SEQ.thy: legacy theorem names
authorhuffman
Fri Aug 19 15:07:10 2011 -0700 (2011-08-19)
changeset 44313d81d57979771
parent 44312 471ff02a8574
child 44314 dbad46932536
SEQ.thy: legacy theorem names
src/HOL/SEQ.thy
     1.1 --- a/src/HOL/SEQ.thy	Fri Aug 19 14:46:45 2011 -0700
     1.2 +++ b/src/HOL/SEQ.thy	Fri Aug 19 15:07:10 2011 -0700
     1.3 @@ -272,9 +272,6 @@
     1.4  lemma LIMSEQ_iff_nz: "X ----> L = (\<forall>r>0. \<exists>no>0. \<forall>n\<ge>no. dist (X n) L < r)"
     1.5    unfolding LIMSEQ_def by (metis Suc_leD zero_less_Suc)
     1.6  
     1.7 -lemma LIMSEQ_Zfun_iff: "((\<lambda>n. X n) ----> L) = Zfun (\<lambda>n. X n - L) sequentially"
     1.8 -by (rule tendsto_Zfun_iff)
     1.9 -
    1.10  lemma metric_LIMSEQ_I:
    1.11    "(\<And>r. 0 < r \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. dist (X n) L < r) \<Longrightarrow> X ----> L"
    1.12  by (simp add: LIMSEQ_def)
    1.13 @@ -293,19 +290,11 @@
    1.14    shows "\<lbrakk>X ----> L; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n - L) < r"
    1.15  by (simp add: LIMSEQ_iff)
    1.16  
    1.17 -lemma LIMSEQ_const: "(\<lambda>n. k) ----> k"
    1.18 -by (rule tendsto_const)
    1.19 -
    1.20  lemma LIMSEQ_const_iff:
    1.21    fixes k l :: "'a::t2_space"
    1.22    shows "(\<lambda>n. k) ----> l \<longleftrightarrow> k = l"
    1.23    using trivial_limit_sequentially by (rule tendsto_const_iff)
    1.24  
    1.25 -lemma LIMSEQ_norm:
    1.26 -  fixes a :: "'a::real_normed_vector"
    1.27 -  shows "X ----> a \<Longrightarrow> (\<lambda>n. norm (X n)) ----> norm a"
    1.28 -by (rule tendsto_norm)
    1.29 -
    1.30  lemma LIMSEQ_ignore_initial_segment:
    1.31    "f ----> a \<Longrightarrow> (\<lambda>n. f (n + k)) ----> a"
    1.32  apply (rule topological_tendstoI)
    1.33 @@ -341,44 +330,11 @@
    1.34    unfolding tendsto_def eventually_sequentially
    1.35    by (metis div_le_dividend div_mult_self1_is_m le_trans nat_mult_commute)
    1.36  
    1.37 -lemma LIMSEQ_add:
    1.38 -  fixes a b :: "'a::real_normed_vector"
    1.39 -  shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n + Y n) ----> a + b"
    1.40 -by (rule tendsto_add)
    1.41 -
    1.42 -lemma LIMSEQ_minus:
    1.43 -  fixes a :: "'a::real_normed_vector"
    1.44 -  shows "X ----> a \<Longrightarrow> (\<lambda>n. - X n) ----> - a"
    1.45 -by (rule tendsto_minus)
    1.46 -
    1.47 -lemma LIMSEQ_minus_cancel:
    1.48 -  fixes a :: "'a::real_normed_vector"
    1.49 -  shows "(\<lambda>n. - X n) ----> - a \<Longrightarrow> X ----> a"
    1.50 -by (rule tendsto_minus_cancel)
    1.51 -
    1.52 -lemma LIMSEQ_diff:
    1.53 -  fixes a b :: "'a::real_normed_vector"
    1.54 -  shows "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n - Y n) ----> a - b"
    1.55 -by (rule tendsto_diff)
    1.56 -
    1.57  lemma LIMSEQ_unique:
    1.58    fixes a b :: "'a::t2_space"
    1.59    shows "\<lbrakk>X ----> a; X ----> b\<rbrakk> \<Longrightarrow> a = b"
    1.60    using trivial_limit_sequentially by (rule tendsto_unique)
    1.61  
    1.62 -lemma (in bounded_linear) LIMSEQ:
    1.63 -  "X ----> a \<Longrightarrow> (\<lambda>n. f (X n)) ----> f a"
    1.64 -by (rule tendsto)
    1.65 -
    1.66 -lemma (in bounded_bilinear) LIMSEQ:
    1.67 -  "\<lbrakk>X ----> a; Y ----> b\<rbrakk> \<Longrightarrow> (\<lambda>n. X n ** Y n) ----> a ** b"
    1.68 -by (rule tendsto)
    1.69 -
    1.70 -lemma LIMSEQ_mult:
    1.71 -  fixes a b :: "'a::real_normed_algebra"
    1.72 -  shows "[| X ----> a; Y ----> b |] ==> (%n. X n * Y n) ----> a * b"
    1.73 -  by (rule tendsto_mult)
    1.74 -
    1.75  lemma increasing_LIMSEQ:
    1.76    fixes f :: "nat \<Rightarrow> real"
    1.77    assumes inc: "!!n. f n \<le> f (Suc n)"
    1.78 @@ -424,33 +380,6 @@
    1.79    shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
    1.80  unfolding Bseq_conv_Bfun by (rule Bfun_inverse)
    1.81  
    1.82 -lemma LIMSEQ_inverse:
    1.83 -  fixes a :: "'a::real_normed_div_algebra"
    1.84 -  shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. inverse (X n)) ----> inverse a"
    1.85 -by (rule tendsto_inverse)
    1.86 -
    1.87 -lemma LIMSEQ_divide:
    1.88 -  fixes a b :: "'a::real_normed_field"
    1.89 -  shows "\<lbrakk>X ----> a; Y ----> b; b \<noteq> 0\<rbrakk> \<Longrightarrow> (\<lambda>n. X n / Y n) ----> a / b"
    1.90 -by (rule tendsto_divide)
    1.91 -
    1.92 -lemma LIMSEQ_pow:
    1.93 -  fixes a :: "'a::{power, real_normed_algebra}"
    1.94 -  shows "X ----> a \<Longrightarrow> (\<lambda>n. (X n) ^ m) ----> a ^ m"
    1.95 -  by (rule tendsto_power)
    1.96 -
    1.97 -lemma LIMSEQ_setsum:
    1.98 -  fixes L :: "'a \<Rightarrow> 'b::real_normed_vector"
    1.99 -  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
   1.100 -  shows "(\<lambda>m. \<Sum>n\<in>S. X n m) ----> (\<Sum>n\<in>S. L n)"
   1.101 -using assms by (rule tendsto_setsum)
   1.102 -
   1.103 -lemma LIMSEQ_setprod:
   1.104 -  fixes L :: "'a \<Rightarrow> 'b::{real_normed_algebra,comm_ring_1}"
   1.105 -  assumes n: "\<And>n. n \<in> S \<Longrightarrow> X n ----> L n"
   1.106 -  shows "(\<lambda>m. \<Prod>n\<in>S. X n m) ----> (\<Prod>n\<in>S. L n)"
   1.107 -  using assms by (rule tendsto_setprod)
   1.108 -
   1.109  lemma LIMSEQ_add_const: (* FIXME: delete *)
   1.110    fixes a :: "'a::real_normed_vector"
   1.111    shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
   1.112 @@ -470,24 +399,12 @@
   1.113  lemma LIMSEQ_diff_approach_zero:
   1.114    fixes L :: "'a::real_normed_vector"
   1.115    shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
   1.116 -by (drule (1) LIMSEQ_add, simp)
   1.117 +  by (drule (1) tendsto_add, simp)
   1.118  
   1.119  lemma LIMSEQ_diff_approach_zero2:
   1.120    fixes L :: "'a::real_normed_vector"
   1.121    shows "f ----> L ==> (%x. f x - g x) ----> 0 ==> g ----> L"
   1.122 -by (drule (1) LIMSEQ_diff, simp)
   1.123 -
   1.124 -text{*A sequence tends to zero iff its abs does*}
   1.125 -lemma LIMSEQ_norm_zero:
   1.126 -  fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.127 -  shows "((\<lambda>n. norm (X n)) ----> 0) \<longleftrightarrow> (X ----> 0)"
   1.128 -  by (rule tendsto_norm_zero_iff)
   1.129 -
