: "G \<le> F" shows "F = G"
1.64 +  assumes G: "G \<noteq> bot"
1.65 +    and sub: "G \<le> F"
1.66 +  shows "F = G"
1.67  proof (rule antisym)
1.68    show "F \<le> G"
1.69      using sub
1.70 @@ -92,10 +96,9 @@
1.71               intro!: eventually_frequently G proper)
1.72  qed fact
1.73
1.74 +
1.75  subsection \<open>Ultrafilter Theorem\<close>
1.76
1.77 -text "A local context makes proof of ultrafilter Theorem more modular"
1.78 -
1.79  lemma ex_max_ultrafilter:
1.80    fixes F :: "'a filter"
1.81    assumes F: "F \<noteq> bot"
1.82 @@ -104,73 +107,77 @@
1.83    let ?X = "{G. G \<noteq> bot \<and> G \<le> F}"
1.84    let ?R = "{(b, a). a \<noteq> bot \<and> a \<le> b \<and> b \<le> F}"
1.85
1.86 -  have bot_notin_R: "\<And>c. c \<in> Chains ?R \<Longrightarrow> bot \<notin> c"
1.87 +  have bot_notin_R: "c \<in> Chains ?R \<Longrightarrow> bot \<notin> c" for c
1.88      by (auto simp: Chains_def)
1.89
1.90    have [simp]: "Field ?R = ?X"
1.91      by (auto simp: Field_def bot_unique)
1.92
1.93 -  have "\<exists>m\<in>Field ?R. \<forall>a\<in>Field ?R. (m, a) \<in> ?R \<longrightarrow> a = m"
1.94 +  have "\<exists>m\<in>Field ?R. \<forall>a\<in>Field ?R. (m, a) \<in> ?R \<longrightarrow> a = m" (is "\<exists>m\<in>?A. ?B m")
1.95    proof (rule Zorns_po_lemma)
1.96      show "Partial_order ?R"
1.97 -      unfolding partial_order_on_def preorder_on_def
1.98 -      by (auto simp: antisym_def refl_on_def trans_def Field_def bot_unique)
1.99 +      by (auto simp: partial_order_on_def preorder_on_def
1.100 +          antisym_def refl_on_def trans_def Field_def bot_unique)
1.101      show "\<forall>C\<in>Chains ?R. \<exists>u\<in>Field ?R. \<forall>a\<in>C. (a, u) \<in> ?R"
1.102      proof (safe intro!: bexI del: notI)
1.103 -      fix c x assume c: "c \<in> Chains ?R"
1.104 +      fix c x
1.105 +      assume c: "c \<in> Chains ?R"
1.106
1.107 -      { assume "c \<noteq> {}"
1.108 -        with c have "Inf c = bot \<longleftrightarrow> (\<exists>x\<in>c. x = bot)"
1.109 +      have Inf_c: "Inf c \<noteq> bot" "Inf c \<le> F" if "c \<noteq> {}"
1.110 +      proof -
1.111 +        from c that have "Inf c = bot \<longleftrightarrow> (\<exists>x\<in>c. x = bot)"
1.112            unfolding trivial_limit_def by (intro eventually_Inf_base) (auto simp: Chains_def)
1.113 -        with c have 1: "Inf c \<noteq> bot"
1.114 +        with c show "Inf c \<noteq> bot"
1.116 -        from \<open>c \<noteq> {}\<close> obtain x where "x \<in> c" by auto
1.117 -        with c have 2: "Inf c \<le> F"
1.118 +        from that obtain x where "x \<in> c" by auto
1.119 +        with c show "Inf c \<le> F"
1.120            by (auto intro!: Inf_lower2[of x] simp: Chains_def)
1.121 -        note 1 2 }
1.122 -      note Inf_c = this
1.123 +      qed
1.124        then have [simp]: "inf F (Inf c) = (if c = {} then F else Inf c)"
1.125          using c by (auto simp add: inf_absorb2)
1.126
1.127 -      show "inf F (Inf c) \<noteq> bot"
1.128 -        using c by (simp add: F Inf_c)
1.129 +      from c show "inf F (Inf c) \<noteq> bot"
1.130 +        by (simp add: F Inf_c)
1.131 +      from c show "inf F (Inf c) \<in> Field ?R"
1.132 +        by (simp add: Chains_def Inf_c F)
1.133
1.134 -      show "inf F (Inf c) \<in> Field ?R"
1.135 -        using c by (simp add: Chains_def Inf_c F)
1.136 -
1.137 -      assume x: "x \<in> c"
1.138 +      assume "x \<in> c"
1.139        with c show "inf F (Inf c) \<le> x" "x \<le> F"
1.140          by (auto intro: Inf_lower simp: Chains_def)
1.141      qed
1.142    qed
1.143 -  then guess U ..
1.144 -  then show ?thesis
1.145 -    by (intro exI[of _ U] conjI max_filter_ultrafilter) auto
1.146 +  then obtain U where U: "U \<in> ?A" "?B U" ..
1.147 +  show ?thesis
1.148 +  proof
1.149 +    from U show "U \<le> F \<and> ultrafilter U"
1.150 +      by (auto intro!: max_filter_ultrafilter)
1.151 +  qed
1.152  qed
1.153
1.154 +
1.155  subsubsection \<open>Free Ultrafilters\<close>
1.156
1.157 -text \<open>There exists a free ultrafilter on any infinite set\<close>
1.158 +text \<open>There exists a free ultrafilter on any infinite set.\<close>
1.159
1.160  locale freeultrafilter = ultrafilter +
1.161    assumes infinite: "eventually P F \<Longrightarrow> infinite {x. P x}"
1.162  begin
1.163
1.164  lemma finite: "finite {x. P x} \<Longrightarrow> \<not> eventually P F"
1.165 -  by (erule contrapos_pn, erule infinite)
1.166 +  by (erule contrapos_pn) (erule infinite)
1.167
1.168  lemma finite': "finite {x. \<not> P x} \<Longrightarrow> eventually P F"
1.169    by (drule finite) (simp add: not_eventually frequently_eq_eventually)
1.170
1.171  lemma le_cofinite: "F \<le> cofinite"
1.172    by (intro filter_leI)
1.173 -     (auto simp add: eventually_cofinite not_eventually frequently_eq_eventually dest!: finite)
1.174 +    (auto simp add: eventually_cofinite not_eventually frequently_eq_eventually dest!: finite)
1.175
1.176  lemma singleton: "\<not> eventually (\<lambda>x. x = a) F"
1.177 -by (rule finite, simp)
1.178 +  by (rule finite) simp
1.179
1.180  lemma singleton': "\<not> eventually (op = a) F"
1.181 -by (rule finite, simp)
1.182 +  by (rule finite) simp
1.183
1.184  lemma ultrafilter: "ultrafilter F" ..
1.185
1.186 @@ -186,7 +193,8 @@
1.187    interpret ultrafilter U by fact
1.188    have "freeultrafilter U"
1.189    proof
1.190 -    fix P assume "eventually P U"
1.191 +    fix P
1.192 +    assume "eventually P U"
1.193      with proper have "frequently P U"
1.194        by (rule eventually_frequently)
1.195      then have "frequently P cofinite"
```
```     2.1 --- a/src/HOL/Nonstandard_Analysis/HDeriv.thy	Mon Oct 31 16:26:36 2016 +0100
2.2 +++ b/src/HOL/Nonstandard_Analysis/HDeriv.thy	Tue Nov 01 00:44:24 2016 +0100
2.3 @@ -4,284 +4,252 @@
2.4      Conversion to Isar and new proofs by Lawrence C Paulson, 2004
2.5  *)
2.6
2.7 -section\<open>Differentiation (Nonstandard)\<close>
2.8 +section \<open>Differentiation (Nonstandard)\<close>
2.9
2.10  theory HDeriv
2.11 -imports HLim
2.12 +  imports HLim
2.13  begin
2.14
2.15 -text\<open>Nonstandard Definitions\<close>
2.16 +text \<open>Nonstandard Definitions.\<close>
2.17
2.18 -definition
2.19 -  nsderiv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
2.20 -          ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60) where
2.21 -  "NSDERIV f x :> D = (\<forall>h \<in> Infinitesimal - {0}.
2.22 -      (( *f* f)(star_of x + h)
2.23 -       - star_of (f x))/h \<approx> star_of D)"
2.24 +definition nsderiv :: "['a::real_normed_field \<Rightarrow> 'a, 'a, 'a] \<Rightarrow> bool"
2.25 +    ("(NSDERIV (_)/ (_)/ :> (_))" [1000, 1000, 60] 60)
2.26 +  where "NSDERIV f x :> D \<longleftrightarrow>
2.27 +    (\<forall>h \<in> Infinitesimal - {0}. (( *f* f)(star_of x + h) - star_of (f x)) / h \<approx> star_of D)"
2.28
2.29 -definition
2.30 -  NSdifferentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
2.31 -    (infixl "NSdifferentiable" 60) where
2.32 -  "f NSdifferentiable x = (\<exists>D. NSDERIV f x :> D)"
2.33 +definition NSdifferentiable :: "['a::real_normed_field \<Rightarrow> 'a, 'a] \<Rightarrow> bool"
2.34 +    (infixl "NSdifferentiable" 60)
2.35 +  where "f NSdifferentiable x \<longleftrightarrow> (\<exists>D. NSDERIV f x :> D)"
2.36
2.37 -definition
2.38 -  increment :: "[real=>real,real,hypreal] => hypreal" where
2.39 -  "increment f x h = (@inc. f NSdifferentiable x &
2.40 -           inc = ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x))"
2.41 +definition increment :: "(real \<Rightarrow> real) \<Rightarrow> real \<Rightarrow> hypreal \<Rightarrow> hypreal"
2.42 +  where "increment f x h =
2.43 +    (SOME inc. f NSdifferentiable x \<and> inc = ( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x))"
2.44
2.45
2.46  subsection \<open>Derivatives\<close>
2.47
2.48 -lemma DERIV_NS_iff:
2.49 -      "(DERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D)"
2.50 -by (simp add: DERIV_def LIM_NSLIM_iff)
2.51 +lemma DERIV_NS_iff: "(DERIV f x :> D) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D"
2.52 +  by (simp add: DERIV_def LIM_NSLIM_iff)
2.53
2.54 -lemma NS_DERIV_D: "DERIV f x :> D ==> (%h. (f(x + h) - f(x))/h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D"
2.55 -by (simp add: DERIV_def LIM_NSLIM_iff)
2.56 +lemma NS_DERIV_D: "DERIV f x :> D \<Longrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D"
2.57 +  by (simp add: DERIV_def LIM_NSLIM_iff)
2.58
2.59 -lemma hnorm_of_hypreal:
2.60 -  "\<And>r. hnorm (( *f* of_real) r::'a::real_normed_div_algebra star) = \<bar>r\<bar>"
2.61 -by transfer (rule norm_of_real)
2.62 +lemma hnorm_of_hypreal: "\<And>r. hnorm (( *f* of_real) r::'a::real_normed_div_algebra star) = \<bar>r\<bar>"
2.63 +  by transfer (rule norm_of_real)
2.64
2.65  lemma Infinitesimal_of_hypreal:
2.66 -  "x \<in> Infinitesimal \<Longrightarrow>
2.67 -   (( *f* of_real) x::'a::real_normed_div_algebra star) \<in> Infinitesimal"
2.68 -apply (rule InfinitesimalI2)
2.69 -apply (drule (1) InfinitesimalD2)
2.71 -done
2.72 +  "x \<in> Infinitesimal \<Longrightarrow> (( *f* of_real) x::'a::real_normed_div_algebra star) \<in> Infinitesimal"
2.73 +  apply (rule InfinitesimalI2)
2.74 +  apply (drule (1) InfinitesimalD2)
2.75 +  apply (simp add: hnorm_of_hypreal)
2.76 +  done
2.77
2.78 -lemma of_hypreal_eq_0_iff:
2.79 -  "\<And>x. (( *f* of_real) x = (0::'a::real_algebra_1 star)) = (x = 0)"
2.80 -by transfer (rule of_real_eq_0_iff)
2.81 +lemma of_hypreal_eq_0_iff: "\<And>x. (( *f* of_real) x = (0::'a::real_algebra_1 star)) = (x = 0)"
2.82 +  by transfer (rule of_real_eq_0_iff)
2.83
2.84 -lemma NSDeriv_unique:
2.85 -     "[| NSDERIV f x :> D; NSDERIV f x :> E |] ==> D = E"
2.86 -apply (subgoal_tac "( *f* of_real) \<epsilon> \<in> Infinitesimal - {0::'a star}")
2.87 -apply (simp only: nsderiv_def)
2.88 -apply (drule (1) bspec)+
2.89 -apply (drule (1) approx_trans3)
2.90 -apply simp
2.91 -apply (simp add: Infinitesimal_of_hypreal Infinitesimal_epsilon)
2.92 -apply (simp add: of_hypreal_eq_0_iff hypreal_epsilon_not_zero)
2.93 -done
2.94 +lemma NSDeriv_unique: "NSDERIV f x :> D \<Longrightarrow> NSDERIV f x :> E \<Longrightarrow> D = E"
2.95 +  apply (subgoal_tac "( *f* of_real) \<epsilon> \<in> Infinitesimal - {0::'a star}")
2.96 +   apply (simp only: nsderiv_def)
2.97 +   apply (drule (1) bspec)+
2.98 +   apply (drule (1) approx_trans3)
2.99 +   apply simp
2.100 +  apply (simp add: Infinitesimal_of_hypreal Infinitesimal_epsilon)
2.101 +  apply (simp add: of_hypreal_eq_0_iff hypreal_epsilon_not_zero)
2.102 +  done
2.103
2.104 -text \<open>First NSDERIV in terms of NSLIM\<close>
2.105 +text \<open>First \<open>NSDERIV\<close> in terms of \<open>NSLIM\<close>.\<close>
2.106
2.107 -text\<open>first equivalence\<close>
2.108 -lemma NSDERIV_NSLIM_iff:
2.109 -      "(NSDERIV f x :> D) = ((%h. (f(x + h) - f(x))/h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D)"
2.110 -apply (simp add: nsderiv_def NSLIM_def, auto)
2.111 -apply (drule_tac x = xa in bspec)
2.112 -apply (rule_tac [3] ccontr)
2.113 -apply (drule_tac [3] x = h in spec)
2.114 -apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
2.115 -done
2.116 +text \<open>First equivalence.\<close>
2.117 +lemma NSDERIV_NSLIM_iff: "(NSDERIV f x :> D) \<longleftrightarrow> (\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D"
2.118 +  apply (auto simp add: nsderiv_def NSLIM_def)
2.119 +   apply (drule_tac x = xa in bspec)
2.120 +    apply (rule_tac [3] ccontr)
2.121 +    apply (drule_tac [3] x = h in spec)
2.122 +    apply (auto simp add: mem_infmal_iff starfun_lambda_cancel)
2.123 +  done
2.124
2.125 -text\<open>second equivalence\<close>
2.126 -lemma NSDERIV_NSLIM_iff2:
2.127 -     "(NSDERIV f x :> D) = ((%z. (f(z) - f(x)) / (z-x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S D)"
2.128 +text \<open>Second equivalence.\<close>
2.129 +lemma NSDERIV_NSLIM_iff2: "(NSDERIV f x :> D) \<longleftrightarrow> (\<lambda>z. (f z - f x) / (z - x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S D"
2.130    by (simp add: NSDERIV_NSLIM_iff DERIV_LIM_iff LIM_NSLIM_iff [symmetric])
2.131
2.132 -(* while we're at it! *)
2.133 -
2.134 +text \<open>While we're at it!\<close>
2.135  lemma NSDERIV_iff2:
2.136 -     "(NSDERIV f x :> D) =
2.137 -      (\<forall>w.
2.138 -        w \<noteq> star_of x & w \<approx> star_of x -->
2.139 -        ( *f* (%z. (f z - f x) / (z-x))) w \<approx> star_of D)"
2.140 -by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
2.141 +  "(NSDERIV f x :> D) \<longleftrightarrow>
2.142 +    (\<forall>w. w \<noteq> star_of x & w \<approx> star_of x \<longrightarrow> ( *f* (%z. (f z - f x) / (z-x))) w \<approx> star_of D)"
2.143 +  by (simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
2.144
2.145 -(*FIXME DELETE*)
2.146 -lemma hypreal_not_eq_minus_iff:
2.147 -  "(x \<noteq> a) = (x - a \<noteq> (0::'a::ab_group_add))"
2.148 -by auto
2.149 +(* FIXME delete *)
2.150 +lemma hypreal_not_eq_minus_iff: "x \<noteq> a \<longleftrightarrow> x - a \<noteq> (0::'a::ab_group_add)"
2.151 +  by auto
2.152
2.153  lemma NSDERIVD5:
2.154 -  "(NSDERIV f x :> D) ==>
2.155 -   (\<forall>u. u \<approx> hypreal_of_real x -->
2.156 -     ( *f* (%z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
2.157 -apply (auto simp add: NSDERIV_iff2)
2.158 -apply (case_tac "u = hypreal_of_real x", auto)
2.159 -apply (drule_tac x = u in spec, auto)
2.160 -apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
2.161 -apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
2.162 -apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
2.164 -         approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
2.165 -         Infinitesimal_subset_HFinite [THEN subsetD])
2.166 -done
2.167 +  "(NSDERIV f x :> D) \<Longrightarrow>
2.168 +   (\<forall>u. u \<approx> hypreal_of_real x \<longrightarrow>
2.169 +     ( *f* (\<lambda>z. f z - f x)) u \<approx> hypreal_of_real D * (u - hypreal_of_real x))"
2.170 +  apply (auto simp add: NSDERIV_iff2)
2.171 +  apply (case_tac "u = hypreal_of_real x", auto)
2.172 +  apply (drule_tac x = u in spec, auto)
2.173 +  apply (drule_tac c = "u - hypreal_of_real x" and b = "hypreal_of_real D" in approx_mult1)
2.174 +   apply (drule_tac [!] hypreal_not_eq_minus_iff [THEN iffD1])
2.175 +   apply (subgoal_tac [2] "( *f* (%z. z-x)) u \<noteq> (0::hypreal) ")
2.176 +    apply (auto simp: approx_minus_iff [THEN iffD1, THEN mem_infmal_iff [THEN iffD2]]
2.177 +      Infinitesimal_subset_HFinite [THEN subsetD])
2.178 +  done
2.179
2.180  lemma NSDERIVD4:
2.181 -     "(NSDERIV f x :> D) ==>
2.182 -      (\<forall>h \<in> Infinitesimal.
2.183 -               (( *f* f)(hypreal_of_real x + h) -
2.184 -                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
2.185 -apply (auto simp add: nsderiv_def)
2.186 -apply (case_tac "h = (0::hypreal) ")
2.187 -apply auto
2.188 -apply (drule_tac x = h in bspec)
2.189 -apply (drule_tac [2] c = h in approx_mult1)
2.190 -apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
2.191 -done
2.192 +  "(NSDERIV f x :> D) \<Longrightarrow>
2.193 +    (\<forall>h \<in> Infinitesimal.
2.194 +      ( *f* f)(hypreal_of_real x + h) - hypreal_of_real (f x) \<approx> hypreal_of_real D * h)"
2.195 +  apply (auto simp add: nsderiv_def)
2.196 +  apply (case_tac "h = 0")
2.197 +   apply auto
2.198 +  apply (drule_tac x = h in bspec)
2.199 +   apply (drule_tac [2] c = h in approx_mult1)
2.200 +    apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
2.201 +  done
2.202
2.203  lemma NSDERIVD3:
2.204 -     "(NSDERIV f x :> D) ==>
2.205 -      (\<forall>h \<in> Infinitesimal - {0}.
2.206 -               (( *f* f)(hypreal_of_real x + h) -
2.207 -                 hypreal_of_real (f x))\<approx> (hypreal_of_real D) * h)"
2.208 -apply (auto simp add: nsderiv_def)
2.209 -apply (rule ccontr, drule_tac x = h in bspec)
2.210 -apply (drule_tac [2] c = h in approx_mult1)
2.211 -apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: mult.assoc)
2.212 -done
2.213 +  "(NSDERIV f x :> D) \<Longrightarrow>
2.214 +    \<forall>h \<in> Infinitesimal - {0}.
2.215 +      (( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)) \<approx> hypreal_of_real D * h"
2.216 +  apply (auto simp add: nsderiv_def)
2.217 +  apply (rule ccontr, drule_tac x = h in bspec)
2.218 +   apply (drule_tac [2] c = h in approx_mult1)
2.219 +    apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: mult.assoc)
2.220 +  done
2.221
2.222 -text\<open>Differentiability implies continuity
2.223 -         nice and simple "algebraic" proof\<close>
2.224 -lemma NSDERIV_isNSCont: "NSDERIV f x :> D ==> isNSCont f x"
2.225 -apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
2.226 -apply (drule approx_minus_iff [THEN iffD1])
2.227 -apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
2.228 -apply (drule_tac x = "xa - star_of x" in bspec)
2.231 -apply (drule_tac c = "xa - star_of x" in approx_mult1)
2.232 -apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD]
2.233 -            simp add: mult.assoc nonzero_mult_div_cancel_right)
2.234 -apply (drule_tac x3=D in
2.235 -           HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult,
2.236 -             THEN mem_infmal_iff [THEN iffD1]])
2.237 -apply (auto simp add: mult.commute
2.238 -            intro: approx_trans approx_minus_iff [THEN iffD2])
2.239 -done
2.240 +text \<open>Differentiability implies continuity nice and simple "algebraic" proof.\<close>
2.241 +lemma NSDERIV_isNSCont: "NSDERIV f x :> D \<Longrightarrow> isNSCont f x"
2.242 +  apply (auto simp add: nsderiv_def isNSCont_NSLIM_iff NSLIM_def)
2.243 +  apply (drule approx_minus_iff [THEN iffD1])
2.244 +  apply (drule hypreal_not_eq_minus_iff [THEN iffD1])
2.245 +  apply (drule_tac x = "xa - star_of x" in bspec)
2.248 +  apply (drule_tac c = "xa - star_of x" in approx_mult1)
2.249 +   apply (auto intro: Infinitesimal_subset_HFinite [THEN subsetD] simp add: mult.assoc)
2.250 +  apply (drule_tac x3=D in
2.251 +      HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult, THEN mem_infmal_iff [THEN iffD1]])
2.252 +  apply (auto simp add: mult.commute intro: approx_trans approx_minus_iff [THEN iffD2])
2.253 +  done
2.254
2.255 -text\<open>Differentiation rules for combinations of functions
2.256 -      follow from clear, straightforard, algebraic
2.257 -      manipulations\<close>
2.258 -text\<open>Constant function\<close>
2.259 +text \<open>Differentiation rules for combinations of functions
2.260 +  follow from clear, straightforard, algebraic manipulations.\<close>
2.261 +
2.262 +text \<open>Constant function.\<close>
2.263
2.264  (* use simple constant nslimit theorem *)
2.265 -lemma NSDERIV_const [simp]: "(NSDERIV (%x. k) x :> 0)"
2.267 -
2.268 -text\<open>Sum of functions- proved easily\<close>
2.269 +lemma NSDERIV_const [simp]: "NSDERIV (\<lambda>x. k) x :> 0"
2.270 +  by (simp add: NSDERIV_NSLIM_iff)
2.271
2.272 -lemma NSDERIV_add: "[| NSDERIV f x :> Da;  NSDERIV g x :> Db |]
2.273 -      ==> NSDERIV (%x. f x + g x) x :> Da + Db"
2.274 -apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
2.276 -apply (drule_tac b = "star_of Da" and d = "star_of Db" in approx_add)
2.277 -apply (auto simp add: ac_simps algebra_simps)
2.278 -done
2.279 -
2.280 -text\<open>Product of functions - Proof is trivial but tedious
2.281 -  and long due to rearrangement of terms\<close>
2.282 +text \<open>Sum of functions- proved easily.\<close>
2.283
2.284 -lemma lemma_nsderiv1:
2.285 -  fixes a b c d :: "'a::comm_ring star"
2.286 -  shows "(a*b) - (c*d) = (b*(a - c)) + (c*(b - d))"
2.287 -by (simp add: right_diff_distrib ac_simps)
2.289 +  "NSDERIV f x :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow> NSDERIV (\<lambda>x. f x + g x) x :> Da + Db"
2.290 +  apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
2.292 +  apply (drule_tac b = "star_of Da" and d = "star_of Db" in approx_add)
2.293 +   apply (auto simp add: ac_simps algebra_simps)
2.294 +  done
2.295
2.296 -lemma lemma_nsderiv2:
2.297 -  fixes x y z :: "'a::real_normed_field star"
2.298 -  shows "[| (x - y) / z = star_of D + yb; z \<noteq> 0;
2.299 -         z \<in> Infinitesimal; yb \<in> Infinitesimal |]
2.300 -      ==> x - y \<approx> 0"
2.302 -apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
2.303 -            simp add: mult.assoc mem_infmal_iff [symmetric])
2.304 -apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
2.305 -done
2.306 +text \<open>Product of functions - Proof is trivial but tedious
2.307 +  and long due to rearrangement of terms.\<close>
2.308 +
2.309 +lemma lemma_nsderiv1: "(a * b) - (c * d) = (b * (a - c)) + (c * (b - d))"
2.310 +  for a b c d :: "'a::comm_ring star"
2.311 +  by (simp add: right_diff_distrib ac_simps)
2.312
2.313 -lemma NSDERIV_mult: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |]
2.314 -      ==> NSDERIV (%x. f x * g x) x :> (Da * g(x)) + (Db * f(x))"
2.315 -apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
2.316 -apply (auto dest!: spec
2.317 -      simp add: starfun_lambda_cancel lemma_nsderiv1)
2.319 -apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
2.320 -apply (auto simp add: times_divide_eq_right [symmetric]
2.321 -            simp del: times_divide_eq_right times_divide_eq_left)
2.322 -apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
2.323 -apply (drule_tac
2.324 -     approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
2.327 -apply (rule_tac b1 = "star_of Db * star_of (f x)"
2.328 -         in add.commute [THEN subst])
2.329 -apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
2.331 -                    Infinitesimal_star_of_mult
2.332 -                    Infinitesimal_star_of_mult2
2.334 -done
2.335 +lemma lemma_nsderiv2: "(x - y) / z = star_of D + yb \<Longrightarrow> z \<noteq> 0 \<Longrightarrow>
2.336 +  z \<in> Infinitesimal \<Longrightarrow> yb \<in> Infinitesimal \<Longrightarrow> x - y \<approx> 0"
2.337 +  for x y z :: "'a::real_normed_field star"
2.338 +  apply (simp add: nonzero_divide_eq_eq)
2.339 +  apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
2.340 +      simp add: mult.assoc mem_infmal_iff [symmetric])
2.341 +  apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
2.342 +  done
2.343
2.344 -text\<open>Multiplying by a constant\<close>
2.345 -lemma NSDERIV_cmult: "NSDERIV f x :> D
2.346 -      ==> NSDERIV (%x. c * f x) x :> c*D"
2.347 -apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
2.348 -                  minus_mult_right right_diff_distrib [symmetric])
2.349 -apply (erule NSLIM_const [THEN NSLIM_mult])
2.350 -done
2.351 +lemma NSDERIV_mult:
2.352 +  "NSDERIV f x :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow>
2.353 +    NSDERIV (\<lambda>x. f x * g x) x :> (Da * g x) + (Db * f x)"
2.354 +  apply (auto simp add: NSDERIV_NSLIM_iff NSLIM_def)
2.355 +  apply (auto dest!: spec simp add: starfun_lambda_cancel lemma_nsderiv1)
2.357 +  apply (drule bex_Infinitesimal_iff2 [THEN iffD2])+
2.358 +  apply (auto simp add: times_divide_eq_right [symmetric]
2.359 +      simp del: times_divide_eq_right times_divide_eq_left)
2.360 +  apply (drule_tac D = Db in lemma_nsderiv2, assumption+)
2.361 +  apply (drule_tac approx_minus_iff [THEN iffD2, THEN bex_Infinitesimal_iff2 [THEN iffD2]])
2.363 +  apply (rule_tac b1 = "star_of Db * star_of (f x)" in add.commute [THEN subst])
2.364 +  apply (auto intro!: Infinitesimal_add_approx_self2 [THEN approx_sym]
2.365 +      Infinitesimal_add Infinitesimal_mult Infinitesimal_star_of_mult Infinitesimal_star_of_mult2
2.367 +  done
2.368
2.369 -text\<open>Negation of function\<close>
2.370 -lemma NSDERIV_minus: "NSDERIV f x :> D ==> NSDERIV (%x. -(f x)) x :> -D"
2.371 +text \<open>Multiplying by a constant.\<close>
2.372 +lemma NSDERIV_cmult: "NSDERIV f x :> D \<Longrightarrow> NSDERIV (\<lambda>x. c * f x) x :> c * D"
2.373 +  apply (simp only: times_divide_eq_right [symmetric] NSDERIV_NSLIM_iff
2.374 +      minus_mult_right right_diff_distrib [symmetric])
2.375 +  apply (erule NSLIM_const [THEN NSLIM_mult])
2.376 +  done
2.377 +
2.378 +text \<open>Negation of function.\<close>
2.379 +lemma NSDERIV_minus: "NSDERIV f x :> D \<Longrightarrow> NSDERIV (\<lambda>x. - f x) x :> - D"
2.381    assume "(\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D"
2.382 -  hence deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S - D"
2.383 +  then have deriv: "(\<lambda>h. - ((f(x+h) - f x) / h)) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S - D"
2.384      by (rule NSLIM_minus)
2.385    have "\<forall>h. - ((f (x + h) - f x) / h) = (- f (x + h) + f x) / h"
2.387 -  with deriv
2.388 -  have "(\<lambda>h. (- f (x + h) + f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S - D" by simp
2.389 -  then show "(\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D \<Longrightarrow>
2.390 -    (\<lambda>h. (f x - f (x + h)) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S - D" by simp
2.391 +  with deriv have "(\<lambda>h. (- f (x + h) + f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S - D"
2.392 +    by simp
2.393 +  then show "(\<lambda>h. (f (x + h) - f x) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S D \<Longrightarrow> (\<lambda>h. (f x - f (x + h)) / h) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S - D"
2.394 +    by simp
2.395  qed
2.396
2.397 -text\<open>Subtraction\<close>
2.398 -lemma NSDERIV_add_minus: "[| NSDERIV f x :> Da; NSDERIV g x :> Db |] ==> NSDERIV (%x. f x + -g x) x :> Da + -Db"
2.399 -by (blast dest: NSDERIV_add NSDERIV_minus)
2.400 +text \<open>Subtraction.\<close>
2.402 +  "NSDERIV f x :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow> NSDERIV (\<lambda>x. f x + - g x) x :> Da + - Db"
2.403 +  by (blast dest: NSDERIV_add NSDERIV_minus)
2.404
2.405  lemma NSDERIV_diff:
2.406 -  "NSDERIV f x :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow> NSDERIV (\<lambda>x. f x - g x) x :> Da-Db"
2.407 +  "NSDERIV f x :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow> NSDERIV (\<lambda>x. f x - g x) x :> Da - Db"
2.408    using NSDERIV_add_minus [of f x Da g Db] by simp
2.409
2.410 -text\<open>Similarly to the above, the chain rule admits an entirely
2.411 -   straightforward derivation. Compare this with Harrison's
2.412 -   HOL proof of the chain rule, which proved to be trickier and
2.413 -   required an alternative characterisation of differentiability-
2.414 -   the so-called Carathedory derivative. Our main problem is
2.415 -   manipulation of terms.\<close>
2.416 +text \<open>Similarly to the above, the chain rule admits an entirely
2.417 +  straightforward derivation. Compare this with Harrison's
2.418 +  HOL proof of the chain rule, which proved to be trickier and
2.419 +  required an alternative characterisation of differentiability-
2.420 +  the so-called Carathedory derivative. Our main problem is
2.421 +  manipulation of terms.\<close>
2.422
2.423 -(* lemmas *)
2.424 +
2.425 +subsection \<open>Lemmas\<close>
2.426
2.427  lemma NSDERIV_zero:
2.428 -      "[| NSDERIV g x :> D;
2.429 -               ( *f* g) (star_of x + xa) = star_of (g x);
2.430 -               xa \<in> Infinitesimal;
2.431 -               xa \<noteq> 0
2.432 -            |] ==> D = 0"
2.434 -apply (drule bspec, auto)
2.435 -done
2.436 +  "NSDERIV g x :> D \<Longrightarrow> ( *f* g) (star_of x + xa) = star_of (g x) \<Longrightarrow>
2.437 +    xa \<in> Infinitesimal \<Longrightarrow> xa \<noteq> 0 \<Longrightarrow> D = 0"
2.438 +  apply (simp add: nsderiv_def)
2.439 +  apply (drule bspec)
2.440 +   apply auto
2.441 +  done
2.442
2.443 -(* can be proved differently using NSLIM_isCont_iff *)
2.444 +text \<open>Can be proved differently using \<open>NSLIM_isCont_iff\<close>.\<close>
2.445  lemma NSDERIV_approx:
2.446 -     "[| NSDERIV f x :> D;  h \<in> Infinitesimal;  h \<noteq> 0 |]
2.447 -      ==> ( *f* f) (star_of x + h) - star_of (f x) \<approx> 0"
2.449 -apply (simp add: mem_infmal_iff [symmetric])
2.450 -apply (rule Infinitesimal_ratio)
2.451 -apply (rule_tac [3] approx_star_of_HFinite, auto)
2.452 -done
2.453 +  "NSDERIV f x :> D \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> h \<noteq> 0 \<Longrightarrow>
2.454 +    ( *f* f) (star_of x + h) - star_of (f x) \<approx> 0"
2.455 +  apply (simp add: nsderiv_def)
2.456 +  apply (simp add: mem_infmal_iff [symmetric])
2.457 +  apply (rule Infinitesimal_ratio)
2.458 +    apply (rule_tac [3] approx_star_of_HFinite, auto)
2.459 +  done
2.460
2.461 -(*---------------------------------------------------------------
2.462 -   from one version of differentiability
2.463 +text \<open>From one version of differentiability
2.464
2.465 -                f(x) - f(a)
2.466 -              --------------- \<approx> Db
2.467 -                  x - a
2.468 - ---------------------------------------------------------------*)
2.469 +        \<open>f x - f a\<close>
2.470 +      \<open>-------------- \<approx> Db\<close>
2.471 +          \<open>x - a\<close>
2.472 +\<close>
2.473
2.474  lemma NSDERIVD1: "[| NSDERIV f (g x) :> Da;
2.475           ( *f* g) (star_of(x) + xa) \<noteq> star_of (g x);
2.476 @@ -290,186 +258,185 @@
2.477                     - star_of (f (g x)))
2.478                / (( *f* g) (star_of(x) + xa) - star_of (g x))
2.479               \<approx> star_of(Da)"
2.480 -by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
2.481 +  by (auto simp add: NSDERIV_NSLIM_iff2 NSLIM_def)
2.482
2.483 -(*--------------------------------------------------------------
2.484 -   from other version of differentiability
2.485 +text \<open>From other version of differentiability
2.486
2.487 -                f(x + h) - f(x)
2.488 -               ----------------- \<approx> Db
2.489 -                       h
2.490 - --------------------------------------------------------------*)
2.491 +      \<open>f (x + h) - f x\<close>
2.492 +     \<open>------------------ \<approx> Db\<close>
2.493 +             \<open>h\<close>
2.494 +\<close>
2.495
2.496  lemma NSDERIVD2: "[| NSDERIV g x :> Db; xa \<in> Infinitesimal; xa \<noteq> 0 |]
2.497        ==> (( *f* g) (star_of(x) + xa) - star_of(g x)) / xa
2.498            \<approx> star_of(Db)"
2.499 -by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
2.500 +  by (auto simp add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff starfun_lambda_cancel)
2.501
2.502 -lemma lemma_chain: "(z::'a::real_normed_field star) \<noteq> 0 ==> x*y = (x*inverse(z))*(z*y)"
2.503 +lemma lemma_chain: "z \<noteq> 0 \<Longrightarrow> x * y = (x * inverse z) * (z * y)"
2.504 +  for x y z :: "'a::real_normed_field star"
2.505  proof -
2.506    assume z: "z \<noteq> 0"
2.507    have "x * y = x * (inverse z * z) * y" by (simp add: z)
2.508 -  thus ?thesis by (simp add: mult.assoc)
2.509 +  then show ?thesis by (simp add: mult.assoc)
2.510  qed
2.511
2.512 -text\<open>This proof uses both definitions of differentiability.\<close>
2.513 -lemma NSDERIV_chain: "[| NSDERIV f (g x) :> Da; NSDERIV g x :> Db |]
2.514 -      ==> NSDERIV (f o g) x :> Da * Db"
2.515 -apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def
2.516 -                mem_infmal_iff [symmetric])
2.517 -apply clarify
2.518 -apply (frule_tac f = g in NSDERIV_approx)
2.519 -apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
2.520 -apply (case_tac "( *f* g) (star_of (x) + xa) = star_of (g x) ")
2.521 -apply (drule_tac g = g in NSDERIV_zero)
2.522 -apply (auto simp add: divide_inverse)
2.523 -apply (rule_tac z1 = "( *f* g) (star_of (x) + xa) - star_of (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
2.524 -apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
2.525 -apply (rule approx_mult_star_of)
2.526 -apply (simp_all add: divide_inverse [symmetric])
2.527 -apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
2.528 -apply (blast intro: NSDERIVD2)
2.529 -done
2.530 +text \<open>This proof uses both definitions of differentiability.\<close>
2.531 +lemma NSDERIV_chain:
2.532 +  "NSDERIV f (g x) :> Da \<Longrightarrow> NSDERIV g x :> Db \<Longrightarrow> NSDERIV (f \<circ> g) x :> Da * Db"
2.533 +  apply (simp (no_asm_simp) add: NSDERIV_NSLIM_iff NSLIM_def mem_infmal_iff [symmetric])
2.534 +  apply clarify
2.535 +  apply (frule_tac f = g in NSDERIV_approx)
2.536 +    apply (auto simp add: starfun_lambda_cancel2 starfun_o [symmetric])
2.537 +  apply (case_tac "( *f* g) (star_of (x) + xa) = star_of (g x) ")
2.538 +   apply (drule_tac g = g in NSDERIV_zero)
2.539 +      apply (auto simp add: divide_inverse)
2.540 +  apply (rule_tac z1 = "( *f* g) (star_of (x) + xa) - star_of (g x) " and y1 = "inverse xa" in lemma_chain [THEN ssubst])
2.541 +   apply (erule hypreal_not_eq_minus_iff [THEN iffD1])
2.542 +  apply (rule approx_mult_star_of)
2.543 +   apply (simp_all add: divide_inverse [symmetric])
2.544 +   apply (blast intro: NSDERIVD1 approx_minus_iff [THEN iffD2])
2.545 +  apply (blast intro: NSDERIVD2)
2.546 +  done
2.547
2.548 -text\<open>Differentiation of natural number powers\<close>
2.549 -lemma NSDERIV_Id [simp]: "NSDERIV (%x. x) x :> 1"
2.550 -by (simp add: NSDERIV_NSLIM_iff NSLIM_def del: divide_self_if)
2.551 +text \<open>Differentiation of natural number powers.\<close>
2.552 +lemma NSDERIV_Id [simp]: "NSDERIV (\<lambda>x. x) x :> 1"
2.553 +  by (simp add: NSDERIV_NSLIM_iff NSLIM_def del: divide_self_if)
2.554
2.555  lemma NSDERIV_cmult_Id [simp]: "NSDERIV (op * c) x :> c"
2.556 -by (cut_tac c = c and x = x in NSDERIV_Id [THEN NSDERIV_cmult], simp)
2.557 +  using NSDERIV_Id [THEN NSDERIV_cmult] by simp
2.558
2.559  lemma NSDERIV_inverse:
2.560    fixes x :: "'a::real_normed_field"
2.561    assumes "x \<noteq> 0" \<comment> \<open>can't get rid of @{term "x \<noteq> 0"} because it isn't continuous at zero\<close>
2.562    shows "NSDERIV (\<lambda>x. inverse x) x :> - (inverse x ^ Suc (Suc 0))"
2.563  proof -
2.564 -  { fix h :: "'a star"
2.565 +  {
2.566 +    fix h :: "'a star"
2.567      assume h_Inf: "h \<in> Infinitesimal"
2.568 -    from this assms have not_0: "star_of x + h \<noteq> 0" by (rule Infinitesimal_add_not_zero)
2.569 +    from this assms have not_0: "star_of x + h \<noteq> 0"
2.571      assume "h \<noteq> 0"
2.572 -    from h_Inf have "h * star_of x \<in> Infinitesimal" by (rule Infinitesimal_HFinite_mult) simp
2.573 +    from h_Inf have "h * star_of x \<in> Infinitesimal"
2.574 +      by (rule Infinitesimal_HFinite_mult) simp
2.575      with assms have "inverse (- (h * star_of x) + - (star_of x * star_of x)) \<approx>
2.576        inverse (- (star_of x * star_of x))"
2.577 -      apply - apply (rule inverse_add_Infinitesimal_approx2)
2.578 -      apply (auto
2.579 -        dest!: hypreal_of_real_HFinite_diff_Infinitesimal
2.580 +      apply -
2.582 +      apply (auto dest!: hypreal_of_real_HFinite_diff_Infinitesimal
2.583          simp add: inverse_minus_eq [symmetric] HFinite_minus_iff)
2.584        done
2.585      moreover from not_0 \<open>h \<noteq> 0\<close> assms
2.586 -      have "inverse (- (h * star_of x) + - (star_of x * star_of x)) =
2.587 -        (inverse (star_of x + h) - inverse (star_of x)) / h"
2.588 +    have "inverse (- (h * star_of x) + - (star_of x * star_of x)) =
2.589 +      (inverse (star_of x + h) - inverse (star_of x)) / h"
2.590        apply (simp add: division_ring_inverse_diff nonzero_inverse_mult_distrib [symmetric]
2.591 -        nonzero_inverse_minus_eq [symmetric] ac_simps ring_distribs)
2.592 +          nonzero_inverse_minus_eq [symmetric] ac_simps ring_distribs)
2.593        apply (subst nonzero_inverse_minus_eq [symmetric])
2.594        using distrib_right [symmetric, of h "star_of x" "star_of x"] apply simp
2.596        done
2.597      ultimately have "(inverse (star_of x + h) - inverse (star_of x)) / h \<approx>
2.598        - (inverse (star_of x) * inverse (star_of x))"
2.599 -      using assms by simp
2.600 -  } then show ?thesis by (simp add: nsderiv_def)
2.601 +      using assms by simp
2.602 +  }
2.603 +  then show ?thesis by (simp add: nsderiv_def)
2.604  qed
2.605
2.606 +
2.607  subsubsection \<open>Equivalence of NS and Standard definitions\<close>
2.608
2.609  lemma divideR_eq_divide: "x /\<^sub>R y = x / y"
2.610 -by (simp add: divide_inverse mult.commute)
2.611 +  by (simp add: divide_inverse mult.commute)
2.612
2.613 -text\<open>Now equivalence between NSDERIV and DERIV\<close>
2.614 -lemma NSDERIV_DERIV_iff: "(NSDERIV f x :> D) = (DERIV f x :> D)"
2.615 -by (simp add: DERIV_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
2.616 +text \<open>Now equivalence between \<open>NSDERIV\<close> and \<open>DERIV\<close>.\<close>
2.617 +lemma NSDERIV_DERIV_iff: "NSDERIV f x :> D \<longleftrightarrow> DERIV f x :> D"
2.618 +  by (simp add: DERIV_def NSDERIV_NSLIM_iff LIM_NSLIM_iff)
2.619
2.620 -(* NS version *)
2.621 -lemma NSDERIV_pow: "NSDERIV (%x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
2.622 -by (simp add: NSDERIV_DERIV_iff DERIV_pow)
2.623 +text \<open>NS version.\<close>
2.624 +lemma NSDERIV_pow: "NSDERIV (\<lambda>x. x ^ n) x :> real n * (x ^ (n - Suc 0))"
2.625 +  by (simp add: NSDERIV_DERIV_iff DERIV_pow)
2.626
2.627 -text\<open>Derivative of inverse\<close>
2.628 -
2.629 +text \<open>Derivative of inverse.\<close>
2.630  lemma NSDERIV_inverse_fun:
2.631 -  fixes x :: "'a::{real_normed_field}"
2.632 -  shows "[| NSDERIV f x :> d; f(x) \<noteq> 0 |]
2.633 -      ==> NSDERIV (%x. inverse(f x)) x :> (- (d * inverse(f(x) ^ Suc (Suc 0))))"
2.634 -by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: power_Suc)
2.635 +  "NSDERIV f x :> d \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow>
2.636 +    NSDERIV (\<lambda>x. inverse (f x)) x :> (- (d * inverse (f x ^ Suc (Suc 0))))"
2.637 +  for x :: "'a::{real_normed_field}"
2.638 +  by (simp add: NSDERIV_DERIV_iff DERIV_inverse_fun del: power_Suc)
2.639
2.640 -text\<open>Derivative of quotient\<close>
2.641 -
2.642 +text \<open>Derivative of quotient.\<close>
2.643  lemma NSDERIV_quotient:
2.644 -  fixes x :: "'a::{real_normed_field}"
2.645 -  shows "[| NSDERIV f x :> d; NSDERIV g x :> e; g(x) \<noteq> 0 |]
2.646 -       ==> NSDERIV (%y. f(y) / (g y)) x :> (d*g(x)
2.647 -                            - (e*f(x))) / (g(x) ^ Suc (Suc 0))"
2.648 -by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: power_Suc)
2.649 +  fixes x :: "'a::real_normed_field"
2.650 +  shows "NSDERIV f x :> d \<Longrightarrow> NSDERIV g x :> e \<Longrightarrow> g x \<noteq> 0 \<Longrightarrow>
2.651 +    NSDERIV (\<lambda>y. f y / g y) x :> (d * g x - (e * f x)) / (g x ^ Suc (Suc 0))"
2.652 +  by (simp add: NSDERIV_DERIV_iff DERIV_quotient del: power_Suc)
2.653
2.654 -lemma CARAT_NSDERIV: "NSDERIV f x :> l ==>
2.655 -      \<exists>g. (\<forall>z. f z - f x = g z * (z-x)) & isNSCont g x & g x = l"
2.656 -by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV
2.657 -                   mult.commute)
2.658 +lemma CARAT_NSDERIV:
2.659 +  "NSDERIV f x :> l \<Longrightarrow> \<exists>g. (\<forall>z. f z - f x = g z * (z - x)) \<and> isNSCont g x \<and> g x = l"
2.660 +  by (auto simp add: NSDERIV_DERIV_iff isNSCont_isCont_iff CARAT_DERIV mult.commute)
2.661
2.662 -lemma hypreal_eq_minus_iff3: "(x = y + z) = (x + -z = (y::hypreal))"
2.663 -by auto
2.664 +lemma hypreal_eq_minus_iff3: "x = y + z \<longleftrightarrow> x + - z = y"
2.665 +  for x y z :: hypreal
2.666 +  by auto
2.667
2.668  lemma CARAT_DERIVD:
2.669 -  assumes all: "\<forall>z. f z - f x = g z * (z-x)"
2.670 -      and nsc: "isNSCont g x"
2.671 +  assumes all: "\<forall>z. f z - f x = g z * (z - x)"
2.672 +    and nsc: "isNSCont g x"
2.673    shows "NSDERIV f x :> g x"
2.674  proof -
2.675 -  from nsc
2.676 -  have "\<forall>w. w \<noteq> star_of x \<and> w \<approx> star_of x \<longrightarrow>
2.677 -         ( *f* g) w * (w - star_of x) / (w - star_of x) \<approx>
2.678 -         star_of (g x)"
2.679 -    by (simp add: isNSCont_def nonzero_mult_div_cancel_right)
2.680 -  thus ?thesis using all
2.681 +  from nsc have "\<forall>w. w \<noteq> star_of x \<and> w \<approx> star_of x \<longrightarrow>
2.682 +       ( *f* g) w * (w - star_of x) / (w - star_of x) \<approx> star_of (g x)"
2.683 +    by (simp add: isNSCont_def)
2.684 +  with all show ?thesis
2.685      by (simp add: NSDERIV_iff2 starfun_if_eq cong: if_cong)
2.686  qed
2.687
2.688 +
2.689  subsubsection \<open>Differentiability predicate\<close>
2.690
2.691 -lemma NSdifferentiableD: "f NSdifferentiable x ==> \<exists>D. NSDERIV f x :> D"
2.693 +lemma NSdifferentiableD: "f NSdifferentiable x \<Longrightarrow> \<exists>D. NSDERIV f x :> D"
2.694 +  by (simp add: NSdifferentiable_def)
2.695
2.696 -lemma NSdifferentiableI: "NSDERIV f x :> D ==> f NSdifferentiable x"
2.697 -by (force simp add: NSdifferentiable_def)
2.698 +lemma NSdifferentiableI: "NSDERIV f x :> D \<Longrightarrow> f NSdifferentiable x"
2.699 +  by (force simp add: NSdifferentiable_def)
2.700
2.701
2.702  subsection \<open>(NS) Increment\<close>
2.703 -lemma incrementI:
2.704 -      "f NSdifferentiable x ==>
2.705 -      increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
2.706 -      hypreal_of_real (f x)"
2.708
2.709 -lemma incrementI2: "NSDERIV f x :> D ==>
2.710 -     increment f x h = ( *f* f) (hypreal_of_real(x) + h) -
2.711 -     hypreal_of_real (f x)"
2.712 -apply (erule NSdifferentiableI [THEN incrementI])
2.713 -done
2.714 +lemma incrementI:
2.715 +  "f NSdifferentiable x \<Longrightarrow>
2.716 +    increment f x h = ( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)"
2.717 +  by (simp add: increment_def)
2.718 +
2.719 +lemma incrementI2:
2.720 +  "NSDERIV f x :> D \<Longrightarrow>
2.721 +    increment f x h = ( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)"
2.722 +  by (erule NSdifferentiableI [THEN incrementI])
2.723
2.724 -(* The Increment theorem -- Keisler p. 65 *)
2.725 -lemma increment_thm: "[| NSDERIV f x :> D; h \<in> Infinitesimal; h \<noteq> 0 |]
2.726 -      ==> \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real(D)*h + e*h"
2.727 -apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
2.728 -apply (drule bspec, auto)
2.729 -apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
2.730 -apply (frule_tac b1 = "hypreal_of_real (D) + y"
2.731 -        in mult_right_cancel [THEN iffD2])
2.732 -apply (erule_tac [2] V = "(( *f* f) (hypreal_of_real (x) + h) - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y" in thin_rl)
2.733 -apply assumption
2.734 -apply (simp add: times_divide_eq_right [symmetric])
2.735 -apply (auto simp add: distrib_right)
2.736 -done
2.737 +text \<open>The Increment theorem -- Keisler p. 65.\<close>
2.738 +lemma increment_thm:
2.739 +  "NSDERIV f x :> D \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> h \<noteq> 0 \<Longrightarrow>
2.740 +    \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real D * h + e * h"
2.741 +  apply (frule_tac h = h in incrementI2, simp add: nsderiv_def)
2.742 +  apply (drule bspec, auto)
2.743 +  apply (drule bex_Infinitesimal_iff2 [THEN iffD2], clarify)
2.744 +  apply (frule_tac b1 = "hypreal_of_real D + y" in mult_right_cancel [THEN iffD2])
2.745 +   apply (erule_tac [2]
2.746 +      V = "(( *f* f) (hypreal_of_real x + h) - hypreal_of_real (f x)) / h = hypreal_of_real (D) + y"
2.747 +      in thin_rl)
2.748 +   apply assumption
2.749 +  apply (simp add: times_divide_eq_right [symmetric])
2.750 +  apply (auto simp add: distrib_right)
2.751 +  done
2.752
2.753  lemma increment_thm2:
2.754 -     "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
2.755 -      ==> \<exists>e \<in> Infinitesimal. increment f x h =
2.756 -              hypreal_of_real(D)*h + e*h"
2.757 -by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
2.758 -
2.759 +  "NSDERIV f x :> D \<Longrightarrow> h \<approx> 0 \<Longrightarrow> h \<noteq> 0 \<Longrightarrow>
2.760 +    \<exists>e \<in> Infinitesimal. increment f x h = hypreal_of_real D * h + e * h"
2.761 +  by (blast dest!: mem_infmal_iff [THEN iffD2] intro!: increment_thm)
2.762
2.763 -lemma increment_approx_zero: "[| NSDERIV f x :> D; h \<approx> 0; h \<noteq> 0 |]
2.764 -      ==> increment f x h \<approx> 0"
2.765 -apply (drule increment_thm2,
2.766 -       auto intro!: Infinitesimal_HFinite_mult2 HFinite_add simp add: distrib_right [symmetric] mem_infmal_iff [symmetric])
2.767 -apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
2.768 -done
2.769 +lemma increment_approx_zero: "NSDERIV f x :> D \<Longrightarrow> h \<approx> 0 \<Longrightarrow> h \<noteq> 0 \<Longrightarrow> increment f x h \<approx> 0"
2.770 +  apply (drule increment_thm2)
2.771 +    apply (auto intro!: Infinitesimal_HFinite_mult2 HFinite_add
2.772 +      simp add: distrib_right [symmetric] mem_infmal_iff [symmetric])
2.773 +  apply (erule Infinitesimal_subset_HFinite [THEN subsetD])
2.774 +  done
2.775
2.776  end
```
```     3.1 --- a/src/HOL/Nonstandard_Analysis/HLim.thy	Mon Oct 31 16:26:36 2016 +0100
3.2 +++ b/src/HOL/Nonstandard_Analysis/HLim.thy	Tue Nov 01 00:44:24 2016 +0100
3.3 @@ -3,330 +3,308 @@
3.4      Author:     Lawrence C Paulson
3.5  *)
3.6
3.7 -section\<open>Limits and Continuity (Nonstandard)\<close>
3.8 +section \<open>Limits and Continuity (Nonstandard)\<close>
3.9
3.10  theory HLim
3.11    imports Star
3.12    abbrevs "--->" = "\<midarrow>\<rightarrow>\<^sub>N\<^sub>S"
3.13  begin
3.14
3.15 -text\<open>Nonstandard Definitions\<close>
3.16 +text \<open>Nonstandard Definitions.\<close>
3.17
3.18 -definition
3.19 -  NSLIM :: "['a::real_normed_vector => 'b::real_normed_vector, 'a, 'b] => bool"
3.20 -            ("((_)/ \<midarrow>(_)/\<rightarrow>\<^sub>N\<^sub>S (_))" [60, 0, 60] 60) where
3.21 -  "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L =
3.22 -    (\<forall>x. (x \<noteq> star_of a & x \<approx> star_of a --> ( *f* f) x \<approx> star_of L))"
3.23 +definition NSLIM :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> bool"
3.24 +    ("((_)/ \<midarrow>(_)/\<rightarrow>\<^sub>N\<^sub>S (_))" [60, 0, 60] 60)
3.25 +  where "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<longleftrightarrow> (\<forall>x. x \<noteq> star_of a \<and> x \<approx> star_of a \<longrightarrow> ( *f* f) x \<approx> star_of L)"
3.26
3.27 -definition
3.28 -  isNSCont :: "['a::real_normed_vector => 'b::real_normed_vector, 'a] => bool" where
3.29 -    \<comment>\<open>NS definition dispenses with limit notions\<close>
3.30 -  "isNSCont f a = (\<forall>y. y \<approx> star_of a -->
3.31 -         ( *f* f) y \<approx> star_of (f a))"
3.32 +definition isNSCont :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> bool"
3.33 +  where  \<comment> \<open>NS definition dispenses with limit notions\<close>
3.34 +    "isNSCont f a \<longleftrightarrow> (\<forall>y. y \<approx> star_of a \<longrightarrow> ( *f* f) y \<approx> star_of (f a))"
3.35
3.36 -definition
3.37 -  isNSUCont :: "['a::real_normed_vector => 'b::real_normed_vector] => bool" where
3.38 -  "isNSUCont f = (\<forall>x y. x \<approx> y --> ( *f* f) x \<approx> ( *f* f) y)"
3.39 +definition isNSUCont :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> bool"
3.40 +  where "isNSUCont f \<longleftrightarrow> (\<forall>x y. x \<approx> y \<longrightarrow> ( *f* f) x \<approx> ( *f* f) y)"
3.41
3.42
3.43  subsection \<open>Limits of Functions\<close>
3.44
3.45 -lemma NSLIM_I:
3.46 -  "(\<And>x. \<lbrakk>x \<noteq> star_of a; x \<approx> star_of a\<rbrakk> \<Longrightarrow> starfun f x \<approx> star_of L)
3.47 -   \<Longrightarrow> f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.49 +lemma NSLIM_I: "(\<And>x. x \<noteq> star_of a \<Longrightarrow> x \<approx> star_of a \<Longrightarrow> starfun f x \<approx> star_of L) \<Longrightarrow> f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.50 +  by (simp add: NSLIM_def)
3.51 +
3.52 +lemma NSLIM_D: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<Longrightarrow> x \<noteq> star_of a \<Longrightarrow> x \<approx> star_of a \<Longrightarrow> starfun f x \<approx> star_of L"
3.53 +  by (simp add: NSLIM_def)
3.54
3.55 -lemma NSLIM_D:
3.56 -  "\<lbrakk>f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L; x \<noteq> star_of a; x \<approx> star_of a\<rbrakk>
3.57 -   \<Longrightarrow> starfun f x \<approx> star_of L"
3.59 +text \<open>Proving properties of limits using nonstandard definition.
