src/HOL/Limits.thy
author wenzelm
Sun Nov 02 18:21:45 2014 +0100 (2014-11-02)
changeset 58889 5b7a9633cfa8
parent 58729 e8ecc79aee43
child 59613 7103019278f0
permissions -rw-r--r--
modernized header uniformly as section;
wenzelm@52265
     1
(*  Title:      HOL/Limits.thy
hoelzl@51526
     2
    Author:     Brian Huffman
hoelzl@51526
     3
    Author:     Jacques D. Fleuriot, University of Cambridge
hoelzl@51526
     4
    Author:     Lawrence C Paulson
hoelzl@51526
     5
    Author:     Jeremy Avigad
huffman@31349
     6
*)
huffman@31349
     7
wenzelm@58889
     8
section {* Limits on Real Vector Spaces *}
huffman@31349
     9
huffman@31349
    10
theory Limits
hoelzl@51524
    11
imports Real_Vector_Spaces
huffman@31349
    12
begin
huffman@31349
    13
hoelzl@51526
    14
subsection {* Filter going to infinity norm *}
hoelzl@51526
    15
hoelzl@50324
    16
definition at_infinity :: "'a::real_normed_vector filter" where
hoelzl@57276
    17
  "at_infinity = (INF r. principal {x. r \<le> norm x})"
hoelzl@50324
    18
hoelzl@57276
    19
lemma eventually_at_infinity: "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> P x)"
hoelzl@57276
    20
  unfolding at_infinity_def
hoelzl@57276
    21
  by (subst eventually_INF_base)
hoelzl@57276
    22
     (auto simp: subset_eq eventually_principal intro!: exI[of _ "max a b" for a b])
huffman@31392
    23
hoelzl@50325
    24
lemma at_infinity_eq_at_top_bot:
hoelzl@50325
    25
  "(at_infinity \<Colon> real filter) = sup at_top at_bot"
hoelzl@57276
    26
  apply (simp add: filter_eq_iff eventually_sup eventually_at_infinity
hoelzl@57276
    27
                   eventually_at_top_linorder eventually_at_bot_linorder)
hoelzl@57276
    28
  apply safe
hoelzl@57276
    29
  apply (rule_tac x="b" in exI, simp)
hoelzl@57276
    30
  apply (rule_tac x="- b" in exI, simp)
hoelzl@57276
    31
  apply (rule_tac x="max (- Na) N" in exI, auto simp: abs_real_def)
hoelzl@57276
    32
  done
hoelzl@50325
    33
hoelzl@57276
    34
lemma at_top_le_at_infinity: "at_top \<le> (at_infinity :: real filter)"
hoelzl@50325
    35
  unfolding at_infinity_eq_at_top_bot by simp
hoelzl@50325
    36
hoelzl@57276
    37
lemma at_bot_le_at_infinity: "at_bot \<le> (at_infinity :: real filter)"
hoelzl@50325
    38
  unfolding at_infinity_eq_at_top_bot by simp
hoelzl@50325
    39
hoelzl@57275
    40
lemma filterlim_at_top_imp_at_infinity:
hoelzl@57275
    41
  fixes f :: "_ \<Rightarrow> real"
hoelzl@57275
    42
  shows "filterlim f at_top F \<Longrightarrow> filterlim f at_infinity F"
hoelzl@57275
    43
  by (rule filterlim_mono[OF _ at_top_le_at_infinity order_refl])
hoelzl@57275
    44
hoelzl@51531
    45
subsubsection {* Boundedness *}
hoelzl@51531
    46
hoelzl@51531
    47
definition Bfun :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a filter \<Rightarrow> bool" where
hoelzl@51531
    48
  Bfun_metric_def: "Bfun f F = (\<exists>y. \<exists>K>0. eventually (\<lambda>x. dist (f x) y \<le> K) F)"
hoelzl@51531
    49
hoelzl@51531
    50
abbreviation Bseq :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
hoelzl@51531
    51
  "Bseq X \<equiv> Bfun X sequentially"
hoelzl@51531
    52
hoelzl@51531
    53
lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially" ..
hoelzl@51531
    54
hoelzl@51531
    55
lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
hoelzl@51531
    56
  unfolding Bfun_metric_def by (subst eventually_sequentially_seg)
hoelzl@51531
    57
hoelzl@51531
    58
lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
hoelzl@51531
    59
  unfolding Bfun_metric_def by (subst (asm) eventually_sequentially_seg)
huffman@31355
    60
hoelzl@51474
    61
lemma Bfun_def:
hoelzl@51474
    62
  "Bfun f F \<longleftrightarrow> (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
hoelzl@51474
    63
  unfolding Bfun_metric_def norm_conv_dist
hoelzl@51474
    64
proof safe
hoelzl@51474
    65
  fix y K assume "0 < K" and *: "eventually (\<lambda>x. dist (f x) y \<le> K) F"
hoelzl@51474
    66
  moreover have "eventually (\<lambda>x. dist (f x) 0 \<le> dist (f x) y + dist 0 y) F"
hoelzl@51474
    67
    by (intro always_eventually) (metis dist_commute dist_triangle)
hoelzl@51474
    68
  with * have "eventually (\<lambda>x. dist (f x) 0 \<le> K + dist 0 y) F"
hoelzl@51474
    69
    by eventually_elim auto
hoelzl@51474
    70
  with `0 < K` show "\<exists>K>0. eventually (\<lambda>x. dist (f x) 0 \<le> K) F"
hoelzl@51474
    71
    by (intro exI[of _ "K + dist 0 y"] add_pos_nonneg conjI zero_le_dist) auto
hoelzl@51474
    72
qed auto
huffman@31355
    73
huffman@31487
    74
lemma BfunI:
huffman@44195
    75
  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
huffman@31355
    76
unfolding Bfun_def
huffman@31355
    77
proof (intro exI conjI allI)
huffman@31355
    78
  show "0 < max K 1" by simp
huffman@31355
    79
next
huffman@44195
    80
  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) F"
huffman@31355
    81
    using K by (rule eventually_elim1, simp)
huffman@31355
    82
qed
huffman@31355
    83
huffman@31355
    84
lemma BfunE:
huffman@44195
    85
  assumes "Bfun f F"
huffman@44195
    86
  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
huffman@31355
    87
using assms unfolding Bfun_def by fast
huffman@31355
    88
hoelzl@51531
    89
lemma Cauchy_Bseq: "Cauchy X \<Longrightarrow> Bseq X"
hoelzl@51531
    90
  unfolding Cauchy_def Bfun_metric_def eventually_sequentially
hoelzl@51531
    91
  apply (erule_tac x=1 in allE)
hoelzl@51531
    92
  apply simp
hoelzl@51531
    93
  apply safe
hoelzl@51531
    94
  apply (rule_tac x="X M" in exI)
hoelzl@51531
    95
  apply (rule_tac x=1 in exI)
hoelzl@51531
    96
  apply (erule_tac x=M in allE)
hoelzl@51531
    97
  apply simp
hoelzl@51531
    98
  apply (rule_tac x=M in exI)
hoelzl@51531
    99
  apply (auto simp: dist_commute)
hoelzl@51531
   100
  done
hoelzl@51531
   101
hoelzl@51531
   102
hoelzl@51531
   103
subsubsection {* Bounded Sequences *}
hoelzl@51531
   104
hoelzl@51531
   105
lemma BseqI': "(\<And>n. norm (X n) \<le> K) \<Longrightarrow> Bseq X"
hoelzl@51531
   106
  by (intro BfunI) (auto simp: eventually_sequentially)
hoelzl@51531
   107
hoelzl@51531
   108
lemma BseqI2': "\<forall>n\<ge>N. norm (X n) \<le> K \<Longrightarrow> Bseq X"
hoelzl@51531
   109
  by (intro BfunI) (auto simp: eventually_sequentially)
hoelzl@51531
   110
hoelzl@51531
   111
lemma Bseq_def: "Bseq X \<longleftrightarrow> (\<exists>K>0. \<forall>n. norm (X n) \<le> K)"
hoelzl@51531
   112
  unfolding Bfun_def eventually_sequentially
hoelzl@51531
   113
proof safe
hoelzl@51531
   114
  fix N K assume "0 < K" "\<forall>n\<ge>N. norm (X n) \<le> K"
hoelzl@51531
   115
  then show "\<exists>K>0. \<forall>n. norm (X n) \<le> K"
haftmann@54863
   116
    by (intro exI[of _ "max (Max (norm ` X ` {..N})) K"] max.strict_coboundedI2)
hoelzl@51531
   117
       (auto intro!: imageI not_less[where 'a=nat, THEN iffD1] Max_ge simp: le_max_iff_disj)
hoelzl@51531
   118
qed auto
hoelzl@51531
   119
hoelzl@51531
   120
lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
hoelzl@51531
   121
unfolding Bseq_def by auto
hoelzl@51531
   122
hoelzl@51531
   123
lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
hoelzl@51531
   124
by (simp add: Bseq_def)
hoelzl@51531
   125
hoelzl@51531
   126
lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
hoelzl@51531
   127
by (auto simp add: Bseq_def)
hoelzl@51531
   128
hoelzl@54263
   129
lemma Bseq_bdd_above: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_above (range X)"
hoelzl@54263
   130
proof (elim BseqE, intro bdd_aboveI2)
hoelzl@54263
   131
  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "X n \<le> K"
hoelzl@54263
   132
    by (auto elim!: allE[of _ n])
hoelzl@54263
   133
qed
hoelzl@54263
   134
hoelzl@54263
   135
lemma Bseq_bdd_below: "Bseq (X::nat \<Rightarrow> real) \<Longrightarrow> bdd_below (range X)"
hoelzl@54263
   136
proof (elim BseqE, intro bdd_belowI2)
hoelzl@54263
   137
  fix K n assume "0 < K" "\<forall>n. norm (X n) \<le> K" then show "- K \<le> X n"
hoelzl@54263
   138
    by (auto elim!: allE[of _ n])
hoelzl@54263
   139
qed
hoelzl@54263
   140
hoelzl@51531
   141
lemma lemma_NBseq_def:
hoelzl@51531
   142
  "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
hoelzl@51531
   143
proof safe
hoelzl@51531
   144
  fix K :: real
hoelzl@51531
   145
  from reals_Archimedean2 obtain n :: nat where "K < real n" ..
hoelzl@51531
   146
  then have "K \<le> real (Suc n)" by auto
hoelzl@51531
   147
  moreover assume "\<forall>m. norm (X m) \<le> K"
hoelzl@51531
   148
  ultimately have "\<forall>m. norm (X m) \<le> real (Suc n)"
hoelzl@51531
   149
    by (blast intro: order_trans)
hoelzl@51531
   150
  then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
hoelzl@51531
   151
qed (force simp add: real_of_nat_Suc)
hoelzl@51531
   152
hoelzl@51531
   153
text{* alternative definition for Bseq *}
hoelzl@51531
   154
lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
hoelzl@51531
   155
apply (simp add: Bseq_def)
hoelzl@51531
   156
apply (simp (no_asm) add: lemma_NBseq_def)
hoelzl@51531
   157
done
hoelzl@51531
   158
hoelzl@51531
   159
lemma lemma_NBseq_def2:
hoelzl@51531
   160
     "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
hoelzl@51531
   161
apply (subst lemma_NBseq_def, auto)
hoelzl@51531
   162
apply (rule_tac x = "Suc N" in exI)
hoelzl@51531
   163
apply (rule_tac [2] x = N in exI)
hoelzl@51531
   164
apply (auto simp add: real_of_nat_Suc)
hoelzl@51531
   165
 prefer 2 apply (blast intro: order_less_imp_le)
hoelzl@51531
   166
apply (drule_tac x = n in spec, simp)
hoelzl@51531
   167
done
hoelzl@51531
   168
hoelzl@51531
   169
(* yet another definition for Bseq *)
hoelzl@51531
   170
lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
hoelzl@51531
   171
by (simp add: Bseq_def lemma_NBseq_def2)
hoelzl@51531
   172
hoelzl@51531
   173
subsubsection{*A Few More Equivalence Theorems for Boundedness*}
hoelzl@51531
   174
hoelzl@51531
   175
text{*alternative formulation for boundedness*}
hoelzl@51531
   176
lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
hoelzl@51531
   177
apply (unfold Bseq_def, safe)
hoelzl@51531
   178
apply (rule_tac [2] x = "k + norm x" in exI)
hoelzl@51531
   179
apply (rule_tac x = K in exI, simp)
hoelzl@51531
   180
apply (rule exI [where x = 0], auto)
hoelzl@51531
   181
apply (erule order_less_le_trans, simp)
haftmann@54230
   182
apply (drule_tac x=n in spec)
hoelzl@51531
   183
apply (drule order_trans [OF norm_triangle_ineq2])
hoelzl@51531
   184
apply simp
hoelzl@51531
   185
done
hoelzl@51531
   186
hoelzl@51531
   187
text{*alternative formulation for boundedness*}
haftmann@53602
   188
lemma Bseq_iff3:
haftmann@53602
   189
  "Bseq X \<longleftrightarrow> (\<exists>k>0. \<exists>N. \<forall>n. norm (X n + - X N) \<le> k)" (is "?P \<longleftrightarrow> ?Q")
haftmann@53602
   190
proof
haftmann@53602
   191
  assume ?P
haftmann@53602
   192
  then obtain K
haftmann@53602
   193
    where *: "0 < K" and **: "\<And>n. norm (X n) \<le> K" by (auto simp add: Bseq_def)
haftmann@53602
   194
  from * have "0 < K + norm (X 0)" by (rule order_less_le_trans) simp
haftmann@54230
   195
  from ** have "\<forall>n. norm (X n - X 0) \<le> K + norm (X 0)"
haftmann@54230
   196
    by (auto intro: order_trans norm_triangle_ineq4)
haftmann@54230
   197
  then have "\<forall>n. norm (X n + - X 0) \<le> K + norm (X 0)"
haftmann@54230
   198
    by simp
haftmann@54230
   199
  with `0 < K + norm (X 0)` show ?Q by blast
haftmann@53602
   200
next
haftmann@53602
   201
  assume ?Q then show ?P by (auto simp add: Bseq_iff2)
haftmann@53602
   202
qed
hoelzl@51531
   203
hoelzl@51531
   204
lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
hoelzl@51531
   205
apply (simp add: Bseq_def)
hoelzl@51531
   206
apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
hoelzl@51531
   207
apply (drule_tac x = n in spec, arith)
hoelzl@51531
   208
done
hoelzl@51531
   209
haftmann@54230
   210
hoelzl@51531
   211
subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
hoelzl@51531
   212
hoelzl@51531
   213
lemma Bseq_minus_iff: "Bseq (%n. -(X n) :: 'a :: real_normed_vector) = Bseq X"
hoelzl@51531
   214
  by (simp add: Bseq_def)
hoelzl@51531
   215
hoelzl@51531
   216
lemma Bseq_eq_bounded: "range f \<subseteq> {a .. b::real} \<Longrightarrow> Bseq f"
hoelzl@51531
   217
  apply (simp add: subset_eq)
hoelzl@51531
   218
  apply (rule BseqI'[where K="max (norm a) (norm b)"])
hoelzl@51531
   219
  apply (erule_tac x=n in allE)
hoelzl@51531
   220
  apply auto
hoelzl@51531
   221
  done
hoelzl@51531
   222
hoelzl@51531
   223
lemma incseq_bounded: "incseq X \<Longrightarrow> \<forall>i. X i \<le> (B::real) \<Longrightarrow> Bseq X"
hoelzl@51531
   224
  by (intro Bseq_eq_bounded[of X "X 0" B]) (auto simp: incseq_def)
hoelzl@51531
   225
hoelzl@51531
   226
lemma decseq_bounded: "decseq X \<Longrightarrow> \<forall>i. (B::real) \<le> X i \<Longrightarrow> Bseq X"
hoelzl@51531
   227
  by (intro Bseq_eq_bounded[of X B "X 0"]) (auto simp: decseq_def)
hoelzl@51531
   228
hoelzl@51531
   229
subsection {* Bounded Monotonic Sequences *}
hoelzl@51531
   230
hoelzl@51531
   231
subsubsection{*A Bounded and Monotonic Sequence Converges*}
hoelzl@51531
   232
hoelzl@51531
   233
(* TODO: delete *)
hoelzl@51531
   234
(* FIXME: one use in NSA/HSEQ.thy *)
hoelzl@51531
   235
lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
hoelzl@51531
   236
  apply (rule_tac x="X m" in exI)
hoelzl@51531
   237
  apply (rule filterlim_cong[THEN iffD2, OF refl refl _ tendsto_const])
hoelzl@51531
   238
  unfolding eventually_sequentially
hoelzl@51531
   239
  apply blast
hoelzl@51531
   240
  done
hoelzl@51531
   241
huffman@31349
   242
subsection {* Convergence to Zero *}
huffman@31349
   243
huffman@44081
   244
definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
huffman@44195
   245
  where "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
huffman@31349
   246
huffman@31349
   247
lemma ZfunI:
huffman@44195
   248
  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
huffman@44081
   249
  unfolding Zfun_def by simp
huffman@31349
   250
huffman@31349
   251
lemma ZfunD:
huffman@44195
   252
  "\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
huffman@44081
   253
  unfolding Zfun_def by simp
huffman@31349
   254
huffman@31355
   255
lemma Zfun_ssubst:
huffman@44195
   256
  "eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
huffman@44081
   257
  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
huffman@31355
   258
huffman@44195
   259
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
huffman@44081
   260
  unfolding Zfun_def by simp
huffman@31349
   261
huffman@44195
   262
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
huffman@44081
   263
  unfolding Zfun_def by simp
huffman@31349
   264
huffman@31349
   265
lemma Zfun_imp_Zfun:
huffman@44195
   266
  assumes f: "Zfun f F"
huffman@44195
   267
  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
huffman@44195
   268
  shows "Zfun (\<lambda>x. g x) F"
huffman@31349
   269
proof (cases)
huffman@31349
   270
  assume K: "0 < K"
huffman@31349
   271
  show ?thesis
huffman@31349
   272
  proof (rule ZfunI)
huffman@31349
   273
    fix r::real assume "0 < r"
nipkow@56541
   274
    hence "0 < r / K" using K by simp
huffman@44195
   275
    then have "eventually (\<lambda>x. norm (f x) < r / K) F"
huffman@31487
   276
      using ZfunD [OF f] by fast
huffman@44195
   277
    with g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   278
    proof eventually_elim
noschinl@46887
   279
      case (elim x)
huffman@31487
   280
      hence "norm (f x) * K < r"
huffman@31349
   281
        by (simp add: pos_less_divide_eq K)
noschinl@46887
   282
      thus ?case
noschinl@46887
   283
        by (simp add: order_le_less_trans [OF elim(1)])
huffman@31349
   284
    qed
huffman@31349
   285
  qed
huffman@31349
   286
next
huffman@31349
   287
  assume "\<not> 0 < K"
huffman@31349
   288
  hence K: "K \<le> 0" by (simp only: not_less)
huffman@31355
   289
  show ?thesis
huffman@31355
   290
  proof (rule ZfunI)
huffman@31355
   291
    fix r :: real
huffman@31355
   292
    assume "0 < r"
huffman@44195
   293
    from g show "eventually (\<lambda>x. norm (g x) < r) F"
noschinl@46887
   294
    proof eventually_elim
noschinl@46887
   295
      case (elim x)
noschinl@46887
   296
      also have "norm (f x) * K \<le> norm (f x) * 0"
huffman@31355
   297
        using K norm_ge_zero by (rule mult_left_mono)
noschinl@46887
   298
      finally show ?case
huffman@31355
   299
        using `0 < r` by simp
huffman@31355
   300
    qed
huffman@31355
   301
  qed
huffman@31349
   302
qed
huffman@31349
   303
huffman@44195
   304
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
huffman@44081
   305
  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
huffman@31349
   306
huffman@31349
   307
lemma Zfun_add:
huffman@44195
   308
  assumes f: "Zfun f F" and g: "Zfun g F"
