src/HOL/Hyperreal/Transcendental.thy
author webertj
Wed, 30 Aug 2006 03:19:08 +0200
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(*  Title       : Transcendental.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998,1999 University of Cambridge
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                  1999,2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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imports NthRoot Fact HSeries EvenOdd Lim
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begin
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definition
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  root :: "[nat,real] => real"
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  "root n x = (SOME u. ((0::real) < x --> 0 < u) & (u ^ n = x))"
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  sqrt :: "real => real"
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  "sqrt x = root 2 x"
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  exp :: "real => real"
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  "exp x = (\<Sum>n. inverse(real (fact n)) * (x ^ n))"
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  sin :: "real => real"
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  "sin x = (\<Sum>n. (if even(n) then 0 else
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             ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
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  diffs :: "(nat => real) => nat => real"
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  "diffs c = (%n. real (Suc n) * c(Suc n))"
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  cos :: "real => real"
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  "cos x = (\<Sum>n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n)) 
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                            else 0) * x ^ n)"
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  ln :: "real => real"
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  "ln x = (SOME u. exp u = x)"
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  pi :: "real"
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  "pi = 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
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  tan :: "real => real"
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  "tan x = (sin x)/(cos x)"
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  arcsin :: "real => real"
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  "arcsin y = (SOME x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
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  arcos :: "real => real"
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  "arcos y = (SOME x. 0 \<le> x & x \<le> pi & cos x = y)"
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  arctan :: "real => real"
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  "arctan y = (SOME x. -(pi/2) < x & x < pi/2 & tan x = y)"
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lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
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apply (simp add: root_def)
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apply (safe intro!: some_equality power_0_Suc elim!: realpow_zero_zero)
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done
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lemma real_root_pow_pos: 
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     "0 < x ==> (root(Suc n) x) ^ (Suc n) = x"
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apply (simp add: root_def)
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apply (drule_tac n = n in realpow_pos_nth2)
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apply (auto intro: someI2)
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done
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lemma real_root_pow_pos2: "0 \<le> x ==> (root(Suc n) x) ^ (Suc n) = x"
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by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)
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lemma real_root_pos: 
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     "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
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apply (simp add: root_def)
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apply (rule some_equality)
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apply (frule_tac [2] n = n in zero_less_power)
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apply (auto simp add: zero_less_mult_iff)
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apply (rule_tac x = u and y = x in linorder_cases)
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apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
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apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
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apply (auto)
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done
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lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
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by (auto dest!: real_le_imp_less_or_eq real_root_pos)
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lemma real_root_pos_pos: 
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     "0 < x ==> 0 \<le> root(Suc n) x"
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apply (simp add: root_def)
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apply (drule_tac n = n in realpow_pos_nth2)
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apply (safe, rule someI2)
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apply (auto intro!: order_less_imp_le dest: zero_less_power 
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            simp add: zero_less_mult_iff)
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done
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lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
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by (auto dest!: real_le_imp_less_or_eq dest: real_root_pos_pos)
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lemma real_root_one [simp]: "root (Suc n) 1 = 1"
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apply (simp add: root_def)
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apply (rule some_equality, auto)
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apply (rule ccontr)
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apply (rule_tac x = u and y = 1 in linorder_cases)
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apply (drule_tac n = n in realpow_Suc_less_one)
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apply (drule_tac [4] n = n in power_gt1_lemma)
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apply (auto)
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done
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subsection{*Square Root*}
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text{*needed because 2 is a binary numeral!*}
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lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
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by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
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         add: nat_numeral_0_eq_0 [symmetric])
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lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
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by (simp add: sqrt_def)
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lemma real_sqrt_one [simp]: "sqrt 1 = 1"
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by (simp add: sqrt_def)
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lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
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apply (simp add: sqrt_def)
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apply (rule iffI) 
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 apply (cut_tac r = "root 2 x" in realpow_two_le)
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 apply (simp add: numeral_2_eq_2)
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apply (subst numeral_2_eq_2)
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apply (erule real_root_pow_pos2)
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done
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lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
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by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])
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lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
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by (simp)
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lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
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by (rule real_sqrt_pow2_iff [THEN iffD2], arith)
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lemma real_pow_sqrt_eq_sqrt_pow: 
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      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
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apply (simp add: sqrt_def)
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apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2)
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done
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lemma real_pow_sqrt_eq_sqrt_abs_pow2:
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     "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
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by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])
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lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
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apply (rule real_sqrt_abs_abs [THEN subst])
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apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
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apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
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apply (assumption, arith)
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done
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   154
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lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
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apply auto
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apply (cut_tac x = x and y = 0 in linorder_less_linear)
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apply (simp add: zero_less_mult_iff)
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   159
done
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   160
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lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
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apply (simp add: sqrt_def root_def)
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apply (drule realpow_pos_nth2 [where n=1], safe)
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apply (rule someI2, auto)
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done
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   166
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lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
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by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
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   169
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lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
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by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 
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(*we need to prove something like this:
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lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
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apply (case_tac n, simp) 
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apply (simp add: root_def) 
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apply (rule someI2 [of _ r], safe)
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apply (auto simp del: realpow_Suc dest: power_inject_base)
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*)
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   181
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lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
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apply (unfold sqrt_def root_def) 
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apply (rule someI2 [of _ r], auto) 
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apply (auto simp add: numeral_2_eq_2 simp del: realpow_Suc 
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            dest: power_inject_base) 
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done
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   188
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lemma real_sqrt_mult_distrib: 
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     "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
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apply (rule sqrt_eqI)
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apply (simp add: power_mult_distrib)  
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apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
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done
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lemma real_sqrt_mult_distrib2:
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     "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
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by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)
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   199
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lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
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by (auto intro!: real_sqrt_ge_zero)
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   202
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lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
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     "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
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by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
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   206
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lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
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     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
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by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)
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   210
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lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
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apply (rule abs_realpow_two [THEN subst])
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apply (rule real_sqrt_abs_abs [THEN subst])
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apply (subst real_pow_sqrt_eq_sqrt_pow)
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apply (auto simp add: numeral_2_eq_2)
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done
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   217
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lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
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   219
apply (rule realpow_two [THEN subst])
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apply (subst numeral_2_eq_2 [symmetric])
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apply (rule real_sqrt_abs)
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   222
done
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   223
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lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
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by simp
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lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
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apply (frule real_sqrt_pow2_gt_zero)
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apply (auto simp add: numeral_2_eq_2)
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done
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   231
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lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
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by (cut_tac n1 = 2 and a1 = "sqrt x" in power_inverse [symmetric], auto)
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   234
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lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
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apply (drule real_le_imp_less_or_eq)
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apply (auto dest: real_sqrt_not_eq_zero)
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   238
done
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   239
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lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
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by (auto simp add: real_sqrt_eq_zero_cancel)
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   242
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lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
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apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
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   245
apply (rule real_le_trans)
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   246
apply (auto simp del: realpow_Suc)
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apply (rule_tac n = 1 in realpow_increasing)
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apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
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done
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   250
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lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
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apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
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   253
done
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   254
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lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
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apply (rule_tac n = 1 in realpow_increasing)
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   257
apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
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            del: realpow_Suc)
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done
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   260
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   261
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subsection{*Exponential Function*}
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   263
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lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
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apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
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apply (cut_tac x = r in reals_Archimedean3, auto)
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apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
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   268
apply (rule_tac N = n and c = r in ratio_test)
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apply (auto simp add: abs_mult mult_assoc [symmetric] simp del: fact_Suc)
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apply (rule mult_right_mono)
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apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
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   272
apply (subst fact_Suc)
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apply (subst real_of_nat_mult)
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   274
apply (auto)
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apply (auto simp add: mult_assoc [symmetric] positive_imp_inverse_positive)
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apply (rule order_less_imp_le)
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   277
apply (rule_tac z1 = "real (Suc na)" in real_mult_less_iff1 [THEN iffD1])
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   278
apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc)
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   279
apply (erule order_less_trans)
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   280
apply (auto simp add: mult_less_cancel_left mult_ac)
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   281
done
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   282
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lemma summable_sin: 
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     "summable (%n.  
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           (if even n then 0  
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           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
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                x ^ n)"
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apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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   289
apply (rule_tac [2] summable_exp)
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   290
apply (rule_tac x = 0 in exI)
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   291
apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
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   292
done
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   293
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lemma summable_cos: 
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      "summable (%n.  
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           (if even n then  
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           (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
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apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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   299
apply (rule_tac [2] summable_exp)
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apply (rule_tac x = 0 in exI)
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apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
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done
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lemma lemma_STAR_sin [simp]:
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     "(if even n then 0  
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   306
       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
15251
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parents: 15241
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   307
by (induct "n", auto)
15229
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parents: 15228
diff changeset
   308
1eb23f805c06 new simprules for abs and for things like a/b<1
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   309
lemma lemma_STAR_cos [simp]:
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   310
     "0 < n -->  
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   311
      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
15251
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parents: 15241
diff changeset
   312
by (induct "n", auto)
15229
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parents: 15228
diff changeset
   313
1eb23f805c06 new simprules for abs and for things like a/b<1
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   314
lemma lemma_STAR_cos1 [simp]:
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   315
     "0 < n -->  
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   316
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
15251
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   317
by (induct "n", auto)
15229
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parents: 15228
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   318
1eb23f805c06 new simprules for abs and for things like a/b<1
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lemma lemma_STAR_cos2 [simp]:
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  "(\<Sum>n=1..<n. if even n then (- 1) ^ (n div 2)/(real (fact n)) *  0 ^ n 
333a88244569 comprehensive cleanup, replacing sumr by setsum
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   321
                         else 0) = 0"
15251
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parents: 15241
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   322
apply (induct "n")
15077
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parents: 15013
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   323
apply (case_tac [2] "n", auto)
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paulson
parents: 15013
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   324
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   325
89840837108e converting Hyperreal/Transcendental to Isar script
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parents: 15013
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   326
lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
15229
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paulson
parents: 15228
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   327
apply (simp add: exp_def)
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parents: 15013
diff changeset
   328
apply (rule summable_exp [THEN summable_sums])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   329
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   330
89840837108e converting Hyperreal/Transcendental to Isar script
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parents: 15013
diff changeset
   331
lemma sin_converges: 
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paulson
parents: 15013
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   332
      "(%n. (if even n then 0  
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paulson
parents: 15013
diff changeset
   333
            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   334
                 x ^ n) sums sin(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
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   335
apply (simp add: sin_def)
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paulson
parents: 15013
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   336
apply (rule summable_sin [THEN summable_sums])
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paulson
parents: 15013
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   337
done
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parents: 15013
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   338
89840837108e converting Hyperreal/Transcendental to Isar script
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   339