3.60 +  The properties hold for standard limits as well!\<close>
3.61
3.62 -text\<open>Proving properties of limits using nonstandard definition.
3.63 -      The properties hold for standard limits as well!\<close>
3.64 -
3.65 -lemma NSLIM_mult:
3.66 -  fixes l m :: "'a::real_normed_algebra"
3.67 -  shows "[| f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l; g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m |]
3.68 -      ==> (%x. f(x) * g(x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l * m)"
3.69 -by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
3.70 +lemma NSLIM_mult: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m \<Longrightarrow> (\<lambda>x. f x * g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l * m)"
3.71 +  for l m :: "'a::real_normed_algebra"
3.72 +  by (auto simp add: NSLIM_def intro!: approx_mult_HFinite)
3.73
3.74 -lemma starfun_scaleR [simp]:
3.75 -  "starfun (\<lambda>x. f x *\<^sub>R g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
3.76 -by transfer (rule refl)
3.77 +lemma starfun_scaleR [simp]: "starfun (\<lambda>x. f x *\<^sub>R g x) = (\<lambda>x. scaleHR (starfun f x) (starfun g x))"
3.78 +  by transfer (rule refl)
3.79
3.80 -lemma NSLIM_scaleR:
3.81 -  "[| f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l; g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m |]
3.82 -      ==> (%x. f(x) *\<^sub>R g(x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l *\<^sub>R m)"
3.83 -by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
3.84 +lemma NSLIM_scaleR: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m \<Longrightarrow> (\<lambda>x. f x *\<^sub>R g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l *\<^sub>R m)"
3.85 +  by (auto simp add: NSLIM_def intro!: approx_scaleR_HFinite)
3.86
3.88 -     "[| f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l; g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m |]
3.89 -      ==> (%x. f(x) + g(x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l + m)"
3.91 +lemma NSLIM_add: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m \<Longrightarrow> (\<lambda>x. f x + g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l + m)"
3.93
3.94 -lemma NSLIM_const [simp]: "(%x. k) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S k"
3.96 +lemma NSLIM_const [simp]: "(\<lambda>x. k) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S k"
3.97 +  by (simp add: NSLIM_def)
3.98
3.99 -lemma NSLIM_minus: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L ==> (%x. -f(x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S -L"
3.101 +lemma NSLIM_minus: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<Longrightarrow> (\<lambda>x. - f x) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S -L"
3.102 +  by (simp add: NSLIM_def)
3.103
3.104 -lemma NSLIM_diff:
3.105 -  "\<lbrakk>f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l; g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m\<rbrakk> \<Longrightarrow> (\<lambda>x. f x - g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l - m)"
3.106 +lemma NSLIM_diff: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m \<Longrightarrow> (\<lambda>x. f x - g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l - m)"
3.108
3.109 -lemma NSLIM_add_minus: "[| f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l; g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m |] ==> (%x. f(x) + -g(x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l + -m)"
3.110 -by (simp only: NSLIM_add NSLIM_minus)
3.111 +lemma NSLIM_add_minus: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S m \<Longrightarrow> (\<lambda>x. f x + - g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S (l + -m)"
3.112 +  by (simp only: NSLIM_add NSLIM_minus)
3.113
3.114 -lemma NSLIM_inverse:
3.115 -  fixes L :: "'a::real_normed_div_algebra"
3.116 -  shows "[| f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L;  L \<noteq> 0 |]
3.117 -      ==> (%x. inverse(f(x))) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (inverse L)"
3.118 -apply (simp add: NSLIM_def, clarify)
3.119 -apply (drule spec)
3.120 -apply (auto simp add: star_of_approx_inverse)
3.121 -done
3.122 +lemma NSLIM_inverse: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<Longrightarrow> L \<noteq> 0 \<Longrightarrow> (\<lambda>x. inverse (f x)) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (inverse L)"
3.123 +  for L :: "'a::real_normed_div_algebra"
3.124 +  apply (simp add: NSLIM_def, clarify)
3.125 +  apply (drule spec)
3.126 +  apply (auto simp add: star_of_approx_inverse)
3.127 +  done
3.128
3.129  lemma NSLIM_zero:
3.130 -  assumes f: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S l" shows "(%x. f(x) - l) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S 0"
3.131 +  assumes f: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S l"
3.132 +  shows "(\<lambda>x. f(x) - l) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S 0"
3.133  proof -
3.134    have "(\<lambda>x. f x - l) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S l - l"
3.135      by (rule NSLIM_diff [OF f NSLIM_const])
3.136 -  thus ?thesis by simp
3.137 +  then show ?thesis by simp
3.138  qed
3.139
3.140 -lemma NSLIM_zero_cancel: "(%x. f(x) - l) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 ==> f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l"
3.141 -apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
3.143 -done
3.144 +lemma NSLIM_zero_cancel: "(\<lambda>x. f x - l) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S l"
3.145 +  apply (drule_tac g = "%x. l" and m = l in NSLIM_add)
3.147 +  done
3.148
3.149 -lemma NSLIM_const_not_eq:
3.150 -  fixes a :: "'a::real_normed_algebra_1"
3.151 -  shows "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.153 -apply (rule_tac x="star_of a + of_hypreal \<epsilon>" in exI)
3.154 -apply (simp add: hypreal_epsilon_not_zero approx_def)
3.155 -done
3.156 +lemma NSLIM_const_not_eq: "k \<noteq> L \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.157 +  for a :: "'a::real_normed_algebra_1"
3.158 +  apply (simp add: NSLIM_def)
3.159 +  apply (rule_tac x="star_of a + of_hypreal \<epsilon>" in exI)
3.160 +  apply (simp add: hypreal_epsilon_not_zero approx_def)
3.161 +  done
3.162
3.163 -lemma NSLIM_not_zero:
3.164 -  fixes a :: "'a::real_normed_algebra_1"
3.165 -  shows "k \<noteq> 0 \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S 0"
3.166 -by (rule NSLIM_const_not_eq)
3.167 +lemma NSLIM_not_zero: "k \<noteq> 0 \<Longrightarrow> \<not> (\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S 0"
3.168 +  for a :: "'a::real_normed_algebra_1"
3.169 +  by (rule NSLIM_const_not_eq)
3.170
3.171 -lemma NSLIM_const_eq:
3.172 -  fixes a :: "'a::real_normed_algebra_1"
3.173 -  shows "(\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<Longrightarrow> k = L"
3.174 -apply (rule ccontr)
3.175 -apply (blast dest: NSLIM_const_not_eq)
3.176 -done
3.177 +lemma NSLIM_const_eq: "(\<lambda>x. k) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<Longrightarrow> k = L"
3.178 +  for a :: "'a::real_normed_algebra_1"
3.179 +  by (rule ccontr) (blast dest: NSLIM_const_not_eq)
3.180 +
3.181 +lemma NSLIM_unique: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<Longrightarrow> f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S M \<Longrightarrow> L = M"
3.182 +  for a :: "'a::real_normed_algebra_1"
3.183 +  by (drule (1) NSLIM_diff) (auto dest!: NSLIM_const_eq)
3.184
3.185 -lemma NSLIM_unique:
3.186 -  fixes a :: "'a::real_normed_algebra_1"
3.187 -  shows "\<lbrakk>f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L; f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S M\<rbrakk> \<Longrightarrow> L = M"
3.188 -apply (drule (1) NSLIM_diff)
3.189 -apply (auto dest!: NSLIM_const_eq)
3.190 -done
3.191 +lemma NSLIM_mult_zero: "f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 \<Longrightarrow> (\<lambda>x. f x * g x) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0"
3.192 +  for f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
3.193 +  by (drule NSLIM_mult) auto
3.194
3.195 -lemma NSLIM_mult_zero:
3.196 -  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_algebra"
3.197 -  shows "[| f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0; g \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0 |] ==> (%x. f(x)*g(x)) \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S 0"
3.198 -by (drule NSLIM_mult, auto)
3.199 +lemma NSLIM_self: "(\<lambda>x. x) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S a"
3.200 +  by (simp add: NSLIM_def)
3.201
3.202 -lemma NSLIM_self: "(%x. x) \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S a"
3.204
3.205  subsubsection \<open>Equivalence of @{term filterlim} and @{term NSLIM}\<close>
3.206
3.207  lemma LIM_NSLIM:
3.208 -  assumes f: "f \<midarrow>a\<rightarrow> L" shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.209 +  assumes f: "f \<midarrow>a\<rightarrow> L"
3.210 +  shows "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.211  proof (rule NSLIM_I)
3.212    fix x
3.213    assume neq: "x \<noteq> star_of a"
3.214    assume approx: "x \<approx> star_of a"
3.215    have "starfun f x - star_of L \<in> Infinitesimal"
3.216    proof (rule InfinitesimalI2)
3.217 -    fix r::real assume r: "0 < r"
3.218 -    from LIM_D [OF f r]
3.219 -    obtain s where s: "0 < s" and
3.220 -      less_r: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < s\<rbrakk> \<Longrightarrow> norm (f x - L) < r"
3.221 +    fix r :: real
3.222 +    assume r: "0 < r"
3.223 +    from LIM_D [OF f r] obtain s
3.224 +      where s: "0 < s" and less_r: "\<And>x. x \<noteq> a \<Longrightarrow> norm (x - a) < s \<Longrightarrow> norm (f x - L) < r"
3.225        by fast
3.226      from less_r have less_r':
3.227 -       "\<And>x. \<lbrakk>x \<noteq> star_of a; hnorm (x - star_of a) < star_of s\<rbrakk>
3.228 -        \<Longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
3.229 +      "\<And>x. x \<noteq> star_of a \<Longrightarrow> hnorm (x - star_of a) < star_of s \<Longrightarrow>
3.230 +        hnorm (starfun f x - star_of L) < star_of r"
3.231        by transfer
3.232      from approx have "x - star_of a \<in> Infinitesimal"
3.233 -      by (unfold approx_def)
3.234 -    hence "hnorm (x - star_of a) < star_of s"
3.235 +      by (simp only: approx_def)
3.236 +    then have "hnorm (x - star_of a) < star_of s"
3.237        using s by (rule InfinitesimalD2)
3.238      with neq show "hnorm (starfun f x - star_of L) < star_of r"
3.239        by (rule less_r')
3.240    qed
3.241 -  thus "starfun f x \<approx> star_of L"
3.242 +  then show "starfun f x \<approx> star_of L"
3.243      by (unfold approx_def)
3.244  qed
3.245
3.246  lemma NSLIM_LIM:
3.247 -  assumes f: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L" shows "f \<midarrow>a\<rightarrow> L"
3.248 +  assumes f: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L"
3.249 +  shows "f \<midarrow>a\<rightarrow> L"
3.250  proof (rule LIM_I)
3.251 -  fix r::real assume r: "0 < r"
3.252 -  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s
3.253 -        \<longrightarrow> hnorm (starfun f x - star_of L) < star_of r"
3.254 +  fix r :: real
3.255 +  assume r: "0 < r"
3.256 +  have "\<exists>s>0. \<forall>x. x \<noteq> star_of a \<and> hnorm (x - star_of a) < s \<longrightarrow>
3.257 +    hnorm (starfun f x - star_of L) < star_of r"
3.258    proof (rule exI, safe)
3.259 -    show "0 < \<epsilon>" by (rule hypreal_epsilon_gt_zero)
3.260 +    show "0 < \<epsilon>"
3.261 +      by (rule hypreal_epsilon_gt_zero)
3.262    next
3.263 -    fix x assume neq: "x \<noteq> star_of a"
3.264 +    fix x
3.265 +    assume neq: "x \<noteq> star_of a"
3.266      assume "hnorm (x - star_of a) < \<epsilon>"
3.267 -    with Infinitesimal_epsilon
3.268 -    have "x - star_of a \<in> Infinitesimal"
3.269 +    with Infinitesimal_epsilon have "x - star_of a \<in> Infinitesimal"
3.270        by (rule hnorm_less_Infinitesimal)
3.271 -    hence "x \<approx> star_of a"
3.272 +    then have "x \<approx> star_of a"
3.273        by (unfold approx_def)
3.274      with f neq have "starfun f x \<approx> star_of L"
3.275        by (rule NSLIM_D)
3.276 -    hence "starfun f x - star_of L \<in> Infinitesimal"
3.277 +    then have "starfun f x - star_of L \<in> Infinitesimal"
3.278        by (unfold approx_def)
3.279 -    thus "hnorm (starfun f x - star_of L) < star_of r"
3.280 +    then show "hnorm (starfun f x - star_of L) < star_of r"
3.281        using r by (rule InfinitesimalD2)
3.282    qed
3.283 -  thus "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
3.284 +  then show "\<exists>s>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < s \<longrightarrow> norm (f x - L) < r"
3.285      by transfer
3.286  qed
3.287
3.288 -theorem LIM_NSLIM_iff: "(f \<midarrow>x\<rightarrow> L) = (f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S L)"
3.289 -by (blast intro: LIM_NSLIM NSLIM_LIM)
3.290 +theorem LIM_NSLIM_iff: "f \<midarrow>x\<rightarrow> L \<longleftrightarrow> f \<midarrow>x\<rightarrow>\<^sub>N\<^sub>S L"
3.291 +  by (blast intro: LIM_NSLIM NSLIM_LIM)
3.292
3.293
3.294  subsection \<open>Continuity\<close>
3.295
3.296 -lemma isNSContD:
3.297 -  "\<lbrakk>isNSCont f a; y \<approx> star_of a\<rbrakk> \<Longrightarrow> ( *f* f) y \<approx> star_of (f a)"
3.299 +lemma isNSContD: "isNSCont f a \<Longrightarrow> y \<approx> star_of a \<Longrightarrow> ( *f* f) y \<approx> star_of (f a)"
3.300 +  by (simp add: isNSCont_def)
3.301 +
3.302 +lemma isNSCont_NSLIM: "isNSCont f a \<Longrightarrow> f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (f a)"
3.303 +  by (simp add: isNSCont_def NSLIM_def)
3.304
3.305 -lemma isNSCont_NSLIM: "isNSCont f a ==> f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (f a) "
3.306 -by (simp add: isNSCont_def NSLIM_def)
3.307 +lemma NSLIM_isNSCont: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (f a) \<Longrightarrow> isNSCont f a"
3.308 +  apply (auto simp add: isNSCont_def NSLIM_def)
3.309 +  apply (case_tac "y = star_of a")
3.310 +   apply auto
3.311 +  done
3.312
3.313 -lemma NSLIM_isNSCont: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (f a) ==> isNSCont f a"
3.314 -apply (simp add: isNSCont_def NSLIM_def, auto)
3.315 -apply (case_tac "y = star_of a", auto)
3.316 -done
3.317 +text \<open>NS continuity can be defined using NS Limit in
3.318 +  similar fashion to standard definition of continuity.\<close>
3.319 +lemma isNSCont_NSLIM_iff: "isNSCont f a \<longleftrightarrow> f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (f a)"
3.320 +  by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
3.321
3.322 -text\<open>NS continuity can be defined using NS Limit in
3.323 -    similar fashion to standard definition of continuity\<close>
3.324 -lemma isNSCont_NSLIM_iff: "(isNSCont f a) = (f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S (f a))"
3.325 -by (blast intro: isNSCont_NSLIM NSLIM_isNSCont)
3.326 +text \<open>Hence, NS continuity can be given in terms of standard limit.\<close>
3.327 +lemma isNSCont_LIM_iff: "(isNSCont f a) = (f \<midarrow>a\<rightarrow> (f a))"
3.328 +  by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
3.329
3.330 -text\<open>Hence, NS continuity can be given
3.331 -  in terms of standard limit\<close>
3.332 -lemma isNSCont_LIM_iff: "(isNSCont f a) = (f \<midarrow>a\<rightarrow> (f a))"
3.333 -by (simp add: LIM_NSLIM_iff isNSCont_NSLIM_iff)
3.334 +text \<open>Moreover, it's trivial now that NS continuity
3.335 +  is equivalent to standard continuity.\<close>
3.336 +lemma isNSCont_isCont_iff: "isNSCont f a \<longleftrightarrow> isCont f a"
3.337 +  by (simp add: isCont_def) (rule isNSCont_LIM_iff)
3.338
3.339 -text\<open>Moreover, it's trivial now that NS continuity
3.340 -  is equivalent to standard continuity\<close>
3.341 -lemma isNSCont_isCont_iff: "(isNSCont f a) = (isCont f a)"
3.343 -apply (rule isNSCont_LIM_iff)
3.344 -done
3.345 +text \<open>Standard continuity \<open>\<Longrightarrow>\<close> NS continuity.\<close>
3.346 +lemma isCont_isNSCont: "isCont f a \<Longrightarrow> isNSCont f a"
3.347 +  by (erule isNSCont_isCont_iff [THEN iffD2])
3.348
3.349 -text\<open>Standard continuity ==> NS continuity\<close>
3.350 -lemma isCont_isNSCont: "isCont f a ==> isNSCont f a"
3.351 -by (erule isNSCont_isCont_iff [THEN iffD2])
3.352 +text \<open>NS continuity \<open>==>\<close> Standard continuity.\<close>
3.353 +lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
3.354 +  by (erule isNSCont_isCont_iff [THEN iffD1])
3.355
3.356 -text\<open>NS continuity ==> Standard continuity\<close>
3.357 -lemma isNSCont_isCont: "isNSCont f a ==> isCont f a"
3.358 -by (erule isNSCont_isCont_iff [THEN iffD1])
3.359 +
3.360 +text \<open>Alternative definition of continuity.\<close>
3.361
3.362 -text\<open>Alternative definition of continuity\<close>
3.363 +text \<open>Prove equivalence between NS limits --
3.364 +  seems easier than using standard definition.\<close>
3.365 +lemma NSLIM_h_iff: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S L"
3.366 +  apply (simp add: NSLIM_def, auto)
3.367 +   apply (drule_tac x = "star_of a + x" in spec)
3.368 +   apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
3.369 +      apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
3.370 +     apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
3.372 +   apply (rule_tac x = x in star_cases)
3.373 +   apply (rule_tac [2] x = x in star_cases)
3.375 +  done
3.376
3.377 -(* Prove equivalence between NS limits - *)
3.378 -(* seems easier than using standard definition  *)
3.379 -lemma NSLIM_h_iff: "(f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S L) = ((%h. f(a + h)) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S L)"
3.380 -apply (simp add: NSLIM_def, auto)
3.381 -apply (drule_tac x = "star_of a + x" in spec)
3.382 -apply (drule_tac [2] x = "- star_of a + x" in spec, safe, simp)
3.383 -apply (erule mem_infmal_iff [THEN iffD2, THEN Infinitesimal_add_approx_self [THEN approx_sym]])
3.384 -apply (erule_tac [3] approx_minus_iff2 [THEN iffD1])
3.386 -apply (rule_tac x = x in star_cases)
3.387 -apply (rule_tac [2] x = x in star_cases)
3.389 -done
3.390 -
3.391 -lemma NSLIM_isCont_iff: "(f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a) = ((%h. f(a + h)) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S f a)"
3.392 +lemma NSLIM_isCont_iff: "f \<midarrow>a\<rightarrow>\<^sub>N\<^sub>S f a \<longleftrightarrow> (\<lambda>h. f (a + h)) \<midarrow>0\<rightarrow>\<^sub>N\<^sub>S f a"
3.393    by (fact NSLIM_h_iff)
3.394
3.395 -lemma isNSCont_minus: "isNSCont f a ==> isNSCont (%x. - f x) a"
3.397 +lemma isNSCont_minus: "isNSCont f a \<Longrightarrow> isNSCont (\<lambda>x. - f x) a"
3.398 +  by (simp add: isNSCont_def)
3.399
3.400 -lemma isNSCont_inverse:
3.401 -  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
3.402 -  shows "[| isNSCont f x; f x \<noteq> 0 |] ==> isNSCont (%x. inverse (f x)) x"
3.403 -by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
3.404 +lemma isNSCont_inverse: "isNSCont f x \<Longrightarrow> f x \<noteq> 0 \<Longrightarrow> isNSCont (\<lambda>x. inverse (f x)) x"
3.405 +  for f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_div_algebra"
3.406 +  by (auto intro: isCont_inverse simp add: isNSCont_isCont_iff)
3.407
3.408 -lemma isNSCont_const [simp]: "isNSCont (%x. k) a"
3.410 +lemma isNSCont_const [simp]: "isNSCont (\<lambda>x. k) a"
3.411 +  by (simp add: isNSCont_def)
3.412
3.413 -lemma isNSCont_abs [simp]: "isNSCont abs (a::real)"
3.415 -apply (auto intro: approx_hrabs simp add: starfun_rabs_hrabs)
3.416 -done
3.417 +lemma isNSCont_abs [simp]: "isNSCont abs a"
3.418 +  for a :: real
3.419 +  by (auto simp: isNSCont_def intro: approx_hrabs simp: starfun_rabs_hrabs)
3.420
3.421
3.422  subsection \<open>Uniform Continuity\<close>
3.423
3.424 -lemma isNSUContD: "[| isNSUCont f; x \<approx> y|] ==> ( *f* f) x \<approx> ( *f* f) y"
3.426 +lemma isNSUContD: "isNSUCont f \<Longrightarrow> x \<approx> y \<Longrightarrow> ( *f* f) x \<approx> ( *f* f) y"
3.427 +  by (simp add: isNSUCont_def)
3.428
3.429  lemma isUCont_isNSUCont:
3.430    fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3.431 -  assumes f: "isUCont f" shows "isNSUCont f"
3.432 -proof (unfold isNSUCont_def, safe)
3.433 +  assumes f: "isUCont f"
3.434 +  shows "isNSUCont f"
3.435 +  unfolding isNSUCont_def
3.436 +proof safe
3.437    fix x y :: "'a star"
3.438    assume approx: "x \<approx> y"
3.439    have "starfun f x - starfun f y \<in> Infinitesimal"
3.440    proof (rule InfinitesimalI2)
3.441 -    fix r::real assume r: "0 < r"
3.442 -    with f obtain s where s: "0 < s" and
3.443 -      less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
3.444 +    fix r :: real
3.445 +    assume r: "0 < r"
3.446 +    with f obtain s where s: "0 < s"
3.447 +      and less_r: "\<And>x y. norm (x - y) < s \<Longrightarrow> norm (f x - f y) < r"
3.448        by (auto simp add: isUCont_def dist_norm)
3.449      from less_r have less_r':
3.450 -       "\<And>x y. hnorm (x - y) < star_of s
3.451 -        \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
3.452 +      "\<And>x y. hnorm (x - y) < star_of s \<Longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
3.453        by transfer
3.454      from approx have "x - y \<in> Infinitesimal"
3.455        by (unfold approx_def)
3.456 -    hence "hnorm (x - y) < star_of s"
3.457 +    then have "hnorm (x - y) < star_of s"
3.458        using s by (rule InfinitesimalD2)
3.459 -    thus "hnorm (starfun f x - starfun f y) < star_of r"
3.460 +    then show "hnorm (starfun f x - starfun f y) < star_of r"
3.461        by (rule less_r')
3.462    qed
3.463 -  thus "starfun f x \<approx> starfun f y"
3.464 +  then show "starfun f x \<approx> starfun f y"
3.465      by (unfold approx_def)
3.466  qed
3.467
3.468  lemma isNSUCont_isUCont:
3.469    fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::real_normed_vector"
3.470 -  assumes f: "isNSUCont f" shows "isUCont f"
3.471 -proof (unfold isUCont_def dist_norm, safe)
3.472 -  fix r::real assume r: "0 < r"
3.473 -  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s
3.474 -        \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
3.475 +  assumes f: "isNSUCont f"
3.476 +  shows "isUCont f"
3.477 +  unfolding isUCont_def dist_norm
3.478 +proof safe
3.479 +  fix r :: real
3.480 +  assume r: "0 < r"
3.481 +  have "\<exists>s>0. \<forall>x y. hnorm (x - y) < s \<longrightarrow> hnorm (starfun f x - starfun f y) < star_of r"
3.482    proof (rule exI, safe)
3.483 -    show "0 < \<epsilon>" by (rule hypreal_epsilon_gt_zero)
3.484 +    show "0 < \<epsilon>"
3.485 +      by (rule hypreal_epsilon_gt_zero)
3.486    next
3.487      fix x y :: "'a star"
3.488      assume "hnorm (x - y) < \<epsilon>"
3.489 -    with Infinitesimal_epsilon
3.490 -    have "x - y \<in> Infinitesimal"
3.491 +    with Infinitesimal_epsilon have "x - y \<in> Infinitesimal"
3.492        by (rule hnorm_less_Infinitesimal)
3.493 -    hence "x \<approx> y"
3.494 +    then have "x \<approx> y"
3.495        by (unfold approx_def)
3.496      with f have "starfun f x \<approx> starfun f y"
3.498 -    hence "starfun f x - starfun f y \<in> Infinitesimal"
3.499 +    then have "starfun f x - starfun f y \<in> Infinitesimal"
3.500        by (unfold approx_def)
3.501 -    thus "hnorm (starfun f x - starfun f y) < star_of r"
3.502 +    then show "hnorm (starfun f x - starfun f y) < star_of r"
3.503        using r by (rule InfinitesimalD2)
3.504    qed
3.505 -  thus "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
3.506 +  then show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
3.507      by transfer
3.508  qed
3.509
```
```     4.1 --- a/src/HOL/Nonstandard_Analysis/HyperDef.thy	Mon Oct 31 16:26:36 2016 +0100
4.2 +++ b/src/HOL/Nonstandard_Analysis/HyperDef.thy	Tue Nov 01 00:44:24 2016 +0100
4.3 @@ -4,50 +4,42 @@
4.4      Conversion to Isar and new proofs by Lawrence C Paulson, 2004
4.5  *)
4.6
4.7 -section\<open>Construction of Hyperreals Using Ultrafilters\<close>
4.8 +section \<open>Construction of Hyperreals Using Ultrafilters\<close>
4.9
4.10  theory HyperDef
4.11 -imports Complex_Main HyperNat
4.12 +  imports Complex_Main HyperNat
4.13  begin
4.14
4.15  type_synonym hypreal = "real star"
4.16
4.17 -abbreviation
4.18 -  hypreal_of_real :: "real => real star" where
4.19 -  "hypreal_of_real == star_of"
4.20 +abbreviation hypreal_of_real :: "real \<Rightarrow> real star"
4.21 +  where "hypreal_of_real \<equiv> star_of"
4.22
4.23 -abbreviation
4.24 -  hypreal_of_hypnat :: "hypnat \<Rightarrow> hypreal" where
4.25 -  "hypreal_of_hypnat \<equiv> of_hypnat"
4.26 +abbreviation hypreal_of_hypnat :: "hypnat \<Rightarrow> hypreal"
4.27 +  where "hypreal_of_hypnat \<equiv> of_hypnat"
4.28
4.29 -definition
4.30 -  omega :: hypreal  ("\<omega>") where
4.31 -   \<comment> \<open>an infinite number \<open>= [<1,2,3,...>]\<close>\<close>
4.32 -  "\<omega> = star_n (\<lambda>n. real (Suc n))"
4.33 +definition omega :: hypreal  ("\<omega>")
4.34 +  where "\<omega> = star_n (\<lambda>n. real (Suc n))"
4.35 +    \<comment> \<open>an infinite number \<open>= [<1, 2, 3, \<dots>>]\<close>\<close>
4.36
4.37 -definition
4.38 -  epsilon :: hypreal  ("\<epsilon>") where
4.39 -   \<comment> \<open>an infinitesimal number \<open>= [<1,1/2,1/3,...>]\<close>\<close>
4.40 -  "\<epsilon> = star_n (\<lambda>n. inverse (real (Suc n)))"
4.41 +definition epsilon :: hypreal  ("\<epsilon>")
4.42 +  where "\<epsilon> = star_n (\<lambda>n. inverse (real (Suc n)))"
4.43 +    \<comment> \<open>an infinitesimal number \<open>= [<1, 1/2, 1/3, \<dots>>]\<close>\<close>
4.44
4.45
4.46  subsection \<open>Real vector class instances\<close>
4.47
4.48  instantiation star :: (scaleR) scaleR
4.49  begin
4.50 -
4.51 -definition
4.52 -  star_scaleR_def [transfer_unfold]: "scaleR r \<equiv> *f* (scaleR r)"
4.53 -
4.54 -instance ..
4.55 -
4.56 +  definition star_scaleR_def [transfer_unfold]: "scaleR r \<equiv> *f* (scaleR r)"
4.57 +  instance ..
4.58  end
4.59
4.60  lemma Standard_scaleR [simp]: "x \<in> Standard \<Longrightarrow> scaleR r x \<in> Standard"
4.62 +  by (simp add: star_scaleR_def)
4.63
4.64  lemma star_of_scaleR [simp]: "star_of (scaleR r x) = scaleR r (star_of x)"
4.65 -by transfer (rule refl)
4.66 +  by transfer (rule refl)
4.67
4.68  instance star :: (real_vector) real_vector
4.69  proof
4.70 @@ -80,230 +72,213 @@
4.71  instance star :: (real_field) real_field ..
4.72
4.73  lemma star_of_real_def [transfer_unfold]: "of_real r = star_of (of_real r)"
4.74 -by (unfold of_real_def, transfer, rule refl)
4.75 +  by (unfold of_real_def, transfer, rule refl)
4.76
4.77  lemma Standard_of_real [simp]: "of_real r \<in> Standard"
4.79 +  by (simp add: star_of_real_def)
4.80
4.81  lemma star_of_of_real [simp]: "star_of (of_real r) = of_real r"
4.82 -by transfer (rule refl)
4.83 +  by transfer (rule refl)
4.84
4.85  lemma of_real_eq_star_of [simp]: "of_real = star_of"
4.86  proof
4.87 -  fix r :: real
4.88 -  show "of_real r = star_of r"
4.89 +  show "of_real r = star_of r" for r :: real
4.90      by transfer simp
4.91  qed
4.92
4.93  lemma Reals_eq_Standard: "(\<real> :: hypreal set) = Standard"
4.94 -by (simp add: Reals_def Standard_def)
4.95 +  by (simp add: Reals_def Standard_def)
4.96
4.97
4.98  subsection \<open>Injection from @{typ hypreal}\<close>
4.99
4.100 -definition
4.101 -  of_hypreal :: "hypreal \<Rightarrow> 'a::real_algebra_1 star" where
4.102 -  [transfer_unfold]: "of_hypreal = *f* of_real"
4.103 +definition of_hypreal :: "hypreal \<Rightarrow> 'a::real_algebra_1 star"
4.104 +  where [transfer_unfold]: "of_hypreal = *f* of_real"
4.105
4.106 -lemma Standard_of_hypreal [simp]:
4.107 -  "r \<in> Standard \<Longrightarrow> of_hypreal r \<in> Standard"
4.109 +lemma Standard_of_hypreal [simp]: "r \<in> Standard \<Longrightarrow> of_hypreal r \<in> Standard"
4.110 +  by (simp add: of_hypreal_def)
4.111
4.112  lemma of_hypreal_0 [simp]: "of_hypreal 0 = 0"
4.113 -by transfer (rule of_real_0)
4.114 +  by transfer (rule of_real_0)
4.115
4.116  lemma of_hypreal_1 [simp]: "of_hypreal 1 = 1"
4.117 -by transfer (rule of_real_1)
4.118 +  by transfer (rule of_real_1)
4.119
4.121 -  "\<And>x y. of_hypreal (x + y) = of_hypreal x + of_hypreal y"
4.123 +lemma of_hypreal_add [simp]: "\<And>x y. of_hypreal (x + y) = of_hypreal x + of_hypreal y"
4.124 +  by transfer (rule of_real_add)
4.125
4.126  lemma of_hypreal_minus [simp]: "\<And>x. of_hypreal (- x) = - of_hypreal x"
4.127 -by transfer (rule of_real_minus)
4.128 +  by transfer (rule of_real_minus)
4.129
4.130 -lemma of_hypreal_diff [simp]:
4.131 -  "\<And>x y. of_hypreal (x - y) = of_hypreal x - of_hypreal y"
4.132 -by transfer (rule of_real_diff)
4.133 +lemma of_hypreal_diff [simp]: "\<And>x y. of_hypreal (x - y) = of_hypreal x - of_hypreal y"
4.134 +  by transfer (rule of_real_diff)
4.135
4.136 -lemma of_hypreal_mult [simp]:
4.137 -  "\<And>x y. of_hypreal (x * y) = of_hypreal x * of_hypreal y"
4.138 -by transfer (rule of_real_mult)
4.139 +lemma of_hypreal_mult [simp]: "\<And>x y. of_hypreal (x * y) = of_hypreal x * of_hypreal y"
4.140 +  by transfer (rule of_real_mult)
4.141
4.142  lemma of_hypreal_inverse [simp]:
4.143    "\<And>x. of_hypreal (inverse x) =
4.144 -   inverse (of_hypreal x :: 'a::{real_div_algebra, division_ring} star)"
4.145 -by transfer (rule of_real_inverse)
4.146 +    inverse (of_hypreal x :: 'a::{real_div_algebra, division_ring} star)"
4.147 +  by transfer (rule of_real_inverse)
4.148
4.149  lemma of_hypreal_divide [simp]:
4.150    "\<And>x y. of_hypreal (x / y) =
4.151 -   (of_hypreal x / of_hypreal y :: 'a::{real_field, field} star)"
4.152 -by transfer (rule of_real_divide)
4.153 +    (of_hypreal x / of_hypreal y :: 'a::{real_field, field} star)"
4.154 +  by transfer (rule of_real_divide)
4.155
4.156 -lemma of_hypreal_eq_iff [simp]:
4.157 -  "\<And>x y. (of_hypreal x = of_hypreal y) = (x = y)"
4.158 -by transfer (rule of_real_eq_iff)
4.159 +lemma of_hypreal_eq_iff [simp]: "\<And>x y. (of_hypreal x = of_hypreal y) = (x = y)"
4.160 +  by transfer (rule of_real_eq_iff)
4.161
4.162 -lemma of_hypreal_eq_0_iff [simp]:
4.163 -  "\<And>x. (of_hypreal x = 0) = (x = 0)"
4.164 -by transfer (rule of_real_eq_0_iff)
4.165 +lemma of_hypreal_eq_0_iff [simp]: "\<And>x. (of_hypreal x = 0) = (x = 0)"
4.166 +  by transfer (rule of_real_eq_0_iff)
4.167
4.168
4.169 -subsection\<open>Properties of @{term starrel}\<close>
4.170 +subsection \<open>Properties of @{term starrel}\<close>
4.171
4.172  lemma lemma_starrel_refl [simp]: "x \<in> starrel `` {x}"
4.174 +  by (simp add: starrel_def)
4.175
4.176  lemma starrel_in_hypreal [simp]: "starrel``{x}:star"
4.177 -by (simp add: star_def starrel_def quotient_def, blast)
4.178 +  by (simp add: star_def starrel_def quotient_def, blast)
4.179
4.180  declare Abs_star_inject [simp] Abs_star_inverse [simp]
4.181  declare equiv_starrel [THEN eq_equiv_class_iff, simp]
4.182
4.183 -subsection\<open>@{term hypreal_of_real}:
4.184 -            the Injection from @{typ real} to @{typ hypreal}\<close>
4.185 +
4.186 +subsection \<open>@{term hypreal_of_real}: the Injection from @{typ real} to @{typ hypreal}\<close>
4.187
4.188  lemma inj_star_of: "inj star_of"
4.189 -by (rule inj_onI, simp)
4.190 +  by (rule inj_onI) simp
4.191
4.192 -lemma mem_Rep_star_iff: "(X \<in> Rep_star x) = (x = star_n X)"
4.193 -by (cases x, simp add: star_n_def)
4.194 +lemma mem_Rep_star_iff: "X \<in> Rep_star x \<longleftrightarrow> x = star_n X"
4.195 +  by (cases x) (simp add: star_n_def)
4.196
4.197 -lemma Rep_star_star_n_iff [simp]:
4.198 -  "(X \<in> Rep_star (star_n Y)) = (eventually (\<lambda>n. Y n = X n) \<U>)"
4.200 +lemma Rep_star_star_n_iff [simp]: "X \<in> Rep_star (star_n Y) \<longleftrightarrow> eventually (\<lambda>n. Y n = X n) \<U>"
4.201 +  by (simp add: star_n_def)
4.202
4.203  lemma Rep_star_star_n: "X \<in> Rep_star (star_n X)"
4.204 -by simp
4.205 +  by simp
4.206
4.207 -subsection\<open>Properties of @{term star_n}\<close>
4.208
4.210 -  "star_n X + star_n Y = star_n (%n. X n + Y n)"
4.211 -by (simp only: star_add_def starfun2_star_n)
4.212 +subsection \<open>Properties of @{term star_n}\<close>
4.213 +
4.214 +lemma star_n_add: "star_n X + star_n Y = star_n (\<lambda>n. X n + Y n)"
4.215 +  by (simp only: star_add_def starfun2_star_n)
4.216
4.217 -lemma star_n_minus:
4.218 -   "- star_n X = star_n (%n. -(X n))"
4.219 -by (simp only: star_minus_def starfun_star_n)
4.220 +lemma star_n_minus: "- star_n X = star_n (\<lambda>n. -(X n))"
4.221 +  by (simp only: star_minus_def starfun_star_n)
4.222
4.223 -lemma star_n_diff:
4.224 -     "star_n X - star_n Y = star_n (%n. X n - Y n)"
4.225 -by (simp only: star_diff_def starfun2_star_n)
4.226 +lemma star_n_diff: "star_n X - star_n Y = star_n (\<lambda>n. X n - Y n)"
4.227 +  by (simp only: star_diff_def starfun2_star_n)
4.228
4.229 -lemma star_n_mult:
4.230 -  "star_n X * star_n Y = star_n (%n. X n * Y n)"
4.231 -by (simp only: star_mult_def starfun2_star_n)
4.232 +lemma star_n_mult: "star_n X * star_n Y = star_n (\<lambda>n. X n * Y n)"
4.233 +  by (simp only: star_mult_def starfun2_star_n)
4.234
4.235 -lemma star_n_inverse:
4.236 -      "inverse (star_n X) = star_n (%n. inverse(X n))"
4.237 -by (simp only: star_inverse_def starfun_star_n)
4.238 +lemma star_n_inverse: "inverse (star_n X) = star_n (\<lambda>n. inverse (X n))"
4.239 +  by (simp only: star_inverse_def starfun_star_n)
4.240
4.241 -lemma star_n_le:
4.242 -      "star_n X \<le> star_n Y = (eventually (\<lambda>n. X n \<le> Y n) FreeUltrafilterNat)"
4.243 -by (simp only: star_le_def starP2_star_n)
4.244 +lemma star_n_le: "star_n X \<le> star_n Y = eventually (\<lambda>n. X n \<le> Y n) FreeUltrafilterNat"
4.245 +  by (simp only: star_le_def starP2_star_n)
4.246 +
4.247 +lemma star_n_less: "star_n X < star_n Y = eventually (\<lambda>n. X n < Y n) FreeUltrafilterNat"
4.248 +  by (simp only: star_less_def starP2_star_n)
4.249
4.250 -lemma star_n_less:
4.251 -      "star_n X < star_n Y = (eventually (\<lambda>n. X n < Y n) FreeUltrafilterNat)"
4.252 -by (simp only: star_less_def starP2_star_n)
4.253 +lemma star_n_zero_num: "0 = star_n (\<lambda>n. 0)"
4.254 +  by (simp only: star_zero_def star_of_def)
4.255
4.256 -lemma star_n_zero_num: "0 = star_n (%n. 0)"
4.257 -by (simp only: star_zero_def star_of_def)
4.258 +lemma star_n_one_num: "1 = star_n (\<lambda>n. 1)"
4.259 +  by (simp only: star_one_def star_of_def)
4.260
4.261 -lemma star_n_one_num: "1 = star_n (%n. 1)"
4.262 -by (simp only: star_one_def star_of_def)
4.263 -
4.264 -lemma star_n_abs: "\<bar>star_n X\<bar> = star_n (%n. \<bar>X n\<bar>)"
4.265 -by (simp only: star_abs_def starfun_star_n)
4.266 +lemma star_n_abs: "\<bar>star_n X\<bar> = star_n (\<lambda>n. \<bar>X n\<bar>)"
4.267 +  by (simp only: star_abs_def starfun_star_n)
4.268
4.269  lemma hypreal_omega_gt_zero [simp]: "0 < \<omega>"
4.270 -by (simp add: omega_def star_n_zero_num star_n_less)
4.271 -
4.272 -subsection\<open>Existence of Infinite Hyperreal Number\<close>
4.273 -
4.274 -text\<open>Existence of infinite number not corresponding to any real number.
4.275 -Use assumption that member @{term FreeUltrafilterNat} is not finite.\<close>
4.276 +  by (simp add: omega_def star_n_zero_num star_n_less)
4.277
4.278
4.279 -text\<open>A few lemmas first\<close>
4.280 +subsection \<open>Existence of Infinite Hyperreal Number\<close>
4.281 +
4.282 +text \<open>Existence of infinite number not corresponding to any real number.
