src/HOL/Hyperreal/Transcendental.thy
author paulson
Mon Jul 26 17:34:52 2004 +0200 (2004-07-26)
changeset 15077 89840837108e
parent 15013 34264f5e4691
child 15079 2ef899e4526d
permissions -rw-r--r--
converting Hyperreal/Transcendental to Isar script
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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 = NthRoot + Fact + HSeries + EvenOdd + Lim:
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(*????FOR RING_AND_FIELD*)
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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 (unfold 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 (unfold 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 (unfold 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 simp add: order_less_irrefl)
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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 (unfold 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 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 (unfold 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 simp add: order_less_irrefl)
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done
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subsection{*Square Root*}
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(*lcp: needed now 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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apply (simp (no_asm) del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 add: nat_numeral_0_eq_0 [symmetric])
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done
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lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
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by (unfold sqrt_def, auto)
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lemma real_sqrt_one [simp]: "sqrt 1 = 1"
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by (unfold sqrt_def, auto)
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lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
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apply (unfold 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 add: real_sqrt_pow2_iff)
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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 (unfold sqrt_def)
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apply (subst numeral_2_eq_2)
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apply (auto intro: real_root_pow_pos2 [THEN ssubst] real_root_pos2 [THEN ssubst] simp del: realpow_Suc)
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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 (unfold sqrt_def root_def)
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apply (subst numeral_2_eq_2)
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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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(*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 (unfold 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: "[|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]: "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: real_sqrt_pow2_iff 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 abs_mult)
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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 order_less_irrefl)
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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: abs_mult mult_assoc [symmetric] simp del: fact_Suc)
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apply (rule mult_right_mono)
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apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
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apply (subst fact_Suc)
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apply (subst real_of_nat_mult)
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apply (auto simp add: abs_mult inverse_mult_distrib)
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apply (auto simp add: mult_assoc [symmetric] abs_eqI2 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 abs_inverse)
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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 (unfold real_divide_def)
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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: power_abs [symmetric] abs_mult 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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   299
apply (unfold real_divide_def)
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   300
apply (rule_tac g = " (%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n) " in summable_comparison_test)
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   301
apply (rule_tac [2] summable_exp)
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   302
apply (rule_tac x = 0 in exI)
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   303
apply (auto simp add: power_abs [symmetric] abs_mult zero_le_mult_iff)
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   304
done
paulson@15077
   305
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   306
lemma lemma_STAR_sin [simp]: "(if even n then 0  
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   307
       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
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   308
apply (induct_tac "n", auto)
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   309
done
paulson@15077
   310
paulson@15077
   311
lemma lemma_STAR_cos [simp]: "0 < n -->  
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   312
      (- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
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   313
apply (induct_tac "n", auto)
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   314
done
paulson@15077
   315
paulson@15077
   316
lemma lemma_STAR_cos1 [simp]: "0 < n -->  
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   317
      (-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
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   318
apply (induct_tac "n", auto)
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   319
done
paulson@15077
   320
paulson@15077
   321
lemma lemma_STAR_cos2 [simp]: "sumr 1 n (%n. if even n  
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   322
                    then (- 1) ^ (n div 2)/(real (fact n)) *  
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   323
                          0 ^ n  
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   324
                    else 0) = 0"
paulson@15077
   325
apply (induct_tac "n")
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   326
apply (case_tac [2] "n", auto)
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   327
done
paulson@15077
   328
paulson@15077
   329
lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
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   330
apply (unfold exp_def)
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   331
apply (rule summable_exp [THEN summable_sums])