huffman@44195
   309
  shows "Zfun (\<lambda>x. f x + g x) F"
huffman@31349
   310
proof (rule ZfunI)
huffman@31349
   311
  fix r::real assume "0 < r"
huffman@31349
   312
  hence r: "0 < r / 2" by simp
huffman@44195
   313
  have "eventually (\<lambda>x. norm (f x) < r/2) F"
huffman@31487
   314
    using f r by (rule ZfunD)
huffman@31349
   315
  moreover
huffman@44195
   316
  have "eventually (\<lambda>x. norm (g x) < r/2) F"
huffman@31487
   317
    using g r by (rule ZfunD)
huffman@31349
   318
  ultimately
huffman@44195
   319
  show "eventually (\<lambda>x. norm (f x + g x) < r) F"
noschinl@46887
   320
  proof eventually_elim
noschinl@46887
   321
    case (elim x)
huffman@31487
   322
    have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
huffman@31349
   323
      by (rule norm_triangle_ineq)
huffman@31349
   324
    also have "\<dots> < r/2 + r/2"
noschinl@46887
   325
      using elim by (rule add_strict_mono)
noschinl@46887
   326
    finally show ?case
huffman@31349
   327
      by simp
huffman@31349
   328
  qed
huffman@31349
   329
qed
huffman@31349
   330
huffman@44195
   331
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
huffman@44081
   332
  unfolding Zfun_def by simp
huffman@31349
   333
huffman@44195
   334
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
haftmann@54230
   335
  using Zfun_add [of f F "\<lambda>x. - g x"] by (simp add: Zfun_minus)
huffman@31349
   336
huffman@31349
   337
lemma (in bounded_linear) Zfun:
huffman@44195
   338
  assumes g: "Zfun g F"
huffman@44195
   339
  shows "Zfun (\<lambda>x. f (g x)) F"
huffman@31349
   340
proof -
huffman@31349
   341
  obtain K where "\<And>x. norm (f x) \<le> norm x * K"
huffman@31349
   342
    using bounded by fast
huffman@44195
   343
  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) F"
huffman@31355
   344
    by simp
huffman@31487
   345
  with g show ?thesis
huffman@31349
   346
    by (rule Zfun_imp_Zfun)
huffman@31349
   347
qed
huffman@31349
   348
huffman@31349
   349
lemma (in bounded_bilinear) Zfun:
huffman@44195
   350
  assumes f: "Zfun f F"
huffman@44195
   351
  assumes g: "Zfun g F"
huffman@44195
   352
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31349
   353
proof (rule ZfunI)
huffman@31349
   354
  fix r::real assume r: "0 < r"
huffman@31349
   355
  obtain K where K: "0 < K"
huffman@31349
   356
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31349
   357
    using pos_bounded by fast
huffman@31349
   358
  from K have K': "0 < inverse K"
huffman@31349
   359
    by (rule positive_imp_inverse_positive)
huffman@44195
   360
  have "eventually (\<lambda>x. norm (f x) < r) F"
huffman@31487
   361
    using f r by (rule ZfunD)
huffman@31349
   362
  moreover
huffman@44195
   363
  have "eventually (\<lambda>x. norm (g x) < inverse K) F"
huffman@31487
   364
    using g K' by (rule ZfunD)
huffman@31349
   365
  ultimately
huffman@44195
   366
  show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
noschinl@46887
   367
  proof eventually_elim
noschinl@46887
   368
    case (elim x)
huffman@31487
   369
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31349
   370
      by (rule norm_le)
huffman@31487
   371
    also have "norm (f x) * norm (g x) * K < r * inverse K * K"
noschinl@46887
   372
      by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero elim K)
huffman@31349
   373
    also from K have "r * inverse K * K = r"
huffman@31349
   374
      by simp
noschinl@46887
   375
    finally show ?case .
huffman@31349
   376
  qed
huffman@31349
   377
qed
huffman@31349
   378
huffman@31349
   379
lemma (in bounded_bilinear) Zfun_left:
huffman@44195
   380
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
huffman@44081
   381
  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
huffman@31349
   382
huffman@31349
   383
lemma (in bounded_bilinear) Zfun_right:
huffman@44195
   384
  "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
huffman@44081
   385
  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
huffman@31349
   386
huffman@44282
   387
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
huffman@44282
   388
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
huffman@44282
   389
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
huffman@31349
   390
huffman@44195
   391
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
huffman@44081
   392
  by (simp only: tendsto_iff Zfun_def dist_norm)
huffman@31349
   393
lp15@56366
   394
lemma tendsto_0_le: "\<lbrakk>(f ---> 0) F; eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F\<rbrakk> 
lp15@56366
   395
                     \<Longrightarrow> (g ---> 0) F"
lp15@56366
   396
  by (simp add: Zfun_imp_Zfun tendsto_Zfun_iff)
lp15@56366
   397
huffman@44205
   398
subsubsection {* Distance and norms *}
huffman@36662
   399
hoelzl@51531
   400
lemma tendsto_dist [tendsto_intros]:
hoelzl@51531
   401
  fixes l m :: "'a :: metric_space"
hoelzl@51531
   402
  assumes f: "(f ---> l) F" and g: "(g ---> m) F"
hoelzl@51531
   403
  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
hoelzl@51531
   404
proof (rule tendstoI)
hoelzl@51531
   405
  fix e :: real assume "0 < e"
hoelzl@51531
   406
  hence e2: "0 < e/2" by simp
hoelzl@51531
   407
  from tendstoD [OF f e2] tendstoD [OF g e2]
hoelzl@51531
   408
  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) F"
hoelzl@51531
   409
  proof (eventually_elim)
hoelzl@51531
   410
    case (elim x)
hoelzl@51531
   411
    then show "dist (dist (f x) (g x)) (dist l m) < e"
hoelzl@51531
   412
      unfolding dist_real_def
hoelzl@51531
   413
      using dist_triangle2 [of "f x" "g x" "l"]
hoelzl@51531
   414
      using dist_triangle2 [of "g x" "l" "m"]
hoelzl@51531
   415
      using dist_triangle3 [of "l" "m" "f x"]
hoelzl@51531
   416
      using dist_triangle [of "f x" "m" "g x"]
hoelzl@51531
   417
      by arith
hoelzl@51531
   418
  qed
hoelzl@51531
   419
qed
hoelzl@51531
   420
hoelzl@51531
   421
lemma continuous_dist[continuous_intros]:
hoelzl@51531
   422
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   423
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   424
  unfolding continuous_def by (rule tendsto_dist)
hoelzl@51531
   425
hoelzl@56371
   426
lemma continuous_on_dist[continuous_intros]:
hoelzl@51531
   427
  fixes f g :: "_ \<Rightarrow> 'a :: metric_space"
hoelzl@51531
   428
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. dist (f x) (g x))"
hoelzl@51531
   429
  unfolding continuous_on_def by (auto intro: tendsto_dist)
hoelzl@51531
   430
huffman@31565
   431
lemma tendsto_norm [tendsto_intros]:
huffman@44195
   432
  "(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
huffman@44081
   433
  unfolding norm_conv_dist by (intro tendsto_intros)
huffman@36662
   434
hoelzl@51478
   435
lemma continuous_norm [continuous_intros]:
hoelzl@51478
   436
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. norm (f x))"
hoelzl@51478
   437
  unfolding continuous_def by (rule tendsto_norm)
hoelzl@51478
   438
hoelzl@56371
   439
lemma continuous_on_norm [continuous_intros]:
hoelzl@51478
   440
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. norm (f x))"
hoelzl@51478
   441
  unfolding continuous_on_def by (auto intro: tendsto_norm)
hoelzl@51478
   442
huffman@36662
   443
lemma tendsto_norm_zero:
huffman@44195
   444
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
huffman@44081
   445
  by (drule tendsto_norm, simp)
huffman@36662
   446
huffman@36662
   447
lemma tendsto_norm_zero_cancel:
huffman@44195
   448
  "((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
huffman@44081
   449
  unfolding tendsto_iff dist_norm by simp
huffman@36662
   450
huffman@36662
   451
lemma tendsto_norm_zero_iff:
huffman@44195
   452
  "((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
huffman@44081
   453
  unfolding tendsto_iff dist_norm by simp
huffman@31349
   454
huffman@44194
   455
lemma tendsto_rabs [tendsto_intros]:
huffman@44195
   456
  "(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
huffman@44194
   457
  by (fold real_norm_def, rule tendsto_norm)
huffman@44194
   458
hoelzl@51478
   459
lemma continuous_rabs [continuous_intros]:
hoelzl@51478
   460
  "continuous F f \<Longrightarrow> continuous F (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   461
  unfolding real_norm_def[symmetric] by (rule continuous_norm)
hoelzl@51478
   462
hoelzl@56371
   463
lemma continuous_on_rabs [continuous_intros]:
hoelzl@51478
   464
  "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. \<bar>f x :: real\<bar>)"
hoelzl@51478
   465
  unfolding real_norm_def[symmetric] by (rule continuous_on_norm)
hoelzl@51478
   466
huffman@44194
   467
lemma tendsto_rabs_zero:
huffman@44195
   468
  "(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
huffman@44194
   469
  by (fold real_norm_def, rule tendsto_norm_zero)
huffman@44194
   470
huffman@44194
   471
lemma tendsto_rabs_zero_cancel:
huffman@44195
   472
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
huffman@44194
   473
  by (fold real_norm_def, rule tendsto_norm_zero_cancel)
huffman@44194
   474
huffman@44194
   475
lemma tendsto_rabs_zero_iff:
huffman@44195
   476
  "((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
huffman@44194
   477
  by (fold real_norm_def, rule tendsto_norm_zero_iff)
huffman@44194
   478
huffman@44194
   479
subsubsection {* Addition and subtraction *}
huffman@44194
   480
huffman@31565
   481
lemma tendsto_add [tendsto_intros]:
huffman@31349
   482
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   483
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
huffman@44081
   484
  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
huffman@31349
   485
hoelzl@51478
   486
lemma continuous_add [continuous_intros]:
hoelzl@51478
   487
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   488
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x + g x)"
hoelzl@51478
   489
  unfolding continuous_def by (rule tendsto_add)
hoelzl@51478
   490
hoelzl@56371
   491
lemma continuous_on_add [continuous_intros]:
hoelzl@51478
   492
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   493
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x + g x)"
hoelzl@51478
   494
  unfolding continuous_on_def by (auto intro: tendsto_add)
hoelzl@51478
   495
huffman@44194
   496
lemma tendsto_add_zero:
hoelzl@51478
   497
  fixes f g :: "_ \<Rightarrow> 'b::real_normed_vector"
huffman@44195
   498
  shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
huffman@44194
   499
  by (drule (1) tendsto_add, simp)
huffman@44194
   500
huffman@31565
   501
lemma tendsto_minus [tendsto_intros]:
huffman@31349
   502
  fixes a :: "'a::real_normed_vector"
huffman@44195
   503
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
huffman@44081
   504
  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
huffman@31349
   505
hoelzl@51478
   506
lemma continuous_minus [continuous_intros]:
hoelzl@51478
   507
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   508
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. - f x)"
hoelzl@51478
   509
  unfolding continuous_def by (rule tendsto_minus)
hoelzl@51478
   510
hoelzl@56371
   511
lemma continuous_on_minus [continuous_intros]:
hoelzl@51478
   512
  fixes f :: "_ \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   513
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. - f x)"
hoelzl@51478
   514
  unfolding continuous_on_def by (auto intro: tendsto_minus)
hoelzl@51478
   515
huffman@31349
   516
lemma tendsto_minus_cancel:
huffman@31349
   517
  fixes a :: "'a::real_normed_vector"
huffman@44195
   518
  shows "((\<lambda>x. - f x) ---> - a) F \<Longrightarrow> (f ---> a) F"
huffman@44081
   519
  by (drule tendsto_minus, simp)
huffman@31349
   520
hoelzl@50330
   521
lemma tendsto_minus_cancel_left:
hoelzl@50330
   522
    "(f ---> - (y::_::real_normed_vector)) F \<longleftrightarrow> ((\<lambda>x. - f x) ---> y) F"
hoelzl@50330
   523
  using tendsto_minus_cancel[of f "- y" F]  tendsto_minus[of f "- y" F]
hoelzl@50330
   524
  by auto
hoelzl@50330
   525
huffman@31565
   526
lemma tendsto_diff [tendsto_intros]:
huffman@31349
   527
  fixes a b :: "'a::real_normed_vector"
huffman@44195
   528
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
haftmann@54230
   529
  using tendsto_add [of f a F "\<lambda>x. - g x" "- b"] by (simp add: tendsto_minus)
huffman@31349
   530
hoelzl@51478
   531
lemma continuous_diff [continuous_intros]:
hoelzl@51478
   532
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   533
  shows "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x - g x)"
hoelzl@51478
   534
  unfolding continuous_def by (rule tendsto_diff)
hoelzl@51478
   535
hoelzl@56371
   536
lemma continuous_on_diff [continuous_intros]:
hoelzl@51478
   537
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   538
  shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
hoelzl@51478
   539
  unfolding continuous_on_def by (auto intro: tendsto_diff)
hoelzl@51478
   540
huffman@31588
   541
lemma tendsto_setsum [tendsto_intros]:
huffman@31588
   542
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
huffman@44195
   543
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
huffman@44195
   544
  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
huffman@31588
   545
proof (cases "finite S")
huffman@31588
   546
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   547
    by (induct, simp, simp add: tendsto_add)
hoelzl@58729
   548
qed simp
huffman@31588
   549
hoelzl@51478
   550
lemma continuous_setsum [continuous_intros]:
hoelzl@51478
   551
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   552
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   553
  unfolding continuous_def by (rule tendsto_setsum)
hoelzl@51478
   554
hoelzl@51478
   555
lemma continuous_on_setsum [continuous_intros]:
hoelzl@51478
   556
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::real_normed_vector"
hoelzl@51478
   557
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Sum>i\<in>S. f i x)"
hoelzl@51478
   558
  unfolding continuous_on_def by (auto intro: tendsto_setsum)
hoelzl@51478
   559
hoelzl@50999
   560
lemmas real_tendsto_sandwich = tendsto_sandwich[where 'b=real]
hoelzl@50999
   561
huffman@44194
   562
subsubsection {* Linear operators and multiplication *}
huffman@44194
   563
huffman@44282
   564
lemma (in bounded_linear) tendsto:
huffman@44195
   565
  "(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
huffman@44081
   566
  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
huffman@31349
   567
hoelzl@51478
   568
lemma (in bounded_linear) continuous:
hoelzl@51478
   569
  "continuous F g \<Longrightarrow> continuous F (\<lambda>x. f (g x))"
hoelzl@51478
   570
  using tendsto[of g _ F] by (auto simp: continuous_def)
hoelzl@51478
   571
hoelzl@51478
   572
lemma (in bounded_linear) continuous_on:
hoelzl@51478
   573
  "continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f (g x))"
hoelzl@51478
   574
  using tendsto[of g] by (auto simp: continuous_on_def)
hoelzl@51478
   575
huffman@44194
   576
lemma (in bounded_linear) tendsto_zero:
huffman@44195
   577
  "(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
huffman@44194
   578
  by (drule tendsto, simp only: zero)
huffman@44194
   579
huffman@44282
   580
lemma (in bounded_bilinear) tendsto:
huffman@44195
   581
  "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
huffman@44081
   582
  by (simp only: tendsto_Zfun_iff prod_diff_prod
huffman@44081
   583
                 Zfun_add Zfun Zfun_left Zfun_right)
huffman@31349
   584
hoelzl@51478
   585
lemma (in bounded_bilinear) continuous:
hoelzl@51478
   586
  "continuous F f \<Longrightarrow> continuous F g \<Longrightarrow> continuous F (\<lambda>x. f x ** g x)"
hoelzl@51478
   587
  using tendsto[of f _ F g] by (auto simp: continuous_def)
hoelzl@51478
   588
hoelzl@51478
   589
lemma (in bounded_bilinear) continuous_on:
hoelzl@51478
   590
  "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x ** g x)"
hoelzl@51478
   591
  using tendsto[of f _ _ g] by (auto simp: continuous_on_def)
hoelzl@51478
   592
huffman@44194
   593
lemma (in bounded_bilinear) tendsto_zero:
huffman@44195
   594
  assumes f: "(f ---> 0) F"
huffman@44195
   595
  assumes g: "(g ---> 0) F"
huffman@44195
   596
  shows "((\<lambda>x. f x ** g x) ---> 0) F"
huffman@44194
   597
  using tendsto [OF f g] by (simp add: zero_left)
huffman@31355
   598
huffman@44194
   599
lemma (in bounded_bilinear) tendsto_left_zero:
huffman@44195
   600
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
huffman@44194
   601
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
huffman@44194
   602
huffman@44194
   603
lemma (in bounded_bilinear) tendsto_right_zero:
huffman@44195
   604
  "(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
huffman@44194
   605
  by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
huffman@44194
   606
huffman@44282
   607
lemmas tendsto_of_real [tendsto_intros] =
huffman@44282