4.283 +  Use assumption that member @{term FreeUltrafilterNat} is not finite.\<close>
4.284 +
4.285 +text \<open>A few lemmas first.\<close>
4.286
4.287  lemma lemma_omega_empty_singleton_disj:
4.288    "{n::nat. x = real n} = {} \<or> (\<exists>y. {n::nat. x = real n} = {y})"
4.289 -by force
4.290 +  by force
4.291
4.292  lemma lemma_finite_omega_set: "finite {n::nat. x = real n}"
4.293    using lemma_omega_empty_singleton_disj [of x] by auto
4.294
4.295 -lemma not_ex_hypreal_of_real_eq_omega:
4.296 -      "~ (\<exists>x. hypreal_of_real x = \<omega>)"
4.297 -apply (simp add: omega_def star_of_def star_n_eq_iff)
4.298 -apply clarify
4.299 -apply (rule_tac x2="x-1" in lemma_finite_omega_set [THEN FreeUltrafilterNat.finite, THEN notE])
4.300 -apply (erule eventually_mono)
4.301 -apply auto
4.302 -done
4.303 +lemma not_ex_hypreal_of_real_eq_omega: "\<nexists>x. hypreal_of_real x = \<omega>"
4.304 +  apply (simp add: omega_def star_of_def star_n_eq_iff)
4.305 +  apply clarify
4.306 +  apply (rule_tac x2="x-1" in lemma_finite_omega_set [THEN FreeUltrafilterNat.finite, THEN notE])
4.307 +  apply (erule eventually_mono)
4.308 +  apply auto
4.309 +  done
4.310
4.311  lemma hypreal_of_real_not_eq_omega: "hypreal_of_real x \<noteq> \<omega>"
4.312 -by (insert not_ex_hypreal_of_real_eq_omega, auto)
4.313 +  using not_ex_hypreal_of_real_eq_omega by auto
4.314
4.315 -text\<open>Existence of infinitesimal number also not corresponding to any
4.316 - real number\<close>
4.317 +text \<open>Existence of infinitesimal number also not corresponding to any real number.\<close>
4.318
4.319  lemma lemma_epsilon_empty_singleton_disj:
4.320 -     "{n::nat. x = inverse(real(Suc n))} = {} |
4.321 -      (\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})"
4.322 -by auto
4.323 +  "{n::nat. x = inverse(real(Suc n))} = {} \<or> (\<exists>y. {n::nat. x = inverse(real(Suc n))} = {y})"
4.324 +  by auto
4.325
4.326 -lemma lemma_finite_epsilon_set: "finite {n. x = inverse(real(Suc n))}"
4.327 -by (cut_tac x = x in lemma_epsilon_empty_singleton_disj, auto)
4.328 +lemma lemma_finite_epsilon_set: "finite {n. x = inverse (real (Suc n))}"
4.329 +  using lemma_epsilon_empty_singleton_disj [of x] by auto
4.330
4.331 -lemma not_ex_hypreal_of_real_eq_epsilon: "~ (\<exists>x. hypreal_of_real x = \<epsilon>)"
4.332 -by (auto simp add: epsilon_def star_of_def star_n_eq_iff
4.333 -                   lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite] simp del: of_nat_Suc)
4.334 +lemma not_ex_hypreal_of_real_eq_epsilon: "\<nexists>x. hypreal_of_real x = \<epsilon>"
4.335 +  by (auto simp: epsilon_def star_of_def star_n_eq_iff
4.336 +      lemma_finite_epsilon_set [THEN FreeUltrafilterNat.finite] simp del: of_nat_Suc)
4.337
4.338  lemma hypreal_of_real_not_eq_epsilon: "hypreal_of_real x \<noteq> \<epsilon>"
4.339 -by (insert not_ex_hypreal_of_real_eq_epsilon, auto)
4.340 +  using not_ex_hypreal_of_real_eq_epsilon by auto
4.341
4.342  lemma hypreal_epsilon_not_zero: "\<epsilon> \<noteq> 0"
4.343 -by (simp add: epsilon_def star_zero_def star_of_def star_n_eq_iff FreeUltrafilterNat.proper
4.344 -         del: star_of_zero)
4.345 +  by (simp add: epsilon_def star_zero_def star_of_def star_n_eq_iff FreeUltrafilterNat.proper
4.346 +      del: star_of_zero)
4.347
4.348  lemma hypreal_epsilon_inverse_omega: "\<epsilon> = inverse \<omega>"
4.349 -by (simp add: epsilon_def omega_def star_n_inverse)
4.350 +  by (simp add: epsilon_def omega_def star_n_inverse)
4.351
4.352  lemma hypreal_epsilon_gt_zero: "0 < \<epsilon>"
4.354 -
4.355 -subsection\<open>Absolute Value Function for the Hyperreals\<close>
4.356 -
4.357 -lemma hrabs_add_less: "[| \<bar>x\<bar> < r; \<bar>y\<bar> < s |] ==> \<bar>x + y\<bar> < r + (s::hypreal)"
4.358 -by (simp add: abs_if split: if_split_asm)
4.359 -
4.360 -lemma hrabs_less_gt_zero: "\<bar>x\<bar> < r ==> (0::hypreal) < r"
4.361 -by (blast intro!: order_le_less_trans abs_ge_zero)
4.362 -
4.363 -lemma hrabs_disj: "\<bar>x\<bar> = (x::'a::abs_if) \<or> \<bar>x\<bar> = -x"
4.365 -
4.366 -lemma hrabs_add_lemma_disj: "(y::hypreal) + - x + (y + - z) = \<bar>x + - z\<bar> ==> y = z | x = y"
4.367 -by (simp add: abs_if split: if_split_asm)
4.368 +  by (simp add: hypreal_epsilon_inverse_omega)
4.369
4.370
4.371 -subsection\<open>Embedding the Naturals into the Hyperreals\<close>
4.372 +subsection \<open>Absolute Value Function for the Hyperreals\<close>
4.373 +
4.374 +lemma hrabs_add_less: "\<bar>x\<bar> < r \<Longrightarrow> \<bar>y\<bar> < s \<Longrightarrow> \<bar>x + y\<bar> < r + s"
4.375 +  for x y r s :: hypreal
4.376 +  by (simp add: abs_if split: if_split_asm)
4.377 +
4.378 +lemma hrabs_less_gt_zero: "\<bar>x\<bar> < r \<Longrightarrow> 0 < r"
4.379 +  for x r :: hypreal
4.380 +  by (blast intro!: order_le_less_trans abs_ge_zero)
4.381
4.382 -abbreviation
4.383 -  hypreal_of_nat :: "nat => hypreal" where
4.384 -  "hypreal_of_nat == of_nat"
4.385 +lemma hrabs_disj: "\<bar>x\<bar> = x \<or> \<bar>x\<bar> = -x"
4.386 +  for x :: "'a::abs_if"
4.387 +  by (simp add: abs_if)
4.388 +
4.389 +lemma hrabs_add_lemma_disj: "y + - x + (y + - z) = \<bar>x + - z\<bar> \<Longrightarrow> y = z \<or> x = y"
4.390 +  for x y z :: hypreal
4.391 +  by (simp add: abs_if split: if_split_asm)
4.392 +
4.393 +
4.394 +subsection \<open>Embedding the Naturals into the Hyperreals\<close>
4.395 +
4.396 +abbreviation hypreal_of_nat :: "nat \<Rightarrow> hypreal"
4.397 +  where "hypreal_of_nat \<equiv> of_nat"
4.398
4.399  lemma SNat_eq: "Nats = {n. \<exists>N. n = hypreal_of_nat N}"
4.400 -by (simp add: Nats_def image_def)
4.401 +  by (simp add: Nats_def image_def)
4.402
4.403 -(*------------------------------------------------------------*)
4.404 -(* naturals embedded in hyperreals                            *)
4.405 -(* is a hyperreal c.f. NS extension                           *)
4.406 -(*------------------------------------------------------------*)
4.407 +text \<open>Naturals embedded in hyperreals: is a hyperreal c.f. NS extension.\<close>
4.408
4.409 -lemma hypreal_of_nat: "hypreal_of_nat m = star_n (%n. real m)"
4.410 -by (simp add: star_of_def [symmetric])
4.411 +lemma hypreal_of_nat: "hypreal_of_nat m = star_n (\<lambda>n. real m)"
4.412 +  by (simp add: star_of_def [symmetric])
4.413
4.414  declaration \<open>
4.415    K (Lin_Arith.add_inj_thms [@{thm star_of_le} RS iffD2,
4.416 @@ -314,75 +289,77 @@
4.417    #> Lin_Arith.add_inj_const (@{const_name "StarDef.star_of"}, @{typ "real \<Rightarrow> hypreal"}))
4.418  \<close>
4.419
4.420 -simproc_setup fast_arith_hypreal ("(m::hypreal) < n" | "(m::hypreal) <= n" | "(m::hypreal) = n") =
4.421 +simproc_setup fast_arith_hypreal ("(m::hypreal) < n" | "(m::hypreal) \<le> n" | "(m::hypreal) = n") =
4.422    \<open>K Lin_Arith.simproc\<close>
4.423
4.424
4.425  subsection \<open>Exponentials on the Hyperreals\<close>
4.426
4.427 -lemma hpowr_0 [simp]:   "r ^ 0       = (1::hypreal)"
4.428 -by (rule power_0)
4.429 -
4.430 -lemma hpowr_Suc [simp]: "r ^ (Suc n) = (r::hypreal) * (r ^ n)"
4.431 -by (rule power_Suc)
4.432 +lemma hpowr_0 [simp]: "r ^ 0 = (1::hypreal)"
4.433 +  for r :: hypreal
4.434 +  by (rule power_0)
4.435
4.436 -lemma hrealpow_two: "(r::hypreal) ^ Suc (Suc 0) = r * r"
4.437 -by simp
4.438 +lemma hpowr_Suc [simp]: "r ^ (Suc n) = r * (r ^ n)"
4.439 +  for r :: hypreal
4.440 +  by (rule power_Suc)
4.441
4.442 -lemma hrealpow_two_le [simp]: "(0::hypreal) \<le> r ^ Suc (Suc 0)"
4.443 -by (auto simp add: zero_le_mult_iff)
4.444 +lemma hrealpow_two: "r ^ Suc (Suc 0) = r * r"
4.445 +  for r :: hypreal
4.446 +  by simp
4.447
4.449 -     "(0::hypreal) \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0)"
4.450 -by (simp only: hrealpow_two_le add_nonneg_nonneg)
4.451 +lemma hrealpow_two_le [simp]: "0 \<le> r ^ Suc (Suc 0)"
4.452 +  for r :: hypreal
4.453 +  by (auto simp add: zero_le_mult_iff)
4.454 +
4.455 +lemma hrealpow_two_le_add_order [simp]: "0 \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0)"
4.456 +  for u v :: hypreal
4.457 +  by (simp only: hrealpow_two_le add_nonneg_nonneg)
4.458
4.460 -     "(0::hypreal) \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)"
4.461 -by (simp only: hrealpow_two_le add_nonneg_nonneg)
4.462 +lemma hrealpow_two_le_add_order2 [simp]: "0 \<le> u ^ Suc (Suc 0) + v ^ Suc (Suc 0) + w ^ Suc (Suc 0)"
4.463 +  for u v w :: hypreal
4.464 +  by (simp only: hrealpow_two_le add_nonneg_nonneg)
4.465
4.467 -     "[| 0 \<le> x; 0 \<le> y |] ==> (x+y = 0) = (x = 0 & y = (0::hypreal))"
4.468 -by arith
4.469 +lemma hypreal_add_nonneg_eq_0_iff: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
4.470 +  for x y :: hypreal
4.471 +  by arith
4.472
4.473
4.474 -text\<open>FIXME: DELETE THESE\<close>
4.476 -     "(x*x + y*y + z*z = 0) = (x = 0 & y = 0 & z = (0::hypreal))"
4.478 -done
4.479 +(* FIXME: DELETE THESE *)
4.480 +lemma hypreal_three_squares_add_zero_iff: "x * x + y * y + z * z = 0 \<longleftrightarrow> x = 0 \<and> y = 0 \<and> z = 0"
4.481 +  for x y z :: hypreal
4.483
4.485 -     "(x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = (0::hypreal)) =
4.486 -      (x = 0 & y = 0 & z = 0)"
4.487 -by (simp only: hypreal_three_squares_add_zero_iff hrealpow_two)
4.488 +  "x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + z ^ Suc (Suc 0) = 0 \<longleftrightarrow> x = 0 \<and> y = 0 \<and> z = 0"
4.489 +  for x y z :: hypreal
4.490 +  by (simp only: hypreal_three_squares_add_zero_iff hrealpow_two)
4.491
4.492  (*FIXME: This and RealPow.abs_realpow_two should be replaced by an abstract
4.493    result proved in Rings or Fields*)
4.494 -lemma hrabs_hrealpow_two [simp]: "\<bar>x ^ Suc (Suc 0)\<bar> = (x::hypreal) ^ Suc (Suc 0)"
4.496 +lemma hrabs_hrealpow_two [simp]: "\<bar>x ^ Suc (Suc 0)\<bar> = x ^ Suc (Suc 0)"
4.497 +  for x :: hypreal
4.498 +  by (simp add: abs_mult)
4.499
4.500  lemma two_hrealpow_ge_one [simp]: "(1::hypreal) \<le> 2 ^ n"
4.501 -by (insert power_increasing [of 0 n "2::hypreal"], simp)
4.502 +  using power_increasing [of 0 n "2::hypreal"] by simp
4.503
4.504 -lemma hrealpow:
4.505 -    "star_n X ^ m = star_n (%n. (X n::real) ^ m)"
4.506 -apply (induct_tac "m")
4.507 -apply (auto simp add: star_n_one_num star_n_mult power_0)
4.508 -done
4.509 +lemma hrealpow: "star_n X ^ m = star_n (\<lambda>n. (X n::real) ^ m)"
4.510 +  by (induct m) (auto simp: star_n_one_num star_n_mult)
4.511
4.512  lemma hrealpow_sum_square_expand:
4.513 -     "(x + (y::hypreal)) ^ Suc (Suc 0) =
4.514 -      x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0)))*x*y"
4.515 -by (simp add: distrib_left distrib_right)
4.516 +  "(x + y) ^ Suc (Suc 0) =
4.517 +    x ^ Suc (Suc 0) + y ^ Suc (Suc 0) + (hypreal_of_nat (Suc (Suc 0))) * x * y"
4.518 +  for x y :: hypreal
4.519 +  by (simp add: distrib_left distrib_right)
4.520
4.521  lemma power_hypreal_of_real_numeral:
4.522 -     "(numeral v :: hypreal) ^ n = hypreal_of_real ((numeral v) ^ n)"
4.523 -by simp
4.524 +  "(numeral v :: hypreal) ^ n = hypreal_of_real ((numeral v) ^ n)"
4.525 +  by simp
4.526  declare power_hypreal_of_real_numeral [of _ "numeral w", simp] for w
4.527
4.528  lemma power_hypreal_of_real_neg_numeral:
4.529 -     "(- numeral v :: hypreal) ^ n = hypreal_of_real ((- numeral v) ^ n)"
4.530 -by simp
4.531 +  "(- numeral v :: hypreal) ^ n = hypreal_of_real ((- numeral v) ^ n)"
4.532 +  by simp
4.533  declare power_hypreal_of_real_neg_numeral [of _ "numeral w", simp] for w
4.534  (*
4.535  lemma hrealpow_HFinite:
4.536 @@ -393,148 +370,119 @@
4.537  done
4.538  *)
4.539
4.540 -subsection\<open>Powers with Hypernatural Exponents\<close>
4.541
4.542 -definition pow :: "['a::power star, nat star] \<Rightarrow> 'a star" (infixr "pow" 80) where
4.543 -  hyperpow_def [transfer_unfold]: "R pow N = ( *f2* op ^) R N"
4.544 -  (* hypernatural powers of hyperreals *)
4.545 +subsection \<open>Powers with Hypernatural Exponents\<close>
4.546
4.547 -lemma Standard_hyperpow [simp]:
4.548 -  "\<lbrakk>r \<in> Standard; n \<in> Standard\<rbrakk> \<Longrightarrow> r pow n \<in> Standard"
4.549 -unfolding hyperpow_def by simp
4.550 +text \<open>Hypernatural powers of hyperreals.\<close>
4.551 +definition pow :: "'a::power star \<Rightarrow> nat star \<Rightarrow> 'a star"  (infixr "pow" 80)
4.552 +  where hyperpow_def [transfer_unfold]: "R pow N = ( *f2* op ^) R N"
4.553
4.554 -lemma hyperpow: "star_n X pow star_n Y = star_n (%n. X n ^ Y n)"
4.555 -by (simp add: hyperpow_def starfun2_star_n)
4.556 -
4.557 -lemma hyperpow_zero [simp]:
4.558 -  "\<And>n. (0::'a::{power,semiring_0} star) pow (n + (1::hypnat)) = 0"
4.559 -by transfer simp
4.560 +lemma Standard_hyperpow [simp]: "r \<in> Standard \<Longrightarrow> n \<in> Standard \<Longrightarrow> r pow n \<in> Standard"
4.561 +  by (simp add: hyperpow_def)
4.562
4.563 -lemma hyperpow_not_zero:
4.564 -  "\<And>r n. r \<noteq> (0::'a::{field} star) ==> r pow n \<noteq> 0"
4.565 -by transfer (rule power_not_zero)
4.566 +lemma hyperpow: "star_n X pow star_n Y = star_n (\<lambda>n. X n ^ Y n)"
4.567 +  by (simp add: hyperpow_def starfun2_star_n)
4.568 +
4.569 +lemma hyperpow_zero [simp]: "\<And>n. (0::'a::{power,semiring_0} star) pow (n + (1::hypnat)) = 0"
4.570 +  by transfer simp
4.571
4.572 -lemma hyperpow_inverse:
4.573 -  "\<And>r n. r \<noteq> (0::'a::field star)
4.574 -   \<Longrightarrow> inverse (r pow n) = (inverse r) pow n"
4.575 -by transfer (rule power_inverse [symmetric])
4.576 +lemma hyperpow_not_zero: "\<And>r n. r \<noteq> (0::'a::{field} star) \<Longrightarrow> r pow n \<noteq> 0"
4.577 +  by transfer (rule power_not_zero)
4.578
4.579 -lemma hyperpow_hrabs:
4.580 -  "\<And>r n. \<bar>r::'a::{linordered_idom} star\<bar> pow n = \<bar>r pow n\<bar>"
4.581 -by transfer (rule power_abs [symmetric])
4.582 +lemma hyperpow_inverse: "\<And>r n. r \<noteq> (0::'a::field star) \<Longrightarrow> inverse (r pow n) = (inverse r) pow n"
4.583 +  by transfer (rule power_inverse [symmetric])
4.584
4.586 -  "\<And>r n m. (r::'a::monoid_mult star) pow (n + m) = (r pow n) * (r pow m)"
4.588 +lemma hyperpow_hrabs: "\<And>r n. \<bar>r::'a::{linordered_idom} star\<bar> pow n = \<bar>r pow n\<bar>"
4.589 +  by transfer (rule power_abs [symmetric])
4.590
4.591 -lemma hyperpow_one [simp]:
4.592 -  "\<And>r. (r::'a::monoid_mult star) pow (1::hypnat) = r"
4.593 -by transfer (rule power_one_right)
4.594 +lemma hyperpow_add: "\<And>r n m. (r::'a::monoid_mult star) pow (n + m) = (r pow n) * (r pow m)"
4.595 +  by transfer (rule power_add)
4.596
4.597 -lemma hyperpow_two:
4.598 -  "\<And>r. (r::'a::monoid_mult star) pow (2::hypnat) = r * r"
4.599 -by transfer (rule power2_eq_square)
4.600 +lemma hyperpow_one [simp]: "\<And>r. (r::'a::monoid_mult star) pow (1::hypnat) = r"
4.601 +  by transfer (rule power_one_right)
4.602
4.603 -lemma hyperpow_gt_zero:
4.604 -  "\<And>r n. (0::'a::{linordered_semidom} star) < r \<Longrightarrow> 0 < r pow n"
4.605 -by transfer (rule zero_less_power)
4.606 +lemma hyperpow_two: "\<And>r. (r::'a::monoid_mult star) pow (2::hypnat) = r * r"
4.607 +  by transfer (rule power2_eq_square)
4.608
4.609 -lemma hyperpow_ge_zero:
4.610 -  "\<And>r n. (0::'a::{linordered_semidom} star) \<le> r \<Longrightarrow> 0 \<le> r pow n"
4.611 -by transfer (rule zero_le_power)
4.612 +lemma hyperpow_gt_zero: "\<And>r n. (0::'a::{linordered_semidom} star) < r \<Longrightarrow> 0 < r pow n"
4.613 +  by transfer (rule zero_less_power)
4.614 +
4.615 +lemma hyperpow_ge_zero: "\<And>r n. (0::'a::{linordered_semidom} star) \<le> r \<Longrightarrow> 0 \<le> r pow n"
4.616 +  by transfer (rule zero_le_power)
4.617
4.618 -lemma hyperpow_le:
4.619 -  "\<And>x y n. \<lbrakk>(0::'a::{linordered_semidom} star) < x; x \<le> y\<rbrakk>
4.620 -   \<Longrightarrow> x pow n \<le> y pow n"
4.621 -by transfer (rule power_mono [OF _ order_less_imp_le])
4.622 +lemma hyperpow_le: "\<And>x y n. (0::'a::{linordered_semidom} star) < x \<Longrightarrow> x \<le> y \<Longrightarrow> x pow n \<le> y pow n"
4.623 +  by transfer (rule power_mono [OF _ order_less_imp_le])
4.624
4.625 -lemma hyperpow_eq_one [simp]:
4.626 -  "\<And>n. 1 pow n = (1::'a::monoid_mult star)"
4.627 -by transfer (rule power_one)
4.628 +lemma hyperpow_eq_one [simp]: "\<And>n. 1 pow n = (1::'a::monoid_mult star)"
4.629 +  by transfer (rule power_one)
4.630
4.631 -lemma hrabs_hyperpow_minus [simp]:
4.632 -  "\<And>(a::'a::{linordered_idom} star) n. \<bar>(-a) pow n\<bar> = \<bar>a pow n\<bar>"
4.633 -by transfer (rule abs_power_minus)
4.634 +lemma hrabs_hyperpow_minus [simp]: "\<And>(a::'a::linordered_idom star) n. \<bar>(-a) pow n\<bar> = \<bar>a pow n\<bar>"
4.635 +  by transfer (rule abs_power_minus)
4.636
4.637 -lemma hyperpow_mult:
4.638 -  "\<And>r s n. (r * s::'a::{comm_monoid_mult} star) pow n
4.639 -   = (r pow n) * (s pow n)"
4.640 -by transfer (rule power_mult_distrib)
4.641 +lemma hyperpow_mult: "\<And>r s n. (r * s::'a::comm_monoid_mult star) pow n = (r pow n) * (s pow n)"
4.642 +  by transfer (rule power_mult_distrib)
4.643
4.644 -lemma hyperpow_two_le [simp]:
4.645 -  "\<And>r. (0::'a::{monoid_mult,linordered_ring_strict} star) \<le> r pow 2"
4.646 -by (auto simp add: hyperpow_two zero_le_mult_iff)
4.647 +lemma hyperpow_two_le [simp]: "\<And>r. (0::'a::{monoid_mult,linordered_ring_strict} star) \<le> r pow 2"
4.648 +  by (auto simp add: hyperpow_two zero_le_mult_iff)
4.649
4.650  lemma hrabs_hyperpow_two [simp]:
4.651 -  "\<bar>x pow 2\<bar> =
4.652 -   (x::'a::{monoid_mult,linordered_ring_strict} star) pow 2"
4.653 -by (simp only: abs_of_nonneg hyperpow_two_le)
4.654 +  "\<bar>x pow 2\<bar> = (x::'a::{monoid_mult,linordered_ring_strict} star) pow 2"
4.655 +  by (simp only: abs_of_nonneg hyperpow_two_le)
4.656
4.657 -lemma hyperpow_two_hrabs [simp]:
4.658 -  "\<bar>x::'a::{linordered_idom} star\<bar> pow 2 = x pow 2"
4.660 +lemma hyperpow_two_hrabs [simp]: "\<bar>x::'a::linordered_idom star\<bar> pow 2 = x pow 2"
4.661 +  by (simp add: hyperpow_hrabs)
4.662
4.663 -text\<open>The precondition could be weakened to @{term "0\<le>x"}\<close>
4.664 -lemma hypreal_mult_less_mono:
4.665 -     "[| u<v;  x<y;  (0::hypreal) < v;  0 < x |] ==> u*x < v* y"
4.666 - by (simp add: mult_strict_mono order_less_imp_le)
4.667 +text \<open>The precondition could be weakened to @{term "0\<le>x"}.\<close>
4.668 +lemma hypreal_mult_less_mono: "u < v \<Longrightarrow> x < y \<Longrightarrow> 0 < v \<Longrightarrow> 0 < x \<Longrightarrow> u * x < v * y"
4.669 +  for u v x y :: hypreal
4.670 +  by (simp add: mult_strict_mono order_less_imp_le)
4.671
4.672 -lemma hyperpow_two_gt_one:
4.673 -  "\<And>r::'a::{linordered_semidom} star. 1 < r \<Longrightarrow> 1 < r pow 2"
4.674 -by transfer simp
4.675 +lemma hyperpow_two_gt_one: "\<And>r::'a::linordered_semidom star. 1 < r \<Longrightarrow> 1 < r pow 2"
4.676 +  by transfer simp
4.677
4.678 -lemma hyperpow_two_ge_one:
4.679 -  "\<And>r::'a::{linordered_semidom} star. 1 \<le> r \<Longrightarrow> 1 \<le> r pow 2"
4.680 -by transfer (rule one_le_power)
4.681 +lemma hyperpow_two_ge_one: "\<And>r::'a::linordered_semidom star. 1 \<le> r \<Longrightarrow> 1 \<le> r pow 2"
4.682 +  by transfer (rule one_le_power)
4.683
4.684  lemma two_hyperpow_ge_one [simp]: "(1::hypreal) \<le> 2 pow n"
4.685 -apply (rule_tac y = "1 pow n" in order_trans)
4.686 -apply (rule_tac [2] hyperpow_le, auto)
4.687 -done
4.688 +  apply (rule_tac y = "1 pow n" in order_trans)
4.689 +   apply (rule_tac [2] hyperpow_le)
4.690 +    apply auto
4.691 +  done
4.692
4.693 -lemma hyperpow_minus_one2 [simp]:
4.694 -     "\<And>n. (- 1) pow (2*n) = (1::hypreal)"
4.695 -by transfer (rule power_minus1_even)
4.696 +lemma hyperpow_minus_one2 [simp]: "\<And>n. (- 1) pow (2 * n) = (1::hypreal)"
4.697 +  by transfer (rule power_minus1_even)
4.698
4.699 -lemma hyperpow_less_le:
4.700 -     "!!r n N. [|(0::hypreal) \<le> r; r \<le> 1; n < N|] ==> r pow N \<le> r pow n"
4.701 -by transfer (rule power_decreasing [OF order_less_imp_le])
4.702 +lemma hyperpow_less_le: "\<And>r n N. (0::hypreal) \<le> r \<Longrightarrow> r \<le> 1 \<Longrightarrow> n < N \<Longrightarrow> r pow N \<le> r pow n"
4.703 +  by transfer (rule power_decreasing [OF order_less_imp_le])
4.704
4.705  lemma hyperpow_SHNat_le:
4.706 -     "[| 0 \<le> r;  r \<le> (1::hypreal);  N \<in> HNatInfinite |]
4.707 -      ==> ALL n: Nats. r pow N \<le> r pow n"
4.708 -by (auto intro!: hyperpow_less_le simp add: HNatInfinite_iff)
4.709 +  "0 \<le> r \<Longrightarrow> r \<le> (1::hypreal) \<Longrightarrow> N \<in> HNatInfinite \<Longrightarrow> \<forall>n\<in>Nats. r pow N \<le> r pow n"
4.710 +  by (auto intro!: hyperpow_less_le simp: HNatInfinite_iff)
4.711
4.712 -lemma hyperpow_realpow:
4.713 -      "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)"
4.714 -by transfer (rule refl)
4.715 +lemma hyperpow_realpow: "(hypreal_of_real r) pow (hypnat_of_nat n) = hypreal_of_real (r ^ n)"
4.716 +  by transfer (rule refl)
4.717
4.718 -lemma hyperpow_SReal [simp]:
4.719 -     "(hypreal_of_real r) pow (hypnat_of_nat n) \<in> \<real>"
4.721 +lemma hyperpow_SReal [simp]: "(hypreal_of_real r) pow (hypnat_of_nat n) \<in> \<real>"
4.722 +  by (simp add: Reals_eq_Standard)
4.723
4.724 -lemma hyperpow_zero_HNatInfinite [simp]:
4.725 -     "N \<in> HNatInfinite ==> (0::hypreal) pow N = 0"
4.726 -by (drule HNatInfinite_is_Suc, auto)
4.727 +lemma hyperpow_zero_HNatInfinite [simp]: "N \<in> HNatInfinite \<Longrightarrow> (0::hypreal) pow N = 0"
4.728 +  by (drule HNatInfinite_is_Suc, auto)
4.729
4.730 -lemma hyperpow_le_le:
4.731 -     "[| (0::hypreal) \<le> r; r \<le> 1; n \<le> N |] ==> r pow N \<le> r pow n"
4.732 -apply (drule order_le_less [of n, THEN iffD1])
4.733 -apply (auto intro: hyperpow_less_le)
4.734 -done
4.735 +lemma hyperpow_le_le: "(0::hypreal) \<le> r \<Longrightarrow> r \<le> 1 \<Longrightarrow> n \<le> N \<Longrightarrow> r pow N \<le> r pow n"
4.736 +  apply (drule order_le_less [of n, THEN iffD1])
4.737 +  apply (auto intro: hyperpow_less_le)
4.738 +  done
4.739
4.740 -lemma hyperpow_Suc_le_self2:
4.741 -     "[| (0::hypreal) \<le> r; r < 1 |] ==> r pow (n + (1::hypnat)) \<le> r"
4.742 -apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le)
4.743 -apply auto
4.744 -done
4.745 +lemma hyperpow_Suc_le_self2: "(0::hypreal) \<le> r \<Longrightarrow> r < 1 \<Longrightarrow> r pow (n + (1::hypnat)) \<le> r"
4.746 +  apply (drule_tac n = " (1::hypnat) " in hyperpow_le_le)
4.747 +    apply auto
4.748 +  done
4.749
4.750  lemma hyperpow_hypnat_of_nat: "\<And>x. x pow hypnat_of_nat n = x ^ n"
4.751 -by transfer (rule refl)
4.752 +  by transfer (rule refl)
4.753
4.754  lemma of_hypreal_hyperpow:
4.755 -  "\<And>x n. of_hypreal (x pow n) =
4.756 -   (of_hypreal x::'a::{real_algebra_1} star) pow n"
4.757 -by transfer (rule of_real_power)
4.758 +  "\<And>x n. of_hypreal (x pow n) = (of_hypreal x::'a::{real_algebra_1} star) pow n"
4.759 +  by transfer (rule of_real_power)
4.760
4.761  end
```
```     5.1 --- a/src/HOL/Nonstandard_Analysis/HyperNat.thy	Mon Oct 31 16:26:36 2016 +0100
5.2 +++ b/src/HOL/Nonstandard_Analysis/HyperNat.thy	Tue Nov 01 00:44:24 2016 +0100
5.3 @@ -5,410 +5,395 @@
5.4  Converted to Isar and polished by lcp
5.5  *)
5.6
5.7 -section\<open>Hypernatural numbers\<close>
5.8 +section \<open>Hypernatural numbers\<close>
5.9
5.10  theory HyperNat
5.11 -imports StarDef
5.12 +  imports StarDef
5.13  begin
5.14
5.15  type_synonym hypnat = "nat star"
5.16
5.17 -abbreviation
5.18 -  hypnat_of_nat :: "nat => nat star" where
5.19 -  "hypnat_of_nat == star_of"
5.20 +abbreviation hypnat_of_nat :: "nat \<Rightarrow> nat star"
5.21 +  where "hypnat_of_nat \<equiv> star_of"
5.22
5.23 -definition
5.24 -  hSuc :: "hypnat => hypnat" where
5.25 -  hSuc_def [transfer_unfold]: "hSuc = *f* Suc"
5.26 +definition hSuc :: "hypnat \<Rightarrow> hypnat"
5.27 +  where hSuc_def [transfer_unfold]: "hSuc = *f* Suc"
5.28
5.29 -subsection\<open>Properties Transferred from Naturals\<close>
5.30 +
5.31 +subsection \<open>Properties Transferred from Naturals\<close>
5.32
5.33  lemma hSuc_not_zero [iff]: "\<And>m. hSuc m \<noteq> 0"
5.34 -by transfer (rule Suc_not_Zero)
5.35 +  by transfer (rule Suc_not_Zero)
5.36
5.37  lemma zero_not_hSuc [iff]: "\<And>m. 0 \<noteq> hSuc m"
5.38 -by transfer (rule Zero_not_Suc)
5.39 +  by transfer (rule Zero_not_Suc)
5.40
5.41 -lemma hSuc_hSuc_eq [iff]: "\<And>m n. (hSuc m = hSuc n) = (m = n)"
5.42 -by transfer (rule nat.inject)
5.43 +lemma hSuc_hSuc_eq [iff]: "\<And>m n. hSuc m = hSuc n \<longleftrightarrow> m = n"
5.44 +  by transfer (rule nat.inject)
5.45
5.46  lemma zero_less_hSuc [iff]: "\<And>n. 0 < hSuc n"
5.47 -by transfer (rule zero_less_Suc)
5.48 +  by transfer (rule zero_less_Suc)
5.49
5.50 -lemma hypnat_minus_zero [simp]: "!!z. z - z = (0::hypnat)"
5.51 -by transfer (rule diff_self_eq_0)
5.52 +lemma hypnat_minus_zero [simp]: "\<And>z::hypnat. z - z = 0"
5.53 +  by transfer (rule diff_self_eq_0)
5.54
5.55 -lemma hypnat_diff_0_eq_0 [simp]: "!!n. (0::hypnat) - n = 0"
5.56 -by transfer (rule diff_0_eq_0)
5.57 +lemma hypnat_diff_0_eq_0 [simp]: "\<And>n::hypnat. 0 - n = 0"
5.58 +  by transfer (rule diff_0_eq_0)
5.59
5.60 -lemma hypnat_add_is_0 [iff]: "!!m n. (m+n = (0::hypnat)) = (m=0 & n=0)"
5.62 +lemma hypnat_add_is_0 [iff]: "\<And>m n::hypnat. m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
5.63 +  by transfer (rule add_is_0)
5.64
5.65 -lemma hypnat_diff_diff_left: "!!i j k. (i::hypnat) - j - k = i - (j+k)"
5.66 -by transfer (rule diff_diff_left)
5.67 +lemma hypnat_diff_diff_left: "\<And>i j k::hypnat. i - j - k = i - (j + k)"
5.68 +  by transfer (rule diff_diff_left)
5.69
5.70 -lemma hypnat_diff_commute: "!!i j k. (i::hypnat) - j - k = i-k-j"
5.71 -by transfer (rule diff_commute)
5.72 +lemma hypnat_diff_commute: "\<And>i j k::hypnat. i - j - k = i - k - j"
5.73 +  by transfer (rule diff_commute)
5.74
5.75 -lemma hypnat_diff_add_inverse [simp]: "!!m n. ((n::hypnat) + m) - n = m"
5.77 +lemma hypnat_diff_add_inverse [simp]: "\<And>m n::hypnat. n + m - n = m"
5.78 +  by transfer (rule diff_add_inverse)
5.79
5.80 -lemma hypnat_diff_add_inverse2 [simp]:  "!!m n. ((m::hypnat) + n) - n = m"
5.82 +lemma hypnat_diff_add_inverse2 [simp]:  "\<And>m n::hypnat. m + n - n = m"
5.83 +  by transfer (rule diff_add_inverse2)
5.84
5.85 -lemma hypnat_diff_cancel [simp]: "!!k m n. ((k::hypnat) + m) - (k+n) = m - n"
5.86 -by transfer (rule diff_cancel)
5.87 +lemma hypnat_diff_cancel [simp]: "\<And>k m n::hypnat. (k + m) - (k + n) = m - n"
5.88 +  by transfer (rule diff_cancel)
5.89
5.90 -lemma hypnat_diff_cancel2 [simp]: "!!k m n. ((m::hypnat) + k) - (n+k) = m - n"
5.91 -by transfer (rule diff_cancel2)
5.92 +lemma hypnat_diff_cancel2 [simp]: "\<And>k m n::hypnat. (m + k) - (n + k) = m - n"
5.93 +  by transfer (rule diff_cancel2)
5.94
5.95 -lemma hypnat_diff_add_0 [simp]: "!!m n. (n::hypnat) - (n+m) = (0::hypnat)"
5.97 +lemma hypnat_diff_add_0 [simp]: "\<And>m n::hypnat. n - (n + m) = 0"
5.98 +  by transfer (rule diff_add_0)
5.99
5.100 -lemma hypnat_diff_mult_distrib: "!!k m n. ((m::hypnat) - n) * k = (m * k) - (n * k)"
5.101 -by transfer (rule diff_mult_distrib)
5.102 +lemma hypnat_diff_mult_distrib: "\<And>k m n::hypnat. (m - n) * k = (m * k) - (n * k)"
5.103 +  by transfer (rule diff_mult_distrib)
5.104
5.105 -lemma hypnat_diff_mult_distrib2: "!!k m n. (k::hypnat) * (m - n) = (k * m) - (k * n)"
5.106 -by transfer (rule diff_mult_distrib2)
5.107 +lemma hypnat_diff_mult_distrib2: "\<And>k m n::hypnat. k * (m - n) = (k * m) - (k * n)"
5.108 +  by transfer (rule diff_mult_distrib2)
5.109
5.110 -lemma hypnat_le_zero_cancel [iff]: "!!n. (n \<le> (0::hypnat)) = (n = 0)"
5.111 -by transfer (rule le_0_eq)
5.112 +lemma hypnat_le_zero_cancel [iff]: "\<And>n::hypnat. n \<le> 0 \<longleftrightarrow> n = 0"
5.113 +  by transfer (rule le_0_eq)
5.114
5.115 -lemma hypnat_mult_is_0 [simp]: "!!m n. (m*n = (0::hypnat)) = (m=0 | n=0)"
5.116 -by transfer (rule mult_is_0)
5.117 +lemma hypnat_mult_is_0 [simp]: "\<And>m n::hypnat. m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0"
5.118 +  by transfer (rule mult_is_0)
5.119
5.120 -lemma hypnat_diff_is_0_eq [simp]: "!!m n. ((m::hypnat) - n = 0) = (m \<le> n)"
5.121 -by transfer (rule diff_is_0_eq)
5.122 +lemma hypnat_diff_is_0_eq [simp]: "\<And>m n::hypnat. m - n = 0 \<longleftrightarrow> m \<le> n"
5.123 +  by transfer (rule diff_is_0_eq)
5.124
5.125 -lemma hypnat_not_less0 [iff]: "!!n. ~ n < (0::hypnat)"
5.126 -by transfer (rule not_less0)
5.127 +lemma hypnat_not_less0 [iff]: "\<And>n::hypnat. \<not> n < 0"
5.128 +  by transfer (rule not_less0)
5.129 +
5.130 +lemma hypnat_less_one [iff]: "\<And>n::hypnat. n < 1 \<longleftrightarrow> n = 0"
5.131 +  by transfer (rule less_one)
5.132
5.133 -lemma hypnat_less_one [iff]:
5.134 -      "!!n. (n < (1::hypnat)) = (n=0)"
5.135 -by transfer (rule less_one)
5.136 +lemma hypnat_add_diff_inverse: "\<And>m n::hypnat. \<not> m < n \<Longrightarrow> n + (m - n) = m"
5.137 +  by transfer (rule add_diff_inverse)
5.138
5.139 -lemma hypnat_add_diff_inverse: "!!m n. ~ m<n ==> n+(m-n) = (m::hypnat)"
5.141 +lemma hypnat_le_add_diff_inverse [simp]: "\<And>m n::hypnat. n \<le> m \<Longrightarrow> n + (m - n) = m"
5.142 +  by transfer (rule le_add_diff_inverse)
5.143
5.144 -lemma hypnat_le_add_diff_inverse [simp]: "!!m n. n \<le> m ==> n+(m-n) = (m::hypnat)"
5.146 -
5.147 -lemma hypnat_le_add_diff_inverse2 [simp]: "!!m n. n\<le>m ==> (m-n)+n = (m::hypnat)"
5.149 +lemma hypnat_le_add_diff_inverse2 [simp]: "\<And>m n::hypnat. n \<le> m \<Longrightarrow> (m - n) + n = m"
5.150 +  by transfer (rule le_add_diff_inverse2)
5.151
5.153
5.154 -lemma hypnat_le0 [iff]: "!!n. (0::hypnat) \<le> n"
5.155 -by transfer (rule le0)
5.156 +lemma hypnat_le0 [iff]: "\<And>n::hypnat. 0 \<le> n"
5.157 +  by transfer (rule le0)
5.158
5.159 -lemma hypnat_le_add1 [simp]: "!!x n. (x::hypnat) \<le> x + n"
5.161 +lemma hypnat_le_add1 [simp]: "\<And>x n::hypnat. x \<le> x + n"
5.162 +  by transfer (rule le_add1)
5.163
5.164 -lemma hypnat_add_self_le [simp]: "!!x n. (x::hypnat) \<le> n + x"
5.166 +lemma hypnat_add_self_le [simp]: "\<And>x n::hypnat. x \<le> n + x"
5.167 +  by transfer (rule le_add2)
5.168
5.169 -lemma hypnat_add_one_self_less [simp]: "(x::hypnat) < x + (1::hypnat)"
5.170 +lemma hypnat_add_one_self_less [simp]: "x < x + 1" for x :: hypnat
5.172
5.173 -lemma hypnat_neq0_conv [iff]: "!!n. (n \<noteq> 0) = (0 < (n::hypnat))"
5.174 -by transfer (rule neq0_conv)
5.175 +lemma hypnat_neq0_conv [iff]: "\<And>n::hypnat. n \<noteq> 0 \<longleftrightarrow> 0 < n"
5.176 +  by transfer (rule neq0_conv)
5.177
5.178 -lemma hypnat_gt_zero_iff: "((0::hypnat) < n) = ((1::hypnat) \<le> n)"
5.179 -by (auto simp add: linorder_not_less [symmetric])
5.180 +lemma hypnat_gt_zero_iff: "0 < n \<longleftrightarrow> 1 \<le> n" for n :: hypnat
5.181 +  by (auto simp add: linorder_not_less [symmetric])
5.182
5.183 -lemma hypnat_gt_zero_iff2: "(0 < n) = (\<exists>m. n = m + (1::hypnat))"
5.184 +lemma hypnat_gt_zero_iff2: "0 < n \<longleftrightarrow> (\<exists>m. n = m + 1)" for n :: hypnat
5.185    by (auto intro!: add_nonneg_pos exI[of _ "n - 1"] simp: hypnat_gt_zero_iff)
5.186
5.187 -lemma hypnat_add_self_not_less: "~ (x + y < (x::hypnat))"
5.189 +lemma hypnat_add_self_not_less: "\<not> x + y < x" for x y :: hypnat
5.191
5.192 -lemma hypnat_diff_split:
5.193 -    "P(a - b::hypnat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
5.194 -    \<comment> \<open>elimination of \<open>-\<close> on \<open>hypnat\<close>\<close>
5.195 -proof (cases "a<b" rule: case_split)
5.196 +lemma hypnat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
5.197 +  for a b :: hypnat
5.198 +  \<comment> \<open>elimination of \<open>-\<close> on \<open>hypnat\<close>\<close>
5.199 +proof (cases "a < b" rule: case_split)
5.200    case True
5.201 -    thus ?thesis
5.203 -                         hypnat_diff_is_0_eq [THEN iffD2])
5.204 +  then show ?thesis
5.206  next
5.207    case False
5.208 -    thus ?thesis
5.209 -      by (auto simp add: linorder_not_less dest: order_le_less_trans)
5.210 +  then show ?thesis
5.211 +    by (auto simp add: linorder_not_less dest: order_le_less_trans)
5.212  qed
5.213
5.214 -subsection\<open>Properties of the set of embedded natural numbers\<close>
5.215 +
5.216 +subsection \<open>Properties of the set of embedded natural numbers\<close>
5.217
5.218  lemma of_nat_eq_star_of [simp]: "of_nat = star_of"
5.219  proof
5.220 -  fix n :: nat
5.221 -  show "of_nat n = star_of n" by transfer simp
5.222 +  show "of_nat n = star_of n" for n
5.223 +    by transfer simp
5.224  qed
5.225
5.226  lemma Nats_eq_Standard: "(Nats :: nat star set) = Standard"
5.227 -by (auto simp add: Nats_def Standard_def)
5.228 +  by (auto simp: Nats_def Standard_def)
5.229
5.230  lemma hypnat_of_nat_mem_Nats [simp]: "hypnat_of_nat n \<in> Nats"
5.232 +  by (simp add: Nats_eq_Standard)
5.233
5.234 -lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = (1::hypnat)"
5.235 -by transfer simp
5.236 +lemma hypnat_of_nat_one [simp]: "hypnat_of_nat (Suc 0) = 1"
5.237 +  by transfer simp
5.238
5.239 -lemma hypnat_of_nat_Suc [simp]:
5.240 -     "hypnat_of_nat (Suc n) = hypnat_of_nat n + (1::hypnat)"
5.241 -by transfer simp
5.242 +lemma hypnat_of_nat_Suc [simp]: "hypnat_of_nat (Suc n) = hypnat_of_nat n + 1"
5.243 +  by transfer simp
5.244
5.246 -     "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"
5.247 -apply (induct n)
5.249 -apply (case_tac x)
5.251 -done
5.252 +lemma of_nat_eq_add [rule_format]: "\<forall>d::hypnat. of_nat m = of_nat n + d --> d \<in> range of_nat"
5.253 +  apply (induct n)
5.255 +  apply (case_tac x)
5.257 +  done
5.258
5.259 -lemma Nats_diff [simp]: "[|a \<in> Nats; b \<in> Nats|] ==> (a-b :: hypnat) \<in> Nats"
5.261 +lemma Nats_diff [simp]: "a \<in> Nats \<Longrightarrow> b \<in> Nats \<Longrightarrow> a - b \<in> Nats" for a b :: hypnat
5.262 +  by (simp add: Nats_eq_Standard)
5.263
5.264
5.265 -subsection\<open>Infinite Hypernatural Numbers -- @{term HNatInfinite}\<close>
5.266 +subsection \<open>Infinite Hypernatural Numbers -- @{term HNatInfinite}\<close>
5.267 +
5.268 +text \<open>The set of infinite hypernatural numbers.\<close>
5.269 +definition HNatInfinite :: "hypnat set"
5.270 +  where "HNatInfinite = {n. n \<notin> Nats}"
5.271
5.272 -definition
5.273 -  (* the set of infinite hypernatural numbers *)
5.274 -  HNatInfinite :: "hypnat set" where
5.275 -  "HNatInfinite = {n. n \<notin> Nats}"
5.276 +lemma Nats_not_HNatInfinite_iff: "x \<in> Nats \<longleftrightarrow> x \<notin> HNatInfinite"
5.277 +  by (simp add: HNatInfinite_def)
5.278
5.279 -lemma Nats_not_HNatInfinite_iff: "(x \<in> Nats) = (x \<notin> HNatInfinite)"
5.281 -
5.282 -lemma HNatInfinite_not_Nats_iff: "(x \<in> HNatInfinite) = (x \<notin> Nats)"
5.284 +lemma HNatInfinite_not_Nats_iff: "x \<in> HNatInfinite \<longleftrightarrow> x \<notin> Nats"
5.285 +  by (simp add: HNatInfinite_def)
5.286
5.287  lemma star_of_neq_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<noteq> N"
5.288 -by (auto simp add: HNatInfinite_def Nats_eq_Standard)
5.289 +  by (auto simp add: HNatInfinite_def Nats_eq_Standard)
5.290
5.291 -lemma star_of_Suc_lessI:
5.292 -  "\<And>N. \<lbrakk>star_of n < N; star_of (Suc n) \<noteq> N\<rbrakk> \<Longrightarrow> star_of (Suc n) < N"
5.293 -by transfer (rule Suc_lessI)
5.294 +lemma star_of_Suc_lessI: "\<And>N. star_of n < N \<Longrightarrow> star_of (Suc n) \<noteq> N \<Longrightarrow> star_of (Suc n) < N"
5.295 +  by transfer (rule Suc_lessI)
5.296
5.297  lemma star_of_less_HNatInfinite:
5.298    assumes N: "N \<in> HNatInfinite"
5.299    shows "star_of n < N"
5.300  proof (induct n)
5.301    case 0
5.302 -  from N have "star_of 0 \<noteq> N" by (rule star_of_neq_HNatInfinite)
5.303 -  thus "star_of 0 < N" by simp
5.304 +  from N have "star_of 0 \<noteq> N"
5.305 +    by (rule star_of_neq_HNatInfinite)
5.306 +  then show ?case by simp
5.307  next
5.308    case (Suc n)
5.309 -  from N have "star_of (Suc n) \<noteq> N" by (rule star_of_neq_HNatInfinite)
5.310 -  with Suc show "star_of (Suc n) < N" by (rule star_of_Suc_lessI)
5.311 +  from N have "star_of (Suc n) \<noteq> N"
5.312 +    by (rule star_of_neq_HNatInfinite)
5.313 +  with Suc show ?case
5.314 +    by (rule star_of_Suc_lessI)
5.315  qed
5.316
5.317  lemma star_of_le_HNatInfinite: "N \<in> HNatInfinite \<Longrightarrow> star_of n \<le> N"
5.318 -by (rule star_of_less_HNatInfinite [THEN order_less_imp_le])
5.319 +  by (rule star_of_less_HNatInfinite [THEN order_less_imp_le])
5.320 +
5.321
5.322  subsubsection \<open>Closure Rules\<close>
5.323
5.324 -lemma Nats_less_HNatInfinite: "\<lbrakk>x \<in> Nats; y \<in> HNatInfinite\<rbrakk> \<Longrightarrow> x < y"
5.325 -by (auto simp add: Nats_def star_of_less_HNatInfinite)
5.326 +lemma Nats_less_HNatInfinite: "x \<in> Nats \<Longrightarrow> y \<in> HNatInfinite \<Longrightarrow> x < y"
5.327 +  by (auto simp add: Nats_def star_of_less_HNatInfinite)
5.328
5.329 -lemma Nats_le_HNatInfinite: "\<lbrakk>x \<in> Nats; y \<in> HNatInfinite\<rbrakk> \<Longrightarrow> x \<le> y"
5.330 -by (rule Nats_less_HNatInfinite [THEN order_less_imp_le])
5.331 +lemma Nats_le_HNatInfinite: "x \<in> Nats \<Longrightarrow> y \<in> HNatInfinite \<Longrightarrow> x \<le> y"
5.332 +  by (rule Nats_less_HNatInfinite [THEN order_less_imp_le])
5.333
5.334  lemma zero_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 0 < x"
5.336 +  by (simp add: Nats_less_HNatInfinite)
5.337
5.338  lemma one_less_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 < x"
5.340 +  by (simp add: Nats_less_HNatInfinite)
5.341
5.342  lemma one_le_HNatInfinite: "x \<in> HNatInfinite \<Longrightarrow> 1 \<le> x"
5.344 +  by (simp add: Nats_le_HNatInfinite)
5.345
5.346  lemma zero_not_mem_HNatInfinite [simp]: "0 \<notin> HNatInfinite"
5.348 +  by (simp add: HNatInfinite_def)
5.349
5.350 -lemma Nats_downward_closed:
5.351 -  "\<lbrakk>x \<in> Nats; (y::hypnat) \<le> x\<rbrakk> \<Longrightarrow> y \<in> Nats"
5.352 -apply (simp only: linorder_not_less [symmetric])
5.353 -apply (erule contrapos_np)
5.354 -apply (drule HNatInfinite_not_Nats_iff [THEN iffD2])
5.355 -apply (erule (1) Nats_less_HNatInfinite)
5.356 -done
5.357 +lemma Nats_downward_closed: "x \<in> Nats \<Longrightarrow> y \<le> x \<Longrightarrow> y \<in> Nats" for x y :: hypnat
5.358 +  apply (simp only: linorder_not_less [symmetric])
5.359 +  apply (erule contrapos_np)
5.360 +  apply (drule HNatInfinite_not_Nats_iff [THEN iffD2])
5.361 +  apply (erule (1) Nats_less_HNatInfinite)
5.362 +  done
5.363
5.364 -lemma HNatInfinite_upward_closed:
5.365 -  "\<lbrakk>x \<in> HNatInfinite; x \<le> y\<rbrakk> \<Longrightarrow> y \<in> HNatInfinite"
5.366 -apply (simp only: HNatInfinite_not_Nats_iff)
5.367 -apply (erule contrapos_nn)
5.368 -apply (erule (1) Nats_downward_closed)
5.369 -done
5.370 +lemma HNatInfinite_upward_closed: "x \<in> HNatInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> y \<in> HNatInfinite"
5.371 +  apply (simp only: HNatInfinite_not_Nats_iff)
5.372 +  apply (erule contrapos_nn)
5.373 +  apply (erule (1) Nats_downward_closed)
5.374 +  done
5.375
5.376  lemma HNatInfinite_add: "x \<in> HNatInfinite \<Longrightarrow> x + y \<in> HNatInfinite"
5.377 -apply (erule HNatInfinite_upward_closed)
5.379 -done
5.380 +  apply (erule HNatInfinite_upward_closed)
5.382 +  done
5.383
5.384  lemma HNatInfinite_add_one: "x \<in> HNatInfinite \<Longrightarrow> x + 1 \<in> HNatInfinite"
5.387
5.388 -lemma HNatInfinite_diff:
5.389 -  "\<lbrakk>x \<in> HNatInfinite; y \<in> Nats\<rbrakk> \<Longrightarrow> x - y \<in> HNatInfinite"
5.390 -apply (frule (1) Nats_le_HNatInfinite)
5.391 -apply (simp only: HNatInfinite_not_Nats_iff)
5.392 -apply (erule contrapos_nn)
5.393 -apply (drule (1) Nats_add, simp)
5.394 -done
5.395 +lemma HNatInfinite_diff: "x \<in> HNatInfinite \<Longrightarrow> y \<in> Nats \<Longrightarrow> x - y \<in> HNatInfinite"
5.396 +  apply (frule (1) Nats_le_HNatInfinite)
5.397 +  apply (simp only: HNatInfinite_not_Nats_iff)
5.398 +  apply (erule contrapos_nn)
5.399 +  apply (drule (1) Nats_add, simp)
5.400 +  done
5.401
5.402 -lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite ==> \<exists>y. x = y + (1::hypnat)"
5.403 -apply (rule_tac x = "x - (1::hypnat) " in exI)
5.405 -done
5.406 +lemma HNatInfinite_is_Suc: "x \<in> HNatInfinite \<Longrightarrow> \<exists>y. x = y + 1" for x :: hypnat
5.407 +  apply (rule_tac x = "x - (1::hypnat) " in exI)
5.408 +  apply (simp add: Nats_le_HNatInfinite)
5.409 +  done
5.410
5.411
5.412 -subsection\<open>Existence of an infinite hypernatural number\<close>
5.413 +subsection \<open>Existence of an infinite hypernatural number\<close>
5.414
5.415 -definition
5.416 -  (* \<omega> is in fact an infinite hypernatural number = [<1,2,3,...>] *)
5.417 -  whn :: hypnat where
5.418 -  hypnat_omega_def: "whn = star_n (%n::nat. n)"
5.419 +text \<open>\<open>\<omega>\<close> is in fact an infinite hypernatural number = \<open>[<1, 2, 3, \<dots>>]\<close>\<close>
5.420 +definition whn :: hypnat
5.421 +  where hypnat_omega_def: "whn = star_n (\<lambda>n::nat. n)"
5.422
5.423  lemma hypnat_of_nat_neq_whn: "hypnat_of_nat n \<noteq> whn"
5.424 -by (simp add: FreeUltrafilterNat.singleton' hypnat_omega_def star_of_def star_n_eq_iff)
5.425 +  by (simp add: FreeUltrafilterNat.singleton' hypnat_omega_def star_of_def star_n_eq_iff)
5.426
5.427  lemma whn_neq_hypnat_of_nat: "whn \<noteq> hypnat_of_nat n"
5.428 -by (simp add: FreeUltrafilterNat.singleton hypnat_omega_def star_of_def star_n_eq_iff)
5.429 +  by (simp add: FreeUltrafilterNat.singleton hypnat_omega_def star_of_def star_n_eq_iff)
5.430
5.431  lemma whn_not_Nats [simp]: "whn \<notin> Nats"
5.432 -by (simp add: Nats_def image_def whn_neq_hypnat_of_nat)
5.433 +  by (simp add: Nats_def image_def whn_neq_hypnat_of_nat)
5.434
5.435  lemma HNatInfinite_whn [simp]: "whn \<in> HNatInfinite"
5.437 +  by (simp add: HNatInfinite_def)
5.438
5.439  lemma lemma_unbounded_set [simp]: "eventually (\<lambda>n::nat. m < n) \<U>"
5.440    by (rule filter_leD[OF FreeUltrafilterNat.le_cofinite])
5.441       (auto simp add: cofinite_eq_sequentially eventually_at_top_dense)
5.442
5.443 -lemma hypnat_of_nat_eq:
5.444 -     "hypnat_of_nat m  = star_n (%n::nat. m)"
5.446 +lemma hypnat_of_nat_eq: "hypnat_of_nat m  = star_n (\<lambda>n::nat. m)"
5.447 +  by (simp add: star_of_def)
5.448
5.449  lemma SHNat_eq: "Nats = {n. \<exists>N. n = hypnat_of_nat N}"
5.450 -by (simp add: Nats_def image_def)
5.451 +  by (simp add: Nats_def image_def)
5.452
5.453  lemma Nats_less_whn: "n \<in> Nats \<Longrightarrow> n < whn"
5.455 +  by (simp add: Nats_less_HNatInfinite)
5.456
5.457  lemma Nats_le_whn: "n \<in> Nats \<Longrightarrow> n \<le> whn"
5.459 +  by (simp add: Nats_le_HNatInfinite)
5.460
5.461  lemma hypnat_of_nat_less_whn [simp]: "hypnat_of_nat n < whn"
5.463 +  by (simp add: Nats_less_whn)
5.464
5.465  lemma hypnat_of_nat_le_whn [simp]: "hypnat_of_nat n \<le> whn"
5.467 +  by (simp add: Nats_le_whn)
5.468
5.469  lemma hypnat_zero_less_hypnat_omega [simp]: "0 < whn"
5.471 +  by (simp add: Nats_less_whn)
5.472
5.473  lemma hypnat_one_less_hypnat_omega [simp]: "1 < whn"
5.475 +  by (simp add: Nats_less_whn)
5.476
5.477
5.478 -subsubsection\<open>Alternative characterization of the set of infinite hypernaturals\<close>
5.479 +subsubsection \<open>Alternative characterization of the set of infinite hypernaturals\<close>
5.480
5.481 -text\<open>@{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}\<close>
5.482 +text \<open>@{term "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"}\<close>
5.483
5.484  (*??delete? similar reasoning in hypnat_omega_gt_SHNat above*)
5.485  lemma HNatInfinite_FreeUltrafilterNat_lemma:
5.486    assumes "\<forall>N::nat. eventually (\<lambda>n. f n \<noteq> N) \<U>"
5.487    shows "eventually (\<lambda>n. N < f n) \<U>"
5.488 -apply (induct N)
5.489 -using assms
5.490 -apply (drule_tac x = 0 in spec, simp)
5.491 -using assms
5.492 -apply (drule_tac x = "Suc N" in spec)
5.493 -apply (auto elim: eventually_elim2)
5.494 -done
5.495 +  apply (induct N)
5.496 +  using assms
5.497 +   apply (drule_tac x = 0 in spec, simp)
5.498 +  using assms
5.499 +  apply (drule_tac x = "Suc N" in spec)
5.500 +  apply (auto elim: eventually_elim2)
5.501 +  done
5.502
5.503  lemma HNatInfinite_iff: "HNatInfinite = {N. \<forall>n \<in> Nats. n < N}"
5.504 -apply (safe intro!: Nats_less_HNatInfinite)
5.505 -apply (auto simp add: HNatInfinite_def)
5.506 -done
5.507 +  apply (safe intro!: Nats_less_HNatInfinite)
5.508 +  apply (auto simp add: HNatInfinite_def)
5.509 +  done
5.510
5.511
5.512 -subsubsection\<open>Alternative Characterization of @{term HNatInfinite} using
5.513 -Free Ultrafilter\<close>
5.514 +subsubsection \<open>Alternative Characterization of @{term HNatInfinite} using Free Ultrafilter\<close>
5.515
5.516  lemma HNatInfinite_FreeUltrafilterNat:
5.517 -     "star_n X \<in> HNatInfinite ==> \<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat"
5.518 -apply (auto simp add: HNatInfinite_iff SHNat_eq)
5.519 -apply (drule_tac x="star_of u" in spec, simp)
5.520 -apply (simp add: star_of_def star_less_def starP2_star_n)
5.521 -done
5.522 +  "star_n X \<in> HNatInfinite \<Longrightarrow> \<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat"
5.523 +  apply (auto simp add: HNatInfinite_iff SHNat_eq)
5.524 +  apply (drule_tac x="star_of u" in spec, simp)
5.525 +  apply (simp add: star_of_def star_less_def starP2_star_n)
5.526 +  done
5.527
5.528  lemma FreeUltrafilterNat_HNatInfinite:
5.529 -     "\<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat ==> star_n X \<in> HNatInfinite"
5.530 -by (auto simp add: star_less_def starP2_star_n HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
5.531 +  "\<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat \<Longrightarrow> star_n X \<in> HNatInfinite"
5.532 +  by (auto simp add: star_less_def starP2_star_n HNatInfinite_iff SHNat_eq hypnat_of_nat_eq)
5.533
5.534  lemma HNatInfinite_FreeUltrafilterNat_iff:
5.535 -     "(star_n X \<in> HNatInfinite) = (\<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat)"
5.536 -by (rule iffI [OF HNatInfinite_FreeUltrafilterNat
5.537 -                 FreeUltrafilterNat_HNatInfinite])
5.538 +  "(star_n X \<in> HNatInfinite) = (\<forall>u. eventually (\<lambda>n. u < X n) FreeUltrafilterNat)"
5.539 +  by (rule iffI [OF HNatInfinite_FreeUltrafilterNat FreeUltrafilterNat_HNatInfinite])
5.540 +
5.541
5.542  subsection \<open>Embedding of the Hypernaturals into other types\<close>
5.543
5.544 -definition
5.545 -  of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star" where
5.546 -  of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat"
5.547 +definition of_hypnat :: "hypnat \<Rightarrow> 'a::semiring_1_cancel star"
5.548 +  where of_hypnat_def [transfer_unfold]: "of_hypnat = *f* of_nat"
5.549
5.550  lemma of_hypnat_0 [simp]: "of_hypnat 0 = 0"
5.551 -by transfer (rule of_nat_0)
5.552 +  by transfer (rule of_nat_0)
5.553
5.554  lemma of_hypnat_1 [simp]: "of_hypnat 1 = 1"
5.555 -by transfer (rule of_nat_1)
5.556 +  by transfer (rule of_nat_1)
5.557
5.558  lemma of_hypnat_hSuc: "\<And>m. of_hypnat (hSuc m) = 1 + of_hypnat m"
5.559 -by transfer (rule of_nat_Suc)
5.560 +  by transfer (rule of_nat_Suc)
5.561
5.563 -  "\<And>m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n"
5.565 +lemma of_hypnat_add [simp]: "\<And>m n. of_hypnat (m + n) = of_hypnat m + of_hypnat n"
5.566 +  by transfer (rule of_nat_add)
5.567
5.568 -lemma of_hypnat_mult [simp]:
5.569 -  "\<And>m n. of_hypnat (m * n) = of_hypnat m * of_hypnat n"
5.570 -by transfer (rule of_nat_mult)
5.571 +lemma of_hypnat_mult [simp]: "\<And>m n. of_hypnat (m * n) = of_hypnat m * of_hypnat n"
5.572 +  by transfer (rule of_nat_mult)
5.573
5.574  lemma of_hypnat_less_iff [simp]:
5.575 -  "\<And>m n. (of_hypnat m < (of_hypnat n::'a::linordered_semidom star)) = (m < n)"
5.576 -by transfer (rule of_nat_less_iff)
5.577 +  "\<And>m n. of_hypnat m < (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m < n"
5.578 +  by transfer (rule of_nat_less_iff)
5.579
5.580  lemma of_hypnat_0_less_iff [simp]:
5.581 -  "\<And>n. (0 < (of_hypnat n::'a::linordered_semidom star)) = (0 < n)"
5.582 -by transfer (rule of_nat_0_less_iff)
5.583 +  "\<And>n. 0 < (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> 0 < n"
5.584 +  by transfer (rule of_nat_0_less_iff)
5.585
5.586 -lemma of_hypnat_less_0_iff [simp]:
5.587 -  "\<And>m. \<not> (of_hypnat m::'a::linordered_semidom star) < 0"
5.588 -by transfer (rule of_nat_less_0_iff)
5.589 +lemma of_hypnat_less_0_iff [simp]: "\<And>m. \<not> (of_hypnat m::'a::linordered_semidom star) < 0"
5.590 +  by transfer (rule of_nat_less_0_iff)
5.591
5.592  lemma of_hypnat_le_iff [simp]:
5.593 -  "\<And>m n. (of_hypnat m \<le> (of_hypnat n::'a::linordered_semidom star)) = (m \<le> n)"
5.594 -by transfer (rule of_nat_le_iff)
5.595 +  "\<And>m n. of_hypnat m \<le> (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m \<le> n"
5.596 +  by transfer (rule of_nat_le_iff)
5.597
5.598 -lemma of_hypnat_0_le_iff [simp]:
5.599 -  "\<And>n. 0 \<le> (of_hypnat n::'a::linordered_semidom star)"
5.600 -by transfer (rule of_nat_0_le_iff)
5.601 +lemma of_hypnat_0_le_iff [simp]: "\<And>n. 0 \<le> (of_hypnat n::'a::linordered_semidom star)"
5.602 +  by transfer (rule of_nat_0_le_iff)
5.603
5.604 -lemma of_hypnat_le_0_iff [simp]:
5.605 -  "\<And>m. ((of_hypnat m::'a::linordered_semidom star) \<le> 0) = (m = 0)"
5.606 -by transfer (rule of_nat_le_0_iff)
5.607 +lemma of_hypnat_le_0_iff [simp]: "\<And>m. (of_hypnat m::'a::linordered_semidom star) \<le> 0 \<longleftrightarrow> m = 0"
5.608 +  by transfer (rule of_nat_le_0_iff)
5.609
5.610  lemma of_hypnat_eq_iff [simp]:
5.611 -  "\<And>m n. (of_hypnat m = (of_hypnat n::'a::linordered_semidom star)) = (m = n)"
5.612 -by transfer (rule of_nat_eq_iff)
5.613 +  "\<And>m n. of_hypnat m = (of_hypnat n::'a::linordered_semidom star) \<longleftrightarrow> m = n"
5.614 +  by transfer (rule of_nat_eq_iff)
5.615
5.616 -lemma of_hypnat_eq_0_iff [simp]:
5.617 -  "\<And>m. ((of_hypnat m::'a::linordered_semidom star) = 0) = (m = 0)"
5.618 -by transfer (rule of_nat_eq_0_iff)
5.619 +lemma of_hypnat_eq_0_iff [simp]: "\<And>m. (of_hypnat m::'a::linordered_semidom star) = 0 \<longleftrightarrow> m = 0"
5.620 +  by transfer (rule of_nat_eq_0_iff)
5.621
5.622  lemma HNatInfinite_of_hypnat_gt_zero:
5.623    "N \<in> HNatInfinite \<Longrightarrow> (0::'a::linordered_semidom star) < of_hypnat N"
5.624 -by (rule ccontr, simp add: linorder_not_less)
5.625 +  by (rule ccontr) (simp add: linorder_not_less)
5.626
5.627  end
```
```     6.1 --- a/src/HOL/Nonstandard_Analysis/NSA.thy	Mon Oct 31 16:26:36 2016 +0100
6.2 +++ b/src/HOL/Nonstandard_Analysis/NSA.thy	Tue Nov 01 00:44:24 2016 +0100
6.3 @@ -3,649 +3,584 @@
6.4      Author:     Lawrence C Paulson, University of Cambridge
6.5  *)
6.6
6.7 -section\<open>Infinite Numbers, Infinitesimals, Infinitely Close Relation\<close>
6.8 +section \<open>Infinite Numbers, Infinitesimals, Infinitely Close Relation\<close>
6.9
6.10  theory NSA
6.11 -imports HyperDef "~~/src/HOL/Library/Lub_Glb"
6.12 +  imports HyperDef "~~/src/HOL/Library/Lub_Glb"
6.13  begin
6.14
6.15 -definition
6.16 -  hnorm :: "'a::real_normed_vector star \<Rightarrow> real star" where
6.17 -  [transfer_unfold]: "hnorm = *f* norm"
6.18 +definition hnorm :: "'a::real_normed_vector star \<Rightarrow> real star"
6.19 +  where [transfer_unfold]: "hnorm = *f* norm"
6.20
6.21 -definition
6.22 -  Infinitesimal  :: "('a::real_normed_vector) star set" where
6.23 -  "Infinitesimal = {x. \<forall>r \<in> Reals. 0 < r --> hnorm x < r}"
6.24 +definition Infinitesimal  :: "('a::real_normed_vector) star set"
6.25 +  where "Infinitesimal = {x. \<forall>r \<in> Reals. 0 < r \<longrightarrow> hnorm x < r}"
6.26
6.27 -definition
6.28 -  HFinite :: "('a::real_normed_vector) star set" where
6.29 -  "HFinite = {x. \<exists>r \<in> Reals. hnorm x < r}"
6.30 +definition HFinite :: "('a::real_normed_vector) star set"
6.31 +  where "HFinite = {x. \<exists>r \<in> Reals. hnorm x < r}"
6.32
6.33 -definition
6.34 -  HInfinite :: "('a::real_normed_vector) star set" where
6.35 -  "HInfinite = {x. \<forall>r \<in> Reals. r < hnorm x}"
6.36 +definition HInfinite :: "('a::real_normed_vector) star set"
6.37 +  where "HInfinite = {x. \<forall>r \<in> Reals. r < hnorm x}"
6.38
6.39 -definition
6.40 -  approx :: "['a::real_normed_vector star, 'a star] => bool"  (infixl "\<approx>" 50) where
6.41 -    \<comment>\<open>the `infinitely close' relation\<close>
6.42 -  "(x \<approx> y) = ((x - y) \<in> Infinitesimal)"
6.43 +definition approx :: "'a::real_normed_vector star \<Rightarrow> 'a star \<Rightarrow> bool"  (infixl "\<approx>" 50)
6.44 +  where "x \<approx> y \<longleftrightarrow> x - y \<in> Infinitesimal"
6.45 +    \<comment> \<open>the ``infinitely close'' relation\<close>
6.46
6.47 -definition
6.48 -  st        :: "hypreal => hypreal" where
6.49 -    \<comment>\<open>the standard part of a hyperreal\<close>
6.50 -  "st = (%x. @r. x \<in> HFinite & r \<in> \<real> & r \<approx> x)"
6.51 +definition st :: "hypreal \<Rightarrow> hypreal"
6.52 +  where "st = (\<lambda>x. SOME r. x \<in> HFinite \<and> r \<in> \<real> \<and> r \<approx> x)"
6.53 +    \<comment> \<open>the standard part of a hyperreal\<close>
6.54
6.55 -definition
6.56 -  monad     :: "'a::real_normed_vector star => 'a star set" where
6.57 -  "monad x = {y. x \<approx> y}"
6.58 +definition monad :: "'a::real_normed_vector star \<Rightarrow> 'a star set"
6.59 +  where "monad x = {y. x \<approx> y}"
6.60
6.61 -definition
6.62 -  galaxy    :: "'a::real_normed_vector star => 'a star set" where
6.63 -  "galaxy x = {y. (x + -y) \<in> HFinite}"
6.64 +definition galaxy :: "'a::real_normed_vector star \<Rightarrow> 'a star set"
6.65 +  where "galaxy x = {y. (x + -y) \<in> HFinite}"
6.66
6.67 -lemma SReal_def: "\<real> == {x. \<exists>r. x = hypreal_of_real r}"
6.68 -by (simp add: Reals_def image_def)
6.69 +lemma SReal_def: "\<real> \<equiv> {x. \<exists>r. x = hypreal_of_real r}"
6.70 +  by (simp add: Reals_def image_def)
6.71 +
6.72
6.73  subsection \<open>Nonstandard Extension of the Norm Function\<close>
6.74
6.75 -definition
6.76 -  scaleHR :: "real star \<Rightarrow> 'a star \<Rightarrow> 'a::real_normed_vector star" where
6.77 -  [transfer_unfold]: "scaleHR = starfun2 scaleR"
6.78 +definition scaleHR :: "real star \<Rightarrow> 'a star \<Rightarrow> 'a::real_normed_vector star"
6.79 +  where [transfer_unfold]: "scaleHR = starfun2 scaleR"
6.80
6.81  lemma Standard_hnorm [simp]: "x \<in> Standard \<Longrightarrow> hnorm x \<in> Standard"
6.83 +  by (simp add: hnorm_def)
6.84
6.85  lemma star_of_norm [simp]: "star_of (norm x) = hnorm (star_of x)"
6.86 -by transfer (rule refl)
6.87 +  by transfer (rule refl)
6.88
6.89 -lemma hnorm_ge_zero [simp]:
6.90 -  "\<And>x::'a::real_normed_vector star. 0 \<le> hnorm x"
6.91 -by transfer (rule norm_ge_zero)
6.92 +lemma hnorm_ge_zero [simp]: "\<And>x::'a::real_normed_vector star. 0 \<le> hnorm x"
6.93 +  by transfer (rule norm_ge_zero)
6.94
6.95 -lemma hnorm_eq_zero [simp]:
6.96 -  "\<And>x::'a::real_normed_vector star. (hnorm x = 0) = (x = 0)"
6.97 -by transfer (rule norm_eq_zero)
6.98 +lemma hnorm_eq_zero [simp]: "\<And>x::'a::real_normed_vector star. hnorm x = 0 \<longleftrightarrow> x = 0"
6.99 +  by transfer (rule norm_eq_zero)
6.100
6.101 -lemma hnorm_triangle_ineq:
6.102 -  "\<And>x y::'a::real_normed_vector star. hnorm (x + y) \<le> hnorm x + hnorm y"
6.103 -by transfer (rule norm_triangle_ineq)
6.104 -
6.105 -lemma hnorm_triangle_ineq3:
6.106 -  "\<And>x y::'a::real_normed_vector star. \<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
6.107 -by transfer (rule norm_triangle_ineq3)
6.108 +lemma hnorm_triangle_ineq: "\<And>x y::'a::real_normed_vector star. hnorm (x + y) \<le> hnorm x + hnorm y"
6.109 +  by transfer (rule norm_triangle_ineq)
6.110
6.111 -lemma hnorm_scaleR:
6.112 -  "\<And>x::'a::real_normed_vector star.