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   332
done
paulson@15077
   333
paulson@15077
   334
lemma sin_converges: 
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   335
      "(%n. (if even n then 0  
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   336
            else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
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   337
                 x ^ n) sums sin(x)"
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   338
apply (unfold sin_def)
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   339
apply (rule summable_sin [THEN summable_sums])
paulson@15077
   340
done
paulson@15077
   341
paulson@15077
   342
lemma cos_converges: 
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   343
      "(%n. (if even n then  
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   344
           (- 1) ^ (n div 2)/(real (fact n))  
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   345
           else 0) * x ^ n) sums cos(x)"
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   346
apply (unfold cos_def)
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   347
apply (rule summable_cos [THEN summable_sums])
paulson@15077
   348
done
paulson@15077
   349
paulson@15077
   350
lemma lemma_realpow_diff [rule_format (no_asm)]: "p \<le> n --> y ^ (Suc n - p) = ((y::real) ^ (n - p)) * y"
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   351
apply (induct_tac "n", auto)
paulson@15077
   352
apply (subgoal_tac "p = Suc n")
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   353
apply (simp (no_asm_simp), auto)
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   354
apply (drule sym)
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   355
apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric] 
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   356
       del: realpow_Suc)
paulson@15077
   357
done
paulson@15077
   358
paulson@15077
   359
paulson@15077
   360
subsection{*Properties of Power Series*}
paulson@15077
   361
paulson@15077
   362
lemma lemma_realpow_diff_sumr:
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   363
     "sumr 0 (Suc n) (%p. (x ^ p) * y ^ ((Suc n) - p)) =  
paulson@15077
   364
      y * sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p)))"
paulson@15077
   365
apply (auto simp add: sumr_mult simp del: sumr_Suc)
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   366
apply (rule sumr_subst)
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   367
apply (intro strip)
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   368
apply (subst lemma_realpow_diff)
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   369
apply (auto simp add: mult_ac)
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   370
done
paulson@15077
   371
paulson@15077
   372
lemma lemma_realpow_diff_sumr2: "x ^ (Suc n) - y ^ (Suc n) =  
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   373
      (x - y) * sumr 0 (Suc n) (%p. (x ^ p) * (y ^(n - p)))"
paulson@15077
   374
apply (induct_tac "n", simp)
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   375
apply (auto simp del: sumr_Suc)
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   376
apply (subst sumr_Suc)
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   377
apply (drule sym)
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   378
apply (auto simp add: lemma_realpow_diff_sumr right_distrib real_diff_def mult_ac simp del: sumr_Suc)
paulson@15077
   379
done
paulson@15077
   380
paulson@15077
   381
lemma lemma_realpow_rev_sumr: "sumr 0 (Suc n) (%p. (x ^ p) * (y ^ (n - p))) =  
paulson@15077
   382
      sumr 0 (Suc n) (%p. (x ^ (n - p)) * (y ^ p))"
paulson@15077
   383
apply (case_tac "x = y")
paulson@15077
   384
apply (auto simp add: mult_commute power_add [symmetric] simp del: sumr_Suc)
paulson@15077
   385
apply (rule_tac c1 = "x - y" in real_mult_left_cancel [THEN iffD1])
paulson@15077
   386
apply (rule_tac [2] minus_minus [THEN subst], simp)
paulson@15077
   387
apply (subst minus_mult_left)
paulson@15077
   388
apply (simp add: lemma_realpow_diff_sumr2 [symmetric] del: sumr_Suc)
paulson@15077
   389
done
paulson@15077
   390
paulson@15077
   391
text{*Power series has a `circle` of convergence, i.e. if it sums for @{term
paulson@15077
   392
x}, then it sums absolutely for @{term z} with @{term "\<bar>z\<bar> < \<bar>x\<bar>"}.*}
paulson@15077
   393
paulson@15077
   394
lemma powser_insidea:
paulson@15077
   395
     "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
paulson@15077
   396
      ==> summable (%n. abs(f(n)) * (z ^ n))"
paulson@15077
   397
apply (drule summable_LIMSEQ_zero)
paulson@15077
   398
apply (drule convergentI)
paulson@15077
   399
apply (simp add: Cauchy_convergent_iff [symmetric])
paulson@15077
   400
apply (drule Cauchy_Bseq)
paulson@15077
   401
apply (simp add: Bseq_def, safe)
paulson@15077
   402
apply (rule_tac g = "%n. K * abs (z ^ n) * inverse (abs (x ^ n))" in summable_comparison_test)
paulson@15077
   403
apply (rule_tac x = 0 in exI, safe)
paulson@15077
   404
apply (subgoal_tac "0 < abs (x ^ n) ")
paulson@15077
   405
apply (rule_tac c="abs (x ^ n)" in mult_right_le_imp_le) 
paulson@15077
   406
apply (auto simp add: mult_assoc power_abs)
paulson@15077
   407
 prefer 2
paulson@15077
   408
 apply (drule_tac x = 0 in spec, force)
paulson@15077
   409
apply (auto simp add: abs_mult power_abs mult_ac)
paulson@15077
   410
apply (rule_tac a2 = "z ^ n" 
paulson@15077
   411
       in abs_ge_zero [THEN real_le_imp_less_or_eq, THEN disjE])
paulson@15077
   412
apply (auto intro!: mult_right_mono simp add: mult_assoc [symmetric] power_abs summable_def power_0_left)
paulson@15077
   413
apply (rule_tac x = "K * inverse (1 - (\<bar>z\<bar> * inverse (\<bar>x\<bar>))) " in exI)
paulson@15077
   414
apply (auto intro!: sums_mult simp add: mult_assoc)
paulson@15077
   415
apply (subgoal_tac "abs (z ^ n) * inverse (\<bar>x\<bar> ^ n) = (\<bar>z\<bar> * inverse (\<bar>x\<bar>)) ^ n")
paulson@15077
   416
apply (auto simp add: power_abs [symmetric])
paulson@15077
   417
apply (subgoal_tac "x \<noteq> 0")
paulson@15077
   418
apply (subgoal_tac [3] "x \<noteq> 0")
paulson@15077
   419
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])
paulson@15077
   420
apply (auto intro!: geometric_sums simp add: power_abs inverse_eq_divide)
paulson@15077
   421
apply (rule_tac c="\<bar>x\<bar>" in mult_right_less_imp_less) 
paulson@15077
   422
apply (auto simp add: abs_mult [symmetric] mult_assoc)
paulson@15077
   423
done
paulson@15077
   424
paulson@15077
   425
lemma powser_inside: "[| summable (%n. f(n) * (x ^ n)); \<bar>z\<bar> < \<bar>x\<bar> |]  
paulson@15077
   426
      ==> summable (%n. f(n) * (z ^ n))"
paulson@15077
   427
apply (drule_tac z = "\<bar>z\<bar>" in powser_insidea)
paulson@15077
   428
apply (auto intro: summable_rabs_cancel simp add: power_abs [symmetric])
paulson@15077
   429
done
paulson@15077
   430
paulson@15077
   431
paulson@15077
   432
subsection{*Differentiation of Power Series*}
paulson@15077
   433
paulson@15077
   434
text{*Lemma about distributing negation over it*}
paulson@15077
   435
lemma diffs_minus: "diffs (%n. - c n) = (%n. - diffs c n)"