   1.130 -lemma LIMSEQ_rabs_zero: "((%n. \<bar>f n\<bar>) ----> 0) = (f ----> (0::real))"
   1.131 -  by (rule tendsto_rabs_zero_iff)
   1.132 -
   1.133 -lemma LIMSEQ_imp_rabs: "f ----> (l::real) ==> (%n. \<bar>f n\<bar>) ----> \<bar>l\<bar>"
   1.134 -  by (rule tendsto_rabs)
   1.135 +  by (drule (1) tendsto_diff, simp)
   1.136  
   1.137  text{*An unbounded sequence's inverse tends to 0*}
   1.138  
   1.139 @@ -517,16 +434,17 @@
   1.140  
   1.141  lemma LIMSEQ_inverse_real_of_nat_add:
   1.142       "(%n. r + inverse(real(Suc n))) ----> r"
   1.143 -by (cut_tac LIMSEQ_add [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
   1.144 +  using tendsto_add [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
   1.145  
   1.146  lemma LIMSEQ_inverse_real_of_nat_add_minus:
   1.147       "(%n. r + -inverse(real(Suc n))) ----> r"
   1.148 -by (cut_tac LIMSEQ_add_minus [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat], auto)
   1.149 +  using LIMSEQ_add_minus [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
   1.150  
   1.151  lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
   1.152       "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
   1.153 -by (cut_tac b=1 in
   1.154 -        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_inverse_real_of_nat_add_minus], auto)
   1.155 +  using tendsto_mult [OF tendsto_const
   1.156 +    LIMSEQ_inverse_real_of_nat_add_minus [of 1]]
   1.157 +  by auto
   1.158  
   1.159  lemma LIMSEQ_le_const:
   1.160    "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. a \<le> X n\<rbrakk> \<Longrightarrow> a \<le> x"
   1.161 @@ -542,7 +460,7 @@
   1.162    "\<lbrakk>X ----> (x::real); \<exists>N. \<forall>n\<ge>N. X n \<le> a\<rbrakk> \<Longrightarrow> x \<le> a"
   1.163  apply (subgoal_tac "- a \<le> - x", simp)
   1.164  apply (rule LIMSEQ_le_const)
   1.165 -apply (erule LIMSEQ_minus)
   1.166 +apply (erule tendsto_minus)
   1.167  apply simp
   1.168  done
   1.169  
   1.170 @@ -550,7 +468,7 @@
   1.171    "\<lbrakk>X ----> x; Y ----> y; \<exists>N. \<forall>n\<ge>N. X n \<le> Y n\<rbrakk> \<Longrightarrow> x \<le> (y::real)"
   1.172  apply (subgoal_tac "0 \<le> y - x", simp)
   1.173  apply (rule LIMSEQ_le_const)
   1.174 -apply (erule (1) LIMSEQ_diff)
   1.175 +apply (erule (1) tendsto_diff)
   1.176  apply (simp add: le_diff_eq)
   1.177  done
   1.178  
   1.179 @@ -572,14 +490,14 @@
   1.180  by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)
   1.181  
   1.182  lemma convergent_const: "convergent (\<lambda>n. c)"
   1.183 -by (rule convergentI, rule LIMSEQ_const)
   1.184 +  by (rule convergentI, rule tendsto_const)
   1.185  
   1.186  lemma convergent_add:
   1.187    fixes X Y :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.188    assumes "convergent (\<lambda>n. X n)"
   1.189    assumes "convergent (\<lambda>n. Y n)"
   1.190    shows "convergent (\<lambda>n. X n + Y n)"
   1.191 -using assms unfolding convergent_def by (fast intro: LIMSEQ_add)
   1.192 +  using assms unfolding convergent_def by (fast intro: tendsto_add)
   1.193  
   1.194  lemma convergent_setsum:
   1.195    fixes X :: "'a \<Rightarrow> nat \<Rightarrow> 'b::real_normed_vector"
   1.196 @@ -593,19 +511,19 @@
   1.197  lemma (in bounded_linear) convergent:
   1.198    assumes "convergent (\<lambda>n. X n)"
   1.199    shows "convergent (\<lambda>n. f (X n))"
   1.200 -using assms unfolding convergent_def by (fast intro: LIMSEQ)
   1.201 +  using assms unfolding convergent_def by (fast intro: tendsto)
   1.202  
   1.203  lemma (in bounded_bilinear) convergent:
   1.204    assumes "convergent (\<lambda>n. X n)" and "convergent (\<lambda>n. Y n)"
   1.205    shows "convergent (\<lambda>n. X n ** Y n)"
   1.206 -using assms unfolding convergent_def by (fast intro: LIMSEQ)
   1.207 +  using assms unfolding convergent_def by (fast intro: tendsto)
   1.208  
   1.209  lemma convergent_minus_iff:
   1.210    fixes X :: "nat \<Rightarrow> 'a::real_normed_vector"
   1.211    shows "convergent X \<longleftrightarrow> convergent (\<lambda>n. - X n)"
   1.212  apply (simp add: convergent_def)
   1.213 -apply (auto dest: LIMSEQ_minus)
   1.214 -apply (drule LIMSEQ_minus, auto)
   1.215 +apply (auto dest: tendsto_minus)
   1.216 +apply (drule tendsto_minus, auto)
   1.217  done
   1.218  
   1.219  lemma lim_le:
   1.220 @@ -661,7 +579,7 @@
   1.221        case True with top_down and `a ----> x` show ?thesis by auto
   1.222      next
   1.223        case False with `monoseq a`[unfolded monoseq_def] have "\<forall> m. \<forall> n \<ge> m. - a m \<le> - a n" by auto
   1.224 -      hence "- a m \<le> - x" using top_down[OF LIMSEQ_minus[OF `a ----> x`]] by blast
   1.225 +      hence "- a m \<le> - x" using top_down[OF tendsto_minus[OF `a ----> x`]] by blast
   1.226        hence False using `a m < x` by auto
   1.227        thus ?thesis ..
   1.228      qed
   1.229 @@ -676,7 +594,7 @@
   1.230        show ?thesis by blast
   1.231      next
   1.232        case False hence "- a m < - x" using `a m \<noteq> x` by auto
   1.233 -      with when_decided[OF LIMSEQ_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
   1.234 +      with when_decided[OF tendsto_minus[OF `a ----> x`] monoseq_minus[OF `monoseq a`], where m2=m]
   1.235        show ?thesis by auto
   1.236      qed
   1.237    qed auto
   1.238 @@ -855,8 +773,8 @@
   1.239      by (blast intro: const [of 0]) 
   1.240    have "X = (\<lambda>n. X 0)"
   1.241      by (blast intro: ext X)
   1.242 -  hence "L = X 0" using LIMSEQ_const [of "X 0"]
   1.243 -    by (auto intro: LIMSEQ_unique lim) 
   1.244 +  hence "L = X 0" using tendsto_const [of "X 0" sequentially]
   1.245 +    by (auto intro: LIMSEQ_unique lim)
   1.246    thus ?thesis
   1.247      by (blast intro: eq_refl X)
   1.248  qed
   1.249 @@ -867,7 +785,7 @@
   1.250    have inc: "incseq (\<lambda>n. - X n)" using dec
   1.251      by (simp add: decseq_eq_incseq)
   1.252    have "- X n \<le> - L" 
   1.253 -    by (blast intro: incseq_le [OF inc] LIMSEQ_minus lim) 
   1.254 +    by (blast intro: incseq_le [OF inc] tendsto_minus lim) 
   1.255    thus ?thesis
   1.256      by simp
   1.257  qed
   1.258 @@ -1187,7 +1105,7 @@