6.113 -   hnorm (a *\<^sub>R x) = \<bar>star_of a\<bar> * hnorm x"
6.114 -by transfer (rule norm_scaleR)
6.115 +lemma hnorm_triangle_ineq3: "\<And>x y::'a::real_normed_vector star. \<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
6.116 +  by transfer (rule norm_triangle_ineq3)
6.117 +
6.118 +lemma hnorm_scaleR: "\<And>x::'a::real_normed_vector star. hnorm (a *\<^sub>R x) = \<bar>star_of a\<bar> * hnorm x"
6.119 +  by transfer (rule norm_scaleR)
6.120
6.121 -lemma hnorm_scaleHR:
6.122 -  "\<And>a (x::'a::real_normed_vector star).
6.123 -   hnorm (scaleHR a x) = \<bar>a\<bar> * hnorm x"
6.124 -by transfer (rule norm_scaleR)
6.125 +lemma hnorm_scaleHR: "\<And>a (x::'a::real_normed_vector star). hnorm (scaleHR a x) = \<bar>a\<bar> * hnorm x"
6.126 +  by transfer (rule norm_scaleR)
6.127
6.128 -lemma hnorm_mult_ineq:
6.129 -  "\<And>x y::'a::real_normed_algebra star. hnorm (x * y) \<le> hnorm x * hnorm y"
6.130 -by transfer (rule norm_mult_ineq)
6.131 +lemma hnorm_mult_ineq: "\<And>x y::'a::real_normed_algebra star. hnorm (x * y) \<le> hnorm x * hnorm y"
6.132 +  by transfer (rule norm_mult_ineq)
6.133
6.134 -lemma hnorm_mult:
6.135 -  "\<And>x y::'a::real_normed_div_algebra star. hnorm (x * y) = hnorm x * hnorm y"
6.136 -by transfer (rule norm_mult)
6.137 +lemma hnorm_mult: "\<And>x y::'a::real_normed_div_algebra star. hnorm (x * y) = hnorm x * hnorm y"
6.138 +  by transfer (rule norm_mult)
6.139
6.140 -lemma hnorm_hyperpow:
6.141 -  "\<And>(x::'a::{real_normed_div_algebra} star) n.
6.142 -   hnorm (x pow n) = hnorm x pow n"
6.143 -by transfer (rule norm_power)
6.144 +lemma hnorm_hyperpow: "\<And>(x::'a::{real_normed_div_algebra} star) n. hnorm (x pow n) = hnorm x pow n"
6.145 +  by transfer (rule norm_power)
6.146 +
6.147 +lemma hnorm_one [simp]: "hnorm (1::'a::real_normed_div_algebra star) = 1"
6.148 +  by transfer (rule norm_one)
6.149
6.150 -lemma hnorm_one [simp]:
6.151 -  "hnorm (1::'a::real_normed_div_algebra star) = 1"
6.152 -by transfer (rule norm_one)
6.153 +lemma hnorm_zero [simp]: "hnorm (0::'a::real_normed_vector star) = 0"
6.154 +  by transfer (rule norm_zero)
6.155
6.156 -lemma hnorm_zero [simp]:
6.157 -  "hnorm (0::'a::real_normed_vector star) = 0"
6.158 -by transfer (rule norm_zero)
6.159 +lemma zero_less_hnorm_iff [simp]: "\<And>x::'a::real_normed_vector star. 0 < hnorm x \<longleftrightarrow> x \<noteq> 0"
6.160 +  by transfer (rule zero_less_norm_iff)
6.161
6.162 -lemma zero_less_hnorm_iff [simp]:
6.163 -  "\<And>x::'a::real_normed_vector star. (0 < hnorm x) = (x \<noteq> 0)"
6.164 -by transfer (rule zero_less_norm_iff)
6.165 +lemma hnorm_minus_cancel [simp]: "\<And>x::'a::real_normed_vector star. hnorm (- x) = hnorm x"
6.166 +  by transfer (rule norm_minus_cancel)
6.167
6.168 -lemma hnorm_minus_cancel [simp]:
6.169 -  "\<And>x::'a::real_normed_vector star. hnorm (- x) = hnorm x"
6.170 -by transfer (rule norm_minus_cancel)
6.171 +lemma hnorm_minus_commute: "\<And>a b::'a::real_normed_vector star. hnorm (a - b) = hnorm (b - a)"
6.172 +  by transfer (rule norm_minus_commute)
6.173
6.174 -lemma hnorm_minus_commute:
6.175 -  "\<And>a b::'a::real_normed_vector star. hnorm (a - b) = hnorm (b - a)"
6.176 -by transfer (rule norm_minus_commute)
6.177 -
6.178 -lemma hnorm_triangle_ineq2:
6.179 -  "\<And>a b::'a::real_normed_vector star. hnorm a - hnorm b \<le> hnorm (a - b)"
6.180 -by transfer (rule norm_triangle_ineq2)
6.181 +lemma hnorm_triangle_ineq2: "\<And>a b::'a::real_normed_vector star. hnorm a - hnorm b \<le> hnorm (a - b)"
6.182 +  by transfer (rule norm_triangle_ineq2)
6.183
6.184 -lemma hnorm_triangle_ineq4:
6.185 -  "\<And>a b::'a::real_normed_vector star. hnorm (a - b) \<le> hnorm a + hnorm b"
6.186 -by transfer (rule norm_triangle_ineq4)
6.187 +lemma hnorm_triangle_ineq4: "\<And>a b::'a::real_normed_vector star. hnorm (a - b) \<le> hnorm a + hnorm b"
6.188 +  by transfer (rule norm_triangle_ineq4)
6.189
6.190 -lemma abs_hnorm_cancel [simp]:
6.191 -  "\<And>a::'a::real_normed_vector star. \<bar>hnorm a\<bar> = hnorm a"
6.192 -by transfer (rule abs_norm_cancel)
6.193 +lemma abs_hnorm_cancel [simp]: "\<And>a::'a::real_normed_vector star. \<bar>hnorm a\<bar> = hnorm a"
6.194 +  by transfer (rule abs_norm_cancel)
6.195
6.196 -lemma hnorm_of_hypreal [simp]:
6.197 -  "\<And>r. hnorm (of_hypreal r::'a::real_normed_algebra_1 star) = \<bar>r\<bar>"
6.198 -by transfer (rule norm_of_real)
6.199 +lemma hnorm_of_hypreal [simp]: "\<And>r. hnorm (of_hypreal r::'a::real_normed_algebra_1 star) = \<bar>r\<bar>"
6.200 +  by transfer (rule norm_of_real)
6.201
6.202  lemma nonzero_hnorm_inverse:
6.203 -  "\<And>a::'a::real_normed_div_algebra star.
6.204 -   a \<noteq> 0 \<Longrightarrow> hnorm (inverse a) = inverse (hnorm a)"
6.205 -by transfer (rule nonzero_norm_inverse)
6.206 +  "\<And>a::'a::real_normed_div_algebra star. a \<noteq> 0 \<Longrightarrow> hnorm (inverse a) = inverse (hnorm a)"
6.207 +  by transfer (rule nonzero_norm_inverse)
6.208
6.209  lemma hnorm_inverse:
6.210 -  "\<And>a::'a::{real_normed_div_algebra, division_ring} star.
6.211 -   hnorm (inverse a) = inverse (hnorm a)"
6.212 -by transfer (rule norm_inverse)
6.213 +  "\<And>a::'a::{real_normed_div_algebra, division_ring} star. hnorm (inverse a) = inverse (hnorm a)"
6.214 +  by transfer (rule norm_inverse)
6.215
6.216 -lemma hnorm_divide:
6.217 -  "\<And>a b::'a::{real_normed_field, field} star.
6.218 -   hnorm (a / b) = hnorm a / hnorm b"
6.219 -by transfer (rule norm_divide)
6.220 +lemma hnorm_divide: "\<And>a b::'a::{real_normed_field, field} star. hnorm (a / b) = hnorm a / hnorm b"
6.221 +  by transfer (rule norm_divide)
6.222
6.223 -lemma hypreal_hnorm_def [simp]:
6.224 -  "\<And>r::hypreal. hnorm r = \<bar>r\<bar>"
6.225 -by transfer (rule real_norm_def)
6.226 +lemma hypreal_hnorm_def [simp]: "\<And>r::hypreal. hnorm r = \<bar>r\<bar>"
6.227 +  by transfer (rule real_norm_def)
6.228
6.230 -  "\<And>(x::'a::real_normed_vector star) y r s.
6.231 -   \<lbrakk>hnorm x < r; hnorm y < s\<rbrakk> \<Longrightarrow> hnorm (x + y) < r + s"
6.233 +  "\<And>(x::'a::real_normed_vector star) y r s. hnorm x < r \<Longrightarrow> hnorm y < s \<Longrightarrow> hnorm (x + y) < r + s"
6.234 +  by transfer (rule norm_add_less)
6.235
6.236  lemma hnorm_mult_less:
6.237 -  "\<And>(x::'a::real_normed_algebra star) y r s.
6.238 -   \<lbrakk>hnorm x < r; hnorm y < s\<rbrakk> \<Longrightarrow> hnorm (x * y) < r * s"
6.239 -by transfer (rule norm_mult_less)
6.240 +  "\<And>(x::'a::real_normed_algebra star) y r s. hnorm x < r \<Longrightarrow> hnorm y < s \<Longrightarrow> hnorm (x * y) < r * s"
6.241 +  by transfer (rule norm_mult_less)
6.242
6.243 -lemma hnorm_scaleHR_less:
6.244 -  "\<lbrakk>\<bar>x\<bar> < r; hnorm y < s\<rbrakk> \<Longrightarrow> hnorm (scaleHR x y) < r * s"
6.245 -apply (simp only: hnorm_scaleHR)
6.247 -done
6.248 +lemma hnorm_scaleHR_less: "\<bar>x\<bar> < r \<Longrightarrow> hnorm y < s \<Longrightarrow> hnorm (scaleHR x y) < r * s"
6.249 + by (simp only: hnorm_scaleHR) (simp add: mult_strict_mono')
6.250 +
6.251 +
6.252 +subsection \<open>Closure Laws for the Standard Reals\<close>
6.253
6.254 -subsection\<open>Closure Laws for the Standard Reals\<close>
6.255 +lemma Reals_minus_iff [simp]: "- x \<in> \<real> \<longleftrightarrow> x \<in> \<real>"
6.256 +  apply auto
6.257 +  apply (drule Reals_minus)
6.258 +  apply auto
6.259 +  done
6.260
6.261 -lemma Reals_minus_iff [simp]: "(-x \<in> \<real>) = (x \<in> \<real>)"
6.262 -apply auto
6.263 -apply (drule Reals_minus, auto)
6.264 -done
6.265 +lemma Reals_add_cancel: "x + y \<in> \<real> \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x \<in> \<real>"
6.266 +  by (drule (1) Reals_diff) simp
6.267
6.268 -lemma Reals_add_cancel: "\<lbrakk>x + y \<in> \<real>; y \<in> \<real>\<rbrakk> \<Longrightarrow> x \<in> \<real>"
6.269 -by (drule (1) Reals_diff, simp)
6.270 -
6.271 -lemma SReal_hrabs: "(x::hypreal) \<in> \<real> ==> \<bar>x\<bar> \<in> \<real>"
6.273 +lemma SReal_hrabs: "x \<in> \<real> \<Longrightarrow> \<bar>x\<bar> \<in> \<real>"
6.274 +  for x :: hypreal
6.275 +  by (simp add: Reals_eq_Standard)
6.276
6.277  lemma SReal_hypreal_of_real [simp]: "hypreal_of_real x \<in> \<real>"
6.279 +  by (simp add: Reals_eq_Standard)
6.280
6.281 -lemma SReal_divide_numeral: "r \<in> \<real> ==> r/(numeral w::hypreal) \<in> \<real>"
6.282 -by simp
6.283 +lemma SReal_divide_numeral: "r \<in> \<real> \<Longrightarrow> r / (numeral w::hypreal) \<in> \<real>"
6.284 +  by simp
6.285
6.286  text \<open>\<open>\<epsilon>\<close> is not in Reals because it is an infinitesimal\<close>
6.287  lemma SReal_epsilon_not_mem: "\<epsilon> \<notin> \<real>"
6.289 -apply (auto simp add: hypreal_of_real_not_eq_epsilon [THEN not_sym])
6.290 -done
6.291 +  by (auto simp: SReal_def hypreal_of_real_not_eq_epsilon [symmetric])
6.292
6.293  lemma SReal_omega_not_mem: "\<omega> \<notin> \<real>"
6.295 -apply (auto simp add: hypreal_of_real_not_eq_omega [THEN not_sym])
6.296 -done
6.297 +  by (auto simp: SReal_def hypreal_of_real_not_eq_omega [symmetric])
6.298
6.299  lemma SReal_UNIV_real: "{x. hypreal_of_real x \<in> \<real>} = (UNIV::real set)"
6.300 -by simp
6.301 +  by simp
6.302
6.303 -lemma SReal_iff: "(x \<in> \<real>) = (\<exists>y. x = hypreal_of_real y)"
6.305 +lemma SReal_iff: "x \<in> \<real> \<longleftrightarrow> (\<exists>y. x = hypreal_of_real y)"
6.306 +  by (simp add: SReal_def)
6.307
6.308  lemma hypreal_of_real_image: "hypreal_of_real `(UNIV::real set) = \<real>"
6.309 -by (simp add: Reals_eq_Standard Standard_def)
6.310 +  by (simp add: Reals_eq_Standard Standard_def)
6.311
6.312  lemma inv_hypreal_of_real_image: "inv hypreal_of_real ` \<real> = UNIV"
6.313 -apply (auto simp add: SReal_def)
6.314 -apply (rule inj_star_of [THEN inv_f_f, THEN subst], blast)
6.315 -done
6.316 +  apply (auto simp add: SReal_def)
6.317 +  apply (rule inj_star_of [THEN inv_f_f, THEN subst], blast)
6.318 +  done
6.319
6.320 -lemma SReal_hypreal_of_real_image:
6.321 -      "[| \<exists>x. x: P; P \<subseteq> \<real> |] ==> \<exists>Q. P = hypreal_of_real ` Q"
6.322 -by (simp add: SReal_def image_def, blast)
6.323 +lemma SReal_hypreal_of_real_image: "\<exists>x. x \<in> P \<Longrightarrow> P \<subseteq> \<real> \<Longrightarrow> \<exists>Q. P = hypreal_of_real ` Q"
6.324 +  unfolding SReal_def image_def by blast
6.325
6.326 -lemma SReal_dense:
6.327 -     "[| (x::hypreal) \<in> \<real>; y \<in> \<real>;  x<y |] ==> \<exists>r \<in> Reals. x<r & r<y"
6.328 -apply (auto simp add: SReal_def)
6.329 -apply (drule dense, auto)
6.330 -done
6.331 +lemma SReal_dense: "x \<in> \<real> \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x < y \<Longrightarrow> \<exists>r \<in> Reals. x < r \<and> r < y"
6.332 +  for x y :: hypreal
6.333 +  apply (auto simp: SReal_def)
6.334 +  apply (drule dense)
6.335 +  apply auto
6.336 +  done
6.337
6.338 -text\<open>Completeness of Reals, but both lemmas are unused.\<close>
6.339 +
6.340 +text \<open>Completeness of Reals, but both lemmas are unused.\<close>
6.341
6.342  lemma SReal_sup_lemma:
6.343 -     "P \<subseteq> \<real> ==> ((\<exists>x \<in> P. y < x) =
6.344 -      (\<exists>X. hypreal_of_real X \<in> P & y < hypreal_of_real X))"
6.345 -by (blast dest!: SReal_iff [THEN iffD1])
6.346 +  "P \<subseteq> \<real> \<Longrightarrow> (\<exists>x \<in> P. y < x) = (\<exists>X. hypreal_of_real X \<in> P \<and> y < hypreal_of_real X)"
6.347 +  by (blast dest!: SReal_iff [THEN iffD1])
6.348
6.349  lemma SReal_sup_lemma2:
6.350 -     "[| P \<subseteq> \<real>; \<exists>x. x \<in> P; \<exists>y \<in> Reals. \<forall>x \<in> P. x < y |]
6.351 -      ==> (\<exists>X. X \<in> {w. hypreal_of_real w \<in> P}) &
6.352 -          (\<exists>Y. \<forall>X \<in> {w. hypreal_of_real w \<in> P}. X < Y)"
6.353 -apply (rule conjI)
6.354 -apply (fast dest!: SReal_iff [THEN iffD1])
6.355 -apply (auto, frule subsetD, assumption)
6.356 -apply (drule SReal_iff [THEN iffD1])
6.357 -apply (auto, rule_tac x = ya in exI, auto)
6.358 -done
6.359 +  "P \<subseteq> \<real> \<Longrightarrow> \<exists>x. x \<in> P \<Longrightarrow> \<exists>y \<in> Reals. \<forall>x \<in> P. x < y \<Longrightarrow>
6.360 +    (\<exists>X. X \<in> {w. hypreal_of_real w \<in> P}) \<and>
6.361 +    (\<exists>Y. \<forall>X \<in> {w. hypreal_of_real w \<in> P}. X < Y)"
6.362 +  apply (rule conjI)
6.363 +   apply (fast dest!: SReal_iff [THEN iffD1])
6.364 +  apply (auto, frule subsetD, assumption)
6.365 +  apply (drule SReal_iff [THEN iffD1])
6.366 +  apply (auto, rule_tac x = ya in exI, auto)
6.367 +  done
6.368
6.369
6.370 -subsection\<open>Set of Finite Elements is a Subring of the Extended Reals\<close>
6.371 +subsection \<open>Set of Finite Elements is a Subring of the Extended Reals\<close>
6.372
6.373 -lemma HFinite_add: "[|x \<in> HFinite; y \<in> HFinite|] ==> (x+y) \<in> HFinite"
6.376 -done
6.377 +lemma HFinite_add: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x + y \<in> HFinite"
6.379
6.380 -lemma HFinite_mult:
6.381 -  fixes x y :: "'a::real_normed_algebra star"
6.382 -  shows "[|x \<in> HFinite; y \<in> HFinite|] ==> x*y \<in> HFinite"
6.384 -apply (blast intro!: Reals_mult hnorm_mult_less)
6.385 -done
6.386 +lemma HFinite_mult: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x * y \<in> HFinite"
6.387 +  for x y :: "'a::real_normed_algebra star"
6.388 +  unfolding HFinite_def by (blast intro!: Reals_mult hnorm_mult_less)
6.389
6.390 -lemma HFinite_scaleHR:
6.391 -  "[|x \<in> HFinite; y \<in> HFinite|] ==> scaleHR x y \<in> HFinite"
6.393 -apply (blast intro!: Reals_mult hnorm_scaleHR_less)
6.394 -done
6.395 +lemma HFinite_scaleHR: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> scaleHR x y \<in> HFinite"
6.396 +  by (auto simp: HFinite_def intro!: Reals_mult hnorm_scaleHR_less)
6.397
6.398 -lemma HFinite_minus_iff: "(-x \<in> HFinite) = (x \<in> HFinite)"
6.400 +lemma HFinite_minus_iff: "- x \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
6.401 +  by (simp add: HFinite_def)
6.402
6.403  lemma HFinite_star_of [simp]: "star_of x \<in> HFinite"
6.405 -apply (rule_tac x="star_of (norm x) + 1" in bexI)
6.406 -apply (transfer, simp)
6.407 -apply (blast intro: Reals_add SReal_hypreal_of_real Reals_1)
6.408 -done
6.409 +  apply (simp add: HFinite_def)
6.410 +  apply (rule_tac x="star_of (norm x) + 1" in bexI)
6.411 +   apply (transfer, simp)
6.412 +  apply (blast intro: Reals_add SReal_hypreal_of_real Reals_1)
6.413 +  done
6.414
6.415  lemma SReal_subset_HFinite: "(\<real>::hypreal set) \<subseteq> HFinite"
6.416 -by (auto simp add: SReal_def)
6.417 +  by (auto simp add: SReal_def)
6.418
6.419 -lemma HFiniteD: "x \<in> HFinite ==> \<exists>t \<in> Reals. hnorm x < t"
6.421 +lemma HFiniteD: "x \<in> HFinite \<Longrightarrow> \<exists>t \<in> Reals. hnorm x < t"
6.422 +  by (simp add: HFinite_def)
6.423
6.424 -lemma HFinite_hrabs_iff [iff]: "(\<bar>x::hypreal\<bar> \<in> HFinite) = (x \<in> HFinite)"
6.426 +lemma HFinite_hrabs_iff [iff]: "\<bar>x\<bar> \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
6.427 +  for x :: hypreal
6.428 +  by (simp add: HFinite_def)
6.429
6.430 -lemma HFinite_hnorm_iff [iff]:
6.431 -  "(hnorm (x::hypreal) \<in> HFinite) = (x \<in> HFinite)"
6.433 +lemma HFinite_hnorm_iff [iff]: "hnorm x \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
6.434 +  for x :: hypreal
6.435 +  by (simp add: HFinite_def)
6.436
6.437  lemma HFinite_numeral [simp]: "numeral w \<in> HFinite"
6.438 -unfolding star_numeral_def by (rule HFinite_star_of)
6.439 +  unfolding star_numeral_def by (rule HFinite_star_of)
6.440
6.441 -(** As always with numerals, 0 and 1 are special cases **)
6.442 +text \<open>As always with numerals, \<open>0\<close> and \<open>1\<close> are special cases.\<close>
6.443
6.444  lemma HFinite_0 [simp]: "0 \<in> HFinite"
6.445 -unfolding star_zero_def by (rule HFinite_star_of)
6.446 +  unfolding star_zero_def by (rule HFinite_star_of)
6.447
6.448  lemma HFinite_1 [simp]: "1 \<in> HFinite"
6.449 -unfolding star_one_def by (rule HFinite_star_of)
6.450 +  unfolding star_one_def by (rule HFinite_star_of)
6.451
6.452 -lemma hrealpow_HFinite:
6.453 -  fixes x :: "'a::{real_normed_algebra,monoid_mult} star"
6.454 -  shows "x \<in> HFinite ==> x ^ n \<in> HFinite"
6.455 -apply (induct n)
6.456 -apply (auto simp add: power_Suc intro: HFinite_mult)
6.457 -done
6.458 +lemma hrealpow_HFinite: "x \<in> HFinite \<Longrightarrow> x ^ n \<in> HFinite"
6.459 +  for x :: "'a::{real_normed_algebra,monoid_mult} star"
6.460 +  by (induct n) (auto simp add: power_Suc intro: HFinite_mult)
6.461
6.462 -lemma HFinite_bounded:
6.463 -  "[|(x::hypreal) \<in> HFinite; y \<le> x; 0 \<le> y |] ==> y \<in> HFinite"
6.464 -apply (cases "x \<le> 0")
6.465 -apply (drule_tac y = x in order_trans)
6.466 -apply (drule_tac [2] order_antisym)
6.467 -apply (auto simp add: linorder_not_le)
6.468 -apply (auto intro: order_le_less_trans simp add: abs_if HFinite_def)
6.469 -done
6.470 +lemma HFinite_bounded: "x \<in> HFinite \<Longrightarrow> y \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<in> HFinite"
6.471 +  for x y :: hypreal
6.472 +  apply (cases "x \<le> 0")
6.473 +   apply (drule_tac y = x in order_trans)
6.474 +    apply (drule_tac [2] order_antisym)
6.475 +     apply (auto simp add: linorder_not_le)
6.476 +  apply (auto intro: order_le_less_trans simp add: abs_if HFinite_def)
6.477 +  done
6.478
6.479
6.480 -subsection\<open>Set of Infinitesimals is a Subring of the Hyperreals\<close>
6.481 +subsection \<open>Set of Infinitesimals is a Subring of the Hyperreals\<close>
6.482
6.483 -lemma InfinitesimalI:
6.484 -  "(\<And>r. \<lbrakk>r \<in> \<real>; 0 < r\<rbrakk> \<Longrightarrow> hnorm x < r) \<Longrightarrow> x \<in> Infinitesimal"
6.486 +lemma InfinitesimalI: "(\<And>r. r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> hnorm x < r) \<Longrightarrow> x \<in> Infinitesimal"
6.487 +  by (simp add: Infinitesimal_def)
6.488
6.489 -lemma InfinitesimalD:
6.490 -      "x \<in> Infinitesimal ==> \<forall>r \<in> Reals. 0 < r --> hnorm x < r"
6.492 +lemma InfinitesimalD: "x \<in> Infinitesimal \<Longrightarrow> \<forall>r \<in> Reals. 0 < r \<longrightarrow> hnorm x < r"
6.493 +  by (simp add: Infinitesimal_def)
6.494
6.495 -lemma InfinitesimalI2:
6.496 -  "(\<And>r. 0 < r \<Longrightarrow> hnorm x < star_of r) \<Longrightarrow> x \<in> Infinitesimal"
6.497 -by (auto simp add: Infinitesimal_def SReal_def)
6.498 +lemma InfinitesimalI2: "(\<And>r. 0 < r \<Longrightarrow> hnorm x < star_of r) \<Longrightarrow> x \<in> Infinitesimal"
6.499 +  by (auto simp add: Infinitesimal_def SReal_def)
6.500
6.501 -lemma InfinitesimalD2:
6.502 -  "\<lbrakk>x \<in> Infinitesimal; 0 < r\<rbrakk> \<Longrightarrow> hnorm x < star_of r"
6.503 -by (auto simp add: Infinitesimal_def SReal_def)
6.504 +lemma InfinitesimalD2: "x \<in> Infinitesimal \<Longrightarrow> 0 < r \<Longrightarrow> hnorm x < star_of r"
6.505 +  by (auto simp add: Infinitesimal_def SReal_def)
6.506
6.507  lemma Infinitesimal_zero [iff]: "0 \<in> Infinitesimal"
6.509 +  by (simp add: Infinitesimal_def)
6.510
6.511 -lemma hypreal_sum_of_halves: "x/(2::hypreal) + x/(2::hypreal) = x"
6.512 -by auto
6.513 +lemma hypreal_sum_of_halves: "x / 2 + x / 2 = x"
6.514 +  for x :: hypreal
6.515 +  by auto
6.516
6.518 -     "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> (x+y) \<in> Infinitesimal"
6.519 -apply (rule InfinitesimalI)
6.520 -apply (rule hypreal_sum_of_halves [THEN subst])
6.521 -apply (drule half_gt_zero)
6.522 -apply (blast intro: hnorm_add_less SReal_divide_numeral dest: InfinitesimalD)
6.523 -done
6.524 +lemma Infinitesimal_add: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x + y \<in> Infinitesimal"
6.525 +  apply (rule InfinitesimalI)
6.526 +  apply (rule hypreal_sum_of_halves [THEN subst])
6.527 +  apply (drule half_gt_zero)
6.528 +  apply (blast intro: hnorm_add_less SReal_divide_numeral dest: InfinitesimalD)
6.529 +  done
6.530
6.531 -lemma Infinitesimal_minus_iff [simp]: "(-x:Infinitesimal) = (x:Infinitesimal)"
6.533 +lemma Infinitesimal_minus_iff [simp]: "- x \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
6.534 +  by (simp add: Infinitesimal_def)
6.535
6.536 -lemma Infinitesimal_hnorm_iff:
6.537 -  "(hnorm x \<in> Infinitesimal) = (x \<in> Infinitesimal)"
6.539 +lemma Infinitesimal_hnorm_iff: "hnorm x \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
6.540 +  by (simp add: Infinitesimal_def)
6.541
6.542 -lemma Infinitesimal_hrabs_iff [iff]:
6.543 -  "(\<bar>x::hypreal\<bar> \<in> Infinitesimal) = (x \<in> Infinitesimal)"
6.545 +lemma Infinitesimal_hrabs_iff [iff]: "\<bar>x\<bar> \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
6.546 +  for x :: hypreal
6.547 +  by (simp add: abs_if)
6.548
6.549  lemma Infinitesimal_of_hypreal_iff [simp]:
6.550 -  "((of_hypreal x::'a::real_normed_algebra_1 star) \<in> Infinitesimal) =
6.551 -   (x \<in> Infinitesimal)"
6.552 -by (subst Infinitesimal_hnorm_iff [symmetric], simp)
6.553 +  "(of_hypreal x::'a::real_normed_algebra_1 star) \<in> Infinitesimal \<longleftrightarrow> x \<in> Infinitesimal"
6.554 +  by (subst Infinitesimal_hnorm_iff [symmetric]) simp
6.555
6.556 -lemma Infinitesimal_diff:
6.557 -     "[| x \<in> Infinitesimal;  y \<in> Infinitesimal |] ==> x-y \<in> Infinitesimal"
6.558 +lemma Infinitesimal_diff: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x - y \<in> Infinitesimal"
6.559    using Infinitesimal_add [of x "- y"] by simp
6.560
6.561 -lemma Infinitesimal_mult:
6.562 -  fixes x y :: "'a::real_normed_algebra star"
6.563 -  shows "[|x \<in> Infinitesimal; y \<in> Infinitesimal|] ==> (x * y) \<in> Infinitesimal"
6.564 -apply (rule InfinitesimalI)
6.565 -apply (subgoal_tac "hnorm (x * y) < 1 * r", simp only: mult_1)
6.566 -apply (rule hnorm_mult_less)
6.568 -done
6.569 +lemma Infinitesimal_mult: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x * y \<in> Infinitesimal"
6.570 +  for x y :: "'a::real_normed_algebra star"
6.571 +  apply (rule InfinitesimalI)
6.572 +  apply (subgoal_tac "hnorm (x * y) < 1 * r")
6.573 +   apply (simp only: mult_1)
6.574 +  apply (rule hnorm_mult_less)
6.575 +   apply (simp_all add: InfinitesimalD)
6.576 +  done
6.577
6.578 -lemma Infinitesimal_HFinite_mult:
6.579 -  fixes x y :: "'a::real_normed_algebra star"
6.580 -  shows "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> (x * y) \<in> Infinitesimal"
6.581 -apply (rule InfinitesimalI)
6.582 -apply (drule HFiniteD, clarify)
6.583 -apply (subgoal_tac "0 < t")
6.584 -apply (subgoal_tac "hnorm (x * y) < (r / t) * t", simp)
6.585 -apply (subgoal_tac "0 < r / t")
6.586 -apply (rule hnorm_mult_less)
6.588 -apply assumption
6.589 -apply simp
6.590 -apply (erule order_le_less_trans [OF hnorm_ge_zero])
6.591 -done
6.592 +lemma Infinitesimal_HFinite_mult: "x \<in> Infinitesimal \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x * y \<in> Infinitesimal"
6.593 +  for x y :: "'a::real_normed_algebra star"
6.594 +  apply (rule InfinitesimalI)
6.595 +  apply (drule HFiniteD, clarify)
6.596 +  apply (subgoal_tac "0 < t")
6.597 +   apply (subgoal_tac "hnorm (x * y) < (r / t) * t", simp)
6.598 +   apply (subgoal_tac "0 < r / t")
6.599 +    apply (rule hnorm_mult_less)
6.600 +     apply (simp add: InfinitesimalD)
6.601 +    apply assumption
6.602 +   apply simp
6.603 +  apply (erule order_le_less_trans [OF hnorm_ge_zero])
6.604 +  done
6.605
6.606  lemma Infinitesimal_HFinite_scaleHR:
6.607 -  "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> scaleHR x y \<in> Infinitesimal"
6.608 -apply (rule InfinitesimalI)
6.609 -apply (drule HFiniteD, clarify)
6.610 -apply (drule InfinitesimalD)
6.612 -apply (subgoal_tac "0 < t")
6.613 -apply (subgoal_tac "\<bar>x\<bar> * hnorm y < (r / t) * t", simp)
6.614 -apply (subgoal_tac "0 < r / t")
6.615 -apply (rule mult_strict_mono', simp_all)
6.616 -apply (erule order_le_less_trans [OF hnorm_ge_zero])
6.617 -done
6.618 +  "x \<in> Infinitesimal \<Longrightarrow> y \<in> HFinite \<Longrightarrow> scaleHR x y \<in> Infinitesimal"
6.619 +  apply (rule InfinitesimalI)
6.620 +  apply (drule HFiniteD, clarify)
6.621 +  apply (drule InfinitesimalD)
6.622 +  apply (simp add: hnorm_scaleHR)
6.623 +  apply (subgoal_tac "0 < t")
6.624 +   apply (subgoal_tac "\<bar>x\<bar> * hnorm y < (r / t) * t", simp)
6.625 +   apply (subgoal_tac "0 < r / t")
6.626 +    apply (rule mult_strict_mono', simp_all)
6.627 +  apply (erule order_le_less_trans [OF hnorm_ge_zero])
6.628 +  done
6.629
6.630  lemma Infinitesimal_HFinite_mult2:
6.631 -  fixes x y :: "'a::real_normed_algebra star"
6.632 -  shows "[| x \<in> Infinitesimal; y \<in> HFinite |] ==> (y * x) \<in> Infinitesimal"
6.633 -apply (rule InfinitesimalI)
6.634 -apply (drule HFiniteD, clarify)
6.635 -apply (subgoal_tac "0 < t")
6.636 -apply (subgoal_tac "hnorm (y * x) < t * (r / t)", simp)
6.637 -apply (subgoal_tac "0 < r / t")
6.638 -apply (rule hnorm_mult_less)
6.639 -apply assumption
6.641 -apply simp
6.642 -apply (erule order_le_less_trans [OF hnorm_ge_zero])
6.643 -done
6.644 +  "x \<in> Infinitesimal \<Longrightarrow> y \<in> HFinite \<Longrightarrow> y * x \<in> Infinitesimal"
6.645 +  for x y :: "'a::real_normed_algebra star"
6.646 +  apply (rule InfinitesimalI)
6.647 +  apply (drule HFiniteD, clarify)
6.648 +  apply (subgoal_tac "0 < t")
6.649 +   apply (subgoal_tac "hnorm (y * x) < t * (r / t)", simp)
6.650 +   apply (subgoal_tac "0 < r / t")
6.651 +    apply (rule hnorm_mult_less)
6.652 +     apply assumption
6.653 +    apply (simp add: InfinitesimalD)
6.654 +   apply simp
6.655 +  apply (erule order_le_less_trans [OF hnorm_ge_zero])
6.656 +  done
6.657
6.658 -lemma Infinitesimal_scaleR2:
6.659 -  "x \<in> Infinitesimal ==> a *\<^sub>R x \<in> Infinitesimal"
6.660 -apply (case_tac "a = 0", simp)
6.661 -apply (rule InfinitesimalI)
6.662 -apply (drule InfinitesimalD)
6.663 -apply (drule_tac x="r / \<bar>star_of a\<bar>" in bspec)
6.665 -apply simp
6.666 -apply (simp add: hnorm_scaleR pos_less_divide_eq mult.commute)
6.667 -done
6.668 +lemma Infinitesimal_scaleR2: "x \<in> Infinitesimal \<Longrightarrow> a *\<^sub>R x \<in> Infinitesimal"
6.669 +  apply (case_tac "a = 0", simp)
6.670 +  apply (rule InfinitesimalI)
6.671 +  apply (drule InfinitesimalD)
6.672 +  apply (drule_tac x="r / \<bar>star_of a\<bar>" in bspec)
6.673 +   apply (simp add: Reals_eq_Standard)
6.674 +  apply simp
6.675 +  apply (simp add: hnorm_scaleR pos_less_divide_eq mult.commute)
6.676 +  done
6.677
6.678  lemma Compl_HFinite: "- HFinite = HInfinite"
6.679 -apply (auto simp add: HInfinite_def HFinite_def linorder_not_less)
6.680 -apply (rule_tac y="r + 1" in order_less_le_trans, simp)
6.681 -apply simp
6.682 -done
6.683 +  apply (auto simp add: HInfinite_def HFinite_def linorder_not_less)
6.684 +  apply (rule_tac y="r + 1" in order_less_le_trans, simp)
6.685 +  apply simp
6.686 +  done
6.687
6.688 -lemma HInfinite_inverse_Infinitesimal:
6.689 -  fixes x :: "'a::real_normed_div_algebra star"
6.690 -  shows "x \<in> HInfinite ==> inverse x \<in> Infinitesimal"
6.691 -apply (rule InfinitesimalI)
6.692 -apply (subgoal_tac "x \<noteq> 0")
6.693 -apply (rule inverse_less_imp_less)
6.695 -apply (simp add: HInfinite_def Reals_inverse)
6.696 -apply assumption
6.697 -apply (clarify, simp add: Compl_HFinite [symmetric])
6.698 -done
6.699 +lemma HInfinite_inverse_Infinitesimal: "x \<in> HInfinite \<Longrightarrow> inverse x \<in> Infinitesimal"
6.700 +  for x :: "'a::real_normed_div_algebra star"
6.701 +  apply (rule InfinitesimalI)
6.702 +  apply (subgoal_tac "x \<noteq> 0")
6.703 +   apply (rule inverse_less_imp_less)
6.704 +    apply (simp add: nonzero_hnorm_inverse)
6.705 +    apply (simp add: HInfinite_def Reals_inverse)
6.706 +   apply assumption
6.707 +  apply (clarify, simp add: Compl_HFinite [symmetric])
6.708 +  done
6.709
6.710  lemma HInfiniteI: "(\<And>r. r \<in> \<real> \<Longrightarrow> r < hnorm x) \<Longrightarrow> x \<in> HInfinite"
6.712 +  by (simp add: HInfinite_def)
6.713
6.714 -lemma HInfiniteD: "\<lbrakk>x \<in> HInfinite; r \<in> \<real>\<rbrakk> \<Longrightarrow> r < hnorm x"
6.716 +lemma HInfiniteD: "x \<in> HInfinite \<Longrightarrow> r \<in> \<real> \<Longrightarrow> r < hnorm x"
6.717 +  by (simp add: HInfinite_def)
6.718
6.719 -lemma HInfinite_mult:
6.720 -  fixes x y :: "'a::real_normed_div_algebra star"
6.721 -  shows "[|x \<in> HInfinite; y \<in> HInfinite|] ==> (x*y) \<in> HInfinite"
6.722 -apply (rule HInfiniteI, simp only: hnorm_mult)
6.723 -apply (subgoal_tac "r * 1 < hnorm x * hnorm y", simp only: mult_1)
6.724 -apply (case_tac "x = 0", simp add: HInfinite_def)
6.725 -apply (rule mult_strict_mono)
6.727 -done
6.728 +lemma HInfinite_mult: "x \<in> HInfinite \<Longrightarrow> y \<in> HInfinite \<Longrightarrow> x * y \<in> HInfinite"
6.729 +  for x y :: "'a::real_normed_div_algebra star"
6.730 +  apply (rule HInfiniteI, simp only: hnorm_mult)
6.731 +  apply (subgoal_tac "r * 1 < hnorm x * hnorm y", simp only: mult_1)
6.732 +  apply (case_tac "x = 0", simp add: HInfinite_def)
6.733 +  apply (rule mult_strict_mono)
6.734 +     apply (simp_all add: HInfiniteD)
6.735 +  done
6.736
6.737 -lemma hypreal_add_zero_less_le_mono: "[|r < x; (0::hypreal) \<le> y|] ==> r < x+y"
6.739 +lemma hypreal_add_zero_less_le_mono: "r < x \<Longrightarrow> 0 \<le> y \<Longrightarrow> r < x + y"
6.740 +  for r x y :: hypreal
6.741 +  by (auto dest: add_less_le_mono)
6.742
6.744 -     "[|(x::hypreal) \<in> HInfinite; 0 \<le> y; 0 \<le> x|] ==> (x + y): HInfinite"
6.747 +lemma HInfinite_add_ge_zero: "x \<in> HInfinite \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> x + y \<in> HInfinite"
6.748 +  for x y :: hypreal
6.749 +  by (auto simp: abs_if add.commute HInfinite_def)
6.750
6.752 -     "[|(x::hypreal) \<in> HInfinite; 0 \<le> y; 0 \<le> x|] ==> (y + x): HInfinite"
6.754 +lemma HInfinite_add_ge_zero2: "x \<in> HInfinite \<Longrightarrow> 0 \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> y + x \<in> HInfinite"
6.755 +  for x y :: hypreal
6.757
6.759 -     "[|(x::hypreal) \<in> HInfinite; 0 < y; 0 < x|] ==> (x + y): HInfinite"
6.760 -by (blast intro: HInfinite_add_ge_zero order_less_imp_le)
6.761 +lemma HInfinite_add_gt_zero: "x \<in> HInfinite \<Longrightarrow> 0 < y \<Longrightarrow> 0 < x \<Longrightarrow> x + y \<in> HInfinite"
6.762 +  for x y :: hypreal
6.763 +  by (blast intro: HInfinite_add_ge_zero order_less_imp_le)
6.764
6.765 -lemma HInfinite_minus_iff: "(-x \<in> HInfinite) = (x \<in> HInfinite)"
6.767 +lemma HInfinite_minus_iff: "- x \<in> HInfinite \<longleftrightarrow> x \<in> HInfinite"
6.768 +  by (simp add: HInfinite_def)
6.769
6.771 -     "[|(x::hypreal) \<in> HInfinite; y \<le> 0; x \<le> 0|] ==> (x + y): HInfinite"
6.772 -apply (drule HInfinite_minus_iff [THEN iffD2])
6.773 -apply (rule HInfinite_minus_iff [THEN iffD1])
6.776 -apply simp_all
6.777 -done
6.778 +lemma HInfinite_add_le_zero: "x \<in> HInfinite \<Longrightarrow> y \<le> 0 \<Longrightarrow> x \<le> 0 \<Longrightarrow> x + y \<in> HInfinite"
6.779 +  for x y :: hypreal
6.780 +  apply (drule HInfinite_minus_iff [THEN iffD2])
6.781 +  apply (rule HInfinite_minus_iff [THEN iffD1])
6.784 +    apply simp_all
6.785 +  done
6.786
6.788 -     "[|(x::hypreal) \<in> HInfinite; y < 0; x < 0|] ==> (x + y): HInfinite"
6.789 -by (blast intro: HInfinite_add_le_zero order_less_imp_le)
6.790 +lemma HInfinite_add_lt_zero: "x \<in> HInfinite \<Longrightarrow> y < 0 \<Longrightarrow> x < 0 \<Longrightarrow> x + y \<in> HInfinite"
6.791 +  for x y :: hypreal
6.792 +  by (blast intro: HInfinite_add_le_zero order_less_imp_le)
6.793
6.794  lemma HFinite_sum_squares:
6.795 -  fixes a b c :: "'a::real_normed_algebra star"
6.796 -  shows "[|a: HFinite; b: HFinite; c: HFinite|]
6.797 -      ==> a*a + b*b + c*c \<in> HFinite"
6.798 -by (auto intro: HFinite_mult HFinite_add)
6.799 +  "a \<in> HFinite \<Longrightarrow> b \<in> HFinite \<Longrightarrow> c \<in> HFinite \<Longrightarrow> a * a + b * b + c * c \<in> HFinite"
6.800 +  for a b c :: "'a::real_normed_algebra star"
6.801 +  by (auto intro: HFinite_mult HFinite_add)
6.802
6.803 -lemma not_Infinitesimal_not_zero: "x \<notin> Infinitesimal ==> x \<noteq> 0"
6.804 -by auto
6.805 +lemma not_Infinitesimal_not_zero: "x \<notin> Infinitesimal \<Longrightarrow> x \<noteq> 0"
6.806 +  by auto
6.807
6.808 -lemma not_Infinitesimal_not_zero2: "x \<in> HFinite - Infinitesimal ==> x \<noteq> 0"
6.809 -by auto
6.810 +lemma not_Infinitesimal_not_zero2: "x \<in> HFinite - Infinitesimal \<Longrightarrow> x \<noteq> 0"
6.811 +  by auto
6.812
6.813  lemma HFinite_diff_Infinitesimal_hrabs:
6.814 -  "(x::hypreal) \<in> HFinite - Infinitesimal ==> \<bar>x\<bar> \<in> HFinite - Infinitesimal"
6.815 -by blast
6.816 +  "x \<in> HFinite - Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<in> HFinite - Infinitesimal"
6.817 +  for x :: hypreal
6.818 +  by blast
6.819
6.820 -lemma hnorm_le_Infinitesimal:
6.821 -  "\<lbrakk>e \<in> Infinitesimal; hnorm x \<le> e\<rbrakk> \<Longrightarrow> x \<in> Infinitesimal"
6.822 -by (auto simp add: Infinitesimal_def abs_less_iff)
6.823 +lemma hnorm_le_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> hnorm x \<le> e \<Longrightarrow> x \<in> Infinitesimal"
6.824 +  by (auto simp: Infinitesimal_def abs_less_iff)
6.825
6.826 -lemma hnorm_less_Infinitesimal:
6.827 -  "\<lbrakk>e \<in> Infinitesimal; hnorm x < e\<rbrakk> \<Longrightarrow> x \<in> Infinitesimal"
6.828 -by (erule hnorm_le_Infinitesimal, erule order_less_imp_le)
6.829 +lemma hnorm_less_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> hnorm x < e \<Longrightarrow> x \<in> Infinitesimal"
6.830 +  by (erule hnorm_le_Infinitesimal, erule order_less_imp_le)
6.831
6.832 -lemma hrabs_le_Infinitesimal:
6.833 -     "[| e \<in> Infinitesimal; \<bar>x::hypreal\<bar> \<le> e |] ==> x \<in> Infinitesimal"
6.834 -by (erule hnorm_le_Infinitesimal, simp)
6.835 +lemma hrabs_le_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<le> e \<Longrightarrow> x \<in> Infinitesimal"
6.836 +  for x :: hypreal
6.837 +  by (erule hnorm_le_Infinitesimal) simp
6.838
6.839 -lemma hrabs_less_Infinitesimal:
6.840 -      "[| e \<in> Infinitesimal; \<bar>x::hypreal\<bar> < e |] ==> x \<in> Infinitesimal"
6.841 -by (erule hnorm_less_Infinitesimal, simp)
6.842 +lemma hrabs_less_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> < e \<Longrightarrow> x \<in> Infinitesimal"
6.843 +  for x :: hypreal
6.844 +  by (erule hnorm_less_Infinitesimal) simp
6.845
6.846  lemma Infinitesimal_interval:
6.847 -      "[| e \<in> Infinitesimal; e' \<in> Infinitesimal; e' < x ; x < e |]
6.848 -       ==> (x::hypreal) \<in> Infinitesimal"
6.849 -by (auto simp add: Infinitesimal_def abs_less_iff)
6.850 +  "e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> e' < x \<Longrightarrow> x < e \<Longrightarrow> x \<in> Infinitesimal"
6.851 +  for x :: hypreal
6.852 +  by (auto simp add: Infinitesimal_def abs_less_iff)
6.853
6.854  lemma Infinitesimal_interval2:
6.855 -     "[| e \<in> Infinitesimal; e' \<in> Infinitesimal;
6.856 -         e' \<le> x ; x \<le> e |] ==> (x::hypreal) \<in> Infinitesimal"
6.857 -by (auto intro: Infinitesimal_interval simp add: order_le_less)
6.858 +  "e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> e' \<le> x \<Longrightarrow> x \<le> e \<Longrightarrow> x \<in> Infinitesimal"
6.859 +  for x :: hypreal
6.860 +  by (auto intro: Infinitesimal_interval simp add: order_le_less)
6.861
6.862
6.863 -lemma lemma_Infinitesimal_hyperpow:
6.864 -     "[| (x::hypreal) \<in> Infinitesimal; 0 < N |] ==> \<bar>x pow N\<bar> \<le> \<bar>x\<bar>"
6.865 -apply (unfold Infinitesimal_def)
6.866 -apply (auto intro!: hyperpow_Suc_le_self2
6.867 -          simp add: hyperpow_hrabs [symmetric] hypnat_gt_zero_iff2 abs_ge_zero)
6.868 -done
6.869 +lemma lemma_Infinitesimal_hyperpow: "x \<in> Infinitesimal \<Longrightarrow> 0 < N \<Longrightarrow> \<bar>x pow N\<bar> \<le> \<bar>x\<bar>"
6.870 +  for x :: hypreal
6.871 +  apply (unfold Infinitesimal_def)
6.872 +  apply (auto intro!: hyperpow_Suc_le_self2
6.873 +      simp: hyperpow_hrabs [symmetric] hypnat_gt_zero_iff2 abs_ge_zero)
6.874 +  done
6.875
6.876 -lemma Infinitesimal_hyperpow:
6.877 -     "[| (x::hypreal) \<in> Infinitesimal; 0 < N |] ==> x pow N \<in> Infinitesimal"
6.878 -apply (rule hrabs_le_Infinitesimal)
6.879 -apply (rule_tac [2] lemma_Infinitesimal_hyperpow, auto)
6.880 -done
6.881 +lemma Infinitesimal_hyperpow: "x \<in> Infinitesimal \<Longrightarrow> 0 < N \<Longrightarrow> x pow N \<in> Infinitesimal"
6.882 +  for x :: hypreal
6.883 +  apply (rule hrabs_le_Infinitesimal)
6.884 +   apply (rule_tac [2] lemma_Infinitesimal_hyperpow)
6.885 +  apply auto
6.886 +  done
6.887
6.888  lemma hrealpow_hyperpow_Infinitesimal_iff:
6.889 -     "(x ^ n \<in> Infinitesimal) = (x pow (hypnat_of_nat n) \<in> Infinitesimal)"
6.890 -by (simp only: hyperpow_hypnat_of_nat)
6.891 +  "(x ^ n \<in> Infinitesimal) \<longleftrightarrow> x pow (hypnat_of_nat n) \<in> Infinitesimal"
6.892 +  by (simp only: hyperpow_hypnat_of_nat)
6.893
6.894 -lemma Infinitesimal_hrealpow:
6.895 -     "[| (x::hypreal) \<in> Infinitesimal; 0 < n |] ==> x ^ n \<in> Infinitesimal"
6.896 -by (simp add: hrealpow_hyperpow_Infinitesimal_iff Infinitesimal_hyperpow)
6.897 +lemma Infinitesimal_hrealpow: "x \<in> Infinitesimal \<Longrightarrow> 0 < n \<Longrightarrow> x ^ n \<in> Infinitesimal"
6.898 +  for x :: hypreal
6.899 +  by (simp add: hrealpow_hyperpow_Infinitesimal_iff Infinitesimal_hyperpow)
6.900
6.901  lemma not_Infinitesimal_mult:
6.902 -  fixes x y :: "'a::real_normed_div_algebra star"
6.903 -  shows "[| x \<notin> Infinitesimal;  y \<notin> Infinitesimal|] ==> (x*y) \<notin>Infinitesimal"
6.904 -apply (unfold Infinitesimal_def, clarify, rename_tac r s)
6.905 -apply (simp only: linorder_not_less hnorm_mult)
6.906 -apply (drule_tac x = "r * s" in bspec)
6.907 -apply (fast intro: Reals_mult)
6.908 -apply (simp)
6.909 -apply (drule_tac c = s and d = "hnorm y" and a = r and b = "hnorm x" in mult_mono)
6.910 -apply (simp_all (no_asm_simp))
6.911 -done
6.912 +  "x \<notin> Infinitesimal \<Longrightarrow> y \<notin> Infinitesimal \<Longrightarrow> x * y \<notin> Infinitesimal"
6.913 +  for x y :: "'a::real_normed_div_algebra star"
6.914 +  apply (unfold Infinitesimal_def, clarify, rename_tac r s)
6.915 +  apply (simp only: linorder_not_less hnorm_mult)
6.916 +  apply (drule_tac x = "r * s" in bspec)
6.917 +   apply (fast intro: Reals_mult)
6.918 +  apply simp
6.919 +  apply (drule_tac c = s and d = "hnorm y" and a = r and b = "hnorm x" in mult_mono)
6.920 +     apply simp_all
6.921 +  done
6.922
6.923 -lemma Infinitesimal_mult_disj:
6.924 -  fixes x y :: "'a::real_normed_div_algebra star"
6.925 -  shows "x*y \<in> Infinitesimal ==> x \<in> Infinitesimal | y \<in> Infinitesimal"
6.926 -apply (rule ccontr)
6.927 -apply (drule de_Morgan_disj [THEN iffD1])
6.928 -apply (fast dest: not_Infinitesimal_mult)
6.929 -done
6.930 +lemma Infinitesimal_mult_disj: "x * y \<in> Infinitesimal \<Longrightarrow> x \<in> Infinitesimal \<or> y \<in> Infinitesimal"
6.931 +  for x y :: "'a::real_normed_div_algebra star"
6.932 +  apply (rule ccontr)
6.933 +  apply (drule de_Morgan_disj [THEN iffD1])
6.934 +  apply (fast dest: not_Infinitesimal_mult)
6.935 +  done
6.936
6.937 -lemma HFinite_Infinitesimal_not_zero: "x \<in> HFinite-Infinitesimal ==> x \<noteq> 0"
6.938 -by blast
6.939 +lemma HFinite_Infinitesimal_not_zero: "x \<in> HFinite-Infinitesimal \<Longrightarrow> x \<noteq> 0"
6.940 +  by blast
6.941
6.942  lemma HFinite_Infinitesimal_diff_mult:
6.943 -  fixes x y :: "'a::real_normed_div_algebra star"
6.944 -  shows "[| x \<in> HFinite - Infinitesimal;
6.945 -                   y \<in> HFinite - Infinitesimal
6.946 -                |] ==> (x*y) \<in> HFinite - Infinitesimal"
6.947 -apply clarify
6.948 -apply (blast dest: HFinite_mult not_Infinitesimal_mult)
6.949 -done
6.950 +  "x \<in> HFinite - Infinitesimal \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> x * y \<in> HFinite - Infinitesimal"
6.951 +  for x y :: "'a::real_normed_div_algebra star"
6.952 +  apply clarify
6.953 +  apply (blast dest: HFinite_mult not_Infinitesimal_mult)
6.954 +  done
6.955
6.956 -lemma Infinitesimal_subset_HFinite:
6.957 -      "Infinitesimal \<subseteq> HFinite"
6.958 -apply (simp add: Infinitesimal_def HFinite_def, auto)
6.959 -apply (rule_tac x = 1 in bexI, auto)
6.960 -done
6.961 +lemma Infinitesimal_subset_HFinite: "Infinitesimal \<subseteq> HFinite"
6.962 +  apply (simp add: Infinitesimal_def HFinite_def)
6.963 +  apply auto
6.964 +  apply (rule_tac x = 1 in bexI)
6.965 +  apply auto
6.966 +  done
6.967
6.968 -lemma Infinitesimal_star_of_mult:
6.969 -  fixes x :: "'a::real_normed_algebra star"
6.970 -  shows "x \<in> Infinitesimal ==> x * star_of r \<in> Infinitesimal"
6.971 -by (erule HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult])
6.972 +lemma Infinitesimal_star_of_mult: "x \<in> Infinitesimal \<Longrightarrow> x * star_of r \<in> Infinitesimal"
6.973 +  for x :: "'a::real_normed_algebra star"
6.974 +  by (erule HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult])
6.975
6.976 -lemma Infinitesimal_star_of_mult2:
6.977 -  fixes x :: "'a::real_normed_algebra star"
6.978 -  shows "x \<in> Infinitesimal ==> star_of r * x \<in> Infinitesimal"
6.979 -by (erule HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult2])
6.980 +lemma Infinitesimal_star_of_mult2: "x \<in> Infinitesimal \<Longrightarrow> star_of r * x \<in> Infinitesimal"
6.981 +  for x :: "'a::real_normed_algebra star"
6.982 +  by (erule HFinite_star_of [THEN [2] Infinitesimal_HFinite_mult2])
6.983
6.984
6.985 -subsection\<open>The Infinitely Close Relation\<close>
6.986 +subsection \<open>The Infinitely Close Relation\<close>
6.987
6.988 -lemma mem_infmal_iff: "(x \<in> Infinitesimal) = (x \<approx> 0)"
6.989 -by (simp add: Infinitesimal_def approx_def)
6.990 +lemma mem_infmal_iff: "x \<in> Infinitesimal \<longleftrightarrow> x \<approx> 0"
6.991 +  by (simp add: Infinitesimal_def approx_def)
6.992
6.993 -lemma approx_minus_iff: " (x \<approx> y) = (x - y \<approx> 0)"
6.995 +lemma approx_minus_iff: "x \<approx> y \<longleftrightarrow> x - y \<approx> 0"
6.996 +  by (simp add: approx_def)
6.997
6.998 -lemma approx_minus_iff2: " (x \<approx> y) = (-y + x \<approx> 0)"
6.1000 +lemma approx_minus_iff2: "x \<approx> y \<longleftrightarrow> - y + x \<approx> 0"
6.1002
6.1003  lemma approx_refl [iff]: "x \<approx> x"
6.1004 -by (simp add: approx_def Infinitesimal_def)
6.1005 +  by (simp add: approx_def Infinitesimal_def)
6.1006
6.1007 -lemma hypreal_minus_distrib1: "-(y + -(x::'a::ab_group_add)) = x + -y"
6.1009 +lemma hypreal_minus_distrib1: "- (y + - x) = x + -y"
6.1010 +  for x y :: "'a::ab_group_add"
6.1012
6.1013 -lemma approx_sym: "x \<approx> y ==> y \<approx> x"
6.1015 -apply (drule Infinitesimal_minus_iff [THEN iffD2])
6.1016 -apply simp
6.1017 -done
6.1018 +lemma approx_sym: "x \<approx> y \<Longrightarrow> y \<approx> x"
6.1019 +  apply (simp add: approx_def)
6.1020 +  apply (drule Infinitesimal_minus_iff [THEN iffD2])
6.1021 +  apply simp
6.1022 +  done
6.1023
6.1024 -lemma approx_trans: "[| x \<approx> y; y \<approx> z |] ==> x \<approx> z"
6.1027 -apply simp
6.1028 -done
6.1029 +lemma approx_trans: "x \<approx> y \<Longrightarrow> y \<approx> z \<Longrightarrow> x \<approx> z"
6.1030 +  apply (simp add: approx_def)
6.1031 +  apply (drule (1) Infinitesimal_add)
6.1032 +  apply simp
6.1033 +  done
6.1034
6.1035 -lemma approx_trans2: "[| r \<approx> x; s \<approx> x |] ==> r \<approx> s"
6.1036 -by (blast intro: approx_sym approx_trans)
6.1037 +lemma approx_trans2: "r \<approx> x \<Longrightarrow> s \<approx> x \<Longrightarrow> r \<approx> s"
6.1038 +  by (blast intro: approx_sym approx_trans)
6.1039
6.1040 -lemma approx_trans3: "[| x \<approx> r; x \<approx> s|] ==> r \<approx> s"
6.1041 -by (blast intro: approx_sym approx_trans)
6.1042 +lemma approx_trans3: "x \<approx> r \<Longrightarrow> x \<approx> s \<Longrightarrow> r \<approx> s"
6.1043 +  by (blast intro: approx_sym approx_trans)
6.1044
6.1045 -lemma approx_reorient: "(x \<approx> y) = (y \<approx> x)"
6.1046 -by (blast intro: approx_sym)
6.1047 +lemma approx_reorient: "x \<approx> y \<longleftrightarrow> y \<approx> x"
6.1048 +  by (blast intro: approx_sym)
6.1049
6.1050 -(*reorientation simplification procedure: reorients (polymorphic)
6.1051 -  0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
6.1052 +text \<open>Reorientation simplification procedure: reorients (polymorphic)
6.1053 +  \<open>0 = x\<close>, \<open>1 = x\<close>, \<open>nnn = x\<close> provided \<open>x\<close> isn't \<open>0\<close>, \<open>1\<close> or a numeral.\<close>
6.1054  simproc_setup approx_reorient_simproc
6.1055    ("0 \<approx> x" | "1 \<approx> y" | "numeral w \<approx> z" | "- 1 \<approx> y" | "- numeral w \<approx> r") =
6.1056  \<open>
6.1057 @@ -658,1133 +593,1067 @@
6.1058    in proc end
6.1059  \<close>
6.1060
6.1061 -lemma Infinitesimal_approx_minus: "(x-y \<in> Infinitesimal) = (x \<approx> y)"
6.1062 -by (simp add: approx_minus_iff [symmetric] mem_infmal_iff)
6.1063 +lemma Infinitesimal_approx_minus: "x - y \<in> Infinitesimal \<longleftrightarrow> x \<approx> y"
6.1064 +  by (simp add: approx_minus_iff [symmetric] mem_infmal_iff)
6.1065
6.1068 -apply (auto dest: approx_sym elim!: approx_trans equalityCE)
6.1069 -done
6.1071 +  by (auto simp add: monad_def dest: approx_sym elim!: approx_trans equalityCE)
6.1072
6.1073 -lemma Infinitesimal_approx:
6.1074 -     "[| x \<in> Infinitesimal; y \<in> Infinitesimal |] ==> x \<approx> y"
6.1076 -apply (blast intro: approx_trans approx_sym)
6.1077 -done
6.1078 +lemma Infinitesimal_approx: "x \<in> Infinitesimal \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x \<approx> y"
6.1079 +  apply (simp add: mem_infmal_iff)
6.1080 +  apply (blast intro: approx_trans approx_sym)
6.1081 +  done
6.1082
6.1083 -lemma approx_add: "[| a \<approx> b; c \<approx> d |] ==> a+c \<approx> b+d"
6.1084 +lemma approx_add: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a + c \<approx> b + d"
6.1085  proof (unfold approx_def)
6.1086    assume inf: "a - b \<in> Infinitesimal" "c - d \<in> Infinitesimal"
6.1087    have "a + c - (b + d) = (a - b) + (c - d)" by simp
6.1088 -  also have "... \<in> Infinitesimal" using inf by (rule Infinitesimal_add)
6.1089 +  also have "... \<in> Infinitesimal"
6.1090 +    using inf by (rule Infinitesimal_add)
6.1091    finally show "a + c - (b + d) \<in> Infinitesimal" .
