src/HOL/Hyperreal/Transcendental.thy
 changeset 15081 32402f5624d1 parent 15079 2ef899e4526d child 15085 5693a977a767
```     1.1 --- a/src/HOL/Hyperreal/Transcendental.thy	Wed Jul 28 16:25:28 2004 +0200
1.2 +++ b/src/HOL/Hyperreal/Transcendental.thy	Wed Jul 28 16:25:40 2004 +0200
1.3 @@ -393,16 +393,16 @@
1.4
1.5  lemma powser_insidea:
1.6       "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]
1.7 -      ==> summable (%n. abs(f(n)) * (z ^ n))"
1.8 +      ==> summable (%n. \<bar>f(n)\<bar> * (z ^ n))"
1.9  apply (drule summable_LIMSEQ_zero)
1.10  apply (drule convergentI)
1.11  apply (simp add: Cauchy_convergent_iff [symmetric])
1.12  apply (drule Cauchy_Bseq)
1.13  apply (simp add: Bseq_def, safe)
1.14 -apply (rule_tac g = "%n. K * abs (z ^ n) * inverse (abs (x ^ n))" in summable_comparison_test)
1.15 +apply (rule_tac g = "%n. K * \<bar>z ^ n\<bar> * inverse (\<bar>x ^ n\<bar>)" in summable_comparison_test)
1.16  apply (rule_tac x = 0 in exI, safe)
1.17 -apply (subgoal_tac "0 < abs (x ^ n) ")
1.18 -apply (rule_tac c="abs (x ^ n)" in mult_right_le_imp_le)
1.19 +apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
1.20 +apply (rule_tac c="\<bar>x ^ n\<bar>" in mult_right_le_imp_le)
1.21  apply (auto simp add: mult_assoc power_abs)
1.22   prefer 2
1.23   apply (drule_tac x = 0 in spec, force)
1.24 @@ -412,7 +412,7 @@
1.25  apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
1.26  apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>))) " in exI)
1.27  apply (auto intro!: sums_mult simp add: mult_assoc)
1.28 -apply (subgoal_tac "abs (z ^ n) * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
1.29 +apply (subgoal_tac "\<bar>z ^ n\<bar> * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
1.30  apply (auto simp add: power_abs [symmetric])
1.31  apply (subgoal_tac "x \<noteq> 0")
1.32  apply (subgoal_tac [3] "x \<noteq> 0")
1.33 @@ -502,7 +502,7 @@
1.34                   del: sumr_Suc realpow_Suc)
1.35  done
1.36
1.37 -lemma lemma_termdiff3: "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; abs (z + h) \<le> K |]
1.38 +lemma lemma_termdiff3: "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<le> K |]
1.39        ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))
1.40            \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
1.41  apply (subst lemma_termdiff2, assumption)
1.42 @@ -532,7 +532,7 @@
1.43
1.44  lemma lemma_termdiff4:
1.45    "[| 0 < k;
1.46 -      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> abs(f h) \<le> K * \<bar>h\<bar>) |]
1.47 +      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]
1.48     ==> f -- 0 --> 0"
1.49  apply (unfold LIM_def, auto)
1.50  apply (subgoal_tac "0 \<le> K")
1.51 @@ -562,7 +562,7 @@
1.52                      (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]
1.53           ==> (%h. suminf(g h)) -- 0 --> 0"
1.54  apply (drule summable_sums)
1.55 -apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> abs (suminf (g h)) \<le> suminf f * \<bar>h\<bar>")
1.56 +apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>suminf (g h)\<bar> \<le> suminf f * \<bar>h\<bar>")
1.57  apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
1.58  apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
1.59   prefer 2
1.60 @@ -573,7 +573,7 @@
1.61  apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
1.62   apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
1.63    apply (rule_tac [2] x = 0 in exI, auto)
1.64 -apply (rule_tac j = "suminf (%n. abs (g h n))" in real_le_trans)
1.65 +apply (rule_tac j = "suminf (%n. \<bar>g h n\<bar>)" in real_le_trans)
1.66  apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
1.67  done
1.68
1.69 @@ -591,7 +591,7 @@
1.70  apply (drule dense, safe)
1.71  apply (frule real_less_sum_gt_zero)
1.72  apply (drule_tac
1.73 -         f = "%n. abs (c n) * real n * real (n - Suc 0) * (r ^ (n - 2))"
1.74 +         f = "%n. \<bar>c n\<bar> * real n * real (n - Suc 0) * (r ^ (n - 2))"
1.75       and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h)
1.76                               - (real n * (x ^ (n - Suc 0))))"
1.77         in lemma_termdiff5)
1.78 @@ -609,7 +609,7 @@
1.79  apply (drule sums_summable)
1.81  apply (simp add: diffs_def mult_ac)
1.82 -apply (subgoal_tac " (%n. real n * (real (Suc n) * (abs (c (Suc n)) * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * abs (c m) * inverse r) n * (r ^ n))")
1.83 +apply (subgoal_tac " (%n. real n * (real (Suc n) * (\<bar>c (Suc n)\<bar> * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * \<bar>c m\<bar> * inverse r) n * (r ^ n))")
1.84  apply auto
1.85    prefer 2
1.86    apply (rule ext)
1.87 @@ -899,7 +899,7 @@
1.88  lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
1.89  by (auto intro: positive_imp_inverse_positive)
1.90
1.91 -lemma abs_exp_cancel [simp]: "abs(exp x) = exp x"
1.92 +lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
1.93  by (auto simp add: abs_eqI2)
1.94
1.95  lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
1.96 @@ -1179,7 +1179,7 @@
1.97  lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
1.98  by arith
1.99
1.100 -lemma abs_sin_le_one [simp]: "abs(sin x) \<le> 1"
1.101 +lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
1.102  apply (auto simp add: linorder_not_less [symmetric])
1.103  apply (drule_tac n = "Suc 0" in power_gt1)
1.104  apply (auto simp del: realpow_Suc)
1.105 @@ -1197,7 +1197,7 @@
1.106  apply (simp add: abs_le_interval_iff del: abs_sin_le_one)
1.107  done
1.108
1.109 -lemma abs_cos_le_one [simp]: "abs(cos x) \<le> 1"
1.110 +lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
1.111  apply (auto simp add: linorder_not_less [symmetric])
1.112  apply (drule_tac n = "Suc 0" in power_gt1)
1.113  apply (auto simp del: realpow_Suc)
1.114 @@ -2065,7 +2065,7 @@
1.115  lemma isCont_exp [simp]: "isCont exp x"
1.116  by (rule DERIV_exp [THEN DERIV_isCont])
1.117
1.118 -lemma sin_zero_abs_cos_one: "sin x = 0 ==> abs(cos x) = 1"
1.119 +lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
1.120  by (auto simp add: sin_zero_iff even_mult_two_ex)
1.121
1.122  lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
1.123 @@ -2550,7 +2550,7 @@
1.124           \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
1.125           \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
1.126         ==> \<exists>e. 0 < e &
1.127 -             (\<forall>y. 0 < abs(y - f(x)) & abs(y - f(x)) < e --> f(g(y)) = y)"
1.128 +             (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
1.129  apply (drule isCont_inj_range)
1.130  prefer 2 apply (assumption, assumption, auto)
1.131  apply (rule_tac x = e in exI, auto)
```