paulson@15077
   436
by (simp add: diffs_def)
paulson@15077
   437
paulson@15077
   438
text{*Show that we can shift the terms down one*}
paulson@15077
   439
lemma lemma_diffs:
paulson@15077
   440
     "sumr 0 n (%n. (diffs c)(n) * (x ^ n)) =  
paulson@15077
   441
      sumr 0 n (%n. real n * c(n) * (x ^ (n - Suc 0))) +  
paulson@15077
   442
      (real n * c(n) * x ^ (n - Suc 0))"
paulson@15077
   443
apply (induct_tac "n")
paulson@15077
   444
apply (auto simp add: mult_assoc add_assoc [symmetric] diffs_def)
paulson@15077
   445
done
paulson@15077
   446
paulson@15077
   447
lemma lemma_diffs2: "sumr 0 n (%n. real n * c(n) * (x ^ (n - Suc 0))) =  
paulson@15077
   448
      sumr 0 n (%n. (diffs c)(n) * (x ^ n)) -  
paulson@15077
   449
      (real n * c(n) * x ^ (n - Suc 0))"
paulson@15077
   450
by (auto simp add: lemma_diffs)
paulson@15077
   451
paulson@15077
   452
paulson@15077
   453
lemma diffs_equiv: "summable (%n. (diffs c)(n) * (x ^ n)) ==>  
paulson@15077
   454
      (%n. real n * c(n) * (x ^ (n - Suc 0))) sums  
paulson@15077
   455
         (suminf(%n. (diffs c)(n) * (x ^ n)))"
paulson@15077
   456
apply (subgoal_tac " (%n. real n * c (n) * (x ^ (n - Suc 0))) ----> 0")
paulson@15077
   457
apply (rule_tac [2] LIMSEQ_imp_Suc)
paulson@15077
   458
apply (drule summable_sums) 
paulson@15077
   459
apply (auto simp add: sums_def)
paulson@15077
   460
apply (drule_tac X="(\<lambda>n. \<Sum>n = 0..<n. diffs c n * x ^ n)" in LIMSEQ_diff)
paulson@15077
   461
apply (auto simp add: lemma_diffs2 [symmetric] diffs_def [symmetric])
paulson@15077
   462
apply (simp add: diffs_def summable_LIMSEQ_zero)
paulson@15077
   463
done
paulson@15077
   464
paulson@15077
   465
paulson@15077
   466
subsection{*Term-by-Term Differentiability of Power Series*}
paulson@15077
   467
paulson@15077
   468
lemma lemma_termdiff1:
paulson@15077
   469
     "sumr 0 m (%p. (((z + h) ^ (m - p)) * (z ^ p)) - (z ^ m)) =  
paulson@15077
   470
        sumr 0 m (%p. (z ^ p) *  
paulson@15077
   471
        (((z + h) ^ (m - p)) - (z ^ (m - p))))"
paulson@15077
   472
apply (rule sumr_subst)
paulson@15077
   473
apply (auto simp add: right_distrib real_diff_def power_add [symmetric] mult_ac)
paulson@15077
   474
done
paulson@15077
   475
paulson@15077
   476
lemma less_add_one: "m < n ==> (\<exists>d. n = m + d + Suc 0)"
paulson@15077
   477
by (simp add: less_iff_Suc_add)
paulson@15077
   478
paulson@15077
   479
lemma sumdiff: "a + b - (c + d) = a - c + b - (d::real)"
paulson@15077
   480
by arith
paulson@15077
   481
paulson@15077
   482
paulson@15077
   483
lemma lemma_termdiff2: " h \<noteq> 0 ==>  
paulson@15077
   484
        (((z + h) ^ n) - (z ^ n)) * inverse h -  
paulson@15077
   485
            real n * (z ^ (n - Suc 0)) =  
paulson@15077
   486
         h * sumr 0 (n - Suc 0) (%p. (z ^ p) *  
paulson@15077
   487
               sumr 0 ((n - Suc 0) - p)  
paulson@15077
   488
                 (%q. ((z + h) ^ q) * (z ^ (((n - 2) - p) - q))))"
paulson@15077
   489
apply (rule real_mult_left_cancel [THEN iffD1], simp (no_asm_simp))
paulson@15077
   490
apply (simp add: right_diff_distrib mult_ac)
paulson@15077
   491
apply (simp add: mult_assoc [symmetric])
paulson@15077
   492
apply (case_tac "n")
paulson@15077
   493
apply (auto simp add: lemma_realpow_diff_sumr2 
paulson@15077
   494
                      right_diff_distrib [symmetric] mult_assoc
paulson@15077
   495
            simp del: realpow_Suc sumr_Suc)
paulson@15077
   496
apply (auto simp add: lemma_realpow_rev_sumr simp del: sumr_Suc)
paulson@15077
   497
apply (auto simp add: real_of_nat_Suc sumr_diff_mult_const left_distrib 
paulson@15077
   498
                sumdiff lemma_termdiff1 sumr_mult)
paulson@15077
   499
apply (auto intro!: sumr_subst simp add: real_diff_def real_add_assoc)
paulson@15077
   500
apply (simp add: diff_minus [symmetric] less_iff_Suc_add) 
paulson@15077
   501
apply (auto simp add: sumr_mult lemma_realpow_diff_sumr2 mult_ac simp
paulson@15077
   502
                 del: sumr_Suc realpow_Suc)
paulson@15077
   503
done
paulson@15077
   504
paulson@15077
   505
lemma lemma_termdiff3: "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; abs (z + h) \<le> K |]  
paulson@15077
   506
      ==> abs (((z + h) ^ n - z ^ n) * inverse h - real n * z ^ (n - Suc 0))  
paulson@15077
   507
          \<le> real n * real (n - Suc 0) * K ^ (n - 2) * \<bar>h\<bar>"
paulson@15077
   508
apply (subst lemma_termdiff2, assumption)
paulson@15077
   509
apply (simp add: abs_mult mult_commute) 
paulson@15077
   510
apply (simp add: mult_commute [of _ "K ^ (n - 2)"]) 
paulson@15077
   511
apply (rule sumr_rabs [THEN real_le_trans])
paulson@15077
   512
apply (simp add: mult_assoc [symmetric])
paulson@15077
   513
apply (simp add: mult_commute [of _ "real (n - Suc 0)"])
paulson@15077
   514
apply (auto intro!: sumr_bound2 simp add: abs_mult)
paulson@15077
   515
apply (case_tac "n", auto)
paulson@15077
   516
apply (drule less_add_one)
paulson@15077
   517
(*CLAIM_SIMP " (a * b * c = a * (c * (b::real))" mult_ac]*)
paulson@15077
   518
apply clarify 
paulson@15077
   519
apply (subgoal_tac "K ^ p * K ^ d * real (Suc (Suc (p + d))) =
paulson@15077
   520
                    K ^ p * (real (Suc (Suc (p + d))) * K ^ d)") 
paulson@15077
   521
apply (simp (no_asm_simp) add: power_add del: sumr_Suc)
paulson@15077
   522
apply (auto intro!: mult_mono simp del: sumr_Suc)
paulson@15077
   523
apply (auto intro!: power_mono simp add: power_abs simp del: sumr_Suc)
paulson@15077
   524
apply (rule_tac j = "real (Suc d) * (K ^ d) " in real_le_trans)
paulson@15077
   525
apply (subgoal_tac [2] "0 \<le> K")
paulson@15077
   526
apply (drule_tac [2] n = d in zero_le_power)
paulson@15077
   527
apply (auto simp del: sumr_Suc)
paulson@15077
   528
apply (rule sumr_rabs [THEN real_le_trans])
paulson@15077
   529
apply (rule sumr_bound2, auto dest!: less_add_one intro!: mult_mono simp add: abs_mult power_add)
paulson@15077
   530
apply (auto intro!: power_mono zero_le_power simp add: power_abs, arith+)
paulson@15077
   531
done
paulson@15077
   532
paulson@15077
   533
lemma lemma_termdiff4: 
paulson@15077
   534
  "[| 0 < k;  
paulson@15077
   535
      (\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> abs(f h) \<le> K * \<bar>h\<bar>) |]  
paulson@15077
   536
   ==> f -- 0 --> 0"
paulson@15077
   537
apply (unfold LIM_def, auto)
paulson@15077
   538
apply (subgoal_tac "0 \<le> K")
paulson@15077
   539
apply (drule_tac [2] x = "k/2" in spec)
paulson@15077
   540
apply (frule_tac [2] real_less_half_sum)
paulson@15077
   541
apply (drule_tac [2] real_gt_half_sum)
paulson@15077
   542
apply (auto simp add: abs_eqI2)
paulson@15077
   543
apply (rule_tac [2] c = "k/2" in mult_right_le_imp_le)
paulson@15077
   544
apply (auto intro: abs_ge_zero [THEN real_le_trans])
paulson@15077
   545
apply (drule real_le_imp_less_or_eq, auto)
paulson@15077
   546
apply (subgoal_tac "0 < (r * inverse K) * inverse 2")
paulson@15077
   547
apply (rule_tac [2] real_mult_order)+
paulson@15077
   548
apply (drule_tac ?d1.0 = "r * inverse K * inverse 2" and ?d2.0 = k in real_lbound_gt_zero)
paulson@15077
   549
