src/HOL/SEQ.thy
author huffman
Tue May 04 10:06:05 2010 -0700 (2010-05-04)
changeset 36660 1cc4ab4b7ff7
parent 36657 f376af79f6b7
child 36662 621122eeb138
permissions -rw-r--r--
make (X ----> L) an abbreviation for (X ---> L) sequentially
     1 (*  Title:      HOL/SEQ.thy
     2     Author:     Jacques D. Fleuriot, University of Cambridge
     3     Author:     Lawrence C Paulson
     4     Author:     Jeremy Avigad
     5     Author:     Brian Huffman
     6 
     7 Convergence of sequences and series.
     8 *)
     9 
    10 header {* Sequences and Convergence *}
    11 
    12 theory SEQ
    13 imports Limits
    14 begin
    15 
    16 abbreviation
    17   LIMSEQ :: "[nat \<Rightarrow> 'a::metric_space, 'a] \<Rightarrow> bool"
    18     ("((_)/ ----> (_))" [60, 60] 60) where
    19   "X ----> L \<equiv> (X ---> L) sequentially"
    20 
    21 definition
    22   lim :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> 'a" where
    23     --{*Standard definition of limit using choice operator*}
    24   "lim X = (THE L. X ----> L)"
    25 
    26 definition
    27   convergent :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
    28     --{*Standard definition of convergence*}
    29   "convergent X = (\<exists>L. X ----> L)"
    30 
    31 definition
    32   Bseq :: "(nat => 'a::real_normed_vector) => bool" where
    33     --{*Standard definition for bounded sequence*}
    34   [code del]: "Bseq X = (\<exists>K>0.\<forall>n. norm (X n) \<le> K)"
    35 
    36 definition
    37   monoseq :: "(nat=>real)=>bool" where
    38     --{*Definition of monotonicity. 
    39         The use of disjunction here complicates proofs considerably. 
    40         One alternative is to add a Boolean argument to indicate the direction. 
    41         Another is to develop the notions of increasing and decreasing first.*}
    42   [code del]: "monoseq X = ((\<forall>m. \<forall>n\<ge>m. X m \<le> X n) | (\<forall>m. \<forall>n\<ge>m. X n \<le> X m))"
    43 
    44 definition
    45   incseq :: "(nat=>real)=>bool" where
    46     --{*Increasing sequence*}
    47   [code del]: "incseq X = (\<forall>m. \<forall>n\<ge>m. X m \<le> X n)"
    48 
    49 definition
    50   decseq :: "(nat=>real)=>bool" where
    51     --{*Increasing sequence*}
    52   [code del]: "decseq X = (\<forall>m. \<forall>n\<ge>m. X n \<le> X m)"
    53 
    54 definition
    55   subseq :: "(nat => nat) => bool" where
    56     --{*Definition of subsequence*}
    57   [code del]:   "subseq f = (\<forall>m. \<forall>n>m. (f m) < (f n))"
    58 
    59 definition
    60   Cauchy :: "(nat \<Rightarrow> 'a::metric_space) \<Rightarrow> bool" where
    61     --{*Standard definition of the Cauchy condition*}
    62   [code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
    63 
    64 
    65 subsection {* Bounded Sequences *}
    66 
    67 lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
    68 unfolding Bseq_def
    69 proof (intro exI conjI allI)
    70   show "0 < max K 1" by simp
    71 next
    72   fix n::nat
    73   have "norm (X n) \<le> K" by (rule K)
    74   thus "norm (X n) \<le> max K 1" by simp
    75 qed
    76 
    77 lemma BseqE: "\<lbrakk>Bseq X; \<And>K. \<lbrakk>0 < K; \<forall>n. norm (X n) \<le> K\<rbrakk> \<Longrightarrow> Q\<rbrakk> \<Longrightarrow> Q"
    78 unfolding Bseq_def by auto
    79 
    80 lemma BseqI2': assumes K: "\<forall>n\<ge>N. norm (X n) \<le> K" shows "Bseq X"
    81 proof (rule BseqI')
    82   let ?A = "norm ` X ` {..N}"
    83   have 1: "finite ?A" by simp
    84   fix n::nat
    85   show "norm (X n) \<le> max K (Max ?A)"
    86   proof (cases rule: linorder_le_cases)
    87     assume "n \<ge> N"
    88     hence "norm (X n) \<le> K" using K by simp
    89     thus "norm (X n) \<le> max K (Max ?A)" by simp
    90   next
    91     assume "n \<le> N"
    92     hence "norm (X n) \<in> ?A" by simp
    93     with 1 have "norm (X n) \<le> Max ?A" by (rule Max_ge)
    94     thus "norm (X n) \<le> max K (Max ?A)" by simp
    95   qed
    96 qed
    97 
    98 lemma Bseq_ignore_initial_segment: "Bseq X \<Longrightarrow> Bseq (\<lambda>n. X (n + k))"
    99 unfolding Bseq_def by auto
   100 
   101 lemma Bseq_offset: "Bseq (\<lambda>n. X (n + k)) \<Longrightarrow> Bseq X"
   102 apply (erule BseqE)
   103 apply (rule_tac N="k" and K="K" in BseqI2')
   104 apply clarify
   105 apply (drule_tac x="n - k" in spec, simp)
   106 done
   107 
   108 lemma Bseq_conv_Bfun: "Bseq X \<longleftrightarrow> Bfun X sequentially"
   109 unfolding Bfun_def eventually_sequentially
   110 apply (rule iffI)
   111 apply (simp add: Bseq_def)
   112 apply (auto intro: BseqI2')
   113 done
   114 
   115 
   116 subsection {* Limits of Sequences *}
   117 
   118 lemma [trans]: "X=Y ==> Y ----> z ==> X ----> z"
   119   by simp
   120 
   121 lemma LIMSEQ_def: "X ----> L = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. dist (X n) L < r)"
   122 unfolding tendsto_iff eventually_sequentially ..
   123 
   124 lemma LIMSEQ_iff:
   125   fixes L :: "'a::real_normed_vector"
   126   shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"
   127 unfolding LIMSEQ_def dist_norm ..