lemma cos_converges: 
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parents: 15013
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   340
      "(%n. (if even n then  
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parents: 15013
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   341
           (- 1) ^ (n div 2)/(real (fact n))  
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parents: 15013
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   342
           else 0) * x ^ n) sums cos(x)"
15229
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parents: 15228
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   343
apply (simp add: cos_def)
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parents: 15013
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   344
apply (rule summable_cos [THEN summable_sums])
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parents: 15013
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   345
done
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parents: 15013
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   346
15229
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lemma lemma_realpow_diff [rule_format (no_asm)]:
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parents: 15228
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   348
     "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
15251
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paulson
parents: 15241
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   349
apply (induct "n", auto)
15077
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parents: 15013
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   350
apply (subgoal_tac "p = Suc n")
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paulson
parents: 15013
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   351
apply (simp (no_asm_simp), auto)
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paulson
parents: 15013
diff changeset
   352
apply (drule sym)
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paulson
parents: 15013
diff changeset
   353
apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
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   354
       del: realpow_Suc)
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parents: 15013
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   355
done
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paulson
parents: 15013
diff changeset
   356
89840837108e converting Hyperreal/Transcendental to Isar script
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parents: 15013
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   357
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parents: 15013
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   358
subsection{*Properties of Power Series*}
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paulson
parents: 15013
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   359
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parents: 15013
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   360
lemma lemma_realpow_diff_sumr:
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   361
     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   362
      y * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18585
diff changeset
   363
by (auto simp add: setsum_right_distrib lemma_realpow_diff mult_ac
16641
fce796ad9c2b Simplified some proofs (thanks to strong_setsum_cong).
berghofe
parents: 15561
diff changeset
   364
  simp del: setsum_op_ivl_Suc cong: strong_setsum_cong)
15077
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paulson
parents: 15013
diff changeset
   365
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
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parents: 15228
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   366
lemma lemma_realpow_diff_sumr2:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
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   367
     "x ^ (Suc n) - y ^ (Suc n) =  
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   368
      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
15251
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paulson
parents: 15241
diff changeset
   369
apply (induct "n", simp)
15561
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nipkow
parents: 15546
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   370
apply (auto simp del: setsum_op_ivl_Suc)
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
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   371
apply (subst setsum_op_ivl_Suc)
15077
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paulson
parents: 15013
diff changeset
   372
apply (drule sym)
15561
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nipkow
parents: 15546
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   373
apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_op_ivl_Suc)
15077
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parents: 15013
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   374
done
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paulson
parents: 15013
diff changeset
   375
15229
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paulson
parents: 15228
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   376
lemma lemma_realpow_rev_sumr:
15539
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parents: 15536
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   377
     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   378
      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p)::real)"
15077
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paulson
parents: 15013
diff changeset
   379
apply (case_tac "x = y")
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   380
apply (auto simp add: mult_commute power_add [symmetric] simp del: setsum_op_ivl_Suc)
15077
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paulson
parents: 15013
diff changeset
   381
apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   382
apply (rule_tac [2] minus_minus [THEN subst], simp)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   383
apply (subst minus_mult_left)
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   384
apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: setsum_op_ivl_Suc)
15077
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paulson
parents: 15013
diff changeset
   385
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   386
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
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   387
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   388
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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paulson
parents: 15013
diff changeset
   389
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   390
lemma powser_insidea:
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paulson
parents: 15013
diff changeset
   391
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   392
      ==> summable (%n. \<bar>f(n)\<bar> * (z ^ n))"
15077
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paulson
parents: 15013
diff changeset
   393
apply (drule summable_LIMSEQ_zero)
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paulson
parents: 15013
diff changeset
   394
apply (drule convergentI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   395
apply (simp add: Cauchy_convergent_iff [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   396
apply (drule Cauchy_Bseq)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   397
apply (simp add: Bseq_def, safe)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   398
apply (rule_tac g = "%n. K * \<bar>z ^ n\<bar> * inverse (\<bar>x ^ n\<bar>)" in summable_comparison_test)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   399
apply (rule_tac x = 0 in exI, safe)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   400
apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   401
apply (rule_tac c="\<bar>x ^ n\<bar>" in mult_right_le_imp_le) 
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   402
apply (auto simp add: mult_assoc power_abs abs_mult)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   403
 prefer 2
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   404
 apply (drule_tac x = 0 in spec, force)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   405
apply (auto simp add: power_abs mult_ac)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   406
apply (rule_tac a2 = "z ^ n" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   407
       in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   408
apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   409
apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>)))" in exI)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   410
apply (auto intro!: sums_mult simp add: mult_assoc)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   411
apply (subgoal_tac "\<bar>z ^ n\<bar> * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   412
apply (auto simp add: power_abs [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   413
apply (subgoal_tac "x \<noteq> 0")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   414
apply (subgoal_tac [3] "x \<noteq> 0")
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   415
apply (auto simp del: abs_inverse 
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   416
            simp add: abs_inverse [symmetric] realpow_not_zero
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   417
            abs_mult [symmetric] power_inverse power_mult_distrib [symmetric])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   418
apply (auto intro!: geometric_sums  simp add: power_abs inverse_eq_divide)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   419
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   420
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   421
lemma powser_inside:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   422
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   423
      ==> summable (%n. f(n) * (z ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   424
apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   425
apply (auto intro: summable_rabs_cancel simp add: abs_mult power_abs [symmetric])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   426
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   427
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   428
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   429
subsection{*Differentiation of Power Series*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   430
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   431
text{*Lemma about distributing negation over it*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   432
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   433
by (simp add: diffs_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   434
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   435
text{*Show that we can shift the terms down one*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   436
lemma lemma_diffs:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   437
     "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   438
      (\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) +  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   439
      (real n * c(n) * x ^ (n - Suc 0))"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
   440
apply (induct "n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   441
apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   442
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   443
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   444
lemma lemma_diffs2:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   445
     "(\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   446
      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   447
      (real n * c(n) * x ^ (n - Suc 0))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   448
by (auto simp add: lemma_diffs)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   449
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   450
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   451
lemma diffs_equiv:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   452
     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   453
      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   454
         (\<Sum>n. (diffs c)(n) * (x ^ n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   455
apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   456
apply (rule_tac [2] LIMSEQ_imp_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   457
apply (drule summable_sums) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   458
apply (auto simp add: sums_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   459
apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   460
apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   461
apply (simp add: diffs_def summable_LIMSEQ_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   462
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   463
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   464
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   465
subsection{*Term-by-Term Differentiability of Power Series*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   466
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   467
lemma lemma_termdiff1:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   468
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   469
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p)))::real)"
16641
fce796ad9c2b Simplified some proofs (thanks to strong_setsum_cong).
