1.1 --- a/NEWS Sun Sep 04 07:15:13 2011 -0700
1.2 +++ b/NEWS Sun Sep 04 09:49:45 2011 -0700
1.3 @@ -296,6 +296,9 @@
1.4 LIMSEQ_norm_zero ~> tendsto_norm_zero_iff
1.5 LIMSEQ_rabs_zero ~> tendsto_rabs_zero_iff
1.6 LIMSEQ_imp_rabs ~> tendsto_rabs
1.7 + LIMSEQ_add_minus ~> tendsto_add [OF _ tendsto_minus]
1.8 + LIMSEQ_add_const ~> tendsto_add [OF _ tendsto_const]
1.9 + LIMSEQ_diff_const ~> tendsto_diff [OF _ tendsto_const]
1.10 LIM_ident ~> tendsto_ident_at
1.11 LIM_const ~> tendsto_const
1.12 LIM_add ~> tendsto_add
2.1 --- a/src/HOL/SEQ.thy Sun Sep 04 07:15:13 2011 -0700
2.2 +++ b/src/HOL/SEQ.thy Sun Sep 04 09:49:45 2011 -0700
2.3 @@ -380,22 +380,6 @@
2.4 shows "\<lbrakk>X ----> a; a \<noteq> 0\<rbrakk> \<Longrightarrow> Bseq (\<lambda>n. inverse (X n))"
2.5 unfolding Bseq_conv_Bfun by (rule Bfun_inverse)
2.6
2.7 -lemma LIMSEQ_add_const: (* FIXME: delete *)
2.8 - fixes a :: "'a::real_normed_vector"
2.9 - shows "f ----> a ==> (%n.(f n + b)) ----> a + b"
2.10 -by (intro tendsto_intros)
2.11 -
2.12 -(* FIXME: delete *)
2.13 -lemma LIMSEQ_add_minus:
2.14 - fixes a b :: "'a::real_normed_vector"
2.15 - shows "[| X ----> a; Y ----> b |] ==> (%n. X n + -Y n) ----> a + -b"
2.16 -by (intro tendsto_intros)
2.17 -
2.18 -lemma LIMSEQ_diff_const: (* FIXME: delete *)
2.19 - fixes a b :: "'a::real_normed_vector"
2.20 - shows "f ----> a ==> (%n.(f n - b)) ----> a - b"
2.21 -by (intro tendsto_intros)
2.22 -
2.23 lemma LIMSEQ_diff_approach_zero:
2.24 fixes L :: "'a::real_normed_vector"
2.25 shows "g ----> L ==> (%x. f x - g x) ----> 0 ==> f ----> L"
2.26 @@ -438,7 +422,8 @@
2.27
2.28 lemma LIMSEQ_inverse_real_of_nat_add_minus:
2.29 "(%n. r + -inverse(real(Suc n))) ----> r"
2.30 - using LIMSEQ_add_minus [OF tendsto_const LIMSEQ_inverse_real_of_nat] by auto
2.31 + using tendsto_add [OF tendsto_const
2.32 + tendsto_minus [OF LIMSEQ_inverse_real_of_nat]] by auto
2.33
2.34 lemma LIMSEQ_inverse_real_of_nat_add_minus_mult:
2.35 "(%n. r*( 1 + -inverse(real(Suc n)))) ----> r"
3.1 --- a/src/HOL/Series.thy Sun Sep 04 07:15:13 2011 -0700
3.2 +++ b/src/HOL/Series.thy Sun Sep 04 09:49:45 2011 -0700
3.3 @@ -122,7 +122,7 @@
3.4 shows "f sums s ==> (\<lambda>n. f(n + k)) sums (s - (SUM i = 0..< k. f i))"
3.5 apply (unfold sums_def)
3.6 apply (simp add: sumr_offset)
3.7 - apply (rule LIMSEQ_diff_const)
3.8 + apply (rule tendsto_diff [OF _ tendsto_const])
3.9 apply (rule LIMSEQ_ignore_initial_segment)
3.10 apply assumption
3.11 done
3.12 @@ -166,7 +166,7 @@
3.13 proof -
3.14 from sumSuc[unfolded sums_def]
3.15 have "(\<lambda>i. \<Sum>n = Suc 0..<Suc i. f n) ----> l" unfolding setsum_reindex[OF inj_Suc] image_Suc_atLeastLessThan[symmetric] comp_def .
3.16 - from LIMSEQ_add_const[OF this, where b="f 0"]
3.17 + from tendsto_add[OF this tendsto_const, where b="f 0"]
3.18 have "(\<lambda>i. \<Sum>n = 0..<Suc i. f n) ----> l + f 0" unfolding add_commute setsum_head_upt_Suc[OF zero_less_Suc] .
3.19 thus ?thesis unfolding sums_def by (rule LIMSEQ_imp_Suc)
3.20 qed
3.21 @@ -625,7 +625,7 @@
3.22 apply (simp add:diff_Suc split:nat.splits)
3.23 apply (blast intro: norm_ratiotest_lemma)
3.24 apply (rule_tac x = "Suc N" in exI, clarify)
3.25 -apply(simp cong:setsum_ivl_cong)
3.26 +apply(simp cong del: setsum_cong cong: setsum_ivl_cong)
3.27 done
3.28
3.29 lemma ratio_test:
4.1 --- a/src/HOL/Transcendental.thy Sun Sep 04 07:15:13 2011 -0700
4.2 +++ b/src/HOL/Transcendental.thy Sun Sep 04 09:49:45 2011 -0700
4.3 @@ -1999,7 +1999,7 @@
4.4 apply (drule tan_total_pos)
4.5 apply (cut_tac [2] y="-y" in tan_total_pos, safe)
4.6 apply (rule_tac [3] x = "-x" in exI)
4.7 -apply (auto intro!: exI)
4.8 +apply (auto del: exI intro!: exI)
4.9 done
4.10
4.11 lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
4.12 @@ -2009,7 +2009,7 @@
4.13 apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
4.14 apply (rule_tac [4] Rolle)
4.15 apply (rule_tac [2] Rolle)
4.16 -apply (auto intro!: DERIV_tan DERIV_isCont exI
4.17 +apply (auto del: exI intro!: DERIV_tan DERIV_isCont exI
4.18 simp add: differentiable_def)
4.19 txt{*Now, simulate TRYALL*}
4.20 apply (rule_tac [!] DERIV_tan asm_rl)
4.21 @@ -2785,7 +2785,7 @@
4.22 have "norm (?diff 1 n - 0) < r" by auto }
4.23 thus "\<exists> N. \<forall> n \<ge> N. norm (?diff 1 n - 0) < r" by blast
4.24 qed
4.25 - from this[unfolded tendsto_rabs_zero_iff diff_minus add_commute[of "arctan 1"], THEN LIMSEQ_add_const, of "- arctan 1", THEN tendsto_minus]
4.26 + from this [unfolded tendsto_rabs_zero_iff, THEN tendsto_add [OF _ tendsto_const], of "- arctan 1", THEN tendsto_minus]
4.27 have "(?c 1) sums (arctan 1)" unfolding sums_def by auto
4.28 hence "arctan 1 = (\<Sum> i. ?c 1 i)" by (rule sums_unique)
4.29