   608
  bounded_linear.tendsto [OF bounded_linear_of_real]
huffman@44282
   609
huffman@44282
   610
lemmas tendsto_scaleR [tendsto_intros] =
huffman@44282
   611
  bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
huffman@44282
   612
huffman@44282
   613
lemmas tendsto_mult [tendsto_intros] =
huffman@44282
   614
  bounded_bilinear.tendsto [OF bounded_bilinear_mult]
huffman@44194
   615
hoelzl@51478
   616
lemmas continuous_of_real [continuous_intros] =
hoelzl@51478
   617
  bounded_linear.continuous [OF bounded_linear_of_real]
hoelzl@51478
   618
hoelzl@51478
   619
lemmas continuous_scaleR [continuous_intros] =
hoelzl@51478
   620
  bounded_bilinear.continuous [OF bounded_bilinear_scaleR]
hoelzl@51478
   621
hoelzl@51478
   622
lemmas continuous_mult [continuous_intros] =
hoelzl@51478
   623
  bounded_bilinear.continuous [OF bounded_bilinear_mult]
hoelzl@51478
   624
hoelzl@56371
   625
lemmas continuous_on_of_real [continuous_intros] =
hoelzl@51478
   626
  bounded_linear.continuous_on [OF bounded_linear_of_real]
hoelzl@51478
   627
hoelzl@56371
   628
lemmas continuous_on_scaleR [continuous_intros] =
hoelzl@51478
   629
  bounded_bilinear.continuous_on [OF bounded_bilinear_scaleR]
hoelzl@51478
   630
hoelzl@56371
   631
lemmas continuous_on_mult [continuous_intros] =
hoelzl@51478
   632
  bounded_bilinear.continuous_on [OF bounded_bilinear_mult]
hoelzl@51478
   633
huffman@44568
   634
lemmas tendsto_mult_zero =
huffman@44568
   635
  bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
huffman@44568
   636
huffman@44568
   637
lemmas tendsto_mult_left_zero =
huffman@44568
   638
  bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
huffman@44568
   639
huffman@44568
   640
lemmas tendsto_mult_right_zero =
huffman@44568
   641
  bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
huffman@44568
   642
huffman@44194
   643
lemma tendsto_power [tendsto_intros]:
huffman@44194
   644
  fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
huffman@44195
   645
  shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
hoelzl@58729
   646
  by (induct n) (simp_all add: tendsto_mult)
huffman@44194
   647
hoelzl@51478
   648
lemma continuous_power [continuous_intros]:
hoelzl@51478
   649
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   650
  shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. (f x)^n)"
hoelzl@51478
   651
  unfolding continuous_def by (rule tendsto_power)
hoelzl@51478
   652
hoelzl@56371
   653
lemma continuous_on_power [continuous_intros]:
hoelzl@51478
   654
  fixes f :: "_ \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51478
   655
  shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. (f x)^n)"
hoelzl@51478
   656
  unfolding continuous_on_def by (auto intro: tendsto_power)
hoelzl@51478
   657
huffman@44194
   658
lemma tendsto_setprod [tendsto_intros]:
huffman@44194
   659
  fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
huffman@44195
   660
  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
huffman@44195
   661
  shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
huffman@44194
   662
proof (cases "finite S")
huffman@44194
   663
  assume "finite S" thus ?thesis using assms
hoelzl@58729
   664
    by (induct, simp, simp add: tendsto_mult)
hoelzl@58729
   665
qed simp
huffman@44194
   666
hoelzl@51478
   667
lemma continuous_setprod [continuous_intros]:
hoelzl@51478
   668
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   669
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous F (f i)) \<Longrightarrow> continuous F (\<lambda>x. \<Prod>i\<in>S. f i x)"
hoelzl@51478
   670
  unfolding continuous_def by (rule tendsto_setprod)
hoelzl@51478
   671
hoelzl@51478
   672
lemma continuous_on_setprod [continuous_intros]:
hoelzl@51478
   673
  fixes f :: "'a \<Rightarrow> _ \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
hoelzl@51478
   674
  shows "(\<And>i. i \<in> S \<Longrightarrow> continuous_on s (f i)) \<Longrightarrow> continuous_on s (\<lambda>x. \<Prod>i\<in>S. f i x)"
hoelzl@51478
   675
  unfolding continuous_on_def by (auto intro: tendsto_setprod)
hoelzl@51478
   676
huffman@44194
   677
subsubsection {* Inverse and division *}
huffman@31355
   678
huffman@31355
   679
lemma (in bounded_bilinear) Zfun_prod_Bfun:
huffman@44195
   680
  assumes f: "Zfun f F"
huffman@44195
   681
  assumes g: "Bfun g F"
huffman@44195
   682
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@31355
   683
proof -
huffman@31355
   684
  obtain K where K: "0 \<le> K"
huffman@31355
   685
    and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
huffman@31355
   686
    using nonneg_bounded by fast
huffman@31355
   687
  obtain B where B: "0 < B"
huffman@44195
   688
    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) F"
huffman@31487
   689
    using g by (rule BfunE)
huffman@44195
   690
  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
noschinl@46887
   691
  using norm_g proof eventually_elim
noschinl@46887
   692
    case (elim x)
huffman@31487
   693
    have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
huffman@31355
   694
      by (rule norm_le)
huffman@31487
   695
    also have "\<dots> \<le> norm (f x) * B * K"
huffman@31487
   696
      by (intro mult_mono' order_refl norm_g norm_ge_zero
noschinl@46887
   697
                mult_nonneg_nonneg K elim)
huffman@31487
   698
    also have "\<dots> = norm (f x) * (B * K)"
haftmann@57512
   699
      by (rule mult.assoc)
huffman@31487
   700
    finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
huffman@31355
   701
  qed
huffman@31487
   702
  with f show ?thesis
huffman@31487
   703
    by (rule Zfun_imp_Zfun)
huffman@31355
   704
qed
huffman@31355
   705
huffman@31355
   706
lemma (in bounded_bilinear) flip:
huffman@31355
   707
  "bounded_bilinear (\<lambda>x y. y ** x)"
huffman@44081
   708
  apply default
huffman@44081
   709
  apply (rule add_right)
huffman@44081
   710
  apply (rule add_left)
huffman@44081
   711
  apply (rule scaleR_right)
huffman@44081
   712
  apply (rule scaleR_left)
haftmann@57512
   713
  apply (subst mult.commute)
huffman@44081
   714
  using bounded by fast
huffman@31355
   715
huffman@31355
   716
lemma (in bounded_bilinear) Bfun_prod_Zfun:
huffman@44195
   717
  assumes f: "Bfun f F"
huffman@44195
   718
  assumes g: "Zfun g F"
huffman@44195
   719
  shows "Zfun (\<lambda>x. f x ** g x) F"
huffman@44081
   720
  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
huffman@31355
   721
huffman@31355
   722
lemma Bfun_inverse_lemma:
huffman@31355
   723
  fixes x :: "'a::real_normed_div_algebra"
huffman@31355
   724
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
huffman@44081
   725
  apply (subst nonzero_norm_inverse, clarsimp)
huffman@44081
   726
  apply (erule (1) le_imp_inverse_le)
huffman@44081
   727
  done
huffman@31355
   728
huffman@31355
   729
lemma Bfun_inverse:
huffman@31355
   730
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   731
  assumes f: "(f ---> a) F"
huffman@31355
   732
  assumes a: "a \<noteq> 0"
huffman@44195
   733
  shows "Bfun (\<lambda>x. inverse (f x)) F"
huffman@31355
   734
proof -
huffman@31355
   735
  from a have "0 < norm a" by simp
huffman@31355
   736
  hence "\<exists>r>0. r < norm a" by (rule dense)
huffman@31355
   737
  then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
huffman@44195
   738
  have "eventually (\<lambda>x. dist (f x) a < r) F"
huffman@31487
   739
    using tendstoD [OF f r1] by fast
huffman@44195
   740
  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
noschinl@46887
   741
  proof eventually_elim
noschinl@46887
   742
    case (elim x)
huffman@31487
   743
    hence 1: "norm (f x - a) < r"
huffman@31355
   744
      by (simp add: dist_norm)
huffman@31487
   745
    hence 2: "f x \<noteq> 0" using r2 by auto
huffman@31487
   746
    hence "norm (inverse (f x)) = inverse (norm (f x))"
huffman@31355
   747
      by (rule nonzero_norm_inverse)
huffman@31355
   748
    also have "\<dots> \<le> inverse (norm a - r)"
huffman@31355
   749
    proof (rule le_imp_inverse_le)
huffman@31355
   750
      show "0 < norm a - r" using r2 by simp
huffman@31355
   751
    next
huffman@31487
   752
      have "norm a - norm (f x) \<le> norm (a - f x)"
huffman@31355
   753
        by (rule norm_triangle_ineq2)
huffman@31487
   754
      also have "\<dots> = norm (f x - a)"
huffman@31355
   755
        by (rule norm_minus_commute)
huffman@31355
   756
      also have "\<dots> < r" using 1 .
huffman@31487
   757
      finally show "norm a - r \<le> norm (f x)" by simp
huffman@31355
   758
    qed
huffman@31487
   759
    finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
huffman@31355
   760
  qed
huffman@31355
   761
  thus ?thesis by (rule BfunI)
huffman@31355
   762
qed
huffman@31355
   763
huffman@31565
   764
lemma tendsto_inverse [tendsto_intros]:
huffman@31355
   765
  fixes a :: "'a::real_normed_div_algebra"
huffman@44195
   766
  assumes f: "(f ---> a) F"
huffman@31355
   767
  assumes a: "a \<noteq> 0"
huffman@44195
   768
  shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
huffman@31355
   769
proof -
huffman@31355
   770
  from a have "0 < norm a" by simp
huffman@44195
   771
  with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
huffman@31355
   772
    by (rule tendstoD)
huffman@44195
   773
  then have "eventually (\<lambda>x. f x \<noteq> 0) F"
huffman@31355
   774
    unfolding dist_norm by (auto elim!: eventually_elim1)
huffman@44627
   775
  with a have "eventually (\<lambda>x. inverse (f x) - inverse a =
huffman@44627
   776
    - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   777
    by (auto elim!: eventually_elim1 simp: inverse_diff_inverse)
huffman@44627
   778
  moreover have "Zfun (\<lambda>x. - (inverse (f x) * (f x - a) * inverse a)) F"
huffman@44627
   779
    by (intro Zfun_minus Zfun_mult_left
huffman@44627
   780
      bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult]
huffman@44627
   781
      Bfun_inverse [OF f a] f [unfolded tendsto_Zfun_iff])
huffman@44627
   782
  ultimately show ?thesis
huffman@44627
   783
    unfolding tendsto_Zfun_iff by (rule Zfun_ssubst)
huffman@31355
   784
qed
huffman@31355
   785
hoelzl@51478
   786
lemma continuous_inverse:
hoelzl@51478
   787
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   788
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   789
  shows "continuous F (\<lambda>x. inverse (f x))"
hoelzl@51478
   790
  using assms unfolding continuous_def by (rule tendsto_inverse)
hoelzl@51478
   791
hoelzl@51478
   792
lemma continuous_at_within_inverse[continuous_intros]:
hoelzl@51478
   793
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   794
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   795
  shows "continuous (at a within s) (\<lambda>x. inverse (f x))"
hoelzl@51478
   796
  using assms unfolding continuous_within by (rule tendsto_inverse)
hoelzl@51478
   797
hoelzl@51478
   798
lemma isCont_inverse[continuous_intros, simp]:
hoelzl@51478
   799
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   800
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   801
  shows "isCont (\<lambda>x. inverse (f x)) a"
hoelzl@51478
   802
  using assms unfolding continuous_at by (rule tendsto_inverse)
hoelzl@51478
   803
hoelzl@56371
   804
lemma continuous_on_inverse[continuous_intros]:
hoelzl@51478
   805
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_div_algebra"
hoelzl@51478
   806
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   807
  shows "continuous_on s (\<lambda>x. inverse (f x))"
hoelzl@51478
   808
  using assms unfolding continuous_on_def by (fast intro: tendsto_inverse)
hoelzl@51478
   809
huffman@31565
   810
lemma tendsto_divide [tendsto_intros]:
huffman@31355
   811
  fixes a b :: "'a::real_normed_field"
huffman@44195
   812
  shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
huffman@44195
   813
    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
huffman@44282
   814
  by (simp add: tendsto_mult tendsto_inverse divide_inverse)
huffman@31355
   815
hoelzl@51478
   816
lemma continuous_divide:
hoelzl@51478
   817
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   818
  assumes "continuous F f" and "continuous F g" and "g (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   819
  shows "continuous F (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   820
  using assms unfolding continuous_def by (rule tendsto_divide)
hoelzl@51478
   821
hoelzl@51478
   822
lemma continuous_at_within_divide[continuous_intros]:
hoelzl@51478
   823
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   824
  assumes "continuous (at a within s) f" "continuous (at a within s) g" and "g a \<noteq> 0"
hoelzl@51478
   825
  shows "continuous (at a within s) (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   826
  using assms unfolding continuous_within by (rule tendsto_divide)
hoelzl@51478
   827
hoelzl@51478
   828
lemma isCont_divide[continuous_intros, simp]:
hoelzl@51478
   829
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   830
  assumes "isCont f a" "isCont g a" "g a \<noteq> 0"
hoelzl@51478
   831
  shows "isCont (\<lambda>x. (f x) / g x) a"
hoelzl@51478
   832
  using assms unfolding continuous_at by (rule tendsto_divide)
hoelzl@51478
   833
hoelzl@56371
   834
lemma continuous_on_divide[continuous_intros]:
hoelzl@51478
   835
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_field"
hoelzl@51478
   836
  assumes "continuous_on s f" "continuous_on s g" and "\<forall>x\<in>s. g x \<noteq> 0"
hoelzl@51478
   837
  shows "continuous_on s (\<lambda>x. (f x) / (g x))"
hoelzl@51478
   838
  using assms unfolding continuous_on_def by (fast intro: tendsto_divide)
hoelzl@51478
   839
huffman@44194
   840
lemma tendsto_sgn [tendsto_intros]:
huffman@44194
   841
  fixes l :: "'a::real_normed_vector"
huffman@44195
   842
  shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
huffman@44194
   843
  unfolding sgn_div_norm by (simp add: tendsto_intros)
huffman@44194
   844
hoelzl@51478
   845
lemma continuous_sgn:
hoelzl@51478
   846
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   847
  assumes "continuous F f" and "f (Lim F (\<lambda>x. x)) \<noteq> 0"
hoelzl@51478
   848
  shows "continuous F (\<lambda>x. sgn (f x))"
hoelzl@51478
   849
  using assms unfolding continuous_def by (rule tendsto_sgn)
hoelzl@51478
   850
hoelzl@51478
   851
lemma continuous_at_within_sgn[continuous_intros]:
hoelzl@51478
   852
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   853
  assumes "continuous (at a within s) f" and "f a \<noteq> 0"
hoelzl@51478
   854
  shows "continuous (at a within s) (\<lambda>x. sgn (f x))"
hoelzl@51478
   855
  using assms unfolding continuous_within by (rule tendsto_sgn)
hoelzl@51478
   856
hoelzl@51478
   857
lemma isCont_sgn[continuous_intros]:
hoelzl@51478
   858
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   859
  assumes "isCont f a" and "f a \<noteq> 0"
hoelzl@51478
   860
  shows "isCont (\<lambda>x. sgn (f x)) a"
hoelzl@51478
   861
  using assms unfolding continuous_at by (rule tendsto_sgn)
hoelzl@51478
   862
hoelzl@56371
   863
lemma continuous_on_sgn[continuous_intros]:
hoelzl@51478
   864
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51478
   865
  assumes "continuous_on s f" and "\<forall>x\<in>s. f x \<noteq> 0"
hoelzl@51478
   866
  shows "continuous_on s (\<lambda>x. sgn (f x))"
hoelzl@51478
   867
  using assms unfolding continuous_on_def by (fast intro: tendsto_sgn)
hoelzl@51478
   868
hoelzl@50325
   869
lemma filterlim_at_infinity:
hoelzl@50325
   870
  fixes f :: "_ \<Rightarrow> 'a\<Colon>real_normed_vector"
hoelzl@50325
   871
  assumes "0 \<le> c"
hoelzl@50325
   872
  shows "(LIM x F. f x :> at_infinity) \<longleftrightarrow> (\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F)"
hoelzl@50325
   873
  unfolding filterlim_iff eventually_at_infinity
hoelzl@50325
   874
proof safe
hoelzl@50325
   875
  fix P :: "'a \<Rightarrow> bool" and b
hoelzl@50325
   876
  assume *: "\<forall>r>c. eventually (\<lambda>x. r \<le> norm (f x)) F"
hoelzl@50325
   877
    and P: "\<forall>x. b \<le> norm x \<longrightarrow> P x"
hoelzl@50325
   878
  have "max b (c + 1) > c" by auto
hoelzl@50325
   879
  with * have "eventually (\<lambda>x. max b (c + 1) \<le> norm (f x)) F"
hoelzl@50325
   880
    by auto
hoelzl@50325
   881
  then show "eventually (\<lambda>x. P (f x)) F"
hoelzl@50325
   882
  proof eventually_elim
hoelzl@50325
   883
    fix x assume "max b (c + 1) \<le> norm (f x)"
hoelzl@50325
   884
    with P show "P (f x)" by auto
hoelzl@50325
   885
  qed
hoelzl@50325
   886
qed force
hoelzl@50325
   887
hoelzl@51529
   888
hoelzl@50347
   889
subsection {* Relate @{const at}, @{const at_left} and @{const at_right} *}
hoelzl@50347
   890
hoelzl@50347
   891
text {*
hoelzl@50347
   892
hoelzl@50347
   893
This lemmas are useful for conversion between @{term "at x"} to @{term "at_left x"} and
hoelzl@50347
   894
@{term "at_right x"} and also @{term "at_right 0"}.