   128 
   129 lemma LIMSEQ_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
   130   unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
   131 
   132 lemma LIMSEQ_Zfun_iff: "((\<lambda>n. X n) ----> L) = Zfun (\<lambda>n. X n - L) sequentially"
   133 by (rule tendsto_Zfun_iff)
   134 
   135 lemma metric_LIMSEQ_I:
   136   "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
   137 by (simp add: LIMSEQ_def)
   138 
   139 lemma metric_LIMSEQ_D:
   140   "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r"
   141 by (simp add: LIMSEQ_def)
   142 
   143 lemma LIMSEQ_I:
   144   fixes L :: "'a::real_normed_vector"
   145   shows "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r) \<Longrightarrow> X ----> L"
   146 by (simp add: LIMSEQ_iff)
   147 
   148 lemma LIMSEQ_D:
   149   fixes L :: "'a::real_normed_vector"
   150   shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
   151 by (simp add: LIMSEQ_iff)
   152 
   153 lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
   154 by (rule tendsto_const)
   155 
   156 lemma LIMSEQ_const_iff: "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
   157 apply (safe intro!: LIMSEQ_const)
   158 apply (rule ccontr)
   159 apply (drule_tac r="dist k l" in metric_LIMSEQ_D)
   160 apply (simp add: zero_less_dist_iff)
   161 apply auto
   162 done
   163 
   164 lemma LIMSEQ_norm:
   165   fixes a :: "'a::real_normed_vector"
   166   shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
   167 by (rule tendsto_norm)
   168 
   169 lemma LIMSEQ_ignore_initial_segment:
   170   "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
   171 apply (rule metric_LIMSEQ_I)
   172 apply (drule (1) metric_LIMSEQ_D)
   173 apply (erule exE, rename_tac N)
   174 apply (rule_tac x=N in exI)
   175 apply simp
   176 done
   177 
   178 lemma LIMSEQ_offset:
   179   "(\<lambda>n. f (n + k)) ----> a \<Longrightarrow> f ----> a"
   180 apply (rule metric_LIMSEQ_I)
   181 apply (drule (1) metric_LIMSEQ_D)
   182 apply (erule exE, rename_tac N)
   183 apply (rule_tac x="N + k" in exI)
   184 apply clarify
   185 apply (drule_tac x="n - k" in spec)
   186 apply (simp add: le_diff_conv2)
   187 done
   188 
   189 lemma LIMSEQ_Suc: "f ----> l \<Longrightarrow> (\<lambda>n. f (Suc n)) ----> l"
   190 by (drule_tac k="Suc 0" in LIMSEQ_ignore_initial_segment, simp)
   191 
   192 lemma LIMSEQ_imp_Suc: "(\<lambda>n. f (Suc n)) ----> l \<Longrightarrow> f ----> l"
   193 by (rule_tac k="Suc 0" in LIMSEQ_offset, simp)
   194 
   195 lemma LIMSEQ_Suc_iff: "(\<lambda>n. f (Suc n)) ----> l = f ----> l"
   196 by (blast intro: LIMSEQ_imp_Suc LIMSEQ_Suc)
   197 
   198 lemma LIMSEQ_linear: "\<lbrakk> X ----> x ; l > 0 \<rbrakk> \<Longrightarrow> (\<lambda> n. X (n * l)) ----> x"
   199   unfolding LIMSEQ_def
   200   by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
   201 
   202 lemma LIMSEQ_add:
   203   fixes a b :: "'a::real_normed_vector"
   204   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
   205 by (rule tendsto_add)
   206 
   207 lemma LIMSEQ_minus:
   208   fixes a :: "'a::real_normed_vector"
   209   shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
   210 by (rule tendsto_minus)
   211 
   212 lemma LIMSEQ_minus_cancel:
   213   fixes a :: "'a::real_normed_vector"
   214   shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
   215 by (rule tendsto_minus_cancel)
   216 
   217 lemma LIMSEQ_diff:
   218   fixes a b :: "'a::real_normed_vector"
   219   shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
   220 by (rule tendsto_diff)
   221 
   222 lemma LIMSEQ_unique: "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
   223 apply (rule ccontr)
   224 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
   225 apply (drule_tac r="dist a b / 2" in metric_LIMSEQ_D, simp add: zero_less_dist_iff)
   226 apply (clarify, rename_tac M N)
   227 apply (subgoal_tac "dist a b < dist a b / 2 + dist a b / 2", simp)
   228 apply (subgoal_tac "dist a b \<le> dist (X (max M N)) a + dist (X (max M N)) b")
   229 apply (erule le_less_trans, rule add_strict_mono, simp, simp)
   230 apply (subst dist_commute, rule dist_triangle)
   231 done
   232 
   233 lemma (in bounded_linear) LIMSEQ:
   234   "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
   235 by (rule tendsto)
   236 
   237 lemma (in bounded_bilinear) LIMSEQ:
   238   "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
   239 by (rule tendsto)
   240 
   241 lemma LIMSEQ_mult:
   242   fixes a b :: "'a::real_normed_algebra"
   243   shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
   244 by (rule mult.tendsto)
   245 
   246 lemma increasing_LIMSEQ:
   247   fixes f :: "nat \<Rightarrow> real"
   248   assumes inc: "!!n. f n \<le> f (Suc n)"
   249       and bdd: "!!n. f n \<le> l"
   250       and en: "!!e. 0 < e \<Longrightarrow> \<exists>n. l \<le> f n + e"
   251   shows "f ----> l"
   252 proof (auto simp add: LIMSEQ_def)
   253   fix e :: real
   254   assume e: "0 < e"
   255   then obtain N where "l \<le> f N + e/2"
   256     by (metis half_gt_zero e en that)
   257   hence N: "l < f N + e" using e
   258     by simp
   259   { fix k
   260     have [simp]: "!!n. \<bar>f n - l\<bar> = l - f n"
   261       by (simp add: bdd) 
   262     have "\<bar>f (N+k) - l\<bar> < e"
   263     proof (induct k)
   264       case 0 show ?case using N
   265         by simp   
   266     next
   267       case (Suc k) thus ?case using N inc [of "N+k"]
   268         by simp
   269     qed 
   270   } note 1 = this
   271   { fix n
   272     have "N \<le> n \<Longrightarrow> \<bar>f n - l\<bar> < e" using 1 [of "n-N"]
   273       by simp 
   274   } note [intro] = this
   275   show " \<exists>no. \<forall>n\<ge>no. dist (f n) l < e"
   276     by (auto simp add: dist_real_def) 
   277   qed
   278 
   279 lemma Bseq_inverse_lemma:
   280   fixes x :: "'a::real_normed_div_algebra"
   281   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
   282 apply (subst nonzero_norm_inverse, clarsimp)
   283 apply (erule (1) le_imp_inverse_le)
   284 done
   285 
   286 lemma Bseq_inverse:
   287   fixes a :: "'a::real_normed_div_algebra"
   288   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
   289 unfolding Bseq_conv_Bfun by (rule Bfun_inverse)
   290 
   291 lemma LIMSEQ_inverse:
   292   fixes a :: "'a::real_normed_div_algebra"
   293   shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
   294 by (rule tendsto_inverse)
   295 
   296 lemma LIMSEQ_divide:
   297   fixes a b :: "'a::real_normed_field"
   298   shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
   299 by (rule tendsto_divide)
   300 
   301 lemma LIMSEQ_pow:
   302   fixes a :: "'a::{power, real_normed_algebra}"
   303   shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
   304 by (induct m) (simp_all add: LIMSEQ_const LIMSEQ_mult)
   305 
   306 lemma LIMSEQ_setsum:
   307   fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
   308   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
   309   shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
   310 using assms by (rule tendsto_setsum)
   311 
   312 lemma LIMSEQ_setprod:
   313   fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
   314   assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
   315   shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
   316 proof (cases "finite S")
   317   case True
   318   thus ?thesis using n
   319   proof (induct)
   320     case empty
   321     show ?case
   322       by (simp add: LIMSEQ_const)
   323   next