apply (auto simp add: positive_imp_inverse_positive zero_less_mult_iff)
paulson@15077
   550
apply (rule_tac [2] y="\<bar>f (k / 2)\<bar> * 2" in order_trans, auto)
paulson@15077
   551
apply (rule_tac x = e in exI, auto)
paulson@15077
   552
apply (rule_tac y = "K * \<bar>x\<bar>" in order_le_less_trans)
paulson@15077
   553
apply (rule_tac [2] y = "K * e" in order_less_trans)
paulson@15077
   554
apply (rule_tac [3] c = "inverse K" in mult_right_less_imp_less, force)
paulson@15077
   555
apply (simp add: mult_less_cancel_left)
paulson@15077
   556
apply (auto simp add: mult_ac)
paulson@15077
   557
done
paulson@15077
   558
paulson@15077
   559
lemma lemma_termdiff5: "[| 0 < k;  
paulson@15077
   560
            summable f;  
paulson@15077
   561
            \<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k -->  
paulson@15077
   562
                    (\<forall>n. abs(g(h) (n::nat)) \<le> (f(n) * \<bar>h\<bar>)) |]  
paulson@15077
   563
         ==> (%h. suminf(g h)) -- 0 --> 0"
paulson@15077
   564
apply (drule summable_sums)
paulson@15077
   565
apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> abs (suminf (g h)) \<le> suminf f * \<bar>h\<bar>")
paulson@15077
   566
apply (auto intro!: lemma_termdiff4 simp add: sums_summable [THEN suminf_mult, symmetric])
paulson@15077
   567
apply (subgoal_tac "summable (%n. f n * \<bar>h\<bar>) ")
paulson@15077
   568
 prefer 2
paulson@15077
   569
 apply (simp add: summable_def) 
paulson@15077
   570
 apply (rule_tac x = "suminf f * \<bar>h\<bar>" in exI)
paulson@15077
   571
 apply (drule_tac c = "\<bar>h\<bar>" in sums_mult)
paulson@15077
   572
 apply (simp add: mult_ac) 
paulson@15077
   573
apply (subgoal_tac "summable (%n. abs (g (h::real) (n::nat))) ")
paulson@15077
   574
 apply (rule_tac [2] g = "%n. f n * \<bar>h\<bar>" in summable_comparison_test)
paulson@15077
   575
  apply (rule_tac [2] x = 0 in exI, auto)
paulson@15077
   576
apply (rule_tac j = "suminf (%n. abs (g h n))" in real_le_trans)
paulson@15077
   577
apply (auto intro: summable_rabs summable_le simp add: sums_summable [THEN suminf_mult])
paulson@15077
   578
done
paulson@15077
   579
paulson@15077
   580
paulson@15077
   581
paulson@15077
   582
text{* FIXME: Long proofs*}
paulson@15077
   583
paulson@15077
   584
lemma termdiffs_aux:
paulson@15077
   585
     "[|summable (\<lambda>n. diffs (diffs c) n * K ^ n); \<bar>x\<bar> < \<bar>K\<bar> |]
paulson@15077
   586
    ==> (\<lambda>h. suminf
paulson@15077
   587
             (\<lambda>n. c n *
paulson@15077
   588
                  (((x + h) ^ n - x ^ n) * inverse h -
paulson@15077
   589
                   real n * x ^ (n - Suc 0))))
paulson@15077
   590
       -- 0 --> 0"
paulson@15077
   591
apply (drule dense, safe)
paulson@15077
   592
apply (frule real_less_sum_gt_zero)
paulson@15077
   593
apply (drule_tac
paulson@15077
   594
         f = "%n. abs (c n) * real n * real (n - Suc 0) * (r ^ (n - 2))" 
paulson@15077
   595
     and g = "%h n. c (n) * ((( ((x + h) ^ n) - (x ^ n)) * inverse h) 
paulson@15077
   596
                             - (real n * (x ^ (n - Suc 0))))" 
paulson@15077
   597
       in lemma_termdiff5)
paulson@15077
   598
apply (auto simp add: add_commute)
paulson@15077
   599
apply (subgoal_tac "summable (%n. \<bar>diffs (diffs c) n\<bar> * (r ^ n))")
paulson@15077
   600
apply (rule_tac [2] x = K in powser_insidea, auto)
paulson@15077
   601
apply (subgoal_tac [2] "\<bar>r\<bar> = r", auto)
paulson@15077
   602
apply (rule_tac [2] y1 = "\<bar>x\<bar>" in order_trans [THEN abs_eqI1], auto)
paulson@15077
   603
apply (simp add: diffs_def mult_assoc [symmetric])
paulson@15077
   604
apply (subgoal_tac
paulson@15077
   605
        "\<forall>n. real (Suc n) * real (Suc (Suc n)) * \<bar>c (Suc (Suc n))\<bar> * (r ^ n) 
paulson@15077
   606
              = diffs (diffs (%n. \<bar>c n\<bar>)) n * (r ^ n) ") 
paulson@15077
   607
apply auto
paulson@15077
   608
apply (drule diffs_equiv)
paulson@15077
   609
apply (drule sums_summable)
paulson@15077
   610
apply (simp_all add: diffs_def) 
paulson@15077
   611
apply (simp add: diffs_def mult_ac)
paulson@15077
   612
apply (subgoal_tac " (%n. real n * (real (Suc n) * (abs (c (Suc n)) * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * abs (c m) * inverse r) n * (r ^ n))")
paulson@15077
   613
apply auto
paulson@15077
   614
  prefer 2
paulson@15077
   615
  apply (rule ext)
paulson@15077
   616
  apply (simp add: diffs_def) 
paulson@15077
   617
  apply (case_tac "n", auto)
paulson@15077
   618
txt{*23*}
paulson@15077
   619
   apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077
   620
   apply (simp add: mult_ac) 
paulson@15077
   621
  apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077
   622
  apply (simp add: mult_ac) 
paulson@15077
   623
 apply (drule diffs_equiv)
paulson@15077
   624
 apply (drule sums_summable)
paulson@15077
   625
apply (subgoal_tac
paulson@15077
   626
          "summable
paulson@15077
   627
            (\<lambda>n. real n * (real (n - Suc 0) * \<bar>c n\<bar> * inverse r) *
paulson@15077
   628
                 r ^ (n - Suc 0)) =
paulson@15077
   629
           summable
paulson@15077
   630
            (\<lambda>n. real n * (\<bar>c n\<bar> * (real (n - Suc 0) * r ^ (n - 2))))")
paulson@15077
   631
apply simp 
paulson@15077
   632
apply (rule_tac f = summable in arg_cong, rule ext)
paulson@15077
   633
txt{*33*}
paulson@15077
   634
apply (case_tac "n", auto)
paulson@15077
   635
apply (case_tac "nat", auto)
paulson@15077
   636
apply (drule abs_ge_zero [THEN order_le_less_trans], auto) 
paulson@15077
   637
apply (drule abs_ge_zero [THEN order_le_less_trans])
paulson@15077
   638
apply (simp add: mult_assoc)
paulson@15077
   639
apply (rule mult_left_mono)
paulson@15077
   640
apply (rule add_commute [THEN subst])
paulson@15077
   641
apply (simp (no_asm) add: mult_assoc [symmetric])
paulson@15077
   642
apply (rule lemma_termdiff3)
paulson@15077
   643
apply (auto intro: abs_triangle_ineq [THEN order_trans], arith)
paulson@15077
   644
done
paulson@15077
   645
paulson@15077
   646
paulson@15077
   647
lemma termdiffs: 
paulson@15077
   648
    "[| summable(%n. c(n) * (K ^ n));  
paulson@15077
   649
        summable(%n. (diffs c)(n) * (K ^ n));  
paulson@15077
   650
        summable(%n. (diffs(diffs c))(n) * (K ^ n));  
paulson@15077
   651
        \<bar>x\<bar> < \<bar>K\<bar> |]  
paulson@15077
   652
     ==> DERIV (%x. suminf (%n. c(n) * (x ^ n)))  x :>  
paulson@15077
   653
             suminf (%n. (diffs c)(n) * (x ^ n))"
paulson@15077
   654
apply (unfold deriv_def)
paulson@15077
   655
apply (rule_tac g = "%h. suminf (%n. ((c (n) * ( (x + h) ^ n)) - (c (n) * (x ^ n))) * inverse h) " in LIM_trans)
paulson@15077
   656
apply (simp add: LIM_def, safe)
paulson@15077
   657
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077
   658
apply (auto simp add: less_diff_eq)
paulson@15077
   659
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077
   660
apply (rule_tac y = 0 in order_le_less_trans, auto)
paulson@15077
   661
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
   662
apply (auto intro!: summable_sums)
paulson@15077
   663
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077