6.1092  qed
6.1093
6.1094 -lemma approx_minus: "a \<approx> b ==> -a \<approx> -b"
6.1095 -apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
6.1096 -apply (drule approx_minus_iff [THEN iffD1])
6.1098 -done
6.1099 +lemma approx_minus: "a \<approx> b \<Longrightarrow> - a \<approx> - b"
6.1100 +  apply (rule approx_minus_iff [THEN iffD2, THEN approx_sym])
6.1101 +  apply (drule approx_minus_iff [THEN iffD1])
6.1103 +  done
6.1104
6.1105 -lemma approx_minus2: "-a \<approx> -b ==> a \<approx> b"
6.1106 -by (auto dest: approx_minus)
6.1107 +lemma approx_minus2: "- a \<approx> - b \<Longrightarrow> a \<approx> b"
6.1108 +  by (auto dest: approx_minus)
6.1109
6.1110 -lemma approx_minus_cancel [simp]: "(-a \<approx> -b) = (a \<approx> b)"
6.1111 -by (blast intro: approx_minus approx_minus2)
6.1112 +lemma approx_minus_cancel [simp]: "- a \<approx> - b \<longleftrightarrow> a \<approx> b"
6.1113 +  by (blast intro: approx_minus approx_minus2)
6.1114
6.1115 -lemma approx_add_minus: "[| a \<approx> b; c \<approx> d |] ==> a + -c \<approx> b + -d"
6.1116 -by (blast intro!: approx_add approx_minus)
6.1117 +lemma approx_add_minus: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a + - c \<approx> b + - d"
6.1118 +  by (blast intro!: approx_add approx_minus)
6.1119
6.1120 -lemma approx_diff: "[| a \<approx> b; c \<approx> d |] ==> a - c \<approx> b - d"
6.1121 +lemma approx_diff: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> a - c \<approx> b - d"
6.1122    using approx_add [of a b "- c" "- d"] by simp
6.1123
6.1124 -lemma approx_mult1:
6.1125 -  fixes a b c :: "'a::real_normed_algebra star"
6.1126 -  shows "[| a \<approx> b; c: HFinite|] ==> a*c \<approx> b*c"
6.1127 -by (simp add: approx_def Infinitesimal_HFinite_mult
6.1128 -              left_diff_distrib [symmetric])
6.1129 +lemma approx_mult1: "a \<approx> b \<Longrightarrow> c \<in> HFinite \<Longrightarrow> a * c \<approx> b * c"
6.1130 +  for a b c :: "'a::real_normed_algebra star"
6.1131 +  by (simp add: approx_def Infinitesimal_HFinite_mult left_diff_distrib [symmetric])
6.1132 +
6.1133 +lemma approx_mult2: "a \<approx> b \<Longrightarrow> c \<in> HFinite \<Longrightarrow> c * a \<approx> c * b"
6.1134 +  for a b c :: "'a::real_normed_algebra star"
6.1135 +  by (simp add: approx_def Infinitesimal_HFinite_mult2 right_diff_distrib [symmetric])
6.1136
6.1137 -lemma approx_mult2:
6.1138 -  fixes a b c :: "'a::real_normed_algebra star"
6.1139 -  shows "[|a \<approx> b; c: HFinite|] ==> c*a \<approx> c*b"
6.1140 -by (simp add: approx_def Infinitesimal_HFinite_mult2
6.1141 -              right_diff_distrib [symmetric])
6.1142 +lemma approx_mult_subst: "u \<approx> v * x \<Longrightarrow> x \<approx> y \<Longrightarrow> v \<in> HFinite \<Longrightarrow> u \<approx> v * y"
6.1143 +  for u v x y :: "'a::real_normed_algebra star"
6.1144 +  by (blast intro: approx_mult2 approx_trans)
6.1145
6.1146 -lemma approx_mult_subst:
6.1147 -  fixes u v x y :: "'a::real_normed_algebra star"
6.1148 -  shows "[|u \<approx> v*x; x \<approx> y; v \<in> HFinite|] ==> u \<approx> v*y"
6.1149 -by (blast intro: approx_mult2 approx_trans)
6.1150 -
6.1151 -lemma approx_mult_subst2:
6.1152 -  fixes u v x y :: "'a::real_normed_algebra star"
6.1153 -  shows "[| u \<approx> x*v; x \<approx> y; v \<in> HFinite |] ==> u \<approx> y*v"
6.1154 -by (blast intro: approx_mult1 approx_trans)
6.1155 +lemma approx_mult_subst2: "u \<approx> x * v \<Longrightarrow> x \<approx> y \<Longrightarrow> v \<in> HFinite \<Longrightarrow> u \<approx> y * v"
6.1156 +  for u v x y :: "'a::real_normed_algebra star"
6.1157 +  by (blast intro: approx_mult1 approx_trans)
6.1158
6.1159 -lemma approx_mult_subst_star_of:
6.1160 -  fixes u x y :: "'a::real_normed_algebra star"
6.1161 -  shows "[| u \<approx> x*star_of v; x \<approx> y |] ==> u \<approx> y*star_of v"
6.1162 -by (auto intro: approx_mult_subst2)
6.1163 +lemma approx_mult_subst_star_of: "u \<approx> x * star_of v \<Longrightarrow> x \<approx> y \<Longrightarrow> u \<approx> y * star_of v"
6.1164 +  for u x y :: "'a::real_normed_algebra star"
6.1165 +  by (auto intro: approx_mult_subst2)
6.1166
6.1167 -lemma approx_eq_imp: "a = b ==> a \<approx> b"
6.1169 +lemma approx_eq_imp: "a = b \<Longrightarrow> a \<approx> b"
6.1170 +  by (simp add: approx_def)
6.1171
6.1172 -lemma Infinitesimal_minus_approx: "x \<in> Infinitesimal ==> -x \<approx> x"
6.1173 -by (blast intro: Infinitesimal_minus_iff [THEN iffD2]
6.1174 -                    mem_infmal_iff [THEN iffD1] approx_trans2)
6.1175 +lemma Infinitesimal_minus_approx: "x \<in> Infinitesimal \<Longrightarrow> - x \<approx> x"
6.1176 +  by (blast intro: Infinitesimal_minus_iff [THEN iffD2] mem_infmal_iff [THEN iffD1] approx_trans2)
6.1177
6.1178 -lemma bex_Infinitesimal_iff: "(\<exists>y \<in> Infinitesimal. x - z = y) = (x \<approx> z)"
6.1180 +lemma bex_Infinitesimal_iff: "(\<exists>y \<in> Infinitesimal. x - z = y) \<longleftrightarrow> x \<approx> z"
6.1181 +  by (simp add: approx_def)
6.1182
6.1183 -lemma bex_Infinitesimal_iff2: "(\<exists>y \<in> Infinitesimal. x = z + y) = (x \<approx> z)"
6.1184 -by (force simp add: bex_Infinitesimal_iff [symmetric])
6.1185 +lemma bex_Infinitesimal_iff2: "(\<exists>y \<in> Infinitesimal. x = z + y) \<longleftrightarrow> x \<approx> z"
6.1186 +  by (force simp add: bex_Infinitesimal_iff [symmetric])
6.1187
6.1188 -lemma Infinitesimal_add_approx: "[| y \<in> Infinitesimal; x + y = z |] ==> x \<approx> z"
6.1189 -apply (rule bex_Infinitesimal_iff [THEN iffD1])
6.1190 -apply (drule Infinitesimal_minus_iff [THEN iffD2])
6.1192 -done
6.1193 +lemma Infinitesimal_add_approx: "y \<in> Infinitesimal \<Longrightarrow> x + y = z \<Longrightarrow> x \<approx> z"
6.1194 +  apply (rule bex_Infinitesimal_iff [THEN iffD1])
6.1195 +  apply (drule Infinitesimal_minus_iff [THEN iffD2])
6.1197 +  done
6.1198
6.1199 -lemma Infinitesimal_add_approx_self: "y \<in> Infinitesimal ==> x \<approx> x + y"
6.1200 -apply (rule bex_Infinitesimal_iff [THEN iffD1])
6.1201 -apply (drule Infinitesimal_minus_iff [THEN iffD2])
6.1203 -done
6.1204 +lemma Infinitesimal_add_approx_self: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> x + y"
6.1205 +  apply (rule bex_Infinitesimal_iff [THEN iffD1])
6.1206 +  apply (drule Infinitesimal_minus_iff [THEN iffD2])
6.1208 +  done
6.1209
6.1210 -lemma Infinitesimal_add_approx_self2: "y \<in> Infinitesimal ==> x \<approx> y + x"
6.1212 +lemma Infinitesimal_add_approx_self2: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> y + x"
6.1214
6.1215 -lemma Infinitesimal_add_minus_approx_self: "y \<in> Infinitesimal ==> x \<approx> x + -y"
6.1216 -by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])
6.1217 +lemma Infinitesimal_add_minus_approx_self: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> x + - y"
6.1218 +  by (blast intro!: Infinitesimal_add_approx_self Infinitesimal_minus_iff [THEN iffD2])
6.1219
6.1220 -lemma Infinitesimal_add_cancel: "[| y \<in> Infinitesimal; x+y \<approx> z|] ==> x \<approx> z"
6.1221 -apply (drule_tac x = x in Infinitesimal_add_approx_self [THEN approx_sym])
6.1222 -apply (erule approx_trans3 [THEN approx_sym], assumption)
6.1223 -done
6.1224 +lemma Infinitesimal_add_cancel: "y \<in> Infinitesimal \<Longrightarrow> x + y \<approx> z \<Longrightarrow> x \<approx> z"
6.1225 +  apply (drule_tac x = x in Infinitesimal_add_approx_self [THEN approx_sym])
6.1226 +  apply (erule approx_trans3 [THEN approx_sym], assumption)
6.1227 +  done
6.1228
6.1230 -     "[| y \<in> Infinitesimal; x \<approx> z + y|] ==> x \<approx> z"
6.1231 -apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
6.1232 -apply (erule approx_trans3 [THEN approx_sym])
6.1234 -apply (erule approx_sym)
6.1235 -done
6.1236 +lemma Infinitesimal_add_right_cancel: "y \<in> Infinitesimal \<Longrightarrow> x \<approx> z + y \<Longrightarrow> x \<approx> z"
6.1237 +  apply (drule_tac x = z in Infinitesimal_add_approx_self2 [THEN approx_sym])
6.1238 +  apply (erule approx_trans3 [THEN approx_sym])
6.1240 +  apply (erule approx_sym)
6.1241 +  done
6.1242
6.1243 -lemma approx_add_left_cancel: "d + b  \<approx> d + c ==> b \<approx> c"
6.1244 -apply (drule approx_minus_iff [THEN iffD1])
6.1245 -apply (simp add: approx_minus_iff [symmetric] ac_simps)
6.1246 -done
6.1247 +lemma approx_add_left_cancel: "d + b  \<approx> d + c \<Longrightarrow> b \<approx> c"
6.1248 +  apply (drule approx_minus_iff [THEN iffD1])
6.1249 +  apply (simp add: approx_minus_iff [symmetric] ac_simps)
6.1250 +  done
6.1251
6.1252 -lemma approx_add_right_cancel: "b + d \<approx> c + d ==> b \<approx> c"
6.1255 -done
6.1256 +lemma approx_add_right_cancel: "b + d \<approx> c + d \<Longrightarrow> b \<approx> c"
6.1259 +  done
6.1260
6.1261 -lemma approx_add_mono1: "b \<approx> c ==> d + b \<approx> d + c"
6.1262 -apply (rule approx_minus_iff [THEN iffD2])
6.1263 -apply (simp add: approx_minus_iff [symmetric] ac_simps)
6.1264 -done
6.1265 +lemma approx_add_mono1: "b \<approx> c \<Longrightarrow> d + b \<approx> d + c"
6.1266 +  apply (rule approx_minus_iff [THEN iffD2])
6.1267 +  apply (simp add: approx_minus_iff [symmetric] ac_simps)
6.1268 +  done
6.1269
6.1270 -lemma approx_add_mono2: "b \<approx> c ==> b + a \<approx> c + a"
6.1272 +lemma approx_add_mono2: "b \<approx> c \<Longrightarrow> b + a \<approx> c + a"
6.1274
6.1275 -lemma approx_add_left_iff [simp]: "(a + b \<approx> a + c) = (b \<approx> c)"
6.1277 +lemma approx_add_left_iff [simp]: "a + b \<approx> a + c \<longleftrightarrow> b \<approx> c"
6.1279
6.1280 -lemma approx_add_right_iff [simp]: "(b + a \<approx> c + a) = (b \<approx> c)"
6.1282 +lemma approx_add_right_iff [simp]: "b + a \<approx> c + a \<longleftrightarrow> b \<approx> c"
6.1284
6.1285 -lemma approx_HFinite: "[| x \<in> HFinite; x \<approx> y |] ==> y \<in> HFinite"
6.1286 -apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
6.1287 -apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
6.1290 -done
6.1291 +lemma approx_HFinite: "x \<in> HFinite \<Longrightarrow> x \<approx> y \<Longrightarrow> y \<in> HFinite"
6.1292 +  apply (drule bex_Infinitesimal_iff2 [THEN iffD2], safe)
6.1293 +  apply (drule Infinitesimal_subset_HFinite [THEN subsetD, THEN HFinite_minus_iff [THEN iffD2]])
6.1296 +  done
6.1297
6.1298 -lemma approx_star_of_HFinite: "x \<approx> star_of D ==> x \<in> HFinite"
6.1299 -by (rule approx_sym [THEN [2] approx_HFinite], auto)
6.1300 +lemma approx_star_of_HFinite: "x \<approx> star_of D \<Longrightarrow> x \<in> HFinite"
6.1301 +  by (rule approx_sym [THEN [2] approx_HFinite], auto)
6.1302
6.1303 -lemma approx_mult_HFinite:
6.1304 -  fixes a b c d :: "'a::real_normed_algebra star"
6.1305 -  shows "[|a \<approx> b; c \<approx> d; b: HFinite; d: HFinite|] ==> a*c \<approx> b*d"
6.1306 -apply (rule approx_trans)
6.1307 -apply (rule_tac [2] approx_mult2)
6.1308 -apply (rule approx_mult1)
6.1309 -prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
6.1310 -done
6.1311 +lemma approx_mult_HFinite: "a \<approx> b \<Longrightarrow> c \<approx> d \<Longrightarrow> b \<in> HFinite \<Longrightarrow> d \<in> HFinite \<Longrightarrow> a * c \<approx> b * d"
6.1312 +  for a b c d :: "'a::real_normed_algebra star"
6.1313 +  apply (rule approx_trans)
6.1314 +   apply (rule_tac [2] approx_mult2)
6.1315 +    apply (rule approx_mult1)
6.1316 +     prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
6.1317 +  done
6.1318
6.1319 -lemma scaleHR_left_diff_distrib:
6.1320 -  "\<And>a b x. scaleHR (a - b) x = scaleHR a x - scaleHR b x"
6.1321 -by transfer (rule scaleR_left_diff_distrib)
6.1322 +lemma scaleHR_left_diff_distrib: "\<And>a b x. scaleHR (a - b) x = scaleHR a x - scaleHR b x"
6.1323 +  by transfer (rule scaleR_left_diff_distrib)
6.1324
6.1325 -lemma approx_scaleR1:
6.1326 -  "[| a \<approx> star_of b; c: HFinite|] ==> scaleHR a c \<approx> b *\<^sub>R c"
6.1327 -apply (unfold approx_def)
6.1328 -apply (drule (1) Infinitesimal_HFinite_scaleHR)
6.1329 -apply (simp only: scaleHR_left_diff_distrib)
6.1330 -apply (simp add: scaleHR_def star_scaleR_def [symmetric])
6.1331 -done
6.1332 +lemma approx_scaleR1: "a \<approx> star_of b \<Longrightarrow> c \<in> HFinite \<Longrightarrow> scaleHR a c \<approx> b *\<^sub>R c"
6.1333 +  apply (unfold approx_def)
6.1334 +  apply (drule (1) Infinitesimal_HFinite_scaleHR)
6.1335 +  apply (simp only: scaleHR_left_diff_distrib)
6.1336 +  apply (simp add: scaleHR_def star_scaleR_def [symmetric])
6.1337 +  done
6.1338
6.1339 -lemma approx_scaleR2:
6.1340 -  "a \<approx> b ==> c *\<^sub>R a \<approx> c *\<^sub>R b"
6.1341 -by (simp add: approx_def Infinitesimal_scaleR2
6.1342 -              scaleR_right_diff_distrib [symmetric])
6.1343 +lemma approx_scaleR2: "a \<approx> b \<Longrightarrow> c *\<^sub>R a \<approx> c *\<^sub>R b"
6.1344 +  by (simp add: approx_def Infinitesimal_scaleR2 scaleR_right_diff_distrib [symmetric])
6.1345 +
6.1346 +lemma approx_scaleR_HFinite: "a \<approx> star_of b \<Longrightarrow> c \<approx> d \<Longrightarrow> d \<in> HFinite \<Longrightarrow> scaleHR a c \<approx> b *\<^sub>R d"
6.1347 +  apply (rule approx_trans)
6.1348 +   apply (rule_tac [2] approx_scaleR2)
6.1349 +   apply (rule approx_scaleR1)
6.1350 +    prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
6.1351 +  done
6.1352
6.1353 -lemma approx_scaleR_HFinite:
6.1354 -  "[|a \<approx> star_of b; c \<approx> d; d: HFinite|] ==> scaleHR a c \<approx> b *\<^sub>R d"
6.1355 -apply (rule approx_trans)
6.1356 -apply (rule_tac [2] approx_scaleR2)
6.1357 -apply (rule approx_scaleR1)
6.1358 -prefer 2 apply (blast intro: approx_HFinite approx_sym, auto)
6.1359 -done
6.1360 +lemma approx_mult_star_of: "a \<approx> star_of b \<Longrightarrow> c \<approx> star_of d \<Longrightarrow> a * c \<approx> star_of b * star_of d"
6.1361 +  for a c :: "'a::real_normed_algebra star"
6.1362 +  by (blast intro!: approx_mult_HFinite approx_star_of_HFinite HFinite_star_of)
6.1363 +
6.1364 +lemma approx_SReal_mult_cancel_zero: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * x \<approx> 0 \<Longrightarrow> x \<approx> 0"
6.1365 +  for a x :: hypreal
6.1366 +  apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
6.1367 +  apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
6.1368 +  done
6.1369
6.1370 -lemma approx_mult_star_of:
6.1371 -  fixes a c :: "'a::real_normed_algebra star"
6.1372 -  shows "[|a \<approx> star_of b; c \<approx> star_of d |]
6.1373 -      ==> a*c \<approx> star_of b*star_of d"
6.1374 -by (blast intro!: approx_mult_HFinite approx_star_of_HFinite HFinite_star_of)
6.1375 +lemma approx_mult_SReal1: "a \<in> \<real> \<Longrightarrow> x \<approx> 0 \<Longrightarrow> x * a \<approx> 0"
6.1376 +  for a x :: hypreal
6.1377 +  by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)
6.1378
6.1379 -lemma approx_SReal_mult_cancel_zero:
6.1380 -     "[| (a::hypreal) \<in> \<real>; a \<noteq> 0; a*x \<approx> 0 |] ==> x \<approx> 0"
6.1381 -apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
6.1382 -apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
6.1383 -done
6.1384 +lemma approx_mult_SReal2: "a \<in> \<real> \<Longrightarrow> x \<approx> 0 \<Longrightarrow> a * x \<approx> 0"
6.1385 +  for a x :: hypreal
6.1386 +  by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)
6.1387
6.1388 -lemma approx_mult_SReal1: "[| (a::hypreal) \<in> \<real>; x \<approx> 0 |] ==> x*a \<approx> 0"
6.1389 -by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult1)
6.1390 -
6.1391 -lemma approx_mult_SReal2: "[| (a::hypreal) \<in> \<real>; x \<approx> 0 |] ==> a*x \<approx> 0"
6.1392 -by (auto dest: SReal_subset_HFinite [THEN subsetD] approx_mult2)
6.1393 +lemma approx_mult_SReal_zero_cancel_iff [simp]: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * x \<approx> 0 \<longleftrightarrow> x \<approx> 0"
6.1394 +  for a x :: hypreal
6.1395 +  by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)
6.1396
6.1397 -lemma approx_mult_SReal_zero_cancel_iff [simp]:
6.1398 -     "[|(a::hypreal) \<in> \<real>; a \<noteq> 0 |] ==> (a*x \<approx> 0) = (x \<approx> 0)"
6.1399 -by (blast intro: approx_SReal_mult_cancel_zero approx_mult_SReal2)
6.1400 +lemma approx_SReal_mult_cancel: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * w \<approx> a * z \<Longrightarrow> w \<approx> z"
6.1401 +  for a w z :: hypreal
6.1402 +  apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
6.1403 +  apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
6.1404 +  done
6.1405
6.1406 -lemma approx_SReal_mult_cancel:
6.1407 -     "[| (a::hypreal) \<in> \<real>; a \<noteq> 0; a* w \<approx> a*z |] ==> w \<approx> z"
6.1408 -apply (drule Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
6.1409 -apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
6.1410 -done
6.1411 +lemma approx_SReal_mult_cancel_iff1 [simp]: "a \<in> \<real> \<Longrightarrow> a \<noteq> 0 \<Longrightarrow> a * w \<approx> a * z \<longleftrightarrow> w \<approx> z"
6.1412 +  for a w z :: hypreal
6.1413 +  by (auto intro!: approx_mult2 SReal_subset_HFinite [THEN subsetD]
6.1414 +      intro: approx_SReal_mult_cancel)
6.1415
6.1416 -lemma approx_SReal_mult_cancel_iff1 [simp]:
6.1417 -     "[| (a::hypreal) \<in> \<real>; a \<noteq> 0|] ==> (a* w \<approx> a*z) = (w \<approx> z)"
6.1418 -by (auto intro!: approx_mult2 SReal_subset_HFinite [THEN subsetD]
6.1419 -         intro: approx_SReal_mult_cancel)
6.1420 +lemma approx_le_bound: "z \<le> f \<Longrightarrow> f \<approx> g \<Longrightarrow> g \<le> z ==> f \<approx> z"
6.1421 +  for z :: hypreal
6.1422 +  apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
6.1423 +  apply (rule_tac x = "g + y - z" in bexI)
6.1424 +   apply simp
6.1425 +  apply (rule Infinitesimal_interval2)
6.1426 +     apply (rule_tac [2] Infinitesimal_zero, auto)
6.1427 +  done
6.1428
6.1429 -lemma approx_le_bound: "[| (z::hypreal) \<le> f; f \<approx> g; g \<le> z |] ==> f \<approx> z"
6.1430 -apply (simp add: bex_Infinitesimal_iff2 [symmetric], auto)
6.1431 -apply (rule_tac x = "g+y-z" in bexI)
6.1432 -apply (simp (no_asm))
6.1433 -apply (rule Infinitesimal_interval2)
6.1434 -apply (rule_tac [2] Infinitesimal_zero, auto)
6.1435 -done
6.1436 -
6.1437 -lemma approx_hnorm:
6.1438 -  fixes x y :: "'a::real_normed_vector star"
6.1439 -  shows "x \<approx> y \<Longrightarrow> hnorm x \<approx> hnorm y"
6.1440 +lemma approx_hnorm: "x \<approx> y \<Longrightarrow> hnorm x \<approx> hnorm y"
6.1441 +  for x y :: "'a::real_normed_vector star"
6.1442  proof (unfold approx_def)
6.1443    assume "x - y \<in> Infinitesimal"
6.1444 -  hence 1: "hnorm (x - y) \<in> Infinitesimal"
6.1445 +  then have "hnorm (x - y) \<in> Infinitesimal"
6.1446      by (simp only: Infinitesimal_hnorm_iff)
6.1447 -  moreover have 2: "(0::real star) \<in> Infinitesimal"
6.1448 +  moreover have "(0::real star) \<in> Infinitesimal"
6.1449      by (rule Infinitesimal_zero)
6.1450 -  moreover have 3: "0 \<le> \<bar>hnorm x - hnorm y\<bar>"
6.1451 +  moreover have "0 \<le> \<bar>hnorm x - hnorm y\<bar>"
6.1452      by (rule abs_ge_zero)
6.1453 -  moreover have 4: "\<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
6.1454 +  moreover have "\<bar>hnorm x - hnorm y\<bar> \<le> hnorm (x - y)"
6.1455      by (rule hnorm_triangle_ineq3)
6.1456    ultimately have "\<bar>hnorm x - hnorm y\<bar> \<in> Infinitesimal"
6.1457      by (rule Infinitesimal_interval2)
6.1458 -  thus "hnorm x - hnorm y \<in> Infinitesimal"
6.1459 +  then show "hnorm x - hnorm y \<in> Infinitesimal"
6.1460      by (simp only: Infinitesimal_hrabs_iff)
6.1461  qed
6.1462
6.1463
6.1464 -subsection\<open>Zero is the Only Infinitesimal that is also a Real\<close>
6.1465 +subsection \<open>Zero is the Only Infinitesimal that is also a Real\<close>
6.1466
6.1467 -lemma Infinitesimal_less_SReal:
6.1468 -     "[| (x::hypreal) \<in> \<real>; y \<in> Infinitesimal; 0 < x |] ==> y < x"
6.1470 -apply (rule abs_ge_self [THEN order_le_less_trans], auto)
6.1471 -done
6.1472 +lemma Infinitesimal_less_SReal: "x \<in> \<real> \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> 0 < x \<Longrightarrow> y < x"
6.1473 +  for x y :: hypreal
6.1474 +  apply (simp add: Infinitesimal_def)
6.1475 +  apply (rule abs_ge_self [THEN order_le_less_trans], auto)
6.1476 +  done
6.1477
6.1478 -lemma Infinitesimal_less_SReal2:
6.1479 -     "(y::hypreal) \<in> Infinitesimal ==> \<forall>r \<in> Reals. 0 < r --> y < r"
6.1480 -by (blast intro: Infinitesimal_less_SReal)
6.1481 +lemma Infinitesimal_less_SReal2: "y \<in> Infinitesimal \<Longrightarrow> \<forall>r \<in> Reals. 0 < r \<longrightarrow> y < r"
6.1482 +  for y :: hypreal
6.1483 +  by (blast intro: Infinitesimal_less_SReal)
6.1484
6.1485 -lemma SReal_not_Infinitesimal:
6.1486 -     "[| 0 < y;  (y::hypreal) \<in> \<real>|] ==> y \<notin> Infinitesimal"
6.1488 -apply (auto simp add: abs_if)
6.1489 -done
6.1490 +lemma SReal_not_Infinitesimal: "0 < y \<Longrightarrow> y \<in> \<real> ==> y \<notin> Infinitesimal"
6.1491 +  for y :: hypreal
6.1492 +  apply (simp add: Infinitesimal_def)
6.1493 +  apply (auto simp add: abs_if)
6.1494 +  done
6.1495
6.1496 -lemma SReal_minus_not_Infinitesimal:
6.1497 -     "[| y < 0;  (y::hypreal) \<in> \<real> |] ==> y \<notin> Infinitesimal"
6.1498 -apply (subst Infinitesimal_minus_iff [symmetric])
6.1499 -apply (rule SReal_not_Infinitesimal, auto)
6.1500 -done
6.1501 +lemma SReal_minus_not_Infinitesimal: "y < 0 \<Longrightarrow> y \<in> \<real> \<Longrightarrow> y \<notin> Infinitesimal"
6.1502 +  for y :: hypreal
6.1503 +  apply (subst Infinitesimal_minus_iff [symmetric])
6.1504 +  apply (rule SReal_not_Infinitesimal, auto)
6.1505 +  done
6.1506
6.1507  lemma SReal_Int_Infinitesimal_zero: "\<real> Int Infinitesimal = {0::hypreal}"
6.1508 -apply auto
6.1509 -apply (cut_tac x = x and y = 0 in linorder_less_linear)
6.1510 -apply (blast dest: SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
6.1511 -done
6.1512 +  apply auto
6.1513 +  apply (cut_tac x = x and y = 0 in linorder_less_linear)
6.1514 +  apply (blast dest: SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
6.1515 +  done
6.1516
6.1517 -lemma SReal_Infinitesimal_zero:
6.1518 -  "[| (x::hypreal) \<in> \<real>; x \<in> Infinitesimal|] ==> x = 0"
6.1519 -by (cut_tac SReal_Int_Infinitesimal_zero, blast)
6.1520 +lemma SReal_Infinitesimal_zero: "x \<in> \<real> \<Longrightarrow> x \<in> Infinitesimal \<Longrightarrow> x = 0"
6.1521 +  for x :: hypreal
6.1522 +  using SReal_Int_Infinitesimal_zero by blast
6.1523
6.1524 -lemma SReal_HFinite_diff_Infinitesimal:
6.1525 -     "[| (x::hypreal) \<in> \<real>; x \<noteq> 0 |] ==> x \<in> HFinite - Infinitesimal"
6.1526 -by (auto dest: SReal_Infinitesimal_zero SReal_subset_HFinite [THEN subsetD])
6.1527 +lemma SReal_HFinite_diff_Infinitesimal: "x \<in> \<real> \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> x \<in> HFinite - Infinitesimal"
6.1528 +  for x :: hypreal
6.1529 +  by (auto dest: SReal_Infinitesimal_zero SReal_subset_HFinite [THEN subsetD])
6.1530
6.1531  lemma hypreal_of_real_HFinite_diff_Infinitesimal:
6.1532 -     "hypreal_of_real x \<noteq> 0 ==> hypreal_of_real x \<in> HFinite - Infinitesimal"
6.1533 -by (rule SReal_HFinite_diff_Infinitesimal, auto)
6.1534 +  "hypreal_of_real x \<noteq> 0 \<Longrightarrow> hypreal_of_real x \<in> HFinite - Infinitesimal"
6.1535 +  by (rule SReal_HFinite_diff_Infinitesimal) auto
6.1536
6.1537 -lemma star_of_Infinitesimal_iff_0 [iff]:
6.1538 -  "(star_of x \<in> Infinitesimal) = (x = 0)"
6.1539 -apply (auto simp add: Infinitesimal_def)
6.1540 -apply (drule_tac x="hnorm (star_of x)" in bspec)
6.1542 -apply (rule_tac x="norm x" in exI, simp)
6.1543 -apply simp
6.1544 -done
6.1545 +lemma star_of_Infinitesimal_iff_0 [iff]: "star_of x \<in> Infinitesimal \<longleftrightarrow> x = 0"
6.1546 +  apply (auto simp add: Infinitesimal_def)
6.1547 +  apply (drule_tac x="hnorm (star_of x)" in bspec)
6.1548 +   apply (simp add: SReal_def)
6.1549 +   apply (rule_tac x="norm x" in exI, simp)
6.1550 +  apply simp
6.1551 +  done
6.1552
6.1553 -lemma star_of_HFinite_diff_Infinitesimal:
6.1554 -     "x \<noteq> 0 ==> star_of x \<in> HFinite - Infinitesimal"
6.1555 -by simp
6.1556 +lemma star_of_HFinite_diff_Infinitesimal: "x \<noteq> 0 \<Longrightarrow> star_of x \<in> HFinite - Infinitesimal"
6.1557 +  by simp
6.1558
6.1559  lemma numeral_not_Infinitesimal [simp]:
6.1560 -     "numeral w \<noteq> (0::hypreal) ==> (numeral w :: hypreal) \<notin> Infinitesimal"
6.1561 -by (fast dest: Reals_numeral [THEN SReal_Infinitesimal_zero])
6.1562 +  "numeral w \<noteq> (0::hypreal) \<Longrightarrow> (numeral w :: hypreal) \<notin> Infinitesimal"
6.1563 +  by (fast dest: Reals_numeral [THEN SReal_Infinitesimal_zero])
6.1564
6.1565 -(*again: 1 is a special case, but not 0 this time*)
6.1566 +text \<open>Again: \<open>1\<close> is a special case, but not \<open>0\<close> this time.\<close>
6.1567  lemma one_not_Infinitesimal [simp]:
6.1568    "(1::'a::{real_normed_vector,zero_neq_one} star) \<notin> Infinitesimal"
6.1569 -apply (simp only: star_one_def star_of_Infinitesimal_iff_0)
6.1570 -apply simp
6.1571 -done
6.1572 +  apply (simp only: star_one_def star_of_Infinitesimal_iff_0)
6.1573 +  apply simp
6.1574 +  done
6.1575
6.1576 -lemma approx_SReal_not_zero:
6.1577 -  "[| (y::hypreal) \<in> \<real>; x \<approx> y; y\<noteq> 0 |] ==> x \<noteq> 0"
6.1578 -apply (cut_tac x = 0 and y = y in linorder_less_linear, simp)
6.1579 -apply (blast dest: approx_sym [THEN mem_infmal_iff [THEN iffD2]] SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
6.1580 -done
6.1581 +lemma approx_SReal_not_zero: "y \<in> \<real> \<Longrightarrow> x \<approx> y \<Longrightarrow> y \<noteq> 0 \<Longrightarrow> x \<noteq> 0"
6.1582 +  for x y :: hypreal
6.1583 +  apply (cut_tac x = 0 and y = y in linorder_less_linear, simp)
6.1584 +  apply (blast dest: approx_sym [THEN mem_infmal_iff [THEN iffD2]]
6.1585 +      SReal_not_Infinitesimal SReal_minus_not_Infinitesimal)
6.1586 +  done
6.1587
6.1588  lemma HFinite_diff_Infinitesimal_approx:
6.1589 -     "[| x \<approx> y; y \<in> HFinite - Infinitesimal |]
6.1590 -      ==> x \<in> HFinite - Infinitesimal"
6.1591 -apply (auto intro: approx_sym [THEN [2] approx_HFinite]
6.1593 -apply (drule approx_trans3, assumption)
6.1594 -apply (blast dest: approx_sym)
6.1595 -done
6.1596 +  "x \<approx> y \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> x \<in> HFinite - Infinitesimal"
6.1597 +  apply (auto intro: approx_sym [THEN [2] approx_HFinite] simp: mem_infmal_iff)
6.1598 +  apply (drule approx_trans3, assumption)
6.1599 +  apply (blast dest: approx_sym)
6.1600 +  done
6.1601
6.1602 -(*The premise y\<noteq>0 is essential; otherwise x/y =0 and we lose the
6.1603 -  HFinite premise.*)
6.1604 +text \<open>The premise \<open>y \<noteq> 0\<close> is essential; otherwise \<open>x / y = 0\<close> and we lose the
6.1605 +  \<open>HFinite\<close> premise.\<close>
6.1606  lemma Infinitesimal_ratio:
6.1607 -  fixes x y :: "'a::{real_normed_div_algebra,field} star"
6.1608 -  shows "[| y \<noteq> 0;  y \<in> Infinitesimal;  x/y \<in> HFinite |]
6.1609 -         ==> x \<in> Infinitesimal"
6.1610 -apply (drule Infinitesimal_HFinite_mult2, assumption)
6.1611 -apply (simp add: divide_inverse mult.assoc)
6.1612 -done
6.1613 +  "y \<noteq> 0 \<Longrightarrow> y \<in> Infinitesimal \<Longrightarrow> x / y \<in> HFinite \<Longrightarrow> x \<in> Infinitesimal"
6.1614 +  for x y :: "'a::{real_normed_div_algebra,field} star"
6.1615 +  apply (drule Infinitesimal_HFinite_mult2, assumption)
6.1616 +  apply (simp add: divide_inverse mult.assoc)
6.1617 +  done
6.1618 +
6.1619 +lemma Infinitesimal_SReal_divide: "x \<in> Infinitesimal \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x / y \<in> Infinitesimal"
6.1620 +  for x y :: hypreal
6.1621 +  apply (simp add: divide_inverse)
6.1622 +  apply (auto intro!: Infinitesimal_HFinite_mult
6.1623 +      dest!: Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
6.1624 +  done
6.1625 +
6.1626 +
6.1627 +section \<open>Standard Part Theorem\<close>
6.1628
6.1629 -lemma Infinitesimal_SReal_divide:
6.1630 -  "[| (x::hypreal) \<in> Infinitesimal; y \<in> \<real> |] ==> x/y \<in> Infinitesimal"
6.1632 -apply (auto intro!: Infinitesimal_HFinite_mult
6.1633 -            dest!: Reals_inverse [THEN SReal_subset_HFinite [THEN subsetD]])
6.1634 -done
6.1635 +text \<open>
6.1636 +  Every finite \<open>x \<in> R*\<close> is infinitely close to a unique real number
6.1637 +  (i.e. a member of \<open>Reals\<close>).