berghofe
parents: 15561
diff changeset
   470
by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac
fce796ad9c2b Simplified some proofs (thanks to strong_setsum_cong).
berghofe
parents: 15561
diff changeset
   471
  cong: strong_setsum_cong)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   472
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   473
lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   474
by (simp add: less_iff_Suc_add)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   475
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   476
lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   477
by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   478
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   479
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   480
lemma lemma_termdiff2:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   481
  "h \<noteq> 0 ==>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   482
   (((z + h) ^ n) - (z ^ n)) * inverse h - real n * (z ^ (n - Suc 0)) =
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   483
   h * (\<Sum>p=0..< n - Suc 0. (z ^ p) *
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   484
       (\<Sum>q=0..< (n - Suc 0) - p. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   485
apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   486
apply (simp add: right_diff_distrib mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   487
apply (simp add: mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   488
apply (case_tac "n")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   489
apply (auto simp add: lemma_realpow_diff_sumr2 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   490
                      right_diff_distrib [symmetric] mult_assoc
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   491
            simp del: realpow_Suc setsum_op_ivl_Suc)
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   492
apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_op_ivl_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   493
apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18585
diff changeset
   494
                sumdiff lemma_termdiff1 setsum_right_distrib)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   495
apply (auto intro!: setsum_cong[OF refl] simp add: diff_minus real_add_assoc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   496
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
19279
48b527d0331b Renamed setsum_mult to setsum_right_distrib.
ballarin
parents: 18585
diff changeset
   497
apply (auto simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac simp
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   498
                 del: setsum_op_ivl_Suc realpow_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   499
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   500
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   501
lemma lemma_termdiff3:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   502
     "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<le> K |]  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   503
      ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   504
          \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   505
apply (subst lemma_termdiff2, assumption)
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   506
apply (simp add: mult_commute abs_mult) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   507
apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   508
apply (rule setsum_abs [THEN real_le_trans])
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   509
apply (simp add: mult_assoc [symmetric] abs_mult)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   510
apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
15542
ee6cd48cf840 more fine tuniung
nipkow
parents: 15539
diff changeset
   511
apply (auto intro!: real_setsum_nat_ivl_bounded)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   512
apply (case_tac "n", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   513
apply (drule less_add_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   514
(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   515
apply clarify 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   516
apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   517
                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   518
apply (simp (no_asm_simp) add: power_add del: setsum_op_ivl_Suc)
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   519
apply (auto intro!: mult_mono simp del: setsum_op_ivl_Suc)
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   520
apply (auto intro!: power_mono simp add: power_abs
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   521
           simp del: setsum_op_ivl_Suc)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   522
apply (rule_tac j = "real (Suc d) * (K ^ d)" in real_le_trans)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   523
apply (subgoal_tac [2] "0 \<le> K")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   524
apply (drule_tac [2] n = d in zero_le_power)
15561
045a07ac35a7 another reorganization of setsums and intervals
nipkow
parents: 15546
diff changeset
   525
apply (auto simp del: setsum_op_ivl_Suc)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   526
apply (rule setsum_abs [THEN real_le_trans])
16924
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   527
apply (rule real_setsum_nat_ivl_bounded)
04246269386e removed the dependence on abs_mult
paulson
parents: 16819
diff changeset
   528
apply (auto dest!: less_add_one intro!: mult_mono simp add: power_add abs_mult)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   529
apply (auto intro!: power_mono zero_le_power simp add: power_abs)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   530
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   531
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   532
lemma lemma_termdiff4: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   533
  "[| 0 < k;  
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   534
      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   535
   ==> f -- 0 --> 0"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   536
apply (simp add: LIM_def, auto)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   537
apply (subgoal_tac "0 \<le> K")
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   538
 prefer 2
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   539
 apply (drule_tac x = "k/2" in spec)
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   540
apply (simp add: ); 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   541
 apply (subgoal_tac "0 \<le> K*k", simp add: zero_le_mult_iff) 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   542
 apply (force intro: order_trans [of _ "\<bar>f (k / 2)\<bar> * 2"]) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   543
apply (drule real_le_imp_less_or_eq, auto)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   544
apply (subgoal_tac "0 < (r * inverse K) / 2")
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   545
apply (drule_tac ?d1.0 = "(r * inverse K) / 2" and ?d2.0 = k in real_lbound_gt_zero)
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   546
apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff zero_less_divide_iff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   547
apply (rule_tac x = e in exI, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   548
apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   549
apply (force ); 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   550
apply (rule_tac y = "K * e" in order_less_trans)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   551
apply (simp add: mult_less_cancel_left)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   552
apply (rule_tac c = "inverse K" in mult_right_less_imp_less)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   553
apply (auto simp add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   554
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   555
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   556
lemma lemma_termdiff5:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   557
     "[| 0 < k;  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   558
            summable f;  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   559
            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   560
                    (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   561
         ==> (%h. suminf(g h)) -- 0 --> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   562
apply (drule summable_sums)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   563
apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>suminf (g h)\<bar> \<le> suminf f * \<bar>h\<bar>")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   564
apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   565
apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   566
 prefer 2
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   567
 apply (simp add: summable_def) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   568
 apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   569
 apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   570
 apply (simp add: mult_ac) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   571
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   572
 apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   573
  apply (rule_tac [2] x = 0 in exI, auto)
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   574
apply (rule_tac j = "\<Sum>n. \<bar>g h n\<bar>" in real_le_trans)
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   575
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult2])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   576
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   577
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   578
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   579
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   580
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   581
ML {* fast_arith_split_limit := 0; *}  (* FIXME: rewrite proofs *)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   582
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   583
lemma termdiffs_aux:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   584
     "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   585
    ==> (\<lambda>h. \<Sum>n. c n *
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   586
                  (((x + h) ^ n - x ^ n) * inverse h -
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   587
                   real n * x ^ (n - Suc 0)))
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   588
       -- 0 --> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   589
apply (drule dense, safe)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   590
apply (frule real_less_sum_gt_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   591
apply (drule_tac
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   592
         f = "%n. \<bar>c n\<bar> * real n * real (n - Suc 0) * (r ^ (n - 2))" 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   593
     and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   594
                             - (real n * (x ^ (n - Suc 0))))" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   595
       in lemma_termdiff5)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   596
apply (auto simp add: add_commute)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   597
apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   598