   664
apply (auto simp add: add_commute)
paulson@15077
   665
apply (drule_tac x="(\<lambda>n. c n * (xa + x) ^ n)" in sums_diff, assumption) 
paulson@15077
   666
apply (drule_tac x = " (%n. c n * (xa + x) ^ n - c n * x ^ n) " and c = "inverse xa" in sums_mult)
paulson@15077
   667
apply (rule sums_unique [symmetric])
paulson@15077
   668
apply (simp add: diff_def real_divide_def add_ac mult_ac)
paulson@15077
   669
apply (rule LIM_zero_cancel)
paulson@15077
   670
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
   671
 prefer 2 apply (blast intro: termdiffs_aux) 
paulson@15077
   672
apply (simp add: LIM_def, safe)
paulson@15077
   673
apply (rule_tac x = "\<bar>K\<bar> - \<bar>x\<bar>" in exI)
paulson@15077
   674
apply (auto simp add: less_diff_eq)
paulson@15077
   675
apply (drule abs_triangle_ineq [THEN order_le_less_trans])
paulson@15077
   676
apply (rule_tac y = 0 in order_le_less_trans, auto)
paulson@15077
   677
apply (subgoal_tac "summable (%n. (diffs c) (n) * (x ^ n))")
paulson@15077
   678
apply (rule_tac [2] powser_inside, auto)
paulson@15077
   679
apply (drule_tac c = c and x = x in diffs_equiv)
paulson@15077
   680
apply (frule sums_unique, auto)
paulson@15077
   681
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
   682
apply safe
paulson@15077
   683
apply (auto intro!: summable_sums)
paulson@15077
   684
apply (rule_tac [2] powser_inside, rule_tac [4] powser_inside)
paulson@15077
   685
apply (auto simp add: add_commute)
paulson@15077
   686
apply (frule_tac x = " (%n. c n * (xa + x) ^ n) " and y = " (%n. c n * x ^ n) " in sums_diff, assumption)
paulson@15077
   687
apply (simp add: suminf_diff [OF sums_summable sums_summable] 
paulson@15077
   688
               right_diff_distrib [symmetric])
paulson@15077
   689
apply (frule_tac x = " (%n. c n * ((xa + x) ^ n - x ^ n))" and c = "inverse xa" in sums_mult)
paulson@15077
   690
apply (simp add: sums_summable [THEN suminf_mult2])
paulson@15077
   691
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
   692
apply assumption
paulson@15077
   693
apply (subst minus_equation_iff, simp (no_asm))  
paulson@15077
   694
apply (simp add: suminf_diff [OF sums_summable sums_summable] add_ac mult_ac)
paulson@15077
   695
apply (rule_tac f = suminf in arg_cong)
paulson@15077
   696
apply (rule ext)
paulson@15077
   697
apply (simp add: diff_def right_distrib add_ac mult_ac)
paulson@15077
   698
done
paulson@15077
   699
paulson@15077
   700
paulson@15077
   701
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*} 
paulson@15077
   702
paulson@15077
   703
lemma exp_fdiffs: 
paulson@15077
   704
      "diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
paulson@15077
   705
apply (unfold diffs_def)
paulson@15077
   706
apply (rule ext)
paulson@15077
   707
apply (subst fact_Suc)
paulson@15077
   708
apply (subst real_of_nat_mult)
paulson@15077
   709
apply (subst inverse_mult_distrib)
paulson@15077
   710
apply (auto simp add: mult_assoc [symmetric])
paulson@15077
   711
done
paulson@15077
   712
paulson@15077
   713
lemma sin_fdiffs: 
paulson@15077
   714
      "diffs(%n. if even n then 0  
paulson@15077
   715
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))  
paulson@15077
   716
       = (%n. if even n then  
paulson@15077
   717
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   718
              else 0)"
paulson@15077
   719
apply (unfold diffs_def real_divide_def)
paulson@15077
   720
apply (rule ext)
paulson@15077
   721
apply (subst fact_Suc)
paulson@15077
   722
apply (subst real_of_nat_mult)
paulson@15077
   723
apply (subst even_nat_Suc)
paulson@15077
   724
apply (subst inverse_mult_distrib, auto)
paulson@15077
   725
done
paulson@15077
   726
paulson@15077
   727
lemma sin_fdiffs2: 
paulson@15077
   728
       "diffs(%n. if even n then 0  
paulson@15077
   729
           else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n  
paulson@15077
   730
       = (if even n then  
paulson@15077
   731
                 (- 1) ^ (n div 2)/(real (fact n))  
paulson@15077
   732
              else 0)"
paulson@15077
   733
apply (unfold diffs_def real_divide_def)
paulson@15077
   734
apply (subst fact_Suc)
paulson@15077
   735
apply (subst real_of_nat_mult)
paulson@15077
   736
apply (subst even_nat_Suc)
paulson@15077
   737
apply (subst inverse_mult_distrib, auto)
paulson@15077
   738
done
paulson@15077
   739
paulson@15077
   740
lemma cos_fdiffs: 
paulson@15077
   741
      "diffs(%n. if even n then  
paulson@15077
   742
                 (- 1) ^ (n div 2)/(real (fact n)) else 0)  
paulson@15077
   743
       = (%n. - (if even n then 0  
paulson@15077
   744
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
paulson@15077
   745
apply (unfold diffs_def real_divide_def)
paulson@15077
   746
apply (rule ext)
paulson@15077
   747
apply (subst fact_Suc)
paulson@15077
   748
apply (subst real_of_nat_mult)
paulson@15077
   749
apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)
paulson@15077
   750
done
paulson@15077
   751
paulson@15077
   752
paulson@15077
   753
lemma cos_fdiffs2: 
paulson@15077
   754
      "diffs(%n. if even n then  
paulson@15077
   755
                 (- 1) ^ (n div 2)/(real (fact n)) else 0) n 
paulson@15077
   756
       = - (if even n then 0  
paulson@15077
   757
           else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
paulson@15077
   758
apply (unfold diffs_def real_divide_def)
paulson@15077
   759
apply (subst fact_Suc)
paulson@15077
   760
apply (subst real_of_nat_mult) 
paulson@15077
   761
apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)
paulson@15077
   762
done
paulson@15077
   763
paulson@15077
   764
text{*Now at last we can get the derivatives of exp, sin and cos*}
paulson@15077
   765
paulson@15077
   766
lemma lemma_sin_minus:
paulson@15077
   767
     "- sin x =
paulson@15077
   768
         suminf(%n. - ((if even n then 0 
paulson@15077
   769
                  else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
paulson@15077
   770
by (auto intro!: sums_unique sums_minus sin_converges)
paulson@15077
   771
paulson@15077
   772
lemma lemma_exp_ext: "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))"
paulson@15077
   773
by (auto intro!: ext simp add: exp_def)
paulson@15077
   774
paulson@15077
   775
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
paulson@15077
   776
apply (unfold exp_def)
paulson@15077
   777
apply (subst lemma_exp_ext)
paulson@15077
   778
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@15077
   779
apply (rule_tac [2] K = "1 + \<bar>x\<bar> " in termdiffs)
paulson@15077
   780
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
paulson@15077
   781
done
paulson@15077
   782
paulson@15077
   783
lemma lemma_sin_ext:
paulson@15077
   784
     "sin = (%x. suminf(%n. 
paulson@15077
   785
                   (if even n then 0  
paulson@15077
   786
                       else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *  
paulson@15077
   787
                   x ^ n))"
paulson@15077
   788
by (auto intro!: ext simp add: sin_def)
paulson@15077
   789
paulson@15077
   790
lemma lemma_cos_ext:
paulson@15077
   791
     "cos = (%x. suminf(%n. 
paulson@15077
   792
                   (if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
paulson@15077
   793
                   x ^ n))"
paulson@15077
   794
by (auto intro!: ext simp add: cos_def)
paulson@15077
   795
paulson@15077
   796
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
paulson@15077
   797
apply (unfold cos_def)
paulson@15077
   798
apply (subst lemma_sin_ext)
paulson@15077
   799
apply (auto simp add: sin_fdiffs2 [symmetric])
paulson@15077
   800
apply (rule_tac K = "1 + \<bar>x\<bar> " in termdiffs)
paulson@15077
   801
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs, arith)
paulson@15077
   802
done
paulson@15077
   803
paulson@15077
   804
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
paulson@15077
   805
apply (subst lemma_cos_ext)
paulson@15077
   806
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
paulson@15077
   807
apply (rule_tac K = "1 + \<bar>x\<bar> " in termdiffs)
paulson@15077
   808
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
   809
done
paulson@15077
   810
paulson@15077
   811
paulson@15077
   812
subsection{*Properties of the Exponential Function*}
paulson@15077
   813
paulson@15077
   814
lemma exp_zero [simp]: "exp 0 = 1"
paulson@15077
   815
proof -
paulson@15077
   816
  have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
paulson@15077
   817
        suminf (\<lambda>n. inverse (real (fact n)) * 0 ^ n)"
paulson@15077
   818
    by (rule series_zero [rule_format, THEN sums_unique],
paulson@15077
   819
        case_tac "m", auto)
paulson@15077
   820
  thus ?thesis by (simp add:  exp_def) 
paulson@15077
   821
qed
paulson@15077
   822
paulson@15077
   823
lemma exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> exp(x)"
paulson@15077
   824
apply (drule real_le_imp_less_or_eq, auto)
paulson@15077
   825
apply (unfold exp_def)
paulson@15077
   826
apply (rule real_le_trans)
paulson@15077
   827
apply (rule_tac [2] n = 2 and f = " (%n. inverse (real (fact n)) * x ^ n) " in series_pos_le)
paulson@15077
   828
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
paulson@15077
   829
done
paulson@15077
   830
paulson@15077
   831
lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
paulson@15077
   832
apply (rule order_less_le_trans)
paulson@15077
   833
apply (rule_tac [2] exp_ge_add_one_self, auto)
paulson@15077
   834
done
paulson@15077
   835
paulson@15077
   836
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
paulson@15077
   837
proof -
paulson@15077
   838
  have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
paulson@15077
   839
    by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const) 
paulson@15077
   840
  thus ?thesis by (simp add: o_def)
paulson@15077
   841
qed
paulson@15077
   842
paulson@15077
   843
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
paulson@15077
   844
proof -
paulson@15077
   845
  have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
paulson@15077
   846
    by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id) 
paulson@15077
   847
  thus ?thesis by (simp add: o_def)
paulson@15077
   848
qed
paulson@15077
   849
paulson@15077
   850
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
paulson@15077
   851
proof -
paulson@15077
   852
  have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
paulson@15077
   853
       :> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
paulson@15077
   854
    by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult) 
paulson@15077
   855
  thus ?thesis by simp
paulson@15077
   856
qed
paulson@15077
   857
paulson@15077
   858
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
paulson@15077
   859
proof -
paulson@15077
   860
  have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
paulson@15077
   861
  hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)" 
paulson@15077
   862
    by (rule DERIV_isconst_all) 
paulson@15077
   863
  thus ?thesis by simp
paulson@15077
   864
qed
paulson@15077
   865
paulson@15077
   866
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
paulson@15077
   867
proof -
paulson@15077
   868
  have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus) 
paulson@15077
   869
  thus ?thesis by simp
paulson@15077
   870
qed
paulson@15077
   871
paulson@15077
   872
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
paulson@15077
   873
by (simp add: mult_commute)
paulson@15077
   874
paulson@15077
   875
paulson@15077
   876
lemma exp_minus: "exp(-x) = inverse(exp(x))"
paulson@15077
   877
by (auto intro: inverse_unique [symmetric])
paulson@15077
   878
paulson@15077
   879
lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
paulson@15077
   880
proof -
paulson@15077
   881
  have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
paulson@15077
   882
  thus ?thesis by (simp (no_asm_simp) add: mult_ac)
paulson@15077
   883
qed
paulson@15077
   884
paulson@15077
   885
text{*Proof: because every exponential can be seen as a square.*}
paulson@15077
   886
lemma exp_ge_zero [simp]: "0 \<le> exp x"
paulson@15077
   887
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
paulson@15077
   888
apply (subst exp_add, auto)
paulson@15077
   889
done
paulson@15077
   890
paulson@15077
   891
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
paulson@15077
   892
apply (cut_tac x = x in exp_mult_minus2)
paulson@15077
   893
apply (auto simp del: exp_mult_minus2)
paulson@15077
   894
done
paulson@15077
   895
paulson@15077
   896
lemma exp_gt_zero [simp]: "0 < exp x"
paulson@15077
   897
by (simp add: order_less_le)
paulson@15077
   898
paulson@15077
   899
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
paulson@15077
   900
by (auto intro: positive_imp_inverse_positive)
paulson@15077
   901
paulson@15077
   902
lemma abs_exp_cancel [simp]: "abs(exp x) = exp x"
paulson@15077
   903
by (auto simp add: abs_eqI2)
paulson@15077
   904
paulson@15077
   905
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
paulson@15077
   906
apply (induct_tac "n")
paulson@15077
   907
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
paulson@15077
   908
done
paulson@15077
   909
paulson@15077
   910
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
paulson@15077
   911
apply (unfold real_diff_def real_divide_def)
paulson@15077
   912
apply (simp (no_asm) add: exp_add exp_minus)
paulson@15077
   913
done
paulson@15077
   914
paulson@15077
   915
paulson@15077
   916
lemma exp_less_mono:
paulson@15077
   917
  assumes xy: "x < y" shows "exp x < exp y"
paulson@15077
   918
proof -
paulson@15077
   919
  have "1 < exp (y + - x)"
paulson@15077
   920
    by (rule real_less_sum_gt_zero [THEN exp_gt_one])
paulson@15077
   921
  hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
paulson@15077
   922
    by (auto simp add: exp_add exp_minus)
paulson@15077
   923
  thus ?thesis
paulson@15077
   924
    by (simp add: divide_inverse [symmetric] pos_less_divide_eq)
paulson@15077
   925
qed
paulson@15077
   926
paulson@15077
   927
lemma exp_less_cancel: "exp x < exp y ==> x < y"
paulson@15077
   928
apply (rule ccontr) 
paulson@15077
   929
apply (simp add: linorder_not_less order_le_less) 
paulson@15077
   930
apply (auto dest: exp_less_mono)
paulson@15077
   931
done
paulson@15077
   932
paulson@15077
   933
lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
paulson@15077
   934
by (auto intro: exp_less_mono exp_less_cancel)
paulson@15077
   935
paulson@15077
   936
lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
paulson@15077
   937
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
   938
paulson@15077
   939
lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
paulson@15077
   940
by (simp add: order_eq_iff)
paulson@15077
   941
paulson@15077
   942
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
paulson@15077
   943
apply (rule IVT)
paulson@15077
   944
apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
paulson@15077
   945
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)") 
paulson@15077
   946
apply simp 
paulson@15077
   947
apply (rule exp_ge_add_one_self, simp)
paulson@15077
   948
done
paulson@15077
   949
paulson@15077
   950
lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
paulson@15077
   951
apply (rule_tac x = 1 and y = y in linorder_cases)
paulson@15077
   952
apply (drule order_less_imp_le [THEN lemma_exp_total])
paulson@15077
   953
apply (rule_tac [2] x = 0 in exI)
paulson@15077
   954
apply (frule_tac [3] real_inverse_gt_one)
paulson@15077
   955
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
paulson@15077
   956
apply (rule_tac x = "-x" in exI)
paulson@15077
   957
apply (simp add: exp_minus)
paulson@15077
   958
done
paulson@15077
   959
paulson@15077
   960
paulson@15077
   961
subsection{*Properties of the Logarithmic Function*}
paulson@15077
   962
paulson@15077
   963
lemma ln_exp[simp]: "ln(exp x) = x"
paulson@15077
   964
by (simp add: ln_def)
paulson@15077
   965
paulson@15077
   966
lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
paulson@15077
   967
apply (auto dest: exp_total)
paulson@15077
   968
apply (erule subst, simp) 
paulson@15077
   969
done
paulson@15077
   970
paulson@15077
   971
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
paulson@15077
   972
apply (rule exp_inj_iff [THEN iffD1])
paulson@15077
   973
apply (frule real_mult_order)
paulson@15077
   974
apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
paulson@15077
   975
done
paulson@15077
   976
paulson@15077
   977
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
paulson@15077
   978
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   979
apply (erule subst)+
paulson@15077
   980
apply simp 
paulson@15077
   981
done
paulson@15077
   982
paulson@15077
   983
lemma ln_one[simp]: "ln 1 = 0"
paulson@15077
   984
by (rule exp_inj_iff [THEN iffD1], auto)
paulson@15077
   985
paulson@15077
   986
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
paulson@15077
   987
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
paulson@15077
   988
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
paulson@15077
   989
done
paulson@15077
   990
paulson@15077
   991
lemma ln_div: 
paulson@15077
   992
    "[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
paulson@15077
   993
apply (unfold real_divide_def)
paulson@15077
   994
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
paulson@15077
   995
done
paulson@15077
   996
paulson@15077
   997
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
paulson@15077
   998
apply (simp only: exp_ln_iff [symmetric])
paulson@15077
   999
apply (erule subst)+
paulson@15077
  1000
apply simp 
paulson@15077
  1001
done
paulson@15077
  1002
paulson@15077
  1003
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
paulson@15077
  1004
by (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1005
paulson@15077
  1006
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
paulson@15077
  1007
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
paulson@15077
  1008
paulson@15077
  1009
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
paulson@15077
  1010
apply (rule ln_exp [THEN subst])
paulson@15077
  1011
apply (rule ln_le_cancel_iff [THEN iffD2], auto)
paulson@15077
  1012
done
paulson@15077
  1013
paulson@15077
  1014
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
paulson@15077
  1015
apply (rule order_less_le_trans)
paulson@15077
  1016
apply (rule_tac [2] ln_add_one_self_le_self)
paulson@15077
  1017
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
paulson@15077
  1018
done
paulson@15077
  1019
paulson@15077
  1020
lemma ln_ge_zero:
paulson@15077
  1021
  assumes x: "1 \<le> x" shows "0 \<le> ln x"
paulson@15077
  1022
proof -
paulson@15077
  1023
  have "0 < x" using x by arith
paulson@15077
  1024
  hence "exp 0 \<le> exp (ln x)"
paulson@15077
  1025
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077
  1026
  thus ?thesis by (simp only: exp_le_cancel_iff)
paulson@15077
  1027
qed
paulson@15077
  1028
paulson@15077
  1029
lemma ln_ge_zero_imp_ge_one:
paulson@15077
  1030
  assumes ln: "0 \<le> ln x" 
paulson@15077
  1031
      and x:  "0 < x"
paulson@15077
  1032
  shows "1 \<le> x"
paulson@15077
  1033
proof -
paulson@15077
  1034
  from ln have "ln 1 \<le> ln x" by simp
paulson@15077
  1035
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
  1036
qed
paulson@15077
  1037
paulson@15077
  1038
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
paulson@15077
  1039
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
paulson@15077
  1040
paulson@15077
  1041
lemma ln_gt_zero:
paulson@15077
  1042
  assumes x: "1 < x" shows "0 < ln x"
paulson@15077
  1043
proof -
paulson@15077
  1044
  have "0 < x" using x by arith
paulson@15077
  1045
  hence "exp 0 < exp (ln x)"
paulson@15077
  1046
    by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
paulson@15077
  1047
  thus ?thesis  by (simp only: exp_less_cancel_iff)
paulson@15077
  1048
qed
paulson@15077
  1049
paulson@15077
  1050
lemma ln_gt_zero_imp_gt_one:
paulson@15077
  1051
  assumes ln: "0 < ln x" 
paulson@15077
  1052
      and x:  "0 < x"
paulson@15077
  1053
  shows "1 < x"
paulson@15077
  1054
proof -
paulson@15077
  1055
  from ln have "ln 1 < ln x" by simp
paulson@15077
  1056
  thus ?thesis by (simp add: x del: ln_one) 
paulson@15077
  1057
qed
paulson@15077
  1058
paulson@15077
  1059
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
paulson@15077
  1060
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
paulson@15077
  1061
paulson@15077
  1062
lemma ln_not_eq_zero [simp]: "[| 0 < x; x \<noteq> 1 |] ==> ln x \<noteq> 0"
paulson@15077
  1063
apply safe
paulson@15077
  1064
apply (drule exp_inj_iff [THEN iffD2])
paulson@15077
  1065
apply (drule exp_ln_iff [THEN iffD2], auto)
paulson@15077
  1066
done
paulson@15077
  1067
paulson@15077
  1068
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
paulson@15077
  1069
apply (rule exp_less_cancel_iff [THEN iffD1])
paulson@15077
  1070
apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
paulson@15077
  1071
done
paulson@15077
  1072
paulson@15077
  1073
lemma exp_ln_eq: "exp u = x ==> ln x = u"
paulson@15077
  1074
by auto
paulson@15077
  1075
paulson@15077
  1076
paulson@15077
  1077
subsection{*Basic Properties of the Trigonometric Functions*}
paulson@15077
  1078
paulson@15077
  1079
lemma sin_zero [simp]: "sin 0 = 0"
paulson@15077
  1080
by (auto intro!: sums_unique [symmetric] LIMSEQ_const 
paulson@15077
  1081
         simp add: sin_def sums_def simp del: power_0_left)
paulson@15077
  1082
paulson@15077
  1083
lemma lemma_series_zero2: "(\<forall>m. n \<le> m --> f m = 0) --> f sums sumr 0 n f"
paulson@15077
  1084
by (auto intro: series_zero)
paulson@15077
  1085
paulson@15077
  1086
lemma cos_zero [simp]: "cos 0 = 1"
paulson@15077
  1087
apply (unfold cos_def)
paulson@15077
  1088
apply (rule sums_unique [symmetric])
paulson@15077
  1089
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
  1090
apply auto
paulson@15077
  1091