6.1638 +\<close>
6.1639
6.1640 -(*------------------------------------------------------------------
6.1641 -       Standard Part Theorem: Every finite x: R* is infinitely
6.1642 -       close to a unique real number (i.e a member of Reals)
6.1643 - ------------------------------------------------------------------*)
6.1644
6.1645 -subsection\<open>Uniqueness: Two Infinitely Close Reals are Equal\<close>
6.1646 +subsection \<open>Uniqueness: Two Infinitely Close Reals are Equal\<close>
6.1647
6.1648 -lemma star_of_approx_iff [simp]: "(star_of x \<approx> star_of y) = (x = y)"
6.1649 -apply safe
6.1651 -apply (simp only: star_of_diff [symmetric])
6.1652 -apply (simp only: star_of_Infinitesimal_iff_0)
6.1653 -apply simp
6.1654 -done
6.1655 +lemma star_of_approx_iff [simp]: "star_of x \<approx> star_of y \<longleftrightarrow> x = y"
6.1656 +  apply safe
6.1657 +  apply (simp add: approx_def)
6.1658 +  apply (simp only: star_of_diff [symmetric])
6.1659 +  apply (simp only: star_of_Infinitesimal_iff_0)
6.1660 +  apply simp
6.1661 +  done
6.1662
6.1663 -lemma SReal_approx_iff:
6.1664 -  "[|(x::hypreal) \<in> \<real>; y \<in> \<real>|] ==> (x \<approx> y) = (x = y)"
6.1665 -apply auto
6.1667 -apply (drule (1) Reals_diff)
6.1668 -apply (drule (1) SReal_Infinitesimal_zero)
6.1669 -apply simp
6.1670 -done
6.1671 +lemma SReal_approx_iff: "x \<in> \<real> \<Longrightarrow> y \<in> \<real> \<Longrightarrow> x \<approx> y \<longleftrightarrow> x = y"
6.1672 +  for x y :: hypreal
6.1673 +  apply auto
6.1674 +  apply (simp add: approx_def)
6.1675 +  apply (drule (1) Reals_diff)
6.1676 +  apply (drule (1) SReal_Infinitesimal_zero)
6.1677 +  apply simp
6.1678 +  done
6.1679
6.1680  lemma numeral_approx_iff [simp]:
6.1681 -     "(numeral v \<approx> (numeral w :: 'a::{numeral,real_normed_vector} star)) =
6.1682 -      (numeral v = (numeral w :: 'a))"
6.1683 -apply (unfold star_numeral_def)
6.1684 -apply (rule star_of_approx_iff)
6.1685 -done
6.1686 +  "(numeral v \<approx> (numeral w :: 'a::{numeral,real_normed_vector} star)) =
6.1687 +    (numeral v = (numeral w :: 'a))"
6.1688 +  apply (unfold star_numeral_def)
6.1689 +  apply (rule star_of_approx_iff)
6.1690 +  done
6.1691
6.1692 -(*And also for 0 \<approx> #nn and 1 \<approx> #nn, #nn \<approx> 0 and #nn \<approx> 1.*)
6.1693 +text \<open>And also for \<open>0 \<approx> #nn\<close> and \<open>1 \<approx> #nn\<close>, \<open>#nn \<approx> 0\<close> and \<open>#nn \<approx> 1\<close>.\<close>
6.1694  lemma [simp]:
6.1695 -  "(numeral w \<approx> (0::'a::{numeral,real_normed_vector} star)) =
6.1696 -   (numeral w = (0::'a))"
6.1697 -  "((0::'a::{numeral,real_normed_vector} star) \<approx> numeral w) =
6.1698 -   (numeral w = (0::'a))"
6.1699 -  "(numeral w \<approx> (1::'b::{numeral,one,real_normed_vector} star)) =
6.1700 -   (numeral w = (1::'b))"
6.1701 -  "((1::'b::{numeral,one,real_normed_vector} star) \<approx> numeral w) =
6.1702 -   (numeral w = (1::'b))"
6.1703 -  "~ (0 \<approx> (1::'c::{zero_neq_one,real_normed_vector} star))"
6.1704 -  "~ (1 \<approx> (0::'c::{zero_neq_one,real_normed_vector} star))"
6.1705 -apply (unfold star_numeral_def star_zero_def star_one_def)
6.1706 -apply (unfold star_of_approx_iff)
6.1707 -by (auto intro: sym)
6.1708 +  "(numeral w \<approx> (0::'a::{numeral,real_normed_vector} star)) = (numeral w = (0::'a))"
6.1709 +  "((0::'a::{numeral,real_normed_vector} star) \<approx> numeral w) = (numeral w = (0::'a))"
6.1710 +  "(numeral w \<approx> (1::'b::{numeral,one,real_normed_vector} star)) = (numeral w = (1::'b))"
6.1711 +  "((1::'b::{numeral,one,real_normed_vector} star) \<approx> numeral w) = (numeral w = (1::'b))"
6.1712 +  "\<not> (0 \<approx> (1::'c::{zero_neq_one,real_normed_vector} star))"
6.1713 +  "\<not> (1 \<approx> (0::'c::{zero_neq_one,real_normed_vector} star))"
6.1714 +       apply (unfold star_numeral_def star_zero_def star_one_def)
6.1715 +       apply (unfold star_of_approx_iff)
6.1716 +       apply (auto intro: sym)
6.1717 +  done
6.1718
6.1719 -lemma star_of_approx_numeral_iff [simp]:
6.1720 -     "(star_of k \<approx> numeral w) = (k = numeral w)"
6.1721 -by (subst star_of_approx_iff [symmetric], auto)
6.1722 +lemma star_of_approx_numeral_iff [simp]: "star_of k \<approx> numeral w \<longleftrightarrow> k = numeral w"
6.1723 +  by (subst star_of_approx_iff [symmetric]) auto
6.1724
6.1725 -lemma star_of_approx_zero_iff [simp]: "(star_of k \<approx> 0) = (k = 0)"
6.1726 -by (simp_all add: star_of_approx_iff [symmetric])
6.1727 +lemma star_of_approx_zero_iff [simp]: "star_of k \<approx> 0 \<longleftrightarrow> k = 0"
6.1728 +  by (simp_all add: star_of_approx_iff [symmetric])
6.1729
6.1730 -lemma star_of_approx_one_iff [simp]: "(star_of k \<approx> 1) = (k = 1)"
6.1731 -by (simp_all add: star_of_approx_iff [symmetric])
6.1732 +lemma star_of_approx_one_iff [simp]: "star_of k \<approx> 1 \<longleftrightarrow> k = 1"
6.1733 +  by (simp_all add: star_of_approx_iff [symmetric])
6.1734
6.1735 -lemma approx_unique_real:
6.1736 -     "[| (r::hypreal) \<in> \<real>; s \<in> \<real>; r \<approx> x; s \<approx> x|] ==> r = s"
6.1737 -by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)
6.1738 +lemma approx_unique_real: "r \<in> \<real> \<Longrightarrow> s \<in> \<real> \<Longrightarrow> r \<approx> x \<Longrightarrow> s \<approx> x \<Longrightarrow> r = s"
6.1739 +  for r s :: hypreal
6.1740 +  by (blast intro: SReal_approx_iff [THEN iffD1] approx_trans2)
6.1741
6.1742
6.1743 -subsection\<open>Existence of Unique Real Infinitely Close\<close>
6.1744 +subsection \<open>Existence of Unique Real Infinitely Close\<close>
6.1745
6.1746 -subsubsection\<open>Lifting of the Ub and Lub Properties\<close>
6.1747 +subsubsection \<open>Lifting of the Ub and Lub Properties\<close>
6.1748
6.1749 -lemma hypreal_of_real_isUb_iff:
6.1750 -      "(isUb \<real> (hypreal_of_real ` Q) (hypreal_of_real Y)) =
6.1751 -       (isUb (UNIV :: real set) Q Y)"
6.1752 -by (simp add: isUb_def setle_def)
6.1753 +lemma hypreal_of_real_isUb_iff: "isUb \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) = isUb UNIV Q Y"
6.1754 +  for Q :: "real set" and Y :: real
6.1755 +  by (simp add: isUb_def setle_def)
6.1756
6.1757 -lemma hypreal_of_real_isLub1:
6.1758 -     "isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y)
6.1759 -      ==> isLub (UNIV :: real set) Q Y"
6.1760 -apply (simp add: isLub_def leastP_def)
6.1761 -apply (auto intro: hypreal_of_real_isUb_iff [THEN iffD2]
6.1762 -            simp add: hypreal_of_real_isUb_iff setge_def)
6.1763 -done
6.1764 +lemma hypreal_of_real_isLub1: "isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) \<Longrightarrow> isLub UNIV Q Y"
6.1765 +  for Q :: "real set" and Y :: real
6.1766 +  apply (simp add: isLub_def leastP_def)
6.1767 +  apply (auto intro: hypreal_of_real_isUb_iff [THEN iffD2]
6.1768 +      simp add: hypreal_of_real_isUb_iff setge_def)
6.1769 +  done
6.1770
6.1771 -lemma hypreal_of_real_isLub2:
6.1772 -      "isLub (UNIV :: real set) Q Y
6.1773 -       ==> isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y)"
6.1774 -apply (auto simp add: isLub_def leastP_def hypreal_of_real_isUb_iff setge_def)
6.1775 -by (metis SReal_iff hypreal_of_real_isUb_iff isUbD2a star_of_le)
6.1776 +lemma hypreal_of_real_isLub2: "isLub UNIV Q Y \<Longrightarrow> isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y)"
6.1777 +  for Q :: "real set" and Y :: real
6.1778 +  apply (auto simp add: isLub_def leastP_def hypreal_of_real_isUb_iff setge_def)
6.1779 +  apply (metis SReal_iff hypreal_of_real_isUb_iff isUbD2a star_of_le)
6.1780 +  done
6.1781
6.1782  lemma hypreal_of_real_isLub_iff:
6.1783 -     "(isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y)) =
6.1784 -      (isLub (UNIV :: real set) Q Y)"
6.1785 -by (blast intro: hypreal_of_real_isLub1 hypreal_of_real_isLub2)
6.1786 +  "isLub \<real> (hypreal_of_real ` Q) (hypreal_of_real Y) = isLub (UNIV :: real set) Q Y"
6.1787 +  for Q :: "real set" and Y :: real
6.1788 +  by (blast intro: hypreal_of_real_isLub1 hypreal_of_real_isLub2)
6.1789
6.1790 -lemma lemma_isUb_hypreal_of_real:
6.1791 -     "isUb \<real> P Y ==> \<exists>Yo. isUb \<real> P (hypreal_of_real Yo)"
6.1792 -by (auto simp add: SReal_iff isUb_def)
6.1793 +lemma lemma_isUb_hypreal_of_real: "isUb \<real> P Y \<Longrightarrow> \<exists>Yo. isUb \<real> P (hypreal_of_real Yo)"
6.1794 +  by (auto simp add: SReal_iff isUb_def)
6.1795 +
6.1796 +lemma lemma_isLub_hypreal_of_real: "isLub \<real> P Y \<Longrightarrow> \<exists>Yo. isLub \<real> P (hypreal_of_real Yo)"
6.1797 +  by (auto simp add: isLub_def leastP_def isUb_def SReal_iff)
6.1798
6.1799 -lemma lemma_isLub_hypreal_of_real:
6.1800 -     "isLub \<real> P Y ==> \<exists>Yo. isLub \<real> P (hypreal_of_real Yo)"
6.1801 -by (auto simp add: isLub_def leastP_def isUb_def SReal_iff)
6.1802 +lemma lemma_isLub_hypreal_of_real2: "\<exists>Yo. isLub \<real> P (hypreal_of_real Yo) \<Longrightarrow> \<exists>Y. isLub \<real> P Y"
6.1803 +  by (auto simp add: isLub_def leastP_def isUb_def)
6.1804
6.1805 -lemma lemma_isLub_hypreal_of_real2:
6.1806 -     "\<exists>Yo. isLub \<real> P (hypreal_of_real Yo) ==> \<exists>Y. isLub \<real> P Y"
6.1807 -by (auto simp add: isLub_def leastP_def isUb_def)
6.1808 +lemma SReal_complete: "P \<subseteq> \<real> \<Longrightarrow> \<exists>x. x \<in> P \<Longrightarrow> \<exists>Y. isUb \<real> P Y \<Longrightarrow> \<exists>t::hypreal. isLub \<real> P t"
6.1809 +  apply (frule SReal_hypreal_of_real_image)
6.1810 +   apply (auto, drule lemma_isUb_hypreal_of_real)
6.1811 +  apply (auto intro!: reals_complete lemma_isLub_hypreal_of_real2
6.1812 +      simp add: hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff)
6.1813 +  done
6.1814 +
6.1815
6.1816 -lemma SReal_complete:
6.1817 -     "[| P \<subseteq> \<real>;  \<exists>x. x \<in> P;  \<exists>Y. isUb \<real> P Y |]
6.1818 -      ==> \<exists>t::hypreal. isLub \<real> P t"
6.1819 -apply (frule SReal_hypreal_of_real_image)
6.1820 -apply (auto, drule lemma_isUb_hypreal_of_real)
6.1821 -apply (auto intro!: reals_complete lemma_isLub_hypreal_of_real2
6.1822 -            simp add: hypreal_of_real_isLub_iff hypreal_of_real_isUb_iff)
6.1823 -done
6.1825
6.1826 -(* lemma about lubs *)
6.1827 +lemma lemma_st_part_ub: "x \<in> HFinite \<Longrightarrow> \<exists>u. isUb \<real> {s. s \<in> \<real> \<and> s < x} u"
6.1828 +  for x :: hypreal
6.1829 +  apply (drule HFiniteD, safe)
6.1830 +  apply (rule exI, rule isUbI)
6.1831 +   apply (auto intro: setleI isUbI simp add: abs_less_iff)
6.1832 +  done
6.1833
6.1834 -lemma lemma_st_part_ub:
6.1835 -     "(x::hypreal) \<in> HFinite ==> \<exists>u. isUb \<real> {s. s \<in> \<real> & s < x} u"
6.1836 -apply (drule HFiniteD, safe)
6.1837 -apply (rule exI, rule isUbI)
6.1838 -apply (auto intro: setleI isUbI simp add: abs_less_iff)
6.1839 -done
6.1840 +lemma lemma_st_part_nonempty: "x \<in> HFinite \<Longrightarrow> \<exists>y. y \<in> {s. s \<in> \<real> \<and> s < x}"
6.1841 +  for x :: hypreal
6.1842 +  apply (drule HFiniteD, safe)
6.1843 +  apply (drule Reals_minus)
6.1844 +  apply (rule_tac x = "-t" in exI)
6.1845 +  apply (auto simp add: abs_less_iff)
6.1846 +  done
6.1847
6.1848 -lemma lemma_st_part_nonempty:
6.1849 -  "(x::hypreal) \<in> HFinite ==> \<exists>y. y \<in> {s. s \<in> \<real> & s < x}"
6.1850 -apply (drule HFiniteD, safe)
6.1851 -apply (drule Reals_minus)
6.1852 -apply (rule_tac x = "-t" in exI)
6.1853 -apply (auto simp add: abs_less_iff)
6.1854 -done
6.1855 -
6.1856 -lemma lemma_st_part_lub:
6.1857 -     "(x::hypreal) \<in> HFinite ==> \<exists>t. isLub \<real> {s. s \<in> \<real> & s < x} t"
6.1858 -by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty Collect_restrict)
6.1859 +lemma lemma_st_part_lub: "x \<in> HFinite \<Longrightarrow> \<exists>t. isLub \<real> {s. s \<in> \<real> \<and> s < x} t"
6.1860 +  for x :: hypreal
6.1861 +  by (blast intro!: SReal_complete lemma_st_part_ub lemma_st_part_nonempty Collect_restrict)
6.1862
6.1863  lemma lemma_st_part_le1:
6.1864 -     "[| (x::hypreal) \<in> HFinite;  isLub \<real> {s. s \<in> \<real> & s < x} t;
6.1865 -         r \<in> \<real>;  0 < r |] ==> x \<le> t + r"
6.1866 -apply (frule isLubD1a)
6.1867 -apply (rule ccontr, drule linorder_not_le [THEN iffD2])
6.1869 -apply (drule_tac y = "r + t" in isLubD1 [THEN setleD], auto)
6.1870 -done
6.1871 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> x \<le> t + r"
6.1872 +  for x r t :: hypreal
6.1873 +  apply (frule isLubD1a)
6.1874 +  apply (rule ccontr, drule linorder_not_le [THEN iffD2])
6.1875 +  apply (drule (1) Reals_add)
6.1876 +  apply (drule_tac y = "r + t" in isLubD1 [THEN setleD], auto)
6.1877 +  done
6.1878
6.1879 -lemma hypreal_setle_less_trans:
6.1880 -     "[| S *<= (x::hypreal); x < y |] ==> S *<= y"
6.1882 -apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
6.1883 -done
6.1884 +lemma hypreal_setle_less_trans: "S *<= x \<Longrightarrow> x < y \<Longrightarrow> S *<= y"
6.1885 +  for x y :: hypreal
6.1886 +  apply (simp add: setle_def)
6.1887 +  apply (auto dest!: bspec order_le_less_trans intro: order_less_imp_le)
6.1888 +  done
6.1889
6.1890 -lemma hypreal_gt_isUb:
6.1891 -     "[| isUb R S (x::hypreal); x < y; y \<in> R |] ==> isUb R S y"
6.1893 -apply (blast intro: hypreal_setle_less_trans)
6.1894 -done
6.1895 +lemma hypreal_gt_isUb: "isUb R S x \<Longrightarrow> x < y \<Longrightarrow> y \<in> R \<Longrightarrow> isUb R S y"
6.1896 +  for x y :: hypreal
6.1897 +  apply (simp add: isUb_def)
6.1898 +  apply (blast intro: hypreal_setle_less_trans)
6.1899 +  done
6.1900
6.1901 -lemma lemma_st_part_gt_ub:
6.1902 -     "[| (x::hypreal) \<in> HFinite; x < y; y \<in> \<real> |]
6.1903 -      ==> isUb \<real> {s. s \<in> \<real> & s < x} y"
6.1904 -by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)
6.1905 +lemma lemma_st_part_gt_ub: "x \<in> HFinite \<Longrightarrow> x < y \<Longrightarrow> y \<in> \<real> \<Longrightarrow> isUb \<real> {s. s \<in> \<real> \<and> s < x} y"
6.1906 +  for x y :: hypreal
6.1907 +  by (auto dest: order_less_trans intro: order_less_imp_le intro!: isUbI setleI)
6.1908
6.1909 -lemma lemma_minus_le_zero: "t \<le> t + -r ==> r \<le> (0::hypreal)"
6.1910 -apply (drule_tac c = "-t" in add_left_mono)
6.1912 -done
6.1913 +lemma lemma_minus_le_zero: "t \<le> t + -r \<Longrightarrow> r \<le> 0"
6.1914 +  for r t :: hypreal
6.1915 +  apply (drule_tac c = "-t" in add_left_mono)
6.1917 +  done
6.1918
6.1919  lemma lemma_st_part_le2:
6.1920 -     "[| (x::hypreal) \<in> HFinite;
6.1921 -         isLub \<real> {s. s \<in> \<real> & s < x} t;
6.1922 -         r \<in> \<real>; 0 < r |]
6.1923 -      ==> t + -r \<le> x"
6.1924 -apply (frule isLubD1a)
6.1925 -apply (rule ccontr, drule linorder_not_le [THEN iffD1])
6.1926 -apply (drule Reals_minus, drule_tac a = t in Reals_add, assumption)
6.1927 -apply (drule lemma_st_part_gt_ub, assumption+)
6.1928 -apply (drule isLub_le_isUb, assumption)
6.1929 -apply (drule lemma_minus_le_zero)
6.1930 -apply (auto dest: order_less_le_trans)
6.1931 -done
6.1932 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> t + -r \<le> x"
6.1933 +  for x r t :: hypreal
6.1934 +  apply (frule isLubD1a)
6.1935 +  apply (rule ccontr, drule linorder_not_le [THEN iffD1])
6.1936 +  apply (drule Reals_minus, drule_tac a = t in Reals_add, assumption)
6.1937 +  apply (drule lemma_st_part_gt_ub, assumption+)
6.1938 +  apply (drule isLub_le_isUb, assumption)
6.1939 +  apply (drule lemma_minus_le_zero)
6.1940 +  apply (auto dest: order_less_le_trans)
6.1941 +  done
6.1942
6.1943  lemma lemma_st_part1a:
6.1944 -     "[| (x::hypreal) \<in> HFinite;
6.1945 -         isLub \<real> {s. s \<in> \<real> & s < x} t;
6.1946 -         r \<in> \<real>; 0 < r |]
6.1947 -      ==> x + -t \<le> r"
6.1948 -apply (subgoal_tac "x \<le> t+r")
6.1949 -apply (auto intro: lemma_st_part_le1)
6.1950 -done
6.1951 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> x + -t \<le> r"
6.1952 +  for x r t :: hypreal
6.1953 +  apply (subgoal_tac "x \<le> t + r")
6.1954 +   apply (auto intro: lemma_st_part_le1)
6.1955 +  done
6.1956
6.1957  lemma lemma_st_part2a:
6.1958 -     "[| (x::hypreal) \<in> HFinite;
6.1959 -         isLub \<real> {s. s \<in> \<real> & s < x} t;
6.1960 -         r \<in> \<real>;  0 < r |]
6.1961 -      ==> -(x + -t) \<le> r"
6.1962 -apply (subgoal_tac "(t + -r \<le> x)")
6.1963 -apply simp
6.1964 -apply (rule lemma_st_part_le2)
6.1965 -apply auto
6.1966 -done
6.1967 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> - (x + -t) \<le> r"
6.1968 +  for x r t :: hypreal
6.1969 +  apply (subgoal_tac "(t + -r \<le> x)")
6.1970 +   apply simp
6.1971 +  apply (rule lemma_st_part_le2)
6.1972 +     apply auto
6.1973 +  done
6.1974
6.1975 -lemma lemma_SReal_ub:
6.1976 -     "(x::hypreal) \<in> \<real> ==> isUb \<real> {s. s \<in> \<real> & s < x} x"
6.1977 -by (auto intro: isUbI setleI order_less_imp_le)
6.1978 +lemma lemma_SReal_ub: "x \<in> \<real> \<Longrightarrow> isUb \<real> {s. s \<in> \<real> \<and> s < x} x"
6.1979 +  for x :: hypreal
6.1980 +  by (auto intro: isUbI setleI order_less_imp_le)
6.1981
6.1982 -lemma lemma_SReal_lub:
6.1983 -     "(x::hypreal) \<in> \<real> ==> isLub \<real> {s. s \<in> \<real> & s < x} x"
6.1984 -apply (auto intro!: isLubI2 lemma_SReal_ub setgeI)
6.1985 -apply (frule isUbD2a)
6.1986 -apply (rule_tac x = x and y = y in linorder_cases)
6.1987 -apply (auto intro!: order_less_imp_le)
6.1988 -apply (drule SReal_dense, assumption, assumption, safe)
6.1989 -apply (drule_tac y = r in isUbD)
6.1990 -apply (auto dest: order_less_le_trans)
6.1991 -done
6.1992 +lemma lemma_SReal_lub: "x \<in> \<real> \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} x"
6.1993 +  for x :: hypreal
6.1994 +  apply (auto intro!: isLubI2 lemma_SReal_ub setgeI)
6.1995 +  apply (frule isUbD2a)
6.1996 +  apply (rule_tac x = x and y = y in linorder_cases)
6.1997 +    apply (auto intro!: order_less_imp_le)
6.1998 +  apply (drule SReal_dense, assumption, assumption, safe)
6.1999 +  apply (drule_tac y = r in isUbD)
6.2000 +   apply (auto dest: order_less_le_trans)
6.2001 +  done
6.2002
6.2003  lemma lemma_st_part_not_eq1:
6.2004 -     "[| (x::hypreal) \<in> HFinite;
6.2005 -         isLub \<real> {s. s \<in> \<real> & s < x} t;
6.2006 -         r \<in> \<real>; 0 < r |]
6.2007 -      ==> x + -t \<noteq> r"
6.2008 -apply auto
6.2009 -apply (frule isLubD1a [THEN Reals_minus])
6.2010 -using Reals_add_cancel [of x "- t"] apply simp
6.2011 -apply (drule_tac x = x in lemma_SReal_lub)
6.2012 -apply (drule isLub_unique, assumption, auto)
6.2013 -done
6.2014 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> x + - t \<noteq> r"
6.2015 +  for x r t :: hypreal
6.2016 +  apply auto
6.2017 +  apply (frule isLubD1a [THEN Reals_minus])
6.2018 +  using Reals_add_cancel [of x "- t"] apply simp
6.2019 +  apply (drule_tac x = x in lemma_SReal_lub)
6.2020 +  apply (drule isLub_unique, assumption, auto)
6.2021 +  done
6.2022
6.2023  lemma lemma_st_part_not_eq2:
6.2024 -     "[| (x::hypreal) \<in> HFinite;
6.2025 -         isLub \<real> {s. s \<in> \<real> & s < x} t;
6.2026 -         r \<in> \<real>; 0 < r |]
6.2027 -      ==> -(x + -t) \<noteq> r"
6.2028 -apply (auto)
6.2029 -apply (frule isLubD1a)
6.2030 -using Reals_add_cancel [of "- x" t] apply simp
6.2031 -apply (drule_tac x = x in lemma_SReal_lub)
6.2032 -apply (drule isLub_unique, assumption, auto)
6.2033 -done
6.2034 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> - (x + -t) \<noteq> r"
6.2035 +  for x r t :: hypreal
6.2036 +  apply (auto)
6.2037 +  apply (frule isLubD1a)
6.2038 +  using Reals_add_cancel [of "- x" t] apply simp
6.2039 +  apply (drule_tac x = x in lemma_SReal_lub)
6.2040 +  apply (drule isLub_unique, assumption, auto)
6.2041 +  done
6.2042
6.2043  lemma lemma_st_part_major:
6.2044 -     "[| (x::hypreal) \<in> HFinite;
6.2045 -         isLub \<real> {s. s \<in> \<real> & s < x} t;
6.2046 -         r \<in> \<real>; 0 < r |]
6.2047 -      ==> \<bar>x - t\<bar> < r"
6.2048 -apply (frule lemma_st_part1a)
6.2049 -apply (frule_tac [4] lemma_st_part2a, auto)
6.2050 -apply (drule order_le_imp_less_or_eq)+
6.2051 -apply auto
6.2052 -using lemma_st_part_not_eq2 apply fastforce
6.2053 -using lemma_st_part_not_eq1 apply fastforce
6.2054 -done
6.2055 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> r \<in> \<real> \<Longrightarrow> 0 < r \<Longrightarrow> \<bar>x - t\<bar> < r"
6.2056 +  for x r t :: hypreal
6.2057 +  apply (frule lemma_st_part1a)
6.2058 +     apply (frule_tac [4] lemma_st_part2a, auto)
6.2059 +  apply (drule order_le_imp_less_or_eq)+
6.2060 +  apply auto
6.2061 +  using lemma_st_part_not_eq2 apply fastforce
6.2062 +  using lemma_st_part_not_eq1 apply fastforce
6.2063 +  done
6.2064
6.2065  lemma lemma_st_part_major2:
6.2066 -     "[| (x::hypreal) \<in> HFinite; isLub \<real> {s. s \<in> \<real> & s < x} t |]
6.2067 -      ==> \<forall>r \<in> Reals. 0 < r --> \<bar>x - t\<bar> < r"
6.2068 -by (blast dest!: lemma_st_part_major)
6.2069 -
6.2070 +  "x \<in> HFinite \<Longrightarrow> isLub \<real> {s. s \<in> \<real> \<and> s < x} t \<Longrightarrow> \<forall>r \<in> Reals. 0 < r \<longrightarrow> \<bar>x - t\<bar> < r"
6.2071 +  for x t :: hypreal
6.2072 +  by (blast dest!: lemma_st_part_major)
6.2073
6.2074
6.2075 -text\<open>Existence of real and Standard Part Theorem\<close>
6.2076 -lemma lemma_st_part_Ex:
6.2077 -     "(x::hypreal) \<in> HFinite
6.2078 -       ==> \<exists>t \<in> Reals. \<forall>r \<in> Reals. 0 < r --> \<bar>x - t\<bar> < r"
6.2079 -apply (frule lemma_st_part_lub, safe)
6.2080 -apply (frule isLubD1a)
6.2081 -apply (blast dest: lemma_st_part_major2)
6.2082 -done
6.2083 +text\<open>Existence of real and Standard Part Theorem.\<close>
6.2084 +
6.2085 +lemma lemma_st_part_Ex: "x \<in> HFinite \<Longrightarrow> \<exists>t\<in>Reals. \<forall>r \<in> Reals. 0 < r \<longrightarrow> \<bar>x - t\<bar> < r"
6.2086 +  for x :: hypreal
6.2087 +  apply (frule lemma_st_part_lub, safe)
6.2088 +  apply (frule isLubD1a)
6.2089 +  apply (blast dest: lemma_st_part_major2)
6.2090 +  done
6.2091
6.2092 -lemma st_part_Ex:
6.2093 -     "(x::hypreal) \<in> HFinite ==> \<exists>t \<in> Reals. x \<approx> t"
6.2094 -apply (simp add: approx_def Infinitesimal_def)
6.2095 -apply (drule lemma_st_part_Ex, auto)
6.2096 -done
6.2097 +lemma st_part_Ex: "x \<in> HFinite \<Longrightarrow> \<exists>t\<in>Reals. x \<approx> t"
6.2098 +  for x :: hypreal
6.2099 +  apply (simp add: approx_def Infinitesimal_def)
6.2100 +  apply (drule lemma_st_part_Ex, auto)
6.2101 +  done
6.2102
6.2103 -text\<open>There is a unique real infinitely close\<close>
6.2104 -lemma st_part_Ex1: "x \<in> HFinite ==> \<exists>!t::hypreal. t \<in> \<real> & x \<approx> t"
6.2105 -apply (drule st_part_Ex, safe)
6.2106 -apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
6.2107 -apply (auto intro!: approx_unique_real)
6.2108 -done
6.2109 +text \<open>There is a unique real infinitely close.\<close>
6.2110 +lemma st_part_Ex1: "x \<in> HFinite \<Longrightarrow> \<exists>!t::hypreal. t \<in> \<real> \<and> x \<approx> t"
6.2111 +  apply (drule st_part_Ex, safe)
6.2112 +   apply (drule_tac [2] approx_sym, drule_tac [2] approx_sym, drule_tac [2] approx_sym)
6.2113 +   apply (auto intro!: approx_unique_real)
6.2114 +  done
6.2115
6.2116 -subsection\<open>Finite, Infinite and Infinitesimal\<close>
6.2117 +
6.2118 +subsection \<open>Finite, Infinite and Infinitesimal\<close>
6.2119
6.2120  lemma HFinite_Int_HInfinite_empty [simp]: "HFinite Int HInfinite = {}"
6.2121 -apply (simp add: HFinite_def HInfinite_def)
6.2122 -apply (auto dest: order_less_trans)
6.2123 -done
6.2124 +  apply (simp add: HFinite_def HInfinite_def)
6.2125 +  apply (auto dest: order_less_trans)
6.2126 +  done
6.2127
6.2128  lemma HFinite_not_HInfinite:
6.2129 -  assumes x: "x \<in> HFinite" shows "x \<notin> HInfinite"
6.2130 +  assumes x: "x \<in> HFinite"
6.2131 +  shows "x \<notin> HInfinite"
6.2132  proof
6.2133    assume x': "x \<in> HInfinite"
6.2134    with x have "x \<in> HFinite \<inter> HInfinite" by blast
6.2135 -  thus False by auto
6.2136 +  then show False by auto
6.2137  qed
6.2138
6.2139 -lemma not_HFinite_HInfinite: "x\<notin> HFinite ==> x \<in> HInfinite"
6.2140 -apply (simp add: HInfinite_def HFinite_def, auto)
6.2141 -apply (drule_tac x = "r + 1" in bspec)
6.2142 -apply (auto)
6.2143 -done
6.2144 +lemma not_HFinite_HInfinite: "x \<notin> HFinite \<Longrightarrow> x \<in> HInfinite"
6.2145 +  apply (simp add: HInfinite_def HFinite_def, auto)
6.2146 +  apply (drule_tac x = "r + 1" in bspec)
6.2147 +   apply (auto)
6.2148 +  done
6.2149
6.2150 -lemma HInfinite_HFinite_disj: "x \<in> HInfinite | x \<in> HFinite"
6.2151 -by (blast intro: not_HFinite_HInfinite)
6.2152 +lemma HInfinite_HFinite_disj: "x \<in> HInfinite \<or> x \<in> HFinite"
6.2153 +  by (blast intro: not_HFinite_HInfinite)
6.2154
6.2155 -lemma HInfinite_HFinite_iff: "(x \<in> HInfinite) = (x \<notin> HFinite)"
6.2156 -by (blast dest: HFinite_not_HInfinite not_HFinite_HInfinite)
6.2157 +lemma HInfinite_HFinite_iff: "x \<in> HInfinite \<longleftrightarrow> x \<notin> HFinite"
6.2158 +  by (blast dest: HFinite_not_HInfinite not_HFinite_HInfinite)
6.2159
6.2160 -lemma HFinite_HInfinite_iff: "(x \<in> HFinite) = (x \<notin> HInfinite)"
6.2162 +lemma HFinite_HInfinite_iff: "x \<in> HFinite \<longleftrightarrow> x \<notin> HInfinite"
6.2163 +  by (simp add: HInfinite_HFinite_iff)
6.2164
6.2165
6.2166  lemma HInfinite_diff_HFinite_Infinitesimal_disj:
6.2167 -     "x \<notin> Infinitesimal ==> x \<in> HInfinite | x \<in> HFinite - Infinitesimal"
6.2168 -by (fast intro: not_HFinite_HInfinite)
6.2169 +  "x \<notin> Infinitesimal \<Longrightarrow> x \<in> HInfinite \<or> x \<in> HFinite - Infinitesimal"
6.2170 +  by (fast intro: not_HFinite_HInfinite)
6.2171
6.2172 -lemma HFinite_inverse:
6.2173 -  fixes x :: "'a::real_normed_div_algebra star"
6.2174 -  shows "[| x \<in> HFinite; x \<notin> Infinitesimal |] ==> inverse x \<in> HFinite"
6.2175 -apply (subgoal_tac "x \<noteq> 0")
6.2176 -apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
6.2177 -apply (auto dest!: HInfinite_inverse_Infinitesimal
6.2179 -done
6.2180 +lemma HFinite_inverse: "x \<in> HFinite \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> inverse x \<in> HFinite"
6.2181 +  for x :: "'a::real_normed_div_algebra star"
6.2182 +  apply (subgoal_tac "x \<noteq> 0")
6.2183 +   apply (cut_tac x = "inverse x" in HInfinite_HFinite_disj)
6.2184 +   apply (auto dest!: HInfinite_inverse_Infinitesimal simp: nonzero_inverse_inverse_eq)
6.2185 +  done
6.2186
6.2187 -lemma HFinite_inverse2:
6.2188 -  fixes x :: "'a::real_normed_div_algebra star"
6.2189 -  shows "x \<in> HFinite - Infinitesimal ==> inverse x \<in> HFinite"
6.2190 -by (blast intro: HFinite_inverse)
6.2191 +lemma HFinite_inverse2: "x \<in> HFinite - Infinitesimal \<Longrightarrow> inverse x \<in> HFinite"
6.2192 +  for x :: "'a::real_normed_div_algebra star"
6.2193 +  by (blast intro: HFinite_inverse)
6.2194
6.2195 -(* stronger statement possible in fact *)
6.2196 -lemma Infinitesimal_inverse_HFinite:
6.2197 -  fixes x :: "'a::real_normed_div_algebra star"
6.2198 -  shows "x \<notin> Infinitesimal ==> inverse(x) \<in> HFinite"
6.2199 -apply (drule HInfinite_diff_HFinite_Infinitesimal_disj)
6.2200 -apply (blast intro: HFinite_inverse HInfinite_inverse_Infinitesimal Infinitesimal_subset_HFinite [THEN subsetD])
6.2201 -done
6.2202 +text \<open>Stronger statement possible in fact.\<close>
6.2203 +lemma Infinitesimal_inverse_HFinite: "x \<notin> Infinitesimal \<Longrightarrow> inverse x \<in> HFinite"
6.2204 +  for x :: "'a::real_normed_div_algebra star"
6.2205 +  apply (drule HInfinite_diff_HFinite_Infinitesimal_disj)
6.2206 +  apply (blast intro: HFinite_inverse HInfinite_inverse_Infinitesimal Infinitesimal_subset_HFinite [THEN subsetD])
6.2207 +  done
6.2208
6.2209  lemma HFinite_not_Infinitesimal_inverse:
6.2210 -  fixes x :: "'a::real_normed_div_algebra star"
6.2211 -  shows "x \<in> HFinite - Infinitesimal ==> inverse x \<in> HFinite - Infinitesimal"
6.2212 -apply (auto intro: Infinitesimal_inverse_HFinite)
6.2213 -apply (drule Infinitesimal_HFinite_mult2, assumption)
6.2215 -done
6.2216 +  "x \<in> HFinite - Infinitesimal \<Longrightarrow> inverse x \<in> HFinite - Infinitesimal"
6.2217 +  for x :: "'a::real_normed_div_algebra star"
6.2218 +  apply (auto intro: Infinitesimal_inverse_HFinite)
6.2219 +  apply (drule Infinitesimal_HFinite_mult2, assumption)
6.2220 +  apply (simp add: not_Infinitesimal_not_zero)
6.2221 +  done
6.2222
6.2223 -lemma approx_inverse:
6.2224 -  fixes x y :: "'a::real_normed_div_algebra star"
6.2225 -  shows
6.2226 -     "[| x \<approx> y; y \<in>  HFinite - Infinitesimal |]
6.2227 -      ==> inverse x \<approx> inverse y"
6.2228 -apply (frule HFinite_diff_Infinitesimal_approx, assumption)
6.2229 -apply (frule not_Infinitesimal_not_zero2)
6.2230 -apply (frule_tac x = x in not_Infinitesimal_not_zero2)
6.2231 -apply (drule HFinite_inverse2)+
6.2232 -apply (drule approx_mult2, assumption, auto)
6.2233 -apply (drule_tac c = "inverse x" in approx_mult1, assumption)
6.2234 -apply (auto intro: approx_sym simp add: mult.assoc)
6.2235 -done
6.2236 +lemma approx_inverse: "x \<approx> y \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> inverse x \<approx> inverse y"
6.2237 +  for x y :: "'a::real_normed_div_algebra star"
6.2238 +  apply (frule HFinite_diff_Infinitesimal_approx, assumption)
6.2239 +  apply (frule not_Infinitesimal_not_zero2)
6.2240 +  apply (frule_tac x = x in not_Infinitesimal_not_zero2)
6.2241 +  apply (drule HFinite_inverse2)+
6.2242 +  apply (drule approx_mult2, assumption, auto)
6.2243 +  apply (drule_tac c = "inverse x" in approx_mult1, assumption)
6.2244 +  apply (auto intro: approx_sym simp add: mult.assoc)
6.2245 +  done
6.2246
6.2247  (*Used for NSLIM_inverse, NSLIMSEQ_inverse*)
6.2248  lemmas star_of_approx_inverse = star_of_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
6.2249  lemmas hypreal_of_real_approx_inverse =  hypreal_of_real_HFinite_diff_Infinitesimal [THEN [2] approx_inverse]
6.2250
6.2252 -  fixes x h :: "'a::real_normed_div_algebra star"
6.2253 -  shows
6.2254 -     "[| x \<in> HFinite - Infinitesimal;
6.2255 -         h \<in> Infinitesimal |] ==> inverse(x + h) \<approx> inverse x"
6.2256 -apply (auto intro: approx_inverse approx_sym Infinitesimal_add_approx_self)
6.2257 -done
6.2258 +  "x \<in> HFinite - Infinitesimal \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> inverse (x + h) \<approx> inverse x"
6.2259 +  for x h :: "'a::real_normed_div_algebra star"
6.2260 +  by (auto intro: approx_inverse approx_sym Infinitesimal_add_approx_self)
6.2261
6.2263 -  fixes x h :: "'a::real_normed_div_algebra star"
6.2264 -  shows
6.2265 -     "[| x \<in> HFinite - Infinitesimal;
6.2266 -         h \<in> Infinitesimal |] ==> inverse(h + x) \<approx> inverse x"
6.2267 -apply (rule add.commute [THEN subst])
6.2269 -done
6.2270 +  "x \<in> HFinite - Infinitesimal \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> inverse (h + x) \<approx> inverse x"
6.2271 +  for x h :: "'a::real_normed_div_algebra star"
6.2272 +  apply (rule add.commute [THEN subst])
6.2273 +  apply (blast intro: inverse_add_Infinitesimal_approx)
6.2274 +  done
6.2275
6.2277 -  fixes x h :: "'a::real_normed_div_algebra star"
6.2278 -  shows
6.2279 -     "[| x \<in> HFinite - Infinitesimal;
6.2280 -         h \<in> Infinitesimal |] ==> inverse(x + h) - inverse x \<approx> h"
6.2281 -apply (rule approx_trans2)
6.2283 -            simp add: mem_infmal_iff approx_minus_iff [symmetric])
6.2284 -done
6.2285 +  "x \<in> HFinite - Infinitesimal \<Longrightarrow> h \<in> Infinitesimal \<Longrightarrow> inverse (x + h) - inverse x \<approx> h"
6.2286 +  for x h :: "'a::real_normed_div_algebra star"
6.2287 +  apply (rule approx_trans2)
6.2288 +   apply (auto intro: inverse_add_Infinitesimal_approx
6.2289 +      simp add: mem_infmal_iff approx_minus_iff [symmetric])
6.2290 +  done
6.2291
6.2292 -lemma Infinitesimal_square_iff:
6.2293 -  fixes x :: "'a::real_normed_div_algebra star"
6.2294 -  shows "(x \<in> Infinitesimal) = (x*x \<in> Infinitesimal)"
6.2295 -apply (auto intro: Infinitesimal_mult)
6.2296 -apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
6.2297 -apply (frule not_Infinitesimal_not_zero)
6.2298 -apply (auto dest: Infinitesimal_HFinite_mult simp add: mult.assoc)
6.2299 -done
6.2300 +lemma Infinitesimal_square_iff: "x \<in> Infinitesimal \<longleftrightarrow> x * x \<in> Infinitesimal"
6.2301 +  for x :: "'a::real_normed_div_algebra star"
6.2302 +  apply (auto intro: Infinitesimal_mult)
6.2303 +  apply (rule ccontr, frule Infinitesimal_inverse_HFinite)
6.2304 +  apply (frule not_Infinitesimal_not_zero)
6.2305 +  apply (auto dest: Infinitesimal_HFinite_mult simp add: mult.assoc)
6.2306 +  done
6.2307  declare Infinitesimal_square_iff [symmetric, simp]
6.2308
6.2309 -lemma HFinite_square_iff [simp]:
6.2310 -  fixes x :: "'a::real_normed_div_algebra star"
6.2311 -  shows "(x*x \<in> HFinite) = (x \<in> HFinite)"
6.2312 -apply (auto intro: HFinite_mult)
6.2313 -apply (auto dest: HInfinite_mult simp add: HFinite_HInfinite_iff)
6.2314 -done
6.2315 +lemma HFinite_square_iff [simp]: "x * x \<in> HFinite \<longleftrightarrow> x \<in> HFinite"
6.2316 +  for x :: "'a::real_normed_div_algebra star"
6.2317 +  apply (auto intro: HFinite_mult)
6.2318 +  apply (auto dest: HInfinite_mult simp add: HFinite_HInfinite_iff)
6.2319 +  done
6.2320
6.2321 -lemma HInfinite_square_iff [simp]:
6.2322 -  fixes x :: "'a::real_normed_div_algebra star"
6.2323 -  shows "(x*x \<in> HInfinite) = (x \<in> HInfinite)"
6.2324 -by (auto simp add: HInfinite_HFinite_iff)
6.2325 +lemma HInfinite_square_iff [simp]: "x * x \<in> HInfinite \<longleftrightarrow> x \<in> HInfinite"
6.2326 +  for x :: "'a::real_normed_div_algebra star"
6.2327 +  by (auto simp add: HInfinite_HFinite_iff)
6.2328
6.2329 -lemma approx_HFinite_mult_cancel:
6.2330 -  fixes a w z :: "'a::real_normed_div_algebra star"
6.2331 -  shows "[| a: HFinite-Infinitesimal; a* w \<approx> a*z |] ==> w \<approx> z"
6.2332 -apply safe
6.2333 -apply (frule HFinite_inverse, assumption)
6.2334 -apply (drule not_Infinitesimal_not_zero)
6.2335 -apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
6.2336 -done
6.2337 +lemma approx_HFinite_mult_cancel: "a \<in> HFinite - Infinitesimal \<Longrightarrow> a * w \<approx> a * z \<Longrightarrow> w \<approx> z"
6.2338 +  for a w z :: "'a::real_normed_div_algebra star"
6.2339 +  apply safe
6.2340 +  apply (frule HFinite_inverse, assumption)
6.2341 +  apply (drule not_Infinitesimal_not_zero)
6.2342 +  apply (auto dest: approx_mult2 simp add: mult.assoc [symmetric])
6.2343 +  done
6.2344
6.2345 -lemma approx_HFinite_mult_cancel_iff1:
6.2346 -  fixes a w z :: "'a::real_normed_div_algebra star"
6.2347 -  shows "a: HFinite-Infinitesimal ==> (a * w \<approx> a * z) = (w \<approx> z)"
6.2348 -by (auto intro: approx_mult2 approx_HFinite_mult_cancel)
6.2349 +lemma approx_HFinite_mult_cancel_iff1: "a \<in> HFinite - Infinitesimal \<Longrightarrow> a * w \<approx> a * z \<longleftrightarrow> w \<approx> z"
6.2350 +  for a w z :: "'a::real_normed_div_algebra star"
6.2351 +  by (auto intro: approx_mult2 approx_HFinite_mult_cancel)
6.2352
6.2354 -     "[| x + y \<in> HInfinite; y \<in> HFinite |] ==> x \<in> HInfinite"
6.2355 -apply (rule ccontr)
6.2356 -apply (drule HFinite_HInfinite_iff [THEN iffD2])
6.2358 -done
6.2359 +lemma HInfinite_HFinite_add_cancel: "x + y \<in> HInfinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x \<in> HInfinite"
6.2360 +  apply (rule ccontr)
6.2361 +  apply (drule HFinite_HInfinite_iff [THEN iffD2])
6.2363 +  done
6.2364
6.2366 -     "[| x \<in> HInfinite; y \<in> HFinite |] ==> x + y \<in> HInfinite"
6.2367 -apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
6.2369 -done
6.2370 +lemma HInfinite_HFinite_add: "x \<in> HInfinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x + y \<in> HInfinite"
6.2371 +  apply (rule_tac y = "-y" in HInfinite_HFinite_add_cancel)
6.2373 +  done
6.2374
6.2375 -lemma HInfinite_ge_HInfinite:
6.2376 -     "[| (x::hypreal) \<in> HInfinite; x \<le> y; 0 \<le> x |] ==> y \<in> HInfinite"
6.2377 -by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)
6.2378 +lemma HInfinite_ge_HInfinite: "x \<in> HInfinite \<Longrightarrow> x \<le> y \<Longrightarrow> 0 \<le> x \<Longrightarrow> y \<in> HInfinite"
6.2379 +  for x y :: hypreal
6.2380 +  by (auto intro: HFinite_bounded simp add: HInfinite_HFinite_iff)
6.2381
6.2382 -lemma Infinitesimal_inverse_HInfinite:
6.2383 -  fixes x :: "'a::real_normed_div_algebra star"
6.2384 -  shows "[| x \<in> Infinitesimal; x \<noteq> 0 |] ==> inverse x \<in> HInfinite"
6.2385 -apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
6.2386 -apply (auto dest: Infinitesimal_HFinite_mult2)
6.2387 -done
6.2388 +lemma Infinitesimal_inverse_HInfinite: "x \<in> Infinitesimal \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> inverse x \<in> HInfinite"
6.2389 +  for x :: "'a::real_normed_div_algebra star"
6.2390 +  apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
6.2391 +  apply (auto dest: Infinitesimal_HFinite_mult2)
6.2392 +  done
6.2393
6.2394  lemma HInfinite_HFinite_not_Infinitesimal_mult:
6.2395 -  fixes x y :: "'a::real_normed_div_algebra star"
6.2396 -  shows "[| x \<in> HInfinite; y \<in> HFinite - Infinitesimal |]
6.2397 -      ==> x * y \<in> HInfinite"
6.2398 -apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
6.2399 -apply (frule HFinite_Infinitesimal_not_zero)
6.2400 -apply (drule HFinite_not_Infinitesimal_inverse)
6.2401 -apply (safe, drule HFinite_mult)
6.2402 -apply (auto simp add: mult.assoc HFinite_HInfinite_iff)
6.2403 -done
6.2404 +  "x \<in> HInfinite \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> x * y \<in> HInfinite"
6.2405 +  for x y :: "'a::real_normed_div_algebra star"
6.2406 +  apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
6.2407 +  apply (frule HFinite_Infinitesimal_not_zero)
6.2408 +  apply (drule HFinite_not_Infinitesimal_inverse)
6.2409 +  apply (safe, drule HFinite_mult)
6.2410 +   apply (auto simp add: mult.assoc HFinite_HInfinite_iff)
6.2411 +  done
6.2412
6.2413  lemma HInfinite_HFinite_not_Infinitesimal_mult2:
6.2414 -  fixes x y :: "'a::real_normed_div_algebra star"
6.2415 -  shows "[| x \<in> HInfinite; y \<in> HFinite - Infinitesimal |]
6.2416 -      ==> y * x \<in> HInfinite"
6.2417 -apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
6.2418 -apply (frule HFinite_Infinitesimal_not_zero)
6.2419 -apply (drule HFinite_not_Infinitesimal_inverse)
6.2420 -apply (safe, drule_tac x="inverse y" in HFinite_mult)
6.2421 -apply assumption
6.2422 -apply (auto simp add: mult.assoc [symmetric] HFinite_HInfinite_iff)
6.2423 -done
6.2424 +  "x \<in> HInfinite \<Longrightarrow> y \<in> HFinite - Infinitesimal \<Longrightarrow> y * x \<in> HInfinite"
6.2425 +  for x y :: "'a::real_normed_div_algebra star"
6.2426 +  apply (rule ccontr, drule HFinite_HInfinite_iff [THEN iffD2])
6.2427 +  apply (frule HFinite_Infinitesimal_not_zero)
6.2428 +  apply (drule HFinite_not_Infinitesimal_inverse)
6.2429 +  apply (safe, drule_tac x="inverse y" in HFinite_mult)
6.2430 +   apply assumption
6.2431 +  apply (auto simp add: mult.assoc [symmetric] HFinite_HInfinite_iff)
6.2432 +  done
6.2433
6.2434 -lemma HInfinite_gt_SReal:
6.2435 -  "[| (x::hypreal) \<in> HInfinite; 0 < x; y \<in> \<real> |] ==> y < x"
6.2436 -by (auto dest!: bspec simp add: HInfinite_def abs_if order_less_imp_le)
6.2437 +lemma HInfinite_gt_SReal: "x \<in> HInfinite \<Longrightarrow> 0 < x \<Longrightarrow> y \<in> \<real> \<Longrightarrow> y < x"
6.2438 +  for x y :: hypreal
6.2439 +  by (auto dest!: bspec simp add: HInfinite_def abs_if order_less_imp_le)
6.2440
6.2441 -lemma HInfinite_gt_zero_gt_one:
6.2442 -  "[| (x::hypreal) \<in> HInfinite; 0 < x |] ==> 1 < x"
6.2443 -by (auto intro: HInfinite_gt_SReal)
6.2444 +lemma HInfinite_gt_zero_gt_one: "x \<in> HInfinite \<Longrightarrow> 0 < x \<Longrightarrow> 1 < x"
6.2445 +  for x :: hypreal
6.2446 +  by (auto intro: HInfinite_gt_SReal)
6.2447
6.2448
6.2449  lemma not_HInfinite_one [simp]: "1 \<notin> HInfinite"
6.2450 -apply (simp (no_asm) add: HInfinite_HFinite_iff)
6.2451 -done
6.2452 +  by (simp add: HInfinite_HFinite_iff)
6.2453
6.2454 -lemma approx_hrabs_disj: "\<bar>x::hypreal\<bar> \<approx> x \<or> \<bar>x\<bar> \<approx> -x"
6.2455 -by (cut_tac x = x in hrabs_disj, auto)
6.2456 +lemma approx_hrabs_disj: "\<bar>x\<bar> \<approx> x \<or> \<bar>x\<bar> \<approx> -x"
6.2457 +  for x :: hypreal
6.2458 +  using hrabs_disj [of x] by auto
6.2459
6.2460
6.2462 -
6.2464 -by (rule_tac x1 = x in hrabs_disj [THEN disjE], auto)
6.2466
6.2470 +  for x :: hypreal
6.2471 +  by (rule hrabs_disj [of x, THEN disjE]) auto
6.2472
6.2477
6.2482 +
6.2485
6.2488 -done
6.2491
6.2493 -apply (rule_tac x1 = x in hrabs_disj [THEN disjE])
6.2495 -done
6.2497 +  for x :: hypreal
6.2498 +  by (rule hrabs_disj [of x, THEN disjE]) (auto simp: monad_zero_minus_iff [symmetric])
6.2499
6.2503
6.2504
6.2505 -subsection\<open>Proof that @{term "x \<approx> y"} implies @{term"\<bar>x\<bar> \<approx> \<bar>y\<bar>"}\<close>
6.2506 +subsection \<open>Proof that @{term "x \<approx> y"} implies @{term"\<bar>x\<bar> \<approx> \<bar>y\<bar>"}\<close>
6.2507
6.2509 -apply (simp (no_asm))
6.2511 -done
6.2512 +lemma approx_subset_monad: "x \<approx> y \<Longrightarrow> {x, y} \<le> monad x"
6.2514
6.2516 -apply (drule approx_sym)
6.2518 -done
6.2519 +lemma approx_subset_monad2: "x \<approx> y \<Longrightarrow> {x, y} \<le> monad y"
6.2520 +  apply (drule approx_sym)
6.2521 +  apply (fast dest: approx_subset_monad)
6.2522 +  done
6.2523
6.2528 +
6.2531
6.2534 -
6.2537 -apply (blast intro!: approx_sym)
6.2538 -done
6.2541 +  apply (blast intro!: approx_sym)
6.2542 +  done
6.2543
6.2546 -apply (fast intro: approx_mem_monad approx_trans)
6.2547 -done
6.2550 +  apply (fast intro: approx_mem_monad approx_trans)
6.2551 +  done
6.2552
6.2553 -lemma Infinitesimal_approx_hrabs:
6.2554 -     "[| x \<approx> y; (x::hypreal) \<in> Infinitesimal |] ==> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2555 -apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
6.2557 -done
6.2558 +lemma Infinitesimal_approx_hrabs: "x \<approx> y \<Longrightarrow> x \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2559 +  for x y :: hypreal
6.2560 +  apply (drule Infinitesimal_monad_zero_iff [THEN iffD1])
6.2563 +  done
6.2564
6.2565 -lemma less_Infinitesimal_less:
6.2566 -     "[| 0 < x;  (x::hypreal) \<notin>Infinitesimal;  e :Infinitesimal |] ==> e < x"
6.2567 -apply (rule ccontr)
6.2568 -apply (auto intro: Infinitesimal_zero [THEN [2] Infinitesimal_interval]
6.2569 -            dest!: order_le_imp_less_or_eq simp add: linorder_not_less)
6.2570 -done
6.2571 +lemma less_Infinitesimal_less: "0 < x \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> e < x"
6.2572 +  for x :: hypreal
6.2573 +  apply (rule ccontr)
6.2574 +  apply (auto intro: Infinitesimal_zero [THEN [2] Infinitesimal_interval]
6.2575 +      dest!: order_le_imp_less_or_eq simp add: linorder_not_less)
6.2576 +  done
6.2577
6.2579 -     "[| 0 < (x::hypreal);  x \<notin> Infinitesimal; u \<in> monad x |] ==> 0 < u"
6.2580 -apply (drule mem_monad_approx [THEN approx_sym])
6.2581 -apply (erule bex_Infinitesimal_iff2 [THEN iffD2, THEN bexE])
6.2582 -apply (drule_tac e = "-xa" in less_Infinitesimal_less, auto)
6.2583 -done
6.2584 +lemma Ball_mem_monad_gt_zero: "0 < x \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> u \<in> monad x \<Longrightarrow> 0 < u"
6.2585 +  for u x :: hypreal
6.2586 +  apply (drule mem_monad_approx [THEN approx_sym])
6.2587 +  apply (erule bex_Infinitesimal_iff2 [THEN iffD2, THEN bexE])
6.2588 +  apply (drule_tac e = "-xa" in less_Infinitesimal_less, auto)
6.2589 +  done
6.2590
6.2592 -     "[| (x::hypreal) < 0; x \<notin> Infinitesimal; u \<in> monad x |] ==> u < 0"
6.2593 -apply (drule mem_monad_approx [THEN approx_sym])
6.2594 -apply (erule bex_Infinitesimal_iff [THEN iffD2, THEN bexE])
6.2595 -apply (cut_tac x = "-x" and e = xa in less_Infinitesimal_less, auto)
6.2596 -done
6.2597 +lemma Ball_mem_monad_less_zero: "x < 0 \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> u \<in> monad x \<Longrightarrow> u < 0"
6.2598 +  for u x :: hypreal
6.2599 +  apply (drule mem_monad_approx [THEN approx_sym])
6.2600 +  apply (erule bex_Infinitesimal_iff [THEN iffD2, THEN bexE])
6.2601 +  apply (cut_tac x = "-x" and e = xa in less_Infinitesimal_less, auto)
6.2602 +  done
6.2603
6.2604 -lemma lemma_approx_gt_zero:
6.2605 -     "[|0 < (x::hypreal); x \<notin> Infinitesimal; x \<approx> y|] ==> 0 < y"
6.2607 +lemma lemma_approx_gt_zero: "0 < x \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> x \<approx> y \<Longrightarrow> 0 < y"
6.2608 +  for x y :: hypreal
6.2610
6.2611 -lemma lemma_approx_less_zero:
6.2612 -     "[|(x::hypreal) < 0; x \<notin> Infinitesimal; x \<approx> y|] ==> y < 0"
6.2614 +lemma lemma_approx_less_zero: "x < 0 \<Longrightarrow> x \<notin> Infinitesimal \<Longrightarrow> x \<approx> y \<Longrightarrow> y < 0"
6.2615 +  for x y :: hypreal
6.2617
6.2618 -theorem approx_hrabs: "(x::hypreal) \<approx> y ==> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2619 -by (drule approx_hnorm, simp)
6.2620 +lemma approx_hrabs: "x \<approx> y \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2621 +  for x y :: hypreal
6.2622 +  by (drule approx_hnorm) simp
6.2623
6.2624 -lemma approx_hrabs_zero_cancel: "\<bar>x::hypreal\<bar> \<approx> 0 ==> x \<approx> 0"
6.2625 -apply (cut_tac x = x in hrabs_disj)
6.2626 -apply (auto dest: approx_minus)
6.2627 -done
6.2628 +lemma approx_hrabs_zero_cancel: "\<bar>x\<bar> \<approx> 0 \<Longrightarrow> x \<approx> 0"
6.2629 +  for x :: hypreal
6.2630 +  using hrabs_disj [of x] by (auto dest: approx_minus)
6.2631
6.2633 -  "(e::hypreal) \<in> Infinitesimal ==> \<bar>x\<bar> \<approx> \<bar>x + e\<bar>"
6.2634 -by (fast intro: approx_hrabs Infinitesimal_add_approx_self)
6.2635 +lemma approx_hrabs_add_Infinitesimal: "e \<in> Infinitesimal \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>x + e\<bar>"
6.2636 +  for e x :: hypreal
6.2637 +  by (fast intro: approx_hrabs Infinitesimal_add_approx_self)
6.2638
6.2640 -     "(e::hypreal) \<in> Infinitesimal ==> \<bar>x\<bar> \<approx> \<bar>x + -e\<bar>"
6.2641 -by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)
6.2642 +lemma approx_hrabs_add_minus_Infinitesimal: "e \<in> Infinitesimal ==> \<bar>x\<bar> \<approx> \<bar>x + -e\<bar>"
6.2643 +  for e x :: hypreal
6.2644 +  by (fast intro: approx_hrabs Infinitesimal_add_minus_approx_self)
6.2645
6.2647 -     "[| (e::hypreal) \<in> Infinitesimal; e' \<in> Infinitesimal;
6.2648 -         \<bar>x + e\<bar> = \<bar>y + e'\<bar>|] ==> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2649 -apply (drule_tac x = x in approx_hrabs_add_Infinitesimal)
6.2650 -apply (drule_tac x = y in approx_hrabs_add_Infinitesimal)
6.2651 -apply (auto intro: approx_trans2)
6.2652 -done
6.2653 +  "e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> \<bar>x + e\<bar> = \<bar>y + e'\<bar> \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2654 +  for e e' x y :: hypreal
6.2655 +  apply (drule_tac x = x in approx_hrabs_add_Infinitesimal)
6.2656 +  apply (drule_tac x = y in approx_hrabs_add_Infinitesimal)
6.2657 +  apply (auto intro: approx_trans2)
6.2658 +  done
6.2659
6.2661 -     "[| (e::hypreal) \<in> Infinitesimal; e' \<in> Infinitesimal;
6.2662 -         \<bar>x + -e\<bar> = \<bar>y + -e'\<bar>|] ==> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2663 -apply (drule_tac x = x in approx_hrabs_add_minus_Infinitesimal)
6.2664 -apply (drule_tac x = y in approx_hrabs_add_minus_Infinitesimal)
6.2665 -apply (auto intro: approx_trans2)
6.2666 -done
6.2667 +  "e \<in> Infinitesimal \<Longrightarrow> e' \<in> Infinitesimal \<Longrightarrow> \<bar>x + -e\<bar> = \<bar>y + -e'\<bar> \<Longrightarrow> \<bar>x\<bar> \<approx> \<bar>y\<bar>"
6.2668 +  for e e' x y :: hypreal
6.2669 +  apply (drule_tac x = x in approx_hrabs_add_minus_Infinitesimal)
6.2670 +  apply (drule_tac x = y in approx_hrabs_add_minus_Infinitesimal)
6.2671 +  apply (auto intro: approx_trans2)
6.2672 +  done
6.2673 +
6.2674
6.2675  subsection \<open>More @{term HFinite} and @{term Infinitesimal} Theorems\<close>
6.2676
6.2677 -(* interesting slightly counterintuitive theorem: necessary
6.2678 -   for proving that an open interval is an NS open set
6.2679 -*)
6.2680 +text \<open>
6.2681 +  Interesting slightly counterintuitive theorem: necessary
6.2682 +  for proving that an open interval is an NS open set.