hoelzl@50347
   895
hoelzl@50347
   896
*}
hoelzl@50347
   897
hoelzl@51471
   898
lemmas filterlim_split_at_real = filterlim_split_at[where 'a=real]
hoelzl@50323
   899
hoelzl@51641
   900
lemma filtermap_homeomorph:
hoelzl@51641
   901
  assumes f: "continuous (at a) f"
hoelzl@51641
   902
  assumes g: "continuous (at (f a)) g"
hoelzl@51641
   903
  assumes bij1: "\<forall>x. f (g x) = x" and bij2: "\<forall>x. g (f x) = x"
hoelzl@51641
   904
  shows "filtermap f (nhds a) = nhds (f a)"
hoelzl@51641
   905
  unfolding filter_eq_iff eventually_filtermap eventually_nhds
hoelzl@51641
   906
proof safe
hoelzl@51641
   907
  fix P S assume S: "open S" "f a \<in> S" and P: "\<forall>x\<in>S. P x"
hoelzl@51641
   908
  from continuous_within_topological[THEN iffD1, rule_format, OF f S] P
hoelzl@51641
   909
  show "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P (f x))" by auto
hoelzl@51641
   910
next
hoelzl@51641
   911
  fix P S assume S: "open S" "a \<in> S" and P: "\<forall>x\<in>S. P (f x)"
hoelzl@51641
   912
  with continuous_within_topological[THEN iffD1, rule_format, OF g, of S] bij2
hoelzl@51641
   913
  obtain A where "open A" "f a \<in> A" "(\<forall>y\<in>A. g y \<in> S)"
hoelzl@51641
   914
    by (metis UNIV_I)
hoelzl@51641
   915
  with P bij1 show "\<exists>S. open S \<and> f a \<in> S \<and> (\<forall>x\<in>S. P x)"
hoelzl@51641
   916
    by (force intro!: exI[of _ A])
hoelzl@51641
   917
qed
hoelzl@50347
   918
hoelzl@51641
   919
lemma filtermap_nhds_shift: "filtermap (\<lambda>x. x - d) (nhds a) = nhds (a - d::'a::real_normed_vector)"
hoelzl@51641
   920
  by (rule filtermap_homeomorph[where g="\<lambda>x. x + d"]) (auto intro: continuous_intros)
hoelzl@50347
   921
hoelzl@51641
   922
lemma filtermap_nhds_minus: "filtermap (\<lambda>x. - x) (nhds a) = nhds (- a::'a::real_normed_vector)"
hoelzl@51641
   923
  by (rule filtermap_homeomorph[where g=uminus]) (auto intro: continuous_minus)
hoelzl@51641
   924
hoelzl@51641
   925
lemma filtermap_at_shift: "filtermap (\<lambda>x. x - d) (at a) = at (a - d::'a::real_normed_vector)"
hoelzl@51641
   926
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50347
   927
hoelzl@50347
   928
lemma filtermap_at_right_shift: "filtermap (\<lambda>x. x - d) (at_right a) = at_right (a - d::real)"
hoelzl@51641
   929
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_shift[symmetric])
hoelzl@50323
   930
hoelzl@50347
   931
lemma at_right_to_0: "at_right (a::real) = filtermap (\<lambda>x. x + a) (at_right 0)"
hoelzl@50347
   932
  using filtermap_at_right_shift[of "-a" 0] by simp
hoelzl@50347
   933
hoelzl@50347
   934
lemma filterlim_at_right_to_0:
hoelzl@50347
   935
  "filterlim f F (at_right (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (x + a)) F (at_right 0)"
hoelzl@50347
   936
  unfolding filterlim_def filtermap_filtermap at_right_to_0[of a] ..
hoelzl@50347
   937
hoelzl@50347
   938
lemma eventually_at_right_to_0:
hoelzl@50347
   939
  "eventually P (at_right (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (x + a)) (at_right 0)"
hoelzl@50347
   940
  unfolding at_right_to_0[of a] by (simp add: eventually_filtermap)
hoelzl@50347
   941
hoelzl@51641
   942
lemma filtermap_at_minus: "filtermap (\<lambda>x. - x) (at a) = at (- a::'a::real_normed_vector)"
hoelzl@51641
   943
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
   944
hoelzl@50347
   945
lemma at_left_minus: "at_left (a::real) = filtermap (\<lambda>x. - x) (at_right (- a))"
hoelzl@51641
   946
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50323
   947
hoelzl@50347
   948
lemma at_right_minus: "at_right (a::real) = filtermap (\<lambda>x. - x) (at_left (- a))"
hoelzl@51641
   949
  by (simp add: filter_eq_iff eventually_filtermap eventually_at_filter filtermap_nhds_minus[symmetric])
hoelzl@50347
   950
hoelzl@50347
   951
lemma filterlim_at_left_to_right:
hoelzl@50347
   952
  "filterlim f F (at_left (a::real)) \<longleftrightarrow> filterlim (\<lambda>x. f (- x)) F (at_right (-a))"
hoelzl@50347
   953
  unfolding filterlim_def filtermap_filtermap at_left_minus[of a] ..
hoelzl@50347
   954
hoelzl@50347
   955
lemma eventually_at_left_to_right:
hoelzl@50347
   956
  "eventually P (at_left (a::real)) \<longleftrightarrow> eventually (\<lambda>x. P (- x)) (at_right (-a))"
hoelzl@50347
   957
  unfolding at_left_minus[of a] by (simp add: eventually_filtermap)
hoelzl@50347
   958
hoelzl@50346
   959
lemma at_top_mirror: "at_top = filtermap uminus (at_bot :: real filter)"
hoelzl@50346
   960
  unfolding filter_eq_iff eventually_filtermap eventually_at_top_linorder eventually_at_bot_linorder
hoelzl@50346
   961
  by (metis le_minus_iff minus_minus)
hoelzl@50346
   962
hoelzl@50346
   963
lemma at_bot_mirror: "at_bot = filtermap uminus (at_top :: real filter)"
hoelzl@50346
   964
  unfolding at_top_mirror filtermap_filtermap by (simp add: filtermap_ident)
hoelzl@50346
   965
hoelzl@50346
   966
lemma filterlim_at_top_mirror: "(LIM x at_top. f x :> F) \<longleftrightarrow> (LIM x at_bot. f (-x::real) :> F)"
hoelzl@50346
   967
  unfolding filterlim_def at_top_mirror filtermap_filtermap ..
hoelzl@50346
   968
hoelzl@50346
   969
lemma filterlim_at_bot_mirror: "(LIM x at_bot. f x :> F) \<longleftrightarrow> (LIM x at_top. f (-x::real) :> F)"
hoelzl@50346
   970
  unfolding filterlim_def at_bot_mirror filtermap_filtermap ..
hoelzl@50346
   971
hoelzl@50323
   972
lemma filterlim_uminus_at_top_at_bot: "LIM x at_bot. - x :: real :> at_top"
hoelzl@50323
   973
  unfolding filterlim_at_top eventually_at_bot_dense
hoelzl@50346
   974
  by (metis leI minus_less_iff order_less_asym)
hoelzl@50323
   975
hoelzl@50323
   976
lemma filterlim_uminus_at_bot_at_top: "LIM x at_top. - x :: real :> at_bot"
hoelzl@50323
   977
  unfolding filterlim_at_bot eventually_at_top_dense
hoelzl@50346
   978
  by (metis leI less_minus_iff order_less_asym)
hoelzl@50323
   979
hoelzl@50346
   980
lemma filterlim_uminus_at_top: "(LIM x F. f x :> at_top) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_bot)"
hoelzl@50346
   981
  using filterlim_compose[OF filterlim_uminus_at_bot_at_top, of f F]
hoelzl@50346
   982
  using filterlim_compose[OF filterlim_uminus_at_top_at_bot, of "\<lambda>x. - f x" F]
hoelzl@50346
   983
  by auto
hoelzl@50346
   984
hoelzl@50346
   985
lemma filterlim_uminus_at_bot: "(LIM x F. f x :> at_bot) \<longleftrightarrow> (LIM x F. - (f x) :: real :> at_top)"
hoelzl@50346
   986
  unfolding filterlim_uminus_at_top by simp
hoelzl@50323
   987
hoelzl@50347
   988
lemma filterlim_inverse_at_top_right: "LIM x at_right (0::real). inverse x :> at_top"
hoelzl@51641
   989
  unfolding filterlim_at_top_gt[where c=0] eventually_at_filter
hoelzl@50347
   990
proof safe
hoelzl@50347
   991
  fix Z :: real assume [arith]: "0 < Z"
hoelzl@50347
   992
  then have "eventually (\<lambda>x. x < inverse Z) (nhds 0)"
hoelzl@50347
   993
    by (auto simp add: eventually_nhds_metric dist_real_def intro!: exI[of _ "\<bar>inverse Z\<bar>"])
hoelzl@51641
   994
  then show "eventually (\<lambda>x. x \<noteq> 0 \<longrightarrow> x \<in> {0<..} \<longrightarrow> Z \<le> inverse x) (nhds 0)"
hoelzl@50347
   995
    by (auto elim!: eventually_elim1 simp: inverse_eq_divide field_simps)
hoelzl@50347
   996
qed
hoelzl@50347
   997
hoelzl@50347
   998
lemma filterlim_inverse_at_top:
hoelzl@50347
   999
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. 0 < f x) F \<Longrightarrow> LIM x F. inverse (f x) :> at_top"
hoelzl@50347
  1000
  by (intro filterlim_compose[OF filterlim_inverse_at_top_right])
hoelzl@51641
  1001
     (simp add: filterlim_def eventually_filtermap eventually_elim1 at_within_def le_principal)
hoelzl@50347
  1002
hoelzl@50347
  1003
lemma filterlim_inverse_at_bot_neg:
hoelzl@50347
  1004
  "LIM x (at_left (0::real)). inverse x :> at_bot"
hoelzl@50347
  1005
  by (simp add: filterlim_inverse_at_top_right filterlim_uminus_at_bot filterlim_at_left_to_right)
hoelzl@50347
  1006
hoelzl@50347
  1007
lemma filterlim_inverse_at_bot:
hoelzl@50347
  1008
  "(f ---> (0 :: real)) F \<Longrightarrow> eventually (\<lambda>x. f x < 0) F \<Longrightarrow> LIM x F. inverse (f x) :> at_bot"
hoelzl@50347
  1009
  unfolding filterlim_uminus_at_bot inverse_minus_eq[symmetric]
hoelzl@50347
  1010
  by (rule filterlim_inverse_at_top) (simp_all add: tendsto_minus_cancel_left[symmetric])
hoelzl@50347
  1011
hoelzl@50325
  1012
lemma tendsto_inverse_0:
hoelzl@50325
  1013
  fixes x :: "_ \<Rightarrow> 'a\<Colon>real_normed_div_algebra"
hoelzl@50325
  1014
  shows "(inverse ---> (0::'a)) at_infinity"
hoelzl@50325
  1015
  unfolding tendsto_Zfun_iff diff_0_right Zfun_def eventually_at_infinity
hoelzl@50325
  1016
proof safe
hoelzl@50325
  1017
  fix r :: real assume "0 < r"
hoelzl@50325
  1018
  show "\<exists>b. \<forall>x. b \<le> norm x \<longrightarrow> norm (inverse x :: 'a) < r"
hoelzl@50325
  1019
  proof (intro exI[of _ "inverse (r / 2)"] allI impI)
hoelzl@50325
  1020
    fix x :: 'a
hoelzl@50325
  1021
    from `0 < r` have "0 < inverse (r / 2)" by simp
hoelzl@50325
  1022
    also assume *: "inverse (r / 2) \<le> norm x"
hoelzl@50325
  1023
    finally show "norm (inverse x) < r"
hoelzl@50325
  1024
      using * `0 < r` by (subst nonzero_norm_inverse) (simp_all add: inverse_eq_divide field_simps)
hoelzl@50325
  1025
  qed
hoelzl@50325
  1026
qed
hoelzl@50325
  1027
hoelzl@50347
  1028
lemma at_right_to_top: "(at_right (0::real)) = filtermap inverse at_top"
hoelzl@50347
  1029
proof (rule antisym)
hoelzl@50347
  1030
  have "(inverse ---> (0::real)) at_top"
hoelzl@50347
  1031
    by (metis tendsto_inverse_0 filterlim_mono at_top_le_at_infinity order_refl)
hoelzl@50347
  1032
  then show "filtermap inverse at_top \<le> at_right (0::real)"
hoelzl@51641
  1033
    by (simp add: le_principal eventually_filtermap eventually_gt_at_top filterlim_def at_within_def)
hoelzl@50347
  1034
next
hoelzl@50347
  1035
  have "filtermap inverse (filtermap inverse (at_right (0::real))) \<le> filtermap inverse at_top"
hoelzl@50347
  1036
    using filterlim_inverse_at_top_right unfolding filterlim_def by (rule filtermap_mono)
hoelzl@50347
  1037
  then show "at_right (0::real) \<le> filtermap inverse at_top"
hoelzl@50347
  1038
    by (simp add: filtermap_ident filtermap_filtermap)
hoelzl@50347
  1039
qed
hoelzl@50347
  1040
hoelzl@50347
  1041
lemma eventually_at_right_to_top:
hoelzl@50347
  1042
  "eventually P (at_right (0::real)) \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) at_top"
hoelzl@50347
  1043
  unfolding at_right_to_top eventually_filtermap ..