   324     case insert
   325     thus ?case
   326       by (simp add: LIMSEQ_mult)
   327   qed
   328 next
   329   case False
   330   thus ?thesis
   331     by (simp add: setprod_def LIMSEQ_const)
   332 qed
   333 
   334 lemma LIMSEQ_add_const: (* FIXME: delete *)
   335   fixes a :: "'a::real_normed_vector"
   336   shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
   337 by (intro tendsto_intros)
   338 
   339 (* FIXME: delete *)
   340 lemma LIMSEQ_add_minus:
   341   fixes a b :: "'a::real_normed_vector"
   342   shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
   343 by (intro tendsto_intros)
   344 
   345 lemma LIMSEQ_diff_const: (* FIXME: delete *)
   346   fixes a b :: "'a::real_normed_vector"
   347   shows "f ----> a ==> (%n.(f n  - b)) ----> a - b"
   348 by (intro tendsto_intros)
   349 
   350 lemma LIMSEQ_diff_approach_zero:
   351   fixes L :: "'a::real_normed_vector"
   352   shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
   353 by (drule (1) LIMSEQ_add, simp)
   354 
   355 lemma LIMSEQ_diff_approach_zero2:
   356   fixes L :: "'a::real_normed_vector"
   357   shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
   358 by (drule (1) LIMSEQ_diff, simp)
   359 
   360 text{*A sequence tends to zero iff its abs does*}
   361 lemma LIMSEQ_norm_zero:
   362   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
   363   shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
   364 by (simp add: LIMSEQ_iff)
   365 
   366 lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
   367 by (simp add: LIMSEQ_iff)
   368 
   369 lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
   370 by (drule LIMSEQ_norm, simp)
   371 
   372 text{*An unbounded sequence's inverse tends to 0*}
   373 
   374 lemma LIMSEQ_inverse_zero:
   375   "\<forall>r::real. \<exists>N. \<forall>n\<ge>N. r < X n \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> 0"
   376 apply (rule LIMSEQ_I)
   377 apply (drule_tac x="inverse r" in spec, safe)
   378 apply (rule_tac x="N" in exI, safe)
   379 apply (drule_tac x="n" in spec, safe)
   380 apply (frule positive_imp_inverse_positive)
   381 apply (frule (1) less_imp_inverse_less)
   382 apply (subgoal_tac "0 < X n", simp)
   383 apply (erule (1) order_less_trans)
   384 done
   385 
   386 text{*The sequence @{term "1/n"} tends to 0 as @{term n} tends to infinity*}
   387 
   388 lemma LIMSEQ_inverse_real_of_nat: "(%n. inverse(real(Suc n))) ----> 0"
   389 apply (rule LIMSEQ_inverse_zero, safe)
   390 apply (cut_tac x = r in reals_Archimedean2)
   391 apply (safe, rule_tac x = n in exI)
   392 apply (auto simp add: real_of_nat_Suc)
   393 done
   394 
   395 text{*The sequence @{term "r + 1/n"} tends to @{term r} as @{term n} tends to
   396 infinity is now easily proved*}
   397 
   398 lemma LIMSEQ_inverse_real_of_nat_add:
   399      "(%n. r + inverse(real(Suc n))) ----> r"
   400 by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
   401 
   402 lemma LIMSEQ_inverse_real_of_nat_add_minus:
   403      "(%n. r + -inverse(real(Suc n))) ----> r"
   404 by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
   405 
   406 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
   407      "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
   408 by (cut_tac b=1 in
   409         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
   410 
   411 lemma LIMSEQ_le_const:
   412   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
   413 apply (rule ccontr, simp only: linorder_not_le)
   414 apply (drule_tac r="a - x" in LIMSEQ_D, simp)
   415 apply clarsimp
   416 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI1)
   417 apply (drule_tac x="max N no" in spec, drule mp, rule le_maxI2)
   418 apply simp
   419 done
   420 
   421 lemma LIMSEQ_le_const2:
   422   "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
   423 apply (subgoal_tac "- a \<le> - x", simp)
   424 apply (rule LIMSEQ_le_const)
   425 apply (erule LIMSEQ_minus)
   426 apply simp
   427 done
   428 
   429 lemma LIMSEQ_le:
   430   "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
   431 apply (subgoal_tac "0 \<le> y - x", simp)
   432 apply (rule LIMSEQ_le_const)
   433 apply (erule (1) LIMSEQ_diff)
   434 apply (simp add: le_diff_eq)
   435 done
   436 
   437 
   438 subsection {* Convergence *}
   439 
   440 lemma limI: "X ----> L ==> lim X = L"
   441 apply (simp add: lim_def)
   442 apply (blast intro: LIMSEQ_unique)
   443 done
   444 
   445 lemma convergentD: "convergent X ==> \<exists>L. (X ----> L)"
   446 by (simp add: convergent_def)
   447 
   448 lemma convergentI: "(X ----> L) ==> convergent X"
   449 by (auto simp add: convergent_def)
   450 
   451 lemma convergent_LIMSEQ_iff: "convergent X = (X ----> lim X)"
   452 by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
   453 
   454 lemma convergent_const: "convergent (\<lambda>n. c)"
   455 by (rule convergentI, rule LIMSEQ_const)
   456 
   457 lemma convergent_add:
   458   fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
   459   assumes "convergent (\<lambda>n. X n)"
   460   assumes "convergent (\<lambda>n. Y n)"
   461   shows "convergent (\<lambda>n. X n + Y n)"
   462 using assms unfolding convergent_def by (fast intro: LIMSEQ_add)
   463 
   464 lemma convergent_setsum:
   465   fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
   466   assumes "finite A" and "\<And>i. i \<in> A \<Longrightarrow> convergent (\<lambda>n. X i n)"
   467   shows "convergent (\<lambda>n. \<Sum>i\<in>A. X i n)"
   468 using assms
   469 by (induct A set: finite, simp_all add: convergent_const convergent_add)
   470 
   471 lemma (in bounded_linear) convergent:
   472   assumes "convergent (\<lambda>n. X n)"
   473   shows "convergent (\<lambda>n. f (X n))"
   474 using assms unfolding convergent_def by (fast intro: LIMSEQ)
   475 
   476 lemma (in bounded_bilinear) convergent:
   477   assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
   478   shows "convergent (\<lambda>n. X n ** Y n)"
   479 using assms unfolding convergent_def by (fast intro: LIMSEQ)
   480 
   481 lemma convergent_minus_iff:
   482   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
   483   shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
   484 apply (simp add: convergent_def)
   485 apply (auto dest: LIMSEQ_minus)
   486 apply (drule LIMSEQ_minus, auto)
   487 done
   488 
   489 lemma lim_le:
   490   fixes x :: real
   491   assumes f: "convergent f" and fn_le: "!!n. f n \<le> x"
   492   shows "lim f \<le> x"
   493 proof (rule classical)
   494   assume "\<not> lim f \<le> x"
   495   hence 0: "0 < lim f - x" by arith
   496   have 1: "f----> lim f"
   497     by (metis convergent_LIMSEQ_iff f) 
   498   thus ?thesis
   499     proof (simp add: LIMSEQ_iff)
   500       assume "\<forall>r>0. \<exists>no. \<forall>n\<ge>no. \<bar>f n - lim f\<bar> < r"
   501       hence "\<exists>no. \<forall>n\<ge>no. \<bar>f n - lim f\<bar> < lim f - x"
   502         by (metis 0)
   503       from this obtain no where "\<forall>n\<ge>no. \<bar>f n - lim f\<bar> < lim f - x"
   504         by blast
   505       thus "lim f \<le> x"
   506         by (metis add_cancel_end add_minus_cancel diff_def linorder_linear 
   507                   linorder_not_le minus_diff_eq abs_diff_less_iff fn_le) 
   508     qed
   509 qed
   510 
   511 text{* Given a binary function @{text "f:: nat \<Rightarrow> 'a \<Rightarrow> 'a"}, its values are uniquely determined by a function g *}
   512 
   513 lemma nat_function_unique: "EX! g. g 0 = e \<and> (\<forall>n. g (Suc n) = f n (g n))"