apply (rule_tac [2] x = K in powser_insidea, auto)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   599
apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   600
apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_of_nonneg], auto)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   601
apply (simp add: diffs_def mult_assoc [symmetric])
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   602
apply (subgoal_tac
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   603
        "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   604
              = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   605
apply (auto simp add: abs_mult)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   606
apply (drule diffs_equiv)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   607
apply (drule sums_summable)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   608
apply (simp_all add: diffs_def) 
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   609
apply (simp add: diffs_def mult_ac)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   610
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))")
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   611
apply auto
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   612
  prefer 2
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   613
  apply (rule ext)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   614
  apply (simp add: diffs_def) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   615
  apply (case_tac "n", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   616
txt{*23*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   617
   apply (drule abs_ge_zero [THEN order_le_less_trans])
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   618
   apply (simp add: mult_ac) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   619
  apply (drule abs_ge_zero [THEN order_le_less_trans])
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   620
  apply (simp add: mult_ac) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   621
 apply (drule diffs_equiv)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   622
 apply (drule sums_summable)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   623
apply (subgoal_tac
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   624
          "summable
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   625
            (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   626
                 r ^ (n - Suc 0)) =
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   627
           summable
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   628
            (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   629
apply simp 
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   630
apply (rule_tac f = summable in arg_cong, rule ext)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   631
txt{*33*}
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   632
apply (case_tac "n", auto)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   633
apply (case_tac "nat", auto)
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   634
apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   635
apply (drule abs_ge_zero [THEN order_le_less_trans])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   636
apply (simp add: mult_assoc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   637
apply (rule mult_left_mono)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   638
 prefer 2 apply arith 
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   639
apply (subst add_commute)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   640
apply (simp (no_asm) add: mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   641
apply (rule lemma_termdiff3)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   642
apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   643
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   644
20256
5024ba0831a6 lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents: 20217
diff changeset
   645
ML {* fast_arith_split_limit := 9; *}  (* FIXME *)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   646
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   647
lemma termdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   648
    "[| summable(%n. c(n) * (K ^ n));  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   649
        summable(%n. (diffs c)(n) * (K ^ n));  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   650
        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   651
        \<bar>x\<bar> < \<bar>K\<bar> |]  
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   652
     ==> DERIV (%x. \<Sum>n. c(n) * (x ^ n))  x :>  
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   653
             (\<Sum>n. (diffs c)(n) * (x ^ n))"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   654
apply (simp add: deriv_def)
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   655
apply (rule_tac g = "%h. \<Sum>n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h" in LIM_trans)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   656
apply (simp add: LIM_def, safe)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   657
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   658
apply (auto simp add: less_diff_eq)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   659
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   660
apply (rule_tac y = 0 in order_le_less_trans, auto)
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   661
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))")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   662
apply (auto intro!: summable_sums)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   663
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   664
apply (auto simp add: add_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   665
apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   666
apply (drule_tac f = "(%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
15085
5693a977a767 removed some [iff] declarations from RealDef.thy, concerning inequalities
paulson
parents: 15081
diff changeset
   667
apply (rule sums_unique)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 15077
diff changeset
   668
apply (simp add: diff_def divide_inverse add_ac mult_ac)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   669
apply (rule LIM_zero_cancel)
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   670
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)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   671
 prefer 2 apply (blast intro: termdiffs_aux) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   672
apply (simp add: LIM_def, safe)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   673
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   674
apply (auto simp add: less_diff_eq)
20432
07ec57376051 lin_arith_prover: splitting reverted because of performance loss
webertj
parents: 20256
diff changeset
   675
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   676
apply (rule_tac y = 0 in order_le_less_trans, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   677
apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   678
apply (rule_tac [2] powser_inside, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   679
apply (drule_tac c = c and x = x in diffs_equiv)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   680
apply (frule sums_unique, auto)
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   681
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))")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   682
apply safe
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   683
apply (auto intro!: summable_sums)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   684
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   685
apply (auto simp add: add_commute)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   686
apply (frule_tac x = "(%n. c n * (xa + x) ^ n) " and y = "(%n. c n * x ^ n)" in sums_diff, assumption)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   687
apply (simp add: suminf_diff [OF sums_summable sums_summable] 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   688
               right_diff_distrib [symmetric])
16819
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   689
apply (subst suminf_diff)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   690
apply (rule summable_mult2)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   691
apply (erule sums_summable)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   692
apply (erule sums_summable)
00d8f9300d13 Additions to the Real (and Hyperreal) libraries:
avigad
parents: 16775
diff changeset
   693
apply (simp add: ring_eq_simps)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   694
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   695
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   696
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   697
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   698
lemma exp_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   699
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   700
by (simp add: diffs_def mult_assoc [symmetric] del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   701
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   702
lemma sin_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   703
      "diffs(%n. if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   704
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   705
       = (%n. if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   706
                 (- 1) ^ (n div 2)/(real (fact n))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   707
              else 0)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   708
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   709
         simp add: diffs_def divide_inverse simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   710
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   711
lemma sin_fdiffs2: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   712
       "diffs(%n. if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   713
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   714
       = (if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   715
                 (- 1) ^ (n div 2)/(real (fact n))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   716
              else 0)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   717
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   718
         simp add: diffs_def divide_inverse simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   719
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   720
lemma cos_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   721
      "diffs(%n. if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   722
                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   723
       = (%n. - (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   724
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   725
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   726
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   727
         simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   728
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   729
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   730
lemma cos_fdiffs2: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   731
      "diffs(%n. if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   732
                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   733
       = - (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   734
