src/HOL/Hyperreal/Transcendental.thy
author nipkow
Mon Feb 21 15:04:10 2005 +0100 (2005-02-21)
changeset 15539 333a88244569
parent 15536 3ce1cb7a24f0
child 15542 ee6cd48cf840
permissions -rw-r--r--
comprehensive cleanup, replacing sumr by setsum
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(*  Title       : Transcendental.thy
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    Author      : Jacques D. Fleuriot
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    Copyright   : 1998,1999 University of Cambridge
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                  1999,2001 University of Edinburgh
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    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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imports NthRoot Fact HSeries EvenOdd Lim
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begin
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constdefs
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    root :: "[nat,real] => real"
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    "root n x == (@u. ((0::real) < x --> 0 < u) & (u ^ n = x))"
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    sqrt :: "real => real"
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    "sqrt x == root 2 x"
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    exp :: "real => real"
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    "exp x == suminf(%n. inverse(real (fact n)) * (x ^ n))"
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    sin :: "real => real"
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    "sin x == suminf(%n. (if even(n) then 0 else
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             ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
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    diffs :: "(nat => real) => nat => real"
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    "diffs c == (%n. real (Suc n) * c(Suc n))"
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    cos :: "real => real"
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    "cos x == suminf(%n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n)) 
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                          else 0) * x ^ n)"
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    ln :: "real => real"
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    "ln x == (@u. exp u = x)"
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    pi :: "real"
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    "pi == 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
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    tan :: "real => real"
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    "tan x == (sin x)/(cos x)"
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    arcsin :: "real => real"
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    "arcsin y == (@x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
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    arcos :: "real => real"
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    "arcos y == (@x. 0 \<le> x & x \<le> pi & cos x = y)"
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    arctan :: "real => real"
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    "arctan y == (@x. -(pi/2) < x & x < pi/2 & tan x = y)"
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lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
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apply (simp add: root_def)
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apply (safe intro!: some_equality power_0_Suc elim!: realpow_zero_zero)
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done
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lemma real_root_pow_pos: 
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     "0 < x ==> (root(Suc n) x) ^ (Suc n) = x"
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apply (simp add: root_def)
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apply (drule_tac n = n in realpow_pos_nth2)
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apply (auto intro: someI2)
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done
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lemma real_root_pow_pos2: "0 \<le> x ==> (root(Suc n) x) ^ (Suc n) = x"
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by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)
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lemma real_root_pos: 
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     "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
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apply (simp add: root_def)
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apply (rule some_equality)
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apply (frule_tac [2] n = n in zero_less_power)
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apply (auto simp add: zero_less_mult_iff)
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apply (rule_tac x = u and y = x in linorder_cases)
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apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
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apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
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apply (auto)
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done
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lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
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by (auto dest!: real_le_imp_less_or_eq real_root_pos)
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lemma real_root_pos_pos: 
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     "0 < x ==> 0 \<le> root(Suc n) x"
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apply (simp add: root_def)
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apply (drule_tac n = n in realpow_pos_nth2)
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apply (safe, rule someI2)
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apply (auto intro!: order_less_imp_le dest: zero_less_power 
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            simp add: zero_less_mult_iff)
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done
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lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
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by (auto dest!: real_le_imp_less_or_eq dest: real_root_pos_pos)
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lemma real_root_one [simp]: "root (Suc n) 1 = 1"
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apply (simp add: root_def)
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apply (rule some_equality, auto)
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apply (rule ccontr)
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apply (rule_tac x = u and y = 1 in linorder_cases)
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apply (drule_tac n = n in realpow_Suc_less_one)
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apply (drule_tac [4] n = n in power_gt1_lemma)
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apply (auto)
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done
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subsection{*Square Root*}
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text{*needed because 2 is a binary numeral!*}
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lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
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by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
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         add: nat_numeral_0_eq_0 [symmetric])
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lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
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by (simp add: sqrt_def)
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lemma real_sqrt_one [simp]: "sqrt 1 = 1"
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by (simp add: sqrt_def)
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lemma real_sqrt_pow2_iff [iff]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
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apply (simp add: sqrt_def)
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apply (rule iffI) 
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 apply (cut_tac r = "root 2 x" in realpow_two_le)
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 apply (simp add: numeral_2_eq_2)
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apply (subst numeral_2_eq_2)
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apply (erule real_root_pow_pos2)
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done
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lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
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by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])
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lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
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by (simp)
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lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
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by (rule real_sqrt_pow2_iff [THEN iffD2], arith)
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lemma real_pow_sqrt_eq_sqrt_pow: 
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      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
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apply (simp add: sqrt_def)
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apply (simp only: numeral_2_eq_2 real_root_pow_pos2 real_root_pos2)
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done
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lemma real_pow_sqrt_eq_sqrt_abs_pow2:
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     "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
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by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])
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lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
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apply (rule real_sqrt_abs_abs [THEN subst])
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apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
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apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
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apply (assumption, arith)
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done
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lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
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apply auto
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apply (cut_tac x = x and y = 0 in linorder_less_linear)
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apply (simp add: zero_less_mult_iff)
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done
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lemma real_mult_self_eq_zero_iff [simp]: "(r * r = 0) = (r = (0::real))"
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by auto
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lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
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apply (simp add: sqrt_def root_def)
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apply (drule realpow_pos_nth2 [where n=1], safe)
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apply (rule someI2, auto)
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done
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lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
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by (auto intro: real_sqrt_gt_zero simp add: order_le_less)
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lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
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by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 
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(*we need to prove something like this:
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lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
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apply (case_tac n, simp) 
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apply (simp add: root_def) 
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apply (rule someI2 [of _ r], safe)
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apply (auto simp del: realpow_Suc dest: power_inject_base)
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*)
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lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
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apply (unfold sqrt_def root_def) 
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apply (rule someI2 [of _ r], auto) 
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apply (auto simp add: numeral_2_eq_2 simp del: realpow_Suc 
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            dest: power_inject_base) 
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done
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lemma real_sqrt_mult_distrib: 
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     "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
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apply (rule sqrt_eqI)
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apply (simp add: power_mult_distrib)  
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apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
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done
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lemma real_sqrt_mult_distrib2:
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     "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
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by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)
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lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
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by (auto intro!: real_sqrt_ge_zero)
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lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
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     "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
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by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)
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lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
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     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
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by (auto simp add: zero_le_mult_iff simp del: realpow_Suc)
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lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
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apply (rule abs_realpow_two [THEN subst])
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apply (rule real_sqrt_abs_abs [THEN subst])
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apply (subst real_pow_sqrt_eq_sqrt_pow)
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apply (auto simp add: numeral_2_eq_2)
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done
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lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
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apply (rule realpow_two [THEN subst])
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apply (subst numeral_2_eq_2 [symmetric])
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apply (rule real_sqrt_abs)
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done
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lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"
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by simp
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lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 0"
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apply (frule real_sqrt_pow2_gt_zero)
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apply (auto simp add: numeral_2_eq_2)
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done
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lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
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by (cut_tac n1 = 2 and a1 = "sqrt x" in power_inverse [symmetric], auto)
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lemma real_sqrt_eq_zero_cancel: "[| 0 \<le> x; sqrt(x) = 0|] ==> x = 0"
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apply (drule real_le_imp_less_or_eq)
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apply (auto dest: real_sqrt_not_eq_zero)
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done
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lemma real_sqrt_eq_zero_cancel_iff [simp]: "0 \<le> x ==> ((sqrt x = 0) = (x=0))"
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by (auto simp add: real_sqrt_eq_zero_cancel)
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lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"
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apply (subgoal_tac "x \<le> 0 | 0 \<le> x", safe)
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apply (rule real_le_trans)
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apply (auto simp del: realpow_Suc)
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apply (rule_tac n = 1 in realpow_increasing)
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apply (auto simp add: numeral_2_eq_2 [symmetric] simp del: realpow_Suc)
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done
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lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(z\<twosuperior> + y\<twosuperior>)"