   514   unfolding Ex1_def
   515   apply (rule_tac x="nat_rec e f" in exI)
   516   apply (rule conjI)+
   517 apply (rule def_nat_rec_0, simp)
   518 apply (rule allI, rule def_nat_rec_Suc, simp)
   519 apply (rule allI, rule impI, rule ext)
   520 apply (erule conjE)
   521 apply (induct_tac x)
   522 apply simp
   523 apply (erule_tac x="n" in allE)
   524 apply (simp)
   525 done
   526 
   527 text{*Subsequence (alternative definition, (e.g. Hoskins)*}
   528 
   529 lemma subseq_Suc_iff: "subseq f = (\<forall>n. (f n) < (f (Suc n)))"
   530 apply (simp add: subseq_def)
   531 apply (auto dest!: less_imp_Suc_add)
   532 apply (induct_tac k)
   533 apply (auto intro: less_trans)
   534 done
   535 
   536 lemma monoseq_Suc:
   537    "monoseq X = ((\<forall>n. X n \<le> X (Suc n))
   538                  | (\<forall>n. X (Suc n) \<le> X n))"
   539 apply (simp add: monoseq_def)
   540 apply (auto dest!: le_imp_less_or_eq)
   541 apply (auto intro!: lessI [THEN less_imp_le] dest!: less_imp_Suc_add)
   542 apply (induct_tac "ka")
   543 apply (auto intro: order_trans)
   544 apply (erule contrapos_np)
   545 apply (induct_tac "k")
   546 apply (auto intro: order_trans)
   547 done
   548 
   549 lemma monoI1: "\<forall>m. \<forall> n \<ge> m. X m \<le> X n ==> monoseq X"
   550 by (simp add: monoseq_def)
   551 
   552 lemma monoI2: "\<forall>m. \<forall> n \<ge> m. X n \<le> X m ==> monoseq X"
   553 by (simp add: monoseq_def)
   554 
   555 lemma mono_SucI1: "\<forall>n. X n \<le> X (Suc n) ==> monoseq X"
   556 by (simp add: monoseq_Suc)
   557 
   558 lemma mono_SucI2: "\<forall>n. X (Suc n) \<le> X n ==> monoseq X"
   559 by (simp add: monoseq_Suc)
   560 
   561 lemma monoseq_minus: assumes "monoseq a"
   562   shows "monoseq (\<lambda> n. - a n)"
   563 proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
   564   case True
   565   hence "\<forall> m. \<forall> n \<ge> m. - a n \<le> - a m" by auto
   566   thus ?thesis by (rule monoI2)
   567 next
   568   case False
   569   hence "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" using `monoseq a`[unfolded monoseq_def] by auto
   570   thus ?thesis by (rule monoI1)
   571 qed
   572 
   573 lemma monoseq_le: assumes "monoseq a" and "a ----> x"
   574   shows "((\<forall> n. a n \<le> x) \<and> (\<forall>m. \<forall>n\<ge>m. a m \<le> a n)) \<or> 
   575          ((\<forall> n. x \<le> a n) \<and> (\<forall>m. \<forall>n\<ge>m. a n \<le> a m))"
   576 proof -
   577   { fix x n fix a :: "nat \<Rightarrow> real"
   578     assume "a ----> x" and "\<forall> m. \<forall> n \<ge> m. a m \<le> a n"
   579     hence monotone: "\<And> m n. m \<le> n \<Longrightarrow> a m \<le> a n" by auto
   580     have "a n \<le> x"
   581     proof (rule ccontr)
   582       assume "\<not> a n \<le> x" hence "x < a n" by auto
   583       hence "0 < a n - x" by auto
   584       from `a ----> x`[THEN LIMSEQ_D, OF this]
   585       obtain no where "\<And>n'. no \<le> n' \<Longrightarrow> norm (a n' - x) < a n - x" by blast
   586       hence "norm (a (max no n) - x) < a n - x" by auto
   587       moreover
   588       { fix n' have "n \<le> n' \<Longrightarrow> x < a n'" using monotone[where m=n and n=n'] and `x < a n` by auto }
   589       hence "x < a (max no n)" by auto
   590       ultimately
   591       have "a (max no n) < a n" by auto
   592       with monotone[where m=n and n="max no n"]
   593       show False by (auto simp:max_def split:split_if_asm)
   594     qed
   595   } note top_down = this
   596   { fix x n m fix a :: "nat \<Rightarrow> real"
   597     assume "a ----> x" and "monoseq a" and "a m < x"
   598     have "a n \<le> x \<and> (\<forall> m. \<forall> n \<ge> m. a m \<le> a n)"
   599     proof (cases "\<forall> m. \<forall> n \<ge> m. a m \<le> a n")
   600       case True with top_down and `a ----> x` show ?thesis by auto
   601     next
   602       case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
   603       hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
   604       hence False using `a m < x` by auto
   605       thus ?thesis ..
   606     qed
   607   } note when_decided = this
   608 
   609   show ?thesis
   610   proof (cases "\<exists> m. a m \<noteq> x")
   611     case True then obtain m where "a m \<noteq> x" by auto
   612     show ?thesis
   613     proof (cases "a m < x")
   614       case True with when_decided[OF `a ----> x` `monoseq a`, where m2=m]
   615       show ?thesis by blast
   616     next
   617       case False hence "- a m < - x" using `a m \<noteq> x` by auto
   618       with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
   619       show ?thesis by auto
   620     qed
   621   qed auto
   622 qed
   623 
   624 text{* for any sequence, there is a mootonic subsequence *}
   625 lemma seq_monosub: "\<exists>f. subseq f \<and> monoseq (\<lambda> n. (s (f n)))"
   626 proof-
   627   {assume H: "\<forall>n. \<exists>p >n. \<forall> m\<ge>p. s m \<le> s p"
   628     let ?P = "\<lambda> p n. p > n \<and> (\<forall>m \<ge> p. s m \<le> s p)"
   629     from nat_function_unique[of "SOME p. ?P p 0" "\<lambda>p n. SOME p. ?P p n"]
   630     obtain f where f: "f 0 = (SOME p. ?P p 0)" "\<forall>n. f (Suc n) = (SOME p. ?P p (f n))" by blast
   631     have "?P (f 0) 0"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p 0"]
   632       using H apply - 
   633       apply (erule allE[where x=0], erule exE, rule_tac x="p" in exI) 
   634       unfolding order_le_less by blast 
   635     hence f0: "f 0 > 0" "\<forall>m \<ge> f 0. s m \<le> s (f 0)" by blast+
   636     {fix n
   637       have "?P (f (Suc n)) (f n)" 
   638         unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
   639         using H apply - 
   640       apply (erule allE[where x="f n"], erule exE, rule_tac x="p" in exI) 
   641       unfolding order_le_less by blast 
   642     hence "f (Suc n) > f n" "\<forall>m \<ge> f (Suc n). s m \<le> s (f (Suc n))" by blast+}
   643   note fSuc = this
   644     {fix p q assume pq: "p \<ge> f q"
   645       have "s p \<le> s(f(q))"  using f0(2)[rule_format, of p] pq fSuc
   646         by (cases q, simp_all) }
   647     note pqth = this
   648     {fix q
   649       have "f (Suc q) > f q" apply (induct q) 
   650         using f0(1) fSuc(1)[of 0] apply simp by (rule fSuc(1))}
   651     note fss = this
   652     from fss have th1: "subseq f" unfolding subseq_Suc_iff ..
   653     {fix a b 
   654       have "f a \<le> f (a + b)"
   655       proof(induct b)
   656         case 0 thus ?case by simp
   657       next
   658         case (Suc b)
   659         from fSuc(1)[of "a + b"] Suc.hyps show ?case by simp
   660       qed}
   661     note fmon0 = this
   662     have "monoseq (\<lambda>n. s (f n))" 
   663     proof-
   664       {fix n
   665         have "s (f n) \<ge> s (f (Suc n))" 
   666         proof(cases n)
   667           case 0
   668           assume n0: "n = 0"
   669           from fSuc(1)[of 0] have th0: "f 0 \<le> f (Suc 0)" by simp
   670           from f0(2)[rule_format, OF th0] show ?thesis  using n0 by simp
   671         next
   672           case (Suc m)
   673           assume m: "n = Suc m"
   674           from fSuc(1)[of n] m have th0: "f (Suc m) \<le> f (Suc (Suc m))" by simp
   675           from m fSuc(2)[rule_format, OF th0] show ?thesis by simp 
   676         qed}
   677       thus "monoseq (\<lambda>n. s (f n))" unfolding monoseq_Suc by blast 
   678     qed
   679     with th1 have ?thesis by blast}
   680   moreover