6.2683 +\<close>
6.2685 -     "[| x < y;  u \<in> Infinitesimal |]
6.2686 -      ==> hypreal_of_real x + u < hypreal_of_real y"
6.2688 -apply (drule_tac x = "hypreal_of_real y + -hypreal_of_real x" in bspec, simp)
6.2690 -done
6.2691 +  "x < y \<Longrightarrow> u \<in> Infinitesimal \<Longrightarrow> hypreal_of_real x + u < hypreal_of_real y"
6.2692 +  apply (simp add: Infinitesimal_def)
6.2693 +  apply (drule_tac x = "hypreal_of_real y + -hypreal_of_real x" in bspec, simp)
6.2694 +  apply (simp add: abs_less_iff)
6.2695 +  done
6.2696
6.2698 -     "[| x \<in> Infinitesimal; \<bar>hypreal_of_real r\<bar> < hypreal_of_real y |]
6.2699 -      ==> \<bar>hypreal_of_real r + x\<bar> < hypreal_of_real y"
6.2700 -apply (drule_tac x = "hypreal_of_real r" in approx_hrabs_add_Infinitesimal)
6.2701 -apply (drule approx_sym [THEN bex_Infinitesimal_iff2 [THEN iffD2]])
6.2703 -            simp del: star_of_abs
6.2704 -            simp add: star_of_abs [symmetric])
6.2705 -done
6.2706 +  "x \<in> Infinitesimal \<Longrightarrow> \<bar>hypreal_of_real r\<bar> < hypreal_of_real y \<Longrightarrow>
6.2707 +    \<bar>hypreal_of_real r + x\<bar> < hypreal_of_real y"
6.2708 +  apply (drule_tac x = "hypreal_of_real r" in approx_hrabs_add_Infinitesimal)
6.2709 +  apply (drule approx_sym [THEN bex_Infinitesimal_iff2 [THEN iffD2]])
6.2710 +  apply (auto intro!: Infinitesimal_add_hypreal_of_real_less
6.2711 +      simp del: star_of_abs simp add: star_of_abs [symmetric])
6.2712 +  done
6.2713
6.2715 -     "[| x \<in> Infinitesimal;  \<bar>hypreal_of_real r\<bar> < hypreal_of_real y |]
6.2716 -      ==> \<bar>x + hypreal_of_real r\<bar> < hypreal_of_real y"
6.2717 -apply (rule add.commute [THEN subst])
6.2719 -done
6.2720 +  "x \<in> Infinitesimal \<Longrightarrow> \<bar>hypreal_of_real r\<bar> < hypreal_of_real y \<Longrightarrow>
6.2721 +    \<bar>x + hypreal_of_real r\<bar> < hypreal_of_real y"
6.2722 +  apply (rule add.commute [THEN subst])
6.2723 +  apply (erule Infinitesimal_add_hrabs_hypreal_of_real_less, assumption)
6.2724 +  done
6.2725
6.2727 -     "[| u \<in> Infinitesimal; v \<in> Infinitesimal;
6.2728 -         hypreal_of_real x + u \<le> hypreal_of_real y + v |]
6.2729 -      ==> hypreal_of_real x \<le> hypreal_of_real y"
6.2730 -apply (simp add: linorder_not_less [symmetric], auto)
6.2731 -apply (drule_tac u = "v-u" in Infinitesimal_add_hypreal_of_real_less)
6.2732 -apply (auto simp add: Infinitesimal_diff)
6.2733 -done
6.2734 +  "u \<in> Infinitesimal \<Longrightarrow> v \<in> Infinitesimal \<Longrightarrow>
6.2735 +    hypreal_of_real x + u \<le> hypreal_of_real y + v \<Longrightarrow>
6.2736 +    hypreal_of_real x \<le> hypreal_of_real y"
6.2737 +  apply (simp add: linorder_not_less [symmetric], auto)
6.2738 +  apply (drule_tac u = "v-u" in Infinitesimal_add_hypreal_of_real_less)
6.2739 +   apply (auto simp add: Infinitesimal_diff)
6.2740 +  done
6.2741
6.2743 -     "[| u \<in> Infinitesimal; v \<in> Infinitesimal;
6.2744 -         hypreal_of_real x + u \<le> hypreal_of_real y + v |]
6.2745 -      ==> x \<le> y"
6.2746 -by (blast intro: star_of_le [THEN iffD1]
6.2748 +  "u \<in> Infinitesimal \<Longrightarrow> v \<in> Infinitesimal \<Longrightarrow>
6.2749 +    hypreal_of_real x + u \<le> hypreal_of_real y + v \<Longrightarrow> x \<le> y"
6.2750 +  by (blast intro: star_of_le [THEN iffD1] intro!: hypreal_of_real_le_add_Infininitesimal_cancel)
6.2751
6.2752  lemma hypreal_of_real_less_Infinitesimal_le_zero:
6.2753 -    "[| hypreal_of_real x < e; e \<in> Infinitesimal |] ==> hypreal_of_real x \<le> 0"
6.2754 -apply (rule linorder_not_less [THEN iffD1], safe)
6.2755 -apply (drule Infinitesimal_interval)
6.2756 -apply (drule_tac [4] SReal_hypreal_of_real [THEN SReal_Infinitesimal_zero], auto)
6.2757 -done
6.2758 +  "hypreal_of_real x < e \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> hypreal_of_real x \<le> 0"
6.2759 +  apply (rule linorder_not_less [THEN iffD1], safe)
6.2760 +  apply (drule Infinitesimal_interval)
6.2761 +     apply (drule_tac [4] SReal_hypreal_of_real [THEN SReal_Infinitesimal_zero], auto)
6.2762 +  done
6.2763
6.2764  (*used once, in Lim/NSDERIV_inverse*)
6.2766 -     "[| h \<in> Infinitesimal; x \<noteq> 0 |] ==> star_of x + h \<noteq> 0"
6.2767 -apply auto
6.2768 -apply (subgoal_tac "h = - star_of x", auto intro: minus_unique [symmetric])
6.2769 -done
6.2770 +lemma Infinitesimal_add_not_zero: "h \<in> Infinitesimal \<Longrightarrow> x \<noteq> 0 \<Longrightarrow> star_of x + h \<noteq> 0"
6.2771 +  apply auto
6.2772 +  apply (subgoal_tac "h = - star_of x")
6.2773 +   apply (auto intro: minus_unique [symmetric])
6.2774 +  done
6.2775
6.2776 -lemma Infinitesimal_square_cancel [simp]:
6.2777 -     "(x::hypreal)*x + y*y \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
6.2778 -apply (rule Infinitesimal_interval2)
6.2779 -apply (rule_tac [3] zero_le_square, assumption)
6.2780 -apply (auto)
6.2781 -done
6.2782 +lemma Infinitesimal_square_cancel [simp]: "x * x + y * y \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
6.2783 +  for x y :: hypreal
6.2784 +  apply (rule Infinitesimal_interval2)
6.2785 +     apply (rule_tac [3] zero_le_square, assumption)
6.2786 +   apply auto
6.2787 +  done
6.2788
6.2789 -lemma HFinite_square_cancel [simp]:
6.2790 -  "(x::hypreal)*x + y*y \<in> HFinite ==> x*x \<in> HFinite"
6.2791 -apply (rule HFinite_bounded, assumption)
6.2792 -apply (auto)
6.2793 -done
6.2794 +lemma HFinite_square_cancel [simp]: "x * x + y * y \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
6.2795 +  for x y :: hypreal
6.2796 +  apply (rule HFinite_bounded, assumption)
6.2797 +   apply auto
6.2798 +  done
6.2799
6.2800 -lemma Infinitesimal_square_cancel2 [simp]:
6.2801 -     "(x::hypreal)*x + y*y \<in> Infinitesimal ==> y*y \<in> Infinitesimal"
6.2802 -apply (rule Infinitesimal_square_cancel)
6.2803 -apply (rule add.commute [THEN subst])
6.2804 -apply (simp (no_asm))
6.2805 -done
6.2806 +lemma Infinitesimal_square_cancel2 [simp]: "x * x + y * y \<in> Infinitesimal \<Longrightarrow> y * y \<in> Infinitesimal"
6.2807 +  for x y :: hypreal
6.2808 +  apply (rule Infinitesimal_square_cancel)
6.2809 +  apply (rule add.commute [THEN subst])
6.2810 +  apply simp
6.2811 +  done
6.2812
6.2813 -lemma HFinite_square_cancel2 [simp]:
6.2814 -  "(x::hypreal)*x + y*y \<in> HFinite ==> y*y \<in> HFinite"
6.2815 -apply (rule HFinite_square_cancel)
6.2816 -apply (rule add.commute [THEN subst])
6.2817 -apply (simp (no_asm))
6.2818 -done
6.2819 +lemma HFinite_square_cancel2 [simp]: "x * x + y * y \<in> HFinite \<Longrightarrow> y * y \<in> HFinite"
6.2820 +  for x y :: hypreal
6.2821 +  apply (rule HFinite_square_cancel)
6.2822 +  apply (rule add.commute [THEN subst])
6.2823 +  apply simp
6.2824 +  done
6.2825
6.2826  lemma Infinitesimal_sum_square_cancel [simp]:
6.2827 -     "(x::hypreal)*x + y*y + z*z \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
6.2828 -apply (rule Infinitesimal_interval2, assumption)
6.2829 -apply (rule_tac [2] zero_le_square, simp)
6.2830 -apply (insert zero_le_square [of y])
6.2831 -apply (insert zero_le_square [of z], simp del:zero_le_square)
6.2832 -done
6.2833 +  "x * x + y * y + z * z \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
6.2834 +  for x y z :: hypreal
6.2835 +  apply (rule Infinitesimal_interval2, assumption)
6.2836 +    apply (rule_tac [2] zero_le_square, simp)
6.2837 +  apply (insert zero_le_square [of y])
6.2838 +  apply (insert zero_le_square [of z], simp del:zero_le_square)
6.2839 +  done
6.2840
6.2841 -lemma HFinite_sum_square_cancel [simp]:
6.2842 -     "(x::hypreal)*x + y*y + z*z \<in> HFinite ==> x*x \<in> HFinite"
6.2843 -apply (rule HFinite_bounded, assumption)
6.2844 -apply (rule_tac [2] zero_le_square)
6.2845 -apply (insert zero_le_square [of y])
6.2846 -apply (insert zero_le_square [of z], simp del:zero_le_square)
6.2847 -done
6.2848 +lemma HFinite_sum_square_cancel [simp]: "x * x + y * y + z * z \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
6.2849 +  for x y z :: hypreal
6.2850 +  apply (rule HFinite_bounded, assumption)
6.2851 +   apply (rule_tac [2] zero_le_square)
6.2852 +  apply (insert zero_le_square [of y])
6.2853 +  apply (insert zero_le_square [of z], simp del:zero_le_square)
6.2854 +  done
6.2855
6.2856  lemma Infinitesimal_sum_square_cancel2 [simp]:
6.2857 -     "(y::hypreal)*y + x*x + z*z \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
6.2858 -apply (rule Infinitesimal_sum_square_cancel)
6.2860 -done
6.2861 +  "y * y + x * x + z * z \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
6.2862 +  for x y z :: hypreal
6.2863 +  apply (rule Infinitesimal_sum_square_cancel)
6.2864 +  apply (simp add: ac_simps)
6.2865 +  done
6.2866
6.2867 -lemma HFinite_sum_square_cancel2 [simp]:
6.2868 -     "(y::hypreal)*y + x*x + z*z \<in> HFinite ==> x*x \<in> HFinite"
6.2869 -apply (rule HFinite_sum_square_cancel)
6.2871 -done
6.2872 +lemma HFinite_sum_square_cancel2 [simp]: "y * y + x * x + z * z \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
6.2873 +  for x y z :: hypreal
6.2874 +  apply (rule HFinite_sum_square_cancel)
6.2875 +  apply (simp add: ac_simps)
6.2876 +  done
6.2877
6.2878  lemma Infinitesimal_sum_square_cancel3 [simp]:
6.2879 -     "(z::hypreal)*z + y*y + x*x \<in> Infinitesimal ==> x*x \<in> Infinitesimal"
6.2880 -apply (rule Infinitesimal_sum_square_cancel)
6.2882 -done
6.2883 +  "z * z + y * y + x * x \<in> Infinitesimal \<Longrightarrow> x * x \<in> Infinitesimal"
6.2884 +  for x y z :: hypreal
6.2885 +  apply (rule Infinitesimal_sum_square_cancel)
6.2886 +  apply (simp add: ac_simps)
6.2887 +  done
6.2888
6.2889 -lemma HFinite_sum_square_cancel3 [simp]:
6.2890 -     "(z::hypreal)*z + y*y + x*x \<in> HFinite ==> x*x \<in> HFinite"
6.2891 -apply (rule HFinite_sum_square_cancel)
6.2893 -done
6.2894 +lemma HFinite_sum_square_cancel3 [simp]: "z * z + y * y + x * x \<in> HFinite \<Longrightarrow> x * x \<in> HFinite"
6.2895 +  for x y z :: hypreal
6.2896 +  apply (rule HFinite_sum_square_cancel)
6.2897 +  apply (simp add: ac_simps)
6.2898 +  done
6.2899
6.2901 -     "[| y \<in> monad x; 0 < hypreal_of_real e |]
6.2902 -      ==> \<bar>y - x\<bar> < hypreal_of_real e"
6.2903 -apply (drule mem_monad_approx [THEN approx_sym])
6.2904 -apply (drule bex_Infinitesimal_iff [THEN iffD2])
6.2905 -apply (auto dest!: InfinitesimalD)
6.2906 -done
6.2907 +lemma monad_hrabs_less: "y \<in> monad x \<Longrightarrow> 0 < hypreal_of_real e \<Longrightarrow> \<bar>y - x\<bar> < hypreal_of_real e"
6.2908 +  apply (drule mem_monad_approx [THEN approx_sym])
6.2909 +  apply (drule bex_Infinitesimal_iff [THEN iffD2])
6.2910 +  apply (auto dest!: InfinitesimalD)
6.2911 +  done
6.2912
6.2914 -     "x \<in> monad (hypreal_of_real  a) ==> x \<in> HFinite"
6.2915 -apply (drule mem_monad_approx [THEN approx_sym])
6.2916 -apply (drule bex_Infinitesimal_iff2 [THEN iffD2])
6.2917 -apply (safe dest!: Infinitesimal_subset_HFinite [THEN subsetD])
6.2918 -apply (erule SReal_hypreal_of_real [THEN SReal_subset_HFinite [THEN subsetD], THEN HFinite_add])
6.2919 -done
6.2920 +lemma mem_monad_SReal_HFinite: "x \<in> monad (hypreal_of_real  a) \<Longrightarrow> x \<in> HFinite"
6.2921 +  apply (drule mem_monad_approx [THEN approx_sym])
6.2922 +  apply (drule bex_Infinitesimal_iff2 [THEN iffD2])
6.2923 +  apply (safe dest!: Infinitesimal_subset_HFinite [THEN subsetD])
6.2924 +  apply (erule SReal_hypreal_of_real [THEN SReal_subset_HFinite [THEN subsetD], THEN HFinite_add])
6.2925 +  done
6.2926
6.2927
6.2929 +subsection \<open>Theorems about Standard Part\<close>
6.2930
6.2931 -lemma st_approx_self: "x \<in> HFinite ==> st x \<approx> x"
6.2933 -apply (frule st_part_Ex, safe)
6.2934 -apply (rule someI2)
6.2935 -apply (auto intro: approx_sym)
6.2936 -done
6.2937 +lemma st_approx_self: "x \<in> HFinite \<Longrightarrow> st x \<approx> x"
6.2938 +  apply (simp add: st_def)
6.2939 +  apply (frule st_part_Ex, safe)
6.2940 +  apply (rule someI2)
6.2941 +   apply (auto intro: approx_sym)
6.2942 +  done
6.2943
6.2944 -lemma st_SReal: "x \<in> HFinite ==> st x \<in> \<real>"
6.2946 -apply (frule st_part_Ex, safe)
6.2947 -apply (rule someI2)
6.2948 -apply (auto intro: approx_sym)
6.2949 -done
6.2950 +lemma st_SReal: "x \<in> HFinite \<Longrightarrow> st x \<in> \<real>"
6.2951 +  apply (simp add: st_def)
6.2952 +  apply (frule st_part_Ex, safe)
6.2953 +  apply (rule someI2)
6.2954 +   apply (auto intro: approx_sym)
6.2955 +  done
6.2956
6.2957 -lemma st_HFinite: "x \<in> HFinite ==> st x \<in> HFinite"
6.2958 -by (erule st_SReal [THEN SReal_subset_HFinite [THEN subsetD]])
6.2959 +lemma st_HFinite: "x \<in> HFinite \<Longrightarrow> st x \<in> HFinite"
6.2960 +  by (erule st_SReal [THEN SReal_subset_HFinite [THEN subsetD]])
6.2961
6.2962 -lemma st_unique: "\<lbrakk>r \<in> \<real>; r \<approx> x\<rbrakk> \<Longrightarrow> st x = r"
6.2963 -apply (frule SReal_subset_HFinite [THEN subsetD])
6.2964 -apply (drule (1) approx_HFinite)
6.2965 -apply (unfold st_def)
6.2966 -apply (rule some_equality)
6.2967 -apply (auto intro: approx_unique_real)
6.2968 -done
6.2969 +lemma st_unique: "r \<in> \<real> \<Longrightarrow> r \<approx> x \<Longrightarrow> st x = r"
6.2970 +  apply (frule SReal_subset_HFinite [THEN subsetD])
6.2971 +  apply (drule (1) approx_HFinite)
6.2972 +  apply (unfold st_def)
6.2973 +  apply (rule some_equality)
6.2974 +   apply (auto intro: approx_unique_real)
6.2975 +  done
6.2976
6.2977 -lemma st_SReal_eq: "x \<in> \<real> ==> st x = x"
6.2978 +lemma st_SReal_eq: "x \<in> \<real> \<Longrightarrow> st x = x"
6.2979    by (metis approx_refl st_unique)
6.2980
6.2981  lemma st_hypreal_of_real [simp]: "st (hypreal_of_real x) = hypreal_of_real x"
6.2982 -by (rule SReal_hypreal_of_real [THEN st_SReal_eq])
6.2983 +  by (rule SReal_hypreal_of_real [THEN st_SReal_eq])
6.2984
6.2985 -lemma st_eq_approx: "[| x \<in> HFinite; y \<in> HFinite; st x = st y |] ==> x \<approx> y"
6.2986 -by (auto dest!: st_approx_self elim!: approx_trans3)
6.2987 +lemma st_eq_approx: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> st x = st y \<Longrightarrow> x \<approx> y"
6.2988 +  by (auto dest!: st_approx_self elim!: approx_trans3)
6.2989
6.2990  lemma approx_st_eq:
6.2991    assumes x: "x \<in> HFinite" and y: "y \<in> HFinite" and xy: "x \<approx> y"
6.2992 @@ -1796,32 +1665,27 @@
6.2993      by (fast elim: approx_trans approx_trans2 SReal_approx_iff [THEN iffD1])
6.2994  qed
6.2995
6.2996 -lemma st_eq_approx_iff:
6.2997 -     "[| x \<in> HFinite; y \<in> HFinite|]
6.2998 -                   ==> (x \<approx> y) = (st x = st y)"
6.2999 -by (blast intro: approx_st_eq st_eq_approx)
6.3000 +lemma st_eq_approx_iff: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x \<approx> y \<longleftrightarrow> st x = st y"
6.3001 +  by (blast intro: approx_st_eq st_eq_approx)
6.3002
6.3004 -     "[| x \<in> \<real>; e \<in> Infinitesimal |] ==> st(x + e) = x"
6.3005 -apply (erule st_unique)
6.3007 -done
6.3008 +lemma st_Infinitesimal_add_SReal: "x \<in> \<real> \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> st (x + e) = x"
6.3009 +  apply (erule st_unique)
6.3011 +  done
6.3012
6.3014 -     "[| x \<in> \<real>; e \<in> Infinitesimal |] ==> st(e + x) = x"
6.3015 -apply (erule st_unique)
6.3017 -done
6.3018 +lemma st_Infinitesimal_add_SReal2: "x \<in> \<real> \<Longrightarrow> e \<in> Infinitesimal \<Longrightarrow> st (e + x) = x"
6.3019 +  apply (erule st_unique)
6.3021 +  done
6.3022
6.3024 -     "x \<in> HFinite ==> \<exists>e \<in> Infinitesimal. x = st(x) + e"
6.3025 -by (blast dest!: st_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])
6.3026 +lemma HFinite_st_Infinitesimal_add: "x \<in> HFinite \<Longrightarrow> \<exists>e \<in> Infinitesimal. x = st(x) + e"
6.3027 +  by (blast dest!: st_approx_self [THEN approx_sym] bex_Infinitesimal_iff2 [THEN iffD2])
6.3028
6.3029 -lemma st_add: "\<lbrakk>x \<in> HFinite; y \<in> HFinite\<rbrakk> \<Longrightarrow> st (x + y) = st x + st y"
6.3031 +lemma st_add: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> st (x + y) = st x + st y"
6.3033
6.3034  lemma st_numeral [simp]: "st (numeral w) = numeral w"
6.3035 -by (rule Reals_numeral [THEN st_SReal_eq])
6.3036 +  by (rule Reals_numeral [THEN st_SReal_eq])
6.3037
6.3038  lemma st_neg_numeral [simp]: "st (- numeral w) = - numeral w"
6.3039  proof -
6.3040 @@ -1831,79 +1695,71 @@
6.3041  qed
6.3042
6.3043  lemma st_0 [simp]: "st 0 = 0"
6.3045 +  by (simp add: st_SReal_eq)
6.3046
6.3047  lemma st_1 [simp]: "st 1 = 1"
6.3049 +  by (simp add: st_SReal_eq)
6.3050
6.3051  lemma st_neg_1 [simp]: "st (- 1) = - 1"
6.3053 +  by (simp add: st_SReal_eq)
6.3054
6.3055  lemma st_minus: "x \<in> HFinite \<Longrightarrow> st (- x) = - st x"
6.3056 -by (simp add: st_unique st_SReal st_approx_self approx_minus)
6.3057 +  by (simp add: st_unique st_SReal st_approx_self approx_minus)
6.3058
6.3059  lemma st_diff: "\<lbrakk>x \<in> HFinite; y \<in> HFinite\<rbrakk> \<Longrightarrow> st (x - y) = st x - st y"
6.3060 -by (simp add: st_unique st_SReal st_approx_self approx_diff)
6.3061 +  by (simp add: st_unique st_SReal st_approx_self approx_diff)
6.3062
6.3063  lemma st_mult: "\<lbrakk>x \<in> HFinite; y \<in> HFinite\<rbrakk> \<Longrightarrow> st (x * y) = st x * st y"
6.3064 -by (simp add: st_unique st_SReal st_approx_self approx_mult_HFinite)
6.3065 +  by (simp add: st_unique st_SReal st_approx_self approx_mult_HFinite)
6.3066
6.3067 -lemma st_Infinitesimal: "x \<in> Infinitesimal ==> st x = 0"
6.3068 -by (simp add: st_unique mem_infmal_iff)
6.3069 +lemma st_Infinitesimal: "x \<in> Infinitesimal \<Longrightarrow> st x = 0"
6.3070 +  by (simp add: st_unique mem_infmal_iff)
6.3071
6.3072 -lemma st_not_Infinitesimal: "st(x) \<noteq> 0 ==> x \<notin> Infinitesimal"
6.3073 +lemma st_not_Infinitesimal: "st(x) \<noteq> 0 \<Longrightarrow> x \<notin> Infinitesimal"
6.3074  by (fast intro: st_Infinitesimal)
6.3075
6.3076 -lemma st_inverse:
6.3077 -     "[| x \<in> HFinite; st x \<noteq> 0 |]
6.3078 -      ==> st(inverse x) = inverse (st x)"
6.3079 -apply (rule_tac c1 = "st x" in mult_left_cancel [THEN iffD1])
6.3080 -apply (auto simp add: st_mult [symmetric] st_not_Infinitesimal HFinite_inverse)
6.3081 -apply (subst right_inverse, auto)
6.3082 -done
6.3083 +lemma st_inverse: "x \<in> HFinite \<Longrightarrow> st x \<noteq> 0 \<Longrightarrow> st (inverse x) = inverse (st x)"
6.3084 +  apply (rule_tac c1 = "st x" in mult_left_cancel [THEN iffD1])
6.3085 +   apply (auto simp add: st_mult [symmetric] st_not_Infinitesimal HFinite_inverse)
6.3086 +  apply (subst right_inverse, auto)
6.3087 +  done
6.3088
6.3089 -lemma st_divide [simp]:
6.3090 -     "[| x \<in> HFinite; y \<in> HFinite; st y \<noteq> 0 |]
6.3091 -      ==> st(x/y) = (st x) / (st y)"
6.3092 -by (simp add: divide_inverse st_mult st_not_Infinitesimal HFinite_inverse st_inverse)
6.3093 +lemma st_divide [simp]: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> st y \<noteq> 0 \<Longrightarrow> st (x / y) = st x / st y"
6.3094 +  by (simp add: divide_inverse st_mult st_not_Infinitesimal HFinite_inverse st_inverse)
6.3095
6.3096 -lemma st_idempotent [simp]: "x \<in> HFinite ==> st(st(x)) = st(x)"
6.3097 -by (blast intro: st_HFinite st_approx_self approx_st_eq)
6.3098 +lemma st_idempotent [simp]: "x \<in> HFinite \<Longrightarrow> st (st x) = st x"
6.3099 +  by (blast intro: st_HFinite st_approx_self approx_st_eq)
6.3100
6.3102 -     "[| x \<in> HFinite; y \<in> HFinite; u \<in> Infinitesimal; st x < st y |]
6.3103 -      ==> st x + u < st y"
6.3104 -apply (drule st_SReal)+
6.3106 -done
6.3107 +  "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> u \<in> Infinitesimal \<Longrightarrow> st x < st y \<Longrightarrow> st x + u < st y"
6.3108 +  apply (drule st_SReal)+
6.3110 +  done
6.3111
6.3113 -     "[| x \<in> HFinite; y \<in> HFinite;
6.3114 -         u \<in> Infinitesimal; st x \<le> st y + u
6.3115 -      |] ==> st x \<le> st y"
6.3116 -apply (simp add: linorder_not_less [symmetric])
6.3118 -done
6.3119 +  "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> u \<in> Infinitesimal \<Longrightarrow>
6.3120 +    st x \<le> st y + u \<Longrightarrow> st x \<le> st y"
6.3121 +  apply (simp add: linorder_not_less [symmetric])
6.3122 +  apply (auto dest: Infinitesimal_add_st_less)
6.3123 +  done
6.3124
6.3125 -lemma st_le: "[| x \<in> HFinite; y \<in> HFinite; x \<le> y |] ==> st(x) \<le> st(y)"
6.3126 -by (metis approx_le_bound approx_sym linear st_SReal st_approx_self st_part_Ex1)
6.3127 +lemma st_le: "x \<in> HFinite \<Longrightarrow> y \<in> HFinite \<Longrightarrow> x \<le> y \<Longrightarrow> st x \<le> st y"
6.3128 +  by (metis approx_le_bound approx_sym linear st_SReal st_approx_self st_part_Ex1)
6.3129
6.3130 -lemma st_zero_le: "[| 0 \<le> x;  x \<in> HFinite |] ==> 0 \<le> st x"
6.3131 -apply (subst st_0 [symmetric])
6.3132 -apply (rule st_le, auto)
6.3133 -done
6.3134 +lemma st_zero_le: "0 \<le> x \<Longrightarrow> x \<in> HFinite \<Longrightarrow> 0 \<le> st x"
6.3135 +  apply (subst st_0 [symmetric])
6.3136 +  apply (rule st_le, auto)
6.3137 +  done
6.3138
6.3139 -lemma st_zero_ge: "[| x \<le> 0;  x \<in> HFinite |] ==> st x \<le> 0"
6.3140 -apply (subst st_0 [symmetric])
6.3141 -apply (rule st_le, auto)
6.3142 -done
6.3143 +lemma st_zero_ge: "x \<le> 0 \<Longrightarrow> x \<in> HFinite \<Longrightarrow> st x \<le> 0"
6.3144 +  apply (subst st_0 [symmetric])
6.3145 +  apply (rule st_le, auto)
6.3146 +  done
6.3147
6.3148 -lemma st_hrabs: "x \<in> HFinite ==> \<bar>st x\<bar> = st \<bar>x\<bar>"
6.3149 -apply (simp add: linorder_not_le st_zero_le abs_if st_minus
6.3150 -   linorder_not_less)
6.3151 -apply (auto dest!: st_zero_ge [OF order_less_imp_le])
6.3152 -done
6.3153 -
6.3154 +lemma st_hrabs: "x \<in> HFinite \<Longrightarrow> \<bar>st x\<bar> = st \<bar>x\<bar>"
6.3155 +  apply (simp add: linorder_not_le st_zero_le abs_if st_minus linorder_not_less)
6.3156 +  apply (auto dest!: st_zero_ge [OF order_less_imp_le])
6.3157 +  done
6.3158
6.3159
6.3160  subsection \<open>Alternative Definitions using Free Ultrafilter\<close>
6.3161 @@ -1911,78 +1767,73 @@
6.3162  subsubsection \<open>@{term HFinite}\<close>
6.3163
6.3164  lemma HFinite_FreeUltrafilterNat:
6.3165 -    "star_n X \<in> HFinite
6.3166 -   ==> \<exists>u. eventually (\<lambda>n. norm (X n) < u) FreeUltrafilterNat"
6.3167 -apply (auto simp add: HFinite_def SReal_def)
6.3168 -apply (rule_tac x=r in exI)
6.3169 -apply (simp add: hnorm_def star_of_def starfun_star_n)
6.3170 -apply (simp add: star_less_def starP2_star_n)
6.3171 -done
6.3172 +  "star_n X \<in> HFinite \<Longrightarrow> \<exists>u. eventually (\<lambda>n. norm (X n) < u) FreeUltrafilterNat"
6.3173 +  apply (auto simp add: HFinite_def SReal_def)
6.3174 +  apply (rule_tac x=r in exI)
6.3175 +  apply (simp add: hnorm_def star_of_def starfun_star_n)
6.3176 +  apply (simp add: star_less_def starP2_star_n)
6.3177 +  done
6.3178
6.3179  lemma FreeUltrafilterNat_HFinite:
6.3180 -     "\<exists>u. eventually (\<lambda>n. norm (X n) < u) FreeUltrafilterNat
6.3181 -       ==>  star_n X \<in> HFinite"
6.3182 -apply (auto simp add: HFinite_def mem_Rep_star_iff)
6.3183 -apply (rule_tac x="star_of u" in bexI)
6.3184 -apply (simp add: hnorm_def starfun_star_n star_of_def)
6.3185 -apply (simp add: star_less_def starP2_star_n)
6.3187 -done
6.3188 +  "\<exists>u. eventually (\<lambda>n. norm (X n) < u) FreeUltrafilterNat \<Longrightarrow> star_n X \<in> HFinite"
6.3189 +  apply (auto simp add: HFinite_def mem_Rep_star_iff)
6.3190 +  apply (rule_tac x="star_of u" in bexI)
6.3191 +   apply (simp add: hnorm_def starfun_star_n star_of_def)
6.3192 +   apply (simp add: star_less_def starP2_star_n)
6.3193 +  apply (simp add: SReal_def)
6.3194 +  done
6.3195
6.3196  lemma HFinite_FreeUltrafilterNat_iff:
6.3197 -     "(star_n X \<in> HFinite) = (\<exists>u. eventually (\<lambda>n. norm (X n) < u) FreeUltrafilterNat)"
6.3198 -by (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)
6.3199 +  "star_n X \<in> HFinite \<longleftrightarrow> (\<exists>u. eventually (\<lambda>n. norm (X n) < u) FreeUltrafilterNat)"
6.3200 +  by (blast intro!: HFinite_FreeUltrafilterNat FreeUltrafilterNat_HFinite)
6.3201 +
6.3202
6.3203  subsubsection \<open>@{term HInfinite}\<close>
6.3204
6.3205  lemma lemma_Compl_eq: "- {n. u < norm (f n)} = {n. norm (f n) \<le> u}"
6.3206 -by auto
6.3207 +  by auto
6.3208
6.3209  lemma lemma_Compl_eq2: "- {n. norm (f n) < u} = {n. u \<le> norm (f n)}"
6.3210 -by auto
6.3211 +  by auto
6.3212
6.3213 -lemma lemma_Int_eq1:
6.3214 -     "{n. norm (f n) \<le> u} Int {n. u \<le> norm (f n)} = {n. norm(f n) = u}"
6.3215 -by auto
6.3216 +lemma lemma_Int_eq1: "{n. norm (f n) \<le> u} Int {n. u \<le> norm (f n)} = {n. norm(f n) = u}"
6.3217 +  by auto
6.3218
6.3219 -lemma lemma_FreeUltrafilterNat_one:
6.3220 -     "{n. norm (f n) = u} \<le> {n. norm (f n) < u + (1::real)}"
6.3221 -by auto
6.3222 +lemma lemma_FreeUltrafilterNat_one: "{n. norm (f n) = u} \<le> {n. norm (f n) < u + (1::real)}"
6.3223 +  by auto
6.3224
6.3225 -(*-------------------------------------
6.3226 -  Exclude this type of sets from free
6.3227 -  ultrafilter for Infinite numbers!
6.3228 - -------------------------------------*)
6.3229 +text \<open>Exclude this type of sets from free ultrafilter for Infinite numbers!\<close>
6.3230  lemma FreeUltrafilterNat_const_Finite:
6.3231 -     "eventually (\<lambda>n. norm (X n) = u) FreeUltrafilterNat ==> star_n X \<in> HFinite"
6.3232 -apply (rule FreeUltrafilterNat_HFinite)
6.3233 -apply (rule_tac x = "u + 1" in exI)
6.3234 -apply (auto elim: eventually_mono)
6.3235 -done
6.3236 +  "eventually (\<lambda>n. norm (X n) = u) FreeUltrafilterNat \<Longrightarrow> star_n X \<in> HFinite"
6.3237 +  apply (rule FreeUltrafilterNat_HFinite)
6.3238 +  apply (rule_tac x = "u + 1" in exI)
6.3239 +  apply (auto elim: eventually_mono)
6.3240 +  done
6.3241
6.3242  lemma HInfinite_FreeUltrafilterNat:
6.3243 -     "star_n X \<in> HInfinite ==> \<forall>u. eventually (\<lambda>n. u < norm (X n)) FreeUltrafilterNat"
6.3244 -apply (drule HInfinite_HFinite_iff [THEN iffD1])
6.3246 -apply (rule allI, drule_tac x="u + 1" in spec)
6.3248 -apply (auto elim: eventually_mono)
6.3249 -done
6.3250 +  "star_n X \<in> HInfinite \<Longrightarrow> \<forall>u. eventually (\<lambda>n. u < norm (X n)) FreeUltrafilterNat"
6.3251 +  apply (drule HInfinite_HFinite_iff [THEN iffD1])
6.3252 +  apply (simp add: HFinite_FreeUltrafilterNat_iff)
6.3253 +  apply (rule allI, drule_tac x="u + 1" in spec)
6.3254 +  apply (simp add: FreeUltrafilterNat.eventually_not_iff[symmetric])
6.3255 +  apply (auto elim: eventually_mono)
6.3256 +  done
6.3257
6.3258 -lemma lemma_Int_HI:
6.3259 -     "{n. norm (Xa n) < u} Int {n. X n = Xa n} \<subseteq> {n. norm (X n) < (u::real)}"
6.3260 -by auto
6.3261 +lemma lemma_Int_HI: "{n. norm (Xa n) < u} \<inter> {n. X n = Xa n} \<subseteq> {n. norm (X n) < u}"
6.3262 +  for u :: real
6.3263 +  by auto
6.3264
6.3265 -lemma lemma_Int_HIa: "{n. u < norm (X n)} Int {n. norm (X n) < u} = {}"
6.3266 -by (auto intro: order_less_asym)
6.3267 +lemma lemma_Int_HIa: "{n. u < norm (X n)} \<inter> {n. norm (X n) < u} = {}"
6.3268 +  by (auto intro: order_less_asym)
6.3269
6.3270  lemma FreeUltrafilterNat_HInfinite:
6.3271 -     "\<forall>u. eventually (\<lambda>n. u < norm (X n)) FreeUltrafilterNat ==> star_n X \<in> HInfinite"
6.3272 -apply (rule HInfinite_HFinite_iff [THEN iffD2])
6.3273 -apply (safe, drule HFinite_FreeUltrafilterNat, safe)
6.3274 -apply (drule_tac x = u in spec)
6.3275 +  "\<forall>u. eventually (\<lambda>n. u < norm (X n)) FreeUltrafilterNat \<Longrightarrow> star_n X \<in> HInfinite"
6.3276 +  apply (rule HInfinite_HFinite_iff [THEN iffD2])
6.3277 +  apply (safe, drule HFinite_FreeUltrafilterNat, safe)
6.3278 +  apply (drule_tac x = u in spec)
6.3279  proof -
6.3280 -  fix u assume "\<forall>\<^sub>Fn in \<U>. norm (X n) < u" "\<forall>\<^sub>Fn in \<U>. u < norm (X n)"
6.3281 +  fix u
6.3282 +  assume "\<forall>\<^sub>Fn in \<U>. norm (X n) < u" "\<forall>\<^sub>Fn in \<U>. u < norm (X n)"
6.3283    then have "\<forall>\<^sub>F x in \<U>. False"
6.3284      by eventually_elim auto
6.3285    then show False
6.3286 @@ -1990,230 +1841,204 @@
6.3287  qed
6.3288
6.3289  lemma HInfinite_FreeUltrafilterNat_iff:
6.3290 -     "(star_n X \<in> HInfinite) = (\<forall>u. eventually (\<lambda>n. u < norm (X n)) FreeUltrafilterNat)"
6.3291 -by (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)
6.3292 +  "star_n X \<in> HInfinite \<longleftrightarrow> (\<forall>u. eventually (\<lambda>n. u < norm (X n)) FreeUltrafilterNat)"
6.3293 +  by (blast intro!: HInfinite_FreeUltrafilterNat FreeUltrafilterNat_HInfinite)
6.3294 +
6.3295
6.3296  subsubsection \<open>@{term Infinitesimal}\<close>
6.3297
6.3298 -lemma ball_SReal_eq: "(\<forall>x::hypreal \<in> Reals. P x) = (\<forall>x::real. P (star_of x))"
6.3299 -by (unfold SReal_def, auto)
6.3300 +lemma ball_SReal_eq: "(\<forall>x::hypreal \<in> Reals. P x) \<longleftrightarrow> (\<forall>x::real. P (star_of x))"
6.3301 +  by (auto simp: SReal_def)
6.3302
6.3303  lemma Infinitesimal_FreeUltrafilterNat:
6.3304 -     "star_n X \<in> Infinitesimal ==> \<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>"
6.3305 -apply (simp add: Infinitesimal_def ball_SReal_eq)
6.3306 -apply (simp add: hnorm_def starfun_star_n star_of_def)
6.3307 -apply (simp add: star_less_def starP2_star_n)
6.3308 -done
6.3309 +  "star_n X \<in> Infinitesimal \<Longrightarrow> \<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>"
6.3310 +  apply (simp add: Infinitesimal_def ball_SReal_eq)
6.3311 +  apply (simp add: hnorm_def starfun_star_n star_of_def)
6.3312 +  apply (simp add: star_less_def starP2_star_n)
6.3313 +  done
6.3314
6.3315  lemma FreeUltrafilterNat_Infinitesimal:
6.3316 -     "\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U> ==> star_n X \<in> Infinitesimal"
6.3317 -apply (simp add: Infinitesimal_def ball_SReal_eq)
6.3318 -apply (simp add: hnorm_def starfun_star_n star_of_def)
6.3319 -apply (simp add: star_less_def starP2_star_n)
6.3320 -done
6.3321 +  "\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U> \<Longrightarrow> star_n X \<in> Infinitesimal"
6.3322 +  apply (simp add: Infinitesimal_def ball_SReal_eq)
6.3323 +  apply (simp add: hnorm_def starfun_star_n star_of_def)
6.3324 +  apply (simp add: star_less_def starP2_star_n)
6.3325 +  done
6.3326
6.3327  lemma Infinitesimal_FreeUltrafilterNat_iff:
6.3328 -     "(star_n X \<in> Infinitesimal) = (\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>)"
6.3329 -by (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)
6.3330 +  "(star_n X \<in> Infinitesimal) = (\<forall>u>0. eventually (\<lambda>n. norm (X n) < u) \<U>)"
6.3331 +  by (blast intro!: Infinitesimal_FreeUltrafilterNat FreeUltrafilterNat_Infinitesimal)
6.3332 +
6.3333
6.3334 -(*------------------------------------------------------------------------
6.3335 -         Infinitesimals as smaller than 1/n for all n::nat (> 0)
6.3336 - ------------------------------------------------------------------------*)
6.3337 +text \<open>Infinitesimals as smaller than \<open>1/n\<close> for all \<open>n::nat (> 0)\<close>.\<close>
6.3338
6.3339 -lemma lemma_Infinitesimal:
6.3340 -     "(\<forall>r. 0 < r --> x < r) = (\<forall>n. x < inverse(real (Suc n)))"
6.3341 -apply (auto simp del: of_nat_Suc)
6.3342 -apply (blast dest!: reals_Archimedean intro: order_less_trans)
6.3343 -done
6.3344 +lemma lemma_Infinitesimal: "(\<forall>r. 0 < r \<longrightarrow> x < r) \<longleftrightarrow> (\<forall>n. x < inverse (real (Suc n)))"
6.3345 +  apply (auto simp del: of_nat_Suc)
6.3346 +  apply (blast dest!: reals_Archimedean intro: order_less_trans)
6.3347 +  done
6.3348
6.3349  lemma lemma_Infinitesimal2:
6.3350 -     "(\<forall>r \<in> Reals. 0 < r --> x < r) =
6.3351 -      (\<forall>n. x < inverse(hypreal_of_nat (Suc n)))"
6.3352 -apply safe
6.3353 - apply (drule_tac x = "inverse (hypreal_of_real (real (Suc n))) " in bspec)
6.3354 -  apply simp_all
6.3355 +  "(\<forall>r \<in> Reals. 0 < r \<longrightarrow> x < r) \<longleftrightarrow> (\<forall>n. x < inverse(hypreal_of_nat (Suc n)))"
6.3356 +  apply safe
6.3357 +   apply (drule_tac x = "inverse (hypreal_of_real (real (Suc n))) " in bspec)
6.3358 +    apply simp_all
6.3359    using less_imp_of_nat_less apply fastforce
6.3360 -apply (auto dest!: reals_Archimedean simp add: SReal_iff simp del: of_nat_Suc)
6.3361 -apply (drule star_of_less [THEN iffD2])
6.3362 -apply simp
6.3363 -apply (blast intro: order_less_trans)
6.3364 -done
6.3365 +  apply (auto dest!: reals_Archimedean simp add: SReal_iff simp del: of_nat_Suc)
6.3366 +  apply (drule star_of_less [THEN iffD2])
6.3367 +  apply simp
6.3368 +  apply (blast intro: order_less_trans)
6.3369 +  done
6.3370
6.3371
6.3372  lemma Infinitesimal_hypreal_of_nat_iff:
6.3373 -     "Infinitesimal = {x. \<forall>n. hnorm x < inverse (hypreal_of_nat (Suc n))}"
6.3375 -apply (auto simp add: lemma_Infinitesimal2)
6.3376 -done
6.3377 +  "Infinitesimal = {x. \<forall>n. hnorm x < inverse (hypreal_of_nat (Suc n))}"
6.3378 +  apply (simp add: Infinitesimal_def)
6.3379 +  apply (auto simp add: lemma_Infinitesimal2)
6.3380 +  done
6.3381
6.3382
6.3383 -subsection\<open>Proof that \<open>\<omega>\<close> is an infinite number\<close>
6.3384 +subsection \<open>Proof that \<open>\<omega>\<close> is an infinite number\<close>
6.3385
6.3386 -text\<open>It will follow that \<open>\<epsilon>\<close> is an infinitesimal number.\<close>
6.3387 +text \<open>It will follow that \<open>\<epsilon>\<close> is an infinitesimal number.\<close>
6.3388
6.3389  lemma Suc_Un_eq: "{n. n < Suc m} = {n. n < m} Un {n. n = m}"
6.3390 -by (auto simp add: less_Suc_eq)
6.3391 +  by (auto simp add: less_Suc_eq)
6.3392
6.3393 -(*-------------------------------------------
6.3394 -  Prove that any segment is finite and hence cannot belong to FreeUltrafilterNat
6.3395 - -------------------------------------------*)
6.3396 +
6.3397 +text \<open>Prove that any segment is finite and hence cannot belong to \<open>FreeUltrafilterNat\<close>.\<close>
6.3398
6.3399  lemma finite_real_of_nat_segment: "finite {n::nat. real n < real (m::nat)}"
6.3400 -  by (auto intro: finite_Collect_less_nat)
6.3401 +  by auto
6.3402
6.3403  lemma finite_real_of_nat_less_real: "finite {n::nat. real n < u}"
6.3404 -apply (cut_tac x = u in reals_Archimedean2, safe)
6.3405 -apply (rule finite_real_of_nat_segment [THEN [2] finite_subset])
6.3406 -apply (auto dest: order_less_trans)
6.3407 -done
6.3408 +  apply (cut_tac x = u in reals_Archimedean2, safe)
6.3409 +  apply (rule finite_real_of_nat_segment [THEN [2] finite_subset])
6.3410 +  apply (auto dest: order_less_trans)
6.3411 +  done
6.3412
6.3413 -lemma lemma_real_le_Un_eq:
6.3414 -     "{n. f n \<le> u} = {n. f n < u} Un {n. u = (f n :: real)}"
6.3415 -by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
6.3416 +lemma lemma_real_le_Un_eq: "{n. f n \<le> u} = {n. f n < u} \<union> {n. u = (f n :: real)}"
6.3417 +  by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
6.3418
6.3419  lemma finite_real_of_nat_le_real: "finite {n::nat. real n \<le> u}"
6.3420 -by (auto simp add: lemma_real_le_Un_eq lemma_finite_omega_set finite_real_of_nat_less_real)
6.3421 +  by (auto simp add: lemma_real_le_Un_eq lemma_finite_omega_set finite_real_of_nat_less_real)
6.3422
6.3423  lemma finite_rabs_real_of_nat_le_real: "finite {n::nat. \<bar>real n\<bar> \<le> u}"
6.3424 -apply (simp (no_asm) add: finite_real_of_nat_le_real)
6.3425 -done
6.3426 +  by (simp add: finite_real_of_nat_le_real)
6.3427
6.3428  lemma rabs_real_of_nat_le_real_FreeUltrafilterNat:
6.3429 -     "\<not> eventually (\<lambda>n. \<bar>real n\<bar> \<le> u) FreeUltrafilterNat"
6.3430 -by (blast intro!: FreeUltrafilterNat.finite finite_rabs_real_of_nat_le_real)
6.3431 +  "\<not> eventually (\<lambda>n. \<bar>real n\<bar> \<le> u) FreeUltrafilterNat"
6.3432 +  by (blast intro!: FreeUltrafilterNat.finite finite_rabs_real_of_nat_le_real)
6.3433
6.3434  lemma FreeUltrafilterNat_nat_gt_real: "eventually (\<lambda>n. u < real n) FreeUltrafilterNat"
6.3435 -apply (rule FreeUltrafilterNat.finite')
6.3436 -apply (subgoal_tac "{n::nat. \<not> u < real n} = {n. real n \<le> u}")
6.3437 -apply (auto simp add: finite_real_of_nat_le_real)
6.3438 -done
6.3439 +  apply (rule FreeUltrafilterNat.finite')
6.3440 +  apply (subgoal_tac "{n::nat. \<not> u < real n} = {n. real n \<le> u}")
6.3441 +   apply (auto simp add: finite_real_of_nat_le_real)
6.3442 +  done
6.3443
6.3444 -(*--------------------------------------------------------------
6.3445 - The complement of {n. \<bar>real n\<bar> \<le> u} =
6.3446 - {n. u < \<bar>real n\<bar>} is in FreeUltrafilterNat
6.3447 - by property of (free) ultrafilters
6.3448 - --------------------------------------------------------------*)
6.3449 +text \<open>The complement of \<open>{n. \<bar>real n\<bar> \<le> u} = {n. u < \<bar>real n\<bar>}\<close> is in
6.3450 +  \<open>FreeUltrafilterNat\<close> by property of (free) ultrafilters.\<close>
6.3451
6.3452  lemma Compl_real_le_eq: "- {n::nat. real n \<le> u} = {n. u < real n}"
6.3453 -by (auto dest!: order_le_less_trans simp add: linorder_not_le)
6.3454 +  by (auto dest!: order_le_less_trans simp add: linorder_not_le)
6.3455
6.3456 -text\<open>@{term \<omega>} is a member of @{term HInfinite}\<close>
6.3457 -
6.3458 +text \<open>@{term \<omega>} is a member of @{term HInfinite}.\<close>
6.3459  theorem HInfinite_omega [simp]: "\<omega> \<in> HInfinite"
6.3461 -apply (rule FreeUltrafilterNat_HInfinite)
6.3462 -apply clarify
6.3463 -apply (rule_tac u1 = "u-1" in eventually_mono [OF FreeUltrafilterNat_nat_gt_real])
6.3464 -apply auto
6.3465 -done
6.3466 +  apply (simp add: omega_def)
6.3467 +  apply (rule FreeUltrafilterNat_HInfinite)
6.3468 +  apply clarify
6.3469 +  apply (rule_tac u1 = "u-1" in eventually_mono [OF FreeUltrafilterNat_nat_gt_real])
6.3470 +  apply auto
6.3471 +  done
6.3472
6.3473 -(*-----------------------------------------------
6.3474 -       Epsilon is a member of Infinitesimal
6.3475 - -----------------------------------------------*)
6.3476 +
6.3477 +text \<open>Epsilon is a member of Infinitesimal.\<close>
6.3478
6.3479  lemma Infinitesimal_epsilon [simp]: "\<epsilon> \<in> Infinitesimal"
6.3480 -by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega simp add: hypreal_epsilon_inverse_omega)
6.3481 +  by (auto intro!: HInfinite_inverse_Infinitesimal HInfinite_omega
6.3483
6.3484  lemma HFinite_epsilon [simp]: "\<epsilon> \<in> HFinite"
6.3485 -by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
6.3486 +  by (auto intro: Infinitesimal_subset_HFinite [THEN subsetD])
6.3487
6.3488  lemma epsilon_approx_zero [simp]: "\<epsilon> \<approx> 0"
6.3489 -apply (simp (no_asm) add: mem_infmal_iff [symmetric])
6.3490 -done
6.3491 -
6.3492 -(*------------------------------------------------------------------------
6.3493 -  Needed for proof that we define a hyperreal [<X(n)] \<approx> hypreal_of_real a given
6.3494 -  that \<forall>n. |X n - a| < 1/n. Used in proof of NSLIM => LIM.
6.3495 - -----------------------------------------------------------------------*)
6.3496 +  by (simp add: mem_infmal_iff [symmetric])
6.3497
6.3498 -lemma real_of_nat_less_inverse_iff:
6.3499 -     "0 < u  ==> (u < inverse (real(Suc n))) = (real(Suc n) < inverse u)"
6.3501 -apply (subst pos_less_divide_eq, assumption)
6.3502 -apply (subst pos_less_divide_eq)
6.3503 - apply simp
6.3505 -done
6.3506 +text \<open>Needed for proof that we define a hyperreal \<open>[<X(n)] \<approx> hypreal_of_real a\<close> given
6.3507 +  that \<open>\<forall>n. |X n - a| < 1/n\<close>. Used in proof of \<open>NSLIM \<Rightarrow> LIM\<close>.\<close>
6.3508 +lemma real_of_nat_less_inverse_iff: "0 < u \<Longrightarrow> u < inverse (real(Suc n)) \<longleftrightarrow> real(Suc n) < inverse u"
6.3509 +  apply (simp add: inverse_eq_divide)
6.3510 +  apply (subst pos_less_divide_eq, assumption)
6.3511 +  apply (subst pos_less_divide_eq)
6.3512 +   apply simp
6.3513 +  apply (simp add: mult.commute)
6.3514 +  done
6.3515
6.3516 -lemma finite_inverse_real_of_posnat_gt_real:
6.3517 -     "0 < u ==> finite {n. u < inverse(real(Suc n))}"
6.3518 +lemma finite_inverse_real_of_posnat_gt_real: "0 < u \<Longrightarrow> finite {n. u < inverse (real (Suc n))}"
6.3519  proof (simp only: real_of_nat_less_inverse_iff)
6.3520    have "{n. 1 + real n < inverse u} = {n. real n < inverse u - 1}"
6.3521      by fastforce
6.3522 -  thus "finite {n. real (Suc n) < inverse u}"
6.3523 -    using finite_real_of_nat_less_real [of "inverse u - 1"] by auto
6.3524 +  then show "finite {n. real (Suc n) < inverse u}"
6.3525 +    using finite_real_of_nat_less_real [of "inverse u - 1"]
6.3526 +    by auto
6.3527  qed
6.3528
6.3529  lemma lemma_real_le_Un_eq2:
6.3530 -     "{n. u \<le> inverse(real(Suc n))} =
6.3531 -     {n. u < inverse(real(Suc n))} Un {n. u = inverse(real(Suc n))}"
6.3532 -by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
6.3533 +  "{n. u \<le> inverse(real(Suc n))} =
6.3534 +    {n. u < inverse(real(Suc n))} \<union> {n. u = inverse(real(Suc n))}"
6.3535 +  by (auto dest: order_le_imp_less_or_eq simp add: order_less_imp_le)
6.3536
6.3537 -lemma finite_inverse_real_of_posnat_ge_real:
6.3538 -     "0 < u ==> finite {n. u \<le> inverse(real(Suc n))}"
6.3539 -by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_epsilon_set finite_inverse_real_of_posnat_gt_real
6.3540 -            simp del: of_nat_Suc)
6.3541 +lemma finite_inverse_real_of_posnat_ge_real: "0 < u \<Longrightarrow> finite {n. u \<le> inverse (real (Suc n))}"
6.3542 +  by (auto simp add: lemma_real_le_Un_eq2 lemma_finite_epsilon_set finite_inverse_real_of_posnat_gt_real
6.3543 +      simp del: of_nat_Suc)
6.3544
6.3545  lemma inverse_real_of_posnat_ge_real_FreeUltrafilterNat:
6.3546 -     "0 < u ==> \<not> eventually (\<lambda>n. u \<le> inverse(real(Suc n))) FreeUltrafilterNat"
6.3547 -by (blast intro!: FreeUltrafilterNat.finite finite_inverse_real_of_posnat_ge_real)
6.3548 +  "0 < u \<Longrightarrow> \<not> eventually (\<lambda>n. u \<le> inverse(real(Suc n))) FreeUltrafilterNat"
6.3549 +  by (blast intro!: FreeUltrafilterNat.finite finite_inverse_real_of_posnat_ge_real)
6.3550
6.3551 -(*--------------------------------------------------------------
6.3552 -    The complement of  {n. u \<le> inverse(real(Suc n))} =
6.3553 -    {n. inverse(real(Suc n)) < u} is in FreeUltrafilterNat
6.3554 -    by property of (free) ultrafilters
6.3555 - --------------------------------------------------------------*)
6.3556 -lemma Compl_le_inverse_eq:
6.3557 -     "- {n. u \<le> inverse(real(Suc n))} = {n. inverse(real(Suc n)) < u}"
6.3558 -by (auto dest!: order_le_less_trans simp add: linorder_not_le)
6.3559 +text \<open>The complement of \<open>{n. u \<le> inverse(real(Suc n))} = {n. inverse (real (Suc n)) < u}\<close>
6.3560 +  is in \<open>FreeUltrafilterNat\<close> by property of (free) ultrafilters.\<close>
6.3561 +lemma Compl_le_inverse_eq: "- {n. u \<le> inverse(real(Suc n))} = {n. inverse(real(Suc n)) < u}"
6.3562 +  by (auto dest!: order_le_less_trans simp add: linorder_not_le)
6.3563
6.3564
6.3565  lemma FreeUltrafilterNat_inverse_real_of_posnat:
6.3566 -     "0 < u ==> eventually (\<lambda>n. inverse(real(Suc n)) < u) FreeUltrafilterNat"
6.3567 -by (drule inverse_real_of_posnat_ge_real_FreeUltrafilterNat)
6.3568 -   (simp add: FreeUltrafilterNat.eventually_not_iff not_le[symmetric])
6.3569 +  "0 < u \<Longrightarrow> eventually (\<lambda>n. inverse(real(Suc n)) < u) FreeUltrafilterNat"
6.3570 +  by (drule inverse_real_of_posnat_ge_real_FreeUltrafilterNat)
6.3571 +    (simp add: FreeUltrafilterNat.eventually_not_iff not_le[symmetric])
6.3572
6.3573 -text\<open>Example of an hypersequence (i.e. an extended standard sequence)
6.3574 -   whose term with an hypernatural suffix is an infinitesimal i.e.