hoelzl@50347
  1044
hoelzl@50347
  1045
lemma filterlim_at_right_to_top:
hoelzl@50347
  1046
  "filterlim f F (at_right (0::real)) \<longleftrightarrow> (LIM x at_top. f (inverse x) :> F)"
hoelzl@50347
  1047
  unfolding filterlim_def at_right_to_top filtermap_filtermap ..
hoelzl@50347
  1048
hoelzl@50347
  1049
lemma at_top_to_right: "at_top = filtermap inverse (at_right (0::real))"
hoelzl@50347
  1050
  unfolding at_right_to_top filtermap_filtermap inverse_inverse_eq filtermap_ident ..
hoelzl@50347
  1051
hoelzl@50347
  1052
lemma eventually_at_top_to_right:
hoelzl@50347
  1053
  "eventually P at_top \<longleftrightarrow> eventually (\<lambda>x. P (inverse x)) (at_right (0::real))"
hoelzl@50347
  1054
  unfolding at_top_to_right eventually_filtermap ..
hoelzl@50347
  1055
hoelzl@50347
  1056
lemma filterlim_at_top_to_right:
hoelzl@50347
  1057
  "filterlim f F at_top \<longleftrightarrow> (LIM x (at_right (0::real)). f (inverse x) :> F)"
hoelzl@50347
  1058
  unfolding filterlim_def at_top_to_right filtermap_filtermap ..
hoelzl@50347
  1059
hoelzl@50325
  1060
lemma filterlim_inverse_at_infinity:
hoelzl@50325
  1061
  fixes x :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
hoelzl@50325
  1062
  shows "filterlim inverse at_infinity (at (0::'a))"
hoelzl@50325
  1063
  unfolding filterlim_at_infinity[OF order_refl]
hoelzl@50325
  1064
proof safe
hoelzl@50325
  1065
  fix r :: real assume "0 < r"
hoelzl@50325
  1066
  then show "eventually (\<lambda>x::'a. r \<le> norm (inverse x)) (at 0)"
hoelzl@50325
  1067
    unfolding eventually_at norm_inverse
hoelzl@50325
  1068
    by (intro exI[of _ "inverse r"])
hoelzl@50325
  1069
       (auto simp: norm_conv_dist[symmetric] field_simps inverse_eq_divide)
hoelzl@50325
  1070
qed
hoelzl@50325
  1071
hoelzl@50325
  1072
lemma filterlim_inverse_at_iff:
hoelzl@50325
  1073
  fixes g :: "'a \<Rightarrow> 'b\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
hoelzl@50325
  1074
  shows "(LIM x F. inverse (g x) :> at 0) \<longleftrightarrow> (LIM x F. g x :> at_infinity)"
hoelzl@50325
  1075
  unfolding filterlim_def filtermap_filtermap[symmetric]
hoelzl@50325
  1076
proof
hoelzl@50325
  1077
  assume "filtermap g F \<le> at_infinity"
hoelzl@50325
  1078
  then have "filtermap inverse (filtermap g F) \<le> filtermap inverse at_infinity"
hoelzl@50325
  1079
    by (rule filtermap_mono)
hoelzl@50325
  1080
  also have "\<dots> \<le> at 0"
hoelzl@51641
  1081
    using tendsto_inverse_0[where 'a='b]
hoelzl@51641
  1082
    by (auto intro!: exI[of _ 1]
hoelzl@51641
  1083
             simp: le_principal eventually_filtermap filterlim_def at_within_def eventually_at_infinity)
hoelzl@50325
  1084
  finally show "filtermap inverse (filtermap g F) \<le> at 0" .
hoelzl@50325
  1085
next
hoelzl@50325
  1086
  assume "filtermap inverse (filtermap g F) \<le> at 0"
hoelzl@50325
  1087
  then have "filtermap inverse (filtermap inverse (filtermap g F)) \<le> filtermap inverse (at 0)"
hoelzl@50325
  1088
    by (rule filtermap_mono)
hoelzl@50325
  1089
  with filterlim_inverse_at_infinity show "filtermap g F \<le> at_infinity"
hoelzl@50325
  1090
    by (auto intro: order_trans simp: filterlim_def filtermap_filtermap)
hoelzl@50325
  1091
qed
hoelzl@50325
  1092
hoelzl@51641
  1093
lemma tendsto_inverse_0_at_top: "LIM x F. f x :> at_top \<Longrightarrow> ((\<lambda>x. inverse (f x) :: real) ---> 0) F"
hoelzl@51641
  1094
 by (metis filterlim_at filterlim_mono[OF _ at_top_le_at_infinity order_refl] filterlim_inverse_at_iff)
hoelzl@50419
  1095
hoelzl@50324
  1096
text {*
hoelzl@50324
  1097
hoelzl@50324
  1098
We only show rules for multiplication and addition when the functions are either against a real
hoelzl@50324
  1099
value or against infinity. Further rules are easy to derive by using @{thm filterlim_uminus_at_top}.
hoelzl@50324
  1100
hoelzl@50324
  1101
*}
hoelzl@50324
  1102
hoelzl@50324
  1103
lemma filterlim_tendsto_pos_mult_at_top: 
hoelzl@50324
  1104
  assumes f: "(f ---> c) F" and c: "0 < c"
hoelzl@50324
  1105
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1106
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1107
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1108
proof safe
hoelzl@50324
  1109
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1110
  from f `0 < c` have "eventually (\<lambda>x. c / 2 < f x) F"
hoelzl@50324
  1111
    by (auto dest!: tendstoD[where e="c / 2"] elim!: eventually_elim1
hoelzl@50324
  1112
             simp: dist_real_def abs_real_def split: split_if_asm)
hoelzl@50346
  1113
  moreover from g have "eventually (\<lambda>x. (Z / c * 2) \<le> g x) F"
hoelzl@50324
  1114
    unfolding filterlim_at_top by auto
hoelzl@50346
  1115
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1116
  proof eventually_elim
hoelzl@50346
  1117
    fix x assume "c / 2 < f x" "Z / c * 2 \<le> g x"
hoelzl@50346
  1118
    with `0 < Z` `0 < c` have "c / 2 * (Z / c * 2) \<le> f x * g x"
hoelzl@50346
  1119
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
  1120
    with `0 < c` show "Z \<le> f x * g x"
hoelzl@50324
  1121
       by simp
hoelzl@50324
  1122
  qed
hoelzl@50324
  1123
qed
hoelzl@50324
  1124
hoelzl@50324
  1125
lemma filterlim_at_top_mult_at_top: 
hoelzl@50324
  1126
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1127
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1128
  shows "LIM x F. (f x * g x :: real) :> at_top"
hoelzl@50324
  1129
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1130
proof safe
hoelzl@50324
  1131
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1132
  from f have "eventually (\<lambda>x. 1 \<le> f x) F"
hoelzl@50324
  1133
    unfolding filterlim_at_top by auto
hoelzl@50346
  1134
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1135
    unfolding filterlim_at_top by auto
hoelzl@50346
  1136
  ultimately show "eventually (\<lambda>x. Z \<le> f x * g x) F"
hoelzl@50324
  1137
  proof eventually_elim
hoelzl@50346
  1138
    fix x assume "1 \<le> f x" "Z \<le> g x"
hoelzl@50346
  1139
    with `0 < Z` have "1 * Z \<le> f x * g x"
hoelzl@50346
  1140
      by (intro mult_mono) (auto simp: zero_le_divide_iff)
hoelzl@50346
  1141
    then show "Z \<le> f x * g x"
hoelzl@50324
  1142
       by simp
hoelzl@50324
  1143
  qed
hoelzl@50324
  1144
qed
hoelzl@50324
  1145
hoelzl@50419
  1146
lemma filterlim_tendsto_pos_mult_at_bot:
hoelzl@50419
  1147
  assumes "(f ---> c) F" "0 < (c::real)" "filterlim g at_bot F"
hoelzl@50419
  1148
  shows "LIM x F. f x * g x :> at_bot"
hoelzl@50419
  1149
  using filterlim_tendsto_pos_mult_at_top[OF assms(1,2), of "\<lambda>x. - g x"] assms(3)
hoelzl@50419
  1150
  unfolding filterlim_uminus_at_bot by simp
hoelzl@50419
  1151
hoelzl@56330
  1152
lemma filterlim_pow_at_top:
hoelzl@56330
  1153
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1154
  assumes "0 < n" and f: "LIM x F. f x :> at_top"
hoelzl@56330
  1155
  shows "LIM x F. (f x)^n :: real :> at_top"
hoelzl@56330
  1156
using `0 < n` proof (induct n)
hoelzl@56330
  1157
  case (Suc n) with f show ?case
hoelzl@56330
  1158
    by (cases "n = 0") (auto intro!: filterlim_at_top_mult_at_top)
hoelzl@56330
  1159
qed simp
hoelzl@56330
  1160
hoelzl@56330
  1161
lemma filterlim_pow_at_bot_even:
hoelzl@56330
  1162
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1163
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> even n \<Longrightarrow> LIM x F. (f x)^n :> at_top"
hoelzl@56330
  1164
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_top)
hoelzl@56330
  1165
hoelzl@56330
  1166
lemma filterlim_pow_at_bot_odd:
hoelzl@56330
  1167
  fixes f :: "real \<Rightarrow> real"
hoelzl@56330
  1168
  shows "0 < n \<Longrightarrow> LIM x F. f x :> at_bot \<Longrightarrow> odd n \<Longrightarrow> LIM x F. (f x)^n :> at_bot"
hoelzl@56330
  1169
  using filterlim_pow_at_top[of n "\<lambda>x. - f x" F] by (simp add: filterlim_uminus_at_bot)
hoelzl@56330
  1170
hoelzl@50324
  1171
lemma filterlim_tendsto_add_at_top: 
hoelzl@50324
  1172
  assumes f: "(f ---> c) F"
hoelzl@50324
  1173
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1174
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1175
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1176
proof safe
hoelzl@50324
  1177
  fix Z :: real assume "0 < Z"
hoelzl@50324
  1178
  from f have "eventually (\<lambda>x. c - 1 < f x) F"
hoelzl@50324
  1179
    by (auto dest!: tendstoD[where e=1] elim!: eventually_elim1 simp: dist_real_def)
hoelzl@50346
  1180
  moreover from g have "eventually (\<lambda>x. Z - (c - 1) \<le> g x) F"
hoelzl@50324
  1181
    unfolding filterlim_at_top by auto
hoelzl@50346
  1182
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1183
    by eventually_elim simp
hoelzl@50324
  1184
qed
hoelzl@50324
  1185
hoelzl@50347
  1186
lemma LIM_at_top_divide:
hoelzl@50347
  1187
  fixes f g :: "'a \<Rightarrow> real"
hoelzl@50347
  1188
  assumes f: "(f ---> a) F" "0 < a"
hoelzl@50347
  1189
  assumes g: "(g ---> 0) F" "eventually (\<lambda>x. 0 < g x) F"
hoelzl@50347
  1190
  shows "LIM x F. f x / g x :> at_top"
hoelzl@50347
  1191
  unfolding divide_inverse
hoelzl@50347
  1192
  by (rule filterlim_tendsto_pos_mult_at_top[OF f]) (rule filterlim_inverse_at_top[OF g])
hoelzl@50347
  1193
hoelzl@50324
  1194
lemma filterlim_at_top_add_at_top: 
hoelzl@50324
  1195
  assumes f: "LIM x F. f x :> at_top"
hoelzl@50324
  1196
  assumes g: "LIM x F. g x :> at_top"
hoelzl@50324
  1197
  shows "LIM x F. (f x + g x :: real) :> at_top"
hoelzl@50324
  1198
  unfolding filterlim_at_top_gt[where c=0]
hoelzl@50324
  1199
proof safe
hoelzl@50324
  1200
  fix Z :: real assume "0 < Z"
hoelzl@50346
  1201
  from f have "eventually (\<lambda>x. 0 \<le> f x) F"
hoelzl@50324
  1202
    unfolding filterlim_at_top by auto
hoelzl@50346
  1203
  moreover from g have "eventually (\<lambda>x. Z \<le> g x) F"
hoelzl@50324
  1204
    unfolding filterlim_at_top by auto
hoelzl@50346
  1205
  ultimately show "eventually (\<lambda>x. Z \<le> f x + g x) F"
hoelzl@50324
  1206
    by eventually_elim simp
hoelzl@50324
  1207
qed
hoelzl@50324
  1208
hoelzl@50331
  1209
lemma tendsto_divide_0:
hoelzl@50331
  1210
  fixes f :: "_ \<Rightarrow> 'a\<Colon>{real_normed_div_algebra, division_ring_inverse_zero}"
hoelzl@50331
  1211
  assumes f: "(f ---> c) F"
hoelzl@50331
  1212
  assumes g: "LIM x F. g x :> at_infinity"
hoelzl@50331
  1213
  shows "((\<lambda>x. f x / g x) ---> 0) F"
hoelzl@50331
  1214
  using tendsto_mult[OF f filterlim_compose[OF tendsto_inverse_0 g]] by (simp add: divide_inverse)
hoelzl@50331
  1215
hoelzl@50331
  1216
lemma linear_plus_1_le_power:
hoelzl@50331
  1217
  fixes x :: real
hoelzl@50331
  1218
  assumes x: "0 \<le> x"
hoelzl@50331
  1219
  shows "real n * x + 1 \<le> (x + 1) ^ n"
hoelzl@50331
  1220
proof (induct n)
hoelzl@50331
  1221
  case (Suc n)
hoelzl@50331
  1222
  have "real (Suc n) * x + 1 \<le> (x + 1) * (real n * x + 1)"
nipkow@56536
  1223
    by (simp add: field_simps real_of_nat_Suc x)
hoelzl@50331
  1224
  also have "\<dots> \<le> (x + 1)^Suc n"
hoelzl@50331
  1225
    using Suc x by (simp add: mult_left_mono)
hoelzl@50331
  1226
  finally show ?case .
hoelzl@50331
  1227
qed simp
hoelzl@50331
  1228
hoelzl@50331
  1229
lemma filterlim_realpow_sequentially_gt1:
hoelzl@50331
  1230
  fixes x :: "'a :: real_normed_div_algebra"
hoelzl@50331
  1231
  assumes x[arith]: "1 < norm x"
hoelzl@50331
  1232
  shows "LIM n sequentially. x ^ n :> at_infinity"
hoelzl@50331
  1233
proof (intro filterlim_at_infinity[THEN iffD2] allI impI)
hoelzl@50331
  1234
  fix y :: real assume "0 < y"
hoelzl@50331
  1235
  have "0 < norm x - 1" by simp
hoelzl@50331
  1236
  then obtain N::nat where "y < real N * (norm x - 1)" by (blast dest: reals_Archimedean3)
hoelzl@50331
  1237
  also have "\<dots> \<le> real N * (norm x - 1) + 1" by simp
hoelzl@50331
  1238
  also have "\<dots> \<le> (norm x - 1 + 1) ^ N" by (rule linear_plus_1_le_power) simp
hoelzl@50331
  1239
  also have "\<dots> = norm x ^ N" by simp
hoelzl@50331
  1240
  finally have "\<forall>n\<ge>N. y \<le> norm x ^ n"
hoelzl@50331
  1241
    by (metis order_less_le_trans power_increasing order_less_imp_le x)
hoelzl@50331
  1242
  then show "eventually (\<lambda>n. y \<le> norm (x ^ n)) sequentially"
hoelzl@50331
  1243
    unfolding eventually_sequentially
hoelzl@50331
  1244
    by (auto simp: norm_power)
hoelzl@50331
  1245
qed simp
hoelzl@50331
  1246
hoelzl@51471
  1247
hoelzl@51526
  1248
subsection {* Limits of Sequences *}
hoelzl@51526
  1249
hoelzl@51526
  1250
lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
hoelzl@51526
  1251
  by simp
hoelzl@51526
  1252
hoelzl@51526
  1253
lemma LIMSEQ_iff:
hoelzl@51526
  1254
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1255
  shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
hoelzl@51526
  1256
unfolding LIMSEQ_def dist_norm ..