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apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
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done
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lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
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apply (rule_tac n = 1 in realpow_increasing)
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apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp 
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            del: realpow_Suc)
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done
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subsection{*Exponential Function*}
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lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
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apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
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apply (cut_tac x = r in reals_Archimedean3, auto)
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apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
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apply (rule_tac N = n and c = r in ratio_test)
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apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
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apply (rule mult_right_mono)
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apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
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apply (subst fact_Suc)
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apply (subst real_of_nat_mult)
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apply (auto)
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apply (auto simp add: mult_assoc [symmetric] positive_imp_inverse_positive)
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apply (rule order_less_imp_le)
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apply (rule_tac z1 = "real (Suc na)" in real_mult_less_iff1 [THEN iffD1])
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apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc)
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apply (erule order_less_trans)
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apply (auto simp add: mult_less_cancel_left mult_ac)
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done
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lemma summable_sin: 
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     "summable (%n.  
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           (if even n then 0  
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           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
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                x ^ n)"
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apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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apply (rule_tac [2] summable_exp)
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apply (rule_tac x = 0 in exI)
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apply (auto simp add: divide_inverse power_abs [symmetric] zero_le_mult_iff)
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done
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lemma summable_cos: 
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      "summable (%n.  
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           (if even n then  
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           (- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
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apply (rule_tac g = "(%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n)" in summable_comparison_test)
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   302
apply (rule_tac [2] summable_exp)
paulson@15077
   303
apply (rule_tac x = 0 in exI)
paulson@15229
   304
apply (auto simp add: divide_inverse power_abs [symmetric] zero_le_mult_iff)
paulson@15077
   305
done
paulson@15077
   306
paulson@15229
   307
lemma lemma_STAR_sin [simp]:
paulson@15229
   308
     "(if even n then 0  
paulson@15077
   309
       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
paulson@15251
   310
by (induct "n", auto)
paulson@15229
   311
paulson@15229
   312
lemma lemma_STAR_cos [simp]:
paulson@15229
   313
     "0 < n -->  
paulson@15077
   314
      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
paulson@15251
   315
by (induct "n", auto)
paulson@15229
   316
paulson@15229
   317
lemma lemma_STAR_cos1 [simp]:
paulson@15229
   318
     "0 < n -->  
paulson@15077
   319
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
paulson@15251
   320
by (induct "n", auto)
paulson@15229
   321
paulson@15229
   322
lemma lemma_STAR_cos2 [simp]:
nipkow@15539
   323
  "(\<Sum>n=1..<n. if even n then (- 1) ^ (n div 2)/(real (fact n)) *  0 ^ n 
nipkow@15539
   324
                         else 0) = 0"
paulson@15251
   325
apply (induct "n")
paulson@15077
   326
apply (case_tac [2] "n", auto)
paulson@15077
   327
done
paulson@15077
   328
paulson@15077
   329
lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
paulson@15229
   330
apply (simp add: exp_def)
paulson@15077
   331
apply (rule summable_exp [THEN summable_sums])
paulson@15077
   332
done
paulson@15077
   333
paulson@15077
   334
lemma sin_converges: 
paulson@15077
   335
      "(%n. (if even n then 0  
paulson@15077
   336
            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
paulson@15077
   337
                 x ^ n) sums sin(x)"
paulson@15229
   338
apply (simp add: sin_def)
paulson@15077
   339
apply (rule summable_sin [THEN summable_sums])
paulson@15077
   340
done
paulson@15077
   341
paulson@15077
   342
lemma cos_converges: 
paulson@15077
   343
      "(%n. (if even n then  
paulson@15077
   344
           (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   345
           else 0) * x ^ n) sums cos(x)"
paulson@15229
   346
apply (simp add: cos_def)
paulson@15077
   347
apply (rule summable_cos [THEN summable_sums])
paulson@15077
   348
done
paulson@15077
   349
paulson@15229
   350
lemma lemma_realpow_diff [rule_format (no_asm)]:
paulson@15229
   351
     "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
paulson@15251
   352
apply (induct "n", auto)
paulson@15077
   353
apply (subgoal_tac "p = Suc n")
paulson@15077
   354
apply (simp (no_asm_simp), auto)
paulson@15077
   355
apply (drule sym)
paulson@15077
   356
apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
paulson@15077
   357
       del: realpow_Suc)
paulson@15077
   358
done
paulson@15077
   359
paulson@15077
   360
paulson@15077
   361
subsection{*Properties of Power Series*}
paulson@15077
   362
paulson@15077
   363
lemma lemma_realpow_diff_sumr:
nipkow@15539
   364
     "(\<Sum>p=0..<Suc n. (x ^ p) * y ^ ((Suc n) - p)) =  
nipkow@15539
   365
      y * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))::real)"
nipkow@15539
   366
apply (auto simp add: setsum_mult simp del: setsum_Suc)
nipkow@15539
   367
apply (rule setsum_cong[OF refl])
paulson@15077
   368
apply (subst lemma_realpow_diff)
paulson@15077
   369
apply (auto simp add: mult_ac)
paulson@15077
   370
done
paulson@15077
   371
paulson@15229
   372
lemma lemma_realpow_diff_sumr2:
paulson@15229
   373
     "x ^ (Suc n) - y ^ (Suc n) =  
nipkow@15539
   374
      (x - y) * (\<Sum>p=0..<Suc n. (x ^ p) * (y ^(n - p))::real)"
paulson@15251
   375
apply (induct "n", simp)
nipkow@15539
   376
apply (auto simp del: setsum_Suc)
nipkow@15539
   377
apply (subst setsum_Suc)
paulson@15077
   378
apply (drule sym)
nipkow@15539
   379
apply (auto simp add: lemma_realpow_diff_sumr right_distrib diff_minus mult_ac simp del: setsum_Suc)
paulson@15077
   380
done
paulson@15077
   381
paulson@15229
   382
lemma lemma_realpow_rev_sumr:
nipkow@15539
   383
     "(\<Sum>p=0..<Suc n. (x ^ p) * (y ^ (n - p))) =  
nipkow@15539
   384
      (\<Sum>p=0..<Suc n. (x ^ (n - p)) * (y ^ p)::real)"
paulson@15077
   385
apply (case_tac "x = y")
nipkow@15539
   386
apply (auto simp add: mult_commute power_add [symmetric] simp del: setsum_Suc)
paulson@15077
   387
apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
paulson@15077
   388
apply (rule_tac [2] minus_minus [THEN subst], simp)
paulson@15077
   389
apply (subst minus_mult_left)
nipkow@15539
   390
apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: setsum_Suc)
paulson@15077
   391
done
paulson@15077
   392
paulson@15077
   393
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
paulson@15077
   394
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
paulson@15077
   395
paulson@15077
   396
lemma powser_insidea:
paulson@15077
   397
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
paulson@15081
   398
      ==> summable (%n. \<bar>f(n)\<bar> * (z ^ n))"
paulson@15077
   399
apply (drule summable_LIMSEQ_zero)
paulson@15077
   400
apply (drule convergentI)
paulson@15077
   401
apply (simp add: Cauchy_convergent_iff [symmetric])
paulson@15077
   402
apply (drule Cauchy_Bseq)
paulson@15077
   403
apply (simp add: Bseq_def, safe)
paulson@15081
   404
apply (rule_tac g = "%n. K * \<bar>z ^ n\<bar> * inverse (\<bar>x ^ n\<bar>)" in summable_comparison_test)
paulson@15077
   405
apply (rule_tac x = 0 in exI, safe)
paulson@15081
   406
apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
paulson@15081
   407
apply (rule_tac c="\<bar>x ^ n\<bar>" in mult_right_le_imp_le) 
paulson@15077
   408
apply (auto simp add: mult_assoc power_abs)
paulson@15077
   409
 prefer 2
paulson@15077
   410
 apply (drule_tac x = 0 in spec, force)
nipkow@15539
   411
apply (auto simp add: power_abs mult_ac)
paulson@15077
   412
apply (rule_tac a2 = "z ^ n" 
paulson@15077
   413
       in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
paulson@15077
   414
apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
paulson@15229
   415
apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>)))" in exI)
paulson@15077
   416
apply (auto intro!: sums_mult simp add: mult_assoc)
paulson@15081
   417
apply (subgoal_tac "\<bar>z ^ n\<bar> * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
paulson@15077
   418
apply (auto simp add: power_abs [symmetric])
paulson@15077
   419
apply (subgoal_tac "x \<noteq> 0")
paulson@15077
   420
apply (subgoal_tac [3] "x \<noteq> 0")
paulson@15077
   421
apply (auto simp del: abs_inverse abs_mult simp add: abs_inverse [symmetric] realpow_not_zero abs_mult [symmetric] power_inverse power_mult_distrib [symmetric])
nipkow@15539
   422
apply (auto intro!: geometric_sums  simp add: power_abs inverse_eq_divide)
paulson@15077
   423
done
paulson@15077
   424
paulson@15229
   425
lemma powser_inside:
paulson@15229
   426
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
paulson@15077
   427
      ==> summable (%n. f(n) * (z ^ n))"
paulson@15077
   428
apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
paulson@15077
   429
apply (auto intro: summable_rabs_cancel simp add: power_abs [symmetric])
paulson@15077
   430
done
paulson@15077
   431
paulson@15077
   432
paulson@15077
   433
subsection{*Differentiation of Power Series*}
paulson@15077
   434
paulson@15077
   435
text{*Lemma about distributing negation over it*}
paulson@15077
   436
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
paulson@15077
   437
by (simp add: diffs_def)
paulson@15077
   438
paulson@15077
   439
text{*Show that we can shift the terms down one*}
paulson@15077
   440
lemma lemma_diffs:
nipkow@15539
   441
     "(\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) =  
nipkow@15539
   442
      (\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) +  
paulson@15077
   443
      (real n * c(n) * x ^ (n - Suc 0))"
paulson@15251
   444
apply (induct "n")
paulson@15077
   445
apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
paulson@15077
   446
done
paulson@15077
   447
paulson@15229
   448
lemma lemma_diffs2:
nipkow@15539
   449
     "(\<Sum>n=0..<n. real n * c(n) * (x ^ (n - Suc 0))) =  
nipkow@15539
   450
      (\<Sum>n=0..<n. (diffs c)(n) * (x ^ n)) -  
paulson@15077
   451
      (real n * c(n) * x ^ (n - Suc 0))"
paulson@15077
   452
by (auto simp add: lemma_diffs)
paulson@15077
   453
paulson@15077
   454
paulson@15229
   455
lemma diffs_equiv:
paulson@15229
   456
     "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
paulson@15077
   457
      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
paulson@15077
   458
         (suminf(%n. (diffs c)(n) * (x ^ n)))"
paulson@15077
   459
apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
paulson@15077
   460
apply (rule_tac [2] LIMSEQ_imp_Suc)
paulson@15077
   461
apply (drule summable_sums) 
paulson@15077
   462
apply (auto simp add: sums_def)
paulson@15077
   463
apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
paulson@15077
   464
apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
paulson@15077
   465
apply (simp add: diffs_def summable_LIMSEQ_zero)
paulson@15077
   466
done
paulson@15077
   467
paulson@15077
   468
paulson@15077
   469
subsection{*Term-by-Term Differentiability of Power Series*}
paulson@15077
   470
paulson@15077
   471
lemma lemma_termdiff1:
nipkow@15539
   472
  "(\<Sum>p=0..<m. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
nipkow@15539
   473
   (\<Sum>p=0..<m. (z ^ p) * (((z + h) ^ (m - p)) - (z ^ (m - p)))::real)"
nipkow@15539
   474
apply (rule setsum_cong[OF refl])
paulson@15229
   475
apply (auto simp add: right_distrib diff_minus power_add [symmetric] mult_ac)
paulson@15077
   476
done
paulson@15077
   477
paulson@15077
   478
lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
paulson@15077
   479
by (simp add: less_iff_Suc_add)
paulson@15077
   480
paulson@15077
   481
lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
paulson@15077
   482
by arith
paulson@15077
   483
paulson@15077
   484
paulson@15229
   485
lemma lemma_termdiff2:
nipkow@15539
   486
  "h \<noteq> 0 ==>
nipkow@15539
   487
   (((z + h) ^ n) - (z ^ n)) * inverse h - real n * (z ^ (n - Suc 0)) =
nipkow@15539
   488
   h * (\<Sum>p=0..< n - Suc 0. (z ^ p) *
nipkow@15539
   489
       (\<Sum>q=0..< (n - Suc 0) - p. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))"
paulson@15077
   490
apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
paulson@15077
   491
apply (simp add: right_diff_distrib mult_ac)
paulson@15077
   492
apply (simp add: mult_assoc [symmetric])
paulson@15077
   493
apply (case_tac "n")
paulson@15077
   494
apply (auto simp add: lemma_realpow_diff_sumr2 
paulson@15077
   495
                      right_diff_distrib [symmetric] mult_assoc
nipkow@15539
   496
            simp del: realpow_Suc setsum_Suc)
nipkow@15539
   497
apply (auto simp add: lemma_realpow_rev_sumr simp del: setsum_Suc)
paulson@15077
   498
apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
nipkow@15536
   499
                sumdiff lemma_termdiff1 setsum_mult)
nipkow@15539
   500
apply (auto intro!: setsum_cong[OF refl] simp add: diff_minus real_add_assoc)
nipkow@15539
   501
apply (simp add: diff_minus [symmetric] less_iff_Suc_add)
nipkow@15536
   502
apply (auto simp add: setsum_mult lemma_realpow_diff_sumr2 mult_ac simp
nipkow@15539
   503
                 del: setsum_Suc realpow_Suc)
paulson@15077
   504
done
paulson@15077
   505
nipkow@15539
   506
(* FIXME move *)
nipkow@15539
   507
lemma sumr_bound2 [rule_format (no_asm)]:
nipkow@15539
   508
     "(\<forall>p. 0 \<le> p & p < n --> (f(p) \<le> K))  
nipkow@15539
   509
      --> setsum f {0..<n::nat} \<le> (real n * K)"
nipkow@15539
   510
using setsum_bounded[where A = "{0..<n}"]
nipkow@15539
   511
by (auto simp:real_of_nat_def)
nipkow@15539
   512
paulson@15229
   513
lemma lemma_termdiff3:
paulson@15229
   514
     "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<le> K |]  
paulson@15077
   515
      ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
paulson@15077
   516
          \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
paulson@15077
   517
apply (subst lemma_termdiff2, assumption)
nipkow@15539
   518
apply (simp add: mult_commute) 
paulson@15077
   519
apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
nipkow@15536
   520
apply (rule setsum_abs [THEN real_le_trans])
paulson@15077
   521
apply (simp add: mult_assoc [symmetric])
paulson@15077
   522
apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
nipkow@15539
   523
apply (auto intro!: sumr_bound2)
paulson@15077
   524
apply (case_tac "n", auto)
paulson@15077
   525
apply (drule less_add_one)
paulson@15077
   526
(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
paulson@15077
   527
apply clarify 
paulson@15077
   528
apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
paulson@15077
   529
                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
nipkow@15539
   530
apply (simp (no_asm_simp) add: power_add del: setsum_Suc)
nipkow@15539
   531
apply (auto intro!: mult_mono simp del: setsum_Suc)
nipkow@15539
   532
apply (auto intro!: power_mono simp add: power_abs simp del: setsum_Suc)
paulson@15229
   533
apply (rule_tac j = "real (Suc d) * (K ^ d)" in real_le_trans)
paulson@15077
   534
apply (subgoal_tac [2] "0 \<le> K")
paulson@15077
   535
apply (drule_tac [2] n = d in zero_le_power)
nipkow@15539
   536
apply (auto simp del: setsum_Suc)
nipkow@15536
   537
apply (rule setsum_abs [THEN real_le_trans])
nipkow@15539
   538
apply (rule sumr_bound2, auto dest!: less_add_one intro!: mult_mono simp add: power_add)
paulson@15077
   539
apply (auto intro!: power_mono zero_le_power simp add: power_abs, arith+)
paulson@15077
   540
done
paulson@15077
   541
paulson@15077
   542
lemma lemma_termdiff4: 
paulson@15077
   543
  "[| 0 < k;  
paulson@15081
   544
      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<le> K * \<bar>h\<bar>) |]  
paulson@15077
   545
   ==> f -- 0 --> 0"
paulson@15229
   546
apply (simp add: LIM_def, auto)
paulson@15077
   547
apply (subgoal_tac "0 \<le> K")
paulson@15229
   548
 prefer 2
paulson@15229
   549
 apply (drule_tac x = "k/2" in spec)
paulson@15229
   550
apply (simp add: ); 
paulson@15229
   551
 apply (subgoal_tac "0 \<le> K*k", simp add: zero_le_mult_iff) 