6.3575 -   the whn'nth term of the hypersequence is a member of Infinitesimal\<close>
6.3576 +text \<open>Example of an hypersequence (i.e. an extended standard sequence)
6.3577 +  whose term with an hypernatural suffix is an infinitesimal i.e.
6.3578 +  the whn'nth term of the hypersequence is a member of Infinitesimal\<close>
6.3579
6.3580 -lemma SEQ_Infinitesimal:
6.3581 -      "( *f* (%n::nat. inverse(real(Suc n)))) whn : Infinitesimal"
6.3582 -by (simp add: hypnat_omega_def starfun_star_n star_n_inverse Infinitesimal_FreeUltrafilterNat_iff
6.3583 -        FreeUltrafilterNat_inverse_real_of_posnat del: of_nat_Suc)
6.3584 +lemma SEQ_Infinitesimal: "( *f* (\<lambda>n::nat. inverse(real(Suc n)))) whn \<in> Infinitesimal"
6.3585 +  by (simp add: hypnat_omega_def starfun_star_n star_n_inverse Infinitesimal_FreeUltrafilterNat_iff
6.3586 +      FreeUltrafilterNat_inverse_real_of_posnat del: of_nat_Suc)
6.3587
6.3588 -text\<open>Example where we get a hyperreal from a real sequence
6.3589 -      for which a particular property holds. The theorem is
6.3590 -      used in proofs about equivalence of nonstandard and
6.3591 -      standard neighbourhoods. Also used for equivalence of
6.3592 -      nonstandard ans standard definitions of pointwise
6.3593 -      limit.\<close>
6.3594 +text \<open>Example where we get a hyperreal from a real sequence
6.3595 +  for which a particular property holds. The theorem is
6.3596 +  used in proofs about equivalence of nonstandard and
6.3597 +  standard neighbourhoods. Also used for equivalence of
6.3598 +  nonstandard ans standard definitions of pointwise
6.3599 +  limit.\<close>
6.3600
6.3601 -(*-----------------------------------------------------
6.3602 -    |X(n) - x| < 1/n ==> [<X n>] - hypreal_of_real x| \<in> Infinitesimal
6.3603 - -----------------------------------------------------*)
6.3604 +text \<open>\<open>|X(n) - x| < 1/n \<Longrightarrow> [<X n>] - hypreal_of_real x| \<in> Infinitesimal\<close>\<close>
6.3605  lemma real_seq_to_hypreal_Infinitesimal:
6.3606 -     "\<forall>n. norm(X n - x) < inverse(real(Suc n))
6.3607 -     ==> star_n X - star_of x \<in> Infinitesimal"
6.3608 -unfolding star_n_diff star_of_def Infinitesimal_FreeUltrafilterNat_iff star_n_inverse
6.3609 -by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
6.3610 -         intro: order_less_trans elim!: eventually_mono)
6.3611 +  "\<forall>n. norm (X n - x) < inverse (real (Suc n)) \<Longrightarrow> star_n X - star_of x \<in> Infinitesimal"
6.3612 +  unfolding star_n_diff star_of_def Infinitesimal_FreeUltrafilterNat_iff star_n_inverse
6.3613 +  by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
6.3614 +      intro: order_less_trans elim!: eventually_mono)
6.3615
6.3616  lemma real_seq_to_hypreal_approx:
6.3617 -     "\<forall>n. norm(X n - x) < inverse(real(Suc n))
6.3618 -      ==> star_n X \<approx> star_of x"
6.3619 -by (metis bex_Infinitesimal_iff real_seq_to_hypreal_Infinitesimal)
6.3620 +  "\<forall>n. norm (X n - x) < inverse (real (Suc n)) \<Longrightarrow> star_n X \<approx> star_of x"
6.3621 +  by (metis bex_Infinitesimal_iff real_seq_to_hypreal_Infinitesimal)
6.3622
6.3623  lemma real_seq_to_hypreal_approx2:
6.3624 -     "\<forall>n. norm(x - X n) < inverse(real(Suc n))
6.3625 -               ==> star_n X \<approx> star_of x"
6.3626 -by (metis norm_minus_commute real_seq_to_hypreal_approx)
6.3627 +  "\<forall>n. norm (x - X n) < inverse(real(Suc n)) \<Longrightarrow> star_n X \<approx> star_of x"
6.3628 +  by (metis norm_minus_commute real_seq_to_hypreal_approx)
6.3629
6.3630  lemma real_seq_to_hypreal_Infinitesimal2:
6.3631 -     "\<forall>n. norm(X n - Y n) < inverse(real(Suc n))
6.3632 -      ==> star_n X - star_n Y \<in> Infinitesimal"
6.3633 -unfolding Infinitesimal_FreeUltrafilterNat_iff star_n_diff
6.3634 -by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
6.3635 -         intro: order_less_trans elim!: eventually_mono)
6.3636 +  "\<forall>n. norm(X n - Y n) < inverse(real(Suc n)) \<Longrightarrow> star_n X - star_n Y \<in> Infinitesimal"
6.3637 +  unfolding Infinitesimal_FreeUltrafilterNat_iff star_n_diff
6.3638 +  by (auto dest!: FreeUltrafilterNat_inverse_real_of_posnat
6.3639 +      intro: order_less_trans elim!: eventually_mono)
6.3640
6.3641  end
```
```     7.1 --- a/src/HOL/Nonstandard_Analysis/NSComplex.thy	Mon Oct 31 16:26:36 2016 +0100
7.2 +++ b/src/HOL/Nonstandard_Analysis/NSComplex.thy	Tue Nov 01 00:44:24 2016 +0100
7.3 @@ -3,654 +3,604 @@
7.4      Author:     Lawrence C Paulson
7.5  *)
7.6
7.7 -section\<open>Nonstandard Complex Numbers\<close>
7.8 +section \<open>Nonstandard Complex Numbers\<close>
7.9
7.10  theory NSComplex
7.11 -imports NSA
7.12 +  imports NSA
7.13  begin
7.14
7.15  type_synonym hcomplex = "complex star"
7.16
7.17 -abbreviation
7.18 -  hcomplex_of_complex :: "complex => complex star" where
7.19 -  "hcomplex_of_complex == star_of"
7.20 +abbreviation hcomplex_of_complex :: "complex \<Rightarrow> complex star"
7.21 +  where "hcomplex_of_complex \<equiv> star_of"
7.22
7.23 -abbreviation
7.24 -  hcmod :: "complex star => real star" where
7.25 -  "hcmod == hnorm"
7.26 +abbreviation hcmod :: "complex star \<Rightarrow> real star"
7.27 +  where "hcmod \<equiv> hnorm"
7.28
7.29
7.30 -  (*--- real and Imaginary parts ---*)
7.31 +subsubsection \<open>Real and Imaginary parts\<close>
7.32 +
7.33 +definition hRe :: "hcomplex \<Rightarrow> hypreal"
7.34 +  where "hRe = *f* Re"
7.35 +
7.36 +definition hIm :: "hcomplex \<Rightarrow> hypreal"
7.37 +  where "hIm = *f* Im"
7.38 +
7.39
7.40 -definition
7.41 -  hRe :: "hcomplex => hypreal" where
7.42 -  "hRe = *f* Re"
7.43 +subsubsection \<open>Imaginary unit\<close>
7.44 +
7.45 +definition iii :: hcomplex
7.46 +  where "iii = star_of \<i>"
7.47
7.48 -definition
7.49 -  hIm :: "hcomplex => hypreal" where
7.50 -  "hIm = *f* Im"
7.51 +
7.52 +subsubsection \<open>Complex conjugate\<close>
7.53 +
7.54 +definition hcnj :: "hcomplex \<Rightarrow> hcomplex"
7.55 +  where "hcnj = *f* cnj"
7.56
7.57
7.58 -  (*------ imaginary unit ----------*)
7.59 +subsubsection \<open>Argand\<close>
7.60
7.61 -definition
7.62 -  iii :: hcomplex where
7.63 -  "iii = star_of \<i>"
7.64 -
7.65 -  (*------- complex conjugate ------*)
7.66 +definition hsgn :: "hcomplex \<Rightarrow> hcomplex"
7.67 +  where "hsgn = *f* sgn"
7.68
7.69 -definition
7.70 -  hcnj :: "hcomplex => hcomplex" where
7.71 -  "hcnj = *f* cnj"
7.72 -
7.73 -  (*------------ Argand -------------*)
7.74 +definition harg :: "hcomplex \<Rightarrow> hypreal"
7.75 +  where "harg = *f* arg"
7.76
7.77 -definition
7.78 -  hsgn :: "hcomplex => hcomplex" where
7.79 -  "hsgn = *f* sgn"
7.80 +definition  \<comment> \<open>abbreviation for \<open>cos a + i sin a\<close>\<close>
7.81 +  hcis :: "hypreal \<Rightarrow> hcomplex"
7.82 +  where "hcis = *f* cis"
7.83
7.84 -definition
7.85 -  harg :: "hcomplex => hypreal" where
7.86 -  "harg = *f* arg"
7.87 +
7.88 +subsubsection \<open>Injection from hyperreals\<close>
7.89
7.90 -definition
7.91 -  (* abbreviation for (cos a + i sin a) *)
7.92 -  hcis :: "hypreal => hcomplex" where
7.93 -  "hcis = *f* cis"
7.94 +abbreviation hcomplex_of_hypreal :: "hypreal \<Rightarrow> hcomplex"
7.95 +  where "hcomplex_of_hypreal \<equiv> of_hypreal"
7.96
7.97 -  (*----- injection from hyperreals -----*)
7.98 +definition  \<comment> \<open>abbreviation for \<open>r * (cos a + i sin a)\<close>\<close>
7.99 +  hrcis :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hcomplex"
7.100 +  where "hrcis = *f2* rcis"
7.101
7.102 -abbreviation
7.103 -  hcomplex_of_hypreal :: "hypreal \<Rightarrow> hcomplex" where
7.104 -  "hcomplex_of_hypreal \<equiv> of_hypreal"
7.105
7.106 -definition
7.107 -  (* abbreviation for r*(cos a + i sin a) *)
7.108 -  hrcis :: "[hypreal, hypreal] => hcomplex" where
7.109 -  "hrcis = *f2* rcis"
7.110 +subsubsection \<open>\<open>e ^ (x + iy)\<close>\<close>
7.111
7.112 -  (*------------ e ^ (x + iy) ------------*)
7.113 -definition
7.114 -  hExp :: "hcomplex => hcomplex" where
7.115 -  "hExp = *f* exp"
7.116 +definition hExp :: "hcomplex \<Rightarrow> hcomplex"
7.117 +  where "hExp = *f* exp"
7.118
7.119 -definition
7.120 -  HComplex :: "[hypreal,hypreal] => hcomplex" where
7.121 -  "HComplex = *f2* Complex"
7.122 +definition HComplex :: "hypreal \<Rightarrow> hypreal \<Rightarrow> hcomplex"
7.123 +  where "HComplex = *f2* Complex"
7.124
7.125  lemmas hcomplex_defs [transfer_unfold] =
7.126    hRe_def hIm_def iii_def hcnj_def hsgn_def harg_def hcis_def
7.127    hrcis_def hExp_def HComplex_def
7.128
7.129  lemma Standard_hRe [simp]: "x \<in> Standard \<Longrightarrow> hRe x \<in> Standard"
7.131 +  by (simp add: hcomplex_defs)
7.132
7.133  lemma Standard_hIm [simp]: "x \<in> Standard \<Longrightarrow> hIm x \<in> Standard"
7.135 +  by (simp add: hcomplex_defs)
7.136
7.137  lemma Standard_iii [simp]: "iii \<in> Standard"
7.139 +  by (simp add: hcomplex_defs)
7.140
7.141  lemma Standard_hcnj [simp]: "x \<in> Standard \<Longrightarrow> hcnj x \<in> Standard"
7.143 +  by (simp add: hcomplex_defs)
7.144
7.145  lemma Standard_hsgn [simp]: "x \<in> Standard \<Longrightarrow> hsgn x \<in> Standard"
7.147 +  by (simp add: hcomplex_defs)
7.148
7.149  lemma Standard_harg [simp]: "x \<in> Standard \<Longrightarrow> harg x \<in> Standard"
7.151 +  by (simp add: hcomplex_defs)
7.152
7.153  lemma Standard_hcis [simp]: "r \<in> Standard \<Longrightarrow> hcis r \<in> Standard"
7.155 +  by (simp add: hcomplex_defs)
7.156
7.157  lemma Standard_hExp [simp]: "x \<in> Standard \<Longrightarrow> hExp x \<in> Standard"
7.159 +  by (simp add: hcomplex_defs)
7.160
7.161 -lemma Standard_hrcis [simp]:
7.162 -  "\<lbrakk>r \<in> Standard; s \<in> Standard\<rbrakk> \<Longrightarrow> hrcis r s \<in> Standard"
7.164 +lemma Standard_hrcis [simp]: "r \<in> Standard \<Longrightarrow> s \<in> Standard \<Longrightarrow> hrcis r s \<in> Standard"
7.165 +  by (simp add: hcomplex_defs)
7.166
7.167 -lemma Standard_HComplex [simp]:
7.168 -  "\<lbrakk>r \<in> Standard; s \<in> Standard\<rbrakk> \<Longrightarrow> HComplex r s \<in> Standard"
7.170 +lemma Standard_HComplex [simp]: "r \<in> Standard \<Longrightarrow> s \<in> Standard \<Longrightarrow> HComplex r s \<in> Standard"
7.171 +  by (simp add: hcomplex_defs)
7.172
7.173  lemma hcmod_def: "hcmod = *f* cmod"
7.174 -by (rule hnorm_def)
7.175 +  by (rule hnorm_def)
7.176
7.177
7.178 -subsection\<open>Properties of Nonstandard Real and Imaginary Parts\<close>
7.179 +subsection \<open>Properties of Nonstandard Real and Imaginary Parts\<close>
7.180
7.181 -lemma hcomplex_hRe_hIm_cancel_iff:
7.182 -     "!!w z. (w=z) = (hRe(w) = hRe(z) & hIm(w) = hIm(z))"
7.183 -by transfer (rule complex_Re_Im_cancel_iff)
7.184 +lemma hcomplex_hRe_hIm_cancel_iff: "\<And>w z. w = z \<longleftrightarrow> hRe w = hRe z \<and> hIm w = hIm z"
7.185 +  by transfer (rule complex_Re_Im_cancel_iff)
7.186
7.187 -lemma hcomplex_equality [intro?]:
7.188 -  "!!z w. hRe z = hRe w ==> hIm z = hIm w ==> z = w"
7.189 -by transfer (rule complex_equality)
7.190 +lemma hcomplex_equality [intro?]: "\<And>z w. hRe z = hRe w \<Longrightarrow> hIm z = hIm w \<Longrightarrow> z = w"
7.191 +  by transfer (rule complex_equality)
7.192
7.193  lemma hcomplex_hRe_zero [simp]: "hRe 0 = 0"
7.194 -by transfer simp
7.195 +  by transfer simp
7.196
7.197  lemma hcomplex_hIm_zero [simp]: "hIm 0 = 0"
7.198 -by transfer simp
7.199 +  by transfer simp
7.200
7.201  lemma hcomplex_hRe_one [simp]: "hRe 1 = 1"
7.202 -by transfer simp
7.203 +  by transfer simp
7.204
7.205  lemma hcomplex_hIm_one [simp]: "hIm 1 = 0"
7.206 -by transfer simp
7.207 +  by transfer simp
7.208 +
7.209 +
7.210 +subsection \<open>Addition for Nonstandard Complex Numbers\<close>
7.211 +
7.212 +lemma hRe_add: "\<And>x y. hRe (x + y) = hRe x + hRe y"
7.213 +  by transfer simp
7.214 +
7.215 +lemma hIm_add: "\<And>x y. hIm (x + y) = hIm x + hIm y"
7.216 +  by transfer simp
7.217
7.218
7.219 -subsection\<open>Addition for Nonstandard Complex Numbers\<close>
7.220 -
7.221 -lemma hRe_add: "!!x y. hRe(x + y) = hRe(x) + hRe(y)"
7.222 -by transfer simp
7.223 +subsection \<open>More Minus Laws\<close>
7.224
7.225 -lemma hIm_add: "!!x y. hIm(x + y) = hIm(x) + hIm(y)"
7.226 -by transfer simp
7.227 +lemma hRe_minus: "\<And>z. hRe (- z) = - hRe z"
7.228 +  by transfer (rule uminus_complex.sel)
7.229
7.230 -subsection\<open>More Minus Laws\<close>
7.231 -
7.232 -lemma hRe_minus: "!!z. hRe(-z) = - hRe(z)"
7.233 -by transfer (rule uminus_complex.sel)
7.234 +lemma hIm_minus: "\<And>z. hIm (- z) = - hIm z"
7.235 +  by transfer (rule uminus_complex.sel)
7.236
7.237 -lemma hIm_minus: "!!z. hIm(-z) = - hIm(z)"
7.238 -by transfer (rule uminus_complex.sel)
7.239 +lemma hcomplex_add_minus_eq_minus: "x + y = 0 \<Longrightarrow> x = - y"
7.240 +  for x y :: hcomplex
7.241 +  apply (drule minus_unique)
7.242 +  apply (simp add: minus_equation_iff [of x y])
7.243 +  done
7.244
7.246 -      "x + y = (0::hcomplex) ==> x = -y"
7.247 -apply (drule minus_unique)
7.248 -apply (simp add: minus_equation_iff [of x y])
7.249 -done
7.250 +lemma hcomplex_i_mult_eq [simp]: "iii * iii = - 1"
7.251 +  by transfer (rule i_squared)
7.252
7.253 -lemma hcomplex_i_mult_eq [simp]: "iii * iii = -1"
7.254 -by transfer (rule i_squared)
7.255 -
7.256 -lemma hcomplex_i_mult_left [simp]: "!!z. iii * (iii * z) = -z"
7.257 -by transfer (rule complex_i_mult_minus)
7.258 +lemma hcomplex_i_mult_left [simp]: "\<And>z. iii * (iii * z) = - z"
7.259 +  by transfer (rule complex_i_mult_minus)
7.260
7.261  lemma hcomplex_i_not_zero [simp]: "iii \<noteq> 0"
7.262 -by transfer (rule complex_i_not_zero)
7.263 +  by transfer (rule complex_i_not_zero)
7.264
7.265
7.266 -subsection\<open>More Multiplication Laws\<close>
7.267 +subsection \<open>More Multiplication Laws\<close>
7.268
7.269 -lemma hcomplex_mult_minus_one: "- 1 * (z::hcomplex) = -z"
7.270 -by simp
7.271 -
7.272 -lemma hcomplex_mult_minus_one_right: "(z::hcomplex) * - 1 = -z"
7.273 -by simp
7.274 +lemma hcomplex_mult_minus_one: "- 1 * z = - z"
7.275 +  for z :: hcomplex
7.276 +  by simp
7.277
7.278 -lemma hcomplex_mult_left_cancel:
7.279 -     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (c*a=c*b) = (a=b)"
7.280 -by simp
7.281 +lemma hcomplex_mult_minus_one_right: "z * - 1 = - z"
7.282 +  for z :: hcomplex
7.283 +  by simp
7.284
7.285 -lemma hcomplex_mult_right_cancel:
7.286 -     "(c::hcomplex) \<noteq> (0::hcomplex) ==> (a*c=b*c) = (a=b)"
7.287 -by simp
7.288 +lemma hcomplex_mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
7.289 +  for a b c :: hcomplex
7.290 +  by simp
7.291 +
7.292 +lemma hcomplex_mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
7.293 +  for a b c :: hcomplex
7.294 +  by simp
7.295
7.296
7.297 -subsection\<open>Subraction and Division\<close>
7.298 +subsection \<open>Subtraction and Division\<close>
7.299
7.300 -lemma hcomplex_diff_eq_eq [simp]: "((x::hcomplex) - y = z) = (x = z + y)"
7.301  (* TODO: delete *)
7.302 -by (rule diff_eq_eq)
7.303 +lemma hcomplex_diff_eq_eq [simp]: "x - y = z \<longleftrightarrow> x = z + y"
7.304 +  for x y z :: hcomplex
7.305 +  by (rule diff_eq_eq)
7.306
7.307
7.308 -subsection\<open>Embedding Properties for @{term hcomplex_of_hypreal} Map\<close>
7.309 +subsection \<open>Embedding Properties for @{term hcomplex_of_hypreal} Map\<close>
7.310 +
7.311 +lemma hRe_hcomplex_of_hypreal [simp]: "\<And>z. hRe (hcomplex_of_hypreal z) = z"
7.312 +  by transfer (rule Re_complex_of_real)
7.313
7.314 -lemma hRe_hcomplex_of_hypreal [simp]: "!!z. hRe(hcomplex_of_hypreal z) = z"
7.315 -by transfer (rule Re_complex_of_real)
7.316 +lemma hIm_hcomplex_of_hypreal [simp]: "\<And>z. hIm (hcomplex_of_hypreal z) = 0"
7.317 +  by transfer (rule Im_complex_of_real)
7.318
7.319 -lemma hIm_hcomplex_of_hypreal [simp]: "!!z. hIm(hcomplex_of_hypreal z) = 0"
7.320 -by transfer (rule Im_complex_of_real)
7.321 +lemma hcomplex_of_hypreal_epsilon_not_zero [simp]: "hcomplex_of_hypreal \<epsilon> \<noteq> 0"
7.322 +  by (simp add: hypreal_epsilon_not_zero)
7.323
7.324 -lemma hcomplex_of_hypreal_epsilon_not_zero [simp]:
7.325 -     "hcomplex_of_hypreal \<epsilon> \<noteq> 0"
7.327 +
7.328 +subsection \<open>\<open>HComplex\<close> theorems\<close>
7.329
7.330 -subsection\<open>HComplex theorems\<close>
7.331 +lemma hRe_HComplex [simp]: "\<And>x y. hRe (HComplex x y) = x"
7.332 +  by transfer simp
7.333
7.334 -lemma hRe_HComplex [simp]: "!!x y. hRe (HComplex x y) = x"
7.335 -by transfer simp
7.336 +lemma hIm_HComplex [simp]: "\<And>x y. hIm (HComplex x y) = y"
7.337 +  by transfer simp
7.338
7.339 -lemma hIm_HComplex [simp]: "!!x y. hIm (HComplex x y) = y"
7.340 -by transfer simp
7.341 -
7.342 -lemma hcomplex_surj [simp]: "!!z. HComplex (hRe z) (hIm z) = z"
7.343 -by transfer (rule complex_surj)
7.344 +lemma hcomplex_surj [simp]: "\<And>z. HComplex (hRe z) (hIm z) = z"
7.345 +  by transfer (rule complex_surj)
7.346
7.347  lemma hcomplex_induct [case_names rect(*, induct type: hcomplex*)]:
7.348 -     "(\<And>x y. P (HComplex x y)) ==> P z"
7.349 -by (rule hcomplex_surj [THEN subst], blast)
7.350 +  "(\<And>x y. P (HComplex x y)) \<Longrightarrow> P z"
7.351 +  by (rule hcomplex_surj [THEN subst]) blast
7.352
7.353
7.354 -subsection\<open>Modulus (Absolute Value) of Nonstandard Complex Number\<close>
7.355 +subsection \<open>Modulus (Absolute Value) of Nonstandard Complex Number\<close>
7.356
7.357  lemma hcomplex_of_hypreal_abs:
7.358 -     "hcomplex_of_hypreal \<bar>x\<bar> =
7.359 -      hcomplex_of_hypreal (hcmod (hcomplex_of_hypreal x))"
7.360 -by simp
7.361 +  "hcomplex_of_hypreal \<bar>x\<bar> = hcomplex_of_hypreal (hcmod (hcomplex_of_hypreal x))"
7.362 +  by simp
7.363
7.364 -lemma HComplex_inject [simp]:
7.365 -  "!!x y x' y'. HComplex x y = HComplex x' y' = (x=x' & y=y')"
7.366 -by transfer (rule complex.inject)
7.367 +lemma HComplex_inject [simp]: "\<And>x y x' y'. HComplex x y = HComplex x' y' \<longleftrightarrow> x = x' \<and> y = y'"
7.368 +  by transfer (rule complex.inject)
7.369
7.371 -  "!!x1 y1 x2 y2. HComplex x1 y1 + HComplex x2 y2 = HComplex (x1+x2) (y1+y2)"
7.373 +  "\<And>x1 y1 x2 y2. HComplex x1 y1 + HComplex x2 y2 = HComplex (x1 + x2) (y1 + y2)"
7.374 +  by transfer (rule complex_add)
7.375
7.376 -lemma HComplex_minus [simp]: "!!x y. - HComplex x y = HComplex (-x) (-y)"
7.377 -by transfer (rule complex_minus)
7.378 +lemma HComplex_minus [simp]: "\<And>x y. - HComplex x y = HComplex (- x) (- y)"
7.379 +  by transfer (rule complex_minus)
7.380
7.381  lemma HComplex_diff [simp]:
7.382 -  "!!x1 y1 x2 y2. HComplex x1 y1 - HComplex x2 y2 = HComplex (x1-x2) (y1-y2)"
7.383 -by transfer (rule complex_diff)
7.384 +  "\<And>x1 y1 x2 y2. HComplex x1 y1 - HComplex x2 y2 = HComplex (x1 - x2) (y1 - y2)"
7.385 +  by transfer (rule complex_diff)
7.386
7.387  lemma HComplex_mult [simp]:
7.388 -  "!!x1 y1 x2 y2. HComplex x1 y1 * HComplex x2 y2 =
7.389 -   HComplex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
7.390 -by transfer (rule complex_mult)
7.391 +  "\<And>x1 y1 x2 y2. HComplex x1 y1 * HComplex x2 y2 = HComplex (x1*x2 - y1*y2) (x1*y2 + y1*x2)"
7.392 +  by transfer (rule complex_mult)
7.393
7.394 -(*HComplex_inverse is proved below*)
7.395 +text \<open>\<open>HComplex_inverse\<close> is proved below.\<close>
7.396
7.397 -lemma hcomplex_of_hypreal_eq: "!!r. hcomplex_of_hypreal r = HComplex r 0"
7.398 -by transfer (rule complex_of_real_def)
7.399 +lemma hcomplex_of_hypreal_eq: "\<And>r. hcomplex_of_hypreal r = HComplex r 0"
7.400 +  by transfer (rule complex_of_real_def)
7.401
7.403 -     "!!x y r. HComplex x y + hcomplex_of_hypreal r = HComplex (x+r) y"
7.405 +  "\<And>x y r. HComplex x y + hcomplex_of_hypreal r = HComplex (x + r) y"
7.406 +  by transfer (rule Complex_add_complex_of_real)
7.407
7.409 -     "!!r x y. hcomplex_of_hypreal r + HComplex x y = HComplex (r+x) y"
7.411 +  "\<And>r x y. hcomplex_of_hypreal r + HComplex x y = HComplex (r + x) y"
7.412 +  by transfer (rule complex_of_real_add_Complex)
7.413
7.414  lemma HComplex_mult_hcomplex_of_hypreal:
7.415 -     "!!x y r. HComplex x y * hcomplex_of_hypreal r = HComplex (x*r) (y*r)"
7.416 -by transfer (rule Complex_mult_complex_of_real)
7.417 +  "\<And>x y r. HComplex x y * hcomplex_of_hypreal r = HComplex (x * r) (y * r)"
7.418 +  by transfer (rule Complex_mult_complex_of_real)
7.419
7.420  lemma hcomplex_of_hypreal_mult_HComplex:
7.421 -     "!!r x y. hcomplex_of_hypreal r * HComplex x y = HComplex (r*x) (r*y)"
7.422 -by transfer (rule complex_of_real_mult_Complex)
7.423 +  "\<And>r x y. hcomplex_of_hypreal r * HComplex x y = HComplex (r * x) (r * y)"
7.424 +  by transfer (rule complex_of_real_mult_Complex)
7.425
7.426 -lemma i_hcomplex_of_hypreal [simp]:
7.427 -     "!!r. iii * hcomplex_of_hypreal r = HComplex 0 r"
7.428 -by transfer (rule i_complex_of_real)
7.429 +lemma i_hcomplex_of_hypreal [simp]: "\<And>r. iii * hcomplex_of_hypreal r = HComplex 0 r"
7.430 +  by transfer (rule i_complex_of_real)
7.431
7.432 -lemma hcomplex_of_hypreal_i [simp]:
7.433 -     "!!r. hcomplex_of_hypreal r * iii = HComplex 0 r"
7.434 -by transfer (rule complex_of_real_i)
7.435 +lemma hcomplex_of_hypreal_i [simp]: "\<And>r. hcomplex_of_hypreal r * iii = HComplex 0 r"
7.436 +  by transfer (rule complex_of_real_i)
7.437
7.438
7.439 -subsection\<open>Conjugation\<close>
7.440 +subsection \<open>Conjugation\<close>
7.441
7.442 -lemma hcomplex_hcnj_cancel_iff [iff]: "!!x y. (hcnj x = hcnj y) = (x = y)"
7.443 -by transfer (rule complex_cnj_cancel_iff)
7.444 +lemma hcomplex_hcnj_cancel_iff [iff]: "\<And>x y. hcnj x = hcnj y \<longleftrightarrow> x = y"
7.445 +  by transfer (rule complex_cnj_cancel_iff)
7.446
7.447 -lemma hcomplex_hcnj_hcnj [simp]: "!!z. hcnj (hcnj z) = z"
7.448 -by transfer (rule complex_cnj_cnj)
7.449 +lemma hcomplex_hcnj_hcnj [simp]: "\<And>z. hcnj (hcnj z) = z"
7.450 +  by transfer (rule complex_cnj_cnj)
7.451
7.452  lemma hcomplex_hcnj_hcomplex_of_hypreal [simp]:
7.453 -     "!!x. hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
7.454 -by transfer (rule complex_cnj_complex_of_real)
7.455 +  "\<And>x. hcnj (hcomplex_of_hypreal x) = hcomplex_of_hypreal x"
7.456 +  by transfer (rule complex_cnj_complex_of_real)
7.457
7.458 -lemma hcomplex_hmod_hcnj [simp]: "!!z. hcmod (hcnj z) = hcmod z"
7.459 -by transfer (rule complex_mod_cnj)
7.460 +lemma hcomplex_hmod_hcnj [simp]: "\<And>z. hcmod (hcnj z) = hcmod z"
7.461 +  by transfer (rule complex_mod_cnj)
7.462
7.463 -lemma hcomplex_hcnj_minus: "!!z. hcnj (-z) = - hcnj z"
7.464 -by transfer (rule complex_cnj_minus)
7.465 +lemma hcomplex_hcnj_minus: "\<And>z. hcnj (- z) = - hcnj z"
7.466 +  by transfer (rule complex_cnj_minus)
7.467
7.468 -lemma hcomplex_hcnj_inverse: "!!z. hcnj(inverse z) = inverse(hcnj z)"
7.469 -by transfer (rule complex_cnj_inverse)
7.470 +lemma hcomplex_hcnj_inverse: "\<And>z. hcnj (inverse z) = inverse (hcnj z)"
7.471 +  by transfer (rule complex_cnj_inverse)
7.472
7.473 -lemma hcomplex_hcnj_add: "!!w z. hcnj(w + z) = hcnj(w) + hcnj(z)"
7.475 +lemma hcomplex_hcnj_add: "\<And>w z. hcnj (w + z) = hcnj w + hcnj z"
7.476 +  by transfer (rule complex_cnj_add)
7.477
7.478 -lemma hcomplex_hcnj_diff: "!!w z. hcnj(w - z) = hcnj(w) - hcnj(z)"
7.479 -by transfer (rule complex_cnj_diff)
7.480 +lemma hcomplex_hcnj_diff: "\<And>w z. hcnj (w - z) = hcnj w - hcnj z"
7.481 +  by transfer (rule complex_cnj_diff)
7.482
7.483 -lemma hcomplex_hcnj_mult: "!!w z. hcnj(w * z) = hcnj(w) * hcnj(z)"
7.484 -by transfer (rule complex_cnj_mult)
7.485 +lemma hcomplex_hcnj_mult: "\<And>w z. hcnj (w * z) = hcnj w * hcnj z"
7.486 +  by transfer (rule complex_cnj_mult)
7.487
7.488 -lemma hcomplex_hcnj_divide: "!!w z. hcnj(w / z) = (hcnj w)/(hcnj z)"
7.489 -by transfer (rule complex_cnj_divide)
7.490 +lemma hcomplex_hcnj_divide: "\<And>w z. hcnj (w / z) = hcnj w / hcnj z"
7.491 +  by transfer (rule complex_cnj_divide)
7.492
7.493  lemma hcnj_one [simp]: "hcnj 1 = 1"
7.494 -by transfer (rule complex_cnj_one)
7.495 +  by transfer (rule complex_cnj_one)
7.496
7.497  lemma hcomplex_hcnj_zero [simp]: "hcnj 0 = 0"
7.498 -by transfer (rule complex_cnj_zero)
7.499 -
7.500 -lemma hcomplex_hcnj_zero_iff [iff]: "!!z. (hcnj z = 0) = (z = 0)"
7.501 -by transfer (rule complex_cnj_zero_iff)
7.502 -
7.503 -lemma hcomplex_mult_hcnj:
7.504 -     "!!z. z * hcnj z = hcomplex_of_hypreal ((hRe z)\<^sup>2 + (hIm z)\<^sup>2)"
7.505 -by transfer (rule complex_mult_cnj)
7.506 -
7.507 -
7.508 -subsection\<open>More Theorems about the Function @{term hcmod}\<close>
7.509 +  by transfer (rule complex_cnj_zero)
7.510
7.511 -lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
7.512 -     "hcmod (hcomplex_of_hypreal(hypreal_of_nat n)) = hypreal_of_nat n"
7.513 -by simp
7.514 -
7.515 -lemma hcmod_hcomplex_of_hypreal_of_hypnat [simp]:
7.516 -     "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
7.517 -by simp
7.518 +lemma hcomplex_hcnj_zero_iff [iff]: "\<And>z. hcnj z = 0 \<longleftrightarrow> z = 0"
7.519 +  by transfer (rule complex_cnj_zero_iff)
7.520
7.521 -lemma hcmod_mult_hcnj: "!!z. hcmod(z * hcnj(z)) = (hcmod z)\<^sup>2"
7.522 -by transfer (rule complex_mod_mult_cnj)
7.523 -
7.524 -lemma hcmod_triangle_ineq2 [simp]:
7.525 -  "!!a b. hcmod(b + a) - hcmod b \<le> hcmod a"
7.526 -by transfer (rule complex_mod_triangle_ineq2)
7.527 -
7.528 -lemma hcmod_diff_ineq [simp]: "!!a b. hcmod(a) - hcmod(b) \<le> hcmod(a + b)"
7.529 -by transfer (rule norm_diff_ineq)
7.530 +lemma hcomplex_mult_hcnj: "\<And>z. z * hcnj z = hcomplex_of_hypreal ((hRe z)\<^sup>2 + (hIm z)\<^sup>2)"
7.531 +  by transfer (rule complex_mult_cnj)
7.532
7.533
7.534 -subsection\<open>Exponentiation\<close>
7.535 +subsection \<open>More Theorems about the Function @{term hcmod}\<close>
7.536 +
7.537 +lemma hcmod_hcomplex_of_hypreal_of_nat [simp]:
7.538 +  "hcmod (hcomplex_of_hypreal (hypreal_of_nat n)) = hypreal_of_nat n"
7.539 +  by simp
7.540 +
7.541 +lemma hcmod_hcomplex_of_hypreal_of_hypnat [simp]:
7.542 +  "hcmod (hcomplex_of_hypreal(hypreal_of_hypnat n)) = hypreal_of_hypnat n"
7.543 +  by simp
7.544 +
7.545 +lemma hcmod_mult_hcnj: "\<And>z. hcmod (z * hcnj z) = (hcmod z)\<^sup>2"
7.546 +  by transfer (rule complex_mod_mult_cnj)
7.547
7.548 -lemma hcomplexpow_0 [simp]:   "z ^ 0       = (1::hcomplex)"
7.549 -by (rule power_0)
7.550 +lemma hcmod_triangle_ineq2 [simp]: "\<And>a b. hcmod (b + a) - hcmod b \<le> hcmod a"
7.551 +  by transfer (rule complex_mod_triangle_ineq2)
7.552 +
7.553 +lemma hcmod_diff_ineq [simp]: "\<And>a b. hcmod a - hcmod b \<le> hcmod (a + b)"
7.554 +  by transfer (rule norm_diff_ineq)
7.555 +
7.556
7.557 -lemma hcomplexpow_Suc [simp]: "z ^ (Suc n) = (z::hcomplex) * (z ^ n)"
7.558 -by (rule power_Suc)
7.559 +subsection \<open>Exponentiation\<close>
7.560 +
7.561 +lemma hcomplexpow_0 [simp]: "z ^ 0 = 1"
7.562 +  for z :: hcomplex
7.563 +  by (rule power_0)
7.564 +
7.565 +lemma hcomplexpow_Suc [simp]: "z ^ (Suc n) = z * (z ^ n)"
7.566 +  for z :: hcomplex
7.567 +  by (rule power_Suc)
7.568
7.569  lemma hcomplexpow_i_squared [simp]: "iii\<^sup>2 = -1"
7.570 -by transfer (rule power2_i)
7.571 +  by transfer (rule power2_i)
7.572
7.573 -lemma hcomplex_of_hypreal_pow:
7.574 -     "!!x. hcomplex_of_hypreal (x ^ n) = (hcomplex_of_hypreal x) ^ n"
7.575 -by transfer (rule of_real_power)
7.576 +lemma hcomplex_of_hypreal_pow: "\<And>x. hcomplex_of_hypreal (x ^ n) = hcomplex_of_hypreal x ^ n"
7.577 +  by transfer (rule of_real_power)
7.578
7.579 -lemma hcomplex_hcnj_pow: "!!z. hcnj(z ^ n) = hcnj(z) ^ n"
7.580 -by transfer (rule complex_cnj_power)
7.581 +lemma hcomplex_hcnj_pow: "\<And>z. hcnj (z ^ n) = hcnj z ^ n"
7.582 +  by transfer (rule complex_cnj_power)
7.583
7.584 -lemma hcmod_hcomplexpow: "!!x. hcmod(x ^ n) = hcmod(x) ^ n"
7.585 -by transfer (rule norm_power)
7.586 +lemma hcmod_hcomplexpow: "\<And>x. hcmod (x ^ n) = hcmod x ^ n"
7.587 +  by transfer (rule norm_power)
7.588
7.589  lemma hcpow_minus:
7.590 -     "!!x n. (-x::hcomplex) pow n =
7.591 -      (if ( *p* even) n then (x pow n) else -(x pow n))"
7.592 -by transfer simp
7.593 +  "\<And>x n. (- x :: hcomplex) pow n = (if ( *p* even) n then (x pow n) else - (x pow n))"
7.594 +  by transfer simp
7.595
7.596 -lemma hcpow_mult:
7.597 -  "((r::hcomplex) * s) pow n = (r pow n) * (s pow n)"
7.598 +lemma hcpow_mult: "(r * s) pow n = (r pow n) * (s pow n)"
7.599 +  for r s :: hcomplex
7.600    by (fact hyperpow_mult)
7.601
7.602 -lemma hcpow_zero2 [simp]:
7.603 -  "\<And>n. 0 pow (hSuc n) = (0::'a::semiring_1 star)"
7.604 +lemma hcpow_zero2 [simp]: "\<And>n. 0 pow (hSuc n) = (0::'a::semiring_1 star)"
7.605    by transfer (rule power_0_Suc)
7.606
7.607 -lemma hcpow_not_zero [simp,intro]:
7.608 -  "!!r n. r \<noteq> 0 ==> r pow n \<noteq> (0::hcomplex)"
7.609 +lemma hcpow_not_zero [simp,intro]: "\<And>r n. r \<noteq> 0 \<Longrightarrow> r pow n \<noteq> (0::hcomplex)"
7.610    by (fact hyperpow_not_zero)
7.611
7.612 -lemma hcpow_zero_zero:
7.613 -  "r pow n = (0::hcomplex) ==> r = 0"
7.614 +lemma hcpow_zero_zero: "r pow n = 0 \<Longrightarrow> r = 0"
7.615 +  for r :: hcomplex
7.616    by (blast intro: ccontr dest: hcpow_not_zero)
7.617
7.618 -subsection\<open>The Function @{term hsgn}\<close>
7.619 +
7.620 +subsection \<open>The Function @{term hsgn}\<close>
7.621
7.622  lemma hsgn_zero [simp]: "hsgn 0 = 0"
7.623 -by transfer (rule sgn_zero)
7.624 +  by transfer (rule sgn_zero)
7.625
7.626  lemma hsgn_one [simp]: "hsgn 1 = 1"
7.627 -by transfer (rule sgn_one)
7.628 +  by transfer (rule sgn_one)
7.629
7.630 -lemma hsgn_minus: "!!z. hsgn (-z) = - hsgn(z)"
7.631 -by transfer (rule sgn_minus)
7.632 +lemma hsgn_minus: "\<And>z. hsgn (- z) = - hsgn z"
7.633 +  by transfer (rule sgn_minus)
7.634
7.635 -lemma hsgn_eq: "!!z. hsgn z = z / hcomplex_of_hypreal (hcmod z)"
7.636 -by transfer (rule sgn_eq)
7.637 +lemma hsgn_eq: "\<And>z. hsgn z = z / hcomplex_of_hypreal (hcmod z)"
7.638 +  by transfer (rule sgn_eq)
7.639
7.640 -lemma hcmod_i: "!!x y. hcmod (HComplex x y) = ( *f* sqrt) (x\<^sup>2 + y\<^sup>2)"
7.641 -by transfer (rule complex_norm)
7.642 +lemma hcmod_i: "\<And>x y. hcmod (HComplex x y) = ( *f* sqrt) (x\<^sup>2 + y\<^sup>2)"
7.643 +  by transfer (rule complex_norm)
7.644
7.645  lemma hcomplex_eq_cancel_iff1 [simp]:
7.646 -     "(hcomplex_of_hypreal xa = HComplex x y) = (xa = x & y = 0)"
7.648 +  "hcomplex_of_hypreal xa = HComplex x y \<longleftrightarrow> xa = x \<and> y = 0"
7.649 +  by (simp add: hcomplex_of_hypreal_eq)
7.650
7.651  lemma hcomplex_eq_cancel_iff2 [simp]:
7.652 -     "(HComplex x y = hcomplex_of_hypreal xa) = (x = xa & y = 0)"
7.654 +  "HComplex x y = hcomplex_of_hypreal xa \<longleftrightarrow> x = xa \<and> y = 0"
7.655 +  by (simp add: hcomplex_of_hypreal_eq)
7.656
7.657 -lemma HComplex_eq_0 [simp]: "!!x y. (HComplex x y = 0) = (x = 0 & y = 0)"
7.658 -by transfer (rule Complex_eq_0)
7.659 +lemma HComplex_eq_0 [simp]: "\<And>x y. HComplex x y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
7.660 +  by transfer (rule Complex_eq_0)
7.661
7.662 -lemma HComplex_eq_1 [simp]: "!!x y. (HComplex x y = 1) = (x = 1 & y = 0)"
7.663 -by transfer (rule Complex_eq_1)
7.664 +lemma HComplex_eq_1 [simp]: "\<And>x y. HComplex x y = 1 \<longleftrightarrow> x = 1 \<and> y = 0"
7.665 +  by transfer (rule Complex_eq_1)
7.666
7.667  lemma i_eq_HComplex_0_1: "iii = HComplex 0 1"
7.668 -by transfer (simp add: complex_eq_iff)
7.669 +  by transfer (simp add: complex_eq_iff)
7.670
7.671 -lemma HComplex_eq_i [simp]: "!!x y. (HComplex x y = iii) = (x = 0 & y = 1)"
7.672 -by transfer (rule Complex_eq_i)
7.673 -
7.674 -lemma hRe_hsgn [simp]: "!!z. hRe(hsgn z) = hRe(z)/hcmod z"
7.675 -by transfer (rule Re_sgn)
7.676 +lemma HComplex_eq_i [simp]: "\<And>x y. HComplex x y = iii \<longleftrightarrow> x = 0 \<and> y = 1"
7.677 +  by transfer (rule Complex_eq_i)
7.678
7.679 -lemma hIm_hsgn [simp]: "!!z. hIm(hsgn z) = hIm(z)/hcmod z"
7.680 -by transfer (rule Im_sgn)
7.681 +lemma hRe_hsgn [simp]: "\<And>z. hRe (hsgn z) = hRe z / hcmod z"
7.682 +  by transfer (rule Re_sgn)
7.683
7.684 -lemma HComplex_inverse:
7.685 -     "!!x y. inverse (HComplex x y) = HComplex (x/(x\<^sup>2 + y\<^sup>2)) (-y/(x\<^sup>2 + y\<^sup>2))"
7.686 -by transfer (rule complex_inverse)
7.687 -
7.688 -lemma hRe_mult_i_eq[simp]:
7.689 -    "!!y. hRe (iii * hcomplex_of_hypreal y) = 0"
7.690 -by transfer simp
7.691 +lemma hIm_hsgn [simp]: "\<And>z. hIm (hsgn z) = hIm z / hcmod z"
7.692 +  by transfer (rule Im_sgn)
7.693
7.694 -lemma hIm_mult_i_eq [simp]:
7.695 -    "!!y. hIm (iii * hcomplex_of_hypreal y) = y"
7.696 -by transfer simp
7.697 +lemma HComplex_inverse: "\<And>x y. inverse (HComplex x y) = HComplex (x / (x\<^sup>2 + y\<^sup>2)) (- y / (x\<^sup>2 + y\<^sup>2))"
7.698 +  by transfer (rule complex_inverse)
7.699
7.700 -lemma hcmod_mult_i [simp]: "!!y. hcmod (iii * hcomplex_of_hypreal y) = \<bar>y\<bar>"
7.701 -by transfer (simp add: norm_complex_def)
7.702 -
7.703 -lemma hcmod_mult_i2 [simp]: "!!y. hcmod (hcomplex_of_hypreal y * iii) = \<bar>y\<bar>"
7.704 -by transfer (simp add: norm_complex_def)
7.705 +lemma hRe_mult_i_eq[simp]: "\<And>y. hRe (iii * hcomplex_of_hypreal y) = 0"
7.706 +  by transfer simp
7.707
7.708 -(*---------------------------------------------------------------------------*)
7.709 -(*  harg                                                                     *)
7.710 -(*---------------------------------------------------------------------------*)
7.711 +lemma hIm_mult_i_eq [simp]: "\<And>y. hIm (iii * hcomplex_of_hypreal y) = y"
7.712 +  by transfer simp
7.713
7.714 -lemma cos_harg_i_mult_zero [simp]:
7.715 -     "!!y. y \<noteq> 0 ==> ( *f* cos) (harg(HComplex 0 y)) = 0"
7.716 -by transfer simp
7.717 +lemma hcmod_mult_i [simp]: "\<And>y. hcmod (iii * hcomplex_of_hypreal y) = \<bar>y\<bar>"
7.718 +  by transfer (simp add: norm_complex_def)
7.719
7.720 -lemma hcomplex_of_hypreal_zero_iff [simp]:
7.721 -     "!!y. (hcomplex_of_hypreal y = 0) = (y = 0)"
7.722 -by transfer (rule of_real_eq_0_iff)
7.723 +lemma hcmod_mult_i2 [simp]: "\<And>y. hcmod (hcomplex_of_hypreal y * iii) = \<bar>y\<bar>"
7.724 +  by transfer (simp add: norm_complex_def)
7.725
7.726
7.727 -subsection\<open>Polar Form for Nonstandard Complex Numbers\<close>
7.728 +subsubsection \<open>\<open>harg\<close>\<close>
7.729 +
7.730 +lemma cos_harg_i_mult_zero [simp]: "\<And>y. y \<noteq> 0 \<Longrightarrow> ( *f* cos) (harg (HComplex 0 y)) = 0"
7.731 +  by transfer simp
7.732
7.733 -lemma complex_split_polar2:
7.734 -     "\<forall>n. \<exists>r a. (z n) =  complex_of_real r * (Complex (cos a) (sin a))"
7.735 -by (auto intro: complex_split_polar)
7.736 +lemma hcomplex_of_hypreal_zero_iff [simp]: "\<And>y. hcomplex_of_hypreal y = 0 \<longleftrightarrow> y = 0"
7.737 +  by transfer (rule of_real_eq_0_iff)
7.738 +
7.739 +
7.740 +subsection \<open>Polar Form for Nonstandard Complex Numbers\<close>
7.741 +
7.742 +lemma complex_split_polar2: "\<forall>n. \<exists>r a. (z n) = complex_of_real r * Complex (cos a) (sin a)"
7.743 +  by (auto intro: complex_split_polar)
7.744
7.745  lemma hcomplex_split_polar:
7.746 -  "!!z. \<exists>r a. z = hcomplex_of_hypreal r * (HComplex(( *f* cos) a)(( *f* sin) a))"
7.747 -by transfer (simp add: complex_split_polar)
7.748 +  "\<And>z. \<exists>r a. z = hcomplex_of_hypreal r * (HComplex (( *f* cos) a) (( *f* sin) a))"
7.749 +  by transfer (simp add: complex_split_polar)
7.750
7.751  lemma hcis_eq:
7.752 -   "!!a. hcis a =
7.753 -    (hcomplex_of_hypreal(( *f* cos) a) +
7.754 -    iii * hcomplex_of_hypreal(( *f* sin) a))"
7.755 -by transfer (simp add: complex_eq_iff)
7.756 +  "\<And>a. hcis a = hcomplex_of_hypreal (( *f* cos) a) + iii * hcomplex_of_hypreal (( *f* sin) a)"
7.757 +  by transfer (simp add: complex_eq_iff)
7.758
7.759 -lemma hrcis_Ex: "!!z. \<exists>r a. z = hrcis r a"
7.760 -by transfer (rule rcis_Ex)
7.761 +lemma hrcis_Ex: "\<And>z. \<exists>r a. z = hrcis r a"
7.762 +  by transfer (rule rcis_Ex)
7.763
7.764  lemma hRe_hcomplex_polar [simp]:
7.765 -  "!!r a. hRe (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) =
7.766 -      r * ( *f* cos) a"
7.767 -by transfer simp
7.768 +  "\<And>r a. hRe (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = r * ( *f* cos) a"
7.769 +  by transfer simp
7.770
7.771 -lemma hRe_hrcis [simp]: "!!r a. hRe(hrcis r a) = r * ( *f* cos) a"
7.772 -by transfer (rule Re_rcis)
7.773 +lemma hRe_hrcis [simp]: "\<And>r a. hRe (hrcis r a) = r * ( *f* cos) a"
7.774 +  by transfer (rule Re_rcis)
7.775
7.776  lemma hIm_hcomplex_polar [simp]:
7.777 -  "!!r a. hIm (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) =
7.778 -      r * ( *f* sin) a"
7.779 -by transfer simp
7.780 +  "\<And>r a. hIm (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = r * ( *f* sin) a"
7.781 +  by transfer simp
7.782
7.783 -lemma hIm_hrcis [simp]: "!!r a. hIm(hrcis r a) = r * ( *f* sin) a"
7.784 -by transfer (rule Im_rcis)
7.785 +lemma hIm_hrcis [simp]: "\<And>r a. hIm (hrcis r a) = r * ( *f* sin) a"
7.786 +  by transfer (rule Im_rcis)
7.787
7.788 -lemma hcmod_unit_one [simp]:
7.789 -     "!!a. hcmod (HComplex (( *f* cos) a) (( *f* sin) a)) = 1"
7.790 -by transfer (simp add: cmod_unit_one)
7.791 +lemma hcmod_unit_one [simp]: "\<And>a. hcmod (HComplex (( *f* cos) a) (( *f* sin) a)) = 1"
7.792 +  by transfer (simp add: cmod_unit_one)
7.793
7.794  lemma hcmod_complex_polar [simp]:
7.795 -  "!!r a. hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = \<bar>r\<bar>"
7.796 -by transfer (simp add: cmod_complex_polar)
7.797 -
7.798 -lemma hcmod_hrcis [simp]: "!!r a. hcmod(hrcis r a) = \<bar>r\<bar>"
7.799 -by transfer (rule complex_mod_rcis)
7.800 +  "\<And>r a. hcmod (hcomplex_of_hypreal r * HComplex (( *f* cos) a) (( *f* sin) a)) = \<bar>r\<bar>"
7.801 +  by transfer (simp add: cmod_complex_polar)
7.802
7.803 -(*---------------------------------------------------------------------------*)
7.804 -(*  (r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b)                *)
7.805 -(*---------------------------------------------------------------------------*)
7.806 +lemma hcmod_hrcis [simp]: "\<And>r a. hcmod(hrcis r a) = \<bar>r\<bar>"
7.807 +  by transfer (rule complex_mod_rcis)
7.808
7.809 -lemma hcis_hrcis_eq: "!!a. hcis a = hrcis 1 a"
7.810 -by transfer (rule cis_rcis_eq)
7.811 +text \<open>\<open>(r1 * hrcis a) * (r2 * hrcis b) = r1 * r2 * hrcis (a + b)\<close>\<close>
7.812 +
7.813 +lemma hcis_hrcis_eq: "\<And>a. hcis a = hrcis 1 a"
7.814 +  by transfer (rule cis_rcis_eq)
7.815  declare hcis_hrcis_eq [symmetric, simp]
7.816
7.817 -lemma hrcis_mult:
7.818 -  "!!a b r1 r2. hrcis r1 a * hrcis r2 b = hrcis (r1*r2) (a + b)"
7.819 -by transfer (rule rcis_mult)
7.820 +lemma hrcis_mult: "\<And>a b r1 r2. hrcis r1 a * hrcis r2 b = hrcis (r1 * r2) (a + b)"
7.821 +  by transfer (rule rcis_mult)
7.822
7.823 -lemma hcis_mult: "!!a b. hcis a * hcis b = hcis (a + b)"
7.824 -by transfer (rule cis_mult)
7.825 +lemma hcis_mult: "\<And>a b. hcis a * hcis b = hcis (a + b)"
7.826 +  by transfer (rule cis_mult)
7.827
7.828  lemma hcis_zero [simp]: "hcis 0 = 1"
7.829 -by transfer (rule cis_zero)
7.830 +  by transfer (rule cis_zero)
7.```