   1.259    "\<lbrakk>0 \<le> (x::real); x < 1\<rbrakk> \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
   1.260  proof (cases)
   1.261    assume "x = 0"
   1.262 -  hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: LIMSEQ_const)
   1.263 +  hence "(\<lambda>n. x ^ Suc n) ----> 0" by (simp add: tendsto_const)
   1.264    thus ?thesis by (rule LIMSEQ_imp_Suc)
   1.265  next
   1.266    assume "0 \<le> x" and "x \<noteq> 0"
   1.267 @@ -1204,14 +1122,14 @@
   1.268    fixes x :: "'a::{real_normed_algebra_1}"
   1.269    shows "norm x < 1 \<Longrightarrow> (\<lambda>n. x ^ n) ----> 0"
   1.270  apply (drule LIMSEQ_realpow_zero [OF norm_ge_zero])
   1.271 -apply (simp only: LIMSEQ_Zfun_iff, erule Zfun_le)
   1.272 +apply (simp only: tendsto_Zfun_iff, erule Zfun_le)
   1.273  apply (simp add: power_abs norm_power_ineq)
   1.274  done
   1.275  
   1.276  lemma LIMSEQ_divide_realpow_zero:
   1.277    "1 < (x::real) ==> (%n. a / (x ^ n)) ----> 0"
   1.278 -apply (cut_tac a = a and x1 = "inverse x" in
   1.279 -        LIMSEQ_mult [OF LIMSEQ_const LIMSEQ_realpow_zero])
   1.280 +using tendsto_mult [OF tendsto_const [of a]
   1.281 +  LIMSEQ_realpow_zero [of "inverse x"]]
   1.282  apply (auto simp add: divide_inverse power_inverse)
   1.283  apply (simp add: inverse_eq_divide pos_divide_less_eq)
   1.284  done
   1.285 @@ -1222,8 +1140,29 @@
   1.286  by (rule LIMSEQ_realpow_zero [OF abs_ge_zero])
   1.287  
   1.288  lemma LIMSEQ_rabs_realpow_zero2: "\<bar>c\<bar> < (1::real) ==> (%n. c ^ n) ----> 0"
   1.289 -apply (rule LIMSEQ_rabs_zero [THEN iffD1])
   1.290 +apply (rule tendsto_rabs_zero_cancel)
   1.291  apply (auto intro: LIMSEQ_rabs_realpow_zero simp add: power_abs)
   1.292  done
   1.293  
   1.294 +subsection {* Legacy theorem names *}
   1.295 +
   1.296 +lemmas LIMSEQ_Zfun_iff = tendsto_Zfun_iff [where F=sequentially]
   1.297 +lemmas LIMSEQ_const = tendsto_const [where F=sequentially]
   1.298 +lemmas LIMSEQ_norm = tendsto_norm [where F=sequentially]
   1.299 +lemmas LIMSEQ_add = tendsto_add [where F=sequentially]
   1.300 +lemmas LIMSEQ_minus = tendsto_minus [where F=sequentially]
   1.301 +lemmas LIMSEQ_minus_cancel = tendsto_minus_cancel [where F=sequentially]
   1.302 +lemmas LIMSEQ_diff = tendsto_diff [where F=sequentially]
   1.303 +lemmas (in bounded_linear) LIMSEQ = tendsto [where F=sequentially]
   1.304 +lemmas (in bounded_bilinear) LIMSEQ = tendsto [where F=sequentially]
   1.305 +lemmas LIMSEQ_mult = tendsto_mult [where F=sequentially]
   1.306 +lemmas LIMSEQ_inverse = tendsto_inverse [where F=sequentially]
   1.307 +lemmas LIMSEQ_divide = tendsto_divide [where F=sequentially]
   1.308 +lemmas LIMSEQ_pow = tendsto_power [where F=sequentially]
   1.309 +lemmas LIMSEQ_setsum = tendsto_setsum [where F=sequentially]
   1.310 +lemmas LIMSEQ_setprod = tendsto_setprod [where F=sequentially]
   1.311 +lemmas LIMSEQ_norm_zero = tendsto_norm_zero_iff [where F=sequentially]
   1.312 +lemmas LIMSEQ_rabs_zero = tendsto_rabs_zero_iff [where F=sequentially]
   1.313 +lemmas LIMSEQ_imp_rabs = tendsto_rabs [where F=sequentially]
   1.314 +
   1.315  end