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   735
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   736
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   737
         simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   738
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   739
text{*Now at last we can get the derivatives of exp, sin and cos*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   740
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   741
lemma lemma_sin_minus:
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   742
     "- sin x = (\<Sum>n. - ((if even n then 0 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   743
                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   744
by (auto intro!: sums_unique sums_minus sin_converges)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   745
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   746
lemma lemma_exp_ext: "exp = (%x. \<Sum>n. inverse (real (fact n)) * x ^ n)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   747
by (auto intro!: ext simp add: exp_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   748
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   749
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   750
apply (simp add: exp_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   751
apply (subst lemma_exp_ext)
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   752
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)")
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   753
apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   754
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   755
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   756
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   757
lemma lemma_sin_ext:
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   758
     "sin = (%x. \<Sum>n. 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   759
                   (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   760
                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   761
                   x ^ n)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   762
by (auto intro!: ext simp add: sin_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   763
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   764
lemma lemma_cos_ext:
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   765
     "cos = (%x. \<Sum>n. 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   766
                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   767
                   x ^ n)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   768
by (auto intro!: ext simp add: cos_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   769
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   770
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   771
apply (simp add: cos_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   772
apply (subst lemma_sin_ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   773
apply (auto simp add: sin_fdiffs2 [symmetric])
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   774
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   775
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   776
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   777
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   778
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   779
apply (subst lemma_cos_ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   780
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   781
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
20217
25b068a99d2b linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents: 19765
diff changeset
   782
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   783
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   784
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   785
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   786
subsection{*Properties of the Exponential Function*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   787
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   788
lemma exp_zero [simp]: "exp 0 = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   789
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   790
  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
15546
5188ce7316b7 suminf -> \<Sum>
nipkow
parents: 15544
diff changeset
   791
        (\<Sum>n. inverse (real (fact n)) * 0 ^ n)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   792
    by (rule series_zero [rule_format, THEN sums_unique],
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   793
        case_tac "m", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   794
  thus ?thesis by (simp add:  exp_def) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   795
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   796
17014
ad5ceb90877d renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents: 16924
diff changeset
   797
lemma exp_ge_add_one_self_aux: "0 \<le> x ==> (1 + x) \<le> exp(x)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   798
apply (drule real_le_imp_less_or_eq, auto)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   799
apply (simp add: exp_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   800
apply (rule real_le_trans)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   801
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   802
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   803
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   804
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   805
lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   806
apply (rule order_less_le_trans)
17014
ad5ceb90877d renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents: 16924
diff changeset
   807
apply (rule_tac [2] exp_ge_add_one_self_aux, auto)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   808
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   809
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   810
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   811
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   812
  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   813
    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   814
  thus ?thesis by (simp add: o_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   815
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   816
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   817
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   818
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   819
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   820
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   821
  thus ?thesis by (simp add: o_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   822
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   823
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   824
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   825
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   826
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   827
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   828
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   829
  thus ?thesis by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   830
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   831
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   832
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   833
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   834
  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   835
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   836
    by (rule DERIV_isconst_all) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   837
  thus ?thesis by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   838
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   839
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   840
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   841
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   842
  have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   843
  thus ?thesis by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   844
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   845
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   846
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   847
by (simp add: mult_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   848
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   849
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   850
lemma exp_minus: "exp(-x) = inverse(exp(x))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   851
by (auto intro: inverse_unique [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   852
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   853
lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   854
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   855
  have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   856
  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   857
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   858
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   859
text{*Proof: because every exponential can be seen as a square.*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   860
lemma exp_ge_zero [simp]: "0 \<le> exp x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   861
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   862
apply (subst exp_add, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   863
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   864
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   865
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   866
apply (cut_tac x = x in exp_mult_minus2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   867
apply (auto simp del: exp_mult_minus2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   868
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   869
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   870
lemma exp_gt_zero [simp]: "0 < exp x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   871
by (simp add: order_less_le)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   872
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   873
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   874
by (auto intro: positive_imp_inverse_positive)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   875
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   876
lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   877
by auto
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   878
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   879
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
   880
apply (induct "n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   881
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   882
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   883
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   884
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   885
apply (simp add: diff_minus divide_inverse)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   886
apply (simp (no_asm) add: exp_add exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   887
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   888
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   889
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   890
lemma exp_less_mono:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   891
  assumes xy: "x < y" shows "exp x < exp y"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   892
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   893
