src/HOL/Hyperreal/Transcendental.thy
author huffman
Tue Oct 03 19:40:34 2006 +0200 (2006-10-03)
changeset 20849 389cd9c8cfe1
parent 20692 6df83a636e67
child 20860 1a8efd618190
permissions -rw-r--r--
rewrite proofs of powser_insidea and termdiffs_aux
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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 Series EvenOdd Lim
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begin
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definition
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  exp :: "real => real"
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  "exp x = (\<Sum>n. inverse(real (fact n)) * (x ^ n))"
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  sin :: "real => real"
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  "sin x = (\<Sum>n. (if even(n) then 0 else
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             ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
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  diffs :: "(nat => real) => nat => real"
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  "diffs c = (%n. real (Suc n) * c(Suc n))"
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  cos :: "real => real"
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  "cos x = (\<Sum>n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n)) 
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                            else 0) * x ^ n)"
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  ln :: "real => real"
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  "ln x = (SOME u. exp u = x)"
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  pi :: "real"
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  "pi = 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
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  tan :: "real => real"
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  "tan x = (sin x)/(cos x)"
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  arcsin :: "real => real"
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  "arcsin y = (SOME x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
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  arcos :: "real => real"
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  "arcos y = (SOME x. 0 \<le> x & x \<le> pi & cos x = y)"
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  arctan :: "real => real"
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  "arctan y = (SOME x. -(pi/2) < x & x < pi/2 & tan x = y)"
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subsection{*Exponential Function*}
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lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
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apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
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apply (cut_tac x = r in reals_Archimedean3, auto)
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apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
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apply (rule_tac N = n and c = r in ratio_test)
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apply (safe, simp add: abs_mult mult_assoc [symmetric] del: fact_Suc)
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apply (rule mult_right_mono)
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apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
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apply (subst fact_Suc)
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apply (subst real_of_nat_mult)
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apply (auto)
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apply (simp add: mult_assoc [symmetric] positive_imp_inverse_positive)
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apply (rule order_less_imp_le)
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apply (rule_tac z1 = "real (Suc na)" in real_mult_less_iff1 [THEN iffD1])
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apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc)
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apply (erule order_less_trans)
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apply (auto simp add: mult_less_cancel_left mult_ac)
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done
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lemma summable_sin: 
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     "summable (%n.  
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           (if even n then 0  
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           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
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                x ^ n)"
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apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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apply (rule_tac [2] summable_exp)
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apply (rule_tac x = 0 in exI)
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apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
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done
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lemma summable_cos: 
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      "summable (%n.  
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           (if even n then  
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           (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
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apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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apply (rule_tac [2] summable_exp)
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apply (rule_tac x = 0 in exI)
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apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
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done
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lemma lemma_STAR_sin [simp]:
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     "(if even n then 0  
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       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
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by (induct "n", auto)
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lemma lemma_STAR_cos [simp]:
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     "0 < n -->  
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      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
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by (induct "n", auto)
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lemma lemma_STAR_cos1 [simp]:
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     "0 < n -->  
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      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
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by (induct "n", auto)
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lemma lemma_STAR_cos2 [simp]:
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  "(\<Sum>n=1..<n. if even n then (- 1) ^ (n div 2)/(real (fact n)) *  0 ^ n 
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                         else 0) = 0"
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apply (induct "n")
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apply (case_tac [2] "n", auto)
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done
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lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
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apply (simp add: exp_def)
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apply (rule summable_exp [THEN summable_sums])
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done
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lemma sin_converges: 
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      "(%n. (if even n then 0  
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            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
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                 x ^ n) sums sin(x)"
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apply (simp add: sin_def)
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apply (rule summable_sin [THEN summable_sums])
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done
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lemma cos_converges: 
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      "(%n. (if even n then  
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           (- 1) ^ (n div 2)/(real (fact n))  
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           else 0) * x ^ n) sums cos(x)"
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apply (simp add: cos_def)
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apply (rule summable_cos [THEN summable_sums])
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done
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lemma lemma_realpow_diff [rule_format (no_asm)]:
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     "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
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apply (induct "n", auto)
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apply (subgoal_tac "p = Suc n")
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apply (simp (no_asm_simp), auto)
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apply (drule sym)
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apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
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       del: realpow_Suc)
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done
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subsection{*Properties of Power Series*}
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lemma lemma_realpow_diff_sumr:
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     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =  
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      y * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
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by (auto simp add: setsum_right_distrib lemma_realpow_diff mult_ac
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  simp del: setsum_op_ivl_Suc cong: strong_setsum_cong)
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lemma lemma_realpow_diff_sumr2:
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     "x ^ (Suc n) - y ^ (Suc n) =  
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      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
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apply (induct "n", simp)
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apply (auto simp del: setsum_op_ivl_Suc)
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apply (subst setsum_op_ivl_Suc)
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apply (drule sym)
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apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_op_ivl_Suc)
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done
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lemma lemma_realpow_rev_sumr:
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     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
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      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p)::real)"
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apply (case_tac "x = y")
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apply (auto simp add: mult_commute power_add [symmetric] simp del: setsum_op_ivl_Suc)
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apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
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apply (rule_tac [2] minus_minus [THEN subst], simp)
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apply (subst minus_mult_left)
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apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: setsum_op_ivl_Suc)
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done
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text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
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x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
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lemma powser_insidea:
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  fixes x z :: real
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  assumes 1: "summable (\<lambda>n. f n * x ^ n)"
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  assumes 2: "\<bar>z\<bar> < \<bar>x\<bar>"
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  shows "summable (\<lambda>n. \<bar>f n\<bar> * z ^ n)"
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proof -
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  from 2 have x_neq_0: "x \<noteq> 0" by clarsimp
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  from 1 have "(\<lambda>n. f n * x ^ n) ----> 0"
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    by (rule summable_LIMSEQ_zero)
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  hence "convergent (\<lambda>n. f n * x ^ n)"
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    by (rule convergentI)
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  hence "Cauchy (\<lambda>n. f n * x ^ n)"
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    by (simp add: Cauchy_convergent_iff)
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  hence "Bseq (\<lambda>n. f n * x ^ n)"
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    by (rule Cauchy_Bseq)
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  then obtain K where 3: "0 < K" and 4: "\<forall>n. \<bar>f n * x ^ n\<bar> \<le> K"
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    by (simp add: Bseq_def, safe)
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  have "\<exists>N. \<forall>n\<ge>N. norm (\<bar>f n\<bar> * z ^ n) \<le> K * \<bar>z ^ n\<bar> * inverse \<bar>x ^ n\<bar>"
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  proof (intro exI allI impI)
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    fix n::nat assume "0 \<le> n"
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    have "norm (\<bar>f n\<bar> * z ^ n) * \<bar>x ^ n\<bar> = \<bar>f n * x ^ n\<bar> * \<bar>z ^ n\<bar>"
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      by (simp add: abs_mult)
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    also have "\<dots> \<le> K * \<bar>z ^ n\<bar>"
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      by (simp only: mult_right_mono 4 abs_ge_zero)
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    also have "\<dots> = K * \<bar>z ^ n\<bar> * (inverse \<bar>x ^ n\<bar> * \<bar>x ^ n\<bar>)"
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      by (simp add: x_neq_0)
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    also have "\<dots> = K * \<bar>z ^ n\<bar> * inverse \<bar>x ^ n\<bar> * \<bar>x ^ n\<bar>"
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      by (simp only: mult_assoc)
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    finally show "norm (\<bar>f n\<bar> * z ^ n) \<le> K * \<bar>z ^ n\<bar> * inverse \<bar>x ^ n\<bar>"
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      by (simp add: mult_le_cancel_right x_neq_0)
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  qed
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  moreover have "summable (\<lambda>n. K * \<bar>z ^ n\<bar> * inverse \<bar>x ^ n\<bar>)"
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  proof -
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    from 2 have "norm \<bar>z * inverse x\<bar> < 1"
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      by (simp add: abs_mult divide_inverse [symmetric])
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    hence "summable (\<lambda>n. \<bar>z * inverse x\<bar> ^ n)"
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      by (rule summable_geometric)
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    hence "summable (\<lambda>n. K * \<bar>z * inverse x\<bar> ^ n)"
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      by (rule summable_mult)
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    thus "summable (\<lambda>n. K * \<bar>z ^ n\<bar> * inverse \<bar>x ^ n\<bar>)"
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      by (simp add: abs_mult power_mult_distrib power_abs
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                    power_inverse mult_assoc)
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  qed
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  ultimately show "summable (\<lambda>n. \<bar>f n\<bar> * z ^ n)"
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    by (rule summable_comparison_test)
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qed
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lemma powser_inside:
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  fixes f :: "nat \<Rightarrow> real" shows
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     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
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      ==> summable (%n. f(n) * (z ^ n))"
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apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea, simp)
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apply (rule summable_rabs_cancel)
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apply (simp add: abs_mult power_abs [symmetric])
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done
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subsection{*Differentiation of Power Series*}
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text{*Lemma about distributing negation over it*}
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lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
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by (simp add: diffs_def)
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text{*Show that we can shift the terms down one*}
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lemma lemma_diffs:
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     "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  
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      (\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) +  
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      (real n * c(n) * x ^ (n - Suc 0))"
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apply (induct "n")
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apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
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done
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lemma lemma_diffs2:
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     "(\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) =  
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      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
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      (real n * c(n) * x ^ (n - Suc 0))"
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by (auto simp add: lemma_diffs)
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lemma diffs_equiv:
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     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
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      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
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         (\<Sum>n. (diffs c)(n) * (x ^ n))"
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apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