   681   {fix N assume N: "\<forall>p >N. \<exists> m\<ge>p. s m > s p"
   682     {fix p assume p: "p \<ge> Suc N" 
   683       hence pN: "p > N" by arith with N obtain m where m: "m \<ge> p" "s m > s p" by blast
   684       have "m \<noteq> p" using m(2) by auto 
   685       with m have "\<exists>m>p. s p < s m" by - (rule exI[where x=m], auto)}
   686     note th0 = this
   687     let ?P = "\<lambda>m x. m > x \<and> s x < s m"
   688     from nat_function_unique[of "SOME x. ?P x (Suc N)" "\<lambda>m x. SOME y. ?P y x"]
   689     obtain f where f: "f 0 = (SOME x. ?P x (Suc N))" 
   690       "\<forall>n. f (Suc n) = (SOME m. ?P m (f n))" by blast
   691     have "?P (f 0) (Suc N)"  unfolding f(1) some_eq_ex[of "\<lambda>p. ?P p (Suc N)"]
   692       using N apply - 
   693       apply (erule allE[where x="Suc N"], clarsimp)
   694       apply (rule_tac x="m" in exI)
   695       apply auto
   696       apply (subgoal_tac "Suc N \<noteq> m")
   697       apply simp
   698       apply (rule ccontr, simp)
   699       done
   700     hence f0: "f 0 > Suc N" "s (Suc N) < s (f 0)" by blast+
   701     {fix n
   702       have "f n > N \<and> ?P (f (Suc n)) (f n)"
   703         unfolding f(2)[rule_format, of n] some_eq_ex[of "\<lambda>p. ?P p (f n)"]
   704       proof (induct n)
   705         case 0 thus ?case
   706           using f0 N apply auto 
   707           apply (erule allE[where x="f 0"], clarsimp) 
   708           apply (rule_tac x="m" in exI, simp)
   709           by (subgoal_tac "f 0 \<noteq> m", auto)
   710       next
   711         case (Suc n)
   712         from Suc.hyps have Nfn: "N < f n" by blast
   713         from Suc.hyps obtain m where m: "m > f n" "s (f n) < s m" by blast
   714         with Nfn have mN: "m > N" by arith
   715         note key = Suc.hyps[unfolded some_eq_ex[of "\<lambda>p. ?P p (f n)", symmetric] f(2)[rule_format, of n, symmetric]]
   716         
   717         from key have th0: "f (Suc n) > N" by simp
   718         from N[rule_format, OF th0]
   719         obtain m' where m': "m' \<ge> f (Suc n)" "s (f (Suc n)) < s m'" by blast
   720         have "m' \<noteq> f (Suc (n))" apply (rule ccontr) using m'(2) by auto
   721         hence "m' > f (Suc n)" using m'(1) by simp
   722         with key m'(2) show ?case by auto
   723       qed}
   724     note fSuc = this
   725     {fix n
   726       have "f n \<ge> Suc N \<and> f(Suc n) > f n \<and> s(f n) < s(f(Suc n))" using fSuc[of n] by auto 
   727       hence "f n \<ge> Suc N" "f(Suc n) > f n" "s(f n) < s(f(Suc n))" by blast+}
   728     note thf = this
   729     have sqf: "subseq f" unfolding subseq_Suc_iff using thf by simp
   730     have "monoseq (\<lambda>n. s (f n))"  unfolding monoseq_Suc using thf
   731       apply -
   732       apply (rule disjI1)
   733       apply auto
   734       apply (rule order_less_imp_le)
   735       apply blast
   736       done
   737     then have ?thesis  using sqf by blast}
   738   ultimately show ?thesis unfolding linorder_not_less[symmetric] by blast
   739 qed
   740 
   741 lemma seq_suble: assumes sf: "subseq f" shows "n \<le> f n"
   742 proof(induct n)
   743   case 0 thus ?case by simp
   744 next
   745   case (Suc n)
   746   from sf[unfolded subseq_Suc_iff, rule_format, of n] Suc.hyps
   747   have "n < f (Suc n)" by arith 
   748   thus ?case by arith
   749 qed
   750 
   751 lemma LIMSEQ_subseq_LIMSEQ:
   752   "\<lbrakk> X ----> L; subseq f \<rbrakk> \<Longrightarrow> (X o f) ----> L"
   753 apply (auto simp add: LIMSEQ_def) 
   754 apply (drule_tac x=r in spec, clarify)  
   755 apply (rule_tac x=no in exI, clarify) 
   756 apply (blast intro: seq_suble le_trans dest!: spec) 
   757 done
   758 
   759 subsection {* Bounded Monotonic Sequences *}
   760 
   761 
   762 text{*Bounded Sequence*}
   763 
   764 lemma BseqD: "Bseq X ==> \<exists>K. 0 < K & (\<forall>n. norm (X n) \<le> K)"
   765 by (simp add: Bseq_def)
   766 
   767 lemma BseqI: "[| 0 < K; \<forall>n. norm (X n) \<le> K |] ==> Bseq X"
   768 by (auto simp add: Bseq_def)
   769 
   770 lemma lemma_NBseq_def:
   771      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) =
   772       (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
   773 proof auto
   774   fix K :: real
   775   from reals_Archimedean2 obtain n :: nat where "K < real n" ..
   776   then have "K \<le> real (Suc n)" by auto
   777   assume "\<forall>m. norm (X m) \<le> K"
   778   have "\<forall>m. norm (X m) \<le> real (Suc n)"
   779   proof
   780     fix m :: 'a
   781     from `\<forall>m. norm (X m) \<le> K` have "norm (X m) \<le> K" ..
   782     with `K \<le> real (Suc n)` show "norm (X m) \<le> real (Suc n)" by auto
   783   qed
   784   then show "\<exists>N. \<forall>n. norm (X n) \<le> real (Suc N)" ..
   785 next
   786   fix N :: nat
   787   have "real (Suc N) > 0" by (simp add: real_of_nat_Suc)
   788   moreover assume "\<forall>n. norm (X n) \<le> real (Suc N)"
   789   ultimately show "\<exists>K>0. \<forall>n. norm (X n) \<le> K" by blast
   790 qed
   791 
   792 
   793 text{* alternative definition for Bseq *}
   794 lemma Bseq_iff: "Bseq X = (\<exists>N. \<forall>n. norm (X n) \<le> real(Suc N))"
   795 apply (simp add: Bseq_def)
   796 apply (simp (no_asm) add: lemma_NBseq_def)
   797 done
   798 
   799 lemma lemma_NBseq_def2:
   800      "(\<exists>K > 0. \<forall>n. norm (X n) \<le> K) = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
   801 apply (subst lemma_NBseq_def, auto)
   802 apply (rule_tac x = "Suc N" in exI)
   803 apply (rule_tac [2] x = N in exI)
   804 apply (auto simp add: real_of_nat_Suc)
   805  prefer 2 apply (blast intro: order_less_imp_le)
   806 apply (drule_tac x = n in spec, simp)
   807 done
   808 
   809 (* yet another definition for Bseq *)
   810 lemma Bseq_iff1a: "Bseq X = (\<exists>N. \<forall>n. norm (X n) < real(Suc N))"
   811 by (simp add: Bseq_def lemma_NBseq_def2)
   812 
   813 subsubsection{*Upper Bounds and Lubs of Bounded Sequences*}
   814 
   815 lemma Bseq_isUb:
   816   "!!(X::nat=>real). Bseq X ==> \<exists>U. isUb (UNIV::real set) {x. \<exists>n. X n = x} U"
   817 by (auto intro: isUbI setleI simp add: Bseq_def abs_le_iff)
   818 
   819 
   820 text{* Use completeness of reals (supremum property)
   821    to show that any bounded sequence has a least upper bound*}
   822 
   823 lemma Bseq_isLub:
   824   "!!(X::nat=>real). Bseq X ==>
   825    \<exists>U. isLub (UNIV::real set) {x. \<exists>n. X n = x} U"
   826 by (blast intro: reals_complete Bseq_isUb)
   827 
   828 subsubsection{*A Bounded and Monotonic Sequence Converges*}
   829 
   830 lemma lemma_converg1:
   831      "!!(X::nat=>real). [| \<forall>m. \<forall> n \<ge> m. X m \<le> X n;
   832                   isLub (UNIV::real set) {x. \<exists>n. X n = x} (X ma)
   833                |] ==> \<forall>n \<ge> ma. X n = X ma"
   834 apply safe
   835 apply (drule_tac y = "X n" in isLubD2)
   836 apply (blast dest: order_antisym)+
   837 done
   838 
   839 text{* The best of both worlds: Easier to prove this result as a standard
   840    theorem and then use equivalence to "transfer" it into the
   841    equivalent nonstandard form if needed!*}
   842 
   843 lemma Bmonoseq_LIMSEQ: "\<forall>n. m \<le> n --> X n = X m ==> \<exists>L. (X ----> L)"
   844 apply (simp add: LIMSEQ_def)
   845 apply (rule_tac x = "X m" in exI, safe)
   846 apply (rule_tac x = m in exI, safe)
   847 apply (drule spec, erule impE, auto)
   848 done
   849 
   850 lemma lemma_converg2:
   851    "!!(X::nat=>real).