paulson@15229
   552
 apply (force intro: order_trans [of _ "\<bar>f (k / 2)\<bar> * 2"]) 
paulson@15077
   553
apply (drule real_le_imp_less_or_eq, auto)
paulson@15229
   554
apply (subgoal_tac "0 < (r * inverse K) / 2")
paulson@15229
   555
apply (drule_tac ?d1.0 = "(r * inverse K) / 2" and ?d2.0 = k in real_lbound_gt_zero)
paulson@15229
   556
apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff zero_less_divide_iff)
paulson@15077
   557
apply (rule_tac x = e in exI, auto)
paulson@15077
   558
apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
paulson@15229
   559
apply (force ); 
paulson@15229
   560
apply (rule_tac y = "K * e" in order_less_trans)
paulson@15077
   561
apply (simp add: mult_less_cancel_left)
paulson@15229
   562
apply (rule_tac c = "inverse K" in mult_right_less_imp_less)
paulson@15077
   563
apply (auto simp add: mult_ac)
paulson@15077
   564
done
paulson@15077
   565
paulson@15229
   566
lemma lemma_termdiff5:
paulson@15229
   567
     "[| 0 < k;  
paulson@15077
   568
            summable f;  
paulson@15077
   569
            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
paulson@15077
   570
                    (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
paulson@15077
   571
         ==> (%h. suminf(g h)) -- 0 --> 0"
paulson@15077
   572
apply (drule summable_sums)
paulson@15081
   573
apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>suminf (g h)\<bar> \<le> suminf f * \<bar>h\<bar>")
paulson@15077
   574
apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
paulson@15077
   575
apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
paulson@15077
   576
 prefer 2
paulson@15077
   577
 apply (simp add: summable_def) 
paulson@15077
   578
 apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
paulson@15077
   579
 apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
paulson@15077
   580
 apply (simp add: mult_ac) 
paulson@15077
   581
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
paulson@15077
   582
 apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
paulson@15077
   583
  apply (rule_tac [2] x = 0 in exI, auto)
paulson@15081
   584
apply (rule_tac j = "suminf (%n. \<bar>g h n\<bar>)" in real_le_trans)
paulson@15077
   585
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
paulson@15077
   586
done
paulson@15077
   587
paulson@15077
   588
paulson@15077
   589
paulson@15077
   590
text{* FIXME: Long proofs*}
paulson@15077
   591
paulson@15077
   592
lemma termdiffs_aux:
paulson@15077
   593
     "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
paulson@15077
   594
    ==> (\<lambda>h. suminf
paulson@15077
   595
             (\<lambda>n. c n *
paulson@15077
   596
                  (((x + h) ^ n - x ^ n) * inverse h -
paulson@15077
   597
                   real n * x ^ (n - Suc 0))))
paulson@15077
   598
       -- 0 --> 0"
paulson@15077
   599
apply (drule dense, safe)
paulson@15077
   600
apply (frule real_less_sum_gt_zero)
paulson@15077
   601
apply (drule_tac
paulson@15081
   602
         f = "%n. \<bar>c n\<bar> * real n * real (n - Suc 0) * (r ^ (n - 2))" 
paulson@15077
   603
     and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) 
paulson@15077
   604
                             - (real n * (x ^ (n - Suc 0))))" 
paulson@15077
   605
       in lemma_termdiff5)
paulson@15077
   606
apply (auto simp add: add_commute)
paulson@15077
   607
apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
paulson@15077
   608
apply (rule_tac [2] x = K in powser_insidea, auto)
paulson@15077
   609
apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
paulson@15229
   610
apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_eq], auto)
paulson@15077
   611
apply (simp add: diffs_def mult_assoc [symmetric])
paulson@15077
   612
apply (subgoal_tac
paulson@15077
   613
        "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
paulson@15077
   614
              = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
paulson@15077
   615
apply auto
paulson@15077
   616
apply (drule diffs_equiv)
paulson@15077
   617
apply (drule sums_summable)
paulson@15077
   618
apply (simp_all add: diffs_def) 
paulson@15077
   619
apply (simp add: diffs_def mult_ac)
paulson@15081
   620
apply (subgoal_tac " (%n. real n * (real (Suc n) * (\<bar>c (Suc n)\<bar> * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * \<bar>c m\<bar> * inverse r) n * (r ^ n))")
paulson@15077
   621
apply auto
paulson@15077
   622
  prefer 2
paulson@15077
   623
  apply (rule ext)
paulson@15077
   624
  apply (simp add: diffs_def) 
paulson@15077
   625
  apply (case_tac "n", auto)
paulson@15077
   626
txt{*23*}
paulson@15077
   627
   apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077
   628
   apply (simp add: mult_ac) 
paulson@15077
   629
  apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077
   630
  apply (simp add: mult_ac) 
paulson@15077
   631
 apply (drule diffs_equiv)
paulson@15077
   632
 apply (drule sums_summable)
paulson@15077
   633
apply (subgoal_tac
paulson@15077
   634
          "summable
paulson@15077
   635
            (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
paulson@15077
   636
                 r ^ (n - Suc 0)) =
paulson@15077
   637
           summable
paulson@15077
   638
            (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
paulson@15077
   639
apply simp 
paulson@15077
   640
apply (rule_tac f = summable in arg_cong, rule ext)
paulson@15077
   641
txt{*33*}
paulson@15077
   642
apply (case_tac "n", auto)
paulson@15077
   643
apply (case_tac "nat", auto)
paulson@15077
   644
apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
paulson@15077
   645
apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077
   646
apply (simp add: mult_assoc)
paulson@15077
   647
apply (rule mult_left_mono)
paulson@15229
   648
 prefer 2 apply arith 
paulson@15229
   649
apply (subst add_commute)
paulson@15077
   650
apply (simp (no_asm) add: mult_assoc [symmetric])
paulson@15077
   651
apply (rule lemma_termdiff3)
paulson@15077
   652
apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
paulson@15077
   653
done
paulson@15077
   654
paulson@15077
   655
paulson@15077
   656
lemma termdiffs: 
paulson@15077
   657
    "[| summable(%n. c(n) * (K ^ n));  
paulson@15077
   658
        summable(%n. (diffs c)(n) * (K ^ n));  
paulson@15077
   659
        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
paulson@15077
   660
        \<bar>x\<bar> < \<bar>K\<bar> |]  
paulson@15077
   661
     ==> DERIV (%x. suminf (%n. c(n) * (x ^ n)))  x :>  
paulson@15077
   662
             suminf (%n. (diffs c)(n) * (x ^ n))"
paulson@15229
   663
apply (simp add: deriv_def)
paulson@15229
   664
apply (rule_tac g = "%h. suminf (%n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h)" in LIM_trans)
paulson@15077
   665
apply (simp add: LIM_def, safe)
paulson@15077
   666
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077
   667
apply (auto simp add: less_diff_eq)
paulson@15077
   668
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077
   669
apply (rule_tac y = 0 in order_le_less_trans, auto)
paulson@15077
   670
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
paulson@15077
   671
apply (auto intro!: summable_sums)
paulson@15077
   672
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077
   673
apply (auto simp add: add_commute)
paulson@15077
   674
apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
paulson@15229
   675
apply (drule_tac x = "(%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
paulson@15085
   676
apply (rule sums_unique)
paulson@15079
   677
apply (simp add: diff_def divide_inverse add_ac mult_ac)
paulson@15077
   678
apply (rule LIM_zero_cancel)
paulson@15229
   679
apply (rule_tac g = "%h. suminf (%n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) - (real n * (x ^ (n - Suc 0)))))" in LIM_trans)
paulson@15077
   680
 prefer 2 apply (blast intro: termdiffs_aux) 
paulson@15077
   681
apply (simp add: LIM_def, safe)
paulson@15077
   682
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077
   683
apply (auto simp add: less_diff_eq)
paulson@15077
   684
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077
   685
apply (rule_tac y = 0 in order_le_less_trans, auto)
paulson@15077
   686
apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
paulson@15077
   687
apply (rule_tac [2] powser_inside, auto)
paulson@15077
   688
apply (drule_tac c = c and x = x in diffs_equiv)
paulson@15077
   689
apply (frule sums_unique, auto)
paulson@15077
   690
apply (subgoal_tac " (%n. (c n) * (x ^ n)) sums (suminf (%n. (c n) * (x ^ n))) & (%n. (c n) * ((x + xa) ^ n)) sums (suminf (%n. (c n) * ( (x + xa) ^ n))) ")
paulson@15077
   691
apply safe
paulson@15077
   692
apply (auto intro!: summable_sums)
paulson@15077
   693
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077
   694
apply (auto simp add: add_commute)
paulson@15229
   695
apply (frule_tac x = "(%n. c n * (xa + x) ^ n) " and y = "(%n. c n * x ^ n)" in sums_diff, assumption)
paulson@15077
   696
apply (simp add: suminf_diff [OF sums_summable sums_summable] 
paulson@15077
   697
               right_diff_distrib [symmetric])
paulson@15229
   698
apply (frule_tac x = "(%n. c n * ((xa + x) ^ n - x ^ n))" and c = "inverse xa" in sums_mult)
paulson@15077
   699
apply (simp add: sums_summable [THEN suminf_mult2])
paulson@15229
   700
apply (frule_tac x = "(%n. inverse xa * (c n * ((xa + x) ^ n - x ^ n))) " and y = "(%n. real n * c n * x ^ (n - Suc 0))" in sums_diff)
paulson@15077
   701
apply assumption
paulson@15077
   702
apply (simp add: suminf_diff [OF sums_summable sums_summable] add_ac mult_ac)
paulson@15077
   703
apply (rule_tac f = suminf in arg_cong)
paulson@15077
   704
apply (rule ext)
paulson@15077
   705
apply (simp add: diff_def right_distrib add_ac mult_ac)
paulson@15077
   706
done
paulson@15077
   707
paulson@15077
   708
paulson@15077
   709
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
paulson@15077
   710
paulson@15077
   711
lemma exp_fdiffs: 
paulson@15077
   712
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
paulson@15229
   713
by (simp add: diffs_def mult_assoc [symmetric] del: mult_Suc)
paulson@15077
   714
paulson@15077
   715
lemma sin_fdiffs: 
paulson@15077
   716
      "diffs(%n. if even n then 0  
paulson@15077
   717
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
paulson@15077
   718
       = (%n. if even n then  
paulson@15077
   719
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   720
              else 0)"
paulson@15229
   721
by (auto intro!: ext 
paulson@15229
   722
         simp add: diffs_def divide_inverse simp del: mult_Suc)
paulson@15077
   723
paulson@15077
   724
lemma sin_fdiffs2: 
paulson@15077
   725
       "diffs(%n. if even n then 0  
paulson@15077
   726
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
paulson@15077
   727
       = (if even n then  
paulson@15077
   728
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   729
              else 0)"
paulson@15229
   730
by (auto intro!: ext 
paulson@15229
   731
         simp add: diffs_def divide_inverse simp del: mult_Suc)
paulson@15077
   732
paulson@15077
   733
lemma cos_fdiffs: 
paulson@15077
   734
      "diffs(%n. if even n then  
paulson@15077
   735
                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
paulson@15077
   736
       = (%n. - (if even n then 0  
paulson@15077
   737
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
paulson@15229
   738
by (auto intro!: ext 
paulson@15229
   739
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
paulson@15229
   740
         simp del: mult_Suc)
paulson@15077
   741
paulson@15077
   742
paulson@15077
   743
lemma cos_fdiffs2: 
paulson@15077
   744
      "diffs(%n. if even n then  
paulson@15077
   745
                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
paulson@15077
   746
       = - (if even n then 0  
paulson@15077
   747
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
paulson@15229
   748
by (auto intro!: ext 
paulson@15229
   749
         simp add: diffs_def divide_inverse odd_Suc_mult_two_ex
paulson@15229
   750
         simp del: mult_Suc)
paulson@15077
   751
paulson@15077
   752
text{*Now at last we can get the derivatives of exp, sin and cos*}
paulson@15077
   753
paulson@15077
   754
lemma lemma_sin_minus:
paulson@15077
   755
     "- sin x =
paulson@15077
   756
         suminf(%n. - ((if even n then 0 
paulson@15077
   757
                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
paulson@15077
   758
by (auto intro!: sums_unique sums_minus sin_converges)
paulson@15077
   759
paulson@15077
   760
lemma lemma_exp_ext: "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))"
paulson@15077
   761
by (auto intro!: ext simp add: exp_def)
paulson@15077
   762
paulson@15077
   763
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15229
   764
apply (simp add: exp_def)
paulson@15077
   765
apply (subst lemma_exp_ext)
paulson@15077
   766
apply (subgoal_tac "DERIV (%u. suminf (%n. inverse (real (fact n)) * u ^ n)) x :> suminf (%n. diffs (%n. inverse (real (fact n))) n * x ^ n) ")
paulson@15229
   767
apply (rule_tac [2] K = "1 + \<bar>x\<bar>" in termdiffs)
paulson@15077
   768
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
paulson@15077
   769
done
paulson@15077
   770
paulson@15077
   771
lemma lemma_sin_ext:
paulson@15077
   772
     "sin = (%x. suminf(%n. 
paulson@15077
   773
                   (if even n then 0  
paulson@15077
   774
                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
paulson@15077
   775
                   x ^ n))"
paulson@15077
   776
by (auto intro!: ext simp add: sin_def)
paulson@15077
   777
paulson@15077
   778
lemma lemma_cos_ext:
paulson@15077
   779
     "cos = (%x. suminf(%n. 
paulson@15077
   780
                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
paulson@15077
   781
                   x ^ n))"
paulson@15077
   782
by (auto intro!: ext simp add: cos_def)
paulson@15077
   783
paulson@15077
   784
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
paulson@15229
   785
apply (simp add: cos_def)
paulson@15077
   786
apply (subst lemma_sin_ext)
paulson@15077
   787
apply (auto simp add: sin_fdiffs2 [symmetric])
paulson@15229
   788
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
paulson@15077
   789
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs, arith)
paulson@15077
   790
done
paulson@15077
   791
paulson@15077
   792
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
paulson@15077
   793
apply (subst lemma_cos_ext)
paulson@15077
   794
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
paulson@15229
   795
apply (rule_tac K = "1 + \<bar>x\<bar>" in termdiffs)
paulson@15077
   796
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs diffs_minus, arith)
paulson@15077
   797
done
paulson@15077
   798
paulson@15077
   799
paulson@15077
   800
subsection{*Properties of the Exponential Function*}
paulson@15077
   801
paulson@15077
   802
lemma exp_zero [simp]: "exp 0 = 1"
paulson@15077
   803
proof -
paulson@15077
   804
  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
paulson@15077
   805
        suminf (\<lambda>n. inverse (real (fact n)) * 0 ^ n)"
paulson@15077
   806
    by (rule series_zero [rule_format, THEN sums_unique],
paulson@15077
   807
        case_tac "m", auto)
paulson@15077
   808
  thus ?thesis by (simp add:  exp_def) 
paulson@15077
   809
qed
paulson@15077
   810
paulson@15077
   811
lemma exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> exp(x)"
paulson@15077
   812
apply (drule real_le_imp_less_or_eq, auto)
paulson@15229
   813
apply (simp add: exp_def)
paulson@15077
   814
apply (rule real_le_trans)
paulson@15229
   815
apply (rule_tac [2] n = 2 and f = "(%n. inverse (real (fact n)) * x ^ n)" in series_pos_le)
paulson@15077
   816
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
paulson@15077
   817
done
paulson@15077
   818
paulson@15077
   819
lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
paulson@15077
   820
apply (rule order_less_le_trans)
paulson@15077
   821
apply (rule_tac [2] exp_ge_add_one_self, auto)
paulson@15077
   822
done
paulson@15077
   823
paulson@15077
   824
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
paulson@15077
   825
proof -
paulson@15077
   826
  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
paulson@15077
   827
    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const) 
paulson@15077
   828
  thus ?thesis by (simp add: o_def)
paulson@15077
   829
qed
paulson@15077
   830
paulson@15077
   831
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
paulson@15077
   832
proof -
paulson@15077
   833
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
paulson@15077
   834
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
paulson@15077
   835
  thus ?thesis by (simp add: o_def)
paulson@15077
   836
qed
paulson@15077
   837
paulson@15077
   838
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
paulson@15077
   839
proof -
paulson@15077
   840
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
paulson@15077
   841
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
paulson@15077
   842
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
paulson@15077
   843
  thus ?thesis by simp
paulson@15077
   844
qed
paulson@15077
   845
paulson@15077
   846
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
paulson@15077
   847
proof -
paulson@15077
   848
  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
paulson@15077
   849
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
paulson@15077
   850
    by (rule DERIV_isconst_all) 
paulson@15077
   851
  thus ?thesis by simp
paulson@15077
   852
qed
paulson@15077
   853
paulson@15077
   854
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
paulson@15077
   855
proof -
paulson@15077
   856
  have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
paulson@15077
   857
  thus ?thesis by simp
paulson@15077
   858
qed
paulson@15077
   859
paulson@15077
   860
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
paulson@15077
   861
by (simp add: mult_commute)
paulson@15077
   862
paulson@15077
   863
paulson@15077
   864
lemma exp_minus: "exp(-x) = inverse(exp(x))"
paulson@15077
   865
by (auto intro: inverse_unique [symmetric])