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apply (rule_tac [2] LIMSEQ_imp_Suc)
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apply (drule summable_sums) 
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apply (auto simp add: sums_def)
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apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
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apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
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apply (simp add: diffs_def summable_LIMSEQ_zero)
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done
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subsection{*Term-by-Term Differentiability of Power Series*}
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lemma lemma_termdiff1:
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  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
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   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p)))::real)"
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by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac
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  cong: strong_setsum_cong)
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lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
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by (simp add: less_iff_Suc_add)
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lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
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by arith
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lemma lemma_termdiff2:
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  "h \<noteq> 0 ==>
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   (((z + h) ^ n) - (z ^ n)) * inverse h - real n * (z ^ (n - Suc 0)) =
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   h * (\<Sum>p=0..< n - Suc 0. (z ^ p) *
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       (\<Sum>q=0..< (n - Suc 0) - p. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))"
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apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
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apply (simp add: right_diff_distrib mult_ac)
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apply (simp add: mult_assoc [symmetric])
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apply (case_tac "n")
paulson@15077
   294
apply (auto simp add: lemma_realpow_diff_sumr2 
paulson@15077
   295
                      right_diff_distrib [symmetric] mult_assoc
nipkow@15561
   296
            simp del: realpow_Suc setsum_op_ivl_Suc)
nipkow@15561
   297
apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_op_ivl_Suc)
paulson@15077
   298
apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
ballarin@19279
   299
                sumdiff lemma_termdiff1 setsum_right_distrib)
nipkow@15539
   300
apply (auto intro!: setsum_cong[OF refl] simp add: diff_minus real_add_assoc)
nipkow@15539
   301
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
ballarin@19279
   302
apply (auto simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac simp
nipkow@15561
   303
                 del: setsum_op_ivl_Suc realpow_Suc)
paulson@15077
   304
done
paulson@15077
   305
paulson@15229
   306
lemma lemma_termdiff3:
paulson@15229
   307
     "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<le> K |]  
paulson@15077
   308
      ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
paulson@15077
   309
          \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
paulson@15077
   310
apply (subst lemma_termdiff2, assumption)
paulson@16924
   311
apply (simp add: mult_commute abs_mult) 
paulson@15077
   312
apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
nipkow@15536
   313
apply (rule setsum_abs [THEN real_le_trans])
paulson@16924
   314
apply (simp add: mult_assoc [symmetric] abs_mult)
paulson@15077
   315
apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
nipkow@15542
   316
apply (auto intro!: real_setsum_nat_ivl_bounded)
paulson@15077
   317
apply (case_tac "n", auto)
paulson@15077
   318
apply (drule less_add_one)
paulson@15077
   319
(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
paulson@15077
   320
apply clarify 
paulson@15077
   321
apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
paulson@15077
   322
                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
nipkow@15561
   323
apply (simp (no_asm_simp) add: power_add del: setsum_op_ivl_Suc)
nipkow@15561
   324
apply (auto intro!: mult_mono simp del: setsum_op_ivl_Suc)
paulson@16924
   325
apply (auto intro!: power_mono simp add: power_abs
paulson@16924
   326
           simp del: setsum_op_ivl_Suc)
paulson@15229
   327
apply (rule_tac j = "real (Suc d) * (K ^ d)" in real_le_trans)
paulson@15077
   328
apply (subgoal_tac [2] "0 \<le> K")
paulson@15077
   329
apply (drule_tac [2] n = d in zero_le_power)
nipkow@15561
   330
apply (auto simp del: setsum_op_ivl_Suc)
nipkow@15536
   331
apply (rule setsum_abs [THEN real_le_trans])
paulson@16924
   332
apply (rule real_setsum_nat_ivl_bounded)
paulson@16924
   333
apply (auto dest!: less_add_one intro!: mult_mono simp add: power_add abs_mult)
webertj@20217
   334
apply (auto intro!: power_mono zero_le_power simp add: power_abs)
paulson@15077
   335
done
paulson@15077
   336
paulson@15077
   337
lemma lemma_termdiff4: 
huffman@20561
   338
  "[| 0 < (k::real);  
paulson@15081
   339
      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]  
paulson@15077
   340
   ==> f -- 0 --> 0"
paulson@15229
   341
apply (simp add: LIM_def, auto)
paulson@15077
   342
apply (subgoal_tac "0 \<le> K")
paulson@15229
   343
 prefer 2
paulson@15229
   344
 apply (drule_tac x = "k/2" in spec)
paulson@15229
   345
apply (simp add: ); 
paulson@15229
   346
 apply (subgoal_tac "0 \<le> K*k", simp add: zero_le_mult_iff) 
paulson@15229
   347
 apply (force intro: order_trans [of _ "\<bar>f (k / 2)\<bar> * 2"]) 
paulson@15077
   348
apply (drule real_le_imp_less_or_eq, auto)
paulson@15229
   349
apply (subgoal_tac "0 < (r * inverse K) / 2")
paulson@15229
   350
apply (drule_tac ?d1.0 = "(r * inverse K) / 2" and ?d2.0 = k in real_lbound_gt_zero)
paulson@15229
   351
apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff zero_less_divide_iff)
paulson@15077
   352
apply (rule_tac x = e in exI, auto)
paulson@15077
   353
apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
paulson@15229
   354
apply (force ); 
paulson@15229
   355
apply (rule_tac y = "K * e" in order_less_trans)
paulson@15077
   356
apply (simp add: mult_less_cancel_left)
paulson@15229
   357
apply (rule_tac c = "inverse K" in mult_right_less_imp_less)
paulson@15077
   358
apply (auto simp add: mult_ac)
paulson@15077
   359
done
paulson@15077
   360
paulson@15229
   361
lemma lemma_termdiff5:
huffman@20692
   362
     "[| 0 < (k::real);  
paulson@15077
   363
            summable f;  
paulson@15077
   364
            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
paulson@15077
   365
                    (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
paulson@15077
   366
         ==> (%h. suminf(g h)) -- 0 --> 0"
paulson@15077
   367
apply (drule summable_sums)
paulson@15081
   368
apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>suminf (g h)\<bar> \<le> suminf f * \<bar>h\<bar>")
paulson@15077
   369
apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
paulson@15077
   370
apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
paulson@15077
   371
 prefer 2
paulson@15077
   372
 apply (simp add: summable_def) 
paulson@15077
   373
 apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
paulson@15077
   374
 apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
paulson@15077
   375
 apply (simp add: mult_ac) 
paulson@15077
   376
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
paulson@15077
   377
 apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
paulson@15077
   378
  apply (rule_tac [2] x = 0 in exI, auto)
nipkow@15546
   379
apply (rule_tac j = "\<Sum>n. \<bar>g h n\<bar>" in real_le_trans)
avigad@16819
   380
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult2])
paulson@15077
   381
done
paulson@15077
   382
paulson@15077
   383
paulson@15077
   384
text{* FIXME: Long proofs*}
paulson@15077
   385
paulson@15077
   386
lemma termdiffs_aux:
huffman@20849
   387
  assumes 1: "summable (\<lambda>n. diffs (diffs c) n * K ^ n)"
huffman@20849
   388
  assumes 2: "\<bar>x\<bar> < \<bar>K\<bar>"
huffman@20849
   389
  shows "(\<lambda>h. \<Sum>n. c n *
paulson@15077
   390
                  (((x + h) ^ n - x ^ n) * inverse h -
nipkow@15546
   391
                   real n * x ^ (n - Suc 0)))
paulson@15077
   392
       -- 0 --> 0"
huffman@20849
   393
proof -
huffman@20849
   394
  from dense [OF 2] obtain r where 3: "\<bar>x\<bar> < r" and 4: "r < \<bar>K\<bar>" by fast
huffman@20849
   395
  from 3 have r_neq_0: "r \<noteq> 0" by auto
huffman@20849
   396
  show "(\<lambda>h. suminf (\<lambda>n. c n * (((x + h) ^ n - x ^ n) * inverse h - real n * x ^ (n - Suc 0)))) -- 0 --> 0"
huffman@20849
   397
  proof (rule lemma_termdiff5)
huffman@20849
   398
    show "0 < r + - \<bar>x\<bar>" using 3 by simp
huffman@20849
   399
  next
huffman@20849
   400
    have A: "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))"
huffman@20849
   401
      apply (rule powser_insidea [OF 1])
huffman@20849
   402
      apply (subgoal_tac "\<bar>r\<bar> = r", erule ssubst, rule 4)
huffman@20849
   403
      apply (rule_tac y1 = "\<bar>x\<bar>" in order_trans [THEN abs_of_nonneg])
huffman@20849
   404
       apply (rule abs_ge_zero)
huffman@20849
   405
      apply (rule order_less_imp_le [OF 3])
huffman@20849
   406
      done
huffman@20849
   407
    have B: "\<forall>n. real (Suc n) * real (Suc (Suc n)) *
huffman@20849
   408
      \<bar>c (Suc (Suc n))\<bar> * (r ^ n) = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n)"
huffman@20849
   409
      by (simp add: diffs_def mult_assoc)
huffman@20849
   410
    have C: "(%n. real n * (real (Suc n) * (\<bar>c (Suc n)\<bar> * (r ^ (n - Suc 0)))))
huffman@20849
   411
      = (%n. diffs (%m. real (m - Suc 0) * \<bar>c m\<bar> * inverse r) n * (r ^ n))"
huffman@20849
   412
      apply (rule ext)
huffman@20849
   413
      apply (simp add: diffs_def)
huffman@20849
   414
      apply (case_tac n, simp_all add: r_neq_0)
huffman@20849
   415
      done
huffman@20849
   416
    have D:
huffman@20849
   417
          "(\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
webertj@20432
   418
                 r ^ (n - Suc 0)) =
huffman@20849
   419
           (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))"
huffman@20849
   420
      apply (rule ext)
huffman@20849
   421
      apply (case_tac "n", simp)
huffman@20849
   422
      apply (case_tac "nat", simp)
huffman@20849
   423
      apply (simp add: r_neq_0)
huffman@20849
   424
      done
huffman@20849
   425
    show "summable (\<lambda>n. \<bar>c n\<bar> * real n * real (n - Suc 0) * r ^ (n - 2))"
huffman@20849
   426
      apply (cut_tac A)
huffman@20849
   427
      apply (simp add: diffs_def mult_assoc [symmetric])
huffman@20849
   428
      apply (simp only: abs_mult abs_real_of_nat_cancel B)
huffman@20849
   429
      apply (drule diffs_equiv)
huffman@20849
   430
      apply (drule sums_summable)
huffman@20849
   431
      apply (simp only: diffs_def mult_ac)
huffman@20849
   432
      apply (simp only: C)
huffman@20849
   433
      apply (drule diffs_equiv)
huffman@20849
   434
      apply (drule sums_summable)
huffman@20849
   435
      apply (simp only: D)
huffman@20849
   436
      done
huffman@20849
   437
  next
huffman@20849
   438
    show "\<forall>h. 0 < \<bar>h\<bar> \<and> \<bar>h\<bar> < r + - \<bar>x\<bar> \<longrightarrow> (\<forall>n. \<bar>c n * (((x + h) ^ n - x ^ n) * inverse h - real n * x ^ (n - Suc 0))\<bar> \<le> \<bar>c n\<bar> * real n * real (n - Suc 0) * r ^ (n - 2) * \<bar>h\<bar>)"
huffman@20849
   439
    proof (clarify)
huffman@20849
   440
      fix h::real and n::nat
huffman@20849
   441
      assume A: "0 < \<bar>h\<bar>" and B: "\<bar>h\<bar> < r + - \<bar>x\<bar>"
huffman@20849
   442
      from A have C: "h \<noteq> 0" by simp
huffman@20849
   443
      show "\<bar>c n * (((x + h) ^ n - x ^ n) * inverse h - real n * x ^ (n - Suc 0))\<bar> \<le> \<bar>c n\<bar> * real n * real (n - Suc 0) * r ^ (n - 2) * \<bar>h\<bar>"
huffman@20849
   444
        apply (cut_tac 3 B C)
huffman@20849
   445
        apply (subst abs_mult)
huffman@20849
   446
        apply (drule abs_ge_zero [THEN order_le_less_trans])
huffman@20849
   447
        apply (simp only: mult_assoc)
huffman@20849
   448
        apply (rule mult_left_mono)
huffman@20849
   449
        prefer 2 apply arith 
huffman@20849
   450
        apply (simp (no_asm) add: mult_assoc [symmetric])
huffman@20849
   451
        apply (rule lemma_termdiff3)
huffman@20849
   452
        apply assumption
huffman@20849
   453
        apply (rule 3 [THEN order_less_imp_le])
huffman@20849
   454
        apply (rule abs_triangle_ineq [THEN order_trans])
huffman@20849
   455
        apply arith
huffman@20849
   456
        done
huffman@20849
   457
    qed
huffman@20849
   458
  qed
huffman@20849
   459
qed
webertj@20217
   460
paulson@15077
   461
lemma termdiffs: 
paulson@15077
   462
    "[| summable(%n. c(n) * (K ^ n));  
paulson@15077
   463
        summable(%n. (diffs c)(n) * (K ^ n));  
paulson@15077
   464
        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
paulson@15077
   465
        \<bar>x\<bar> < \<bar>K\<bar> |]  
nipkow@15546
   466
     ==> DERIV (%x. \<Sum>n. c(n) * (x ^ n))  x :>  
nipkow@15546
   467
             (\<Sum>n. (diffs c)(n) * (x ^ n))"
paulson@15229
   468
apply (simp add: deriv_def)
nipkow@15546
   469
apply (rule_tac g = "%h. \<Sum>n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h" in LIM_trans)
paulson@15077
   470
apply (simp add: LIM_def, safe)
paulson@15077
   471
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077
   472
apply (auto simp add: less_diff_eq)
webertj@20432
   473
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077
   474
apply (rule_tac y = 0 in order_le_less_trans, auto)
nipkow@15546
   475
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (\<Sum>n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (\<Sum>n. (c n) * ( (x + xa) ^ n))")
paulson@15077
   476
apply (auto intro!: summable_sums)
paulson@15077
   477
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077
   478
apply (auto simp add: add_commute)
paulson@15077
   479
apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
avigad@16819
   480
apply (drule_tac f = "(%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
paulson@15085
   481
apply (rule sums_unique)
paulson@15079
   482
apply (simp add: diff_def divide_inverse add_ac mult_ac)
paulson@15077
   483
apply (rule LIM_zero_cancel)
nipkow@15546
   484
apply (rule_tac g = "%h. \<Sum>n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0))))" in LIM_trans)
webertj@20432
   485
 prefer 2 apply (blast intro: termdiffs_aux) 
paulson@15077
   486
apply (simp add: LIM_def, safe)
paulson@15077
   487
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077
   488
apply (auto simp add: less_diff_eq)
webertj@20432
   489
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077
   490
apply (rule_tac y = 0 in order_le_less_trans, auto)
paulson@15077
   491
apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
paulson@15077
   492
apply (rule_tac [2] powser_inside, auto)
paulson@15077
   493
apply (drule_tac c = c and x = x in diffs_equiv)
paulson@15077
   494
apply (frule sums_unique, auto)
nipkow@15546
   495
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (\<Sum>n. (c n) * (x ^ n)) & (%n. (c n) * ((x + xa) ^ n)) sums (\<Sum>n. (c n) * ( (x + xa) ^ n))")
paulson@15077
   496
apply safe
paulson@15077
   497
apply (auto intro!: summable_sums)
paulson@15077
   498
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077
   499
apply (auto simp add: add_commute)
paulson@15229
   500
apply (frule_tac x = "(%n. c n * (xa + x) ^ n) " and y = "(%n. c n * x ^ n)" in sums_diff, assumption)
paulson@15077
   501
apply (simp add: suminf_diff [OF sums_summable sums_summable] 
paulson@15077
   502
               right_diff_distrib [symmetric])
avigad@16819
   503
apply (subst suminf_diff)
avigad@16819
   504
apply (rule summable_mult2)
avigad@16819
   505
apply (erule sums_summable)
avigad@16819
   506
apply (erule sums_summable)
avigad@16819
   507
apply (simp add: ring_eq_simps)
paulson@15077
   508
done
paulson@15077
   509
paulson@15077
   510
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
paulson@15077
   511
paulson@15077
   512
lemma exp_fdiffs: 
paulson@15077
   513
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
paulson@15229
   514
by (simp add: diffs_def mult_assoc [symmetric] del: mult_Suc)
paulson@15077
   515
paulson@15077
   516
lemma sin_fdiffs: 
paulson@15077
   517
      "diffs(%n. if even n then 0  
paulson@15077
   518
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
paulson@15077
   519
       = (%n. if even n then  
paulson@15077
   520
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   521
              else 0)"
paulson@15229
   522
by (auto intro!: ext 
paulson@15229
   523
         simp add: diffs_def divide_inverse simp del: mult_Suc)
paulson@15077
   524
paulson@15077
   525
lemma sin_fdiffs2: 
paulson@15077
   526
       "diffs(%n. if even n then 0  
paulson@15077
   527
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
paulson@15077
   528
       = (if even n then  
paulson@15077
   529
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   530
              else 0)"
paulson@15229
   531
by (auto intro!: ext 
paulson@15229
   532
         simp add: diffs_def divide_inverse simp del: mult_Suc)
paulson@15077
   533
paulson@15077
   534
lemma cos_fdiffs: 
paulson@15077
   535
      "diffs(%n. if even n then  
paulson@15077
   536
                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
paulson@15077
   537
       = (%n. - (if even n then 0  
paulson@15077
   538
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
paulson@15229
   539
by (auto intro!: ext 
paulson@15229
   540
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
paulson@15229
   541
         simp del: mult_Suc)
paulson@15077
   542
paulson@15077
   543
paulson@15077
   544
lemma cos_fdiffs2: 
paulson@15077
   545
      "diffs(%n. if even n then  
paulson@15077
   546
                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
paulson@15077
   547
       = - (if even n then 0  
paulson@15077
   548
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
paulson@15229
   549
by (auto intro!: ext 
paulson@15229
   550
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
paulson@15229
   551
         simp del: mult_Suc)
paulson@15077
   552
paulson@15077
   553
text{*Now at last we can get the derivatives of exp, sin and cos*}
paulson@15077
   554
paulson@15077
   555
lemma lemma_sin_minus:
nipkow@15546
   556
     "- sin x = (\<Sum>n. - ((if even n then 0 
paulson@15077
   557
                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
paulson@15077
   558
by (auto intro!: sums_unique sums_minus sin_converges)
paulson@15077
   559
nipkow@15546
   560
lemma lemma_exp_ext: "exp = (%x. \<Sum>n. inverse (real (fact n)) * x ^ n)"
paulson@15077
   561
by (auto intro!: ext simp add: exp_def)
paulson@15077
   562
paulson@15077
   563
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15229
   564
apply (simp add: exp_def)
paulson@15077
   565
apply (subst lemma_exp_ext)
nipkow@15546
   566
apply (subgoal_tac "DERIV (%u. \<Sum>n. inverse (real (fact n)) * u ^ n) x :> (\<Sum>n. diffs (%n. inverse (real (fact n))) n * x ^ n)")
paulson@15229
   567
apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   568
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs)
paulson@15077
   569
done
paulson@15077
   570
paulson@15077
   571
lemma lemma_sin_ext:
nipkow@15546
   572
     "sin = (%x. \<Sum>n. 
paulson@15077
   573
                   (if even n then 0  
paulson@15077
   574
                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
nipkow@15546
   575
                   x ^ n)"
paulson@15077
   576
by (auto intro!: ext simp add: sin_def)
paulson@15077
   577
paulson@15077
   578
lemma lemma_cos_ext:
nipkow@15546
   579
     "cos = (%x. \<Sum>n. 
paulson@15077
   580
                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
nipkow@15546
   581
                   x ^ n)"
paulson@15077
   582
by (auto intro!: ext simp add: cos_def)
paulson@15077
   583
paulson@15077
   584
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
paulson@15229
   585
apply (simp add: cos_def)
paulson@15077
   586
apply (subst lemma_sin_ext)
paulson@15077
   587
apply (auto simp add: sin_fdiffs2 [symmetric])
paulson@15229
   588
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   589
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
paulson@15077
   590
done
paulson@15077
   591
paulson@15077
   592
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
paulson@15077
   593
apply (subst lemma_cos_ext)
paulson@15077
   594
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
paulson@15229
   595
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217
   596
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus)
paulson@15077
   597
done
paulson@15077
   598
paulson@15077
   599
paulson@15077
   600
subsection{*Properties of the Exponential Function*}
paulson@15077
   601
paulson@15077
   602
lemma exp_zero [simp]: "exp 0 = 1"
paulson@15077
   603
proof -
paulson@15077
   604
  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
nipkow@15546
   605
        (\<Sum>n. inverse (real (fact n)) * 0 ^ n)"
paulson@15077
   606
    by (rule series_zero [rule_format, THEN sums_unique],
paulson@15077
   607
        case_tac "m", auto)
paulson@15077
   608
  thus ?thesis by (simp add:  exp_def) 
paulson@15077
   609
qed
paulson@15077
   610
avigad@17014
   611
lemma exp_ge_add_one_self_aux: "0 \<le> x ==> (1 + x) \<le> exp(x)"
paulson@15077
   612
apply (drule real_le_imp_less_or_eq, auto)
paulson@15229
   613
apply (simp add: exp_def)
paulson@15077
   614
apply (rule real_le_trans)
paulson@15229
   615
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
paulson@15077
   616
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
paulson@15077
   617
done
paulson@15077
   618
paulson@15077
   619
lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
paulson@15077
   620
apply (rule order_less_le_trans)
avigad@17014
   621
apply (rule_tac [2] exp_ge_add_one_self_aux, auto)
paulson@15077
   622
done
paulson@15077
   623
paulson@15077
   624
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
paulson@15077
   625
proof -
paulson@15077
   626
  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
paulson@15077
   627
    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const) 
paulson@15077
   628
  thus ?thesis by (simp add: o_def)
paulson@15077
   629
qed
paulson@15077
   630
paulson@15077
   631
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
paulson@15077
   632
proof -
paulson@15077
   633
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
paulson@15077
   634
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
paulson@15077
   635
  thus ?thesis by (simp add: o_def)
paulson@15077
   636
qed
paulson@15077
   637
paulson@15077
   638
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
paulson@15077
   639
proof -
paulson@15077
   640
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
paulson@15077
   641
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
paulson@15077
   642
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
paulson@15077
   643
  thus ?thesis by simp
paulson@15077
   644
qed
paulson@15077
   645
paulson@15077
   646
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
paulson@15077
   647
proof -
paulson@15077
   648
  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
paulson@15077
   649
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
paulson@15077
   650
    by (rule DERIV_isconst_all) 
paulson@15077
   651
  thus ?thesis by simp
paulson@15077
   652
qed
paulson@15077
   653
paulson@15077
   654
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
paulson@15077
   655
proof -
paulson@15077
   656
  have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
paulson@15077
   657
  thus ?thesis by simp
paulson@15077
   658
qed
paulson@15077
   659
paulson@15077
   660
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
paulson@15077
   661
by (simp add: mult_commute)
paulson@15077
   662
paulson@15077
   663
paulson@15077
   664
lemma exp_minus: "exp(-x) = inverse(exp(x))"
paulson@15077
   665
by (auto intro: inverse_unique [symmetric])
paulson@15077
   666
paulson@15077
   667
lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
paulson@15077
   668
proof -
paulson@15077
   669
  have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
paulson@15077
   670
  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
paulson@15077
   671
qed
paulson@15077
   672
paulson@15077
   673
text{*Proof: because every exponential can be seen as a square.*}
paulson@15077
   674
lemma exp_ge_zero [simp]: "0 \<le> exp x"
paulson@15077
   675
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
paulson@15077
   676
apply (subst exp_add, auto)
paulson@15077
   677
done
paulson@15077
   678
paulson@15077
   679
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
paulson@15077
   680
apply (cut_tac x = x in exp_mult_minus2)
paulson@15077
   681
apply (auto simp del: exp_mult_minus2)
paulson@15077
   682
done
paulson@15077
   683
paulson@15077
   684
lemma exp_gt_zero [simp]: "0 < exp x"
paulson@15077
   685
by (simp add: order_less_le)
paulson@15077
   686
paulson@15077
   687
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
paulson@15077
   688
by (auto intro: positive_imp_inverse_positive)
paulson@15077
   689
paulson@15081
   690
lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
paulson@15229
   691
by auto
paulson@15077
   692
paulson@15077
   693
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
   694
apply (induct "n")
paulson@15077
   695
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
   696
done
paulson@15077
   697
paulson@15077
   698
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
paulson@15229
   699
apply (simp add: diff_minus divide_inverse)
paulson@15077
   700
apply (simp (no_asm) add: exp_add exp_minus)
paulson@15077
   701
done
paulson@15077
   702
paulson@15077
   703
paulson@15077
   704
lemma exp_less_mono:
paulson@15077
   705
  assumes xy: "x < y" shows "exp x < exp y"
paulson@15077
   706
proof -
paulson@15077
   707
  have "1 < exp (y + - x)"
paulson@15077
   708
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
paulson@15077
   709
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
paulson@15077
   710
    by (auto simp add: exp_add exp_minus)
paulson@15077
   711
  thus ?thesis
nipkow@15539
   712
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq
paulson@15228
   713
             del: divide_self_if)
paulson@15077
   714
qed
paulson@15077
   715
paulson@15077
   716
lemma exp_less_cancel: "exp x < exp y ==> x < y"
paulson@15228
   717
apply (simp add: linorder_not_le [symmetric]) 
paulson@15228
   718
apply (auto simp add: order_le_less exp_less_mono) 
paulson@15077
   719
done
paulson@15077
   720
paulson@15077
   721
lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
paulson@15077
   722
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
   723
paulson@15077
   724
lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
paulson@15077
   725
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   726
paulson@15077
   727
lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
paulson@15077
   728
by (simp add: order_eq_iff)
paulson@15077
   729
paulson@15077
   730
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
paulson@15077
   731
apply (rule IVT)
paulson@15077
   732
apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
paulson@15077
   733
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
paulson@15077
   734
apply simp 
avigad@17014
   735
apply (rule exp_ge_add_one_self_aux, simp)
paulson@15077
   736
done
paulson@15077
   737
paulson@15077
   738
lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
paulson@15077
   739
apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077
   740
apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077
   741
apply (rule_tac [2] x = 0 in exI)
paulson@15077
   742
apply (frule_tac [3] real_inverse_gt_one)
paulson@15077
   743
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077
   744
apply (rule_tac x = "-x" in exI)
paulson@15077
   745
apply (simp add: exp_minus)
paulson@15077
   746
done
paulson@15077
   747
paulson@15077
   748
paulson@15077
   749
subsection{*Properties of the Logarithmic Function*}
paulson@15077
   750
paulson@15077
   751
lemma ln_exp[simp]: "ln(exp x) = x"
paulson@15077
   752
by (simp add: ln_def)
paulson@15077
   753
paulson@15077
   754
lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
paulson@15077
   755
apply (auto dest: exp_total)
paulson@15077
   756
apply (erule subst, simp) 
paulson@15077
   757
done
paulson@15077
   758
paulson@15077
   759
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
paulson@15077
   760
apply (rule exp_inj_iff [THEN iffD1])
paulson@15077
   761
apply (frule real_mult_order)
paulson@15077
   762
apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
paulson@15077
   763
done
paulson@15077
   764
paulson@15077
   765
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
paulson@15077
   766
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   767
apply (erule subst)+
paulson@15077
   768
apply simp 
paulson@15077
   769
done
paulson@15077
   770
paulson@15077
   771
lemma ln_one[simp]: "ln 1 = 0"
paulson@15077
   772
by (rule exp_inj_iff [THEN iffD1], auto)
paulson@15077
   773
paulson@15077
   774
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
paulson@15077
   775
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
paulson@15077
   776
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
paulson@15077
   777
done
paulson@15077
   778
paulson@15077
   779
lemma ln_div: 
paulson@15077
   780
    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
paulson@15229
   781
apply (simp add: divide_inverse)
paulson@15077
   782
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
paulson@15077
   783
done
paulson@15077
   784
paulson@15077
   785
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
paulson@15077
   786
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   787
apply (erule subst)+
paulson@15077
   788
apply simp 
paulson@15077
   789
done
paulson@15077
   790
paulson@15077
   791
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
paulson@15077
   792
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   793
paulson@15077
   794
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
paulson@15077
   795
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
paulson@15077
   796
paulson@15077
   797
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
paulson@15077
   798
apply (rule ln_exp [THEN subst])
avigad@17014
   799
apply (rule ln_le_cancel_iff [THEN iffD2]) 
avigad@17014
   800
apply (auto simp add: exp_ge_add_one_self_aux)
paulson@15077
   801
done
paulson@15077
   802
paulson@15077
   803
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
paulson@15077
   804
apply (rule order_less_le_trans)
paulson@15077
   805
apply (rule_tac [2] ln_add_one_self_le_self)
paulson@15077
   806
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
paulson@15077
   807
done
paulson@15077
   808
paulson@15234
   809
lemma ln_ge_zero [simp]:
paulson@15077
   810
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077
   811
proof -
paulson@15077
   812
  have "0 < x" using x by arith
paulson@15077
   813
  hence "exp 0 \<le> exp (ln x)"
paulson@15077
   814
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077
   815
  thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077
   816
qed
paulson@15077
   817
paulson@15077
   818
lemma ln_ge_zero_imp_ge_one:
paulson@15077
   819
  assumes ln: "0 \<le> ln x" 
paulson@15077
   820
      and x:  "0 < x"
paulson@15077
   821
  shows "1 \<le> x"
paulson@15077
   822
proof -
paulson@15077
   823
  from ln have "ln 1 \<le> ln x" by simp
paulson@15077
   824
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
   825
qed
paulson@15077
   826
paulson@15077
   827
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077
   828
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077
   829
paulson@15234
   830
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
paulson@15234
   831
by (insert ln_ge_zero_iff [of x], arith)
paulson@15234
   832
paulson@15077
   833
lemma ln_gt_zero:
paulson@15077
   834
  assumes x: "1 < x" shows "0 < ln x"
paulson@15077
   835
proof -
paulson@15077
   836
  have "0 < x" using x by arith
paulson@15077
   837
  hence "exp 0 < exp (ln x)"
paulson@15077
   838
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077
   839
  thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077
   840
qed
paulson@15077
   841
paulson@15077
   842
lemma ln_gt_zero_imp_gt_one:
paulson@15077
   843
  assumes ln: "0 < ln x" 
paulson@15077
   844
      and x:  "0 < x"
paulson@15077
   845
  shows "1 < x"
paulson@15077
   846
proof -
paulson@15077
   847
  from ln have "ln 1 < ln x" by simp
paulson@15077
   848
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
   849
qed
paulson@15077
   850
paulson@15077
   851
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077
   852
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077
   853
paulson@15234
   854
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
paulson@15234
   855
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
paulson@15077
   856
paulson@15077
   857
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15234
   858
by simp
paulson@15077
   859
paulson@15077
   860
lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077
   861
by auto
paulson@15077
   862
paulson@15077
   863
paulson@15077
   864
subsection{*Basic Properties of the Trigonometric Functions*}
paulson@15077
   865
paulson@15077
   866
lemma sin_zero [simp]: "sin 0 = 0"
paulson@15077
   867
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
paulson@15077
   868
         simp add: sin_def sums_def simp del: power_0_left)
paulson@15077
   869
nipkow@15539
   870
lemma lemma_series_zero2:
nipkow@15539
   871
 "(\<forall>m. n \<le> m --> f m = 0) --> f sums setsum f {0..<n}"
paulson@15077
   872
by (auto intro: series_zero)
paulson@15077
   873
paulson@15077
   874
lemma cos_zero [simp]: "cos 0 = 1"
paulson@15229
   875
apply (simp add: cos_def)
paulson@15077
   876
apply (rule sums_unique [symmetric])
paulson@15229
   877
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)
paulson@15077
   878
apply auto
paulson@15077
   879
done
paulson@15077
   880
paulson@15077
   881
lemma DERIV_sin_sin_mult [simp]:
paulson@15077
   882
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
   883
by (rule DERIV_mult, auto)
paulson@15077
   884
paulson@15077
   885
lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077
   886
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077
   887
apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077
   888
apply (auto simp add: mult_assoc)
paulson@15077
   889
done
paulson@15077
   890
paulson@15077
   891
lemma DERIV_sin_realpow2 [simp]:
paulson@15077
   892
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
   893
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
   894
paulson@15077
   895
lemma DERIV_sin_realpow2a [simp]:
paulson@15077
   896
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077
   897
by (auto simp add: numeral_2_eq_2)
paulson@15077
   898
paulson@15077
   899
lemma DERIV_cos_cos_mult [simp]:
paulson@15077
   900
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
   901
by (rule DERIV_mult, auto)
paulson@15077
   902
paulson@15077
   903
lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077
   904
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077
   905
apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077
   906
apply (auto simp add: mult_ac)
paulson@15077
   907
done
paulson@15077
   908
paulson@15077
   909
lemma DERIV_cos_realpow2 [simp]:
paulson@15077
   910
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
   911
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
   912
paulson@15077
   913
lemma DERIV_cos_realpow2a [simp]:
paulson@15077
   914
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077
   915
by (auto simp add: numeral_2_eq_2)
paulson@15077
   916
paulson@15077
   917
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077
   918
by auto
paulson@15077
   919
paulson@15077
   920
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
paulson@15077
   921
apply (rule lemma_DERIV_subst)
paulson@15077
   922
apply (rule DERIV_cos_realpow2a, auto)
paulson@15077
   923
done
paulson@15077
   924
paulson@15077
   925
(* most useful *)
paulson@15229
   926
lemma DERIV_cos_cos_mult3 [simp]:
paulson@15229
   927
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
paulson@15077
   928
apply (rule lemma_DERIV_subst)
paulson@15077
   929
apply (rule DERIV_cos_cos_mult2, auto)
paulson@15077
   930
done
paulson@15077
   931
paulson@15077
   932
lemma DERIV_sin_circle_all: 
paulson@15077
   933
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077
   934
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
paulson@15229
   935
apply (simp only: diff_minus, safe)
paulson@15229
   936
apply (rule DERIV_add) 
paulson@15077
   937
apply (auto simp add: numeral_2_eq_2)
paulson@15077
   938
done
paulson@15077
   939
paulson@15229
   940
lemma DERIV_sin_circle_all_zero [simp]:
paulson@15229
   941
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077
   942
by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077
   943
paulson@15077
   944
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077
   945
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077
   946
apply (auto simp add: numeral_2_eq_2)
paulson@15077
   947
done
paulson@15077
   948
paulson@15077
   949
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
paulson@15077
   950
apply (subst real_add_commute)
paulson@15077
   951
apply (simp (no_asm) del: realpow_Suc)
paulson@15077
   952
done
paulson@15077
   953
paulson@15077
   954
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077
   955
apply (cut_tac x = x in sin_cos_squared_add2)
paulson@15077
   956
apply (auto simp add: numeral_2_eq_2)
paulson@15077
   957
done
paulson@15077
   958
paulson@15077
   959
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15229
   960
apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
   961
apply (simp del: realpow_Suc)
paulson@15077
   962
done
paulson@15077
   963
paulson@15077
   964
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077
   965
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
   966
apply (simp del: realpow_Suc)
paulson@15077
   967
done
paulson@15077
   968
paulson@15077
   969
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
paulson@15077
   970
by arith
paulson@15077
   971
paulson@15081
   972
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
paulson@15077
   973
apply (auto simp add: linorder_not_less [symmetric])
paulson@15077
   974
apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077
   975
apply (auto simp del: realpow_Suc)
paulson@15077
   976
apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077
   977
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
   978
done
paulson@15077
   979
paulson@15077
   980
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
paulson@15077
   981
apply (insert abs_sin_le_one [of x]) 
paulson@15077
   982
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077
   983
done
paulson@15077
   984
paulson@15077
   985
lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077
   986
apply (insert abs_sin_le_one [of x]) 
paulson@15077
   987
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077
   988
done
paulson@15077
   989
paulson@15081
   990
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
paulson@15077
   991
apply (auto simp add: linorder_not_less [symmetric])
paulson@15077
   992
apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077
   993
apply (auto simp del: realpow_Suc)
paulson@15077
   994
apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077
   995
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
   996
done
paulson@15077
   997
paulson@15077
   998
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
paulson@15077
   999
apply (insert abs_cos_le_one [of x]) 
paulson@15077
  1000
apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
paulson@15077
  1001
done
paulson@15077
  1002
paulson@15077
  1003
lemma cos_le_one [simp]: "cos x \<le> 1"
paulson@15077
  1004
apply (insert abs_cos_le_one [of x]) 
paulson@15077
  1005
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
paulson@15077
  1006
done
paulson@15077
  1007
paulson@15077
  1008
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077
  1009
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
paulson@15077
  1010
apply (rule lemma_DERIV_subst)
paulson@15229
  1011
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
paulson@15077
  1012
apply (rule DERIV_pow, auto)
paulson@15077
  1013
done
paulson@15077
  1014
paulson@15229
  1015
lemma DERIV_fun_exp:
paulson@15229
  1016
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077
  1017
apply (rule lemma_DERIV_subst)
paulson@15077
  1018
apply (rule_tac f = exp in DERIV_chain2)
paulson@15077
  1019
apply (rule DERIV_exp, auto)
paulson@15077
  1020
done
paulson@15077
  1021
paulson@15229
  1022
lemma DERIV_fun_sin:
paulson@15229
  1023
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077
  1024
apply (rule lemma_DERIV_subst)
paulson@15077
  1025
apply (rule_tac f = sin in DERIV_chain2)
paulson@15077
  1026
apply (rule DERIV_sin, auto)
paulson@15077
  1027
done
paulson@15077
  1028
paulson@15229
  1029
lemma DERIV_fun_cos:
paulson@15229
  1030
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077
  1031
apply (rule lemma_DERIV_subst)
paulson@15077
  1032
apply (rule_tac f = cos in DERIV_chain2)
paulson@15077
  1033
apply (rule DERIV_cos, auto)
paulson@15077
  1034
done
paulson@15077
  1035
paulson@15077
  1036
lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
paulson@15077
  1037
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
paulson@15077
  1038
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
paulson@15077
  1039
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
paulson@15077
  1040
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
paulson@15077
  1041
paulson@15077
  1042
(* lemma *)
paulson@15229
  1043
lemma lemma_DERIV_sin_cos_add:
paulson@15229
  1044
     "\<forall>x.  
paulson@15077
  1045
         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1046
               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
paulson@15077
  1047
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1048
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1049
  --{*replaces the old @{text DERIV_tac}*}
paulson@15229
  1050
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1051
done
paulson@15077
  1052
paulson@15077
  1053
lemma sin_cos_add [simp]:
paulson@15077
  1054
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1055
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077
  1056
apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077
  1057
       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077
  1058
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1059
done
paulson@15077
  1060
paulson@15077
  1061
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
paulson@15077
  1062
apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077
  1063
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085
  1064
            simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
paulson@15077
  1065
done
paulson@15077
  1066
paulson@15077
  1067
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077
  1068
apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077
  1069
apply (auto dest!: real_sum_squares_cancel_a
paulson@15085
  1070
            simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
paulson@15077
  1071
done
paulson@15077
  1072
paulson@15085
  1073
lemma lemma_DERIV_sin_cos_minus:
paulson@15085
  1074
    "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
paulson@15077
  1075
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1076
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15229
  1077
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1078
done
paulson@15077
  1079
paulson@15085
  1080
lemma sin_cos_minus [simp]: 
paulson@15085
  1081
    "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15085
  1082
apply (cut_tac y = 0 and x = x 
paulson@15085
  1083
       in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
paulson@15077
  1084
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1085
done
paulson@15077
  1086
paulson@15077
  1087
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
paulson@15077
  1088
apply (cut_tac x = x in sin_cos_minus)
paulson@15085
  1089
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085
  1090
       simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_minus)
paulson@15077
  1091
done
paulson@15077
  1092
paulson@15077
  1093
lemma cos_minus [simp]: "cos (-x) = cos(x)"
paulson@15077
  1094
apply (cut_tac x = x in sin_cos_minus)
paulson@15085
  1095
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085
  1096
                   simp add: numeral_2_eq_2 simp del: sin_cos_minus)
paulson@15077
  1097
done
paulson@15077
  1098
paulson@15077
  1099
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
paulson@15229
  1100
apply (simp add: diff_minus)
paulson@15077
  1101
apply (simp (no_asm) add: sin_add)
paulson@15077
  1102
done
paulson@15077
  1103
paulson@15077
  1104
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077
  1105
by (simp add: sin_diff mult_commute)
paulson@15077
  1106
paulson@15077
  1107
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
paulson@15229
  1108
apply (simp add: diff_minus)
paulson@15077
  1109
apply (simp (no_asm) add: cos_add)
paulson@15077
  1110
done
paulson@15077
  1111
paulson@15077
  1112
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077
  1113
by (simp add: cos_diff mult_commute)
paulson@15077
  1114
paulson@15077
  1115
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
paulson@15077
  1116
by (cut_tac x = x and y = x in sin_add, auto)
paulson@15077
  1117
paulson@15077
  1118
paulson@15077
  1119
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
paulson@15077
  1120
apply (cut_tac x = x and y = x in cos_add)
paulson@15077
  1121
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1122
done
paulson@15077
  1123
paulson@15077
  1124
paulson@15077
  1125
subsection{*The Constant Pi*}
paulson@15077
  1126
paulson@15077
  1127
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077
  1128
   hence define pi.*}
paulson@15077
  1129
paulson@15077
  1130
lemma sin_paired:
paulson@15077
  1131
     "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077
  1132
      sums  sin x"
paulson@15077
  1133
proof -
paulson@15077
  1134
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1135
            (if even k then 0
paulson@15077
  1136
             else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
paulson@15077
  1137
            x ^ k) 
paulson@15077
  1138
	sums
nipkow@15546
  1139
	(\<Sum>n. (if even n then 0
paulson@15077
  1140
		     else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
paulson@15077
  1141
	            x ^ n)" 
paulson@15077
  1142
    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1143
  thus ?thesis by (simp add: mult_ac sin_def)
paulson@15077
  1144
qed
paulson@15077
  1145
paulson@15077
  1146
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1147
apply (subgoal_tac 
paulson@15077
  1148
       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1149
              (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
nipkow@15546
  1150
     sums (\<Sum>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077
  1151
 prefer 2
paulson@15077
  1152
 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1153
apply (rotate_tac 2)
paulson@15077
  1154
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
paulson@15077
  1155
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1156
apply (frule sums_unique)
paulson@15077
  1157
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1158
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
paulson@15077
  1159
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1160
apply (erule sums_summable)
paulson@15077
  1161
apply (case_tac "m=0")
paulson@15077
  1162
apply (simp (no_asm_simp))
paulson@15234
  1163
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") 
nipkow@15539
  1164
apply (simp only: mult_less_cancel_left, simp)  
nipkow@15539
  1165
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077
  1166
apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077
  1167
apply (rule mult_strict_mono)
paulson@15085
  1168
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
paulson@15077
  1169
apply (subst fact_Suc)
paulson@15077
  1170
apply (subst fact_Suc)
paulson@15077
  1171
apply (subst fact_Suc)
paulson@15077
  1172
apply (subst fact_Suc)
paulson@15077
  1173
apply (subst real_of_nat_mult)
paulson@15077
  1174
apply (subst real_of_nat_mult)
paulson@15077
  1175
apply (subst real_of_nat_mult)
paulson@15077
  1176
apply (subst real_of_nat_mult)
nipkow@15539
  1177
apply (simp (no_asm) add: divide_inverse del: fact_Suc)
paulson@15077
  1178
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077
  1179
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
paulson@15077
  1180
apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077
  1181
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
paulson@15077
  1182
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077
  1183
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077
  1184
apply (erule ssubst)+
paulson@15077
  1185
apply (auto simp del: fact_Suc)
paulson@15077
  1186
apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077
  1187
 prefer 2 apply (simp only: zero_less_power) 
paulson@15077
  1188
apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077
  1189
apply (rule mult_strict_mono)
paulson@15077
  1190
apply (simp_all (no_asm_simp))
paulson@15077
  1191
done
paulson@15077
  1192
paulson@15077
  1193
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1194
by (auto intro: sin_gt_zero)
paulson@15077
  1195
paulson@15077
  1196
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077
  1197
apply (cut_tac x = x in sin_gt_zero1)
paulson@15077
  1198
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1199
done
paulson@15077
  1200
paulson@15077
  1201
lemma cos_paired:
paulson@15077
  1202
     "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1203
proof -
paulson@15077
  1204
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1205
            (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
paulson@15077
  1206
            x ^ k) 
paulson@15077
  1207
        sums
nipkow@15546
  1208
	(\<Sum>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
paulson@15077
  1209
	      x ^ n)" 
paulson@15077
  1210
    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1211
  thus ?thesis by (simp add: mult_ac cos_def)
paulson@15077
  1212
qed
paulson@15077
  1213
paulson@15077
  1214
declare zero_less_power [simp]
paulson@15077
  1215
paulson@15077
  1216
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077
  1217
by simp
paulson@15077
  1218
paulson@15077
  1219
lemma cos_two_less_zero: "cos (2) < 0"
paulson@15077
  1220
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1221
apply (drule sums_minus)
paulson@15077
  1222
apply (rule neg_less_iff_less [THEN iffD1]) 
nipkow@15539
  1223
apply (frule sums_unique, auto)
nipkow@15539
  1224
apply (rule_tac y =
nipkow@15539
  1225
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - ((- 1) ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1226
       in order_less_trans)
paulson@15077
  1227
apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
nipkow@15561
  1228
apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077
  1229
apply (rule sumr_pos_lt_pair)
paulson@15077
  1230
apply (erule sums_summable, safe)
paulson@15085
  1231
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] 
paulson@15085
  1232
            del: fact_Suc)
paulson@15077
  1233
apply (rule real_mult_inverse_cancel2)
paulson@15077
  1234
apply (rule real_of_nat_fact_gt_zero)+
paulson@15077
  1235
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1236
apply (subst fact_lemma) 
paulson@15481
  1237
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1238
apply (simp only: real_of_nat_mult)
paulson@15077
  1239
apply (rule real_mult_less_mono, force)
paulson@15481
  1240
  apply (rule_tac [3] real_of_nat_fact_gt_zero)
paulson@15481
  1241
 prefer 2 apply force
paulson@15077
  1242
apply (rule real_of_nat_less_iff [THEN iffD2])
paulson@15077
  1243
apply (rule fact_less_mono, auto)
paulson@15077
  1244
done
paulson@15077
  1245
declare cos_two_less_zero [simp]
paulson@15077
  1246
declare cos_two_less_zero [THEN real_not_refl2, simp]
paulson@15077
  1247
declare cos_two_less_zero [THEN order_less_imp_le, simp]
paulson@15077
  1248
paulson@15077
  1249
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077
  1250
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077
  1251
apply (rule_tac [2] IVT2)
paulson@15077
  1252
apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077
  1253
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1254
apply (rule ccontr)
paulson@15077
  1255
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077
  1256
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077
  1257
apply (drule_tac f = cos in Rolle)
paulson@15077
  1258
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1259
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@15077
  1260
apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
paulson@15077
  1261
apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
paulson@15077
  1262
apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
paulson@15077
  1263
done
paulson@15077
  1264
    
paulson@15077
  1265
lemma pi_half: "pi/2 = (@x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1266
by (simp add: pi_def)
paulson@15077
  1267
paulson@15077
  1268
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
paulson@15077
  1269
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1270
apply (auto intro!: someI2 simp add: pi_half)
paulson@15077
  1271
done
paulson@15077
  1272
paulson@15077
  1273
lemma pi_half_gt_zero: "0 < pi / 2"
paulson@15077
  1274
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1275
apply (auto simp add: pi_half)
paulson@15077
  1276
apply (rule someI2, blast, safe)
paulson@15077
  1277
apply (drule_tac y = xa in real_le_imp_less_or_eq)
paulson@15077
  1278
apply (safe, simp)
paulson@15077
  1279
done
paulson@15077
  1280
declare pi_half_gt_zero [simp]
paulson@15077
  1281
declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
paulson@15077
  1282
declare pi_half_gt_zero [THEN order_less_imp_le, simp]
paulson@15077
  1283
paulson@15077
  1284
lemma pi_half_less_two: "pi / 2 < 2"
paulson@15077
  1285
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1286
apply (auto simp add: pi_half)
paulson@15077
  1287
apply (rule someI2, blast, safe)
paulson@15077
  1288
apply (drule_tac x = xa in order_le_imp_less_or_eq)
paulson@15077
  1289
apply (safe, simp)
paulson@15077
  1290
done
paulson@15077
  1291
declare pi_half_less_two [simp]
paulson@15077
  1292
declare pi_half_less_two [THEN real_not_refl2, simp]
paulson@15077
  1293
declare pi_half_less_two [THEN order_less_imp_le, simp]
paulson@15077
  1294
paulson@15077
  1295
lemma pi_gt_zero [simp]: "0 < pi"
paulson@15229
  1296
apply (insert pi_half_gt_zero) 
paulson@15229
  1297
apply (simp add: ); 
paulson@15077
  1298
done
paulson@15077
  1299
paulson@15077
  1300
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
paulson@15077
  1301
by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])
paulson@15077
  1302
paulson@15077
  1303
lemma pi_not_less_zero [simp]: "~ (pi < 0)"
paulson@15077
  1304
apply (insert pi_gt_zero)
paulson@15077
  1305
apply (blast elim: order_less_asym) 
paulson@15077
  1306
done
paulson@15077
  1307
paulson@15077
  1308
lemma pi_ge_zero [simp]: "0 \<le> pi"
paulson@15077
  1309
by (auto intro: order_less_imp_le)
paulson@15077
  1310
paulson@15077
  1311
lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
paulson@15077
  1312
by auto
paulson@15077
  1313
paulson@15077
  1314
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1315
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1316
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
paulson@15077
  1317
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1318
done
paulson@15077
  1319
paulson@15077
  1320
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1321
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1322
paulson@15077
  1323
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1324
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1325
paulson@15077
  1326
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15229
  1327
by (simp add: diff_minus cos_add)
paulson@15077
  1328
paulson@15077
  1329
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1330
by (simp add: cos_add)
paulson@15077
  1331
declare minus_sin_cos_eq [symmetric, simp]
paulson@15077
  1332
paulson@15077
  1333
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15229
  1334
by (simp add: diff_minus sin_add)
paulson@15077
  1335
declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]
paulson@15077
  1336
paulson@15077
  1337
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1338
by (simp add: sin_add)
paulson@15077
  1339
paulson@15077
  1340
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1341
by (simp add: sin_add)
paulson@15077
  1342
paulson@15077
  1343
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1344
by (simp add: cos_add)
paulson@15077
  1345
paulson@15077
  1346
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1347
by (simp add: sin_add cos_double)
paulson@15077
  1348
paulson@15077
  1349
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1350
by (simp add: cos_add cos_double)
paulson@15077
  1351
paulson@15077
  1352
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1353
apply (induct "n")
paulson@15077
  1354
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1355
done
paulson@15077
  1356
paulson@15383
  1357
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1358
proof -
paulson@15383
  1359
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
paulson@15383
  1360
  also have "... = -1 ^ n" by (rule cos_npi) 
paulson@15383
  1361
  finally show ?thesis .
paulson@15383
  1362
qed
paulson@15383
  1363
paulson@15077
  1364
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1365
apply (induct "n")
paulson@15077
  1366
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1367
done
paulson@15077
  1368
paulson@15077
  1369
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15383
  1370
by (simp add: mult_commute [of pi]) 
paulson@15077
  1371
paulson@15077
  1372
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1373
by (simp add: cos_double)
paulson@15077
  1374
paulson@15077
  1375
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1376
by simp
paulson@15077
  1377
paulson@15077
  1378
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1379
apply (rule sin_gt_zero, assumption)
paulson@15077
  1380
apply (rule order_less_trans, assumption)
paulson@15077
  1381
apply (rule pi_half_less_two)
paulson@15077
  1382
done
paulson@15077
  1383
paulson@15077
  1384
lemma sin_less_zero: 
paulson@15077
  1385
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1386
proof -
paulson@15077
  1387
  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077
  1388
  thus ?thesis by simp
paulson@15077
  1389
qed
paulson@15077
  1390
paulson@15077
  1391
lemma pi_less_4: "pi < 4"
paulson@15077
  1392
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1393
paulson@15077
  1394
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1395
apply (cut_tac pi_less_4)
paulson@15077
  1396
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1397
apply (force intro: DERIV_isCont DERIV_cos)
paulson@15077
  1398
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1399
apply (rename_tac y z)
paulson@15077
  1400
apply (drule_tac x = y in spec)
paulson@15077
  1401
apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077
  1402
done
paulson@15077
  1403
paulson@15077
  1404
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1405
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1406
apply (rule cos_minus [THEN subst])
paulson@15077
  1407
apply (rule cos_gt_zero)
paulson@15077
  1408
apply (auto intro: cos_gt_zero)
paulson@15077
  1409
done
paulson@15077
  1410
 
paulson@15077
  1411
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1412
apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077
  1413
apply (subgoal_tac "x = pi/2", auto) 
paulson@15077
  1414
done
paulson@15077
  1415
paulson@15077
  1416
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077
  1417
apply (subst sin_cos_eq)
paulson@15077
  1418
apply (rotate_tac 1)
paulson@15077
  1419
apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077
  1420
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077
  1421
done
paulson@15077
  1422
paulson@15077
  1423
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1424
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1425
paulson@15077
  1426
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077
  1427
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077
  1428
apply (rule_tac [2] IVT2)
paulson@15077
  1429
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077
  1430
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1431
apply (rule ccontr, auto)
paulson@15077
  1432
apply (drule_tac f = cos in Rolle)
paulson@15077
  1433
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1434
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077
  1435
            dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077
  1436
            simp add: differentiable_def)
paulson@15077
  1437
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077
  1438
done
paulson@15077
  1439
paulson@15077
  1440
lemma sin_total:
paulson@15077
  1441
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1442
apply (rule ccontr)
paulson@15077
  1443
apply (subgoal_tac "\<forall>x. (- (pi/2) \<le> x & x \<le> pi/2 & (sin x = y)) = (0 \<le> (x + pi/2) & (x + pi/2) \<le> pi & (cos (x + pi/2) = -y))")
wenzelm@18585
  1444
apply (erule contrapos_np)
paulson@15077
  1445
apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077
  1446
apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077
  1447
apply (erule ex1E)
paulson@15229
  1448
apply (rule_tac a = "x - (pi/2)" in ex1I)
paulson@15077
  1449
apply (simp (no_asm) add: real_add_assoc)
paulson@15077
  1450
apply (rotate_tac 3)
paulson@15077
  1451
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077
  1452
done
paulson@15077
  1453
paulson@15077
  1454
lemma reals_Archimedean4:
paulson@15077
  1455
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1456
apply (auto dest!: reals_Archimedean3)
paulson@15077
  1457
apply (drule_tac x = x in spec, clarify) 
paulson@15077
  1458
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077
  1459
 prefer 2 apply (erule LeastI) 
paulson@15077
  1460
apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077
  1461
apply (subgoal_tac "~ x < real nat * y")
paulson@15077
  1462
 prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077
  1463
done
paulson@15077
  1464
paulson@15077
  1465
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077
  1466
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1467
lemma cos_zero_lemma:
paulson@15229
  1468
     "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077
  1469
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1470
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15086
  1471
apply (subgoal_tac "0 \<le> x - real n * pi & 
paulson@15086
  1472
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
paulson@15086
  1473
apply (auto simp add: compare_rls) 
paulson@15077
  1474
  prefer 3 apply (simp add: cos_diff) 
paulson@15077
  1475
 prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
paulson@15077
  1476
apply (simp add: cos_diff)
paulson@15077
  1477
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1478
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1479
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1480
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1481
apply (simp add: cos_diff)
paulson@15229
  1482
apply (rule_tac x = "Suc (2 * n)" in exI)
paulson@15077
  1483
apply (simp add: real_of_nat_Suc left_distrib, auto)
paulson@15077
  1484
done
paulson@15077
  1485
paulson@15229
  1486
lemma sin_zero_lemma:
paulson@15229
  1487
     "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077
  1488
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1489
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1490
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1491
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085
  1492
apply (rule cos_zero_lemma)
paulson@15085
  1493
apply (simp_all add: add_increasing)  
paulson@15077
  1494
done
paulson@15077
  1495
paulson@15077
  1496
paulson@15229
  1497
lemma cos_zero_iff:
paulson@15229
  1498
     "(cos x = 0) =  
paulson@15077
  1499
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077
  1500
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1501
apply (rule iffI)
paulson@15077
  1502
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1503
apply (drule cos_zero_lemma, assumption+)
paulson@15077
  1504
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077
  1505
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1506
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
nipkow@15539
  1507
apply (auto simp add: cos_add)
paulson@15077
  1508
done
paulson@15077
  1509
paulson@15077
  1510
(* ditto: but to a lesser extent *)
paulson@15229
  1511
lemma sin_zero_iff:
paulson@15229
  1512
     "(sin x = 0) =  
paulson@15077
  1513
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077
  1514
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1515
apply (rule iffI)
paulson@15077
  1516
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1517
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1518
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077
  1519
apply (force simp add: minus_equation_iff [of x]) 
nipkow@15539
  1520
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1521
done
paulson@15077
  1522
paulson@15077
  1523
paulson@15077
  1524
subsection{*Tangent*}
paulson@15077
  1525
paulson@15077
  1526
lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077
  1527
by (simp add: tan_def)
paulson@15077
  1528
paulson@15077
  1529
lemma tan_pi [simp]: "tan pi = 0"
paulson@15077
  1530
by (simp add: tan_def)
paulson@15077
  1531
paulson@15077
  1532
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077
  1533
by (simp add: tan_def)
paulson@15077
  1534
paulson@15077
  1535
lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077
  1536
by (simp add: tan_def minus_mult_left)
paulson@15077
  1537
paulson@15077
  1538
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077
  1539
by (simp add: tan_def)
paulson@15077
  1540
paulson@15077
  1541
lemma lemma_tan_add1: 
paulson@15077
  1542
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1543
        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15229
  1544
apply (simp add: tan_def divide_inverse)
paulson@15229
  1545
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1546
            simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077
  1547
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15229
  1548
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1549
            simp add: mult_assoc left_diff_distrib cos_add)
paulson@15234
  1550
done  
paulson@15077
  1551
paulson@15077
  1552
lemma add_tan_eq: 
paulson@15077
  1553
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1554
       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15229
  1555
apply (simp add: tan_def)
paulson@15077
  1556
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  1557
apply (auto simp add: mult_assoc left_distrib)
nipkow@15539
  1558
apply (simp add: sin_add)
paulson@15077
  1559
done
paulson@15077
  1560
paulson@15229
  1561
lemma tan_add:
paulson@15229
  1562
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
paulson@15077
  1563
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
paulson@15077
  1564
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
paulson@15077
  1565
apply (simp add: tan_def)
paulson@15077
  1566
done
paulson@15077
  1567
paulson@15229
  1568
lemma tan_double:
paulson@15229
  1569
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077
  1570
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077
  1571
apply (insert tan_add [of x x]) 
paulson@15077
  1572
apply (simp add: mult_2 [symmetric])  
paulson@15077
  1573
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1574
done
paulson@15077
  1575
paulson@15077
  1576
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15229
  1577
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 
paulson@15077
  1578
paulson@15077
  1579
lemma tan_less_zero: 
paulson@15077
  1580
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  1581
proof -
paulson@15077
  1582
  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077
  1583
  thus ?thesis by simp
paulson@15077
  1584
qed
paulson@15077
  1585
paulson@15077
  1586
lemma lemma_DERIV_tan:
paulson@15077
  1587
     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1588
apply (rule lemma_DERIV_subst)
paulson@15077
  1589
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15079
  1590
apply (auto simp add: divide_inverse numeral_2_eq_2)
paulson@15077
  1591
done
paulson@15077
  1592
paulson@15077
  1593
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1594
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077
  1595
paulson@15077
  1596
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077
  1597
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15229
  1598
apply (simp add: divide_inverse [symmetric])
paulson@15077
  1599
apply (rule LIM_mult2)
paulson@15077
  1600
apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077
  1601
apply (rule_tac [2] LIM_inverse)
paulson@15077
  1602
apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077
  1603
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077
  1604
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077
  1605
done
paulson@15077
  1606
paulson@15077
  1607
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  1608
apply (cut_tac LIM_cos_div_sin)
paulson@15077
  1609
apply (simp only: LIM_def)
paulson@15077
  1610
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  1611
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  1612
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  1613
apply (simp (no_asm_simp))
paulson@15229
  1614
apply (drule_tac x = "(pi/2) - e" in spec)
paulson@15229
  1615
apply (auto simp add: tan_def)
paulson@15077
  1616
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  1617
apply (auto simp add: divide_inverse)
paulson@15229
  1618
apply (rule real_mult_order) 
paulson@15229
  1619
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
paulson@15229
  1620
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) 
paulson@15077
  1621
done
paulson@15077
  1622
paulson@15077
  1623
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
paulson@15077
  1624
apply (frule real_le_imp_less_or_eq, safe)
paulson@15077
  1625
 prefer 2 apply force
paulson@15077
  1626
apply (drule lemma_tan_total, safe)
paulson@15077
  1627
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  1628
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  1629
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  1630
apply (auto dest: cos_gt_zero)
paulson@15077
  1631
done
paulson@15077
  1632
paulson@15077
  1633
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1634
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  1635
apply (drule tan_total_pos)
paulson@15077
  1636
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  1637
apply (rule_tac [3] x = "-x" in exI)
paulson@15077
  1638
apply (auto intro!: exI)
paulson@15077
  1639
done
paulson@15077
  1640
paulson@15077
  1641
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1642
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  1643
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  1644
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  1645
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  1646
apply (rule_tac [4] Rolle)
paulson@15077
  1647
apply (rule_tac [2] Rolle)
paulson@15077
  1648
apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077
  1649
            simp add: differentiable_def)
paulson@15077
  1650
txt{*Now, simulate TRYALL*}
paulson@15077
  1651
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  1652
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
paulson@15077
  1653
	    simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym]) 
paulson@15077
  1654
done
paulson@15077
  1655
paulson@15229
  1656
lemma arcsin_pi:
paulson@15229
  1657
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1658
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
paulson@15077
  1659
apply (drule sin_total, assumption)
paulson@15077
  1660
apply (erule ex1E)
paulson@15229
  1661
apply (simp add: arcsin_def)
paulson@15077
  1662
apply (rule someI2, blast) 
paulson@15077
  1663
apply (force intro: order_trans) 
paulson@15077
  1664
done
paulson@15077
  1665
paulson@15229
  1666
lemma arcsin:
paulson@15229
  1667
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1668
      ==> -(pi/2) \<le> arcsin y &  
paulson@15077
  1669
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
paulson@15077
  1670
apply (unfold arcsin_def)
paulson@15077
  1671
apply (drule sin_total, assumption)
paulson@15077
  1672
apply (fast intro: someI2)
paulson@15077
  1673
done
paulson@15077
  1674
paulson@15077
  1675
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  1676
by (blast dest: arcsin)
paulson@15077
  1677
      
paulson@15077
  1678
lemma arcsin_bounded:
paulson@15077
  1679
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  1680
by (blast dest: arcsin)
paulson@15077
  1681
paulson@15077
  1682
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  1683
by (blast dest: arcsin)
paulson@15077
  1684
paulson@15077
  1685
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  1686
by (blast dest: arcsin)
paulson@15077
  1687
paulson@15077
  1688
lemma arcsin_lt_bounded:
paulson@15077
  1689
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  1690
apply (frule order_less_imp_le)
paulson@15077
  1691
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1692
apply (frule arcsin_bounded)
paulson@15077
  1693
apply (safe, simp)
paulson@15077
  1694
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  1695
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  1696
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  1697
done
paulson@15077
  1698
paulson@15077
  1699
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  1700
apply (unfold arcsin_def)
paulson@15077
  1701
apply (rule some1_equality)
paulson@15077
  1702
apply (rule sin_total, auto)
paulson@15077
  1703
done
paulson@15077
  1704
paulson@15229
  1705
lemma arcos:
paulson@15229
  1706
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1707
      ==> 0 \<le> arcos y & arcos y \<le> pi & cos(arcos y) = y"
paulson@15229
  1708
apply (simp add: arcos_def)
paulson@15077
  1709
apply (drule cos_total, assumption)
paulson@15077
  1710
apply (fast intro: someI2)
paulson@15077
  1711
done
paulson@15077
  1712
paulson@15077
  1713
lemma cos_arcos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arcos y) = y"
paulson@15077
  1714
by (blast dest: arcos)
paulson@15077
  1715
      
paulson@15077
  1716
lemma arcos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y & arcos y \<le> pi"
paulson@15077
  1717
by (blast dest: arcos)
paulson@15077
  1718
paulson@15077
  1719
lemma arcos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y"
paulson@15077
  1720
by (blast dest: arcos)
paulson@15077
  1721
paulson@15077
  1722
lemma arcos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcos y \<le> pi"
paulson@15077
  1723
by (blast dest: arcos)
paulson@15077
  1724
paulson@15229
  1725
lemma arcos_lt_bounded:
paulson@15229
  1726
     "[| -1 < y; y < 1 |]  
paulson@15077
  1727
      ==> 0 < arcos y & arcos y < pi"
paulson@15077
  1728
apply (frule order_less_imp_le)
paulson@15077
  1729
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1730
apply (frule arcos_bounded, auto)
paulson@15077
  1731
apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
paulson@15077
  1732
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  1733
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  1734
done
paulson@15077
  1735
paulson@15077
  1736
lemma arcos_cos: "[|0 \<le> x; x \<le> pi |] ==> arcos(cos x) = x"
paulson@15229
  1737
apply (simp add: arcos_def)
paulson@15077
  1738
apply (auto intro!: some1_equality cos_total)
paulson@15077
  1739
done
paulson@15077
  1740
paulson@15077
  1741
lemma arcos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arcos(cos x) = -x"
paulson@15229
  1742
apply (simp add: arcos_def)
paulson@15077
  1743
apply (auto intro!: some1_equality cos_total)
paulson@15077
  1744
done
paulson@15077
  1745
paulson@15077
  1746
lemma arctan [simp]:
paulson@15077
  1747
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
paulson@15077
  1748
apply (cut_tac y = y in tan_total)
paulson@15229
  1749
apply (simp add: arctan_def)
paulson@15077
  1750
apply (fast intro: someI2)
paulson@15077
  1751
done
paulson@15077
  1752
paulson@15077
  1753
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  1754
by auto
paulson@15077
  1755
paulson@15077
  1756
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  1757
by (auto simp only: arctan)
paulson@15077
  1758
paulson@15077
  1759
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  1760
by auto
paulson@15077
  1761
paulson@15077
  1762
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  1763
by (auto simp only: arctan)
paulson@15077
  1764
paulson@15077
  1765
lemma arctan_tan: 
paulson@15077
  1766
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077
  1767
apply (unfold arctan_def)
paulson@15077
  1768
apply (rule some1_equality)
paulson@15077
  1769
apply (rule tan_total, auto)
paulson@15077
  1770
done
paulson@15077
  1771
paulson@15077
  1772
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
paulson@15077
  1773
by (insert arctan_tan [of 0], simp)
paulson@15077
  1774
paulson@15077
  1775
lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
paulson@15077
  1776
apply (auto simp add: cos_zero_iff)
paulson@15077
  1777
apply (case_tac "n")
paulson@15077
  1778
apply (case_tac [3] "n")
paulson@15077
  1779
apply (cut_tac [2] y = x in arctan_ubound)
paulson@15077
  1780
apply (cut_tac [4] y = x in arctan_lbound) 
paulson@15077
  1781
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
paulson@15077
  1782
done
paulson@15077
  1783
paulson@15077
  1784
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077
  1785
apply (rule power_inverse [THEN subst])
paulson@15077
  1786
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
paulson@15077
  1787
apply (auto dest: realpow_not_zero 
huffman@20516
  1788
        simp add: power_mult_distrib left_distrib power_divide tan_def 
paulson@15077
  1789
                  mult_assoc power_inverse [symmetric] 
paulson@15077
  1790
        simp del: realpow_Suc)
paulson@15077
  1791
done
paulson@15077
  1792
paulson@15085
  1793
text{*NEEDED??*}
paulson@15229
  1794
lemma [simp]:
paulson@15229
  1795
     "sin (x + 1 / 2 * real (Suc m) * pi) =  
paulson@15229
  1796
      cos (x + 1 / 2 * real  (m) * pi)"
paulson@15229
  1797
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  1798
paulson@15085
  1799
text{*NEEDED??*}
paulson@15229
  1800
lemma [simp]:
paulson@15229
  1801
     "sin (x + real (Suc m) * pi / 2) =  
paulson@15229
  1802
      cos (x + real (m) * pi / 2)"
paulson@15229
  1803
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
paulson@15077
  1804
paulson@15077
  1805
lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
paulson@15077
  1806
apply (rule lemma_DERIV_subst)
paulson@15077
  1807
apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  1808
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  1809
apply (simp (no_asm))
paulson@15077
  1810
done
paulson@15077
  1811
paulson@15383
  1812
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
paulson@15383
  1813
proof -
paulson@15383
  1814
  have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
paulson@15383
  1815
    by (auto simp add: right_distrib sin_add left_distrib mult_ac)
paulson@15383
  1816
  thus ?thesis
paulson@15383
  1817
    by (simp add: real_of_nat_Suc left_distrib add_divide_distrib 
paulson@15383
  1818
                  mult_commute [of pi])
paulson@15383
  1819
qed
paulson@15077
  1820
paulson@15077
  1821
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
paulson@15077
  1822
by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
paulson@15077
  1823
paulson@15077
  1824
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
paulson@15077
  1825
apply (subgoal_tac "3/2 = (1+1 / 2::real)")
paulson@15077
  1826
apply (simp only: left_distrib) 
paulson@15077
  1827
apply (auto simp add: cos_add mult_ac)
paulson@15077
  1828
done
paulson@15077
  1829
paulson@15077
  1830
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
paulson@15077
  1831
by (auto simp add: mult_assoc)
paulson@15077
  1832
paulson@15077
  1833
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
paulson@15077
  1834
apply (subgoal_tac "3/2 = (1+1 / 2::real)")
paulson@15077
  1835
apply (simp only: left_distrib) 
paulson@15077
  1836
apply (auto simp add: sin_add mult_ac)
paulson@15077
  1837
done
paulson@15077
  1838
paulson@15077
  1839
(*NEEDED??*)
paulson@15229
  1840
lemma [simp]:
paulson@15229
  1841
     "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
paulson@15077
  1842
apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
paulson@15077
  1843
done
paulson@15077
  1844
paulson@15077
  1845
(*NEEDED??*)
paulson@15077
  1846
lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
paulson@15229
  1847
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
paulson@15077
  1848
paulson@15077
  1849
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
paulson@15229
  1850
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
paulson@15077
  1851
paulson@15077
  1852
lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
paulson@15077
  1853
apply (rule lemma_DERIV_subst)
paulson@15077
  1854
apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  1855
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  1856
apply (simp (no_asm))
paulson@15077
  1857
done
paulson@15077
  1858
paulson@15077
  1859
lemma isCont_cos [simp]: "isCont cos x"
paulson@15077
  1860
by (rule DERIV_cos [THEN DERIV_isCont])
paulson@15077
  1861
paulson@15077
  1862
lemma isCont_sin [simp]: "isCont sin x"
paulson@15077
  1863
by (rule DERIV_sin [THEN DERIV_isCont])
paulson@15077
  1864
paulson@15077
  1865
lemma isCont_exp [simp]: "isCont exp x"
paulson@15077
  1866
by (rule DERIV_exp [THEN DERIV_isCont])
paulson@15077
  1867
paulson@15081
  1868
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
nipkow@15539
  1869
by (auto simp add: sin_zero_iff even_mult_two_ex)
paulson@15077
  1870
paulson@15077
  1871
lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
paulson@15077
  1872
apply auto
paulson@15077
  1873
apply (drule_tac f = ln in arg_cong, auto)
paulson@15077
  1874
done
paulson@15077
  1875
paulson@15077
  1876
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
paulson@15077
  1877
by (cut_tac x = x in sin_cos_squared_add3, auto)
paulson@15077
  1878
paulson@15077
  1879
paulson@15229
  1880
lemma real_root_less_mono:
paulson@15229
  1881
     "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
paulson@15077
  1882
apply (frule order_le_less_trans, assumption)
paulson@15077
  1883
apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
paulson@15077
  1884
apply (rotate_tac 1, assumption)
paulson@15077
  1885
apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
paulson@15077
  1886
apply (rotate_tac 3, assumption)
paulson@15077
  1887
apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
paulson@15077
  1888
apply (frule_tac n = n in real_root_pos_pos_le)
paulson@15077
  1889
apply (frule_tac n = n in real_root_pos_pos)
paulson@15077
  1890
apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
paulson@15077
  1891
apply (assumption, assumption)
paulson@15077
  1892
apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
paulson@15077
  1893
apply auto
paulson@15229
  1894
apply (drule_tac f = "%x. x ^ (Suc n)" in arg_cong)
paulson@15077
  1895
apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
paulson@15077
  1896
done
paulson@15077
  1897
paulson@15229
  1898
lemma real_root_le_mono:
paulson@15229
  1899
     "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
paulson@15077
  1900
apply (drule_tac y = y in order_le_imp_less_or_eq)
paulson@15077
  1901
apply (auto dest: real_root_less_mono intro: order_less_imp_le)
paulson@15077
  1902
done
paulson@15077
  1903
paulson@15229
  1904
lemma real_root_less_iff [simp]:
paulson@15229
  1905
     "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
paulson@15077
  1906
apply (auto intro: real_root_less_mono)
paulson@15077
  1907
apply (rule ccontr, drule linorder_not_less [THEN iffD1])
paulson@15077
  1908
apply (drule_tac x = y and n = n in real_root_le_mono, auto)
paulson@15077
  1909
done
paulson@15077
  1910
paulson@15229
  1911
lemma real_root_le_iff [simp]:
paulson@15229
  1912
     "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)"
paulson@15077
  1913
apply (auto intro: real_root_le_mono)
paulson@15077
  1914
apply (simp (no_asm) add: linorder_not_less [symmetric])
paulson@15077
  1915
apply auto
paulson@15077
  1916
apply (drule_tac x = y and n = n in real_root_less_mono, auto)
paulson@15077
  1917
done
paulson@15077
  1918
paulson@15229
  1919
lemma real_root_eq_iff [simp]:
paulson@15229
  1920
     "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
paulson@15077
  1921
apply (auto intro!: order_antisym)
paulson@15077
  1922
apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
paulson@15077
  1923
apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
paulson@15077
  1924
done
paulson@15077
  1925
paulson@15229
  1926
lemma real_root_pos_unique:
paulson@15229
  1927
     "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
paulson@15077
  1928
by (auto dest: real_root_pos2 simp del: realpow_Suc)
paulson@15077
  1929
paulson@15229
  1930
lemma real_root_mult:
paulson@15229
  1931
     "[| 0 \<le> x; 0 \<le> y |] 
paulson@15077
  1932
      ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
paulson@15077
  1933
apply (rule real_root_pos_unique)
paulson@15234
  1934
apply (auto intro!: real_root_pos_pos_le 
paulson@15234
  1935
            simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 
paulson@15234
  1936
            simp del: realpow_Suc)
paulson@15077
  1937
done
paulson@15077
  1938
paulson@15229
  1939
lemma real_root_inverse:
paulson@15229
  1940
     "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
paulson@15077
  1941
apply (rule real_root_pos_unique)
paulson@15234
  1942
apply (auto intro: real_root_pos_pos_le 
paulson@15234
  1943
            simp add: power_inverse [symmetric] real_root_pow_pos2 
paulson@15234
  1944
            simp del: realpow_Suc)
paulson@15077
  1945
done
paulson@15077
  1946
paulson@15077
  1947
lemma real_root_divide: 
paulson@15077
  1948
     "[| 0 \<le> x; 0 \<le> y |]  
paulson@15077
  1949
      ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
paulson@15229
  1950
apply (simp add: divide_inverse)
paulson@15077
  1951
apply (auto simp add: real_root_mult real_root_inverse)
paulson@15077
  1952
done
paulson@15077
  1953
paulson@15077
  1954
lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
paulson@15229
  1955
by (simp add: sqrt_def)
paulson@15077
  1956
paulson@15077
  1957
lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
paulson@15229
  1958
by (simp add: sqrt_def)
paulson@15229
  1959
paulson@15229
  1960
lemma real_sqrt_less_iff [simp]:
paulson@15229
  1961
     "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
paulson@15229
  1962
by (simp add: sqrt_def)
paulson@15229
  1963
paulson@15229
  1964
lemma real_sqrt_le_iff [simp]:
paulson@15229
  1965
     "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
paulson@15229
  1966
by (simp add: sqrt_def)
paulson@15229
  1967
paulson@15229
  1968
lemma real_sqrt_eq_iff [simp]:
paulson@15229
  1969
     "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
paulson@15229
  1970
by (simp add: sqrt_def)
paulson@15077
  1971
paulson@15077
  1972
lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
paulson@15077
  1973
apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077
  1974
apply (rule_tac [2] real_sqrt_less_mono)
paulson@15077
  1975
apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
paulson@15077
  1976
done
paulson@15077
  1977
paulson@15077
  1978
lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
paulson@15077
  1979
apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077
  1980
apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
paulson@15077
  1981
done
paulson@15077
  1982
paulson@15077
  1983
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
paulson@15229
  1984
apply (simp add: divide_inverse)
paulson@15077
  1985
apply (case_tac "r=0")
nipkow@15539
  1986
apply (auto simp add: mult_ac)
paulson@15077
  1987
done
paulson@15077
  1988
paulson@15077
  1989
paulson@15077
  1990
subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}
paulson@15077
  1991
paulson@15228
  1992
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
paulson@15228
  1993
proof (rule order_trans)
paulson@15228
  1994
  show "x \<le> sqrt(x*x)" by (simp add: abs_if) 
paulson@15228
  1995
  show "sqrt (x * x) \<le> sqrt (x * x + y * y)"
paulson@15228
  1996
    by (rule real_sqrt_le_mono, auto) 
paulson@15228
  1997
qed
paulson@15228
  1998
paulson@15228
  1999
lemma minus_le_real_sqrt_sumsq [simp]: "-x \<le> sqrt (x * x + y * y)"
paulson@15228
  2000
proof (rule order_trans)
paulson@15228
  2001
  show "-x \<le> sqrt(x*x)" by (simp add: abs_if) 
paulson@15228
  2002
  show "sqrt (x * x) \<le> sqrt (x * x + y * y)"
paulson@15228
  2003
    by (rule real_sqrt_le_mono, auto) 
paulson@15228
  2004
qed
paulson@15077
  2005
paulson@15077
  2006
lemma lemma_real_divide_sqrt_ge_minus_one:
paulson@15228
  2007
     "0 < x ==> -1 \<le> x/(sqrt (x * x + y * y))" 
paulson@15228
  2008
by (simp add: divide_const_simps linorder_not_le [symmetric])
paulson@15077
  2009
paulson@15077
  2010
lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
paulson@15077
  2011
apply (rule real_sqrt_gt_zero)
paulson@15077
  2012
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077
  2013
apply (auto simp add: zero_less_mult_iff)
paulson@15077
  2014
done
paulson@15077
  2015
paulson@15077
  2016
lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
paulson@15077
  2017
apply (rule real_sqrt_gt_zero)
paulson@15077
  2018
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077
  2019
apply (auto simp add: zero_less_mult_iff)
paulson@15077
  2020
done
paulson@15077
  2021
paulson@15077
  2022
lemma real_sqrt_sum_squares_gt_zero3: "x \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2023
apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077
  2024
apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
paulson@15077
  2025
done
paulson@15077
  2026
paulson@15077
  2027
lemma real_sqrt_sum_squares_gt_zero3a: "y \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2028
apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
paulson@15077
  2029
apply (auto simp add: real_add_commute)
paulson@15077
  2030
done
paulson@15077
  2031
paulson@15544
  2032
lemma real_sqrt_sum_squares_eq_cancel: "sqrt(x\<twosuperior> + y\<twosuperior>) = x ==> y = 0"
paulson@15077
  2033
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, auto)
paulson@15077
  2034
paulson@15544
  2035
lemma real_sqrt_sum_squares_eq_cancel2: "sqrt(x\<twosuperior> + y\<twosuperior>) = y ==> x = 0"
paulson@15077
  2036
apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
paulson@15077
  2037
apply (simp add: real_add_commute)
paulson@15077
  2038
done
paulson@15077
  2039
paulson@15077
  2040
lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077
  2041
by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)
paulson@15077
  2042
paulson@15077
  2043
lemma lemma_real_divide_sqrt_ge_minus_one2:
paulson@15077
  2044
     "x < 0 ==> -1 \<le> x/(sqrt (x * x + y * y))"
paulson@15229
  2045
apply (simp add: divide_const_simps) 
paulson@15229
  2046
apply (insert minus_le_real_sqrt_sumsq [of x y], arith)
paulson@15077
  2047
done
paulson@15077
  2048
paulson@15077
  2049
lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077
  2050
by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)
paulson@15077
  2051
paulson@15228
  2052
lemma minus_sqrt_le: "- sqrt (x * x + y * y) \<le> x"
paulson@15228
  2053
by (insert minus_le_real_sqrt_sumsq [of x y], arith) 
paulson@15228
  2054
paulson@15228
  2055
lemma minus_sqrt_le2: "- sqrt (x * x + y * y) \<le> y"
paulson@15228
  2056
by (subst add_commute, simp add: minus_sqrt_le) 
paulson@15228
  2057
paulson@15228
  2058
lemma not_neg_sqrt_sumsq: "~ sqrt (x * x + y * y) < 0"
paulson@15228
  2059
by (simp add: linorder_not_less)
paulson@15077
  2060
paulson@15077
  2061
lemma cos_x_y_ge_minus_one: "-1 \<le> x / sqrt (x * x + y * y)"
paulson@15229
  2062
by (simp add: minus_sqrt_le not_neg_sqrt_sumsq divide_const_simps)
paulson@15077
  2063
paulson@15077
  2064
lemma cos_x_y_ge_minus_one1a [simp]: "-1 \<le> y / sqrt (x * x + y * y)"
paulson@15229
  2065
by (subst add_commute, simp add: cos_x_y_ge_minus_one)
paulson@15077
  2066
paulson@15228
  2067
lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) \<le> 1" 
paulson@15077
  2068
apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
paulson@15077
  2069
apply (rule lemma_real_divide_sqrt_le_one)
paulson@15077
  2070
apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
paulson@15077
  2071
done
paulson@15077
  2072
paulson@15077
  2073
lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) \<le> 1"
paulson@15077
  2074
apply (cut_tac x = y and y = x in cos_x_y_le_one)
paulson@15077
  2075
apply (simp add: real_add_commute)
paulson@15077
  2076
done
paulson@15077
  2077
paulson@15077
  2078
declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2079
declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2080
paulson@15077
  2081
declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077
  2082
declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077
  2083
paulson@15077
  2084
lemma cos_abs_x_y_ge_minus_one [simp]:
paulson@15077
  2085
     "-1 \<le> \<bar>x\<bar> / sqrt (x * x + y * y)"
paulson@15228
  2086
by (auto simp add: divide_const_simps abs_if linorder_not_le [symmetric]) 
paulson@15077
  2087
paulson@15077
  2088
lemma cos_abs_x_y_le_one [simp]: "\<bar>x\<bar> / sqrt (x * x + y * y) \<le> 1"
paulson@15228
  2089
apply (insert minus_le_real_sqrt_sumsq [of x y] le_real_sqrt_sumsq [of x y]) 
paulson@15228
  2090
apply (auto simp add: divide_const_simps abs_if linorder_neq_iff) 
paulson@15077
  2091
done
paulson@15077
  2092
paulson@15077
  2093
declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077
  2094
declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077
  2095
paulson@15077
  2096
lemma minus_pi_less_zero: "-pi < 0"
paulson@15228
  2097
by simp
paulson@15228
  2098
paulson@15077
  2099
declare minus_pi_less_zero [simp]
paulson@15077
  2100
declare minus_pi_less_zero [THEN order_less_imp_le, simp]
paulson@15077
  2101
paulson@15077
  2102
lemma arcos_ge_minus_pi: "[| -1 \<le> y; y \<le> 1 |] ==> -pi \<le> arcos y"
paulson@15077
  2103
apply (rule real_le_trans)
paulson@15077
  2104
apply (rule_tac [2] arcos_lbound, auto)
paulson@15077
  2105
done
paulson@15077
  2106
paulson@15077
  2107
declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2108
paulson@15077
  2109
(* How tedious! *)
paulson@15077
  2110
lemma lemma_divide_rearrange:
paulson@15077
  2111
     "[| x + (y::real) \<noteq> 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
paulson@15077
  2112
apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
paulson@15077
  2113
apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
paulson@15077
  2114
prefer 2 apply assumption
paulson@15077
  2115
apply (rotate_tac [2] 2)
paulson@15077
  2116
apply (drule_tac [2] mult_assoc [THEN subst])
paulson@15077
  2117
apply (rotate_tac [2] 2)
paulson@15077
  2118
apply (frule_tac [2] left_inverse [THEN subst])
paulson@15077
  2119
prefer 2 apply assumption
paulson@15229
  2120
apply (erule_tac [2] V = "(1 - z) * (x + y) = x / (x + y) * (x + y)" in thin_rl)
paulson@15229
  2121
apply (erule_tac [2] V = "1 - z = x / (x + y)" in thin_rl)
paulson@15077
  2122
apply (auto simp add: mult_assoc)
paulson@15077
  2123
apply (auto simp add: right_distrib left_diff_distrib)
paulson@15077
  2124
done
paulson@15077
  2125
paulson@15077
  2126
lemma lemma_cos_sin_eq:
paulson@15077
  2127
     "[| 0 < x * x + y * y;  
paulson@15077
  2128
         1 - (sin xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2 |] 
paulson@15077
  2129
      ==> (sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2"
paulson@15077
  2130
by (auto intro: lemma_divide_rearrange
huffman@20516
  2131
         simp add: power_divide power2_eq_square [symmetric])
paulson@15077
  2132
paulson@15077
  2133
paulson@15077
  2134
lemma lemma_sin_cos_eq:
paulson@15077
  2135
     "[| 0 < x * x + y * y;  
paulson@15077
  2136
         1 - (cos xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2 |]
paulson@15077
  2137
      ==> (cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2"
huffman@20516
  2138
apply (auto simp add: power_divide power2_eq_square [symmetric])
paulson@15085
  2139
apply (subst add_commute)
paulson@15085
  2140
apply (rule lemma_divide_rearrange, simp add: real_add_eq_0_iff)
paulson@15077
  2141
apply (simp add: add_commute)
paulson@15077
  2142
done
paulson@15077
  2143
paulson@15077
  2144
lemma sin_x_y_disj:
paulson@15077
  2145
     "[| x \<noteq> 0;  
paulson@15077
  2146
         cos xa = x / sqrt (x * x + y * y)  
paulson@15077
  2147
      |] ==>  sin xa = y / sqrt (x * x + y * y) |  
paulson@15077
  2148
              sin xa = - y / sqrt (x * x + y * y)"
paulson@15077
  2149
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077
  2150
apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077
  2151
apply (simp add: cos_squared_eq)
paulson@15077
  2152
apply (subgoal_tac "(sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2")
paulson@15077
  2153
apply (rule_tac [2] lemma_cos_sin_eq)
paulson@15077
  2154
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077
  2155
done
paulson@15077
  2156
paulson@15077
  2157
lemma lemma_cos_not_eq_zero: "x \<noteq> 0 ==> x / sqrt (x * x + y * y) \<noteq> 0"
paulson@15229
  2158
apply (simp add: divide_inverse)
paulson@15077
  2159
apply (frule_tac y3 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym, THEN nonzero_imp_inverse_nonzero])
paulson@15077
  2160
apply (auto simp add: power2_eq_square)
paulson@15077
  2161
done
paulson@15077
  2162
paulson@15229
  2163
lemma cos_x_y_disj:
paulson@15229
  2164
     "[| x \<noteq> 0;  
paulson@15077
  2165
         sin xa = y / sqrt (x * x + y * y)  
paulson@15077
  2166
      |] ==>  cos xa = x / sqrt (x * x + y * y) |  
paulson@15077
  2167
              cos xa = - x / sqrt (x * x + y * y)"
paulson@15077
  2168
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077
  2169
apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077
  2170
apply (simp add: sin_squared_eq del: realpow_Suc)
paulson@15077
  2171
apply (subgoal_tac "(cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2")
paulson@15077
  2172
apply (rule_tac [2] lemma_sin_cos_eq)
paulson@15077
  2173
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077
  2174
done
paulson@15077
  2175
paulson@15077
  2176
lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
paulson@15229
  2177
apply (case_tac "x = 0", auto)
paulson@15077
  2178
apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
paulson@15079
  2179
apply (auto simp add: zero_less_mult_iff divide_inverse power2_eq_square)
paulson@15077
  2180
done
paulson@15077
  2181
paulson@15229
  2182
lemma polar_ex1:
paulson@15229
  2183
     "[| x \<noteq> 0; 0 < y |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15229
  2184
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
paulson@15077
  2185
apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
paulson@15077
  2186
apply auto
paulson@15077
  2187
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2188
apply (auto simp add: power2_eq_square)
paulson@15229
  2189
apply (simp add: arcos_def)
paulson@15077
  2190
apply (cut_tac x1 = x and y1 = y 
paulson@15077
  2191
       in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
paulson@15077
  2192
apply (rule someI2_ex, blast)
paulson@15229
  2193
apply (erule_tac V = "EX! xa. 0 \<le> xa & xa \<le> pi & cos xa = x / sqrt (x * x + y * y)" in thin_rl)
paulson@15077
  2194
apply (frule sin_x_y_disj, blast)
paulson@15077
  2195
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2196
apply (auto simp add: power2_eq_square)
paulson@15077
  2197
apply (drule sin_ge_zero, assumption)
paulson@15077
  2198
apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
paulson@15077
  2199
done
paulson@15077
  2200
paulson@15077
  2201
lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
paulson@15085
  2202
by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)
paulson@15077
  2203
paulson@15229
  2204
lemma polar_ex2:
paulson@15229
  2205
     "[| x \<noteq> 0; y < 0 |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2206
apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
paulson@15228
  2207
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
paulson@15228
  2208
apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI) 
paulson@15085
  2209
apply (auto dest: real_sum_squares_cancel2a 
paulson@15085
  2210
            simp add: power2_eq_square real_0_le_add_iff real_add_eq_0_iff)
paulson@15077
  2211
apply (unfold arcsin_def)
paulson@15077
  2212
apply (cut_tac x1 = x and y1 = y 
paulson@15077
  2213
       in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
paulson@15077
  2214
apply (rule someI2_ex, blast)
paulson@15228
  2215
apply (erule_tac V = "EX! v. ?P v" in thin_rl)
paulson@15085
  2216
apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast)
paulson@15085
  2217
apply (auto simp add: real_0_le_add_iff real_add_eq_0_iff)
paulson@15077
  2218
apply (drule cos_ge_zero, force)
paulson@15077
  2219
apply (drule_tac x = y in real_sqrt_divide_less_zero)
paulson@15085
  2220
apply (auto simp add: add_commute)
paulson@15077
  2221
apply (insert polar_ex1 [of x "-y"], simp, clarify) 
paulson@15077
  2222
apply (rule_tac x = r in exI)
paulson@15077
  2223
apply (rule_tac x = "-a" in exI, simp) 
paulson@15077
  2224
done
paulson@15077
  2225
paulson@15077
  2226
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2227
apply (case_tac "x = 0", auto)
paulson@15077
  2228
apply (rule_tac x = y in exI)
paulson@15077
  2229
apply (rule_tac x = "pi/2" in exI, auto)
paulson@15077
  2230
apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
paulson@15077
  2231
apply (rule_tac [2] x = x in exI)
paulson@15077
  2232
apply (rule_tac [2] x = 0 in exI, auto)
paulson@15077
  2233
apply (blast intro: polar_ex1 polar_ex2)+
paulson@15077
  2234
done
paulson@15077
  2235
paulson@15077
  2236
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2237
apply (rule_tac n = 1 in realpow_increasing)
avigad@16775
  2238
apply (auto simp add: numeral_2_eq_2 [symmetric] power2_abs)
paulson@15077
  2239
done
paulson@15077
  2240
paulson@15077
  2241
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2242
apply (rule real_add_commute [THEN subst])
paulson@15077
  2243
apply (rule real_sqrt_ge_abs1)
paulson@15077
  2244
done
paulson@15077
  2245
declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]
paulson@15077
  2246
paulson@15077
  2247
lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
paulson@15077
  2248
by (auto intro: real_sqrt_gt_zero)
paulson@15077
  2249
paulson@15077
  2250
lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
paulson@15077
  2251
by (auto intro: real_sqrt_ge_zero)
paulson@15077
  2252
paulson@15077
  2253
lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
paulson@15077
  2254
apply (rule order_less_le_trans [of _ "7/5"], simp) 
paulson@15077
  2255
apply (rule_tac n = 1 in realpow_increasing)
paulson@15077
  2256
  prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
nipkow@15539
  2257
apply (simp_all add: numeral_2_eq_2)
paulson@15077
  2258
done
paulson@15077
  2259
paulson@15077
  2260
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
paulson@15234
  2261
by (simp add: divide_less_eq mult_compare_simps) 
paulson@15077
  2262
paulson@15077
  2263
lemma four_x_squared: 
paulson@15077
  2264
  fixes x::real
paulson@15077
  2265
  shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>"
paulson@15077
  2266
by (simp add: power2_eq_square)
paulson@15077
  2267
paulson@15077
  2268
paulson@15077
  2269
text{*Needed for the infinitely close relation over the nonstandard
paulson@15077
  2270
    complex numbers*}
paulson@15077
  2271
lemma lemma_sqrt_hcomplex_capprox:
paulson@15077
  2272
     "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
paulson@15077
  2273
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
paulson@15077
  2274
apply (erule_tac [2] lemma_real_divide_sqrt_less)
paulson@15077
  2275
apply (rule_tac n = 1 in realpow_increasing)
huffman@20516
  2276
apply (auto simp add: real_0_le_divide_iff power_divide numeral_2_eq_2 [symmetric] 
paulson@15077
  2277
        simp del: realpow_Suc)
paulson@15077
  2278
apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst])
paulson@15077
  2279
apply (rule add_mono)
paulson@15077
  2280
apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono)
paulson@15077
  2281
done
paulson@15077
  2282
avigad@16775
  2283
declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]
paulson@15077
  2284
paulson@15077
  2285
paulson@15077
  2286
subsection{*A Few Theorems Involving Ln, Derivatives, etc.*}
paulson@15077
  2287
paulson@15077
  2288
lemma lemma_DERIV_ln:
paulson@15077
  2289
     "DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l"
paulson@15077
  2290
by (erule DERIV_fun_exp)
paulson@15077
  2291
paulson@15077
  2292
lemma STAR_exp_ln: "0 < z ==> ( *f* (%x. exp (ln x))) z = z"
huffman@17318
  2293
apply (cases z)
huffman@17318
  2294
apply (auto simp add: starfun star_n_zero_num star_n_less star_n_eq_iff)
paulson@15077
  2295
done
paulson@15077
  2296
paulson@15229
  2297
lemma hypreal_add_Infinitesimal_gt_zero:
paulson@15229
  2298
     "[|e : Infinitesimal; 0 < x |] ==> 0 < hypreal_of_real x + e"
paulson@15077
  2299
apply (rule_tac c1 = "-e" in add_less_cancel_right [THEN iffD1])
paulson@15077
  2300
apply (auto intro: Infinitesimal_less_SReal)
paulson@15077
  2301
done
paulson@15077
  2302
paulson@15077
  2303
lemma NSDERIV_exp_ln_one: "0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1"
paulson@15229
  2304
apply (simp add: nsderiv_def NSLIM_def, auto)
paulson@15077
  2305
apply (rule ccontr)
paulson@15077
  2306
apply (subgoal_tac "0 < hypreal_of_real z + h")
paulson@15077
  2307
apply (drule STAR_exp_ln)
paulson@15077
  2308
apply (rule_tac [2] hypreal_add_Infinitesimal_gt_zero)
paulson@15077
  2309
apply (subgoal_tac "h/h = 1")
paulson@15077
  2310
apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
paulson@15077
  2311
done
paulson@15077
  2312
paulson@15077
  2313
lemma DERIV_exp_ln_one: "0 < z ==> DERIV (%x. exp (ln x)) z :> 1"
paulson@15077
  2314
by (auto intro: NSDERIV_exp_ln_one simp add: NSDERIV_DERIV_iff [symmetric])
paulson@15077
  2315
paulson@15229
  2316
lemma lemma_DERIV_ln2:
paulson@15229
  2317
     "[| 0 < z; DERIV ln z :> l |] ==>  exp (ln z) * l = 1"
paulson@15077
  2318
apply (rule DERIV_unique)
paulson@15077
  2319
apply (rule lemma_DERIV_ln)
paulson@15077
  2320
apply (rule_tac [2] DERIV_exp_ln_one, auto)
paulson@15077
  2321
done
paulson@15077
  2322
paulson@15229
  2323
lemma lemma_DERIV_ln3:
paulson@15229
  2324
     "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/(exp (ln z))"
paulson@15229
  2325
apply (rule_tac c1 = "exp (ln z)" in real_mult_left_cancel [THEN iffD1])
paulson@15077
  2326
apply (auto intro: lemma_DERIV_ln2)
paulson@15077
  2327
done
paulson@15077
  2328
paulson@15077
  2329
lemma lemma_DERIV_ln4: "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/z"
paulson@15077
  2330
apply (rule_tac t = z in exp_ln_iff [THEN iffD2, THEN subst])
paulson@15077
  2331
apply (auto intro: lemma_DERIV_ln3)
paulson@15077
  2332
done
paulson@15077
  2333
paulson@15077
  2334
(* need to rename second isCont_inverse *)
paulson@15077
  2335
paulson@15229
  2336
lemma isCont_inv_fun:
huffman@20561
  2337
  fixes f g :: "real \<Rightarrow> real"
huffman@20561
  2338
  shows "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077
  2339
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077
  2340
      ==> isCont g (f x)"
paulson@15077
  2341
apply (simp (no_asm) add: isCont_iff LIM_def)
paulson@15077
  2342
apply safe
paulson@15077
  2343
apply (drule_tac ?d1.0 = r in real_lbound_gt_zero)
paulson@15077
  2344
apply (assumption, safe)
paulson@15077
  2345
apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> (g (f z) = z) ")
paulson@15077
  2346
prefer 2 apply force
paulson@15077
  2347
apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> isCont f z")
paulson@15077
  2348
prefer 2 apply force
paulson@15077
  2349
apply (drule_tac d = e in isCont_inj_range)
paulson@15077
  2350
prefer 2 apply (assumption, assumption, safe)
paulson@15077
  2351
apply (rule_tac x = ea in exI, auto)
paulson@15085
  2352
apply (drule_tac x = "f (x) + xa" and P = "%y. \<bar>y - f x\<bar> \<le> ea \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)" in spec)
paulson@15077
  2353
apply auto
webertj@20217
  2354
apply (drule sym, auto)
paulson@15077
  2355
done
paulson@15077
  2356
paulson@15077
  2357
lemma isCont_inv_fun_inv:
huffman@20552
  2358
  fixes f g :: "real \<Rightarrow> real"
huffman@20552
  2359
  shows "[| 0 < d;  
paulson@15077
  2360
         \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077
  2361
         \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077
  2362
       ==> \<exists>e. 0 < e &  
paulson@15081
  2363
             (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
paulson@15077
  2364
apply (drule isCont_inj_range)
paulson@15077
  2365
prefer 2 apply (assumption, assumption, auto)
paulson@15077
  2366
apply (rule_tac x = e in exI, auto)
paulson@15077
  2367
apply (rotate_tac 2)
paulson@15077
  2368
apply (drule_tac x = y in spec, auto)
paulson@15077
  2369
done
paulson@15077
  2370
paulson@15077
  2371
paulson@15077
  2372
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
paulson@15229
  2373
lemma LIM_fun_gt_zero:
huffman@20552
  2374
     "[| f -- c --> (l::real); 0 < l |]  
huffman@20561
  2375
         ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
paulson@15077
  2376
apply (auto simp add: LIM_def)
paulson@15077
  2377
apply (drule_tac x = "l/2" in spec, safe, force)
paulson@15077
  2378
apply (rule_tac x = s in exI)
paulson@15077
  2379
apply (auto simp only: abs_interval_iff)
paulson@15077
  2380
done
paulson@15077
  2381
paulson@15229
  2382
lemma LIM_fun_less_zero:
huffman@20552
  2383
     "[| f -- c --> (l::real); l < 0 |]  
huffman@20561
  2384
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
paulson@15077
  2385
apply (auto simp add: LIM_def)
paulson@15077
  2386
apply (drule_tac x = "-l/2" in spec, safe, force)
paulson@15077
  2387
apply (rule_tac x = s in exI)
paulson@15077
  2388
apply (auto simp only: abs_interval_iff)
paulson@15077
  2389
done
paulson@15077
  2390
paulson@15077
  2391
paulson@15077
  2392
lemma LIM_fun_not_zero:
huffman@20552
  2393
     "[| f -- c --> (l::real); l \<noteq> 0 |] 
huffman@20561
  2394
      ==> \<exists>r. 0 < r & (\<forall>x::real. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
paulson@15077
  2395
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
paulson@15077
  2396
apply (drule LIM_fun_less_zero)
paulson@15241
  2397
apply (drule_tac [3] LIM_fun_gt_zero)
paulson@15241
  2398
apply force+
paulson@15077
  2399
done
webertj@20432
  2400
  
paulson@12196
  2401
end