   852     [| \<forall>m. X m ~= U;  isLub UNIV {x. \<exists>n. X n = x} U |] ==> \<forall>m. X m < U"
   853 apply safe
   854 apply (drule_tac y = "X m" in isLubD2)
   855 apply (auto dest!: order_le_imp_less_or_eq)
   856 done
   857 
   858 lemma lemma_converg3: "!!(X ::nat=>real). \<forall>m. X m \<le> U ==> isUb UNIV {x. \<exists>n. X n = x} U"
   859 by (rule setleI [THEN isUbI], auto)
   860 
   861 text{* FIXME: @{term "U - T < U"} is redundant *}
   862 lemma lemma_converg4: "!!(X::nat=> real).
   863                [| \<forall>m. X m ~= U;
   864                   isLub UNIV {x. \<exists>n. X n = x} U;
   865                   0 < T;
   866                   U + - T < U
   867                |] ==> \<exists>m. U + -T < X m & X m < U"
   868 apply (drule lemma_converg2, assumption)
   869 apply (rule ccontr, simp)
   870 apply (simp add: linorder_not_less)
   871 apply (drule lemma_converg3)
   872 apply (drule isLub_le_isUb, assumption)
   873 apply (auto dest: order_less_le_trans)
   874 done
   875 
   876 text{*A standard proof of the theorem for monotone increasing sequence*}
   877 
   878 lemma Bseq_mono_convergent:
   879      "[| Bseq X; \<forall>m. \<forall>n \<ge> m. X m \<le> X n |] ==> convergent (X::nat=>real)"
   880 apply (simp add: convergent_def)
   881 apply (frule Bseq_isLub, safe)
   882 apply (case_tac "\<exists>m. X m = U", auto)
   883 apply (blast dest: lemma_converg1 Bmonoseq_LIMSEQ)
   884 (* second case *)
   885 apply (rule_tac x = U in exI)
   886 apply (subst LIMSEQ_iff, safe)
   887 apply (frule lemma_converg2, assumption)
   888 apply (drule lemma_converg4, auto)
   889 apply (rule_tac x = m in exI, safe)
   890 apply (subgoal_tac "X m \<le> X n")
   891  prefer 2 apply blast
   892 apply (drule_tac x=n and P="%m. X m < U" in spec, arith)
   893 done
   894 
   895 lemma Bseq_minus_iff: "Bseq (%n. -(X n)) = Bseq X"
   896 by (simp add: Bseq_def)
   897 
   898 text{*Main monotonicity theorem*}
   899 lemma Bseq_monoseq_convergent: "[| Bseq X; monoseq X |] ==> convergent X"
   900 apply (simp add: monoseq_def, safe)
   901 apply (rule_tac [2] convergent_minus_iff [THEN ssubst])
   902 apply (drule_tac [2] Bseq_minus_iff [THEN ssubst])
   903 apply (auto intro!: Bseq_mono_convergent)
   904 done
   905 
   906 subsubsection{*Increasing and Decreasing Series*}
   907 
   908 lemma incseq_imp_monoseq:  "incseq X \<Longrightarrow> monoseq X"
   909   by (simp add: incseq_def monoseq_def) 
   910 
   911 lemma incseq_le: assumes inc: "incseq X" and lim: "X ----> L" shows "X n \<le> L"
   912   using monoseq_le [OF incseq_imp_monoseq [OF inc] lim]
   913 proof
   914   assume "(\<forall>n. X n \<le> L) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X m \<le> X n)"
   915   thus ?thesis by simp
   916 next
   917   assume "(\<forall>n. L \<le> X n) \<and> (\<forall>m n. m \<le> n \<longrightarrow> X n \<le> X m)"
   918   hence const: "(!!m n. m \<le> n \<Longrightarrow> X n = X m)" using inc
   919     by (auto simp add: incseq_def intro: order_antisym)
   920   have X: "!!n. X n = X 0"
   921     by (blast intro: const [of 0]) 
   922   have "X = (\<lambda>n. X 0)"
   923     by (blast intro: ext X)
   924   hence "L = X 0" using LIMSEQ_const [of "X 0"]
   925     by (auto intro: LIMSEQ_unique lim) 
   926   thus ?thesis
   927     by (blast intro: eq_refl X)
   928 qed
   929 
   930 lemma incseq_SucI:
   931   assumes "\<And>n. X n \<le> X (Suc n)"
   932   shows "incseq X" unfolding incseq_def
   933 proof safe
   934   fix m n :: nat
   935   { fix d m :: nat
   936     have "X m \<le> X (m + d)"
   937     proof (induct d)
   938       case (Suc d)
   939       also have "X (m + d) \<le> X (m + Suc d)"
   940         using assms by simp
   941       finally show ?case .
   942     qed simp }
   943   note this[of m "n - m"]
   944   moreover assume "m \<le> n"
   945   ultimately show "X m \<le> X n" by simp
   946 qed
   947 
   948 lemma decseq_imp_monoseq:  "decseq X \<Longrightarrow> monoseq X"
   949   by (simp add: decseq_def monoseq_def)
   950 
   951 lemma decseq_eq_incseq: "decseq X = incseq (\<lambda>n. - X n)" 
   952   by (simp add: decseq_def incseq_def)
   953 
   954 
   955 lemma decseq_le: assumes dec: "decseq X" and lim: "X ----> L" shows "L \<le> X n"
   956 proof -
   957   have inc: "incseq (\<lambda>n. - X n)" using dec
   958     by (simp add: decseq_eq_incseq)
   959   have "- X n \<le> - L" 
   960     by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim) 
   961   thus ?thesis
   962     by simp
   963 qed
   964 
   965 subsubsection{*A Few More Equivalence Theorems for Boundedness*}
   966 
   967 text{*alternative formulation for boundedness*}
   968 lemma Bseq_iff2: "Bseq X = (\<exists>k > 0. \<exists>x. \<forall>n. norm (X(n) + -x) \<le> k)"
   969 apply (unfold Bseq_def, safe)
   970 apply (rule_tac [2] x = "k + norm x" in exI)
   971 apply (rule_tac x = K in exI, simp)
   972 apply (rule exI [where x = 0], auto)
   973 apply (erule order_less_le_trans, simp)
   974 apply (drule_tac x=n in spec, fold diff_def)
   975 apply (drule order_trans [OF norm_triangle_ineq2])
   976 apply simp
   977 done
   978 
   979 text{*alternative formulation for boundedness*}
   980 lemma Bseq_iff3: "Bseq X = (\<exists>k > 0. \<exists>N. \<forall>n. norm(X(n) + -X(N)) \<le> k)"
   981 apply safe
   982 apply (simp add: Bseq_def, safe)
   983 apply (rule_tac x = "K + norm (X N)" in exI)
   984 apply auto
   985 apply (erule order_less_le_trans, simp)
   986 apply (rule_tac x = N in exI, safe)
   987 apply (drule_tac x = n in spec)
   988 apply (rule order_trans [OF norm_triangle_ineq], simp)
   989 apply (auto simp add: Bseq_iff2)
   990 done
   991 
   992 lemma BseqI2: "(\<forall>n. k \<le> f n & f n \<le> (K::real)) ==> Bseq f"
   993 apply (simp add: Bseq_def)
   994 apply (rule_tac x = " (\<bar>k\<bar> + \<bar>K\<bar>) + 1" in exI, auto)
   995 apply (drule_tac x = n in spec, arith)
   996 done
   997 
   998 
   999 subsection {* Cauchy Sequences *}
  1000 
  1001 lemma metric_CauchyI:
  1002   "(\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e) \<Longrightarrow> Cauchy X"
  1003 by (simp add: Cauchy_def)
  1004 
  1005 lemma metric_CauchyD:
  1006   "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e"
  1007 by (simp add: Cauchy_def)
  1008 
  1009 lemma Cauchy_iff:
  1010   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1011   shows "Cauchy X \<longleftrightarrow> (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
  1012 unfolding Cauchy_def dist_norm ..
  1013 
  1014 lemma Cauchy_iff2:
  1015      "Cauchy X =
  1016       (\<forall>j. (\<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. \<bar>X m - X n\<bar> < inverse(real (Suc j))))"
  1017 apply (simp add: Cauchy_iff, auto)
  1018 apply (drule reals_Archimedean, safe)
  1019 apply (drule_tac x = n in spec, auto)
  1020 apply (rule_tac x = M in exI, auto)
  1021 apply (drule_tac x = m in spec, simp)
  1022 apply (drule_tac x = na in spec, auto)
  1023 done
  1024 
  1025 lemma CauchyI:
  1026   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1027   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"
  1028 by (simp add: Cauchy_iff)
  1029 
  1030 lemma CauchyD:
  1031   fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
  1032   shows "\<lbrakk>Cauchy X; 0 < e\<rbrakk> \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e"
  1033 by (simp add: Cauchy_iff)
  1034 
  1035 lemma Cauchy_subseq_Cauchy:
  1036   "\<lbrakk> Cauchy X; subseq f \<rbrakk> \<Longrightarrow> Cauchy (X o f)"
  1037 apply (auto simp add: Cauchy_def)
  1038 apply (drule_tac x=e in spec, clarify)
  1039 apply (rule_tac x=M in exI, clarify)
  1040 apply (blast intro: le_trans [OF _ seq_suble] dest!: spec)
  1041 done
  1042 
  1043 subsubsection {* Cauchy Sequences are Bounded *}
  1044 
  1045 text{*A Cauchy sequence is bounded -- this is the standard
  1046   proof mechanization rather than the nonstandard proof*}
  1047 
  1048 lemma lemmaCauchy: "\<forall>n \<ge> M. norm (X M - X n) < (1::real)
  1049           ==>  \<forall>n \<ge> M. norm (X n :: 'a::real_normed_vector) < 1 + norm (X M)"
  1050 apply (clarify, drule spec, drule (1) mp)
  1051 apply (simp only: norm_minus_commute)
  1052 apply (drule order_le_less_trans [OF norm_triangle_ineq2])
  1053 apply simp
  1054 done
  1055 
  1056 lemma Cauchy_Bseq: "Cauchy X ==> Bseq X"
  1057 apply (simp add: Cauchy_iff)
  1058 apply (drule spec, drule mp, rule zero_less_one, safe)
  1059 apply (drule_tac x="M" in spec, simp)
  1060 apply (drule lemmaCauchy)
  1061 apply (rule_tac k="M" in Bseq_offset)
  1062 apply (simp add: Bseq_def)
  1063 apply (rule_tac x="1 + norm (X M)" in exI)
  1064 apply (rule conjI, rule order_less_le_trans [OF zero_less_one], simp)
  1065 apply (simp add: order_less_imp_le)
  1066 done
  1067 
  1068 subsubsection {* Cauchy Sequences are Convergent *}
  1069 
  1070 class complete_space =
  1071   assumes Cauchy_convergent: "Cauchy X \<Longrightarrow> convergent X"
  1072 
  1073 class banach = real_normed_vector + complete_space
  1074 
  1075 theorem LIMSEQ_imp_Cauchy:
  1076   assumes X: "X ----> a" shows "Cauchy X"
  1077 proof (rule metric_CauchyI)
  1078   fix e::real assume "0 < e"
  1079   hence "0 < e/2" by simp
  1080   with X have "\<exists>N. \<forall>n\<ge>N. dist (X n) a < e/2" by (rule metric_LIMSEQ_D)
  1081   then obtain N where N: "\<forall>n\<ge>N. dist (X n) a < e/2" ..
  1082   show "\<exists>N. \<forall>m\<ge>N. \<forall>n\<ge>N. dist (X m) (X n) < e"
  1083   proof (intro exI allI impI)
  1084     fix m assume "N \<le> m"
  1085     hence m: "dist (X m) a < e/2" using N by fast
  1086     fix n assume "N \<le> n"
  1087     hence n: "dist (X n) a < e/2" using N by fast
  1088     have "dist (X m) (X n) \<le> dist (X m) a + dist (X n) a"
  1089       by (rule dist_triangle2)
  1090     also from m n have "\<dots> < e" by simp
  1091     finally show "dist (X m) (X n) < e" .
  1092   qed
  1093 qed
  1094 
  1095 lemma convergent_Cauchy: "convergent X \<Longrightarrow> Cauchy X"
  1096 unfolding convergent_def
  1097 by (erule exE, erule LIMSEQ_imp_Cauchy)
  1098 
  1099 lemma Cauchy_convergent_iff:
  1100   fixes X :: "nat \<Rightarrow> 'a::complete_space"
  1101   shows "Cauchy X = convergent X"
  1102 by (fast intro: Cauchy_convergent convergent_Cauchy)
  1103 
  1104 lemma convergent_subseq_convergent:
  1105   fixes X :: "nat \<Rightarrow> 'a::complete_space"
  1106   shows "\<lbrakk> convergent X; subseq f \<rbrakk> \<Longrightarrow> convergent (X o f)"
  1107   by (simp add: Cauchy_subseq_Cauchy Cauchy_convergent_iff [symmetric])
  1108 
  1109 text {*
  1110 Proof that Cauchy sequences converge based on the one from
  1111 http://pirate.shu.edu/~wachsmut/ira/numseq/proofs/cauconv.html
  1112 *}
  1113 
  1114 text {*
  1115   If sequence @{term "X"} is Cauchy, then its limit is the lub of
  1116   @{term "{r::real. \<exists>N. \<forall>n\<ge>N. r < X n}"}
  1117 *}
  1118 
  1119 lemma isUb_UNIV_I: "(\<And>y. y \<in> S \<Longrightarrow> y \<le> u) \<Longrightarrow> isUb UNIV S u"
  1120 by (simp add: isUbI setleI)
  1121 
  1122 locale real_Cauchy =
  1123   fixes X :: "nat \<Rightarrow> real"
  1124   assumes X: "Cauchy X"
  1125   fixes S :: "real set"
  1126   defines S_def: "S \<equiv> {x::real. \<exists>N. \<forall>n\<ge>N. x < X n}"
  1127 
  1128 lemma real_CauchyI:
  1129   assumes "Cauchy X"
  1130   shows "real_Cauchy X"
  1131   proof qed (fact assms)
  1132 
  1133 lemma (in real_Cauchy) mem_S: "\<forall>n\<ge>N. x < X n \<Longrightarrow> x \<in> S"
  1134 by (unfold S_def, auto)
  1135 
  1136 lemma (in real_Cauchy) bound_isUb:
  1137   assumes N: "\<forall>n\<ge>N. X n < x"
  1138   shows "isUb UNIV S x"
  1139 proof (rule isUb_UNIV_I)
  1140   fix y::real assume "y \<in> S"
  1141   hence "\<exists>M. \<forall>n\<ge>M. y < X n"
  1142     by (simp add: S_def)
  1143   then obtain M where "\<forall>n\<ge>M. y < X n" ..
