src/HOL/Hyperreal/Transcendental.thy
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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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constdefs
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    root :: "[nat,real] => real"
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    "root n x == (@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 == suminf(%n. inverse(real (fact n)) * (x ^ n))"
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    sin :: "real => real"
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    "sin x == suminf(%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 == suminf(%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 == (@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 == (@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 == (@x. 0 \<le> x & x \<le> pi & cos x = y)"
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    arctan :: "real => real"
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    "arctan y == (@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_mult_self_eq_zero_iff [simp]: "(r * r = 0) = (r = (0::real))"
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   162
by auto
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   163
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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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   169
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lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
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   171
by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
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   172
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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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   184
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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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   191
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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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   198
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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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   202
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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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   205
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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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   208
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
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   209
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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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   213
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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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   216
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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   219
done
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   220
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lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
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   222
apply (rule realpow_two [THEN subst])
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   223
apply (subst numeral_2_eq_2 [symmetric])
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   224
apply (rule real_sqrt_abs)
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   225
done
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   226
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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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   229
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lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
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   231
apply (frule real_sqrt_pow2_gt_zero)
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apply (auto simp add: numeral_2_eq_2)
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   233
done
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   234
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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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   237
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lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
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   239
apply (drule real_le_imp_less_or_eq)
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apply (auto dest: real_sqrt_not_eq_zero)
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   241
done
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   242
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lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
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   244
by (auto simp add: real_sqrt_eq_zero_cancel)
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   245
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lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
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   247
apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
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   248
apply (rule real_le_trans)
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   249
apply (auto simp del: realpow_Suc)
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   250
apply (rule_tac n = 1 in realpow_increasing)
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   251
apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
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   252
done
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   253
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   254
lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
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   255
apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
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   256
done
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   257
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   258
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
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   259
apply (rule_tac n = 1 in realpow_increasing)
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   260
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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   262
done
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   263
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   264
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subsection{*Exponential Function*}
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   266
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lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
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   268
apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
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   269
apply (cut_tac x = r in reals_Archimedean3, auto)
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   270
apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
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   271
apply (rule_tac N = n and c = r in ratio_test)
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apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
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   273
apply (rule mult_right_mono)
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   274
apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
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   275
apply (subst fact_Suc)
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   276
apply (subst real_of_nat_mult)
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   277
apply (auto)
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   278
apply (auto simp add: mult_assoc [symmetric] positive_imp_inverse_positive)
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   279
apply (rule order_less_imp_le)
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   280
apply (rule_tac z1 = "real (Suc na)" in real_mult_less_iff1 [THEN iffD1])
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   281
apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc)
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   282
apply (erule order_less_trans)
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   283
apply (auto simp add: mult_less_cancel_left mult_ac)
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   284
done
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   285
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   286
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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   290
                x ^ n)"
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   291
apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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   292
apply (rule_tac [2] summable_exp)
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   293
apply (rule_tac x = 0 in exI)
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parents: 15228
diff changeset
   294
apply (auto simp add: divide_inverse power_abs [symmetric] zero_le_mult_iff)
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   295
done
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parents: 15013
diff changeset
   296
89840837108e converting Hyperreal/Transcendental to Isar script
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   297
lemma summable_cos: 
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   298
      "summable (%n.  