  have "1 < exp (y + - x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   894
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   895
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   896
    by (auto simp add: exp_add exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   897
  thus ?thesis
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   898
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15140
diff changeset
   899
             del: divide_self_if)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   900
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   901
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   902
lemma exp_less_cancel: "exp x < exp y ==> x < y"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15140
diff changeset
   903
apply (simp add: linorder_not_le [symmetric]) 
4d332d10fa3d revised simprules for division
paulson
parents: 15140
diff changeset
   904
apply (auto simp add: order_le_less exp_less_mono) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   905
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   906
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   907
lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   908
by (auto intro: exp_less_mono exp_less_cancel)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   909
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   910
lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   911
by (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   912
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   913
lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   914
by (simp add: order_eq_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   915
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   916
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   917
apply (rule IVT)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   918
apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   919
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   920
apply simp 
17014
ad5ceb90877d renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents: 16924
diff changeset
   921
apply (rule exp_ge_add_one_self_aux, simp)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   922
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   923
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   924
lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   925
apply (rule_tac x = 1 and y = y in linorder_cases)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   926
apply (drule order_less_imp_le [THEN lemma_exp_total])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   927
apply (rule_tac [2] x = 0 in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   928
apply (frule_tac [3] real_inverse_gt_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   929
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   930
apply (rule_tac x = "-x" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   931
apply (simp add: exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   932
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   933
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   934
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   935
subsection{*Properties of the Logarithmic Function*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   936
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   937
lemma ln_exp[simp]: "ln(exp x) = x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   938
by (simp add: ln_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   939
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   940
lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   941
apply (auto dest: exp_total)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   942
apply (erule subst, simp) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   943
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   944
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   945
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   946
apply (rule exp_inj_iff [THEN iffD1])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   947
apply (frule real_mult_order)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   948
apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   949
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   950
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   951
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   952
apply (simp only: exp_ln_iff [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   953
apply (erule subst)+
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   954
apply simp 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   955
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   956
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   957
lemma ln_one[simp]: "ln 1 = 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   958
by (rule exp_inj_iff [THEN iffD1], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   959
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   960
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   961
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   962
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   963
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   964
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   965
lemma ln_div: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   966
    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   967
apply (simp add: divide_inverse)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   968
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   969
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   970
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   971
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   972
apply (simp only: exp_ln_iff [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   973
apply (erule subst)+
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   974
apply simp 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   975
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   976
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   977
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   978
by (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   979
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   980
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   981
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   982
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   983
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   984
apply (rule ln_exp [THEN subst])
17014
ad5ceb90877d renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents: 16924
diff changeset
   985
apply (rule ln_le_cancel_iff [THEN iffD2]) 
ad5ceb90877d renamed exp_ge_add_one_self to exp_ge_add_one_self_aux
avigad
parents: 16924
diff changeset
   986
apply (auto simp add: exp_ge_add_one_self_aux)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   987
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   988
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   989
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   990
apply (rule order_less_le_trans)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   991
apply (rule_tac [2] ln_add_one_self_le_self)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   992
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   993
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   994
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
   995
lemma ln_ge_zero [simp]:
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   996
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   997
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   998
  have "0 < x" using x by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   999
  hence "exp 0 \<le> exp (ln x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1000
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1001
  thus ?thesis by (simp only: exp_le_cancel_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1002
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1003
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1004
lemma ln_ge_zero_imp_ge_one:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1005
  assumes ln: "0 \<le> ln x" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1006
      and x:  "0 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1007
  shows "1 \<le> x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1008
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1009
  from ln have "ln 1 \<le> ln x" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1010
  thus ?thesis by (simp add: x del: ln_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1011
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1012
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1013
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1014
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1015
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1016
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1017
by (insert ln_ge_zero_iff [of x], arith)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1018
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1019
lemma ln_gt_zero:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1020
  assumes x: "1 < x" shows "0 < ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1021
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1022
  have "0 < x" using x by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1023
  hence "exp 0 < exp (ln x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1024
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1025
  thus ?thesis  by (simp only: exp_less_cancel_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1026
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1027
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1028
lemma ln_gt_zero_imp_gt_one:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1029
  assumes ln: "0 < ln x" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1030
      and x:  "0 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1031
  shows "1 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1032
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1033
  from ln have "ln 1 < ln x" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1034
  thus ?thesis by (simp add: x del: ln_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1035
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1036
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1037
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1038
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1039
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1040
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1041
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1042
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1043
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1044
by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1045
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1046
lemma exp_ln_eq: "exp u = x ==> ln x = u"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1047
by auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1048
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1049
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1050
subsection{*Basic Properties of the Trigonometric Functions*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1051
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1052
lemma sin_zero [simp]: "sin 0 = 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1053
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1054