  1144   hence "y < X (max M N)" by simp
  1145   also have "\<dots> < x" using N by simp
  1146   finally show "y \<le> x"
  1147     by (rule order_less_imp_le)
  1148 qed
  1149 
  1150 lemma (in real_Cauchy) isLub_ex: "\<exists>u. isLub UNIV S u"
  1151 proof (rule reals_complete)
  1152   obtain N where "\<forall>m\<ge>N. \<forall>n\<ge>N. norm (X m - X n) < 1"
  1153     using CauchyD [OF X zero_less_one] by auto
  1154   hence N: "\<forall>n\<ge>N. norm (X n - X N) < 1" by simp
  1155   show "\<exists>x. x \<in> S"
  1156   proof
  1157     from N have "\<forall>n\<ge>N. X N - 1 < X n"
  1158       by (simp add: abs_diff_less_iff)
  1159     thus "X N - 1 \<in> S" by (rule mem_S)
  1160   qed
  1161   show "\<exists>u. isUb UNIV S u"
  1162   proof
  1163     from N have "\<forall>n\<ge>N. X n < X N + 1"
  1164       by (simp add: abs_diff_less_iff)
  1165     thus "isUb UNIV S (X N + 1)"
  1166       by (rule bound_isUb)
  1167   qed
  1168 qed
  1169 
  1170 lemma (in real_Cauchy) isLub_imp_LIMSEQ:
  1171   assumes x: "isLub UNIV S x"
  1172   shows "X ----> x"
  1173 proof (rule LIMSEQ_I)
  1174   fix r::real assume "0 < r"
  1175   hence r: "0 < r/2" by simp
  1176   obtain N where "\<forall>n\<ge>N. \<forall>m\<ge>N. norm (X n - X m) < r/2"
  1177     using CauchyD [OF X r] by auto
  1178   hence "\<forall>n\<ge>N. norm (X n - X N) < r/2" by simp
  1179   hence N: "\<forall>n\<ge>N. X N - r/2 < X n \<and> X n < X N + r/2"
  1180     by (simp only: real_norm_def abs_diff_less_iff)
  1181 
  1182   from N have "\<forall>n\<ge>N. X N - r/2 < X n" by fast
  1183   hence "X N - r/2 \<in> S" by (rule mem_S)
  1184   hence 1: "X N - r/2 \<le> x" using x isLub_isUb isUbD by fast
  1185 
  1186   from N have "\<forall>n\<ge>N. X n < X N + r/2" by fast
  1187   hence "isUb UNIV S (X N + r/2)" by (rule bound_isUb)
  1188   hence 2: "x \<le> X N + r/2" using x isLub_le_isUb by fast
  1189 
  1190   show "\<exists>N. \<forall>n\<ge>N. norm (X n - x) < r"
  1191   proof (intro exI allI impI)
  1192     fix n assume n: "N \<le> n"
  1193     from N n have "X n < X N + r/2" and "X N - r/2 < X n" by simp+
  1194     thus "norm (X n - x) < r" using 1 2
  1195       by (simp add: abs_diff_less_iff)
  1196   qed
  1197 qed
  1198 
  1199 lemma (in real_Cauchy) LIMSEQ_ex: "\<exists>x. X ----> x"
  1200 proof -
  1201   obtain x where "isLub UNIV S x"
  1202     using isLub_ex by fast
  1203   hence "X ----> x"
  1204     by (rule isLub_imp_LIMSEQ)
  1205   thus ?thesis ..
  1206 qed
  1207 
  1208 lemma real_Cauchy_convergent:
  1209   fixes X :: "nat \<Rightarrow> real"
  1210   shows "Cauchy X \<Longrightarrow> convergent X"
  1211 unfolding convergent_def
  1212 by (rule real_Cauchy.LIMSEQ_ex)
  1213  (rule real_CauchyI)
  1214 
  1215 instance real :: banach
  1216 by intro_classes (rule real_Cauchy_convergent)
  1217 
  1218 
  1219 subsection {* Power Sequences *}
  1220 
  1221 text{*The sequence @{term "x^n"} tends to 0 if @{term "0\<le>x"} and @{term
  1222 "x<1"}.  Proof will use (NS) Cauchy equivalence for convergence and
  1223   also fact that bounded and monotonic sequence converges.*}
  1224 
  1225 lemma Bseq_realpow: "[| 0 \<le> (x::real); x \<le> 1 |] ==> Bseq (%n. x ^ n)"
  1226 apply (simp add: Bseq_def)
  1227 apply (rule_tac x = 1 in exI)
  1228 apply (simp add: power_abs)
  1229 apply (auto dest: power_mono)
  1230 done
  1231 
  1232 lemma monoseq_realpow: "[| 0 \<le> x; x \<le> 1 |] ==> monoseq (%n. x ^ n)"
  1233 apply (clarify intro!: mono_SucI2)
  1234 apply (cut_tac n = n and N = "Suc n" and a = x in power_decreasing, auto)
  1235 done
  1236 
  1237 lemma convergent_realpow:
  1238   "[| 0 \<le> (x::real); x \<le> 1 |] ==> convergent (%n. x ^ n)"
  1239 by (blast intro!: Bseq_monoseq_convergent Bseq_realpow monoseq_realpow)
  1240 
  1241 lemma LIMSEQ_inverse_realpow_zero_lemma:
  1242   fixes x :: real
  1243   assumes x: "0 \<le> x"
  1244   shows "real n * x + 1 \<le> (x + 1) ^ n"
  1245 apply (induct n)
  1246 apply simp
  1247 apply simp
  1248 apply (rule order_trans)
  1249 prefer 2
  1250 apply (erule mult_left_mono)
  1251 apply (rule add_increasing [OF x], simp)
  1252 apply (simp add: real_of_nat_Suc)
  1253 apply (simp add: ring_distribs)
  1254 apply (simp add: mult_nonneg_nonneg x)
  1255 done
  1256 
  1257 lemma LIMSEQ_inverse_realpow_zero:
  1258   "1 < (x::real) \<Longrightarrow> (\<lambda>n. inverse (x ^ n)) ----> 0"
  1259 proof (rule LIMSEQ_inverse_zero [rule_format])
  1260   fix y :: real
  1261   assume x: "1 < x"
  1262   hence "0 < x - 1" by simp
  1263   hence "\<forall>y. \<exists>N::nat. y < real N * (x - 1)"
  1264     by (rule reals_Archimedean3)
  1265   hence "\<exists>N::nat. y < real N * (x - 1)" ..
  1266   then obtain N::nat where "y < real N * (x - 1)" ..
  1267   also have "\<dots> \<le> real N * (x - 1) + 1" by simp
  1268   also have "\<dots> \<le> (x - 1 + 1) ^ N"
  1269     by (rule LIMSEQ_inverse_realpow_zero_lemma, cut_tac x, simp)
  1270   also have "\<dots> = x ^ N" by simp
  1271   finally have "y < x ^ N" .
  1272   hence "\<forall>n\<ge>N. y < x ^ n"
  1273     apply clarify
  1274     apply (erule order_less_le_trans)
  1275     apply (erule power_increasing)
  1276     apply (rule order_less_imp_le [OF x])
  1277     done
  1278   thus "\<exists>N. \<forall>n\<ge>N. y < x ^ n" ..
  1279 qed
  1280 
  1281 lemma LIMSEQ_realpow_zero:
  1282   "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
  1283 proof (cases)
  1284   assume "x = 0"
  1285   hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
  1286   thus ?thesis by (rule LIMSEQ_imp_Suc)
  1287 next
  1288   assume "0 \<le> x" and "x \<noteq> 0"
  1289   hence x0: "0 < x" by simp
  1290   assume x1: "x < 1"
  1291   from x0 x1 have "1 < inverse x"
  1292     by (rule real_inverse_gt_one)
  1293   hence "(\<lambda>n. inverse (inverse x ^ n)) ----> 0"
  1294     by (rule LIMSEQ_inverse_realpow_zero)
  1295   thus ?thesis by (simp add: power_inverse)
  1296 qed
  1297 
  1298 lemma LIMSEQ_power_zero:
  1299   fixes x :: "'a::{real_normed_algebra_1}"
  1300   shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
  1301 apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
  1302 apply (simp only: LIMSEQ_Zfun_iff, erule Zfun_le)
  1303 apply (simp add: power_abs norm_power_ineq)
  1304 done
  1305 
  1306 lemma LIMSEQ_divide_realpow_zero:
  1307   "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
  1308 apply (cut_tac a = a and x1 = "inverse x" in
  1309         LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
  1310 apply (auto simp add: divide_inverse power_inverse)
  1311 apply (simp add: inverse_eq_divide pos_divide_less_eq)
  1312 done
  1313 
  1314 text{*Limit of @{term "c^n"} for @{term"\<bar>c\<bar> < 1"}*}
  1315 
  1316 lemma LIMSEQ_rabs_realpow_zero: "\<bar>c\<bar> < (1::real) ==> (%n. \<bar>c\<bar> ^ n) ----> 0"
  1317 by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
  1318 
  1319 lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
  1320 apply (rule LIMSEQ_rabs_zero [THEN iffD1])
  1321 apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
  1322 done
  1323 
  1324 end