paulson@12196: (*  Title       : Transcendental.thy
paulson@12196:     Author      : Jacques D. Fleuriot
paulson@12196:     Copyright   : 1998,1999 University of Cambridge
paulson@13958:                   1999,2001 University of Edinburgh
paulson@15077:     Conversion to Isar and new proofs by Lawrence C Paulson, 2004
paulson@12196: *)
paulson@12196: 
paulson@15077: header{*Power Series, Transcendental Functions etc.*}
paulson@15077: 
nipkow@15131: theory Transcendental
nipkow@15140: imports NthRoot Fact HSeries EvenOdd Lim
nipkow@15131: begin
paulson@15077: 
wenzelm@19765: definition
wenzelm@19765:   root :: "[nat,real] => real"
wenzelm@19765:   "root n x = (SOME u. ((0::real) < x --> 0 < u) & (u ^ n = x))"
wenzelm@19765: 
wenzelm@19765:   sqrt :: "real => real"
wenzelm@19765:   "sqrt x = root 2 x"
wenzelm@19765: 
wenzelm@19765:   exp :: "real => real"
wenzelm@19765:   "exp x = (\<Sum>n. inverse(real (fact n)) * (x ^ n))"
wenzelm@19765: 
wenzelm@19765:   sin :: "real => real"
wenzelm@19765:   "sin x = (\<Sum>n. (if even(n) then 0 else
wenzelm@19765:              ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
paulson@12196:  
wenzelm@19765:   diffs :: "(nat => real) => nat => real"
wenzelm@19765:   "diffs c = (%n. real (Suc n) * c(Suc n))"
wenzelm@19765: 
wenzelm@19765:   cos :: "real => real"
wenzelm@19765:   "cos x = (\<Sum>n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n)) 
wenzelm@19765:                             else 0) * x ^ n)"
paulson@12196:   
wenzelm@19765:   ln :: "real => real"
wenzelm@19765:   "ln x = (SOME u. exp u = x)"
wenzelm@19765: 
wenzelm@19765:   pi :: "real"
wenzelm@19765:   "pi = 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
wenzelm@19765: 
wenzelm@19765:   tan :: "real => real"
wenzelm@19765:   "tan x = (sin x)/(cos x)"
wenzelm@19765: 
wenzelm@19765:   arcsin :: "real => real"
wenzelm@19765:   "arcsin y = (SOME x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
wenzelm@19765: 
wenzelm@19765:   arcos :: "real => real"
wenzelm@19765:   "arcos y = (SOME x. 0 \<le> x & x \<le> pi & cos x = y)"
paulson@12196:      
wenzelm@19765:   arctan :: "real => real"
wenzelm@19765:   "arctan y = (SOME x. -(pi/2) < x & x < pi/2 & tan x = y)"
paulson@15077: 
paulson@15077: 
paulson@15077: lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
paulson@15229: apply (simp add: root_def)
paulson@15077: apply (safe intro!: some_equality power_0_Suc elim!: realpow_zero_zero)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_root_pow_pos: 
paulson@15077:      "0 < x ==> (root(Suc n) x) ^ (Suc n) = x"
paulson@15229: apply (simp add: root_def)
paulson@15077: apply (drule_tac n = n in realpow_pos_nth2)
paulson@15077: apply (auto intro: someI2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_root_pow_pos2: "0 \<le> x ==> (root(Suc n) x) ^ (Suc n) = x"
paulson@15077: by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)
paulson@15077: 
paulson@15077: lemma real_root_pos: 
paulson@15077:      "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
paulson@15229: apply (simp add: root_def)
paulson@15077: apply (rule some_equality)
paulson@15077: apply (frule_tac [2] n = n in zero_less_power)
paulson@15077: apply (auto simp add: zero_less_mult_iff)
paulson@15077: apply (rule_tac x = u and y = x in linorder_cases)
paulson@15077: apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
paulson@15077: apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
nipkow@15539: apply (auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
paulson@15077: by (auto dest!: real_le_imp_less_or_eq real_root_pos)
paulson@15077: 
paulson@15077: lemma real_root_pos_pos: 
paulson@15077:      "0 < x ==> 0 \<le> root(Suc n) x"
paulson@15229: apply (simp add: root_def)
paulson@15077: apply (drule_tac n = n in realpow_pos_nth2)
paulson@15077: apply (safe, rule someI2)
paulson@15229: apply (auto intro!: order_less_imp_le dest: zero_less_power 
paulson@15229:             simp add: zero_less_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
paulson@15077: by (auto dest!: real_le_imp_less_or_eq dest: real_root_pos_pos)
paulson@15077: 
paulson@15077: lemma real_root_one [simp]: "root (Suc n) 1 = 1"
paulson@15229: apply (simp add: root_def)
paulson@15077: apply (rule some_equality, auto)
paulson@15077: apply (rule ccontr)
paulson@15077: apply (rule_tac x = u and y = 1 in linorder_cases)
paulson@15077: apply (drule_tac n = n in realpow_Suc_less_one)
paulson@15077: apply (drule_tac [4] n = n in power_gt1_lemma)
nipkow@15539: apply (auto)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Square Root*}
paulson@15077: 
paulson@15229: text{*needed because 2 is a binary numeral!*}
paulson@15077: lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
paulson@15229: by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
paulson@15229:          add: nat_numeral_0_eq_0 [symmetric])
paulson@15077: 
paulson@15077: lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
paulson@15229: by (simp add: sqrt_def)
paulson@15077: 
paulson@15077: lemma real_sqrt_one [simp]: "sqrt 1 = 1"
paulson@15229: by (simp add: sqrt_def)
paulson@15077: 
nipkow@15539: lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
paulson@15229: apply (simp add: sqrt_def)
paulson@15077: apply (rule iffI) 
paulson@15077:  apply (cut_tac r = "root 2 x" in realpow_two_le)
paulson@15077:  apply (simp add: numeral_2_eq_2)
paulson@15077: apply (subst numeral_2_eq_2)
paulson@15077: apply (erule real_root_pow_pos2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
paulson@15077: by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])
paulson@15077: 
paulson@15077: lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
nipkow@15539: by (simp)
paulson@15077: 
paulson@15077: lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
paulson@15077: by (rule real_sqrt_pow2_iff [THEN iffD2], arith)
paulson@15077: 
paulson@15077: lemma real_pow_sqrt_eq_sqrt_pow: 
paulson@15077:       "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
paulson@15229: apply (simp add: sqrt_def)
paulson@15481: apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_pow_sqrt_eq_sqrt_abs_pow2:
paulson@15077:      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
paulson@15077: by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])
paulson@15077: 
paulson@15077: lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
paulson@15077: apply (rule real_sqrt_abs_abs [THEN subst])
paulson@15077: apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
paulson@15077: apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
paulson@15077: apply (assumption, arith)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
paulson@15077: apply auto
paulson@15077: apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077: apply (simp add: zero_less_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
paulson@15229: apply (simp add: sqrt_def root_def)
paulson@15077: apply (drule realpow_pos_nth2 [where n=1], safe)
paulson@15077: apply (rule someI2, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
paulson@15077: by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
paulson@15077: 
paulson@15228: lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
paulson@15228: by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 
paulson@15228: 
paulson@15077: 
paulson@15077: (*we need to prove something like this:
paulson@15077: lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
paulson@15077: apply (case_tac n, simp) 
paulson@15229: apply (simp add: root_def) 
paulson@15077: apply (rule someI2 [of _ r], safe)
paulson@15077: apply (auto simp del: realpow_Suc dest: power_inject_base)
paulson@15077: *)
paulson@15077: 
paulson@15077: lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
paulson@15077: apply (unfold sqrt_def root_def) 
paulson@15077: apply (rule someI2 [of _ r], auto) 
paulson@15077: apply (auto simp add: numeral_2_eq_2 simp del: realpow_Suc 
paulson@15077:             dest: power_inject_base) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_mult_distrib: 
paulson@15077:      "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
paulson@15077: apply (rule sqrt_eqI)
paulson@15077: apply (simp add: power_mult_distrib)  
paulson@15077: apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_sqrt_mult_distrib2:
paulson@15229:      "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
paulson@15077: by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077: by (auto intro!: real_sqrt_ge_zero)
paulson@15077: 
paulson@15229: lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
paulson@15229:      "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
paulson@15077: by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
paulson@15077:      "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
nipkow@15539: by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)
paulson@15077: 
paulson@15077: lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
paulson@15077: apply (rule abs_realpow_two [THEN subst])
paulson@15077: apply (rule real_sqrt_abs_abs [THEN subst])
paulson@15077: apply (subst real_pow_sqrt_eq_sqrt_pow)
nipkow@15539: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
paulson@15077: apply (rule realpow_two [THEN subst])
paulson@15077: apply (subst numeral_2_eq_2 [symmetric])
paulson@15077: apply (rule real_sqrt_abs)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
paulson@15077: by simp
paulson@15077: 
paulson@15077: lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
paulson@15077: apply (frule real_sqrt_pow2_gt_zero)
nipkow@15539: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
paulson@15077: by (cut_tac n1 = 2 and a1 = "sqrt x" in power_inverse [symmetric], auto)
paulson@15077: 
paulson@15077: lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
paulson@15077: apply (drule real_le_imp_less_or_eq)
paulson@15077: apply (auto dest: real_sqrt_not_eq_zero)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
paulson@15077: by (auto simp add: real_sqrt_eq_zero_cancel)
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077: apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
paulson@15077: apply (rule real_le_trans)
paulson@15077: apply (auto simp del: realpow_Suc)
paulson@15077: apply (rule_tac n = 1 in realpow_increasing)
paulson@15077: apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
paulson@15077: apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
paulson@15077: apply (rule_tac n = 1 in realpow_increasing)
paulson@15077: apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
paulson@15077:             del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Exponential Function*}
paulson@15077: 
paulson@15077: lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
paulson@15077: apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
paulson@15077: apply (cut_tac x = r in reals_Archimedean3, auto)
paulson@15077: apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
paulson@15077: apply (rule_tac N = n and c = r in ratio_test)
paulson@16924: apply (auto simp add: abs_mult mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077: apply (rule mult_right_mono)
paulson@15077: apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
paulson@15077: apply (subst fact_Suc)
paulson@15077: apply (subst real_of_nat_mult)
nipkow@15539: apply (auto)
paulson@15229: apply (auto simp add: mult_assoc [symmetric] positive_imp_inverse_positive)
paulson@15077: apply (rule order_less_imp_le)
paulson@15229: apply (rule_tac z1 = "real (Suc na)" in real_mult_less_iff1 [THEN iffD1])
nipkow@15539: apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc)
paulson@15077: apply (erule order_less_trans)
paulson@15077: apply (auto simp add: mult_less_cancel_left mult_ac)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma summable_sin: 
paulson@15077:      "summable (%n.  
paulson@15077:            (if even n then 0  
paulson@15077:            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
paulson@15077:                 x ^ n)"
paulson@15229: apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
paulson@15077: apply (rule_tac [2] summable_exp)
paulson@15077: apply (rule_tac x = 0 in exI)
paulson@16924: apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma summable_cos: 
paulson@15077:       "summable (%n.  
paulson@15077:            (if even n then  
paulson@15077:            (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
paulson@15229: apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
paulson@15077: apply (rule_tac [2] summable_exp)
paulson@15077: apply (rule_tac x = 0 in exI)
paulson@16924: apply (auto simp add: divide_inverse abs_mult power_abs [symmetric] zero_le_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_STAR_sin [simp]:
paulson@15229:      "(if even n then 0  
paulson@15077:        else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
paulson@15251: by (induct "n", auto)
paulson@15229: 
paulson@15229: lemma lemma_STAR_cos [simp]:
paulson@15229:      "0 < n -->  
paulson@15077:       (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
paulson@15251: by (induct "n", auto)
paulson@15229: 
paulson@15229: lemma lemma_STAR_cos1 [simp]:
paulson@15229:      "0 < n -->  
paulson@15077:       (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
paulson@15251: by (induct "n", auto)
paulson@15229: 
paulson@15229: lemma lemma_STAR_cos2 [simp]:
nipkow@15539:   "(\<Sum>n=1..<n. if even n then (- 1) ^ (n div 2)/(real (fact n)) *  0 ^ n 
nipkow@15539:                          else 0) = 0"
paulson@15251: apply (induct "n")
paulson@15077: apply (case_tac [2] "n", auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
paulson@15229: apply (simp add: exp_def)
paulson@15077: apply (rule summable_exp [THEN summable_sums])
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_converges: 
paulson@15077:       "(%n. (if even n then 0  
paulson@15077:             else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
paulson@15077:                  x ^ n) sums sin(x)"
paulson@15229: apply (simp add: sin_def)
paulson@15077: apply (rule summable_sin [THEN summable_sums])
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_converges: 
paulson@15077:       "(%n. (if even n then  
paulson@15077:            (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077:            else 0) * x ^ n) sums cos(x)"
paulson@15229: apply (simp add: cos_def)
paulson@15077: apply (rule summable_cos [THEN summable_sums])
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_realpow_diff [rule_format (no_asm)]:
paulson@15229:      "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
paulson@15251: apply (induct "n", auto)
paulson@15077: apply (subgoal_tac "p = Suc n")
paulson@15077: apply (simp (no_asm_simp), auto)
paulson@15077: apply (drule sym)
paulson@15077: apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
paulson@15077:        del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Properties of Power Series*}
paulson@15077: 
paulson@15077: lemma lemma_realpow_diff_sumr:
nipkow@15539:      "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =  
nipkow@15539:       y * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
ballarin@19279: by (auto simp add: setsum_right_distrib lemma_realpow_diff mult_ac
berghofe@16641:   simp del: setsum_op_ivl_Suc cong: strong_setsum_cong)
paulson@15077: 
paulson@15229: lemma lemma_realpow_diff_sumr2:
paulson@15229:      "x ^ (Suc n) - y ^ (Suc n) =  
nipkow@15539:       (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
paulson@15251: apply (induct "n", simp)
nipkow@15561: apply (auto simp del: setsum_op_ivl_Suc)
nipkow@15561: apply (subst setsum_op_ivl_Suc)
paulson@15077: apply (drule sym)
nipkow@15561: apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_op_ivl_Suc)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_realpow_rev_sumr:
nipkow@15539:      "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
nipkow@15539:       (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p)::real)"
paulson@15077: apply (case_tac "x = y")
nipkow@15561: apply (auto simp add: mult_commute power_add [symmetric] simp del: setsum_op_ivl_Suc)
paulson@15077: apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
paulson@15077: apply (rule_tac [2] minus_minus [THEN subst], simp)
paulson@15077: apply (subst minus_mult_left)
nipkow@15561: apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: setsum_op_ivl_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
paulson@15077: x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
paulson@15077: 
paulson@15077: lemma powser_insidea:
paulson@15077:      "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
paulson@15081:       ==> summable (%n. \<bar>f(n)\<bar> * (z ^ n))"
paulson@15077: apply (drule summable_LIMSEQ_zero)
paulson@15077: apply (drule convergentI)
paulson@15077: apply (simp add: Cauchy_convergent_iff [symmetric])
paulson@15077: apply (drule Cauchy_Bseq)
paulson@15077: apply (simp add: Bseq_def, safe)
paulson@15081: apply (rule_tac g = "%n. K * \<bar>z ^ n\<bar> * inverse (\<bar>x ^ n\<bar>)" in summable_comparison_test)
paulson@15077: apply (rule_tac x = 0 in exI, safe)
paulson@15081: apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
paulson@15081: apply (rule_tac c="\<bar>x ^ n\<bar>" in mult_right_le_imp_le) 
paulson@16924: apply (auto simp add: mult_assoc power_abs abs_mult)
webertj@20432:  prefer 2
webertj@20432:  apply (drule_tac x = 0 in spec, force)
nipkow@15539: apply (auto simp add: power_abs mult_ac)
paulson@15077: apply (rule_tac a2 = "z ^ n" 
paulson@15077:        in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
paulson@15077: apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
paulson@15229: apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>)))" in exI)
paulson@15077: apply (auto intro!: sums_mult simp add: mult_assoc)
paulson@15081: apply (subgoal_tac "\<bar>z ^ n\<bar> * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
paulson@15077: apply (auto simp add: power_abs [symmetric])
paulson@15077: apply (subgoal_tac "x \<noteq> 0")
paulson@15077: apply (subgoal_tac [3] "x \<noteq> 0")
paulson@16924: apply (auto simp del: abs_inverse 
paulson@16924:             simp add: abs_inverse [symmetric] realpow_not_zero
paulson@16924:             abs_mult [symmetric] power_inverse power_mult_distrib [symmetric])
nipkow@15539: apply (auto intro!: geometric_sums  simp add: power_abs inverse_eq_divide)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma powser_inside:
paulson@15229:      "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
paulson@15077:       ==> summable (%n. f(n) * (z ^ n))"
paulson@15077: apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
paulson@16924: apply (auto intro: summable_rabs_cancel simp add: abs_mult power_abs [symmetric])
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Differentiation of Power Series*}
paulson@15077: 
paulson@15077: text{*Lemma about distributing negation over it*}
paulson@15077: lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
paulson@15077: by (simp add: diffs_def)
paulson@15077: 
paulson@15077: text{*Show that we can shift the terms down one*}
paulson@15077: lemma lemma_diffs:
nipkow@15539:      "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  
nipkow@15539:       (\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) +  
paulson@15077:       (real n * c(n) * x ^ (n - Suc 0))"
paulson@15251: apply (induct "n")
paulson@15077: apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_diffs2:
nipkow@15539:      "(\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) =  
nipkow@15539:       (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
paulson@15077:       (real n * c(n) * x ^ (n - Suc 0))"
paulson@15077: by (auto simp add: lemma_diffs)
paulson@15077: 
paulson@15077: 
paulson@15229: lemma diffs_equiv:
paulson@15229:      "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
paulson@15077:       (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
nipkow@15546:          (\<Sum>n. (diffs c)(n) * (x ^ n))"
paulson@15077: apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
paulson@15077: apply (rule_tac [2] LIMSEQ_imp_Suc)
paulson@15077: apply (drule summable_sums) 
paulson@15077: apply (auto simp add: sums_def)
paulson@15077: apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
paulson@15077: apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
paulson@15077: apply (simp add: diffs_def summable_LIMSEQ_zero)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Term-by-Term Differentiability of Power Series*}
paulson@15077: 
paulson@15077: lemma lemma_termdiff1:
nipkow@15539:   "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
nipkow@15539:    (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p)))::real)"
berghofe@16641: by (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac
berghofe@16641:   cong: strong_setsum_cong)
paulson@15077: 
paulson@15077: lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
paulson@15077: by (simp add: less_iff_Suc_add)
paulson@15077: 
paulson@15077: lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
paulson@15077: by arith
paulson@15077: 
paulson@15077: 
paulson@15229: lemma lemma_termdiff2:
nipkow@15539:   "h \<noteq> 0 ==>
nipkow@15539:    (((z + h) ^ n) - (z ^ n)) * inverse h - real n * (z ^ (n - Suc 0)) =
nipkow@15539:    h * (\<Sum>p=0..< n - Suc 0. (z ^ p) *
nipkow@15539:        (\<Sum>q=0..< (n - Suc 0) - p. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))"
paulson@15077: apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
paulson@15077: apply (simp add: right_diff_distrib mult_ac)
paulson@15077: apply (simp add: mult_assoc [symmetric])
paulson@15077: apply (case_tac "n")
paulson@15077: apply (auto simp add: lemma_realpow_diff_sumr2 
paulson@15077:                       right_diff_distrib [symmetric] mult_assoc
nipkow@15561:             simp del: realpow_Suc setsum_op_ivl_Suc)
nipkow@15561: apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_op_ivl_Suc)
paulson@15077: apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
ballarin@19279:                 sumdiff lemma_termdiff1 setsum_right_distrib)
nipkow@15539: apply (auto intro!: setsum_cong[OF refl] simp add: diff_minus real_add_assoc)
nipkow@15539: apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
ballarin@19279: apply (auto simp add: setsum_right_distrib lemma_realpow_diff_sumr2 mult_ac simp
nipkow@15561:                  del: setsum_op_ivl_Suc realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_termdiff3:
paulson@15229:      "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<le> K |]  
paulson@15077:       ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
paulson@15077:           \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
paulson@15077: apply (subst lemma_termdiff2, assumption)
paulson@16924: apply (simp add: mult_commute abs_mult) 
paulson@15077: apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
nipkow@15536: apply (rule setsum_abs [THEN real_le_trans])
paulson@16924: apply (simp add: mult_assoc [symmetric] abs_mult)
paulson@15077: apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
nipkow@15542: apply (auto intro!: real_setsum_nat_ivl_bounded)
paulson@15077: apply (case_tac "n", auto)
paulson@15077: apply (drule less_add_one)
paulson@15077: (*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
paulson@15077: apply clarify 
paulson@15077: apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
paulson@15077:                     K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
nipkow@15561: apply (simp (no_asm_simp) add: power_add del: setsum_op_ivl_Suc)
nipkow@15561: apply (auto intro!: mult_mono simp del: setsum_op_ivl_Suc)
paulson@16924: apply (auto intro!: power_mono simp add: power_abs
paulson@16924:            simp del: setsum_op_ivl_Suc)
paulson@15229: apply (rule_tac j = "real (Suc d) * (K ^ d)" in real_le_trans)
paulson@15077: apply (subgoal_tac [2] "0 \<le> K")
paulson@15077: apply (drule_tac [2] n = d in zero_le_power)
nipkow@15561: apply (auto simp del: setsum_op_ivl_Suc)
nipkow@15536: apply (rule setsum_abs [THEN real_le_trans])
paulson@16924: apply (rule real_setsum_nat_ivl_bounded)
paulson@16924: apply (auto dest!: less_add_one intro!: mult_mono simp add: power_add abs_mult)
webertj@20217: apply (auto intro!: power_mono zero_le_power simp add: power_abs)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_termdiff4: 
paulson@15077:   "[| 0 < k;  
paulson@15081:       (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]  
paulson@15077:    ==> f -- 0 --> 0"
paulson@15229: apply (simp add: LIM_def, auto)
paulson@15077: apply (subgoal_tac "0 \<le> K")
paulson@15229:  prefer 2
paulson@15229:  apply (drule_tac x = "k/2" in spec)
paulson@15229: apply (simp add: ); 
paulson@15229:  apply (subgoal_tac "0 \<le> K*k", simp add: zero_le_mult_iff) 
paulson@15229:  apply (force intro: order_trans [of _ "\<bar>f (k / 2)\<bar> * 2"]) 
paulson@15077: apply (drule real_le_imp_less_or_eq, auto)
paulson@15229: apply (subgoal_tac "0 < (r * inverse K) / 2")
paulson@15229: apply (drule_tac ?d1.0 = "(r * inverse K) / 2" and ?d2.0 = k in real_lbound_gt_zero)
paulson@15229: apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff zero_less_divide_iff)
paulson@15077: apply (rule_tac x = e in exI, auto)
paulson@15077: apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
paulson@15229: apply (force ); 
paulson@15229: apply (rule_tac y = "K * e" in order_less_trans)
paulson@15077: apply (simp add: mult_less_cancel_left)
paulson@15229: apply (rule_tac c = "inverse K" in mult_right_less_imp_less)
paulson@15077: apply (auto simp add: mult_ac)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_termdiff5:
paulson@15229:      "[| 0 < k;  
paulson@15077:             summable f;  
paulson@15077:             \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
paulson@15077:                     (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
paulson@15077:          ==> (%h. suminf(g h)) -- 0 --> 0"
paulson@15077: apply (drule summable_sums)
paulson@15081: 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: apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
paulson@15077: apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
paulson@15077:  prefer 2
paulson@15077:  apply (simp add: summable_def) 
paulson@15077:  apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
paulson@15077:  apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
paulson@15077:  apply (simp add: mult_ac) 
paulson@15077: apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
paulson@15077:  apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
paulson@15077:   apply (rule_tac [2] x = 0 in exI, auto)
nipkow@15546: apply (rule_tac j = "\<Sum>n. \<bar>g h n\<bar>" in real_le_trans)
avigad@16819: apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult2])
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: text{* FIXME: Long proofs*}
paulson@15077: 
webertj@20432: ML {* fast_arith_split_limit := 0; *}  (* FIXME: rewrite proofs *)
webertj@20432: 
paulson@15077: lemma termdiffs_aux:
paulson@15077:      "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
nipkow@15546:     ==> (\<lambda>h. \<Sum>n. c n *
paulson@15077:                   (((x + h) ^ n - x ^ n) * inverse h -
nipkow@15546:                    real n * x ^ (n - Suc 0)))
paulson@15077:        -- 0 --> 0"
paulson@15077: apply (drule dense, safe)
paulson@15077: apply (frule real_less_sum_gt_zero)
paulson@15077: apply (drule_tac
paulson@15081:          f = "%n. \<bar>c n\<bar> * real n * real (n - Suc 0) * (r ^ (n - 2))" 
paulson@15077:      and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) 
paulson@15077:                              - (real n * (x ^ (n - Suc 0))))" 
paulson@15077:        in lemma_termdiff5)
webertj@20432: apply (auto simp add: add_commute)
webertj@20432: apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
webertj@20432: apply (rule_tac [2] x = K in powser_insidea, auto)
webertj@20432: apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
webertj@20432: apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_of_nonneg], auto)
webertj@20432: apply (simp add: diffs_def mult_assoc [symmetric])
webertj@20432: apply (subgoal_tac
webertj@20432:         "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
webertj@20432:               = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
webertj@20432: apply (auto simp add: abs_mult)
webertj@20432: apply (drule diffs_equiv)
webertj@20432: apply (drule sums_summable)
webertj@20432: apply (simp_all add: diffs_def) 
webertj@20432: apply (simp add: diffs_def mult_ac)
webertj@20432: apply (subgoal_tac " (%n. real n * (real (Suc n) * (\<bar>c (Suc n)\<bar> * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * \<bar>c m\<bar> * inverse r) n * (r ^ n))")
webertj@20432: apply auto
paulson@15077:   prefer 2
paulson@15077:   apply (rule ext)
webertj@20432:   apply (simp add: diffs_def) 
paulson@15077:   apply (case_tac "n", auto)
paulson@15077: txt{*23*}
paulson@15077:    apply (drule abs_ge_zero [THEN order_le_less_trans])
webertj@20432:    apply (simp add: mult_ac) 
paulson@15077:   apply (drule abs_ge_zero [THEN order_le_less_trans])
webertj@20432:   apply (simp add: mult_ac) 
paulson@15077:  apply (drule diffs_equiv)
paulson@15077:  apply (drule sums_summable)
webertj@20432: apply (subgoal_tac
webertj@20432:           "summable
webertj@20432:             (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
webertj@20432:                  r ^ (n - Suc 0)) =
webertj@20432:            summable
webertj@20432:             (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
webertj@20432: apply simp 
webertj@20432: apply (rule_tac f = summable in arg_cong, rule ext)
paulson@15077: txt{*33*}
webertj@20432: apply (case_tac "n", auto)
webertj@20432: apply (case_tac "nat", auto)
webertj@20432: apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
paulson@15077: apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077: apply (simp add: mult_assoc)
paulson@15077: apply (rule mult_left_mono)
webertj@20432:  prefer 2 apply arith 
paulson@15229: apply (subst add_commute)
paulson@15077: apply (simp (no_asm) add: mult_assoc [symmetric])
paulson@15077: apply (rule lemma_termdiff3)
webertj@20432: apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
paulson@15077: done
paulson@15077: 
webertj@20256: ML {* fast_arith_split_limit := 9; *}  (* FIXME *)
webertj@20217: 
paulson@15077: lemma termdiffs: 
paulson@15077:     "[| summable(%n. c(n) * (K ^ n));  
paulson@15077:         summable(%n. (diffs c)(n) * (K ^ n));  
paulson@15077:         summable(%n. (diffs(diffs c))(n) * (K ^ n));  
paulson@15077:         \<bar>x\<bar> < \<bar>K\<bar> |]  
nipkow@15546:      ==> DERIV (%x. \<Sum>n. c(n) * (x ^ n))  x :>  
nipkow@15546:              (\<Sum>n. (diffs c)(n) * (x ^ n))"
paulson@15229: apply (simp add: deriv_def)
nipkow@15546: apply (rule_tac g = "%h. \<Sum>n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h" in LIM_trans)
paulson@15077: apply (simp add: LIM_def, safe)
paulson@15077: apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077: apply (auto simp add: less_diff_eq)
webertj@20432: apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077: apply (rule_tac y = 0 in order_le_less_trans, auto)
nipkow@15546: 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: apply (auto intro!: summable_sums)
paulson@15077: apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077: apply (auto simp add: add_commute)
paulson@15077: apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
avigad@16819: apply (drule_tac f = "(%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
paulson@15085: apply (rule sums_unique)
paulson@15079: apply (simp add: diff_def divide_inverse add_ac mult_ac)
paulson@15077: apply (rule LIM_zero_cancel)
nipkow@15546: 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:  prefer 2 apply (blast intro: termdiffs_aux) 
paulson@15077: apply (simp add: LIM_def, safe)
paulson@15077: apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077: apply (auto simp add: less_diff_eq)
webertj@20432: apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077: apply (rule_tac y = 0 in order_le_less_trans, auto)
paulson@15077: apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
paulson@15077: apply (rule_tac [2] powser_inside, auto)
paulson@15077: apply (drule_tac c = c and x = x in diffs_equiv)
paulson@15077: apply (frule sums_unique, auto)
nipkow@15546: 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: apply safe
paulson@15077: apply (auto intro!: summable_sums)
paulson@15077: apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077: apply (auto simp add: add_commute)
paulson@15229: apply (frule_tac x = "(%n. c n * (xa + x) ^ n) " and y = "(%n. c n * x ^ n)" in sums_diff, assumption)
paulson@15077: apply (simp add: suminf_diff [OF sums_summable sums_summable] 
paulson@15077:                right_diff_distrib [symmetric])
avigad@16819: apply (subst suminf_diff)
avigad@16819: apply (rule summable_mult2)
avigad@16819: apply (erule sums_summable)
avigad@16819: apply (erule sums_summable)
avigad@16819: apply (simp add: ring_eq_simps)
paulson@15077: done
paulson@15077: 
paulson@15077: subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
paulson@15077: 
paulson@15077: lemma exp_fdiffs: 
paulson@15077:       "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
paulson@15229: by (simp add: diffs_def mult_assoc [symmetric] del: mult_Suc)
paulson@15077: 
paulson@15077: lemma sin_fdiffs: 
paulson@15077:       "diffs(%n. if even n then 0  
paulson@15077:            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
paulson@15077:        = (%n. if even n then  
paulson@15077:                  (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077:               else 0)"
paulson@15229: by (auto intro!: ext 
paulson@15229:          simp add: diffs_def divide_inverse simp del: mult_Suc)
paulson@15077: 
paulson@15077: lemma sin_fdiffs2: 
paulson@15077:        "diffs(%n. if even n then 0  
paulson@15077:            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
paulson@15077:        = (if even n then  
paulson@15077:                  (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077:               else 0)"
paulson@15229: by (auto intro!: ext 
paulson@15229:          simp add: diffs_def divide_inverse simp del: mult_Suc)
paulson@15077: 
paulson@15077: lemma cos_fdiffs: 
paulson@15077:       "diffs(%n. if even n then  
paulson@15077:                  (- 1) ^ (n div 2)/(real (fact n)) else 0)  
paulson@15077:        = (%n. - (if even n then 0  
paulson@15077:            else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
paulson@15229: by (auto intro!: ext 
paulson@15229:          simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
paulson@15229:          simp del: mult_Suc)
paulson@15077: 
paulson@15077: 
paulson@15077: lemma cos_fdiffs2: 
paulson@15077:       "diffs(%n. if even n then  
paulson@15077:                  (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
paulson@15077:        = - (if even n then 0  
paulson@15077:            else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
paulson@15229: by (auto intro!: ext 
paulson@15229:          simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
paulson@15229:          simp del: mult_Suc)
paulson@15077: 
paulson@15077: text{*Now at last we can get the derivatives of exp, sin and cos*}
paulson@15077: 
paulson@15077: lemma lemma_sin_minus:
nipkow@15546:      "- sin x = (\<Sum>n. - ((if even n then 0 
paulson@15077:                   else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
paulson@15077: by (auto intro!: sums_unique sums_minus sin_converges)
paulson@15077: 
nipkow@15546: lemma lemma_exp_ext: "exp = (%x. \<Sum>n. inverse (real (fact n)) * x ^ n)"
paulson@15077: by (auto intro!: ext simp add: exp_def)
paulson@15077: 
paulson@15077: lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15229: apply (simp add: exp_def)
paulson@15077: apply (subst lemma_exp_ext)
nipkow@15546: 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: apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217: apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_sin_ext:
nipkow@15546:      "sin = (%x. \<Sum>n. 
paulson@15077:                    (if even n then 0  
paulson@15077:                        else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
nipkow@15546:                    x ^ n)"
paulson@15077: by (auto intro!: ext simp add: sin_def)
paulson@15077: 
paulson@15077: lemma lemma_cos_ext:
nipkow@15546:      "cos = (%x. \<Sum>n. 
paulson@15077:                    (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
nipkow@15546:                    x ^ n)"
paulson@15077: by (auto intro!: ext simp add: cos_def)
paulson@15077: 
paulson@15077: lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
paulson@15229: apply (simp add: cos_def)
paulson@15077: apply (subst lemma_sin_ext)
paulson@15077: apply (auto simp add: sin_fdiffs2 [symmetric])
paulson@15229: apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217: apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
paulson@15077: apply (subst lemma_cos_ext)
paulson@15077: apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
paulson@15229: apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
webertj@20217: 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: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Properties of the Exponential Function*}
paulson@15077: 
paulson@15077: lemma exp_zero [simp]: "exp 0 = 1"
paulson@15077: proof -
paulson@15077:   have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
nipkow@15546:         (\<Sum>n. inverse (real (fact n)) * 0 ^ n)"
paulson@15077:     by (rule series_zero [rule_format, THEN sums_unique],
paulson@15077:         case_tac "m", auto)
paulson@15077:   thus ?thesis by (simp add:  exp_def) 
paulson@15077: qed
paulson@15077: 
avigad@17014: lemma exp_ge_add_one_self_aux: "0 \<le> x ==> (1 + x) \<le> exp(x)"
paulson@15077: apply (drule real_le_imp_less_or_eq, auto)
paulson@15229: apply (simp add: exp_def)
paulson@15077: apply (rule real_le_trans)
paulson@15229: apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
paulson@15077: apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
paulson@15077: apply (rule order_less_le_trans)
avigad@17014: apply (rule_tac [2] exp_ge_add_one_self_aux, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
paulson@15077: proof -
paulson@15077:   have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
paulson@15077:     by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const) 
paulson@15077:   thus ?thesis by (simp add: o_def)
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
paulson@15077: proof -
paulson@15077:   have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
paulson@15077:     by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
paulson@15077:   thus ?thesis by (simp add: o_def)
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
paulson@15077: proof -
paulson@15077:   have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
paulson@15077:        :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
paulson@15077:     by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
paulson@15077:   thus ?thesis by simp
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
paulson@15077: proof -
paulson@15077:   have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
paulson@15077:   hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
paulson@15077:     by (rule DERIV_isconst_all) 
paulson@15077:   thus ?thesis by simp
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
paulson@15077: proof -
paulson@15077:   have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
paulson@15077:   thus ?thesis by simp
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
paulson@15077: by (simp add: mult_commute)
paulson@15077: 
paulson@15077: 
paulson@15077: lemma exp_minus: "exp(-x) = inverse(exp(x))"
paulson@15077: by (auto intro: inverse_unique [symmetric])
paulson@15077: 
paulson@15077: lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
paulson@15077: proof -
paulson@15077:   have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
paulson@15077:   thus ?thesis by (simp (no_asm_simp) add: mult_ac)
paulson@15077: qed
paulson@15077: 
paulson@15077: text{*Proof: because every exponential can be seen as a square.*}
paulson@15077: lemma exp_ge_zero [simp]: "0 \<le> exp x"
paulson@15077: apply (rule_tac t = x in real_sum_of_halves [THEN subst])
paulson@15077: apply (subst exp_add, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
paulson@15077: apply (cut_tac x = x in exp_mult_minus2)
paulson@15077: apply (auto simp del: exp_mult_minus2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_gt_zero [simp]: "0 < exp x"
paulson@15077: by (simp add: order_less_le)
paulson@15077: 
paulson@15077: lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
paulson@15077: by (auto intro: positive_imp_inverse_positive)
paulson@15077: 
paulson@15081: lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
paulson@15229: by auto
paulson@15077: 
paulson@15077: lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251: apply (induct "n")
paulson@15077: apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
paulson@15229: apply (simp add: diff_minus divide_inverse)
paulson@15077: apply (simp (no_asm) add: exp_add exp_minus)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: lemma exp_less_mono:
paulson@15077:   assumes xy: "x < y" shows "exp x < exp y"
paulson@15077: proof -
paulson@15077:   have "1 < exp (y + - x)"
paulson@15077:     by (rule real_less_sum_gt_zero [THEN exp_gt_one])
paulson@15077:   hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
paulson@15077:     by (auto simp add: exp_add exp_minus)
paulson@15077:   thus ?thesis
nipkow@15539:     by (simp add: divide_inverse [symmetric] pos_less_divide_eq
paulson@15228:              del: divide_self_if)
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma exp_less_cancel: "exp x < exp y ==> x < y"
paulson@15228: apply (simp add: linorder_not_le [symmetric]) 
paulson@15228: apply (auto simp add: order_le_less exp_less_mono) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
paulson@15077: by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077: 
paulson@15077: lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
paulson@15077: by (auto simp add: linorder_not_less [symmetric])
paulson@15077: 
paulson@15077: lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
paulson@15077: by (simp add: order_eq_iff)
paulson@15077: 
paulson@15077: lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
paulson@15077: apply (rule IVT)
paulson@15077: apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
paulson@15077: apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
paulson@15077: apply simp 
avigad@17014: apply (rule exp_ge_add_one_self_aux, simp)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
paulson@15077: apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077: apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077: apply (rule_tac [2] x = 0 in exI)
paulson@15077: apply (frule_tac [3] real_inverse_gt_one)
paulson@15077: apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077: apply (rule_tac x = "-x" in exI)
paulson@15077: apply (simp add: exp_minus)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Properties of the Logarithmic Function*}
paulson@15077: 
paulson@15077: lemma ln_exp[simp]: "ln(exp x) = x"
paulson@15077: by (simp add: ln_def)
paulson@15077: 
paulson@15077: lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
paulson@15077: apply (auto dest: exp_total)
paulson@15077: apply (erule subst, simp) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
paulson@15077: apply (rule exp_inj_iff [THEN iffD1])
paulson@15077: apply (frule real_mult_order)
paulson@15077: apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
paulson@15077: apply (simp only: exp_ln_iff [symmetric])
paulson@15077: apply (erule subst)+
paulson@15077: apply simp 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_one[simp]: "ln 1 = 0"
paulson@15077: by (rule exp_inj_iff [THEN iffD1], auto)
paulson@15077: 
paulson@15077: lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
paulson@15077: apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
paulson@15077: apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_div: 
paulson@15077:     "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
paulson@15229: apply (simp add: divide_inverse)
paulson@15077: apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
paulson@15077: apply (simp only: exp_ln_iff [symmetric])
paulson@15077: apply (erule subst)+
paulson@15077: apply simp 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
paulson@15077: by (auto simp add: linorder_not_less [symmetric])
paulson@15077: 
paulson@15077: lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
paulson@15077: by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
paulson@15077: 
paulson@15077: lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
paulson@15077: apply (rule ln_exp [THEN subst])
avigad@17014: apply (rule ln_le_cancel_iff [THEN iffD2]) 
avigad@17014: apply (auto simp add: exp_ge_add_one_self_aux)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma ln_less_self [simp]: "0 < x ==> ln x < x"
paulson@15077: apply (rule order_less_le_trans)
paulson@15077: apply (rule_tac [2] ln_add_one_self_le_self)
paulson@15077: apply (rule ln_less_cancel_iff [THEN iffD2], auto)
paulson@15077: done
paulson@15077: 
paulson@15234: lemma ln_ge_zero [simp]:
paulson@15077:   assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077: proof -
paulson@15077:   have "0 < x" using x by arith
paulson@15077:   hence "exp 0 \<le> exp (ln x)"
paulson@15077:     by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077:   thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma ln_ge_zero_imp_ge_one:
paulson@15077:   assumes ln: "0 \<le> ln x" 
paulson@15077:       and x:  "0 < x"
paulson@15077:   shows "1 \<le> x"
paulson@15077: proof -
paulson@15077:   from ln have "ln 1 \<le> ln x" by simp
paulson@15077:   thus ?thesis by (simp add: x del: ln_one) 
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077: by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077: 
paulson@15234: lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
paulson@15234: by (insert ln_ge_zero_iff [of x], arith)
paulson@15234: 
paulson@15077: lemma ln_gt_zero:
paulson@15077:   assumes x: "1 < x" shows "0 < ln x"
paulson@15077: proof -
paulson@15077:   have "0 < x" using x by arith
paulson@15077:   hence "exp 0 < exp (ln x)"
paulson@15077:     by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077:   thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma ln_gt_zero_imp_gt_one:
paulson@15077:   assumes ln: "0 < ln x" 
paulson@15077:       and x:  "0 < x"
paulson@15077:   shows "1 < x"
paulson@15077: proof -
paulson@15077:   from ln have "ln 1 < ln x" by simp
paulson@15077:   thus ?thesis by (simp add: x del: ln_one) 
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077: by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077: 
paulson@15234: lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
paulson@15234: by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
paulson@15077: 
paulson@15077: lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15234: by simp
paulson@15077: 
paulson@15077: lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077: by auto
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Basic Properties of the Trigonometric Functions*}
paulson@15077: 
paulson@15077: lemma sin_zero [simp]: "sin 0 = 0"
paulson@15077: by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
paulson@15077:          simp add: sin_def sums_def simp del: power_0_left)
paulson@15077: 
nipkow@15539: lemma lemma_series_zero2:
nipkow@15539:  "(\<forall>m. n \<le> m --> f m = 0) --> f sums setsum f {0..<n}"
paulson@15077: by (auto intro: series_zero)
paulson@15077: 
paulson@15077: lemma cos_zero [simp]: "cos 0 = 1"
paulson@15229: apply (simp add: cos_def)
paulson@15077: apply (rule sums_unique [symmetric])
paulson@15229: 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: apply auto
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_sin_sin_mult [simp]:
paulson@15077:      "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077: by (rule DERIV_mult, auto)
paulson@15077: 
paulson@15077: lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077:      "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077: apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077: apply (auto simp add: mult_assoc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_sin_realpow2 [simp]:
paulson@15077:      "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077: by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077: 
paulson@15077: lemma DERIV_sin_realpow2a [simp]:
paulson@15077:      "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077: by (auto simp add: numeral_2_eq_2)
paulson@15077: 
paulson@15077: lemma DERIV_cos_cos_mult [simp]:
paulson@15077:      "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077: by (rule DERIV_mult, auto)
paulson@15077: 
paulson@15077: lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077:      "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077: apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077: apply (auto simp add: mult_ac)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_cos_realpow2 [simp]:
paulson@15077:      "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077: by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077: 
paulson@15077: lemma DERIV_cos_realpow2a [simp]:
paulson@15077:      "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077: by (auto simp add: numeral_2_eq_2)
paulson@15077: 
paulson@15077: lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077: by auto
paulson@15077: 
paulson@15077: lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule DERIV_cos_realpow2a, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: (* most useful *)
paulson@15229: lemma DERIV_cos_cos_mult3 [simp]:
paulson@15229:      "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule DERIV_cos_cos_mult2, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_sin_circle_all: 
paulson@15077:      "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077:              (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
paulson@15229: apply (simp only: diff_minus, safe)
paulson@15229: apply (rule DERIV_add) 
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma DERIV_sin_circle_all_zero [simp]:
paulson@15229:      "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077: by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077: 
paulson@15077: lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077: apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
paulson@15077: apply (subst real_add_commute)
paulson@15077: apply (simp (no_asm) del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077: apply (cut_tac x = x in sin_cos_squared_add2)
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15229: apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077: apply (simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077: apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077: apply (simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
paulson@15077: by arith
paulson@15077: 
paulson@15081: lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
paulson@15077: apply (auto simp add: linorder_not_less [symmetric])
paulson@15077: apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077: apply (auto simp del: realpow_Suc)
paulson@15077: apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077: apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
paulson@15077: apply (insert abs_sin_le_one [of x]) 
paulson@15077: apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077: apply (insert abs_sin_le_one [of x]) 
paulson@15077: apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077: done
paulson@15077: 
paulson@15081: lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
paulson@15077: apply (auto simp add: linorder_not_less [symmetric])
paulson@15077: apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077: apply (auto simp del: realpow_Suc)
paulson@15077: apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077: apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
paulson@15077: apply (insert abs_cos_le_one [of x]) 
paulson@15077: apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_le_one [simp]: "cos x \<le> 1"
paulson@15077: apply (insert abs_cos_le_one [of x]) 
paulson@15077: apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077:       DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15229: apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
paulson@15077: apply (rule DERIV_pow, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma DERIV_fun_exp:
paulson@15229:      "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule_tac f = exp in DERIV_chain2)
paulson@15077: apply (rule DERIV_exp, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma DERIV_fun_sin:
paulson@15229:      "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule_tac f = sin in DERIV_chain2)
paulson@15077: apply (rule DERIV_sin, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma DERIV_fun_cos:
paulson@15229:      "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule_tac f = cos in DERIV_chain2)
paulson@15077: apply (rule DERIV_cos, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
paulson@15077:                     DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
paulson@15077:                     DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
paulson@15077:                     DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
paulson@15077:                     DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
paulson@15077: 
paulson@15077: (* lemma *)
paulson@15229: lemma lemma_DERIV_sin_cos_add:
paulson@15229:      "\<forall>x.  
paulson@15077:          DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077:                (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
paulson@15077: apply (safe, rule lemma_DERIV_subst)
paulson@15077: apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077:   --{*replaces the old @{text DERIV_tac}*}
paulson@15229: apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_cos_add [simp]:
paulson@15077:      "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077:       (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077: apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077:        in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
paulson@15077: apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077: apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085:             simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077: apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077: apply (auto dest!: real_sum_squares_cancel_a
paulson@15085:             simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
paulson@15077: done
paulson@15077: 
paulson@15085: lemma lemma_DERIV_sin_cos_minus:
paulson@15085:     "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
paulson@15077: apply (safe, rule lemma_DERIV_subst)
paulson@15077: apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15229: apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077: done
paulson@15077: 
paulson@15085: lemma sin_cos_minus [simp]: 
paulson@15085:     "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15085: apply (cut_tac y = 0 and x = x 
paulson@15085:        in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_minus [simp]: "sin (-x) = -sin(x)"
paulson@15077: apply (cut_tac x = x in sin_cos_minus)
paulson@15085: apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085:        simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_minus)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_minus [simp]: "cos (-x) = cos(x)"
paulson@15077: apply (cut_tac x = x in sin_cos_minus)
paulson@15085: apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085:                    simp add: numeral_2_eq_2 simp del: sin_cos_minus)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
paulson@15229: apply (simp add: diff_minus)
paulson@15077: apply (simp (no_asm) add: sin_add)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077: by (simp add: sin_diff mult_commute)
paulson@15077: 
paulson@15077: lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
paulson@15229: apply (simp add: diff_minus)
paulson@15077: apply (simp (no_asm) add: cos_add)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077: by (simp add: cos_diff mult_commute)
paulson@15077: 
paulson@15077: lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
paulson@15077: by (cut_tac x = x and y = x in sin_add, auto)
paulson@15077: 
paulson@15077: 
paulson@15077: lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
paulson@15077: apply (cut_tac x = x and y = x in cos_add)
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*The Constant Pi*}
paulson@15077: 
paulson@15077: text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077:    hence define pi.*}
paulson@15077: 
paulson@15077: lemma sin_paired:
paulson@15077:      "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077:       sums  sin x"
paulson@15077: proof -
paulson@15077:   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077:             (if even k then 0
paulson@15077:              else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
paulson@15077:             x ^ k) 
paulson@15077: 	sums
nipkow@15546: 	(\<Sum>n. (if even n then 0
paulson@15077: 		     else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
paulson@15077: 	            x ^ n)" 
paulson@15077:     by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077:   thus ?thesis by (simp add: mult_ac sin_def)
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077: apply (subgoal_tac 
paulson@15077:        "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077:               (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
nipkow@15546:      sums (\<Sum>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077:  prefer 2
paulson@15077:  apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077: apply (rotate_tac 2)
paulson@15077: apply (drule sin_paired [THEN sums_unique, THEN ssubst])
paulson@15077: apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077: apply (frule sums_unique)
paulson@15077: apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077: apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
paulson@15077: apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077: apply (erule sums_summable)
paulson@15077: apply (case_tac "m=0")
paulson@15077: apply (simp (no_asm_simp))
paulson@15234: apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") 
nipkow@15539: apply (simp only: mult_less_cancel_left, simp)  
nipkow@15539: apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077: apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077: apply (rule mult_strict_mono)
paulson@15085: apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
paulson@15077: apply (subst fact_Suc)
paulson@15077: apply (subst fact_Suc)
paulson@15077: apply (subst fact_Suc)
paulson@15077: apply (subst fact_Suc)
paulson@15077: apply (subst real_of_nat_mult)
paulson@15077: apply (subst real_of_nat_mult)
paulson@15077: apply (subst real_of_nat_mult)
paulson@15077: apply (subst real_of_nat_mult)
nipkow@15539: apply (simp (no_asm) add: divide_inverse del: fact_Suc)
paulson@15077: apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077: apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
paulson@15077: apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077: apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
paulson@15077: apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077: apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077: apply (erule ssubst)+
paulson@15077: apply (auto simp del: fact_Suc)
paulson@15077: apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077:  prefer 2 apply (simp only: zero_less_power) 
paulson@15077: apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077: apply (rule mult_strict_mono)
paulson@15077: apply (simp_all (no_asm_simp))
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077: by (auto intro: sin_gt_zero)
paulson@15077: 
paulson@15077: lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077: apply (cut_tac x = x in sin_gt_zero1)
paulson@15077: apply (auto simp add: cos_squared_eq cos_double)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_paired:
paulson@15077:      "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077: proof -
paulson@15077:   have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077:             (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
paulson@15077:             x ^ k) 
paulson@15077:         sums
nipkow@15546: 	(\<Sum>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
paulson@15077: 	      x ^ n)" 
paulson@15077:     by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077:   thus ?thesis by (simp add: mult_ac cos_def)
paulson@15077: qed
paulson@15077: 
paulson@15077: declare zero_less_power [simp]
paulson@15077: 
paulson@15077: lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077: by simp
paulson@15077: 
paulson@15077: lemma cos_two_less_zero: "cos (2) < 0"
paulson@15077: apply (cut_tac x = 2 in cos_paired)
paulson@15077: apply (drule sums_minus)
paulson@15077: apply (rule neg_less_iff_less [THEN iffD1]) 
nipkow@15539: apply (frule sums_unique, auto)
nipkow@15539: apply (rule_tac y =
nipkow@15539:  "\<Sum>n=0..< Suc(Suc(Suc 0)). - ((- 1) ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481:        in order_less_trans)
paulson@15077: apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
nipkow@15561: apply (simp (no_asm) add: mult_assoc del: setsum_op_ivl_Suc)
paulson@15077: apply (rule sumr_pos_lt_pair)
paulson@15077: apply (erule sums_summable, safe)
paulson@15085: apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] 
paulson@15085:             del: fact_Suc)
paulson@15077: apply (rule real_mult_inverse_cancel2)
paulson@15077: apply (rule real_of_nat_fact_gt_zero)+
paulson@15077: apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077: apply (subst fact_lemma) 
paulson@15481: apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481: apply (simp only: real_of_nat_mult)
paulson@15077: apply (rule real_mult_less_mono, force)
paulson@15481:   apply (rule_tac [3] real_of_nat_fact_gt_zero)
paulson@15481:  prefer 2 apply force
paulson@15077: apply (rule real_of_nat_less_iff [THEN iffD2])
paulson@15077: apply (rule fact_less_mono, auto)
paulson@15077: done
paulson@15077: declare cos_two_less_zero [simp]
paulson@15077: declare cos_two_less_zero [THEN real_not_refl2, simp]
paulson@15077: declare cos_two_less_zero [THEN order_less_imp_le, simp]
paulson@15077: 
paulson@15077: lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077: apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077: apply (rule_tac [2] IVT2)
paulson@15077: apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077: apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077: apply (rule ccontr)
paulson@15077: apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077: apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077: apply (drule_tac f = cos in Rolle)
paulson@15077: apply (drule_tac [5] f = cos in Rolle)
paulson@15077: apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@15077: apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
paulson@15077: apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
paulson@15077: apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
paulson@15077: done
paulson@15077:     
paulson@15077: lemma pi_half: "pi/2 = (@x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077: by (simp add: pi_def)
paulson@15077: 
paulson@15077: lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
paulson@15077: apply (rule cos_is_zero [THEN ex1E])
paulson@15077: apply (auto intro!: someI2 simp add: pi_half)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma pi_half_gt_zero: "0 < pi / 2"
paulson@15077: apply (rule cos_is_zero [THEN ex1E])
paulson@15077: apply (auto simp add: pi_half)
paulson@15077: apply (rule someI2, blast, safe)
paulson@15077: apply (drule_tac y = xa in real_le_imp_less_or_eq)
paulson@15077: apply (safe, simp)
paulson@15077: done
paulson@15077: declare pi_half_gt_zero [simp]
paulson@15077: declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
paulson@15077: declare pi_half_gt_zero [THEN order_less_imp_le, simp]
paulson@15077: 
paulson@15077: lemma pi_half_less_two: "pi / 2 < 2"
paulson@15077: apply (rule cos_is_zero [THEN ex1E])
paulson@15077: apply (auto simp add: pi_half)
paulson@15077: apply (rule someI2, blast, safe)
paulson@15077: apply (drule_tac x = xa in order_le_imp_less_or_eq)
paulson@15077: apply (safe, simp)
paulson@15077: done
paulson@15077: declare pi_half_less_two [simp]
paulson@15077: declare pi_half_less_two [THEN real_not_refl2, simp]
paulson@15077: declare pi_half_less_two [THEN order_less_imp_le, simp]
paulson@15077: 
paulson@15077: lemma pi_gt_zero [simp]: "0 < pi"
paulson@15229: apply (insert pi_half_gt_zero) 
paulson@15229: apply (simp add: ); 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma pi_neq_zero [simp]: "pi \<noteq> 0"
paulson@15077: by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])
paulson@15077: 
paulson@15077: lemma pi_not_less_zero [simp]: "~ (pi < 0)"
paulson@15077: apply (insert pi_gt_zero)
paulson@15077: apply (blast elim: order_less_asym) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma pi_ge_zero [simp]: "0 \<le> pi"
paulson@15077: by (auto intro: order_less_imp_le)
paulson@15077: 
paulson@15077: lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
paulson@15077: by auto
paulson@15077: 
paulson@15077: lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077: apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077: apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539: by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077: 
paulson@15077: lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539: by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077: 
paulson@15077: lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15229: by (simp add: diff_minus cos_add)
paulson@15077: 
paulson@15077: lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229: by (simp add: cos_add)
paulson@15077: declare minus_sin_cos_eq [symmetric, simp]
paulson@15077: 
paulson@15077: lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15229: by (simp add: diff_minus sin_add)
paulson@15077: declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]
paulson@15077: 
paulson@15077: lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229: by (simp add: sin_add)
paulson@15077: 
paulson@15077: lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229: by (simp add: sin_add)
paulson@15077: 
paulson@15077: lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229: by (simp add: cos_add)
paulson@15077: 
paulson@15077: lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077: by (simp add: sin_add cos_double)
paulson@15077: 
paulson@15077: lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077: by (simp add: cos_add cos_double)
paulson@15077: 
paulson@15077: lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251: apply (induct "n")
paulson@15077: apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077: done
paulson@15077: 
paulson@15383: lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383: proof -
paulson@15383:   have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
paulson@15383:   also have "... = -1 ^ n" by (rule cos_npi) 
paulson@15383:   finally show ?thesis .
paulson@15383: qed
paulson@15383: 
paulson@15077: lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251: apply (induct "n")
paulson@15077: apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15383: by (simp add: mult_commute [of pi]) 
paulson@15077: 
paulson@15077: lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077: by (simp add: cos_double)
paulson@15077: 
paulson@15077: lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229: by simp
paulson@15077: 
paulson@15077: lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077: apply (rule sin_gt_zero, assumption)
paulson@15077: apply (rule order_less_trans, assumption)
paulson@15077: apply (rule pi_half_less_two)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_less_zero: 
paulson@15077:   assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077: proof -
paulson@15077:   have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077:   thus ?thesis by simp
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma pi_less_4: "pi < 4"
paulson@15077: by (cut_tac pi_half_less_two, auto)
paulson@15077: 
paulson@15077: lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077: apply (cut_tac pi_less_4)
paulson@15077: apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077: apply (force intro: DERIV_isCont DERIV_cos)
paulson@15077: apply (cut_tac cos_is_zero, safe)
paulson@15077: apply (rename_tac y z)
paulson@15077: apply (drule_tac x = y in spec)
paulson@15077: apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077: apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077: apply (rule cos_minus [THEN subst])
paulson@15077: apply (rule cos_gt_zero)
paulson@15077: apply (auto intro: cos_gt_zero)
paulson@15077: done
paulson@15077:  
paulson@15077: lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077: apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077: apply (subgoal_tac "x = pi/2", auto) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077: apply (subst sin_cos_eq)
paulson@15077: apply (rotate_tac 1)
paulson@15077: apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077: apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077: by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077: 
paulson@15077: lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077: apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077: apply (rule_tac [2] IVT2)
paulson@15077: apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077: apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077: apply (rule ccontr, auto)
paulson@15077: apply (drule_tac f = cos in Rolle)
paulson@15077: apply (drule_tac [5] f = cos in Rolle)
paulson@15077: apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077:             dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077:             simp add: differentiable_def)
paulson@15077: apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_total:
paulson@15077:      "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077: apply (rule ccontr)
paulson@15077: 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: apply (erule contrapos_np)
paulson@15077: apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077: apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077: apply (erule ex1E)
paulson@15229: apply (rule_tac a = "x - (pi/2)" in ex1I)
paulson@15077: apply (simp (no_asm) add: real_add_assoc)
paulson@15077: apply (rotate_tac 3)
paulson@15077: apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma reals_Archimedean4:
paulson@15077:      "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077: apply (auto dest!: reals_Archimedean3)
paulson@15077: apply (drule_tac x = x in spec, clarify) 
paulson@15077: apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077:  prefer 2 apply (erule LeastI) 
paulson@15077: apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077: apply (subgoal_tac "~ x < real nat * y")
paulson@15077:  prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077: done
paulson@15077: 
paulson@15077: (* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077:    now causes some unwanted re-arrangements of literals!   *)
paulson@15229: lemma cos_zero_lemma:
paulson@15229:      "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077:       \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077: apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15086: apply (subgoal_tac "0 \<le> x - real n * pi & 
paulson@15086:                     (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
paulson@15086: apply (auto simp add: compare_rls) 
paulson@15077:   prefer 3 apply (simp add: cos_diff) 
paulson@15077:  prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
paulson@15077: apply (simp add: cos_diff)
paulson@15077: apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077: apply (rule_tac [2] cos_total, safe)
paulson@15077: apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077: apply (drule_tac x = "pi/2" in spec)
paulson@15077: apply (simp add: cos_diff)
paulson@15229: apply (rule_tac x = "Suc (2 * n)" in exI)
paulson@15077: apply (simp add: real_of_nat_Suc left_distrib, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma sin_zero_lemma:
paulson@15229:      "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077:       \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077: apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077:  apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077:  apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085: apply (rule cos_zero_lemma)
paulson@15085: apply (simp_all add: add_increasing)  
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15229: lemma cos_zero_iff:
paulson@15229:      "(cos x = 0) =  
paulson@15077:       ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077:        (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077: apply (rule iffI)
paulson@15077: apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077: apply (drule cos_zero_lemma, assumption+)
paulson@15077: apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077: apply (force simp add: minus_equation_iff [of x]) 
paulson@15077: apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
nipkow@15539: apply (auto simp add: cos_add)
paulson@15077: done
paulson@15077: 
paulson@15077: (* ditto: but to a lesser extent *)
paulson@15229: lemma sin_zero_iff:
paulson@15229:      "(sin x = 0) =  
paulson@15077:       ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077:        (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077: apply (rule iffI)
paulson@15077: apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077: apply (drule sin_zero_lemma, assumption+)
paulson@15077: apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077: apply (force simp add: minus_equation_iff [of x]) 
nipkow@15539: apply (auto simp add: even_mult_two_ex)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Tangent*}
paulson@15077: 
paulson@15077: lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077: by (simp add: tan_def)
paulson@15077: 
paulson@15077: lemma tan_pi [simp]: "tan pi = 0"
paulson@15077: by (simp add: tan_def)
paulson@15077: 
paulson@15077: lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077: by (simp add: tan_def)
paulson@15077: 
paulson@15077: lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077: by (simp add: tan_def minus_mult_left)
paulson@15077: 
paulson@15077: lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077: by (simp add: tan_def)
paulson@15077: 
paulson@15077: lemma lemma_tan_add1: 
paulson@15077:       "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077:         ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15229: apply (simp add: tan_def divide_inverse)
paulson@15229: apply (auto simp del: inverse_mult_distrib 
paulson@15229:             simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077: apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15229: apply (auto simp del: inverse_mult_distrib 
paulson@15229:             simp add: mult_assoc left_diff_distrib cos_add)
paulson@15234: done  
paulson@15077: 
paulson@15077: lemma add_tan_eq: 
paulson@15077:       "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077:        ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15229: apply (simp add: tan_def)
paulson@15077: apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077: apply (auto simp add: mult_assoc left_distrib)
nipkow@15539: apply (simp add: sin_add)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma tan_add:
paulson@15229:      "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
paulson@15077:       ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
paulson@15077: apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
paulson@15077: apply (simp add: tan_def)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma tan_double:
paulson@15229:      "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077:       ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077: apply (insert tan_add [of x x]) 
paulson@15077: apply (simp add: mult_2 [symmetric])  
paulson@15077: apply (auto simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15229: by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 
paulson@15077: 
paulson@15077: lemma tan_less_zero: 
paulson@15077:   assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077: proof -
paulson@15077:   have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077:   thus ?thesis by simp
paulson@15077: qed
paulson@15077: 
paulson@15077: lemma lemma_DERIV_tan:
paulson@15077:      "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15079: apply (auto simp add: divide_inverse numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077: by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077: 
paulson@15077: lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077: apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15229: apply (simp add: divide_inverse [symmetric])
paulson@15077: apply (rule LIM_mult2)
paulson@15077: apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077: apply (rule_tac [2] LIM_inverse)
paulson@15077: apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077: apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077: apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077: apply (cut_tac LIM_cos_div_sin)
paulson@15077: apply (simp only: LIM_def)
paulson@15077: apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077: apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229: apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077: apply (simp (no_asm_simp))
paulson@15229: apply (drule_tac x = "(pi/2) - e" in spec)
paulson@15229: apply (auto simp add: tan_def)
paulson@15077: apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079: apply (auto simp add: divide_inverse)
paulson@15229: apply (rule real_mult_order) 
paulson@15229: apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
paulson@15229: apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
paulson@15077: apply (frule real_le_imp_less_or_eq, safe)
paulson@15077:  prefer 2 apply force
paulson@15077: apply (drule lemma_tan_total, safe)
paulson@15077: apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077: apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077: apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077: apply (auto dest: cos_gt_zero)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077: apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077: apply (drule tan_total_pos)
paulson@15077: apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077: apply (rule_tac [3] x = "-x" in exI)
paulson@15077: apply (auto intro!: exI)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077: apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077: apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077: apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077: apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077: apply (rule_tac [4] Rolle)
paulson@15077: apply (rule_tac [2] Rolle)
paulson@15077: apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077:             simp add: differentiable_def)
paulson@15077: txt{*Now, simulate TRYALL*}
paulson@15077: apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077: apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
paulson@15077: 	    simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym]) 
paulson@15077: done
paulson@15077: 
paulson@15229: lemma arcsin_pi:
paulson@15229:      "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077:       ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
paulson@15077: apply (drule sin_total, assumption)
paulson@15077: apply (erule ex1E)
paulson@15229: apply (simp add: arcsin_def)
paulson@15077: apply (rule someI2, blast) 
paulson@15077: apply (force intro: order_trans) 
paulson@15077: done
paulson@15077: 
paulson@15229: lemma arcsin:
paulson@15229:      "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077:       ==> -(pi/2) \<le> arcsin y &  
paulson@15077:            arcsin y \<le> pi/2 & sin(arcsin y) = y"
paulson@15077: apply (unfold arcsin_def)
paulson@15077: apply (drule sin_total, assumption)
paulson@15077: apply (fast intro: someI2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077: by (blast dest: arcsin)
paulson@15077:       
paulson@15077: lemma arcsin_bounded:
paulson@15077:      "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077: by (blast dest: arcsin)
paulson@15077: 
paulson@15077: lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077: by (blast dest: arcsin)
paulson@15077: 
paulson@15077: lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077: by (blast dest: arcsin)
paulson@15077: 
paulson@15077: lemma arcsin_lt_bounded:
paulson@15077:      "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077: apply (frule order_less_imp_le)
paulson@15077: apply (frule_tac y = y in order_less_imp_le)
paulson@15077: apply (frule arcsin_bounded)
paulson@15077: apply (safe, simp)
paulson@15077: apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077: apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077: apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077: apply (unfold arcsin_def)
paulson@15077: apply (rule some1_equality)
paulson@15077: apply (rule sin_total, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma arcos:
paulson@15229:      "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077:       ==> 0 \<le> arcos y & arcos y \<le> pi & cos(arcos y) = y"
paulson@15229: apply (simp add: arcos_def)
paulson@15077: apply (drule cos_total, assumption)
paulson@15077: apply (fast intro: someI2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_arcos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arcos y) = y"
paulson@15077: by (blast dest: arcos)
paulson@15077:       
paulson@15077: lemma arcos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y & arcos y \<le> pi"
paulson@15077: by (blast dest: arcos)
paulson@15077: 
paulson@15077: lemma arcos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y"
paulson@15077: by (blast dest: arcos)
paulson@15077: 
paulson@15077: lemma arcos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcos y \<le> pi"
paulson@15077: by (blast dest: arcos)
paulson@15077: 
paulson@15229: lemma arcos_lt_bounded:
paulson@15229:      "[| -1 < y; y < 1 |]  
paulson@15077:       ==> 0 < arcos y & arcos y < pi"
paulson@15077: apply (frule order_less_imp_le)
paulson@15077: apply (frule_tac y = y in order_less_imp_le)
paulson@15077: apply (frule arcos_bounded, auto)
paulson@15077: apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
paulson@15077: apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077: apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma arcos_cos: "[|0 \<le> x; x \<le> pi |] ==> arcos(cos x) = x"
paulson@15229: apply (simp add: arcos_def)
paulson@15077: apply (auto intro!: some1_equality cos_total)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma arcos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arcos(cos x) = -x"
paulson@15229: apply (simp add: arcos_def)
paulson@15077: apply (auto intro!: some1_equality cos_total)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma arctan [simp]:
paulson@15077:      "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
paulson@15077: apply (cut_tac y = y in tan_total)
paulson@15229: apply (simp add: arctan_def)
paulson@15077: apply (fast intro: someI2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma tan_arctan: "tan(arctan y) = y"
paulson@15077: by auto
paulson@15077: 
paulson@15077: lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077: by (auto simp only: arctan)
paulson@15077: 
paulson@15077: lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077: by auto
paulson@15077: 
paulson@15077: lemma arctan_ubound: "arctan y < pi/2"
paulson@15077: by (auto simp only: arctan)
paulson@15077: 
paulson@15077: lemma arctan_tan: 
paulson@15077:       "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077: apply (unfold arctan_def)
paulson@15077: apply (rule some1_equality)
paulson@15077: apply (rule tan_total, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma arctan_zero_zero [simp]: "arctan 0 = 0"
paulson@15077: by (insert arctan_tan [of 0], simp)
paulson@15077: 
paulson@15077: lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
paulson@15077: apply (auto simp add: cos_zero_iff)
paulson@15077: apply (case_tac "n")
paulson@15077: apply (case_tac [3] "n")
paulson@15077: apply (cut_tac [2] y = x in arctan_ubound)
paulson@15077: apply (cut_tac [4] y = x in arctan_lbound) 
paulson@15077: apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077: apply (rule power_inverse [THEN subst])
paulson@15077: apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
paulson@15077: apply (auto dest: realpow_not_zero 
paulson@15077:         simp add: power_mult_distrib left_distrib realpow_divide tan_def 
paulson@15077:                   mult_assoc power_inverse [symmetric] 
paulson@15077:         simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15085: text{*NEEDED??*}
paulson@15229: lemma [simp]:
paulson@15229:      "sin (x + 1 / 2 * real (Suc m) * pi) =  
paulson@15229:       cos (x + 1 / 2 * real  (m) * pi)"
paulson@15229: by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077: 
paulson@15085: text{*NEEDED??*}
paulson@15229: lemma [simp]:
paulson@15229:      "sin (x + real (Suc m) * pi / 2) =  
paulson@15229:       cos (x + real (m) * pi / 2)"
paulson@15229: by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
paulson@15077: 
paulson@15077: lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
paulson@15077: apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077: apply (simp (no_asm))
paulson@15077: done
paulson@15077: 
paulson@15383: lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
paulson@15383: proof -
paulson@15383:   have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
paulson@15383:     by (auto simp add: right_distrib sin_add left_distrib mult_ac)
paulson@15383:   thus ?thesis
paulson@15383:     by (simp add: real_of_nat_Suc left_distrib add_divide_distrib 
paulson@15383:                   mult_commute [of pi])
paulson@15383: qed
paulson@15077: 
paulson@15077: lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
paulson@15077: by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
paulson@15077: 
paulson@15077: lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
paulson@15077: apply (subgoal_tac "3/2 = (1+1 / 2::real)")
paulson@15077: apply (simp only: left_distrib) 
paulson@15077: apply (auto simp add: cos_add mult_ac)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
paulson@15077: by (auto simp add: mult_assoc)
paulson@15077: 
paulson@15077: lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
paulson@15077: apply (subgoal_tac "3/2 = (1+1 / 2::real)")
paulson@15077: apply (simp only: left_distrib) 
paulson@15077: apply (auto simp add: sin_add mult_ac)
paulson@15077: done
paulson@15077: 
paulson@15077: (*NEEDED??*)
paulson@15229: lemma [simp]:
paulson@15229:      "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
paulson@15077: apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: (*NEEDED??*)
paulson@15077: lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
paulson@15229: by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
paulson@15077: 
paulson@15077: lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
paulson@15229: by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
paulson@15077: 
paulson@15077: lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
paulson@15077: apply (rule lemma_DERIV_subst)
paulson@15077: apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
paulson@15077: apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077: apply (simp (no_asm))
paulson@15077: done
paulson@15077: 
paulson@15077: lemma isCont_cos [simp]: "isCont cos x"
paulson@15077: by (rule DERIV_cos [THEN DERIV_isCont])
paulson@15077: 
paulson@15077: lemma isCont_sin [simp]: "isCont sin x"
paulson@15077: by (rule DERIV_sin [THEN DERIV_isCont])
paulson@15077: 
paulson@15077: lemma isCont_exp [simp]: "isCont exp x"
paulson@15077: by (rule DERIV_exp [THEN DERIV_isCont])
paulson@15077: 
paulson@15081: lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
nipkow@15539: by (auto simp add: sin_zero_iff even_mult_two_ex)
paulson@15077: 
paulson@15077: lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
paulson@15077: apply auto
paulson@15077: apply (drule_tac f = ln in arg_cong, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
paulson@15077: by (cut_tac x = x in sin_cos_squared_add3, auto)
paulson@15077: 
paulson@15077: 
paulson@15229: lemma real_root_less_mono:
paulson@15229:      "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
paulson@15077: apply (frule order_le_less_trans, assumption)
paulson@15077: apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
paulson@15077: apply (rotate_tac 1, assumption)
paulson@15077: apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
paulson@15077: apply (rotate_tac 3, assumption)
paulson@15077: apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
paulson@15077: apply (frule_tac n = n in real_root_pos_pos_le)
paulson@15077: apply (frule_tac n = n in real_root_pos_pos)
paulson@15077: apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
paulson@15077: apply (assumption, assumption)
paulson@15077: apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
paulson@15077: apply auto
paulson@15229: apply (drule_tac f = "%x. x ^ (Suc n)" in arg_cong)
paulson@15077: apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_root_le_mono:
paulson@15229:      "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
paulson@15077: apply (drule_tac y = y in order_le_imp_less_or_eq)
paulson@15077: apply (auto dest: real_root_less_mono intro: order_less_imp_le)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_root_less_iff [simp]:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
paulson@15077: apply (auto intro: real_root_less_mono)
paulson@15077: apply (rule ccontr, drule linorder_not_less [THEN iffD1])
paulson@15077: apply (drule_tac x = y and n = n in real_root_le_mono, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_root_le_iff [simp]:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)"
paulson@15077: apply (auto intro: real_root_le_mono)
paulson@15077: apply (simp (no_asm) add: linorder_not_less [symmetric])
paulson@15077: apply auto
paulson@15077: apply (drule_tac x = y and n = n in real_root_less_mono, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_root_eq_iff [simp]:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
paulson@15077: apply (auto intro!: order_antisym)
paulson@15077: apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
paulson@15077: apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_root_pos_unique:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
paulson@15077: by (auto dest: real_root_pos2 simp del: realpow_Suc)
paulson@15077: 
paulson@15229: lemma real_root_mult:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] 
paulson@15077:       ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
paulson@15077: apply (rule real_root_pos_unique)
paulson@15234: apply (auto intro!: real_root_pos_pos_le 
paulson@15234:             simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 
paulson@15234:             simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma real_root_inverse:
paulson@15229:      "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
paulson@15077: apply (rule real_root_pos_unique)
paulson@15234: apply (auto intro: real_root_pos_pos_le 
paulson@15234:             simp add: power_inverse [symmetric] real_root_pow_pos2 
paulson@15234:             simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_root_divide: 
paulson@15077:      "[| 0 \<le> x; 0 \<le> y |]  
paulson@15077:       ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
paulson@15229: apply (simp add: divide_inverse)
paulson@15077: apply (auto simp add: real_root_mult real_root_inverse)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
paulson@15229: by (simp add: sqrt_def)
paulson@15077: 
paulson@15077: lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
paulson@15229: by (simp add: sqrt_def)
paulson@15229: 
paulson@15229: lemma real_sqrt_less_iff [simp]:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
paulson@15229: by (simp add: sqrt_def)
paulson@15229: 
paulson@15229: lemma real_sqrt_le_iff [simp]:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
paulson@15229: by (simp add: sqrt_def)
paulson@15229: 
paulson@15229: lemma real_sqrt_eq_iff [simp]:
paulson@15229:      "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
paulson@15229: by (simp add: sqrt_def)
paulson@15077: 
paulson@15077: lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
paulson@15077: apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077: apply (rule_tac [2] real_sqrt_less_mono)
paulson@15077: apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
paulson@15077: apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077: apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
paulson@15229: apply (simp add: divide_inverse)
paulson@15077: apply (case_tac "r=0")
nipkow@15539: apply (auto simp add: mult_ac)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}
paulson@15077: 
paulson@15228: lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
paulson@15228: proof (rule order_trans)
paulson@15228:   show "x \<le> sqrt(x*x)" by (simp add: abs_if) 
paulson@15228:   show "sqrt (x * x) \<le> sqrt (x * x + y * y)"
paulson@15228:     by (rule real_sqrt_le_mono, auto) 
paulson@15228: qed
paulson@15228: 
paulson@15228: lemma minus_le_real_sqrt_sumsq [simp]: "-x \<le> sqrt (x * x + y * y)"
paulson@15228: proof (rule order_trans)
paulson@15228:   show "-x \<le> sqrt(x*x)" by (simp add: abs_if) 
paulson@15228:   show "sqrt (x * x) \<le> sqrt (x * x + y * y)"
paulson@15228:     by (rule real_sqrt_le_mono, auto) 
paulson@15228: qed
paulson@15077: 
paulson@15077: lemma lemma_real_divide_sqrt_ge_minus_one:
paulson@15228:      "0 < x ==> -1 \<le> x/(sqrt (x * x + y * y))" 
paulson@15228: by (simp add: divide_const_simps linorder_not_le [symmetric])
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
paulson@15077: apply (rule real_sqrt_gt_zero)
paulson@15077: apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077: apply (auto simp add: zero_less_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
paulson@15077: apply (rule real_sqrt_gt_zero)
paulson@15077: apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077: apply (auto simp add: zero_less_mult_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_gt_zero3: "x \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077: apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077: apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_sum_squares_gt_zero3a: "y \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077: apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
paulson@15077: apply (auto simp add: real_add_commute)
paulson@15077: done
paulson@15077: 
paulson@15544: lemma real_sqrt_sum_squares_eq_cancel: "sqrt(x\<twosuperior> + y\<twosuperior>) = x ==> y = 0"
paulson@15077: by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, auto)
paulson@15077: 
paulson@15544: lemma real_sqrt_sum_squares_eq_cancel2: "sqrt(x\<twosuperior> + y\<twosuperior>) = y ==> x = 0"
paulson@15077: apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
paulson@15077: apply (simp add: real_add_commute)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077: by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)
paulson@15077: 
paulson@15077: lemma lemma_real_divide_sqrt_ge_minus_one2:
paulson@15077:      "x < 0 ==> -1 \<le> x/(sqrt (x * x + y * y))"
paulson@15229: apply (simp add: divide_const_simps) 
paulson@15229: apply (insert minus_le_real_sqrt_sumsq [of x y], arith)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077: by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)
paulson@15077: 
paulson@15228: lemma minus_sqrt_le: "- sqrt (x * x + y * y) \<le> x"
paulson@15228: by (insert minus_le_real_sqrt_sumsq [of x y], arith) 
paulson@15228: 
paulson@15228: lemma minus_sqrt_le2: "- sqrt (x * x + y * y) \<le> y"
paulson@15228: by (subst add_commute, simp add: minus_sqrt_le) 
paulson@15228: 
paulson@15228: lemma not_neg_sqrt_sumsq: "~ sqrt (x * x + y * y) < 0"
paulson@15228: by (simp add: linorder_not_less)
paulson@15077: 
paulson@15077: lemma cos_x_y_ge_minus_one: "-1 \<le> x / sqrt (x * x + y * y)"
paulson@15229: by (simp add: minus_sqrt_le not_neg_sqrt_sumsq divide_const_simps)
paulson@15077: 
paulson@15077: lemma cos_x_y_ge_minus_one1a [simp]: "-1 \<le> y / sqrt (x * x + y * y)"
paulson@15229: by (subst add_commute, simp add: cos_x_y_ge_minus_one)
paulson@15077: 
paulson@15228: lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) \<le> 1" 
paulson@15077: apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
paulson@15077: apply (rule lemma_real_divide_sqrt_le_one)
paulson@15077: apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) \<le> 1"
paulson@15077: apply (cut_tac x = y and y = x in cos_x_y_le_one)
paulson@15077: apply (simp add: real_add_commute)
paulson@15077: done
paulson@15077: 
paulson@15077: declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077: declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077: 
paulson@15077: declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077: declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077: 
paulson@15077: lemma cos_abs_x_y_ge_minus_one [simp]:
paulson@15077:      "-1 \<le> \<bar>x\<bar> / sqrt (x * x + y * y)"
paulson@15228: by (auto simp add: divide_const_simps abs_if linorder_not_le [symmetric]) 
paulson@15077: 
paulson@15077: lemma cos_abs_x_y_le_one [simp]: "\<bar>x\<bar> / sqrt (x * x + y * y) \<le> 1"
paulson@15228: apply (insert minus_le_real_sqrt_sumsq [of x y] le_real_sqrt_sumsq [of x y]) 
paulson@15228: apply (auto simp add: divide_const_simps abs_if linorder_neq_iff) 
paulson@15077: done
paulson@15077: 
paulson@15077: declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077: declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077: 
paulson@15077: lemma minus_pi_less_zero: "-pi < 0"
paulson@15228: by simp
paulson@15228: 
paulson@15077: declare minus_pi_less_zero [simp]
paulson@15077: declare minus_pi_less_zero [THEN order_less_imp_le, simp]
paulson@15077: 
paulson@15077: lemma arcos_ge_minus_pi: "[| -1 \<le> y; y \<le> 1 |] ==> -pi \<le> arcos y"
paulson@15077: apply (rule real_le_trans)
paulson@15077: apply (rule_tac [2] arcos_lbound, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077: 
paulson@15077: (* How tedious! *)
paulson@15077: lemma lemma_divide_rearrange:
paulson@15077:      "[| x + (y::real) \<noteq> 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
paulson@15077: apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
paulson@15077: apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
paulson@15077: prefer 2 apply assumption
paulson@15077: apply (rotate_tac [2] 2)
paulson@15077: apply (drule_tac [2] mult_assoc [THEN subst])
paulson@15077: apply (rotate_tac [2] 2)
paulson@15077: apply (frule_tac [2] left_inverse [THEN subst])
paulson@15077: prefer 2 apply assumption
paulson@15229: apply (erule_tac [2] V = "(1 - z) * (x + y) = x / (x + y) * (x + y)" in thin_rl)
paulson@15229: apply (erule_tac [2] V = "1 - z = x / (x + y)" in thin_rl)
paulson@15077: apply (auto simp add: mult_assoc)
paulson@15077: apply (auto simp add: right_distrib left_diff_distrib)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_cos_sin_eq:
paulson@15077:      "[| 0 < x * x + y * y;  
paulson@15077:          1 - (sin xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2 |] 
paulson@15077:       ==> (sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2"
paulson@15077: by (auto intro: lemma_divide_rearrange
paulson@15077:          simp add: realpow_divide power2_eq_square [symmetric])
paulson@15077: 
paulson@15077: 
paulson@15077: lemma lemma_sin_cos_eq:
paulson@15077:      "[| 0 < x * x + y * y;  
paulson@15077:          1 - (cos xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2 |]
paulson@15077:       ==> (cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2"
paulson@15077: apply (auto simp add: realpow_divide power2_eq_square [symmetric])
paulson@15085: apply (subst add_commute)
paulson@15085: apply (rule lemma_divide_rearrange, simp add: real_add_eq_0_iff)
paulson@15077: apply (simp add: add_commute)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma sin_x_y_disj:
paulson@15077:      "[| x \<noteq> 0;  
paulson@15077:          cos xa = x / sqrt (x * x + y * y)  
paulson@15077:       |] ==>  sin xa = y / sqrt (x * x + y * y) |  
paulson@15077:               sin xa = - y / sqrt (x * x + y * y)"
paulson@15077: apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077: apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077: apply (simp add: cos_squared_eq)
paulson@15077: apply (subgoal_tac "(sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2")
paulson@15077: apply (rule_tac [2] lemma_cos_sin_eq)
paulson@15077: apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_cos_not_eq_zero: "x \<noteq> 0 ==> x / sqrt (x * x + y * y) \<noteq> 0"
paulson@15229: apply (simp add: divide_inverse)
paulson@15077: 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: apply (auto simp add: power2_eq_square)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma cos_x_y_disj:
paulson@15229:      "[| x \<noteq> 0;  
paulson@15077:          sin xa = y / sqrt (x * x + y * y)  
paulson@15077:       |] ==>  cos xa = x / sqrt (x * x + y * y) |  
paulson@15077:               cos xa = - x / sqrt (x * x + y * y)"
paulson@15077: apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077: apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077: apply (simp add: sin_squared_eq del: realpow_Suc)
paulson@15077: apply (subgoal_tac "(cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2")
paulson@15077: apply (rule_tac [2] lemma_sin_cos_eq)
paulson@15077: apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
paulson@15229: apply (case_tac "x = 0", auto)
paulson@15077: apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
paulson@15079: apply (auto simp add: zero_less_mult_iff divide_inverse power2_eq_square)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma polar_ex1:
paulson@15229:      "[| x \<noteq> 0; 0 < y |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15229: apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
paulson@15077: apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
paulson@15077: apply auto
paulson@15077: apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077: apply (auto simp add: power2_eq_square)
paulson@15229: apply (simp add: arcos_def)
paulson@15077: apply (cut_tac x1 = x and y1 = y 
paulson@15077:        in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
paulson@15077: apply (rule someI2_ex, blast)
paulson@15229: 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: apply (frule sin_x_y_disj, blast)
paulson@15077: apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077: apply (auto simp add: power2_eq_square)
paulson@15077: apply (drule sin_ge_zero, assumption)
paulson@15077: apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
paulson@15085: by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)
paulson@15077: 
paulson@15229: lemma polar_ex2:
paulson@15229:      "[| x \<noteq> 0; y < 0 |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077: apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
paulson@15228: apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
paulson@15228: apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI) 
paulson@15085: apply (auto dest: real_sum_squares_cancel2a 
paulson@15085:             simp add: power2_eq_square real_0_le_add_iff real_add_eq_0_iff)
paulson@15077: apply (unfold arcsin_def)
paulson@15077: apply (cut_tac x1 = x and y1 = y 
paulson@15077:        in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
paulson@15077: apply (rule someI2_ex, blast)
paulson@15228: apply (erule_tac V = "EX! v. ?P v" in thin_rl)
paulson@15085: apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast)
paulson@15085: apply (auto simp add: real_0_le_add_iff real_add_eq_0_iff)
paulson@15077: apply (drule cos_ge_zero, force)
paulson@15077: apply (drule_tac x = y in real_sqrt_divide_less_zero)
paulson@15085: apply (auto simp add: add_commute)
paulson@15077: apply (insert polar_ex1 [of x "-y"], simp, clarify) 
paulson@15077: apply (rule_tac x = r in exI)
paulson@15077: apply (rule_tac x = "-a" in exI, simp) 
paulson@15077: done
paulson@15077: 
paulson@15077: lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077: apply (case_tac "x = 0", auto)
paulson@15077: apply (rule_tac x = y in exI)
paulson@15077: apply (rule_tac x = "pi/2" in exI, auto)
paulson@15077: apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
paulson@15077: apply (rule_tac [2] x = x in exI)
paulson@15077: apply (rule_tac [2] x = 0 in exI, auto)
paulson@15077: apply (blast intro: polar_ex1 polar_ex2)+
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077: apply (rule_tac n = 1 in realpow_increasing)
avigad@16775: apply (auto simp add: numeral_2_eq_2 [symmetric] power2_abs)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077: apply (rule real_add_commute [THEN subst])
paulson@15077: apply (rule real_sqrt_ge_abs1)
paulson@15077: done
paulson@15077: declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]
paulson@15077: 
paulson@15077: lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
paulson@15077: by (auto intro: real_sqrt_gt_zero)
paulson@15077: 
paulson@15077: lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
paulson@15077: by (auto intro: real_sqrt_ge_zero)
paulson@15077: 
paulson@15077: lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
paulson@15077: apply (rule order_less_le_trans [of _ "7/5"], simp) 
paulson@15077: apply (rule_tac n = 1 in realpow_increasing)
paulson@15077:   prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
nipkow@15539: apply (simp_all add: numeral_2_eq_2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
paulson@15234: by (simp add: divide_less_eq mult_compare_simps) 
paulson@15077: 
paulson@15077: lemma four_x_squared: 
paulson@15077:   fixes x::real
paulson@15077:   shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>"
paulson@15077: by (simp add: power2_eq_square)
paulson@15077: 
paulson@15077: 
paulson@15077: text{*Needed for the infinitely close relation over the nonstandard
paulson@15077:     complex numbers*}
paulson@15077: lemma lemma_sqrt_hcomplex_capprox:
paulson@15077:      "[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
paulson@15077: apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
paulson@15077: apply (erule_tac [2] lemma_real_divide_sqrt_less)
paulson@15077: apply (rule_tac n = 1 in realpow_increasing)
paulson@15077: apply (auto simp add: real_0_le_divide_iff realpow_divide numeral_2_eq_2 [symmetric] 
paulson@15077:         simp del: realpow_Suc)
paulson@15077: apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst])
paulson@15077: apply (rule add_mono)
paulson@15077: apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono)
paulson@15077: done
paulson@15077: 
avigad@16775: declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]
paulson@15077: 
paulson@15077: 
paulson@15077: subsection{*A Few Theorems Involving Ln, Derivatives, etc.*}
paulson@15077: 
paulson@15077: lemma lemma_DERIV_ln:
paulson@15077:      "DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l"
paulson@15077: by (erule DERIV_fun_exp)
paulson@15077: 
paulson@15077: lemma STAR_exp_ln: "0 < z ==> ( *f* (%x. exp (ln x))) z = z"
huffman@17318: apply (cases z)
huffman@17318: apply (auto simp add: starfun star_n_zero_num star_n_less star_n_eq_iff)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma hypreal_add_Infinitesimal_gt_zero:
paulson@15229:      "[|e : Infinitesimal; 0 < x |] ==> 0 < hypreal_of_real x + e"
paulson@15077: apply (rule_tac c1 = "-e" in add_less_cancel_right [THEN iffD1])
paulson@15077: apply (auto intro: Infinitesimal_less_SReal)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma NSDERIV_exp_ln_one: "0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1"
paulson@15229: apply (simp add: nsderiv_def NSLIM_def, auto)
paulson@15077: apply (rule ccontr)
paulson@15077: apply (subgoal_tac "0 < hypreal_of_real z + h")
paulson@15077: apply (drule STAR_exp_ln)
paulson@15077: apply (rule_tac [2] hypreal_add_Infinitesimal_gt_zero)
paulson@15077: apply (subgoal_tac "h/h = 1")
paulson@15077: apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma DERIV_exp_ln_one: "0 < z ==> DERIV (%x. exp (ln x)) z :> 1"
paulson@15077: by (auto intro: NSDERIV_exp_ln_one simp add: NSDERIV_DERIV_iff [symmetric])
paulson@15077: 
paulson@15229: lemma lemma_DERIV_ln2:
paulson@15229:      "[| 0 < z; DERIV ln z :> l |] ==>  exp (ln z) * l = 1"
paulson@15077: apply (rule DERIV_unique)
paulson@15077: apply (rule lemma_DERIV_ln)
paulson@15077: apply (rule_tac [2] DERIV_exp_ln_one, auto)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma lemma_DERIV_ln3:
paulson@15229:      "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/(exp (ln z))"
paulson@15229: apply (rule_tac c1 = "exp (ln z)" in real_mult_left_cancel [THEN iffD1])
paulson@15077: apply (auto intro: lemma_DERIV_ln2)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma lemma_DERIV_ln4: "[| 0 < z; DERIV ln z :> l |] ==>  l = 1/z"
paulson@15077: apply (rule_tac t = z in exp_ln_iff [THEN iffD2, THEN subst])
paulson@15077: apply (auto intro: lemma_DERIV_ln3)
paulson@15077: done
paulson@15077: 
paulson@15077: (* need to rename second isCont_inverse *)
paulson@15077: 
paulson@15229: lemma isCont_inv_fun:
paulson@15229:      "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077:          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077:       ==> isCont g (f x)"
paulson@15077: apply (simp (no_asm) add: isCont_iff LIM_def)
paulson@15077: apply safe
paulson@15077: apply (drule_tac ?d1.0 = r in real_lbound_gt_zero)
paulson@15077: apply (assumption, safe)
paulson@15077: apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> (g (f z) = z) ")
paulson@15077: prefer 2 apply force
paulson@15077: apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> isCont f z")
paulson@15077: prefer 2 apply force
paulson@15077: apply (drule_tac d = e in isCont_inj_range)
paulson@15077: prefer 2 apply (assumption, assumption, safe)
paulson@15077: apply (rule_tac x = ea in exI, auto)
paulson@15085: 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: apply auto
webertj@20217: apply (drule sym, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: lemma isCont_inv_fun_inv:
paulson@15077:      "[| 0 < d;  
paulson@15077:          \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;  
paulson@15077:          \<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]  
paulson@15077:        ==> \<exists>e. 0 < e &  
paulson@15081:              (\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
paulson@15077: apply (drule isCont_inj_range)
paulson@15077: prefer 2 apply (assumption, assumption, auto)
paulson@15077: apply (rule_tac x = e in exI, auto)
paulson@15077: apply (rotate_tac 2)
paulson@15077: apply (drule_tac x = y in spec, auto)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
paulson@15229: lemma LIM_fun_gt_zero:
paulson@15229:      "[| f -- c --> l; 0 < l |]  
paulson@15077:          ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
paulson@15077: apply (auto simp add: LIM_def)
paulson@15077: apply (drule_tac x = "l/2" in spec, safe, force)
paulson@15077: apply (rule_tac x = s in exI)
paulson@15077: apply (auto simp only: abs_interval_iff)
paulson@15077: done
paulson@15077: 
paulson@15229: lemma LIM_fun_less_zero:
paulson@15229:      "[| f -- c --> l; l < 0 |]  
paulson@15229:       ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
paulson@15077: apply (auto simp add: LIM_def)
paulson@15077: apply (drule_tac x = "-l/2" in spec, safe, force)
paulson@15077: apply (rule_tac x = s in exI)
paulson@15077: apply (auto simp only: abs_interval_iff)
paulson@15077: done
paulson@15077: 
paulson@15077: 
paulson@15077: lemma LIM_fun_not_zero:
paulson@15077:      "[| f -- c --> l; l \<noteq> 0 |] 
paulson@15077:       ==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
paulson@15077: apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
paulson@15077: apply (drule LIM_fun_less_zero)
paulson@15241: apply (drule_tac [3] LIM_fun_gt_zero)
paulson@15241: apply force+
paulson@15077: done
webertj@20432:   
paulson@12196: end