paulson@15077
   866
paulson@15077
   867
lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
paulson@15077
   868
proof -
paulson@15077
   869
  have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
paulson@15077
   870
  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
paulson@15077
   871
qed
paulson@15077
   872
paulson@15077
   873
text{*Proof: because every exponential can be seen as a square.*}
paulson@15077
   874
lemma exp_ge_zero [simp]: "0 \<le> exp x"
paulson@15077
   875
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
paulson@15077
   876
apply (subst exp_add, auto)
paulson@15077
   877
done
paulson@15077
   878
paulson@15077
   879
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
paulson@15077
   880
apply (cut_tac x = x in exp_mult_minus2)
paulson@15077
   881
apply (auto simp del: exp_mult_minus2)
paulson@15077
   882
done
paulson@15077
   883
paulson@15077
   884
lemma exp_gt_zero [simp]: "0 < exp x"
paulson@15077
   885
by (simp add: order_less_le)
paulson@15077
   886
paulson@15077
   887
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
paulson@15077
   888
by (auto intro: positive_imp_inverse_positive)
paulson@15077
   889
paulson@15081
   890
lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
paulson@15229
   891
by auto
paulson@15077
   892
paulson@15077
   893
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15251
   894
apply (induct "n")
paulson@15077
   895
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
   896
done
paulson@15077
   897
paulson@15077
   898
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
paulson@15229
   899
apply (simp add: diff_minus divide_inverse)
paulson@15077
   900
apply (simp (no_asm) add: exp_add exp_minus)
paulson@15077
   901
done
paulson@15077
   902
paulson@15077
   903
paulson@15077
   904
lemma exp_less_mono:
paulson@15077
   905
  assumes xy: "x < y" shows "exp x < exp y"
paulson@15077
   906
proof -
paulson@15077
   907
  have "1 < exp (y + - x)"
paulson@15077
   908
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
paulson@15077
   909
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
paulson@15077
   910
    by (auto simp add: exp_add exp_minus)
paulson@15077
   911
  thus ?thesis
nipkow@15539
   912
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq
paulson@15228
   913
             del: divide_self_if)
paulson@15077
   914
qed
paulson@15077
   915
paulson@15077
   916
lemma exp_less_cancel: "exp x < exp y ==> x < y"
paulson@15228
   917
apply (simp add: linorder_not_le [symmetric]) 
paulson@15228
   918
apply (auto simp add: order_le_less exp_less_mono) 
paulson@15077
   919
done
paulson@15077
   920
paulson@15077
   921
lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
paulson@15077
   922
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
   923
paulson@15077
   924
lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
paulson@15077
   925
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   926
paulson@15077
   927
lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
paulson@15077
   928
by (simp add: order_eq_iff)
paulson@15077
   929
paulson@15077
   930
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
paulson@15077
   931
apply (rule IVT)
paulson@15077
   932
apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
paulson@15077
   933
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
paulson@15077
   934
apply simp 
paulson@15077
   935
apply (rule exp_ge_add_one_self, simp)
paulson@15077
   936
done
paulson@15077
   937
paulson@15077
   938
lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
paulson@15077
   939
apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077
   940
apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077
   941
apply (rule_tac [2] x = 0 in exI)
paulson@15077
   942
apply (frule_tac [3] real_inverse_gt_one)
paulson@15077
   943
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077
   944
apply (rule_tac x = "-x" in exI)
paulson@15077
   945
apply (simp add: exp_minus)
paulson@15077
   946
done
paulson@15077
   947
paulson@15077
   948
paulson@15077
   949
subsection{*Properties of the Logarithmic Function*}
paulson@15077
   950
paulson@15077
   951
lemma ln_exp[simp]: "ln(exp x) = x"
paulson@15077
   952
by (simp add: ln_def)
paulson@15077
   953
paulson@15077
   954
lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
paulson@15077
   955
apply (auto dest: exp_total)
paulson@15077
   956
apply (erule subst, simp) 
paulson@15077
   957
done
paulson@15077
   958
paulson@15077
   959
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
paulson@15077
   960
apply (rule exp_inj_iff [THEN iffD1])
paulson@15077
   961
apply (frule real_mult_order)
paulson@15077
   962
apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
paulson@15077
   963
done
paulson@15077
   964
paulson@15077
   965
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
paulson@15077
   966
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   967
apply (erule subst)+
paulson@15077
   968
apply simp 
paulson@15077
   969
done
paulson@15077
   970
paulson@15077
   971
lemma ln_one[simp]: "ln 1 = 0"
paulson@15077
   972
by (rule exp_inj_iff [THEN iffD1], auto)
paulson@15077
   973
paulson@15077
   974
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
paulson@15077
   975
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
paulson@15077
   976
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
paulson@15077
   977
done
paulson@15077
   978
paulson@15077
   979
lemma ln_div: 
paulson@15077
   980
    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
paulson@15229
   981
apply (simp add: divide_inverse)
paulson@15077
   982
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
paulson@15077
   983
done
paulson@15077
   984
paulson@15077
   985
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
paulson@15077
   986
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   987
apply (erule subst)+
paulson@15077
   988
apply simp 
paulson@15077
   989
done
paulson@15077
   990
paulson@15077
   991
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
paulson@15077
   992
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   993
paulson@15077
   994
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
paulson@15077
   995
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
paulson@15077
   996
paulson@15077
   997
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
paulson@15077
   998
apply (rule ln_exp [THEN subst])
paulson@15077
   999
apply (rule ln_le_cancel_iff [THEN iffD2], auto)
paulson@15077
  1000
done
paulson@15077
  1001
paulson@15077
  1002
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
paulson@15077
  1003
apply (rule order_less_le_trans)
paulson@15077
  1004
apply (rule_tac [2] ln_add_one_self_le_self)
paulson@15077
  1005
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
paulson@15077
  1006
done
paulson@15077
  1007
paulson@15234
  1008
lemma ln_ge_zero [simp]:
paulson@15077
  1009
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077
  1010
proof -
paulson@15077
  1011
  have "0 < x" using x by arith
paulson@15077
  1012
  hence "exp 0 \<le> exp (ln x)"
paulson@15077
  1013
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077
  1014
  thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077
  1015
qed
paulson@15077
  1016
paulson@15077
  1017
lemma ln_ge_zero_imp_ge_one:
paulson@15077
  1018
  assumes ln: "0 \<le> ln x" 
paulson@15077
  1019
      and x:  "0 < x"
paulson@15077
  1020
  shows "1 \<le> x"
paulson@15077
  1021
proof -
paulson@15077
  1022
  from ln have "ln 1 \<le> ln x" by simp
paulson@15077
  1023
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
  1024
qed
paulson@15077
  1025
paulson@15077
  1026
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077
  1027
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077
  1028
paulson@15234
  1029
lemma ln_less_zero_iff [simp]: "0 < x ==> (ln x < 0) = (x < 1)"
paulson@15234
  1030
by (insert ln_ge_zero_iff [of x], arith)
paulson@15234
  1031
paulson@15077
  1032
lemma ln_gt_zero:
paulson@15077
  1033
  assumes x: "1 < x" shows "0 < ln x"
paulson@15077
  1034
proof -
paulson@15077
  1035
  have "0 < x" using x by arith
paulson@15077
  1036
  hence "exp 0 < exp (ln x)"
paulson@15077
  1037
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077
  1038
  thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077
  1039
qed
paulson@15077
  1040
paulson@15077
  1041
lemma ln_gt_zero_imp_gt_one:
paulson@15077
  1042
  assumes ln: "0 < ln x" 
paulson@15077
  1043
      and x:  "0 < x"
paulson@15077
  1044
  shows "1 < x"
paulson@15077
  1045
proof -
paulson@15077
  1046
  from ln have "ln 1 < ln x" by simp
paulson@15077
  1047
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
  1048
qed
paulson@15077
  1049
paulson@15077
  1050
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077
  1051
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077
  1052
paulson@15234
  1053
lemma ln_eq_zero_iff [simp]: "0 < x ==> (ln x = 0) = (x = 1)"
paulson@15234
  1054
by (insert ln_less_zero_iff [of x] ln_gt_zero_iff [of x], arith)
paulson@15077
  1055
paulson@15077
  1056
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15234
  1057
by simp
paulson@15077
  1058
paulson@15077
  1059
lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077
  1060
by auto
paulson@15077
  1061
paulson@15077
  1062
paulson@15077
  1063
subsection{*Basic Properties of the Trigonometric Functions*}
paulson@15077
  1064
paulson@15077
  1065
lemma sin_zero [simp]: "sin 0 = 0"
paulson@15077
  1066
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
paulson@15077
  1067
         simp add: sin_def sums_def simp del: power_0_left)
paulson@15077
  1068
nipkow@15539
  1069
lemma lemma_series_zero2:
nipkow@15539
  1070
 "(\<forall>m. n \<le> m --> f m = 0) --> f sums setsum f {0..<n}"
paulson@15077
  1071
by (auto intro: series_zero)
paulson@15077
  1072
paulson@15077
  1073
lemma cos_zero [simp]: "cos 0 = 1"
paulson@15229
  1074
apply (simp add: cos_def)
paulson@15077
  1075
apply (rule sums_unique [symmetric])
paulson@15229
  1076
apply (cut_tac n = 1 and f = "(%n. (if even n then (- 1) ^ (n div 2) / (real (fact n)) else 0) * 0 ^ n)" in lemma_series_zero2)
paulson@15077
  1077
apply auto
paulson@15077
  1078
done
paulson@15077
  1079
paulson@15077
  1080
lemma DERIV_sin_sin_mult [simp]:
paulson@15077
  1081
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1082
by (rule DERIV_mult, auto)
paulson@15077
  1083
paulson@15077
  1084
lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077
  1085
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1086
apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077
  1087
apply (auto simp add: mult_assoc)
paulson@15077
  1088
done
paulson@15077
  1089
paulson@15077
  1090
lemma DERIV_sin_realpow2 [simp]:
paulson@15077
  1091
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1092
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1093
paulson@15077
  1094
lemma DERIV_sin_realpow2a [simp]:
paulson@15077
  1095
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1096
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1097
paulson@15077
  1098
lemma DERIV_cos_cos_mult [simp]:
paulson@15077
  1099
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1100
by (rule DERIV_mult, auto)
paulson@15077
  1101
paulson@15077
  1102
lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077
  1103
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1104
apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077
  1105
apply (auto simp add: mult_ac)
paulson@15077
  1106
done
paulson@15077
  1107
paulson@15077
  1108
lemma DERIV_cos_realpow2 [simp]:
paulson@15077
  1109
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1110
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1111
paulson@15077
  1112
lemma DERIV_cos_realpow2a [simp]:
paulson@15077
  1113
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1114
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1115
paulson@15077
  1116
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077
  1117
by auto
paulson@15077
  1118
paulson@15077
  1119
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1120
apply (rule lemma_DERIV_subst)
paulson@15077
  1121
apply (rule DERIV_cos_realpow2a, auto)
paulson@15077
  1122
done
paulson@15077
  1123
paulson@15077
  1124
(* most useful *)
paulson@15229
  1125
lemma DERIV_cos_cos_mult3 [simp]:
paulson@15229
  1126
     "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1127
apply (rule lemma_DERIV_subst)
paulson@15077
  1128
apply (rule DERIV_cos_cos_mult2, auto)
paulson@15077
  1129
done
paulson@15077
  1130
paulson@15077
  1131
lemma DERIV_sin_circle_all: 
paulson@15077
  1132
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077
  1133
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
paulson@15229
  1134
apply (simp only: diff_minus, safe)
paulson@15229
  1135
apply (rule DERIV_add) 
paulson@15077
  1136
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1137
done
paulson@15077
  1138
paulson@15229
  1139
lemma DERIV_sin_circle_all_zero [simp]:
paulson@15229
  1140
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077
  1141
by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077
  1142
paulson@15077
  1143
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077
  1144
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077
  1145
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1146
done
paulson@15077
  1147
paulson@15077
  1148
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
paulson@15077
  1149
apply (subst real_add_commute)
paulson@15077
  1150
apply (simp (no_asm) del: realpow_Suc)
paulson@15077
  1151
done
paulson@15077
  1152
paulson@15077
  1153
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077
  1154
apply (cut_tac x = x in sin_cos_squared_add2)
paulson@15077
  1155
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1156
done
paulson@15077
  1157
paulson@15077
  1158
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15229
  1159
apply (rule_tac a1 = "(cos x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
  1160
apply (simp del: realpow_Suc)
paulson@15077
  1161
done
paulson@15077
  1162
paulson@15077
  1163
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077
  1164
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
  1165
apply (simp del: realpow_Suc)
paulson@15077
  1166
done
paulson@15077
  1167
paulson@15077
  1168
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
paulson@15077
  1169
by arith
paulson@15077
  1170
paulson@15081
  1171
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
paulson@15077
  1172
apply (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1173
apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077
  1174
apply (auto simp del: realpow_Suc)
paulson@15077
  1175
apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077
  1176
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
  1177
done
paulson@15077
  1178
paulson@15077
  1179
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
paulson@15077
  1180
apply (insert abs_sin_le_one [of x]) 
paulson@15077
  1181
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077
  1182
done
paulson@15077
  1183
paulson@15077
  1184
lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077
  1185
apply (insert abs_sin_le_one [of x]) 
paulson@15077
  1186
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077
  1187
done
paulson@15077
  1188
paulson@15081
  1189
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
paulson@15077
  1190
apply (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1191
apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077
  1192
apply (auto simp del: realpow_Suc)
paulson@15077
  1193
apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077
  1194
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
  1195
done
paulson@15077
  1196
paulson@15077
  1197
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
paulson@15077
  1198
apply (insert abs_cos_le_one [of x]) 
paulson@15077
  1199
apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
paulson@15077
  1200
done
paulson@15077
  1201
paulson@15077
  1202
lemma cos_le_one [simp]: "cos x \<le> 1"
paulson@15077
  1203
apply (insert abs_cos_le_one [of x]) 
paulson@15077
  1204
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
paulson@15077
  1205
done
paulson@15077
  1206
paulson@15077
  1207
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077
  1208
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
paulson@15077
  1209
apply (rule lemma_DERIV_subst)
paulson@15229
  1210
apply (rule_tac f = "(%x. x ^ n)" in DERIV_chain2)
paulson@15077
  1211
apply (rule DERIV_pow, auto)
paulson@15077
  1212
done
paulson@15077
  1213
paulson@15229
  1214
lemma DERIV_fun_exp:
paulson@15229
  1215
     "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077
  1216
apply (rule lemma_DERIV_subst)
paulson@15077
  1217
apply (rule_tac f = exp in DERIV_chain2)
paulson@15077
  1218
apply (rule DERIV_exp, auto)
paulson@15077
  1219
done
paulson@15077
  1220
paulson@15229
  1221
lemma DERIV_fun_sin:
paulson@15229
  1222
     "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077
  1223
apply (rule lemma_DERIV_subst)
paulson@15077
  1224
apply (rule_tac f = sin in DERIV_chain2)
paulson@15077
  1225
apply (rule DERIV_sin, auto)
paulson@15077
  1226
done
paulson@15077
  1227
paulson@15229
  1228
lemma DERIV_fun_cos:
paulson@15229
  1229
     "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077
  1230
apply (rule lemma_DERIV_subst)
paulson@15077
  1231
apply (rule_tac f = cos in DERIV_chain2)
paulson@15077
  1232
apply (rule DERIV_cos, auto)
paulson@15077
  1233
done
paulson@15077
  1234
paulson@15077
  1235
lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
paulson@15077
  1236
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
paulson@15077
  1237
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
paulson@15077
  1238
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
paulson@15077
  1239
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
paulson@15077
  1240
paulson@15077
  1241
(* lemma *)
paulson@15229
  1242
lemma lemma_DERIV_sin_cos_add:
paulson@15229
  1243
     "\<forall>x.  
paulson@15077
  1244
         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1245
               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
paulson@15077
  1246
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1247
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1248
  --{*replaces the old @{text DERIV_tac}*}
paulson@15229
  1249
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1250
done
paulson@15077
  1251
paulson@15077
  1252
lemma sin_cos_add [simp]:
paulson@15077
  1253
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1254
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077
  1255
apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077
  1256
       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077
  1257
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1258
done
paulson@15077
  1259
paulson@15077
  1260
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
paulson@15077
  1261
apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077
  1262
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085
  1263
            simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
paulson@15077
  1264
done
paulson@15077
  1265
paulson@15077
  1266
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077
  1267
apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077
  1268
apply (auto dest!: real_sum_squares_cancel_a
paulson@15085
  1269
            simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
paulson@15077
  1270
done
paulson@15077
  1271
paulson@15085
  1272
lemma lemma_DERIV_sin_cos_minus:
paulson@15085
  1273
    "\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
paulson@15077
  1274
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1275
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15229
  1276
apply (auto simp add: diff_minus left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1277
done
paulson@15077
  1278
paulson@15085
  1279
lemma sin_cos_minus [simp]: 
paulson@15085
  1280
    "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15085
  1281
apply (cut_tac y = 0 and x = x 
paulson@15085
  1282
       in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
paulson@15077
  1283
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1284
done
paulson@15077
  1285
paulson@15077
  1286
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
paulson@15077
  1287
apply (cut_tac x = x in sin_cos_minus)
paulson@15085
  1288
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085
  1289
       simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_minus)
paulson@15077
  1290
done
paulson@15077
  1291
paulson@15077
  1292
lemma cos_minus [simp]: "cos (-x) = cos(x)"
paulson@15077
  1293
apply (cut_tac x = x in sin_cos_minus)
paulson@15085
  1294
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15085
  1295
                   simp add: numeral_2_eq_2 simp del: sin_cos_minus)
paulson@15077
  1296
done
paulson@15077
  1297
paulson@15077
  1298
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
paulson@15229
  1299
apply (simp add: diff_minus)
paulson@15077
  1300
apply (simp (no_asm) add: sin_add)
paulson@15077
  1301
done
paulson@15077
  1302
paulson@15077
  1303
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077
  1304
by (simp add: sin_diff mult_commute)
paulson@15077
  1305
paulson@15077
  1306
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
paulson@15229
  1307
apply (simp add: diff_minus)
paulson@15077
  1308
apply (simp (no_asm) add: cos_add)
paulson@15077
  1309
done
paulson@15077
  1310
paulson@15077
  1311
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077
  1312
by (simp add: cos_diff mult_commute)
paulson@15077
  1313
paulson@15077
  1314
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
paulson@15077
  1315
by (cut_tac x = x and y = x in sin_add, auto)
paulson@15077
  1316
paulson@15077
  1317
paulson@15077
  1318
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
paulson@15077
  1319
apply (cut_tac x = x and y = x in cos_add)
paulson@15077
  1320
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1321
done
paulson@15077
  1322
paulson@15077
  1323
paulson@15077
  1324
subsection{*The Constant Pi*}
paulson@15077
  1325
paulson@15077
  1326
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077
  1327
   hence define pi.*}
paulson@15077
  1328
paulson@15077
  1329
lemma sin_paired:
paulson@15077
  1330
     "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077
  1331
      sums  sin x"
paulson@15077
  1332
proof -
paulson@15077
  1333
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1334
            (if even k then 0
paulson@15077
  1335
             else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
paulson@15077
  1336
            x ^ k) 
paulson@15077
  1337
	sums
paulson@15077
  1338
	suminf (\<lambda>n. (if even n then 0
paulson@15077
  1339
		     else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
paulson@15077
  1340
	            x ^ n)" 
paulson@15077
  1341
    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1342
  thus ?thesis by (simp add: mult_ac sin_def)
paulson@15077
  1343
qed
paulson@15077
  1344
paulson@15077
  1345
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1346
apply (subgoal_tac 
paulson@15077
  1347
       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1348
              (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
paulson@15077
  1349
     sums suminf (\<lambda>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077
  1350
 prefer 2
paulson@15077
  1351
 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1352
apply (rotate_tac 2)
paulson@15077
  1353
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
paulson@15077
  1354
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1355
apply (frule sums_unique)
paulson@15077
  1356
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1357
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
paulson@15077
  1358
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1359
apply (erule sums_summable)
paulson@15077
  1360
apply (case_tac "m=0")
paulson@15077
  1361
apply (simp (no_asm_simp))
paulson@15234
  1362
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x") 
nipkow@15539
  1363
apply (simp only: mult_less_cancel_left, simp)  
nipkow@15539
  1364
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077
  1365
apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077
  1366
apply (rule mult_strict_mono)
paulson@15085
  1367
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
paulson@15077
  1368
apply (subst fact_Suc)
paulson@15077
  1369
apply (subst fact_Suc)
paulson@15077
  1370
apply (subst fact_Suc)
paulson@15077
  1371
apply (subst fact_Suc)
paulson@15077
  1372
apply (subst real_of_nat_mult)
paulson@15077
  1373
apply (subst real_of_nat_mult)
paulson@15077
  1374
apply (subst real_of_nat_mult)
paulson@15077
  1375
apply (subst real_of_nat_mult)
nipkow@15539
  1376
apply (simp (no_asm) add: divide_inverse del: fact_Suc)
paulson@15077
  1377
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077
  1378
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
paulson@15077
  1379
apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077
  1380
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
paulson@15077
  1381
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077
  1382
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077
  1383
apply (erule ssubst)+
paulson@15077
  1384
apply (auto simp del: fact_Suc)
paulson@15077
  1385
apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077
  1386
 prefer 2 apply (simp only: zero_less_power) 
paulson@15077
  1387
apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077
  1388
apply (rule mult_strict_mono)
paulson@15077
  1389
apply (simp_all (no_asm_simp))
paulson@15077
  1390
done
paulson@15077
  1391
paulson@15077
  1392
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1393
by (auto intro: sin_gt_zero)
paulson@15077
  1394
paulson@15077
  1395
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077
  1396
apply (cut_tac x = x in sin_gt_zero1)
paulson@15077
  1397
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1398
done
paulson@15077
  1399
paulson@15077
  1400
lemma cos_paired:
paulson@15077
  1401
     "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1402
proof -
paulson@15077
  1403
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1404
            (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
paulson@15077
  1405
            x ^ k) 
paulson@15077
  1406
        sums
paulson@15077
  1407
	suminf
paulson@15077
  1408
	 (\<lambda>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
paulson@15077
  1409
	      x ^ n)" 
paulson@15077
  1410
    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1411
  thus ?thesis by (simp add: mult_ac cos_def)
paulson@15077
  1412
qed
paulson@15077
  1413
paulson@15077
  1414
declare zero_less_power [simp]
paulson@15077
  1415
paulson@15077
  1416
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077
  1417
by simp
paulson@15077
  1418
paulson@15077
  1419
lemma cos_two_less_zero: "cos (2) < 0"
paulson@15077
  1420
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1421
apply (drule sums_minus)
paulson@15077
  1422
apply (rule neg_less_iff_less [THEN iffD1]) 
nipkow@15539
  1423
apply (frule sums_unique, auto)
nipkow@15539
  1424
apply (rule_tac y =
nipkow@15539
  1425
 "\<Sum>n=0..< Suc(Suc(Suc 0)). - ((- 1) ^ n / (real(fact (2*n))) * 2 ^ (2*n))"
paulson@15481
  1426
       in order_less_trans)
paulson@15077
  1427
apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
nipkow@15539
  1428
apply (simp (no_asm) add: mult_assoc del: setsum_Suc)
paulson@15077
  1429
apply (rule sumr_pos_lt_pair)
paulson@15077
  1430
apply (erule sums_summable, safe)
paulson@15085
  1431
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric] 
paulson@15085
  1432
            del: fact_Suc)
paulson@15077
  1433
apply (rule real_mult_inverse_cancel2)
paulson@15077
  1434
apply (rule real_of_nat_fact_gt_zero)+
paulson@15077
  1435
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1436
apply (subst fact_lemma) 
paulson@15481
  1437
apply (subst fact_Suc [of "Suc (Suc (Suc (Suc (Suc (Suc (Suc (4 * d)))))))"])
paulson@15481
  1438
apply (simp only: real_of_nat_mult)
paulson@15077
  1439
apply (rule real_mult_less_mono, force)
paulson@15481
  1440
  apply (rule_tac [3] real_of_nat_fact_gt_zero)
paulson@15481
  1441
 prefer 2 apply force
paulson@15077
  1442
apply (rule real_of_nat_less_iff [THEN iffD2])
paulson@15077
  1443
apply (rule fact_less_mono, auto)
paulson@15077
  1444
done
paulson@15077
  1445
declare cos_two_less_zero [simp]
paulson@15077
  1446
declare cos_two_less_zero [THEN real_not_refl2, simp]
paulson@15077
  1447
declare cos_two_less_zero [THEN order_less_imp_le, simp]
paulson@15077
  1448
paulson@15077
  1449
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077
  1450
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077
  1451
apply (rule_tac [2] IVT2)
paulson@15077
  1452
apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077
  1453
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1454
apply (rule ccontr)
paulson@15077
  1455
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077
  1456
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077
  1457
apply (drule_tac f = cos in Rolle)
paulson@15077
  1458
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1459
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@15077
  1460
apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
paulson@15077
  1461
apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
paulson@15077
  1462
apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
paulson@15077
  1463
done
paulson@15077
  1464
    
paulson@15077
  1465
lemma pi_half: "pi/2 = (@x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1466
by (simp add: pi_def)
paulson@15077
  1467
paulson@15077
  1468
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
paulson@15077
  1469
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1470
apply (auto intro!: someI2 simp add: pi_half)
paulson@15077
  1471
done
paulson@15077
  1472
paulson@15077
  1473
lemma pi_half_gt_zero: "0 < pi / 2"
paulson@15077
  1474
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1475
apply (auto simp add: pi_half)
paulson@15077
  1476
apply (rule someI2, blast, safe)
paulson@15077
  1477
apply (drule_tac y = xa in real_le_imp_less_or_eq)
paulson@15077
  1478
apply (safe, simp)
paulson@15077
  1479
done
paulson@15077
  1480
declare pi_half_gt_zero [simp]
paulson@15077
  1481
declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
paulson@15077
  1482
declare pi_half_gt_zero [THEN order_less_imp_le, simp]
paulson@15077
  1483
paulson@15077
  1484
lemma pi_half_less_two: "pi / 2 < 2"
paulson@15077
  1485
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1486
apply (auto simp add: pi_half)
paulson@15077
  1487
apply (rule someI2, blast, safe)
paulson@15077
  1488
apply (drule_tac x = xa in order_le_imp_less_or_eq)
paulson@15077
  1489
apply (safe, simp)
paulson@15077
  1490
done
paulson@15077
  1491
declare pi_half_less_two [simp]
paulson@15077
  1492
declare pi_half_less_two [THEN real_not_refl2, simp]
paulson@15077
  1493
declare pi_half_less_two [THEN order_less_imp_le, simp]
paulson@15077
  1494
paulson@15077
  1495
lemma pi_gt_zero [simp]: "0 < pi"
paulson@15229
  1496
apply (insert pi_half_gt_zero) 
paulson@15229
  1497
apply (simp add: ); 
paulson@15077
  1498
done
paulson@15077
  1499
paulson@15077
  1500
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
paulson@15077
  1501
by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])
paulson@15077
  1502
paulson@15077
  1503
lemma pi_not_less_zero [simp]: "~ (pi < 0)"
paulson@15077
  1504
apply (insert pi_gt_zero)
paulson@15077
  1505
apply (blast elim: order_less_asym) 
paulson@15077
  1506
done
paulson@15077
  1507
paulson@15077
  1508
lemma pi_ge_zero [simp]: "0 \<le> pi"
paulson@15077
  1509
by (auto intro: order_less_imp_le)
paulson@15077
  1510
paulson@15077
  1511
lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
paulson@15077
  1512
by auto
paulson@15077
  1513
paulson@15077
  1514
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1515
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1516
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
paulson@15077
  1517
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1518
done
paulson@15077
  1519
paulson@15077
  1520
lemma cos_pi [simp]: "cos pi = -1"
nipkow@15539
  1521
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1522
paulson@15077
  1523
lemma sin_pi [simp]: "sin pi = 0"
nipkow@15539
  1524
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1525
paulson@15077
  1526
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15229
  1527
by (simp add: diff_minus cos_add)
paulson@15077
  1528
paulson@15077
  1529
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15229
  1530
by (simp add: cos_add)
paulson@15077
  1531
declare minus_sin_cos_eq [symmetric, simp]
paulson@15077
  1532
paulson@15077
  1533
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15229
  1534
by (simp add: diff_minus sin_add)
paulson@15077
  1535
declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]
paulson@15077
  1536
paulson@15077
  1537
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15229
  1538
by (simp add: sin_add)
paulson@15077
  1539
paulson@15077
  1540
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15229
  1541
by (simp add: sin_add)
paulson@15077
  1542
paulson@15077
  1543
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15229
  1544
by (simp add: cos_add)
paulson@15077
  1545
paulson@15077
  1546
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1547
by (simp add: sin_add cos_double)
paulson@15077
  1548
paulson@15077
  1549
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1550
by (simp add: cos_add cos_double)
paulson@15077
  1551
paulson@15077
  1552
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15251
  1553
apply (induct "n")
paulson@15077
  1554
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1555
done
paulson@15077
  1556
paulson@15383
  1557
lemma cos_npi2 [simp]: "cos (pi * real n) = -1 ^ n"
paulson@15383
  1558
proof -
paulson@15383
  1559
  have "cos (pi * real n) = cos (real n * pi)" by (simp only: mult_commute)
paulson@15383
  1560
  also have "... = -1 ^ n" by (rule cos_npi) 
paulson@15383
  1561
  finally show ?thesis .
paulson@15383
  1562
qed
paulson@15383
  1563
paulson@15077
  1564
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15251
  1565
apply (induct "n")
paulson@15077
  1566
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1567
done
paulson@15077
  1568
paulson@15077
  1569
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15383
  1570
by (simp add: mult_commute [of pi]) 
paulson@15077
  1571
paulson@15077
  1572
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1573
by (simp add: cos_double)
paulson@15077
  1574
paulson@15077
  1575
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15229
  1576
by simp
paulson@15077
  1577
paulson@15077
  1578
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1579
apply (rule sin_gt_zero, assumption)
paulson@15077
  1580
apply (rule order_less_trans, assumption)
paulson@15077
  1581
apply (rule pi_half_less_two)
paulson@15077
  1582
done
paulson@15077
  1583
paulson@15077
  1584
lemma sin_less_zero: 
paulson@15077
  1585
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1586
proof -
paulson@15077
  1587
  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077
  1588
  thus ?thesis by simp
paulson@15077
  1589
qed
paulson@15077
  1590
paulson@15077
  1591
lemma pi_less_4: "pi < 4"
paulson@15077
  1592
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1593
paulson@15077
  1594
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1595
apply (cut_tac pi_less_4)
paulson@15077
  1596
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1597
apply (force intro: DERIV_isCont DERIV_cos)
paulson@15077
  1598
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1599
apply (rename_tac y z)
paulson@15077
  1600
apply (drule_tac x = y in spec)
paulson@15077
  1601
apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077
  1602
done
paulson@15077
  1603
paulson@15077
  1604
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1605
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1606
apply (rule cos_minus [THEN subst])
paulson@15077
  1607
apply (rule cos_gt_zero)
paulson@15077
  1608
apply (auto intro: cos_gt_zero)
paulson@15077
  1609
done
paulson@15077
  1610
 
paulson@15077
  1611
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1612
apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077
  1613
apply (subgoal_tac "x = pi/2", auto) 
paulson@15077
  1614
done
paulson@15077
  1615
paulson@15077
  1616
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077
  1617
apply (subst sin_cos_eq)
paulson@15077
  1618
apply (rotate_tac 1)
paulson@15077
  1619
apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077
  1620
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077
  1621
done
paulson@15077
  1622
paulson@15077
  1623
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1624
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1625
paulson@15077
  1626
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077
  1627
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077
  1628
apply (rule_tac [2] IVT2)
paulson@15077
  1629
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077
  1630
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1631
apply (rule ccontr, auto)
paulson@15077
  1632
apply (drule_tac f = cos in Rolle)
paulson@15077
  1633
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1634
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077
  1635
            dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077
  1636
            simp add: differentiable_def)
paulson@15077
  1637
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077
  1638
done
paulson@15077
  1639
paulson@15077
  1640
lemma sin_total:
paulson@15077
  1641
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1642
apply (rule ccontr)
paulson@15077
  1643
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))")
paulson@15077
  1644
apply (erule swap)
paulson@15077
  1645
apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077
  1646
apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077
  1647
apply (erule ex1E)
paulson@15229
  1648
apply (rule_tac a = "x - (pi/2)" in ex1I)
paulson@15077
  1649
apply (simp (no_asm) add: real_add_assoc)
paulson@15077
  1650
apply (rotate_tac 3)
paulson@15077
  1651
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077
  1652
done
paulson@15077
  1653
paulson@15077
  1654
lemma reals_Archimedean4:
paulson@15077
  1655
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1656
apply (auto dest!: reals_Archimedean3)
paulson@15077
  1657
apply (drule_tac x = x in spec, clarify) 
paulson@15077
  1658
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077
  1659
 prefer 2 apply (erule LeastI) 
paulson@15077
  1660
apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077
  1661
apply (subgoal_tac "~ x < real nat * y")
paulson@15077
  1662
 prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077
  1663
done
paulson@15077
  1664
paulson@15077
  1665
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077
  1666
   now causes some unwanted re-arrangements of literals!   *)
paulson@15229
  1667
lemma cos_zero_lemma:
paulson@15229
  1668
     "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077
  1669
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1670
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15086
  1671
apply (subgoal_tac "0 \<le> x - real n * pi & 
paulson@15086
  1672
                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
paulson@15086
  1673
apply (auto simp add: compare_rls) 
paulson@15077
  1674
  prefer 3 apply (simp add: cos_diff) 
paulson@15077
  1675
 prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
paulson@15077
  1676
apply (simp add: cos_diff)
paulson@15077
  1677
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1678
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1679
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1680
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1681
apply (simp add: cos_diff)
paulson@15229
  1682
apply (rule_tac x = "Suc (2 * n)" in exI)
paulson@15077
  1683
apply (simp add: real_of_nat_Suc left_distrib, auto)
paulson@15077
  1684
done
paulson@15077
  1685
paulson@15229
  1686
lemma sin_zero_lemma:
paulson@15229
  1687
     "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077
  1688
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1689
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1690
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1691
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15085
  1692
apply (rule cos_zero_lemma)
paulson@15085
  1693
apply (simp_all add: add_increasing)  
paulson@15077
  1694
done
paulson@15077
  1695
paulson@15077
  1696
paulson@15229
  1697
lemma cos_zero_iff:
paulson@15229
  1698
     "(cos x = 0) =  
paulson@15077
  1699
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077
  1700
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1701
apply (rule iffI)
paulson@15077
  1702
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1703
apply (drule cos_zero_lemma, assumption+)
paulson@15077
  1704
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077
  1705
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1706
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
nipkow@15539
  1707
apply (auto simp add: cos_add)
paulson@15077
  1708
done
paulson@15077
  1709
paulson@15077
  1710
(* ditto: but to a lesser extent *)
paulson@15229
  1711
lemma sin_zero_iff:
paulson@15229
  1712
     "(sin x = 0) =  
paulson@15077
  1713
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077
  1714
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1715
apply (rule iffI)
paulson@15077
  1716
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1717
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1718
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077
  1719
apply (force simp add: minus_equation_iff [of x]) 
nipkow@15539
  1720
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1721
done
paulson@15077
  1722
paulson@15077
  1723
paulson@15077
  1724
subsection{*Tangent*}
paulson@15077
  1725
paulson@15077
  1726
lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077
  1727
by (simp add: tan_def)
paulson@15077
  1728
paulson@15077
  1729
lemma tan_pi [simp]: "tan pi = 0"
paulson@15077
  1730
by (simp add: tan_def)
paulson@15077
  1731
paulson@15077
  1732
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077
  1733
by (simp add: tan_def)
paulson@15077
  1734
paulson@15077
  1735
lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077
  1736
by (simp add: tan_def minus_mult_left)
paulson@15077
  1737
paulson@15077
  1738
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077
  1739
by (simp add: tan_def)
paulson@15077
  1740
paulson@15077
  1741
lemma lemma_tan_add1: 
paulson@15077
  1742
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1743
        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15229
  1744
apply (simp add: tan_def divide_inverse)
paulson@15229
  1745
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1746
            simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077
  1747
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15229
  1748
apply (auto simp del: inverse_mult_distrib 
paulson@15229
  1749
            simp add: mult_assoc left_diff_distrib cos_add)
paulson@15234
  1750
done  
paulson@15077
  1751
paulson@15077
  1752
lemma add_tan_eq: 
paulson@15077
  1753
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1754
       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15229
  1755
apply (simp add: tan_def)
paulson@15077
  1756
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  1757
apply (auto simp add: mult_assoc left_distrib)
nipkow@15539
  1758
apply (simp add: sin_add)
paulson@15077
  1759
done
paulson@15077
  1760
paulson@15229
  1761
lemma tan_add:
paulson@15229
  1762
     "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]  
paulson@15077
  1763
      ==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
paulson@15077
  1764
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
paulson@15077
  1765
apply (simp add: tan_def)
paulson@15077
  1766
done
paulson@15077
  1767
paulson@15229
  1768
lemma tan_double:
paulson@15229
  1769
     "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077
  1770
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077
  1771
apply (insert tan_add [of x x]) 
paulson@15077
  1772
apply (simp add: mult_2 [symmetric])  
paulson@15077
  1773
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1774
done
paulson@15077
  1775
paulson@15077
  1776
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15229
  1777
by (simp add: tan_def zero_less_divide_iff sin_gt_zero2 cos_gt_zero_pi) 
paulson@15077
  1778
paulson@15077
  1779
lemma tan_less_zero: 
paulson@15077
  1780
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  1781
proof -
paulson@15077
  1782
  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077
  1783
  thus ?thesis by simp
paulson@15077
  1784
qed
paulson@15077
  1785
paulson@15077
  1786
lemma lemma_DERIV_tan:
paulson@15077
  1787
     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1788
apply (rule lemma_DERIV_subst)
paulson@15077
  1789
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15079
  1790
apply (auto simp add: divide_inverse numeral_2_eq_2)
paulson@15077
  1791
done
paulson@15077
  1792
paulson@15077
  1793
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1794
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077
  1795
paulson@15077
  1796
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077
  1797
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15229
  1798
apply (simp add: divide_inverse [symmetric])
paulson@15077
  1799
apply (rule LIM_mult2)
paulson@15077
  1800
apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077
  1801
apply (rule_tac [2] LIM_inverse)
paulson@15077
  1802
apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077
  1803
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077
  1804
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077
  1805
done
paulson@15077
  1806
paulson@15077
  1807
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  1808
apply (cut_tac LIM_cos_div_sin)
paulson@15077
  1809
apply (simp only: LIM_def)
paulson@15077
  1810
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  1811
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15229
  1812
apply (rule_tac x = "(pi/2) - e" in exI)
paulson@15077
  1813
apply (simp (no_asm_simp))
paulson@15229
  1814
apply (drule_tac x = "(pi/2) - e" in spec)
paulson@15229
  1815
apply (auto simp add: tan_def)
paulson@15077
  1816
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15079
  1817
apply (auto simp add: divide_inverse)
paulson@15229
  1818
apply (rule real_mult_order) 
paulson@15229
  1819
apply (subgoal_tac [3] "0 < sin e & 0 < cos e")
paulson@15229
  1820
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: mult_commute) 
paulson@15077
  1821
done
paulson@15077
  1822
paulson@15077
  1823
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
paulson@15077
  1824
apply (frule real_le_imp_less_or_eq, safe)
paulson@15077
  1825
 prefer 2 apply force
paulson@15077
  1826
apply (drule lemma_tan_total, safe)
paulson@15077
  1827
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  1828
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  1829
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  1830
apply (auto dest: cos_gt_zero)
paulson@15077
  1831
done
paulson@15077
  1832
paulson@15077
  1833
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1834
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  1835
apply (drule tan_total_pos)
paulson@15077
  1836
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  1837
apply (rule_tac [3] x = "-x" in exI)
paulson@15077
  1838
apply (auto intro!: exI)
paulson@15077
  1839
done
paulson@15077
  1840
paulson@15077
  1841
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1842
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  1843
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  1844
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  1845
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  1846
apply (rule_tac [4] Rolle)
paulson@15077
  1847
apply (rule_tac [2] Rolle)
paulson@15077
  1848
apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077
  1849
            simp add: differentiable_def)
paulson@15077
  1850
txt{*Now, simulate TRYALL*}
paulson@15077
  1851
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  1852
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
paulson@15077
  1853
	    simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym]) 
paulson@15077
  1854
done
paulson@15077
  1855
paulson@15229
  1856
lemma arcsin_pi:
paulson@15229
  1857
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1858
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
paulson@15077
  1859
apply (drule sin_total, assumption)
paulson@15077
  1860
apply (erule ex1E)
paulson@15229
  1861
apply (simp add: arcsin_def)
paulson@15077
  1862
apply (rule someI2, blast) 
paulson@15077
  1863
apply (force intro: order_trans) 
paulson@15077
  1864
done
paulson@15077
  1865
paulson@15229
  1866
lemma arcsin:
paulson@15229
  1867
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1868
      ==> -(pi/2) \<le> arcsin y &  
paulson@15077
  1869
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
paulson@15077
  1870
apply (unfold arcsin_def)
paulson@15077
  1871
apply (drule sin_total, assumption)
paulson@15077
  1872
apply (fast intro: someI2)
paulson@15077
  1873
done
paulson@15077
  1874
paulson@15077
  1875
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  1876
by (blast dest: arcsin)
paulson@15077
  1877
      
paulson@15077
  1878
lemma arcsin_bounded:
paulson@15077
  1879
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  1880
by (blast dest: arcsin)
paulson@15077
  1881
paulson@15077
  1882
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  1883
by (blast dest: arcsin)
paulson@15077
  1884
paulson@15077
  1885
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  1886
by (blast dest: arcsin)
paulson@15077
  1887
paulson@15077
  1888
lemma arcsin_lt_bounded:
paulson@15077
  1889
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  1890
apply (frule order_less_imp_le)
paulson@15077
  1891
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1892
apply (frule arcsin_bounded)
paulson@15077
  1893
apply (safe, simp)
paulson@15077
  1894
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  1895
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  1896
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  1897
done
paulson@15077
  1898
paulson@15077
  1899
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  1900
apply (unfold arcsin_def)
paulson@15077
  1901
apply (rule some1_equality)
paulson@15077
  1902
apply (rule sin_total, auto)
paulson@15077
  1903
done
paulson@15077
  1904
paulson@15229
  1905
lemma arcos:
paulson@15229
  1906
     "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1907
      ==> 0 \<le> arcos y & arcos y \<le> pi & cos(arcos y) = y"
paulson@15229
  1908
apply (simp add: arcos_def)
paulson@15077
  1909
apply (drule cos_total, assumption)
paulson@15077
  1910
apply (fast intro: someI2)
paulson@15077
  1911
done
paulson@15077
  1912
paulson@15077
  1913
lemma cos_arcos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arcos y) = y"
paulson@15077
  1914
by (blast dest: arcos)
paulson@15077
  1915
      
paulson@15077
  1916
lemma arcos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y & arcos y \<le> pi"
paulson@15077
  1917
by (blast dest: arcos)
paulson@15077
  1918
paulson@15077
  1919
lemma arcos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y"
paulson@15077
  1920
by (blast dest: arcos)
paulson@15077
  1921
paulson@15077
  1922
lemma arcos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcos y \<le> pi"
paulson@15077
  1923
by (blast dest: arcos)
paulson@15077
  1924
paulson@15229
  1925
lemma arcos_lt_bounded:
paulson@15229
  1926
     "[| -1 < y; y < 1 |]  
paulson@15077
  1927
      ==> 0 < arcos y & arcos y < pi"
paulson@15077
  1928
apply (frule order_less_imp_le)
paulson@15077
  1929
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1930
apply (frule arcos_bounded, auto)
paulson@15077
  1931
apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
paulson@15077
  1932
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  1933
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  1934
done
paulson@15077
  1935
paulson@15077
  1936
lemma arcos_cos: "[|0 \<le> x; x \<le> pi |] ==> arcos(cos x) = x"
paulson@15229
  1937
apply (simp add: arcos_def)
paulson@15077
  1938
apply (auto intro!: some1_equality cos_total)
paulson@15077
  1939
done
paulson@15077
  1940
paulson@15077
  1941
lemma arcos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arcos(cos x) = -x"
paulson@15229
  1942
apply (simp add: arcos_def)
paulson@15077
  1943
apply (auto intro!: some1_equality cos_total)
paulson@15077
  1944
done
paulson@15077
  1945
paulson@15077
  1946
lemma arctan [simp]:
paulson@15077
  1947
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
paulson@15077
  1948
apply (cut_tac y = y in tan_total)
paulson@15229
  1949
apply (simp add: arctan_def)
paulson@15077
  1950
apply (fast intro: someI2)
paulson@15077
  1951
done
paulson@15077
  1952
paulson@15077
  1953
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  1954
by auto
paulson@15077
  1955
paulson@15077
  1956
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  1957
by (auto simp only: arctan)
paulson@15077
  1958
paulson@15077
  1959
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  1960
by auto
paulson@15077
  1961
paulson@15077
  1962
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  1963
by (auto simp only: arctan)
paulson@15077
  1964
paulson@15077
  1965
lemma arctan_tan: 
paulson@15077
  1966
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077
  1967
apply (unfold arctan_def)
paulson@15077
  1968
apply (rule some1_equality)
paulson@15077
  1969
apply (rule tan_total, auto)
paulson@15077
  1970
done
paulson@15077
  1971
paulson@15077
  1972
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
paulson@15077
  1973
by (insert arctan_tan [of 0], simp)
paulson@15077
  1974
paulson@15077
  1975
lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
paulson@15077
  1976
apply (auto simp add: cos_zero_iff)
paulson@15077
  1977
apply (case_tac "n")
paulson@15077
  1978
apply (case_tac [3] "n")
paulson@15077
  1979
apply (cut_tac [2] y = x in arctan_ubound)
paulson@15077
  1980
apply (cut_tac [4] y = x in arctan_lbound) 
paulson@15077
  1981
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
paulson@15077
  1982
done
paulson@15077
  1983
paulson@15077
  1984
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077
  1985
apply (rule power_inverse [THEN subst])
paulson@15077
  1986
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
paulson@15077
  1987
apply (auto dest: realpow_not_zero 
paulson@15077
  1988
        simp add: power_mult_distrib left_distrib realpow_divide tan_def 
paulson@15077
  1989
                  mult_assoc power_inverse [symmetric] 
paulson@15077
  1990
        simp del: realpow_Suc)
paulson@15077
  1991
done
paulson@15077
  1992
paulson@15085
  1993
text{*NEEDED??*}
paulson@15229
  1994
lemma [simp]:
paulson@15229
  1995
     "sin (x + 1 / 2 * real (Suc m) * pi) =  
paulson@15229
  1996
      cos (x + 1 / 2 * real  (m) * pi)"
paulson@15229
  1997
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  1998
paulson@15085
  1999
text{*NEEDED??*}
paulson@15229
  2000
lemma [simp]:
paulson@15229
  2001
     "sin (x + real (Suc m) * pi / 2) =  
paulson@15229
  2002
      cos (x + real (m) * pi / 2)"
paulson@15229
  2003
by (simp only: cos_add sin_add real_of_nat_Suc add_divide_distrib left_distrib, auto)
paulson@15077
  2004
paulson@15077
  2005
lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
paulson@15077
  2006
apply (rule lemma_DERIV_subst)
paulson@15077
  2007
apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  2008
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  2009
apply (simp (no_asm))
paulson@15077
  2010
done
paulson@15077
  2011
paulson@15383
  2012
lemma sin_cos_npi [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
paulson@15383
  2013
proof -
paulson@15383
  2014
  have "sin ((real n + 1/2) * pi) = cos (real n * pi)"
paulson@15383
  2015
    by (auto simp add: right_distrib sin_add left_distrib mult_ac)
paulson@15383
  2016
  thus ?thesis
paulson@15383
  2017
    by (simp add: real_of_nat_Suc left_distrib add_divide_distrib 
paulson@15383
  2018
                  mult_commute [of pi])
paulson@15383
  2019
qed
paulson@15077
  2020
paulson@15077
  2021
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
paulson@15077
  2022
by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
paulson@15077
  2023
paulson@15077
  2024
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
paulson@15077
  2025
apply (subgoal_tac "3/2 = (1+1 / 2::real)")
paulson@15077
  2026
apply (simp only: left_distrib) 
paulson@15077
  2027
apply (auto simp add: cos_add mult_ac)
paulson@15077
  2028
done
paulson@15077
  2029
paulson@15077
  2030
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
paulson@15077
  2031
by (auto simp add: mult_assoc)
paulson@15077
  2032
paulson@15077
  2033
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
paulson@15077
  2034
apply (subgoal_tac "3/2 = (1+1 / 2::real)")
paulson@15077
  2035
apply (simp only: left_distrib) 
paulson@15077
  2036
apply (auto simp add: sin_add mult_ac)
paulson@15077
  2037
done
paulson@15077
  2038
paulson@15077
  2039
(*NEEDED??*)
paulson@15229
  2040
lemma [simp]:
paulson@15229
  2041
     "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
paulson@15077
  2042
apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
paulson@15077
  2043
done
paulson@15077
  2044
paulson@15077
  2045
(*NEEDED??*)
paulson@15077
  2046
lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
paulson@15229
  2047
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib add_divide_distrib, auto)
paulson@15077
  2048
paulson@15077
  2049
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
paulson@15229
  2050
by (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib add_divide_distrib, auto)
paulson@15077
  2051
paulson@15077
  2052
lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
paulson@15077
  2053
apply (rule lemma_DERIV_subst)
paulson@15077
  2054
apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
paulson@15077
  2055
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
paulson@15077
  2056
apply (simp (no_asm))
paulson@15077
  2057
done
paulson@15077
  2058
paulson@15077
  2059
lemma isCont_cos [simp]: "isCont cos x"
paulson@15077
  2060
by (rule DERIV_cos [THEN DERIV_isCont])
paulson@15077
  2061
paulson@15077
  2062
lemma isCont_sin [simp]: "isCont sin x"
paulson@15077
  2063
by (rule DERIV_sin [THEN DERIV_isCont])
paulson@15077
  2064
paulson@15077
  2065
lemma isCont_exp [simp]: "isCont exp x"
paulson@15077
  2066
by (rule DERIV_exp [THEN DERIV_isCont])
paulson@15077
  2067
paulson@15081
  2068
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
nipkow@15539
  2069
by (auto simp add: sin_zero_iff even_mult_two_ex)
paulson@15077
  2070
paulson@15077
  2071
lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
paulson@15077
  2072
apply auto
paulson@15077
  2073
apply (drule_tac f = ln in arg_cong, auto)
paulson@15077
  2074
done
paulson@15077
  2075
paulson@15077
  2076
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
paulson@15077
  2077
by (cut_tac x = x in sin_cos_squared_add3, auto)
paulson@15077
  2078
paulson@15077
  2079
paulson@15229
  2080
lemma real_root_less_mono:
paulson@15229
  2081
     "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
paulson@15077
  2082
apply (frule order_le_less_trans, assumption)
paulson@15077
  2083
apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
paulson@15077
  2084
apply (rotate_tac 1, assumption)
paulson@15077
  2085
apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
paulson@15077
  2086
apply (rotate_tac 3, assumption)
paulson@15077
  2087
apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
paulson@15077
  2088
apply (frule_tac n = n in real_root_pos_pos_le)
paulson@15077
  2089
apply (frule_tac n = n in real_root_pos_pos)
paulson@15077
  2090
apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
paulson@15077
  2091
apply (assumption, assumption)
paulson@15077
  2092
apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
paulson@15077
  2093
apply auto
paulson@15229
  2094
apply (drule_tac f = "%x. x ^ (Suc n)" in arg_cong)
paulson@15077
  2095
apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
paulson@15077
  2096
done
paulson@15077
  2097
paulson@15229
  2098
lemma real_root_le_mono:
paulson@15229
  2099
     "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
paulson@15077
  2100
apply (drule_tac y = y in order_le_imp_less_or_eq)
paulson@15077
  2101
apply (auto dest: real_root_less_mono intro: order_less_imp_le)
paulson@15077
  2102
done
paulson@15077
  2103
paulson@15229
  2104
lemma real_root_less_iff [simp]:
paulson@15229
  2105
     "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
paulson@15077
  2106
apply (auto intro: real_root_less_mono)
paulson@15077
  2107
apply (rule ccontr, drule linorder_not_less [THEN iffD1])
paulson@15077
  2108
apply (drule_tac x = y and n = n in real_root_le_mono, auto)
paulson@15077
  2109
done
paulson@15077
  2110
paulson@15229
  2111
lemma real_root_le_iff [simp]:
paulson@15229
  2112
     "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)"
paulson@15077
  2113
apply (auto intro: real_root_le_mono)
paulson@15077
  2114
apply (simp (no_asm) add: linorder_not_less [symmetric])
paulson@15077
  2115
apply auto
paulson@15077
  2116
apply (drule_tac x = y and n = n in real_root_less_mono, auto)
paulson@15077
  2117
done
paulson@15077
  2118
paulson@15229
  2119
lemma real_root_eq_iff [simp]:
paulson@15229
  2120
     "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
paulson@15077
  2121
apply (auto intro!: order_antisym)
paulson@15077
  2122
apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
paulson@15077
  2123
apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
paulson@15077
  2124
done
paulson@15077
  2125
paulson@15229
  2126
lemma real_root_pos_unique:
paulson@15229
  2127
     "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
paulson@15077
  2128
by (auto dest: real_root_pos2 simp del: realpow_Suc)
paulson@15077
  2129
paulson@15229
  2130
lemma real_root_mult:
paulson@15229
  2131
     "[| 0 \<le> x; 0 \<le> y |] 
paulson@15077
  2132
      ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
paulson@15077
  2133
apply (rule real_root_pos_unique)
paulson@15234
  2134
apply (auto intro!: real_root_pos_pos_le 
paulson@15234
  2135
            simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 
paulson@15234
  2136
            simp del: realpow_Suc)
paulson@15077
  2137
done
paulson@15077
  2138
paulson@15229
  2139
lemma real_root_inverse:
paulson@15229
  2140
     "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
paulson@15077
  2141
apply (rule real_root_pos_unique)
paulson@15234
  2142
apply (auto intro: real_root_pos_pos_le 
paulson@15234
  2143
            simp add: power_inverse [symmetric] real_root_pow_pos2 
paulson@15234
  2144
            simp del: realpow_Suc)
paulson@15077
  2145
done
paulson@15077
  2146
paulson@15077
  2147
lemma real_root_divide: 
paulson@15077
  2148
     "[| 0 \<le> x; 0 \<le> y |]  
paulson@15077
  2149
      ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
paulson@15229
  2150
apply (simp add: divide_inverse)
paulson@15077
  2151
apply (auto simp add: real_root_mult real_root_inverse)
paulson@15077
  2152
done
paulson@15077
  2153
paulson@15077
  2154
lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
paulson@15229
  2155
by (simp add: sqrt_def)
paulson@15077
  2156
paulson@15077
  2157
lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
paulson@15229
  2158
by (simp add: sqrt_def)
paulson@15229
  2159
paulson@15229
  2160
lemma real_sqrt_less_iff [simp]:
paulson@15229
  2161
     "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
paulson@15229
  2162
by (simp add: sqrt_def)
paulson@15229
  2163
paulson@15229
  2164
lemma real_sqrt_le_iff [simp]:
paulson@15229
  2165
     "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
paulson@15229
  2166
by (simp add: sqrt_def)
paulson@15229
  2167
paulson@15229
  2168
lemma real_sqrt_eq_iff [simp]:
paulson@15229
  2169
     "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
paulson@15229
  2170
by (simp add: sqrt_def)
paulson@15077
  2171
paulson@15077
  2172
lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
paulson@15077
  2173
apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077
  2174
apply (rule_tac [2] real_sqrt_less_mono)
paulson@15077
  2175
apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
paulson@15077
  2176
done
paulson@15077
  2177
paulson@15077
  2178
lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
paulson@15077
  2179
apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077
  2180
apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
paulson@15077
  2181
done
paulson@15077
  2182
paulson@15077
  2183
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
paulson@15229
  2184
apply (simp add: divide_inverse)
paulson@15077
  2185
apply (case_tac "r=0")
nipkow@15539
  2186
apply (auto simp add: mult_ac)
paulson@15077
  2187
done
paulson@15077
  2188
paulson@15077
  2189
paulson@15077
  2190
subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}
paulson@15077
  2191
paulson@15228
  2192
lemma le_real_sqrt_sumsq [simp]: "x \<le> sqrt (x * x + y * y)"
paulson@15228
  2193
proof (rule order_trans)
paulson@15228
  2194
  show "x \<le> sqrt(x*x)" by (simp add: abs_if) 
paulson@15228
  2195
  show "sqrt (x * x) \<le> sqrt (x * x + y * y)"
paulson@15228
  2196
    by (rule real_sqrt_le_mono, auto) 
paulson@15228
  2197
qed
paulson@15228
  2198
paulson@15228
  2199
lemma minus_le_real_sqrt_sumsq [simp]: "-x \<le> sqrt (x * x + y * y)"
paulson@15228
  2200
proof (rule order_trans)
paulson@15228
  2201
  show "-x \<le> sqrt(x*x)" by (simp add: abs_if) 
paulson@15228
  2202
  show "sqrt (x * x) \<le> sqrt (x * x + y * y)"
paulson@15228
  2203
    by (rule real_sqrt_le_mono, auto) 
paulson@15228
  2204
qed
paulson@15077
  2205
paulson@15077
  2206
lemma lemma_real_divide_sqrt_ge_minus_one:
paulson@15228
  2207
     "0 < x ==> -1 \<le> x/(sqrt (x * x + y * y))" 
paulson@15228
  2208
by (simp add: divide_const_simps linorder_not_le [symmetric])
paulson@15077
  2209
paulson@15077
  2210
lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
paulson@15077
  2211
apply (rule real_sqrt_gt_zero)
paulson@15077
  2212
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077
  2213
apply (auto simp add: zero_less_mult_iff)
paulson@15077
  2214
done
paulson@15077
  2215
paulson@15077
  2216
lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
paulson@15077
  2217
apply (rule real_sqrt_gt_zero)
paulson@15077
  2218
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077
  2219
apply (auto simp add: zero_less_mult_iff)
paulson@15077
  2220
done
paulson@15077
  2221
paulson@15077
  2222
lemma real_sqrt_sum_squares_gt_zero3: "x \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2223
apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077
  2224
apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
paulson@15077
  2225
done
paulson@15077
  2226
paulson@15077
  2227
lemma real_sqrt_sum_squares_gt_zero3a: "y \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2228
apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
paulson@15077
  2229
apply (auto simp add: real_add_commute)
paulson@15077
  2230
done
paulson@15077
  2231
paulson@15077
  2232
lemma real_sqrt_sum_squares_eq_cancel [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = x ==> y = 0"
paulson@15077
  2233
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, auto)
paulson@15077
  2234
paulson@15077
  2235
lemma real_sqrt_sum_squares_eq_cancel2 [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = y ==> x = 0"
paulson@15077
  2236
apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
paulson@15077
  2237
apply (simp add: real_add_commute)
paulson@15077
  2238
done
paulson@15077
  2239
paulson@15077
  2240
lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077
  2241
by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)
paulson@15077
  2242
paulson@15077
  2243
lemma lemma_real_divide_sqrt_ge_minus_one2:
paulson@15077
  2244
     "x < 0 ==> -1 \<le> x/(sqrt (x * x + y * y))"
paulson@15229
  2245
apply (simp add: divide_const_simps) 
paulson@15229
  2246
apply (insert minus_le_real_sqrt_sumsq [of x y], arith)
paulson@15077
  2247
done
paulson@15077
  2248
paulson@15077
  2249
lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077
  2250
by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)
paulson@15077
  2251
paulson@15228
  2252
lemma minus_sqrt_le: "- sqrt (x * x + y * y) \<le> x"
paulson@15228
  2253
by (insert minus_le_real_sqrt_sumsq [of x y], arith) 
paulson@15228
  2254
paulson@15228
  2255
lemma minus_sqrt_le2: "- sqrt (x * x + y * y) \<le> y"
paulson@15228
  2256
by (subst add_commute, simp add: minus_sqrt_le) 
paulson@15228
  2257
paulson@15228
  2258
lemma not_neg_sqrt_sumsq: "~ sqrt (x * x + y * y) < 0"
paulson@15228
  2259
by (simp add: linorder_not_less)
paulson@15077
  2260
paulson@15077
  2261
lemma cos_x_y_ge_minus_one: "-1 \<le> x / sqrt (x * x + y * y)"
paulson@15229
  2262
by (simp add: minus_sqrt_le not_neg_sqrt_sumsq divide_const_simps)
paulson@15077
  2263
paulson@15077
  2264
lemma cos_x_y_ge_minus_one1a [simp]: "-1 \<le> y / sqrt (x * x + y * y)"
paulson@15229
  2265
by (subst add_commute, simp add: cos_x_y_ge_minus_one)
paulson@15077
  2266
paulson@15228
  2267
lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) \<le> 1" 
paulson@15077
  2268
apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
paulson@15077
  2269
apply (rule lemma_real_divide_sqrt_le_one)
paulson@15077
  2270
apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
paulson@15077
  2271
done
paulson@15077
  2272
paulson@15077
  2273
lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) \<le> 1"
paulson@15077
  2274
apply (cut_tac x = y and y = x in cos_x_y_le_one)
paulson@15077
  2275
apply (simp add: real_add_commute)
paulson@15077
  2276
done
paulson@15077
  2277
paulson@15077
  2278
declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2279
declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2280
paulson@15077
  2281
declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077
  2282
declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077
  2283
paulson@15077
  2284
lemma cos_abs_x_y_ge_minus_one [simp]:
paulson@15077
  2285
     "-1 \<le> \<bar>x\<bar> / sqrt (x * x + y * y)"
paulson@15228
  2286
by (auto simp add: divide_const_simps abs_if linorder_not_le [symmetric]) 
paulson@15077
  2287
paulson@15077
  2288
lemma cos_abs_x_y_le_one [simp]: "\<bar>x\<bar> / sqrt (x * x + y * y) \<le> 1"
paulson@15228
  2289
apply (insert minus_le_real_sqrt_sumsq [of x y] le_real_sqrt_sumsq [of x y]) 
paulson@15228
  2290
apply (auto simp add: divide_const_simps abs_if linorder_neq_iff) 
paulson@15077
  2291
done
paulson@15077
  2292
paulson@15077
  2293
declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077
  2294
declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077
  2295
paulson@15077
  2296
lemma minus_pi_less_zero: "-pi < 0"
paulson@15228
  2297
by simp
paulson@15228
  2298
paulson@15077
  2299
declare minus_pi_less_zero [simp]
paulson@15077
  2300
declare minus_pi_less_zero [THEN order_less_imp_le, simp]
paulson@15077
  2301
paulson@15077
  2302
lemma arcos_ge_minus_pi: "[| -1 \<le> y; y \<le> 1 |] ==> -pi \<le> arcos y"
paulson@15077
  2303
apply (rule real_le_trans)
paulson@15077
  2304
apply (rule_tac [2] arcos_lbound, auto)
paulson@15077
  2305
done
paulson@15077
  2306
paulson@15077
  2307
declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2308
paulson@15077
  2309
(* How tedious! *)
paulson@15077
  2310
lemma lemma_divide_rearrange:
paulson@15077
  2311
     "[| x + (y::real) \<noteq> 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
paulson@15077
  2312
apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
paulson@15077
  2313
apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
paulson@15077
  2314
prefer 2 apply assumption
paulson@15077
  2315
apply (rotate_tac [2] 2)
paulson@15077
  2316
apply (drule_tac [2] mult_assoc [THEN subst])
paulson@15077
  2317
apply (rotate_tac [2] 2)
paulson@15077
  2318
apply (frule_tac [2] left_inverse [THEN subst])
paulson@15077
  2319
prefer 2 apply assumption
paulson@15229
  2320
apply (erule_tac [2] V = "(1 - z) * (x + y) = x / (x + y) * (x + y)" in thin_rl)
paulson@15229
  2321
apply (erule_tac [2] V = "1 - z = x / (x + y)" in thin_rl)
paulson@15077
  2322
apply (auto simp add: mult_assoc)
paulson@15077
  2323
apply (auto simp add: right_distrib left_diff_distrib)
paulson@15077
  2324
done
paulson@15077
  2325
paulson@15077
  2326
lemma lemma_cos_sin_eq:
paulson@15077
  2327
     "[| 0 < x * x + y * y;  
paulson@15077
  2328
         1 - (sin xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2 |] 
paulson@15077
  2329
      ==> (sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2"
paulson@15077
  2330
by (auto intro: lemma_divide_rearrange
paulson@15077
  2331
         simp add: realpow_divide power2_eq_square [symmetric])
paulson@15077
  2332
paulson@15077
  2333
paulson@15077
  2334
lemma lemma_sin_cos_eq:
paulson@15077
  2335
     "[| 0 < x * x + y * y;  
paulson@15077
  2336
         1 - (cos xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2 |]
paulson@15077
  2337
      ==> (cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2"
paulson@15077
  2338
apply (auto simp add: realpow_divide power2_eq_square [symmetric])
paulson@15085
  2339
apply (subst add_commute)
paulson@15085
  2340
apply (rule lemma_divide_rearrange, simp add: real_add_eq_0_iff)
paulson@15077
  2341
apply (simp add: add_commute)
paulson@15077
  2342
done
paulson@15077
  2343
paulson@15077
  2344
lemma sin_x_y_disj:
paulson@15077
  2345
     "[| x \<noteq> 0;  
paulson@15077
  2346
         cos xa = x / sqrt (x * x + y * y)  
paulson@15077
  2347
      |] ==>  sin xa = y / sqrt (x * x + y * y) |  
paulson@15077
  2348
              sin xa = - y / sqrt (x * x + y * y)"
paulson@15077
  2349
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077
  2350
apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077
  2351
apply (simp add: cos_squared_eq)
paulson@15077
  2352
apply (subgoal_tac "(sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2")
paulson@15077
  2353
apply (rule_tac [2] lemma_cos_sin_eq)
paulson@15077
  2354
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077
  2355
done
paulson@15077
  2356
paulson@15077
  2357
lemma lemma_cos_not_eq_zero: "x \<noteq> 0 ==> x / sqrt (x * x + y * y) \<noteq> 0"
paulson@15229
  2358
apply (simp add: divide_inverse)
paulson@15077
  2359
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
  2360
apply (auto simp add: power2_eq_square)
paulson@15077
  2361
done
paulson@15077
  2362
paulson@15229
  2363
lemma cos_x_y_disj:
paulson@15229
  2364
     "[| x \<noteq> 0;  
paulson@15077
  2365
         sin xa = y / sqrt (x * x + y * y)  
paulson@15077
  2366
      |] ==>  cos xa = x / sqrt (x * x + y * y) |  
paulson@15077
  2367
              cos xa = - x / sqrt (x * x + y * y)"
paulson@15077
  2368
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077
  2369
apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077
  2370
apply (simp add: sin_squared_eq del: realpow_Suc)
paulson@15077
  2371
apply (subgoal_tac "(cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2")
paulson@15077
  2372
apply (rule_tac [2] lemma_sin_cos_eq)
paulson@15077
  2373
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077
  2374
done
paulson@15077
  2375
paulson@15077
  2376
lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
paulson@15229
  2377
apply (case_tac "x = 0", auto)
paulson@15077
  2378
apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
paulson@15079
  2379
apply (auto simp add: zero_less_mult_iff divide_inverse power2_eq_square)
paulson@15077
  2380
done
paulson@15077
  2381
paulson@15229
  2382
lemma polar_ex1:
paulson@15229
  2383
     "[| x \<noteq> 0; 0 < y |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15229
  2384
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
paulson@15077
  2385
apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
paulson@15077
  2386
apply auto
paulson@15077
  2387
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2388
apply (auto simp add: power2_eq_square)
paulson@15229
  2389
apply (simp add: arcos_def)
paulson@15077
  2390
apply (cut_tac x1 = x and y1 = y 
paulson@15077
  2391
       in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
paulson@15077
  2392
apply (rule someI2_ex, blast)
paulson@15229
  2393
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
  2394
apply (frule sin_x_y_disj, blast)
paulson@15077
  2395
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2396
apply (auto simp add: power2_eq_square)
paulson@15077
  2397
apply (drule sin_ge_zero, assumption)
paulson@15077
  2398
apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
paulson@15077
  2399
done
paulson@15077
  2400
paulson@15077
  2401
lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
paulson@15085
  2402
by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)
paulson@15077
  2403
paulson@15229
  2404
lemma polar_ex2:
paulson@15229
  2405
     "[| x \<noteq> 0; y < 0 |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2406
apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
paulson@15228
  2407
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>)" in exI)
paulson@15228
  2408
apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI) 
paulson@15085
  2409
apply (auto dest: real_sum_squares_cancel2a 
paulson@15085
  2410
            simp add: power2_eq_square real_0_le_add_iff real_add_eq_0_iff)
paulson@15077
  2411
apply (unfold arcsin_def)
paulson@15077
  2412
apply (cut_tac x1 = x and y1 = y 
paulson@15077
  2413
       in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
paulson@15077
  2414
apply (rule someI2_ex, blast)
paulson@15228
  2415
apply (erule_tac V = "EX! v. ?P v" in thin_rl)
paulson@15085
  2416
apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast)
paulson@15085
  2417
apply (auto simp add: real_0_le_add_iff real_add_eq_0_iff)
paulson@15077
  2418
apply (drule cos_ge_zero, force)
paulson@15077
  2419
apply (drule_tac x = y in real_sqrt_divide_less_zero)
paulson@15085
  2420
apply (auto simp add: add_commute)
paulson@15077
  2421
apply (insert polar_ex1 [of x "-y"], simp, clarify) 
paulson@15077
  2422
apply (rule_tac x = r in exI)
paulson@15077
  2423
apply (rule_tac x = "-a" in exI, simp) 
paulson@15077
  2424
done
paulson@15077
  2425
paulson@15077
  2426
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2427
apply (case_tac "x = 0", auto)
paulson@15077
  2428
apply (rule_tac x = y in exI)
paulson@15077
  2429
apply (rule_tac x = "pi/2" in exI, auto)
paulson@15077
  2430
apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
paulson@15077
  2431
apply (rule_tac [2] x = x in exI)
paulson@15077
  2432
apply (rule_tac [2] x = 0 in exI, auto)
paulson@15077
  2433
apply (blast intro: polar_ex1 polar_ex2)+
paulson@15077
  2434
done
paulson@15077
  2435
paulson@15077
  2436
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2437
apply (rule_tac n = 1 in realpow_increasing)
paulson@15077
  2438
apply (auto simp add: numeral_2_eq_2 [symmetric])
paulson@15077
  2439
done
paulson@15077
  2440
paulson@15077
  2441
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2442
apply (rule real_add_commute [THEN subst])
paulson@15077
  2443
apply (rule real_sqrt_ge_abs1)
paulson@15077
  2444
done
paulson@15077
  2445
declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]
paulson@15077
  2446
paulson@15077
  2447
lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
paulson@15077
  2448
by (auto intro: real_sqrt_gt_zero)
paulson@15077
  2449
paulson@15077
  2450
lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
paulson@15077
  2451
by (auto intro: real_sqrt_ge_zero)
paulson@15077
  2452
paulson@15077
  2453
lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
paulson@15077
  2454
apply (rule order_less_le_trans [of _ "7/5"], simp) 
paulson@15077
  2455
apply (rule_tac n = 1 in realpow_increasing)
paulson@15077
  2456
  prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
nipkow@15539
  2457
apply (simp_all add: numeral_2_eq_2)
paulson@15077
  2458
done
paulson@15077
  2459