         simp add: sin_def sums_def simp del: power_0_left)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1055
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
  1056
lemma lemma_series_zero2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
  1057
 "(\<forall>m. n \<le> m --> f m = 0) --> f sums setsum f {0..<n}"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1058
by (auto intro: series_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1059
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1060
lemma cos_zero [simp]: "cos 0 = 1"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1061
apply (simp add: cos_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1062
apply (rule sums_unique [symmetric])
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1063
apply (cut_tac n = 1 and f = "(%n. (if even n then (- 1) ^ (n div 2) / (real (fact n)) else 0) * 0 ^ n)" in lemma_series_zero2)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1064
apply auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1065
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1066
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1067
lemma DERIV_sin_sin_mult [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1068
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1069
by (rule DERIV_mult, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1070
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1071
lemma DERIV_sin_sin_mult2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1072
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1073
apply (cut_tac x = x in DERIV_sin_sin_mult)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1074
apply (auto simp add: mult_assoc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1075
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1076
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1077
lemma DERIV_sin_realpow2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1078
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1079
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1080
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1081
lemma DERIV_sin_realpow2a [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1082
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1083
by (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1084
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1085
lemma DERIV_cos_cos_mult [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1086
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1087
by (rule DERIV_mult, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1088
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1089
lemma DERIV_cos_cos_mult2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1090
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1091
apply (cut_tac x = x in DERIV_cos_cos_mult)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1092
apply (auto simp add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1093
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1094
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1095
lemma DERIV_cos_realpow2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1096
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1097
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1098
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1099
lemma DERIV_cos_realpow2a [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1100
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1101
by (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1102
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1103
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1104
by auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1105
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1106
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1107
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1108
apply (rule DERIV_cos_realpow2a, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1109
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1110
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1111
(* most useful *)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1112
lemma DERIV_cos_cos_mult3 [simp]:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1113
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1114
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1115
apply (rule DERIV_cos_cos_mult2, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1116
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1117
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1118
lemma DERIV_sin_circle_all: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1119
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1120
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1121
apply (simp only: diff_minus, safe)
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1122
apply (rule DERIV_add) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1123
apply (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1124
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1125
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1126
lemma DERIV_sin_circle_all_zero [simp]:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1127
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1128
by (cut_tac DERIV_sin_circle_all, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1129
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1130
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1131
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1132
apply (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1133
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1134
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1135
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1136
apply (subst real_add_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1137
apply (simp (no_asm) del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1138
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1139
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1140
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1141
apply (cut_tac x = x in sin_cos_squared_add2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1142
apply (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1143
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1144
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1145
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1146
apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1147
apply (simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1148
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1149
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1150
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1151
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1152
apply (simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1153
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1154
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1155
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1156
by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1157
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
  1158
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1159
apply (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1160
apply (drule_tac n = "Suc 0" in power_gt1)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1161
apply (auto simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1162
apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1163
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1164
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1165
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1166
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1167
apply (insert abs_sin_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1168
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1169
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1170
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1171
lemma sin_le_one [simp]: "sin x \<le> 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1172
apply (insert abs_sin_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1173
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1174
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1175
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
  1176
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1177
apply (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1178
apply (drule_tac n = "Suc 0" in power_gt1)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1179
apply (auto simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1180
apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1181
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1182
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1183
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1184
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1185
apply (insert abs_cos_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1186
apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1187
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1188
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1189
lemma cos_le_one [simp]: "cos x \<le> 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1190
apply (insert abs_cos_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1191
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1192
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1193
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1194
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1195
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1196
apply (rule lemma_DERIV_subst)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1197
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1198
apply (rule DERIV_pow, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1199
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1200
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1201
lemma DERIV_fun_exp:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1202
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1203
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1204
apply (rule_tac f = exp in DERIV_chain2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1205
apply (rule DERIV_exp, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1206
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1207
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1208
lemma DERIV_fun_sin:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1209
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1210
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1211
apply (rule_tac f = sin in DERIV_chain2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1212
apply (rule DERIV_sin, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1213
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1214
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1215
lemma DERIV_fun_cos:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1216
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1217
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1218
apply (rule_tac f = cos in DERIV_chain2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1219
apply (rule DERIV_cos, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1220
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1221
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1222
lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1223
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1224
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1225
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1226
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos