--- a/src/HOL/Hyperreal/Transcendental.thy Tue Feb 22 18:42:22 2005 +0100
+++ b/src/HOL/Hyperreal/Transcendental.thy Wed Feb 23 10:23:22 2005 +0100
@@ -19,18 +19,18 @@
"sqrt x == root 2 x"
exp :: "real => real"
- "exp x == suminf(%n. inverse(real (fact n)) * (x ^ n))"
+ "exp x == \<Sum>n. inverse(real (fact n)) * (x ^ n)"
sin :: "real => real"
- "sin x == suminf(%n. (if even(n) then 0 else
- ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
+ "sin x == \<Sum>n. (if even(n) then 0 else
+ ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n"
diffs :: "(nat => real) => nat => real"
"diffs c == (%n. real (Suc n) * c(Suc n))"
cos :: "real => real"
- "cos x == suminf(%n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n))
- else 0) * x ^ n)"
+ "cos x == \<Sum>n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n))
+ else 0) * x ^ n"
ln :: "real => real"
"ln x == (@u. exp u = x)"
@@ -452,7 +452,7 @@
lemma diffs_equiv:
"summable (%n. (diffs c)(n) * (x ^ n)) ==>
(%n. real n * c(n) * (x ^ (n - Suc 0))) sums
- (suminf(%n. (diffs c)(n) * (x ^ n)))"
+ (\<Sum>n. (diffs c)(n) * (x ^ n))"
apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
apply (rule_tac [2] LIMSEQ_imp_Suc)
apply (drule summable_sums)
@@ -571,7 +571,7 @@
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
apply (rule_tac [2] x = 0 in exI, auto)
-apply (rule_tac j = "suminf (%n. \<bar>g h n\<bar>)" in real_le_trans)
+apply (rule_tac j = "\<Sum>n. \<bar>g h n\<bar>" in real_le_trans)
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
done
@@ -581,10 +581,9 @@
lemma termdiffs_aux:
"[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
- ==> (\<lambda>h. suminf
- (\<lambda>n. c n *
+ ==> (\<lambda>h. \<Sum>n. c n *
(((x + h) ^ n - x ^ n) * inverse h -
- real n * x ^ (n - Suc 0))))
+ real n * x ^ (n - Suc 0)))
-- 0 --> 0"
apply (drule dense, safe)
apply (frule real_less_sum_gt_zero)
@@ -648,16 +647,16 @@
summable(%n. (diffs c)(n) * (K ^ n));
summable(%n. (diffs(diffs c))(n) * (K ^ n));
\<bar>x\<bar> < \<bar>K\<bar> |]
- ==> DERIV (%x. suminf (%n. c(n) * (x ^ n))) x :>
- suminf (%n. (diffs c)(n) * (x ^ n))"
+ ==> DERIV (%x. \<Sum>n. c(n) * (x ^ n)) x :>
+ (\<Sum>n. (diffs c)(n) * (x ^ n))"
apply (simp add: deriv_def)
-apply (rule_tac g = "%h. suminf (%n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h)" in LIM_trans)
+apply (rule_tac g = "%h. \<Sum>n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h" in LIM_trans)
apply (simp add: LIM_def, safe)
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
apply (auto simp add: less_diff_eq)
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
apply (rule_tac y = 0 in order_le_less_trans, auto)
-apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
+apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (\<Sum>n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (\<Sum>n. (c n) * ( (x + xa) ^ n))")
apply (auto intro!: summable_sums)
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
apply (auto simp add: add_commute)
@@ -666,7 +665,7 @@
apply (rule sums_unique)
apply (simp add: diff_def divide_inverse add_ac mult_ac)
apply (rule LIM_zero_cancel)
-apply (rule_tac g = "%h. suminf (%n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0)))))" in LIM_trans)
+apply (rule_tac g = "%h. \<Sum>n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0))))" in LIM_trans)
prefer 2 apply (blast intro: termdiffs_aux)
apply (simp add: LIM_def, safe)
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
@@ -677,7 +676,7 @@
apply (rule_tac [2] powser_inside, auto)
apply (drule_tac c = c and x = x in diffs_equiv)
apply (frule sums_unique, auto)
-apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
+apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (\<Sum>n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (\<Sum>n. (c n) * ( (x + xa) ^ n))")
apply safe
apply (auto intro!: summable_sums)
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
@@ -742,33 +741,32 @@
text{*Now at last we can get the derivatives of exp, sin and cos*}
lemma lemma_sin_minus:
- "- sin x =
- suminf(%n. - ((if even n then 0
+ "- sin x = (\<Sum>n. - ((if even n then 0
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
by (auto intro!: sums_unique sums_minus sin_converges)
-lemma lemma_exp_ext: "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))"
+lemma lemma_exp_ext: "exp = (%x. \<Sum>n. inverse (real (fact n)) * x ^ n)"
by (auto intro!: ext simp add: exp_def)
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
apply (simp add: exp_def)
apply (subst lemma_exp_ext)
-apply (subgoal_tac "DERIV (%u. suminf (%n. inverse (real (fact n)) * u ^ n)) x :> suminf (%n. diffs (%n. inverse (real (fact n))) n * x ^ n) ")
+apply (subgoal_tac "DERIV (%u. \<Sum>n. inverse (real (fact n)) * u ^ n) x :> (\<Sum>n. diffs (%n. inverse (real (fact n))) n * x ^ n)")
apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
done
lemma lemma_sin_ext:
- "sin = (%x. suminf(%n.
+ "sin = (%x. \<Sum>n.
(if even n then 0
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *
- x ^ n))"
+ x ^ n)"
by (auto intro!: ext simp add: sin_def)
lemma lemma_cos_ext:
- "cos = (%x. suminf(%n.
+ "cos = (%x. \<Sum>n.
(if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
- x ^ n))"
+ x ^ n)"
by (auto intro!: ext simp add: cos_def)
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
@@ -792,7 +790,7 @@
lemma exp_zero [simp]: "exp 0 = 1"
proof -
have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
- suminf (\<lambda>n. inverse (real (fact n)) * 0 ^ n)"
+ (\<Sum>n. inverse (real (fact n)) * 0 ^ n)"
by (rule series_zero [rule_format, THEN sums_unique],
case_tac "m", auto)
thus ?thesis by (simp add: exp_def)
@@ -1325,7 +1323,7 @@
else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
x ^ k)
sums
- suminf (\<lambda>n. (if even n then 0
+ (\<Sum>n. (if even n then 0
else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
x ^ n)"
by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
@@ -1336,7 +1334,7 @@
apply (subgoal_tac
"(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
(- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
- sums suminf (\<lambda>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
+ sums (\<Sum>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
prefer 2
apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
apply (rotate_tac 2)
@@ -1394,8 +1392,7 @@
(if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
x ^ k)
sums
- suminf
- (\<lambda>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
+ (\<Sum>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
x ^ n)"
by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
thus ?thesis by (simp add: mult_ac cos_def)