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parents: 15013
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   299
           (if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
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parents: 15013
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   300
           (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
15229
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   301
apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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   302
apply (rule_tac [2] summable_exp)
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parents: 15013
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   303
apply (rule_tac x = 0 in exI)
15229
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parents: 15228
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   304
apply (auto simp add: divide_inverse power_abs [symmetric] zero_le_mult_iff)
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parents: 15013
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   305
done
89840837108e converting Hyperreal/Transcendental to Isar script
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parents: 15013
diff changeset
   306
15229
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   307
lemma lemma_STAR_sin [simp]:
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   308
     "(if even n then 0  
15077
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paulson
parents: 15013
diff changeset
   309
       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
15251
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   310
by (induct "n", auto)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
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parents: 15228
diff changeset
   311
1eb23f805c06 new simprules for abs and for things like a/b<1
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   312
lemma lemma_STAR_cos [simp]:
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parents: 15228
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   313
     "0 < n -->  
15077
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parents: 15013
diff changeset
   314
      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
   315
by (induct "n", auto)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   316
1eb23f805c06 new simprules for abs and for things like a/b<1
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parents: 15228
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   317
lemma lemma_STAR_cos1 [simp]:
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parents: 15228
diff changeset
   318
     "0 < n -->  
15077
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parents: 15013
diff changeset
   319
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
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   320
by (induct "n", auto)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   321
1eb23f805c06 new simprules for abs and for things like a/b<1
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parents: 15228
diff changeset
   322
lemma lemma_STAR_cos2 [simp]:
15539
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parents: 15536
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   323
  "(\<Sum>n=1..<n. if even n then (- 1) ^ (n div 2)/(real (fact n)) *  0 ^ n 
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   324
                         else 0) = 0"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
   325
apply (induct "n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   326
apply (case_tac [2] "n", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   327
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   328
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   329
lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   330
apply (simp add: exp_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   331
apply (rule summable_exp [THEN summable_sums])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   332
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   333
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   334
lemma sin_converges: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   335
      "(%n. (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   336
            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   337
                 x ^ n) sums sin(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   338
apply (simp add: sin_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   339
apply (rule summable_sin [THEN summable_sums])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   340
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   341
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   342
lemma cos_converges: 
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paulson
parents: 15013
diff changeset
   343
      "(%n. (if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   344
           (- 1) ^ (n div 2)/(real (fact n))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   345
           else 0) * x ^ n) sums cos(x)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   346
apply (simp add: cos_def)
15077
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paulson
parents: 15013
diff changeset
   347
apply (rule summable_cos [THEN summable_sums])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   348
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   349
15229
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parents: 15228
diff changeset
   350
lemma lemma_realpow_diff [rule_format (no_asm)]:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   351
     "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
15251
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paulson
parents: 15241
diff changeset
   352
apply (induct "n", auto)
15077
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paulson
parents: 15013
diff changeset
   353
apply (subgoal_tac "p = Suc n")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   354
apply (simp (no_asm_simp), auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   355
apply (drule sym)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   356
apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   357
       del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   358
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   359
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   360
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   361
subsection{*Properties of Power Series*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   362
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   363
lemma lemma_realpow_diff_sumr:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   364
     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   365
      y * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   366
apply (auto simp add: setsum_mult simp del: setsum_Suc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   367
apply (rule setsum_cong[OF refl])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   368
apply (subst lemma_realpow_diff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   369
apply (auto simp add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   370
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   371
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   372
lemma lemma_realpow_diff_sumr2:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   373
     "x ^ (Suc n) - y ^ (Suc n) =  
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   374
      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
   375
apply (induct "n", simp)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   376
apply (auto simp del: setsum_Suc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   377
apply (subst setsum_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   378
apply (drule sym)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   379
apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   380
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   381
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   382
lemma lemma_realpow_rev_sumr:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   383
     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   384
      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p)::real)"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   385
apply (case_tac "x = y")
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   386
apply (auto simp add: mult_commute power_add [symmetric] simp del: setsum_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   387
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
   388
apply (rule_tac [2] minus_minus [THEN subst], simp)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   389
apply (subst minus_mult_left)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   390
apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: setsum_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   391
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   392
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   393
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
   394
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   395
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   396
lemma powser_insidea:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   397
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   398
      ==> summable (%n. \<bar>f(n)\<bar> * (z ^ n))"
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   399
apply (drule summable_LIMSEQ_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   400
apply (drule convergentI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   401
apply (simp add: Cauchy_convergent_iff [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   402
apply (drule Cauchy_Bseq)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   403
apply (simp add: Bseq_def, safe)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   404
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
   405
apply (rule_tac x = 0 in exI, safe)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   406
apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   407
apply (rule_tac c="\<bar>x ^ n\<bar>" in mult_right_le_imp_le) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   408
apply (auto simp add: mult_assoc power_abs)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   409
 prefer 2
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   410
 apply (drule_tac x = 0 in spec, force)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   411
apply (auto simp add: power_abs mult_ac)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   412
apply (rule_tac a2 = "z ^ n" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   413
       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
   414
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
   415
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
   416
apply (auto intro!: sums_mult simp add: mult_assoc)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   417
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
   418
apply (auto simp add: power_abs [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   419
apply (subgoal_tac "x \<noteq> 0")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   420
apply (subgoal_tac [3] "x \<noteq> 0")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   421
apply (auto simp del: abs_inverse abs_mult simp add: abs_inverse [symmetric] realpow_not_zero abs_mult [symmetric] power_inverse power_mult_distrib [symmetric])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   422
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
   423
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   424
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   425
lemma powser_inside:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   426
     "[| 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
   427
      ==> summable (%n. f(n) * (z ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   428
apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   429
apply (auto intro: summable_rabs_cancel simp add: power_abs [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   430
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   431
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   432
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   433
subsection{*Differentiation of Power Series*}
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{*Lemma about distributing negation over it*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   436
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   437
by (simp add: diffs_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   438
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   439
text{*Show that we can shift the terms down one*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   440
lemma lemma_diffs:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   441
     "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   442
      (\<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
   443
      (real n * c(n) * x ^ (n - Suc 0))"
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15241
diff changeset
   444
apply (induct "n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   445
apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   446
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   447
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   448
lemma lemma_diffs2:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   449
     "(\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   450
      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   451
      (real n * c(n) * x ^ (n - Suc 0))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   452
by (auto simp add: lemma_diffs)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   453
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   454
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   455
lemma diffs_equiv:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   456
     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   457
      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   458
         (suminf(%n. (diffs c)(n) * (x ^ n)))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   459
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
   460
apply (rule_tac [2] LIMSEQ_imp_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   461
apply (drule summable_sums) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   462
apply (auto simp add: sums_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   463
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
   464
apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   465
apply (simp add: diffs_def summable_LIMSEQ_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   466
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   467
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   468
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   469
subsection{*Term-by-Term Differentiability of Power Series*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   470
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   471
lemma lemma_termdiff1:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   472
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   473
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p)))::real)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   474
apply (rule setsum_cong[OF refl])
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   475
apply (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   476
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   477
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   478
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
   479
by (simp add: less_iff_Suc_add)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   480
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   481
lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   482
by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   483
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   484
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   485
lemma lemma_termdiff2:
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   486
  "h \<noteq> 0 ==>
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   487
   (((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
   488
   h * (\<Sum>p=0..< n - Suc 0. (z ^ p) *
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   489
       (\<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
   490
apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   491
apply (simp add: right_diff_distrib mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   492
apply (simp add: mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   493
apply (case_tac "n")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   494
apply (auto simp add: lemma_realpow_diff_sumr2 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   495
                      right_diff_distrib [symmetric] mult_assoc
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   496
            simp del: realpow_Suc setsum_Suc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   497
apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   498
apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   499
                sumdiff lemma_termdiff1 setsum_mult)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   500
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
   501
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   502
apply (auto simp add: setsum_mult lemma_realpow_diff_sumr2 mult_ac simp
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   503
                 del: setsum_Suc realpow_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   504
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   505
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   506
(* FIXME move *)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   507
lemma sumr_bound2 [rule_format (no_asm)]:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   508
     "(\<forall>p. 0 \<le> p & p < n --> (f(p) \<le> K))  
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   509
      --> setsum f {0..<n::nat} \<le> (real n * K)"
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   510
using setsum_bounded[where A = "{0..<n}"]
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   511
by (auto simp:real_of_nat_def)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   512
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   513
lemma lemma_termdiff3:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   514
     "[| 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
   515
      ==> 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
   516
          \<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
   517
apply (subst lemma_termdiff2, assumption)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   518
apply (simp add: mult_commute) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   519
apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   520
apply (rule setsum_abs [THEN real_le_trans])
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   521
apply (simp add: mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   522
apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   523
apply (auto intro!: sumr_bound2)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   524
apply (case_tac "n", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   525
apply (drule less_add_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   526
(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   527
apply clarify 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   528
apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   529
                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   530
apply (simp (no_asm_simp) add: power_add del: setsum_Suc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   531
apply (auto intro!: mult_mono simp del: setsum_Suc)
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   532
apply (auto intro!: power_mono simp add: power_abs simp del: setsum_Suc)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   533
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
   534
apply (subgoal_tac [2] "0 \<le> K")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   535
apply (drule_tac [2] n = d in zero_le_power)
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   536
apply (auto simp del: setsum_Suc)
15536
3ce1cb7a24f0 starting to get rid of sumr
nipkow
parents: 15481
diff changeset
   537
apply (rule setsum_abs [THEN real_le_trans])
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   538
apply (rule sumr_bound2, auto dest!: less_add_one intro!: mult_mono simp add: power_add)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   539
apply (auto intro!: power_mono zero_le_power simp add: power_abs, arith+)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   540
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   541
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   542
lemma lemma_termdiff4: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   543
  "[| 0 < k;  
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   544
      (\<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
   545
   ==> f -- 0 --> 0"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   546
apply (simp add: LIM_def, auto)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   547
apply (subgoal_tac "0 \<le> K")
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   548
 prefer 2
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   549
 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
   550
apply (simp add: ); 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   551
 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
   552
 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
   553
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
   554
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
   555
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
   556
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
   557
apply (rule_tac x = e in exI, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   558
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
   559
apply (force ); 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   560
apply (rule_tac y = "K * e" in order_less_trans)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   561
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
   562
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
   563
apply (auto simp add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   564
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   565
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   566
lemma lemma_termdiff5:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   567
     "[| 0 < k;  
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   568
            summable f;  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   569
            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   570
                    (\<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
   571
         ==> (%h. suminf(g h)) -- 0 --> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   572
apply (drule summable_sums)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   573
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
   574
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
   575
apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   576
 prefer 2
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   577
 apply (simp add: summable_def) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   578
 apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   579
 apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   580
 apply (simp add: mult_ac) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   581
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   582
 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
   583
  apply (rule_tac [2] x = 0 in exI, auto)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   584
apply (rule_tac j = "suminf (%n. \<bar>g h n\<bar>)" in real_le_trans)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   585
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   586
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   587
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   588
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   589
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   590
text{* FIXME: Long proofs*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   591
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   592
lemma termdiffs_aux:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   593
     "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   594
    ==> (\<lambda>h. suminf
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   595
             (\<lambda>n. c n *
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   596
                  (((x + h) ^ n - x ^ n) * inverse h -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   597
                   real n * x ^ (n - Suc 0))))
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   598
       -- 0 --> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   599
apply (drule dense, safe)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   600
apply (frule real_less_sum_gt_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   601
apply (drule_tac
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   602
         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
   603
     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
   604
                             - (real n * (x ^ (n - Suc 0))))" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   605
       in lemma_termdiff5)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   606
apply (auto simp add: add_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   607
apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   608
apply (rule_tac [2] x = K in powser_insidea, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   609
apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   610
apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_eq], auto)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   611
apply (simp add: diffs_def mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   612
apply (subgoal_tac
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   613
        "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   614
              = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   615
apply auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   616
apply (drule diffs_equiv)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   617
apply (drule sums_summable)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   618
apply (simp_all add: diffs_def) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   619
apply (simp add: diffs_def mult_ac)
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   620
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))")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   621
apply auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   622
  prefer 2
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   623
  apply (rule ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   624
  apply (simp add: diffs_def) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   625
  apply (case_tac "n", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   626
txt{*23*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   627
   apply (drule abs_ge_zero [THEN order_le_less_trans])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   628
   apply (simp add: mult_ac) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   629
  apply (drule abs_ge_zero [THEN order_le_less_trans])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   630
  apply (simp add: mult_ac) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   631
 apply (drule diffs_equiv)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   632
 apply (drule sums_summable)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   633
apply (subgoal_tac
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   634
          "summable
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   635
            (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   636
                 r ^ (n - Suc 0)) =
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   637
           summable
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   638
            (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   639
apply simp 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   640
apply (rule_tac f = summable in arg_cong, rule ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   641
txt{*33*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   642
apply (case_tac "n", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   643
apply (case_tac "nat", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   644
apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   645
apply (drule abs_ge_zero [THEN order_le_less_trans])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   646
apply (simp add: mult_assoc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   647
apply (rule mult_left_mono)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   648
 prefer 2 apply arith 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   649
apply (subst add_commute)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   650
apply (simp (no_asm) add: mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   651
apply (rule lemma_termdiff3)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   652
apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   653
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   654
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   655
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   656
lemma termdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   657
    "[| summable(%n. c(n) * (K ^ n));  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   658
        summable(%n. (diffs c)(n) * (K ^ n));  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   659
        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   660
        \<bar>x\<bar> < \<bar>K\<bar> |]  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   661
     ==> DERIV (%x. suminf (%n. c(n) * (x ^ n)))  x :>  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   662
             suminf (%n. (diffs c)(n) * (x ^ n))"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   663
apply (simp add: deriv_def)
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   664
apply (rule_tac g = "%h. suminf (%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
   665
apply (simp add: LIM_def, safe)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   666
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   667
apply (auto simp add: less_diff_eq)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   668
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   669
apply (rule_tac y = 0 in order_le_less_trans, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   670
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   671
apply (auto intro!: summable_sums)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   672
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   673
apply (auto simp add: add_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   674
apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   675
apply (drule_tac x = "(%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
   676
apply (rule sums_unique)
15079
2ef899e4526d conversion of Hyperreal/MacLaurin_lemmas to Isar script
paulson
parents: 15077
diff changeset
   677
apply (simp add: diff_def divide_inverse add_ac mult_ac)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   678
apply (rule LIM_zero_cancel)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   679
apply (rule_tac g = "%h. suminf (%n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0)))))" in LIM_trans)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   680
 prefer 2 apply (blast intro: termdiffs_aux) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   681
apply (simp add: LIM_def, safe)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   682
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   683
apply (auto simp add: less_diff_eq)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   684
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   685
apply (rule_tac y = 0 in order_le_less_trans, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   686
apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   687
apply (rule_tac [2] powser_inside, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   688
apply (drule_tac c = c and x = x in diffs_equiv)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   689
apply (frule sums_unique, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   690
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   691
apply safe
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   692
apply (auto intro!: summable_sums)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   693
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   694
apply (auto simp add: add_commute)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   695
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
   696
apply (simp add: suminf_diff [OF sums_summable sums_summable] 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   697
               right_diff_distrib [symmetric])
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   698
apply (frule_tac x = "(%n. c n * ((xa + x) ^ n - x ^ n))" and c = "inverse xa" in sums_mult)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   699
apply (simp add: sums_summable [THEN suminf_mult2])
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   700
apply (frule_tac x = "(%n. inverse xa * (c n * ((xa + x) ^ n - x ^ n))) " and y = "(%n. real n * c n * x ^ (n - Suc 0))" in sums_diff)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   701
apply assumption
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   702
apply (simp add: suminf_diff [OF sums_summable sums_summable] add_ac mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   703
apply (rule_tac f = suminf in arg_cong)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   704
apply (rule ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   705
apply (simp add: diff_def right_distrib add_ac mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   706
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   707
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   708
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   709
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
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 exp_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   712
      "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
   713
by (simp add: diffs_def mult_assoc [symmetric] del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   714
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   715
lemma sin_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   716
      "diffs(%n. if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   717
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   718
       = (%n. if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   719
                 (- 1) ^ (n div 2)/(real (fact n))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   720
              else 0)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   721
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   722
         simp add: diffs_def divide_inverse simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   723
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   724
lemma sin_fdiffs2: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   725
       "diffs(%n. if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   726
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   727
       = (if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   728
                 (- 1) ^ (n div 2)/(real (fact n))  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   729
              else 0)"
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   730
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   731
         simp add: diffs_def divide_inverse simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   732
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   733
lemma cos_fdiffs: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   734
      "diffs(%n. if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   735
                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   736
       = (%n. - (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   737
           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
   738
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   739
         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
   740
         simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   741
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   742
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   743
lemma cos_fdiffs2: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   744
      "diffs(%n. if even n then  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   745
                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   746
       = - (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   747
           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
   748
by (auto intro!: ext 
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   749
         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
   750
         simp del: mult_Suc)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   751
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   752
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
   753
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   754
lemma lemma_sin_minus:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   755
     "- sin x =
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   756
         suminf(%n. - ((if even n then 0 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   757
                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   758
by (auto intro!: sums_unique sums_minus sin_converges)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   759
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   760
lemma lemma_exp_ext: "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   761
by (auto intro!: ext simp add: exp_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   762
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   763
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
   764
apply (simp add: exp_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   765
apply (subst lemma_exp_ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   766
apply (subgoal_tac "DERIV (%u. suminf (%n. inverse (real (fact n)) * u ^ n)) x :> suminf (%n. diffs (%n. inverse (real (fact n))) n * x ^ n) ")
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   767
apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   768
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   769
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   770
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   771
lemma lemma_sin_ext:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   772
     "sin = (%x. suminf(%n. 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   773
                   (if even n then 0  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   774
                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   775
                   x ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   776
by (auto intro!: ext simp add: sin_def)
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 lemma_cos_ext:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   779
     "cos = (%x. suminf(%n. 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   780
                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   781
                   x ^ n))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   782
by (auto intro!: ext simp add: cos_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   783
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   784
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
   785
apply (simp add: cos_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   786
apply (subst lemma_sin_ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   787
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
   788
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   789
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs, arith)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   790
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   791
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   792
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   793
apply (subst lemma_cos_ext)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   794
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
   795
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   796
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus, arith)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   797
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   798
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   799
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   800
subsection{*Properties of the Exponential Function*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   801
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   802
lemma exp_zero [simp]: "exp 0 = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   803
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   804
  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   805
        suminf (\<lambda>n. inverse (real (fact n)) * 0 ^ n)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   806
    by (rule series_zero [rule_format, THEN sums_unique],
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   807
        case_tac "m", auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   808
  thus ?thesis by (simp add:  exp_def) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   809
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   810
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   811
lemma exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> exp(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   812
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
   813
apply (simp add: exp_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   814
apply (rule real_le_trans)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
   815
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
   816
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
   817
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   818
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   819
lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   820
apply (rule order_less_le_trans)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   821
apply (rule_tac [2] exp_ge_add_one_self, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   822
done
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_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
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 (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   827
    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
   828
  thus ?thesis by (simp add: o_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   829
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   830
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   831
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   832
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   833
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   834
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   835
  thus ?thesis by (simp add: o_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   836
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   837
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   838
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
   839
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   840
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   841
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   842
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
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_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   847
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   848
  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
   849
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   850
    by (rule DERIV_isconst_all) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   851
  thus ?thesis by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   852
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   853
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   854
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   855
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   856
  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
   857
  thus ?thesis by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   858
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   859
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   860
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   861
by (simp add: mult_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   862
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   863
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   864
lemma exp_minus: "exp(-x) = inverse(exp(x))"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   865
by (auto intro: inverse_unique [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   866
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   867
lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   868
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   869
  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
   870
  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   871
qed
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
text{*Proof: because every exponential can be seen as a square.*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   874
lemma exp_ge_zero [simp]: "0 \<le> exp x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   875
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   876
apply (subst exp_add, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   877
done
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_not_eq_zero [simp]: "exp x \<noteq> 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   880
apply (cut_tac x = x in exp_mult_minus2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   881
apply (auto simp del: exp_mult_minus2)
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_gt_zero [simp]: "0 < exp x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   885
by (simp add: order_less_le)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   886
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   887
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   888
by (auto intro: positive_imp_inverse_positive)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   889
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
   890
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
   891
by auto
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   892
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   893
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
   894
apply (induct "n")
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   895
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
   896
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   897
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   898
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
   899
apply (simp add: diff_minus divide_inverse)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   900
apply (simp (no_asm) add: exp_add exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   901
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   902
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   903
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   904
lemma exp_less_mono:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   905
  assumes xy: "x < y" shows "exp x < exp y"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   906
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   907
  have "1 < exp (y + - x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   908
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   909
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   910
    by (auto simp add: exp_add exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   911
  thus ?thesis
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
   912
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15140
diff changeset
   913
             del: divide_self_if)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   914
qed
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 exp_less_cancel: "exp x < exp y ==> x < y"
15228
4d332d10fa3d revised simprules for division
paulson
parents: 15140
diff changeset
   917
apply (simp add: linorder_not_le [symmetric]) 
4d332d10fa3d revised simprules for division
paulson
parents: 15140
diff changeset
   918
apply (auto simp add: order_le_less exp_less_mono) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   919
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   920
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   921
lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   922
by (auto intro: exp_less_mono exp_less_cancel)
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_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   925
by (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   926
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   927
lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   928
by (simp add: order_eq_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   929
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   930
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
   931
apply (rule IVT)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   932
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
   933
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   934
apply simp 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   935
apply (rule exp_ge_add_one_self, simp)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   936
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   937
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   938
lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   939
apply (rule_tac x = 1 and y = y in linorder_cases)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   940
apply (drule order_less_imp_le [THEN lemma_exp_total])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   941
apply (rule_tac [2] x = 0 in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   942
apply (frule_tac [3] real_inverse_gt_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   943
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
   944
apply (rule_tac x = "-x" in exI)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   945
apply (simp add: exp_minus)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   946
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   947
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   948
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   949
subsection{*Properties of the Logarithmic Function*}
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_exp[simp]: "ln(exp x) = x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   952
by (simp add: ln_def)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   953
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   954
lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   955
apply (auto dest: exp_total)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   956
apply (erule subst, simp) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   957
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   958
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   959
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
   960
apply (rule exp_inj_iff [THEN iffD1])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   961
apply (frule real_mult_order)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   962
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
   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_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
   966
apply (simp only: exp_ln_iff [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   967
apply (erule subst)+
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   968
apply simp 
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_one[simp]: "ln 1 = 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   972
by (rule exp_inj_iff [THEN iffD1], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   973
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   974
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   975
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   976
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   977
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   978
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   979
lemma ln_div: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   980
    "[|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
   981
apply (simp add: divide_inverse)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   982
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   983
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   984
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   985
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
   986
apply (simp only: exp_ln_iff [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   987
apply (erule subst)+
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   988
apply simp 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   989
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   990
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   991
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
   992
by (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   993
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   994
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   995
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
   996
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   997
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
   998
apply (rule ln_exp [THEN subst])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
   999
apply (rule ln_le_cancel_iff [THEN iffD2], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1000
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1001
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1002
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1003
apply (rule order_less_le_trans)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1004
apply (rule_tac [2] ln_add_one_self_le_self)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1005
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1006
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1007
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1008
lemma ln_ge_zero [simp]:
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1009
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1010
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1011
  have "0 < x" using x by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1012
  hence "exp 0 \<le> exp (ln x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1013
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1014
  thus ?thesis by (simp only: exp_le_cancel_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1015
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1016
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1017
lemma ln_ge_zero_imp_ge_one:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1018
  assumes ln: "0 \<le> ln x" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1019
      and x:  "0 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1020
  shows "1 \<le> 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
  from ln have "ln 1 \<le> ln x" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1023
  thus ?thesis by (simp add: x del: ln_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1024
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1025
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1026
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
  1027
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1028
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1029
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
  1030
by (insert ln_ge_zero_iff [of x], arith)
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1031
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1032
lemma ln_gt_zero:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1033
  assumes x: "1 < x" shows "0 < ln x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1034
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1035
  have "0 < x" using x by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1036
  hence "exp 0 < exp (ln x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1037
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1038
  thus ?thesis  by (simp only: exp_less_cancel_iff)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1039
qed
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1040
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1041
lemma ln_gt_zero_imp_gt_one:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1042
  assumes ln: "0 < ln x" 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1043
      and x:  "0 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1044
  shows "1 < x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1045
proof -
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1046
  from ln have "ln 1 < ln x" by simp
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1047
  thus ?thesis by (simp add: x del: ln_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1048
qed
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
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
  1051
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1052
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1053
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
  1054
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
  1055
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1056
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
15234
ec91a90c604e simplification tweaks for better arithmetic reasoning
paulson
parents: 15229
diff changeset
  1057
by simp
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1058
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1059
lemma exp_ln_eq: "exp u = x ==> ln x = u"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1060
by auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1061
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1062
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1063
subsection{*Basic Properties of the Trigonometric Functions*}
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1064
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1065
lemma sin_zero [simp]: "sin 0 = 0"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1066
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1067
         simp add: sin_def sums_def simp del: power_0_left)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1068
15539
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
  1069
lemma lemma_series_zero2:
333a88244569 comprehensive cleanup, replacing sumr by setsum
nipkow
parents: 15536
diff changeset
  1070
 "(\<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
  1071
by (auto intro: series_zero)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1072
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1073
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
  1074
apply (simp add: cos_def)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1075
apply (rule sums_unique [symmetric])
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1076
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
  1077
apply auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1078
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1079
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1080
lemma DERIV_sin_sin_mult [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1081
     "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
  1082
by (rule DERIV_mult, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1083
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1084
lemma DERIV_sin_sin_mult2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1085
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1086
apply (cut_tac x = x in DERIV_sin_sin_mult)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1087
apply (auto simp add: mult_assoc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1088
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1089
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1090
lemma DERIV_sin_realpow2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1091
     "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
  1092
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1093
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1094
lemma DERIV_sin_realpow2a [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1095
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1096
by (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1097
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1098
lemma DERIV_cos_cos_mult [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1099
     "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
  1100
by (rule DERIV_mult, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1101
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1102
lemma DERIV_cos_cos_mult2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1103
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1104
apply (cut_tac x = x in DERIV_cos_cos_mult)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1105
apply (auto simp add: mult_ac)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1106
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1107
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1108
lemma DERIV_cos_realpow2 [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1109
     "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
  1110
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1111
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1112
lemma DERIV_cos_realpow2a [simp]:
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1113
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1114
by (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1115
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1116
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
  1117
by auto
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1118
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1119
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
  1120
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1121
apply (rule DERIV_cos_realpow2a, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1122
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1123
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1124
(* most useful *)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1125
lemma DERIV_cos_cos_mult3 [simp]:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1126
     "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
  1127
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1128
apply (rule DERIV_cos_cos_mult2, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1129
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1130
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1131
lemma DERIV_sin_circle_all: 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1132
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1133
             (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
  1134
apply (simp only: diff_minus, safe)
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1135
apply (rule DERIV_add) 
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1136
apply (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1137
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1138
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1139
lemma DERIV_sin_circle_all_zero [simp]:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1140
     "\<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
  1141
by (cut_tac DERIV_sin_circle_all, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1142
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1143
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
  1144
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
  1145
apply (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1146
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1147
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1148
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
  1149
apply (subst real_add_commute)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1150
apply (simp (no_asm) del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1151
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1152
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1153
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
  1154
apply (cut_tac x = x in sin_cos_squared_add2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1155
apply (auto simp add: numeral_2_eq_2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1156
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1157
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1158
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
  1159
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
  1160
apply (simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1161
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1162
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1163
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1164
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
  1165
apply (simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1166
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1167
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1168
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
  1169
by arith
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1170
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
  1171
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
  1172
apply (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1173
apply (drule_tac n = "Suc 0" in power_gt1)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1174
apply (auto simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1175
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
  1176
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1177
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1178
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1179
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1180
apply (insert abs_sin_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1181
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
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 sin_le_one [simp]: "sin x \<le> 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1185
apply (insert abs_sin_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_sin_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
15081
32402f5624d1 abs notation
paulson
parents: 15079
diff changeset
  1189
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
  1190
apply (auto simp add: linorder_not_less [symmetric])
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1191
apply (drule_tac n = "Suc 0" in power_gt1)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1192
apply (auto simp del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1193
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
  1194
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1195
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1196
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1197
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1198
apply (insert abs_cos_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1199
apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1200
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1201
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1202
lemma cos_le_one [simp]: "cos x \<le> 1"
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1203
apply (insert abs_cos_le_one [of x]) 
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1204
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1205
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1206
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1207
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1208
      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
  1209
apply (rule lemma_DERIV_subst)
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1210
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
15077
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1211
apply (rule DERIV_pow, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1212
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1213
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1214
lemma DERIV_fun_exp:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1215
     "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
  1216
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1217
apply (rule_tac f = exp in DERIV_chain2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1218
apply (rule DERIV_exp, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1219
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1220
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1221
lemma DERIV_fun_sin:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1222
     "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
  1223
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1224
apply (rule_tac f = sin in DERIV_chain2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1225
apply (rule DERIV_sin, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1226
done
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1227
15229
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1228
lemma DERIV_fun_cos:
1eb23f805c06 new simprules for abs and for things like a/b<1
paulson
parents: 15228
diff changeset
  1229
     "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
  1230
apply (rule lemma_DERIV_subst)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1231
apply (rule_tac f = cos in DERIV_chain2)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1232
apply (rule DERIV_cos, auto)
89840837108e converting Hyperreal/Transcendental to Isar script
paulson
parents: 15013
diff changeset
  1233
done