hoelzl@51526
  1257
hoelzl@51526
  1258
lemma LIMSEQ_I:
hoelzl@51526
  1259
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1260
  shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
hoelzl@51526
  1261
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1262
hoelzl@51526
  1263
lemma LIMSEQ_D:
hoelzl@51526
  1264
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1265
  shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
hoelzl@51526
  1266
by (simp add: LIMSEQ_iff)
hoelzl@51526
  1267
hoelzl@51526
  1268
lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
hoelzl@51526
  1269
  unfolding tendsto_def eventually_sequentially
haftmann@57512
  1270
  by (metis div_le_dividend div_mult_self1_is_m le_trans mult.commute)
hoelzl@51526
  1271
hoelzl@51526
  1272
lemma Bseq_inverse_lemma:
hoelzl@51526
  1273
  fixes x :: "'a::real_normed_div_algebra"
hoelzl@51526
  1274
  shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
hoelzl@51526
  1275
apply (subst nonzero_norm_inverse, clarsimp)
hoelzl@51526
  1276
apply (erule (1) le_imp_inverse_le)
hoelzl@51526
  1277
done
hoelzl@51526
  1278
hoelzl@51526
  1279
lemma Bseq_inverse:
hoelzl@51526
  1280
  fixes a :: "'a::real_normed_div_algebra"
hoelzl@51526
  1281
  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
hoelzl@51526
  1282
  by (rule Bfun_inverse)
hoelzl@51526
  1283
hoelzl@51526
  1284
lemma LIMSEQ_diff_approach_zero:
hoelzl@51526
  1285
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1286
  shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
hoelzl@51526
  1287
  by (drule (1) tendsto_add, simp)
hoelzl@51526
  1288
hoelzl@51526
  1289
lemma LIMSEQ_diff_approach_zero2:
hoelzl@51526
  1290
  fixes L :: "'a::real_normed_vector"
hoelzl@51526
  1291
  shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
hoelzl@51526
  1292
  by (drule (1) tendsto_diff, simp)
hoelzl@51526
  1293
hoelzl@51526
  1294
text{*An unbounded sequence's inverse tends to 0*}
hoelzl@51526
  1295
hoelzl@51526
  1296
lemma LIMSEQ_inverse_zero:
hoelzl@51526
  1297
  "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
hoelzl@51526
  1298
  apply (rule filterlim_compose[OF tendsto_inverse_0])
hoelzl@51526
  1299
  apply (simp add: filterlim_at_infinity[OF order_refl] eventually_sequentially)
hoelzl@51526
  1300
  apply (metis abs_le_D1 linorder_le_cases linorder_not_le)
hoelzl@51526
  1301
  done
hoelzl@51526
  1302
hoelzl@51526
  1303
text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
hoelzl@51526
  1304
hoelzl@51526
  1305
lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
hoelzl@51526
  1306
  by (metis filterlim_compose tendsto_inverse_0 filterlim_mono order_refl filterlim_Suc
hoelzl@51526
  1307
            filterlim_compose[OF filterlim_real_sequentially] at_top_le_at_infinity)
hoelzl@51526
  1308
hoelzl@51526
  1309
text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
hoelzl@51526
  1310
infinity is now easily proved*}
hoelzl@51526
  1311
hoelzl@51526
  1312
lemma LIMSEQ_inverse_real_of_nat_add:
hoelzl@51526
  1313
     "(%n. r + inverse(real(Suc n))) ----> r"
hoelzl@51526
  1314
  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
hoelzl@51526
  1315
hoelzl@51526
  1316
lemma LIMSEQ_inverse_real_of_nat_add_minus:
hoelzl@51526
  1317
     "(%n. r + -inverse(real(Suc n))) ----> r"
hoelzl@51526
  1318
  using tendsto_add [OF tendsto_const tendsto_minus [OF LIMSEQ_inverse_real_of_nat]]
hoelzl@51526
  1319
  by auto
hoelzl@51526
  1320
hoelzl@51526
  1321
lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
hoelzl@51526
  1322
     "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
hoelzl@51526
  1323
  using tendsto_mult [OF tendsto_const LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
hoelzl@51526
  1324
  by auto
hoelzl@51526
  1325
hoelzl@51526
  1326
subsection {* Convergence on sequences *}
hoelzl@51526
  1327
hoelzl@51526
  1328
lemma convergent_add:
hoelzl@51526
  1329
  fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1330
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1331
  assumes "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1332
  shows "convergent (\<lambda>n. X n + Y n)"
hoelzl@51526
  1333
  using assms unfolding convergent_def by (fast intro: tendsto_add)
hoelzl@51526
  1334
hoelzl@51526
  1335
lemma convergent_setsum:
hoelzl@51526
  1336
  fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1337
  assumes "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
hoelzl@51526
  1338
  shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
hoelzl@51526
  1339
proof (cases "finite A")
hoelzl@51526
  1340
  case True from this and assms show ?thesis
hoelzl@51526
  1341
    by (induct A set: finite) (simp_all add: convergent_const convergent_add)
hoelzl@51526
  1342
qed (simp add: convergent_const)
hoelzl@51526
  1343
hoelzl@51526
  1344
lemma (in bounded_linear) convergent:
hoelzl@51526
  1345
  assumes "convergent (\<lambda>n. X n)"
hoelzl@51526
  1346
  shows "convergent (\<lambda>n. f (X n))"
hoelzl@51526
  1347
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1348
hoelzl@51526
  1349
lemma (in bounded_bilinear) convergent:
hoelzl@51526
  1350
  assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
hoelzl@51526
  1351
  shows "convergent (\<lambda>n. X n ** Y n)"
hoelzl@51526
  1352
  using assms unfolding convergent_def by (fast intro: tendsto)
hoelzl@51526
  1353
hoelzl@51526
  1354
lemma convergent_minus_iff:
hoelzl@51526
  1355
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1356
  shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
hoelzl@51526
  1357
apply (simp add: convergent_def)
hoelzl@51526
  1358
apply (auto dest: tendsto_minus)
hoelzl@51526
  1359
apply (drule tendsto_minus, auto)
hoelzl@51526
  1360
done
hoelzl@51526
  1361
hoelzl@51526
  1362
hoelzl@51526
  1363
text {* A monotone sequence converges to its least upper bound. *}
hoelzl@51526
  1364
hoelzl@54263
  1365
lemma LIMSEQ_incseq_SUP:
hoelzl@54263
  1366
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1367
  assumes u: "bdd_above (range X)"
hoelzl@54263
  1368
  assumes X: "incseq X"
hoelzl@54263
  1369
  shows "X ----> (SUP i. X i)"
hoelzl@54263
  1370
  by (rule order_tendstoI)
hoelzl@54263
  1371
     (auto simp: eventually_sequentially u less_cSUP_iff intro: X[THEN incseqD] less_le_trans cSUP_lessD[OF u])
hoelzl@51526
  1372
hoelzl@54263
  1373
lemma LIMSEQ_decseq_INF:
hoelzl@54263
  1374
  fixes X :: "nat \<Rightarrow> 'a::{conditionally_complete_linorder, linorder_topology}"
hoelzl@54263
  1375
  assumes u: "bdd_below (range X)"
hoelzl@54263
  1376
  assumes X: "decseq X"
hoelzl@54263
  1377
  shows "X ----> (INF i. X i)"
hoelzl@54263
  1378
  by (rule order_tendstoI)
hoelzl@54263
  1379
     (auto simp: eventually_sequentially u cINF_less_iff intro: X[THEN decseqD] le_less_trans less_cINF_D[OF u])
hoelzl@51526
  1380
hoelzl@51526
  1381
text{*Main monotonicity theorem*}
hoelzl@51526
  1382
hoelzl@51526
  1383
lemma Bseq_monoseq_convergent: "Bseq X \<Longrightarrow> monoseq X \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1384
  by (auto simp: monoseq_iff convergent_def intro: LIMSEQ_decseq_INF LIMSEQ_incseq_SUP dest: Bseq_bdd_above Bseq_bdd_below)
hoelzl@54263
  1385
hoelzl@54263
  1386
lemma Bseq_mono_convergent: "Bseq X \<Longrightarrow> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n) \<Longrightarrow> convergent (X::nat\<Rightarrow>real)"
hoelzl@54263
  1387
  by (auto intro!: Bseq_monoseq_convergent incseq_imp_monoseq simp: incseq_def)
hoelzl@51526
  1388
hoelzl@51526
  1389
lemma Cauchy_iff:
hoelzl@51526
  1390
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1391
  shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
hoelzl@51526
  1392
  unfolding Cauchy_def dist_norm ..
hoelzl@51526
  1393
hoelzl@51526
  1394
lemma CauchyI:
hoelzl@51526
  1395
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1396
  shows "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e) \<Longrightarrow> Cauchy X"
hoelzl@51526
  1397
by (simp add: Cauchy_iff)
hoelzl@51526
  1398
hoelzl@51526
  1399
lemma CauchyD:
hoelzl@51526
  1400
  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
hoelzl@51526
  1401
  shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
hoelzl@51526
  1402
by (simp add: Cauchy_iff)
hoelzl@51526
  1403
hoelzl@51526
  1404
lemma incseq_convergent:
hoelzl@51526
  1405
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1406
  assumes "incseq X" and "\<forall>i. X i \<le> B"
hoelzl@51526
  1407
  obtains L where "X ----> L" "\<forall>i. X i \<le> L"
hoelzl@51526
  1408
proof atomize_elim
hoelzl@51526
  1409
  from incseq_bounded[OF assms] `incseq X` Bseq_monoseq_convergent[of X]
hoelzl@51526
  1410
  obtain L where "X ----> L"
hoelzl@51526
  1411
    by (auto simp: convergent_def monoseq_def incseq_def)
hoelzl@51526
  1412
  with `incseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. X i \<le> L)"
hoelzl@51526
  1413
    by (auto intro!: exI[of _ L] incseq_le)
hoelzl@51526
  1414
qed
hoelzl@51526
  1415
hoelzl@51526
  1416
lemma decseq_convergent:
hoelzl@51526
  1417
  fixes X :: "nat \<Rightarrow> real"
hoelzl@51526
  1418
  assumes "decseq X" and "\<forall>i. B \<le> X i"
hoelzl@51526
  1419
  obtains L where "X ----> L" "\<forall>i. L \<le> X i"
hoelzl@51526
  1420
proof atomize_elim
hoelzl@51526
  1421
  from decseq_bounded[OF assms] `decseq X` Bseq_monoseq_convergent[of X]
hoelzl@51526
  1422
  obtain L where "X ----> L"
hoelzl@51526
  1423
    by (auto simp: convergent_def monoseq_def decseq_def)
hoelzl@51526
  1424
  with `decseq X` show "\<exists>L. X ----> L \<and> (\<forall>i. L \<le> X i)"
hoelzl@51526
  1425
    by (auto intro!: exI[of _ L] decseq_le)
hoelzl@51526
  1426
qed
hoelzl@51526
  1427
hoelzl@51526
  1428
subsubsection {* Cauchy Sequences are Bounded *}
hoelzl@51526
  1429
hoelzl@51526
  1430
text{*A Cauchy sequence is bounded -- this is the standard
hoelzl@51526
  1431
  proof mechanization rather than the nonstandard proof*}
hoelzl@51526
  1432
hoelzl@51526
  1433
lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
hoelzl@51526
  1434
          ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
hoelzl@51526
  1435
apply (clarify, drule spec, drule (1) mp)
hoelzl@51526
  1436
apply (simp only: norm_minus_commute)
hoelzl@51526
  1437
apply (drule order_le_less_trans [OF norm_triangle_ineq2])
hoelzl@51526
  1438
apply simp
hoelzl@51526
  1439
done
hoelzl@51526
  1440
hoelzl@51526
  1441
subsection {* Power Sequences *}
hoelzl@51526
  1442
hoelzl@51526
  1443
text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
hoelzl@51526
  1444
"x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
hoelzl@51526
  1445
  also fact that bounded and monotonic sequence converges.*}
hoelzl@51526
  1446
hoelzl@51526
  1447
lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
hoelzl@51526
  1448
apply (simp add: Bseq_def)
hoelzl@51526
  1449
apply (rule_tac x = 1 in exI)
hoelzl@51526
  1450
apply (simp add: power_abs)
hoelzl@51526
  1451
apply (auto dest: power_mono)
hoelzl@51526
  1452
done
hoelzl@51526
  1453
hoelzl@51526
  1454
lemma monoseq_realpow: fixes x :: real shows "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
hoelzl@51526
  1455
apply (clarify intro!: mono_SucI2)
hoelzl@51526
  1456
apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
hoelzl@51526
  1457
done
hoelzl@51526
  1458
hoelzl@51526
  1459
lemma convergent_realpow:
hoelzl@51526
  1460
  "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
hoelzl@51526
  1461
by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
hoelzl@51526
  1462
hoelzl@51526
  1463
lemma LIMSEQ_inverse_realpow_zero: "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
hoelzl@51526
  1464
  by (rule filterlim_compose[OF tendsto_inverse_0 filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1465
hoelzl@51526
  1466
lemma LIMSEQ_realpow_zero:
hoelzl@51526
  1467
  "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1468
proof cases
hoelzl@51526
  1469
  assume "0 \<le> x" and "x \<noteq> 0"
hoelzl@51526
  1470
  hence x0: "0 < x" by simp
hoelzl@51526
  1471
  assume x1: "x < 1"
hoelzl@51526
  1472
  from x0 x1 have "1 < inverse x"
hoelzl@51526
  1473
    by (rule one_less_inverse)
hoelzl@51526
  1474
  hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
hoelzl@51526
  1475
    by (rule LIMSEQ_inverse_realpow_zero)
hoelzl@51526
  1476
  thus ?thesis by (simp add: power_inverse)
hoelzl@58729
  1477
qed (rule LIMSEQ_imp_Suc, simp)
hoelzl@51526
  1478
hoelzl@51526
  1479
lemma LIMSEQ_power_zero:
hoelzl@51526
  1480
  fixes x :: "'a::{real_normed_algebra_1}"
hoelzl@51526
  1481
  shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
hoelzl@51526
  1482
apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
hoelzl@51526
  1483
apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
hoelzl@51526
  1484
apply (simp add: power_abs norm_power_ineq)
hoelzl@51526
  1485
done
hoelzl@51526
  1486
hoelzl@51526
  1487
lemma LIMSEQ_divide_realpow_zero: "1 < x \<Longrightarrow> (\<lambda>n. a / (x ^ n) :: real) ----> 0"
hoelzl@51526
  1488
  by (rule tendsto_divide_0 [OF tendsto_const filterlim_realpow_sequentially_gt1]) simp
hoelzl@51526
  1489
hoelzl@51526
  1490
text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
hoelzl@51526
  1491
hoelzl@51526
  1492
lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. \<bar>c\<bar> ^ n :: real) ----> 0"
hoelzl@51526
  1493
  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
hoelzl@51526
  1494
hoelzl@51526
  1495
lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < 1 \<Longrightarrow> (\<lambda>n. c ^ n :: real) ----> 0"
hoelzl@51526
  1496
  by (rule LIMSEQ_power_zero) simp
hoelzl@51526
  1497
hoelzl@51526
  1498
hoelzl@51526
  1499
subsection {* Limits of Functions *}
hoelzl@51526
  1500
hoelzl@51526
  1501
lemma LIM_eq:
hoelzl@51526
  1502
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1503
  shows "f -- a --> L =
hoelzl@51526
  1504
     (\<forall>r>0.\<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)"
hoelzl@51526
  1505
by (simp add: LIM_def dist_norm)
hoelzl@51526
  1506
hoelzl@51526
  1507
lemma LIM_I:
hoelzl@51526
  1508
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1509
  shows "(!!r. 0<r ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r)
hoelzl@51526
  1510
      ==> f -- a --> L"
hoelzl@51526
  1511
by (simp add: LIM_eq)
hoelzl@51526
  1512
hoelzl@51526
  1513
lemma LIM_D:
hoelzl@51526
  1514
  fixes a :: "'a::real_normed_vector" and L :: "'b::real_normed_vector"
hoelzl@51526
  1515
  shows "[| f -- a --> L; 0<r |]
hoelzl@51526
  1516
      ==> \<exists>s>0.\<forall>x. x \<noteq> a & norm (x-a) < s --> norm (f x - L) < r"
hoelzl@51526
  1517
by (simp add: LIM_eq)
hoelzl@51526
  1518
hoelzl@51526
  1519
lemma LIM_offset:
hoelzl@51526
  1520
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1521
  shows "f -- a --> L \<Longrightarrow> (\<lambda>x. f (x + k)) -- a - k --> L"
hoelzl@51641
  1522
  unfolding filtermap_at_shift[symmetric, of a k] filterlim_def filtermap_filtermap by simp
hoelzl@51526
  1523
hoelzl@51526
  1524
lemma LIM_offset_zero:
hoelzl@51526
  1525
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1526
  shows "f -- a --> L \<Longrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
haftmann@57512
  1527
by (drule_tac k="a" in LIM_offset, simp add: add.commute)
hoelzl@51526
  1528
hoelzl@51526
  1529
lemma LIM_offset_zero_cancel:
hoelzl@51526
  1530
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1531
  shows "(\<lambda>h. f (a + h)) -- 0 --> L \<Longrightarrow> f -- a --> L"
hoelzl@51526
  1532
by (drule_tac k="- a" in LIM_offset, simp)
hoelzl@51526
  1533
hoelzl@51642
  1534
lemma LIM_offset_zero_iff:
hoelzl@51642
  1535
  fixes f :: "'a :: real_normed_vector \<Rightarrow> _"
hoelzl@51642
  1536
  shows  "f -- a --> L \<longleftrightarrow> (\<lambda>h. f (a + h)) -- 0 --> L"
hoelzl@51642
  1537
  using LIM_offset_zero_cancel[of f a L] LIM_offset_zero[of f L a] by auto
hoelzl@51642
  1538
hoelzl@51526
  1539
lemma LIM_zero:
hoelzl@51526
  1540
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1541
  shows "(f ---> l) F \<Longrightarrow> ((\<lambda>x. f x - l) ---> 0) F"
hoelzl@51526
  1542
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1543
hoelzl@51526
  1544
lemma LIM_zero_cancel:
hoelzl@51526
  1545
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1546
  shows "((\<lambda>x. f x - l) ---> 0) F \<Longrightarrow> (f ---> l) F"
hoelzl@51526
  1547
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1548
hoelzl@51526
  1549
lemma LIM_zero_iff:
hoelzl@51526
  1550
  fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1551
  shows "((\<lambda>x. f x - l) ---> 0) F = (f ---> l) F"
hoelzl@51526
  1552
unfolding tendsto_iff dist_norm by simp
hoelzl@51526
  1553
hoelzl@51526
  1554
lemma LIM_imp_LIM:
hoelzl@51526
  1555
  fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1556
  fixes g :: "'a::topological_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1557
  assumes f: "f -- a --> l"
hoelzl@51526
  1558
  assumes le: "\<And>x. x \<noteq> a \<Longrightarrow> norm (g x - m) \<le> norm (f x - l)"
hoelzl@51526
  1559
  shows "g -- a --> m"
hoelzl@51526
  1560
  by (rule metric_LIM_imp_LIM [OF f],
hoelzl@51526
  1561
    simp add: dist_norm le)
hoelzl@51526
  1562
hoelzl@51526
  1563
lemma LIM_equal2:
hoelzl@51526
  1564
  fixes f g :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1565
  assumes 1: "0 < R"
hoelzl@51526
  1566
  assumes 2: "\<And>x. \<lbrakk>x \<noteq> a; norm (x - a) < R\<rbrakk> \<Longrightarrow> f x = g x"
hoelzl@51526
  1567
  shows "g -- a --> l \<Longrightarrow> f -- a --> l"
hoelzl@51526
  1568
by (rule metric_LIM_equal2 [OF 1 2], simp_all add: dist_norm)
hoelzl@51526
  1569
hoelzl@51526
  1570
lemma LIM_compose2:
hoelzl@51526
  1571
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1572
  assumes f: "f -- a --> b"
hoelzl@51526
  1573
  assumes g: "g -- b --> c"
hoelzl@51526
  1574
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> b"
hoelzl@51526
  1575
  shows "(\<lambda>x. g (f x)) -- a --> c"
hoelzl@51526
  1576
by (rule metric_LIM_compose2 [OF f g inj [folded dist_norm]])
hoelzl@51526
  1577
hoelzl@51526
  1578
lemma real_LIM_sandwich_zero:
hoelzl@51526
  1579
  fixes f g :: "'a::topological_space \<Rightarrow> real"
hoelzl@51526
  1580
  assumes f: "f -- a --> 0"
hoelzl@51526
  1581
  assumes 1: "\<And>x. x \<noteq> a \<Longrightarrow> 0 \<le> g x"
hoelzl@51526
  1582
  assumes 2: "\<And>x. x \<noteq> a \<Longrightarrow> g x \<le> f x"
hoelzl@51526
  1583
  shows "g -- a --> 0"
hoelzl@51526
  1584
proof (rule LIM_imp_LIM [OF f]) (* FIXME: use tendsto_sandwich *)
hoelzl@51526
  1585
  fix x assume x: "x \<noteq> a"
hoelzl@51526
  1586
  have "norm (g x - 0) = g x" by (simp add: 1 x)
hoelzl@51526
  1587
  also have "g x \<le> f x" by (rule 2 [OF x])
hoelzl@51526
  1588
  also have "f x \<le> \<bar>f x\<bar>" by (rule abs_ge_self)
hoelzl@51526
  1589
  also have "\<bar>f x\<bar> = norm (f x - 0)" by simp
hoelzl@51526
  1590
  finally show "norm (g x - 0) \<le> norm (f x - 0)" .
hoelzl@51526
  1591
qed
hoelzl@51526
  1592
hoelzl@51526
  1593
hoelzl@51526
  1594
subsection {* Continuity *}
hoelzl@51526
  1595
hoelzl@51526
  1596
lemma LIM_isCont_iff:
hoelzl@51526
  1597
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1598
  shows "(f -- a --> f a) = ((\<lambda>h. f (a + h)) -- 0 --> f a)"
hoelzl@51526
  1599
by (rule iffI [OF LIM_offset_zero LIM_offset_zero_cancel])
hoelzl@51526
  1600
hoelzl@51526
  1601
lemma isCont_iff:
hoelzl@51526
  1602
  fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::topological_space"
hoelzl@51526
  1603
  shows "isCont f x = (\<lambda>h. f (x + h)) -- 0 --> f x"
hoelzl@51526
  1604
by (simp add: isCont_def LIM_isCont_iff)
hoelzl@51526
  1605
hoelzl@51526
  1606
lemma isCont_LIM_compose2:
hoelzl@51526
  1607
  fixes a :: "'a::real_normed_vector"
hoelzl@51526
  1608
  assumes f [unfolded isCont_def]: "isCont f a"
hoelzl@51526
  1609
  assumes g: "g -- f a --> l"
hoelzl@51526
  1610
  assumes inj: "\<exists>d>0. \<forall>x. x \<noteq> a \<and> norm (x - a) < d \<longrightarrow> f x \<noteq> f a"
hoelzl@51526
  1611
  shows "(\<lambda>x. g (f x)) -- a --> l"
hoelzl@51526
  1612
by (rule LIM_compose2 [OF f g inj])
hoelzl@51526
  1613
hoelzl@51526
  1614
hoelzl@51526
  1615
lemma isCont_norm [simp]:
hoelzl@51526
  1616
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1617
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. norm (f x)) a"
hoelzl@51526
  1618
  by (fact continuous_norm)
hoelzl@51526
  1619
hoelzl@51526
  1620
lemma isCont_rabs [simp]:
hoelzl@51526
  1621
  fixes f :: "'a::t2_space \<Rightarrow> real"
hoelzl@51526
  1622
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. \<bar>f x\<bar>) a"
hoelzl@51526
  1623
  by (fact continuous_rabs)
hoelzl@51526
  1624
hoelzl@51526
  1625
lemma isCont_add [simp]:
hoelzl@51526
  1626
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1627
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x + g x) a"
hoelzl@51526
  1628
  by (fact continuous_add)
hoelzl@51526
  1629
hoelzl@51526
  1630
lemma isCont_minus [simp]:
hoelzl@51526
  1631
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1632
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. - f x) a"
hoelzl@51526
  1633
  by (fact continuous_minus)
hoelzl@51526
  1634
hoelzl@51526
  1635
lemma isCont_diff [simp]:
hoelzl@51526
  1636
  fixes f :: "'a::t2_space \<Rightarrow> 'b::real_normed_vector"
hoelzl@51526
  1637
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x - g x) a"
hoelzl@51526
  1638
  by (fact continuous_diff)
hoelzl@51526
  1639
hoelzl@51526
  1640
lemma isCont_mult [simp]:
hoelzl@51526
  1641
  fixes f g :: "'a::t2_space \<Rightarrow> 'b::real_normed_algebra"
hoelzl@51526
  1642
  shows "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x * g x) a"
hoelzl@51526
  1643
  by (fact continuous_mult)
hoelzl@51526
  1644
hoelzl@51526
  1645
lemma (in bounded_linear) isCont:
hoelzl@51526
  1646
  "isCont g a \<Longrightarrow> isCont (\<lambda>x. f (g x)) a"
hoelzl@51526
  1647
  by (fact continuous)
hoelzl@51526
  1648
hoelzl@51526
  1649
lemma (in bounded_bilinear) isCont:
hoelzl@51526
  1650
  "\<lbrakk>isCont f a; isCont g a\<rbrakk> \<Longrightarrow> isCont (\<lambda>x. f x ** g x) a"
hoelzl@51526
  1651
  by (fact continuous)
hoelzl@51526
  1652
hoelzl@51526
  1653
lemmas isCont_scaleR [simp] = 
hoelzl@51526
  1654
  bounded_bilinear.isCont [OF bounded_bilinear_scaleR]
hoelzl@51526
  1655
hoelzl@51526
  1656
lemmas isCont_of_real [simp] =
hoelzl@51526
  1657
  bounded_linear.isCont [OF bounded_linear_of_real]
hoelzl@51526
  1658
hoelzl@51526
  1659
lemma isCont_power [simp]:
hoelzl@51526
  1660
  fixes f :: "'a::t2_space \<Rightarrow> 'b::{power,real_normed_algebra}"
hoelzl@51526
  1661
  shows "isCont f a \<Longrightarrow> isCont (\<lambda>x. f x ^ n) a"
hoelzl@51526
  1662
  by (fact continuous_power)
hoelzl@51526
  1663
hoelzl@51526
  1664
lemma isCont_setsum [simp]:
hoelzl@51526
  1665
  fixes f :: "'a \<Rightarrow> 'b::t2_space \<Rightarrow> 'c::real_normed_vector"
hoelzl@51526
  1666
  shows "\<forall>i\<in>A. isCont (f i) a \<Longrightarrow> isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
hoelzl@51526
  1667
  by (auto intro: continuous_setsum)
hoelzl@51526
  1668
hoelzl@51526
  1669
subsection {* Uniform Continuity *}
hoelzl@51526
  1670
hoelzl@51531
  1671
definition
hoelzl@51531
  1672
  isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
hoelzl@51531
  1673
  "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
hoelzl@51531
  1674
hoelzl@51531
  1675
lemma isUCont_isCont: "isUCont f ==> isCont f x"
hoelzl@51531
  1676
by (simp add: isUCont_def isCont_def LIM_def, force)
hoelzl@51531
  1677
hoelzl@51531
  1678
lemma isUCont_Cauchy:
hoelzl@51531
  1679
  "\<lbrakk>isUCont f; Cauchy X\<rbrakk> \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51531
  1680
unfolding isUCont_def
hoelzl@51531
  1681
apply (rule metric_CauchyI)
hoelzl@51531
  1682
apply (drule_tac x=e in spec, safe)
hoelzl@51531
  1683
apply (drule_tac e=s in metric_CauchyD, safe)
hoelzl@51531
  1684
apply (rule_tac x=M in exI, simp)
hoelzl@51531
  1685
done
hoelzl@51531
  1686
hoelzl@51526
  1687
lemma (in bounded_linear) isUCont: "isUCont f"
hoelzl@51526
  1688
unfolding isUCont_def dist_norm
hoelzl@51526
  1689
proof (intro allI impI)
hoelzl@51526
  1690
  fix r::real assume r: "0 < r"
hoelzl@51526
  1691
  obtain K where K: "0 < K" and norm_le: "\<And>x. norm (f x) \<le> norm x * K"
hoelzl@51526
  1692
    using pos_bounded by fast
hoelzl@51526
  1693
  show "\<exists>s>0. \<forall>x y. norm (x - y) < s \<longrightarrow> norm (f x - f y) < r"
hoelzl@51526
  1694
  proof (rule exI, safe)
nipkow@56541
  1695
    from r K show "0 < r / K" by simp
hoelzl@51526
  1696
  next
hoelzl@51526
  1697
    fix x y :: 'a
hoelzl@51526
  1698
    assume xy: "norm (x - y) < r / K"
hoelzl@51526
  1699
    have "norm (f x - f y) = norm (f (x - y))" by (simp only: diff)
hoelzl@51526
  1700
    also have "\<dots> \<le> norm (x - y) * K" by (rule norm_le)
hoelzl@51526
  1701
    also from K xy have "\<dots> < r" by (simp only: pos_less_divide_eq)
hoelzl@51526
  1702
    finally show "norm (f x - f y) < r" .
hoelzl@51526
  1703
  qed
hoelzl@51526
  1704
qed
hoelzl@51526
  1705
hoelzl@51526
  1706
lemma (in bounded_linear) Cauchy: "Cauchy X \<Longrightarrow> Cauchy (\<lambda>n. f (X n))"
hoelzl@51526
  1707
by (rule isUCont [THEN isUCont_Cauchy])
hoelzl@51526
  1708
hoelzl@51526
  1709
lemma LIM_less_bound: 
hoelzl@51526
  1710
  fixes f :: "real \<Rightarrow> real"
hoelzl@51526
  1711
  assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
hoelzl@51526
  1712
  shows "0 \<le> f x"
hoelzl@51526
  1713
proof (rule tendsto_le_const)
hoelzl@51526
  1714
  show "(f ---> f x) (at_left x)"
hoelzl@51526
  1715
    using `isCont f x` by (simp add: filterlim_at_split isCont_def)
hoelzl@51526
  1716
  show "eventually (\<lambda>x. 0 \<le> f x) (at_left x)"
hoelzl@51641
  1717
    using ev by (auto simp: eventually_at dist_real_def intro!: exI[of _ "x - b"])
hoelzl@51526
  1718
qed simp
hoelzl@51471
  1719
hoelzl@51529
  1720
hoelzl@51529
  1721
subsection {* Nested Intervals and Bisection -- Needed for Compactness *}
hoelzl@51529
  1722
hoelzl@51529
  1723
lemma nested_sequence_unique:
hoelzl@51529
  1724
  assumes "\<forall>n. f n \<le> f (Suc n)" "\<forall>n. g (Suc n) \<le> g n" "\<forall>n. f n \<le> g n" "(\<lambda>n. f n - g n) ----> 0"
hoelzl@51529
  1725
  shows "\<exists>l::real. ((\<forall>n. f n \<le> l) \<and> f ----> l) \<and> ((\<forall>n. l \<le> g n) \<and> g ----> l)"
hoelzl@51529
  1726
proof -
hoelzl@51529
  1727
  have "incseq f" unfolding incseq_Suc_iff by fact
hoelzl@51529
  1728
  have "decseq g" unfolding decseq_Suc_iff by fact
hoelzl@51529
  1729
hoelzl@51529
  1730
  { fix n
hoelzl@51529
  1731
    from `decseq g` have "g n \<le> g 0" by (rule decseqD) simp
hoelzl@51529
  1732
    with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f n \<le> g 0" by auto }
hoelzl@51529
  1733
  then obtain u where "f ----> u" "\<forall>i. f i \<le> u"
hoelzl@51529
  1734
    using incseq_convergent[OF `incseq f`] by auto
hoelzl@51529
  1735
  moreover
hoelzl@51529
  1736
  { fix n
hoelzl@51529
  1737
    from `incseq f` have "f 0 \<le> f n" by (rule incseqD) simp
hoelzl@51529
  1738
    with `\<forall>n. f n \<le> g n`[THEN spec, of n] have "f 0 \<le> g n" by simp }
hoelzl@51529
  1739
  then obtain l where "g ----> l" "\<forall>i. l \<le> g i"
hoelzl@51529
  1740
    using decseq_convergent[OF `decseq g`] by auto
hoelzl@51529
  1741
  moreover note LIMSEQ_unique[OF assms(4) tendsto_diff[OF `f ----> u` `g ----> l`]]
hoelzl@51529
  1742
  ultimately show ?thesis by auto
hoelzl@51529
  1743
qed
hoelzl@51529
  1744
hoelzl@51529
  1745
lemma Bolzano[consumes 1, case_names trans local]:
hoelzl@51529
  1746
  fixes P :: "real \<Rightarrow> real \<Rightarrow> bool"
hoelzl@51529
  1747
  assumes [arith]: "a \<le> b"
hoelzl@51529
  1748
  assumes trans: "\<And>a b c. \<lbrakk>P a b; P b c; a \<le> b; b \<le> c\<rbrakk> \<Longrightarrow> P a c"
hoelzl@51529
  1749
  assumes local: "\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> \<exists>d>0. \<forall>a b. a \<le> x \<and> x \<le> b \<and> b - a < d \<longrightarrow> P a b"
hoelzl@51529
  1750
  shows "P a b"
hoelzl@51529
  1751
proof -
blanchet@55415
  1752
  def bisect \<equiv> "rec_nat (a, b) (\<lambda>n (x, y). if P x ((x+y) / 2) then ((x+y)/2, y) else (x, (x+y)/2))"
hoelzl@51529
  1753
  def l \<equiv> "\<lambda>n. fst (bisect n)" and u \<equiv> "\<lambda>n. snd (bisect n)"
hoelzl@51529
  1754
  have l[simp]: "l 0 = a" "\<And>n. l (Suc n) = (if P (l n) ((l n + u n) / 2) then (l n + u n) / 2 else l n)"
hoelzl@51529
  1755
    and u[simp]: "u 0 = b" "\<And>n. u (Suc n) = (if P (l n) ((l n + u n) / 2) then u n else (l n + u n) / 2)"
hoelzl@51529
  1756
    by (simp_all add: l_def u_def bisect_def split: prod.split)
hoelzl@51529
  1757
hoelzl@51529
  1758
  { fix n have "l n \<le> u n" by (induct n) auto } note this[simp]
hoelzl@51529
  1759
hoelzl@51529
  1760
  have "\<exists>x. ((\<forall>n. l n \<le> x) \<and> l ----> x) \<and> ((\<forall>n. x \<le> u n) \<and> u ----> x)"
hoelzl@51529
  1761
  proof (safe intro!: nested_sequence_unique)
hoelzl@51529
  1762
    fix n show "l n \<le> l (Suc n)" "u (Suc n) \<le> u n" by (induct n) auto
hoelzl@51529
  1763
  next
hoelzl@51529
  1764
    { fix n have "l n - u n = (a - b) / 2^n" by (induct n) (auto simp: field_simps) }
hoelzl@51529
  1765
    then show "(\<lambda>n. l n - u n) ----> 0" by (simp add: LIMSEQ_divide_realpow_zero)
hoelzl@51529
  1766
  qed fact
hoelzl@51529
  1767
  then obtain x where x: "\<And>n. l n \<le> x" "\<And>n. x \<le> u n" and "l ----> x" "u ----> x" by auto
hoelzl@51529
  1768
  obtain d where "0 < d" and d: "\<And>a b. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> b - a < d \<Longrightarrow> P a b"
hoelzl@51529
  1769
    using `l 0 \<le> x` `x \<le> u 0` local[of x] by auto
hoelzl@51529
  1770
hoelzl@51529
  1771
  show "P a b"
hoelzl@51529
  1772
  proof (rule ccontr)
hoelzl@51529
  1773
    assume "\<not> P a b" 
hoelzl@51529
  1774
    { fix n have "\<not> P (l n) (u n)"
hoelzl@51529
  1775
      proof (induct n)
hoelzl@51529
  1776
        case (Suc n) with trans[of "l n" "(l n + u n) / 2" "u n"] show ?case by auto
hoelzl@51529
  1777
      qed (simp add: `\<not> P a b`) }
hoelzl@51529
  1778
    moreover
hoelzl@51529
  1779
    { have "eventually (\<lambda>n. x - d / 2 < l n) sequentially"
hoelzl@51529
  1780
        using `0 < d` `l ----> x` by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  1781
      moreover have "eventually (\<lambda>n. u n < x + d / 2) sequentially"
hoelzl@51529
  1782
        using `0 < d` `u ----> x` by (intro order_tendstoD[of _ x]) auto
hoelzl@51529
  1783
      ultimately have "eventually (\<lambda>n. P (l n) (u n)) sequentially"
hoelzl@51529
  1784
      proof eventually_elim
hoelzl@51529
  1785
        fix n assume "x - d / 2 < l n" "u n < x + d / 2"
hoelzl@51529
  1786
        from add_strict_mono[OF this] have "u n - l n < d" by simp
hoelzl@51529
  1787
        with x show "P (l n) (u n)" by (rule d)
hoelzl@51529
  1788
      qed }
hoelzl@51529
  1789
    ultimately show False by simp
hoelzl@51529
  1790
  qed
hoelzl@51529
  1791
qed
hoelzl@51529
  1792
hoelzl@51529
  1793
lemma compact_Icc[simp, intro]: "compact {a .. b::real}"
hoelzl@51529
  1794
proof (cases "a \<le> b", rule compactI)
hoelzl@51529
  1795
  fix C assume C: "a \<le> b" "\<forall>t\<in>C. open t" "{a..b} \<subseteq> \<Union>C"
hoelzl@51529
  1796
  def T == "{a .. b}"
hoelzl@51529
  1797
  from C(1,3) show "\<exists>C'\<subseteq>C. finite C' \<and> {a..b} \<subseteq> \<Union>C'"
hoelzl@51529
  1798
  proof (induct rule: Bolzano)
hoelzl@51529
  1799
    case (trans a b c)
hoelzl@51529
  1800
    then have *: "{a .. c} = {a .. b} \<union> {b .. c}" by auto
hoelzl@51529
  1801
    from trans obtain C1 C2 where "C1\<subseteq>C \<and> finite C1 \<and> {a..b} \<subseteq> \<Union>C1" "C2\<subseteq>C \<and> finite C2 \<and> {b..c} \<subseteq> \<Union>C2"
hoelzl@51529
  1802
      by (auto simp: *)
hoelzl@51529
  1803
    with trans show ?case
hoelzl@51529
  1804
      unfolding * by (intro exI[of _ "C1 \<union> C2"]) auto
hoelzl@51529
  1805
  next
hoelzl@51529
  1806
    case (local x)
hoelzl@51529
  1807
    then have "x \<in> \<Union>C" using C by auto
hoelzl@51529
  1808
    with C(2) obtain c where "x \<in> c" "open c" "c \<in> C" by auto
hoelzl@51529
  1809
    then obtain e where "0 < e" "{x - e <..< x + e} \<subseteq> c"
hoelzl@51529
  1810
      by (auto simp: open_real_def dist_real_def subset_eq Ball_def abs_less_iff)
hoelzl@51529
  1811
    with `c \<in> C` show ?case
hoelzl@51529
  1812
      by (safe intro!: exI[of _ "e/2"] exI[of _ "{c}"]) auto
hoelzl@51529
  1813
  qed
hoelzl@51529
  1814
qed simp
hoelzl@51529
  1815
hoelzl@51529
  1816
hoelzl@57447
  1817
lemma continuous_image_closed_interval:
hoelzl@57447
  1818
  fixes a b and f :: "real \<Rightarrow> real"
hoelzl@57447
  1819
  defines "S \<equiv> {a..b}"
hoelzl@57447
  1820
  assumes "a \<le> b" and f: "continuous_on S f"
hoelzl@57447
  1821
  shows "\<exists>c d. f`S = {c..d} \<and> c \<le> d"
hoelzl@57447
  1822
proof -
hoelzl@57447
  1823
  have S: "compact S" "S \<noteq> {}"
hoelzl@57447
  1824
    using `a \<le> b` by (auto simp: S_def)
hoelzl@57447
  1825
  obtain c where "c \<in> S" "\<forall>d\<in>S. f d \<le> f c"
hoelzl@57447
  1826
    using continuous_attains_sup[OF S f] by auto
hoelzl@57447
  1827
  moreover obtain d where "d \<in> S" "\<forall>c\<in>S. f d \<le> f c"
hoelzl@57447
  1828
    using continuous_attains_inf[OF S f] by auto
hoelzl@57447
  1829
  moreover have "connected (f`S)"
hoelzl@57447
  1830
    using connected_continuous_image[OF f] connected_Icc by (auto simp: S_def)
hoelzl@57447
  1831
  ultimately have "f ` S = {f d .. f c} \<and> f d \<le> f c"
hoelzl@57447
  1832
    by (auto simp: connected_iff_interval)
hoelzl@57447
  1833
  then show ?thesis
hoelzl@57447
  1834
    by auto
hoelzl@57447
  1835
qed
hoelzl@57447
  1836
hoelzl@51529
  1837
subsection {* Boundedness of continuous functions *}
hoelzl@51529
  1838
hoelzl@51529
  1839
text{*By bisection, function continuous on closed interval is bounded above*}
hoelzl@51529
  1840
hoelzl@51529
  1841
lemma isCont_eq_Ub:
hoelzl@51529
  1842
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  1843
  shows "a \<le> b \<Longrightarrow> \<forall>x::real. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  1844
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
hoelzl@51529
  1845
  using continuous_attains_sup[of "{a .. b}" f]
hoelzl@51529
  1846
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  1847
hoelzl@51529
  1848
lemma isCont_eq_Lb:
hoelzl@51529
  1849
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  1850
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  1851
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> M \<le> f x) \<and> (\<exists>x. a \<le> x \<and> x \<le> b \<and> f x = M)"
hoelzl@51529
  1852
  using continuous_attains_inf[of "{a .. b}" f]
hoelzl@51529
  1853
  by (auto simp add: continuous_at_imp_continuous_on Ball_def Bex_def)
hoelzl@51529
  1854
hoelzl@51529
  1855
lemma isCont_bounded:
hoelzl@51529
  1856
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  1857
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow> \<exists>M. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M"
hoelzl@51529
  1858
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  1859
hoelzl@51529
  1860
lemma isCont_has_Ub:
hoelzl@51529
  1861
  fixes f :: "real \<Rightarrow> 'a::linorder_topology"
hoelzl@51529
  1862
  shows "a \<le> b \<Longrightarrow> \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x \<Longrightarrow>
hoelzl@51529
  1863
    \<exists>M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> M) \<and> (\<forall>N. N < M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> N < f x))"
hoelzl@51529
  1864
  using isCont_eq_Ub[of a b f] by auto
hoelzl@51529
  1865
hoelzl@51529
  1866
(*HOL style here: object-level formulations*)
hoelzl@51529
  1867
lemma IVT_objl: "(f(a::real) \<le> (y::real) & y \<le> f(b) & a \<le> b &
hoelzl@51529
  1868
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  1869
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  1870
  by (blast intro: IVT)
hoelzl@51529
  1871
hoelzl@51529
  1872
lemma IVT2_objl: "(f(b::real) \<le> (y::real) & y \<le> f(a) & a \<le> b &
hoelzl@51529
  1873
      (\<forall>x. a \<le> x & x \<le> b --> isCont f x))
hoelzl@51529
  1874
      --> (\<exists>x. a \<le> x & x \<le> b & f(x) = y)"
hoelzl@51529
  1875
  by (blast intro: IVT2)
hoelzl@51529
  1876
hoelzl@51529
  1877
lemma isCont_Lb_Ub:
hoelzl@51529
  1878
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  1879
  assumes "a \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> isCont f x"
hoelzl@51529
  1880
  shows "\<exists>L M. (\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> L \<le> f x \<and> f x \<le> M) \<and> 
hoelzl@51529
  1881
               (\<forall>y. L \<le> y \<and> y \<le> M \<longrightarrow> (\<exists>x. a \<le> x \<and> x \<le> b \<and> (f x = y)))"
hoelzl@51529
  1882
proof -
hoelzl@51529
  1883
  obtain M where M: "a \<le> M" "M \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f x \<le> f M"
hoelzl@51529
  1884
    using isCont_eq_Ub[OF assms] by auto
hoelzl@51529
  1885
  obtain L where L: "a \<le> L" "L \<le> b" "\<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> f L \<le> f x"
hoelzl@51529
  1886
    using isCont_eq_Lb[OF assms] by auto
hoelzl@51529
  1887
  show ?thesis
hoelzl@51529
  1888
    using IVT[of f L _ M] IVT2[of f L _ M] M L assms
hoelzl@51529
  1889
    apply (rule_tac x="f L" in exI)
hoelzl@51529
  1890
    apply (rule_tac x="f M" in exI)
hoelzl@51529
  1891
    apply (cases "L \<le> M")
hoelzl@51529
  1892
    apply (simp, metis order_trans)
hoelzl@51529
  1893
    apply (simp, metis order_trans)
hoelzl@51529
  1894
    done
hoelzl@51529
  1895
qed
hoelzl@51529
  1896
hoelzl@51529
  1897
hoelzl@51529
  1898
text{*Continuity of inverse function*}
hoelzl@51529
  1899
hoelzl@51529
  1900
lemma isCont_inverse_function:
hoelzl@51529
  1901
  fixes f g :: "real \<Rightarrow> real"
hoelzl@51529
  1902
  assumes d: "0 < d"
hoelzl@51529
  1903
      and inj: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> g (f z) = z"
hoelzl@51529
  1904
      and cont: "\<forall>z. \<bar>z-x\<bar> \<le> d \<longrightarrow> isCont f z"
hoelzl@51529
  1905
  shows "isCont g (f x)"
hoelzl@51529
  1906
proof -
hoelzl@51529
  1907
  let ?A = "f (x - d)" and ?B = "f (x + d)" and ?D = "{x - d..x + d}"
hoelzl@51529
  1908
hoelzl@51529
  1909
  have f: "continuous_on ?D f"
hoelzl@51529
  1910
    using cont by (intro continuous_at_imp_continuous_on ballI) auto
hoelzl@51529
  1911
  then have g: "continuous_on (f`?D) g"
hoelzl@51529
  1912
    using inj by (intro continuous_on_inv) auto
hoelzl@51529
  1913
hoelzl@51529
  1914
  from d f have "{min ?A ?B <..< max ?A ?B} \<subseteq> f ` ?D"
hoelzl@51529
  1915
    by (intro connected_contains_Ioo connected_continuous_image) (auto split: split_min split_max)
hoelzl@51529
  1916
  with g have "continuous_on {min ?A ?B <..< max ?A ?B} g"
hoelzl@51529
  1917
    by (rule continuous_on_subset)
hoelzl@51529
  1918
  moreover
hoelzl@51529
  1919
  have "(?A < f x \<and> f x < ?B) \<or> (?B < f x \<and> f x < ?A)"
hoelzl@51529
  1920
    using d inj by (intro continuous_inj_imp_mono[OF _ _ f] inj_on_imageI2[of g, OF inj_onI]) auto
hoelzl@51529
  1921
  then have "f x \<in> {min ?A ?B <..< max ?A ?B}"
hoelzl@51529
  1922
    by auto
hoelzl@51529
  1923
  ultimately
hoelzl@51529
  1924
  show ?thesis
hoelzl@51529
  1925
    by (simp add: continuous_on_eq_continuous_at)
hoelzl@51529
  1926
qed
hoelzl@51529
  1927
hoelzl@51529
  1928
lemma isCont_inverse_function2:
hoelzl@51529
  1929
  fixes f g :: "real \<Rightarrow> real" shows
hoelzl@51529
  1930
  "\<lbrakk>a < x; x < b;
hoelzl@51529
  1931
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> g (f z) = z;
hoelzl@51529
  1932
    \<forall>z. a \<le> z \<and> z \<le> b \<longrightarrow> isCont f z\<rbrakk>
hoelzl@51529
  1933
   \<Longrightarrow> isCont g (f x)"
hoelzl@51529
  1934
apply (rule isCont_inverse_function
hoelzl@51529
  1935
       [where f=f and d="min (x - a) (b - x)"])
hoelzl@51529
  1936
apply (simp_all add: abs_le_iff)
hoelzl@51529
  1937
done
hoelzl@51529
  1938
hoelzl@51529
  1939
(* need to rename second isCont_inverse *)
hoelzl@51529
  1940
hoelzl@51529
  1941
lemma isCont_inv_fun:
hoelzl@51529
  1942
  fixes f g :: "real \<Rightarrow> real"
hoelzl@51529
  1943
  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
hoelzl@51529
  1944
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
hoelzl@51529
  1945
      ==> isCont g (f x)"
hoelzl@51529
  1946
by (rule isCont_inverse_function)
hoelzl@51529
  1947
hoelzl@51529
  1948
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
hoelzl@51529
  1949
lemma LIM_fun_gt_zero:
hoelzl@51529
  1950
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  1951
  shows "f -- c --> l \<Longrightarrow> 0 < l \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> 0 < f x)"
hoelzl@51529
  1952
apply (drule (1) LIM_D, clarify)
hoelzl@51529
  1953
apply (rule_tac x = s in exI)
hoelzl@51529
  1954
apply (simp add: abs_less_iff)
hoelzl@51529
  1955
done
hoelzl@51529
  1956
hoelzl@51529
  1957
lemma LIM_fun_less_zero:
hoelzl@51529
  1958
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  1959
  shows "f -- c --> l \<Longrightarrow> l < 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x < 0)"
hoelzl@51529
  1960
apply (drule LIM_D [where r="-l"], simp, clarify)
hoelzl@51529
  1961
apply (rule_tac x = s in exI)
hoelzl@51529
  1962
apply (simp add: abs_less_iff)
hoelzl@51529
  1963
done
hoelzl@51529
  1964
hoelzl@51529
  1965
lemma LIM_fun_not_zero:
hoelzl@51529
  1966
  fixes f :: "real \<Rightarrow> real"
hoelzl@51529
  1967
  shows "f -- c --> l \<Longrightarrow> l \<noteq> 0 \<Longrightarrow> \<exists>r. 0 < r \<and> (\<forall>x. x \<noteq> c \<and> \<bar>c - x\<bar> < r \<longrightarrow> f x \<noteq> 0)"
hoelzl@51529
  1968
  using LIM_fun_gt_zero[of f l c] LIM_fun_less_zero[of f l c] by (auto simp add: neq_iff)
hoelzl@51531
  1969
huffman@31349
  1970
end
hoelzl@50324
  1971