done
paulson@15077
  1092
paulson@15077
  1093
lemma DERIV_sin_sin_mult [simp]:
paulson@15077
  1094
     "DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1095
by (rule DERIV_mult, auto)
paulson@15077
  1096
paulson@15077
  1097
lemma DERIV_sin_sin_mult2 [simp]:
paulson@15077
  1098
     "DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1099
apply (cut_tac x = x in DERIV_sin_sin_mult)
paulson@15077
  1100
apply (auto simp add: mult_assoc)
paulson@15077
  1101
done
paulson@15077
  1102
paulson@15077
  1103
lemma DERIV_sin_realpow2 [simp]:
paulson@15077
  1104
     "DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
paulson@15077
  1105
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1106
paulson@15077
  1107
lemma DERIV_sin_realpow2a [simp]:
paulson@15077
  1108
     "DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
paulson@15077
  1109
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1110
paulson@15077
  1111
lemma DERIV_cos_cos_mult [simp]:
paulson@15077
  1112
     "DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1113
by (rule DERIV_mult, auto)
paulson@15077
  1114
paulson@15077
  1115
lemma DERIV_cos_cos_mult2 [simp]:
paulson@15077
  1116
     "DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1117
apply (cut_tac x = x in DERIV_cos_cos_mult)
paulson@15077
  1118
apply (auto simp add: mult_ac)
paulson@15077
  1119
done
paulson@15077
  1120
paulson@15077
  1121
lemma DERIV_cos_realpow2 [simp]:
paulson@15077
  1122
     "DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
paulson@15077
  1123
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
paulson@15077
  1124
paulson@15077
  1125
lemma DERIV_cos_realpow2a [simp]:
paulson@15077
  1126
     "DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
paulson@15077
  1127
by (auto simp add: numeral_2_eq_2)
paulson@15077
  1128
paulson@15077
  1129
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
paulson@15077
  1130
by auto
paulson@15077
  1131
paulson@15077
  1132
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1133
apply (rule lemma_DERIV_subst)
paulson@15077
  1134
apply (rule DERIV_cos_realpow2a, auto)
paulson@15077
  1135
done
paulson@15077
  1136
paulson@15077
  1137
(* most useful *)
paulson@15077
  1138
lemma DERIV_cos_cos_mult3 [simp]: "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
paulson@15077
  1139
apply (rule lemma_DERIV_subst)
paulson@15077
  1140
apply (rule DERIV_cos_cos_mult2, auto)
paulson@15077
  1141
done
paulson@15077
  1142
paulson@15077
  1143
lemma DERIV_sin_circle_all: 
paulson@15077
  1144
     "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>  
paulson@15077
  1145
             (2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
paulson@15077
  1146
apply (unfold real_diff_def, safe)
paulson@15077
  1147
apply (rule DERIV_add)
paulson@15077
  1148
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1149
done
paulson@15077
  1150
paulson@15077
  1151
lemma DERIV_sin_circle_all_zero [simp]: "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
paulson@15077
  1152
by (cut_tac DERIV_sin_circle_all, auto)
paulson@15077
  1153
paulson@15077
  1154
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
paulson@15077
  1155
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
paulson@15077
  1156
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1157
done
paulson@15077
  1158
paulson@15077
  1159
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
paulson@15077
  1160
apply (subst real_add_commute)
paulson@15077
  1161
apply (simp (no_asm) del: realpow_Suc)
paulson@15077
  1162
done
paulson@15077
  1163
paulson@15077
  1164
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
paulson@15077
  1165
apply (cut_tac x = x in sin_cos_squared_add2)
paulson@15077
  1166
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1167
done
paulson@15077
  1168
paulson@15077
  1169
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
paulson@15077
  1170
apply (rule_tac a1 = "(cos x)\<twosuperior> " in add_right_cancel [THEN iffD1])
paulson@15077
  1171
apply (simp del: realpow_Suc)
paulson@15077
  1172
done
paulson@15077
  1173
paulson@15077
  1174
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
paulson@15077
  1175
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
paulson@15077
  1176
apply (simp del: realpow_Suc)
paulson@15077
  1177
done
paulson@15077
  1178
paulson@15077
  1179
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
paulson@15077
  1180
by arith
paulson@15077
  1181
paulson@15077
  1182
lemma abs_sin_le_one [simp]: "abs(sin x) \<le> 1"
paulson@15077
  1183
apply (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1184
apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077
  1185
apply (auto simp del: realpow_Suc)
paulson@15077
  1186
apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077
  1187
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
  1188
done
paulson@15077
  1189
paulson@15077
  1190
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
paulson@15077
  1191
apply (insert abs_sin_le_one [of x]) 
paulson@15077
  1192
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077
  1193
done
paulson@15077
  1194
paulson@15077
  1195
lemma sin_le_one [simp]: "sin x \<le> 1"
paulson@15077
  1196
apply (insert abs_sin_le_one [of x]) 
paulson@15077
  1197
apply (simp add: abs_le_interval_iff del: abs_sin_le_one) 
paulson@15077
  1198
done
paulson@15077
  1199
paulson@15077
  1200
lemma abs_cos_le_one [simp]: "abs(cos x) \<le> 1"
paulson@15077
  1201
apply (auto simp add: linorder_not_less [symmetric])
paulson@15077
  1202
apply (drule_tac n = "Suc 0" in power_gt1)
paulson@15077
  1203
apply (auto simp del: realpow_Suc)
paulson@15077
  1204
apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
paulson@15077
  1205
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
  1206
done
paulson@15077
  1207
paulson@15077
  1208
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
paulson@15077
  1209
apply (insert abs_cos_le_one [of x]) 
paulson@15077
  1210
apply (simp add: abs_le_interval_iff del: abs_cos_le_one) 
paulson@15077
  1211
done
paulson@15077
  1212
paulson@15077
  1213
lemma cos_le_one [simp]: "cos x \<le> 1"
paulson@15077
  1214
apply (insert abs_cos_le_one [of x]) 
paulson@15077
  1215
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
paulson@15077
  1216
done
paulson@15077
  1217
paulson@15077
  1218
lemma DERIV_fun_pow: "DERIV g x :> m ==>  
paulson@15077
  1219
      DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
paulson@15077
  1220
apply (rule lemma_DERIV_subst)
paulson@15077
  1221
apply (rule_tac f = " (%x. x ^ n) " in DERIV_chain2)
paulson@15077
  1222
apply (rule DERIV_pow, auto)
paulson@15077
  1223
done
paulson@15077
  1224
paulson@15077
  1225
lemma DERIV_fun_exp: "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
paulson@15077
  1226
apply (rule lemma_DERIV_subst)
paulson@15077
  1227
apply (rule_tac f = exp in DERIV_chain2)
paulson@15077
  1228
apply (rule DERIV_exp, auto)
paulson@15077
  1229
done
paulson@15077
  1230
paulson@15077
  1231
lemma DERIV_fun_sin: "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
paulson@15077
  1232
apply (rule lemma_DERIV_subst)
paulson@15077
  1233
apply (rule_tac f = sin in DERIV_chain2)
paulson@15077
  1234
apply (rule DERIV_sin, auto)
paulson@15077
  1235
done
paulson@15077
  1236
paulson@15077
  1237
lemma DERIV_fun_cos: "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
paulson@15077
  1238
apply (rule lemma_DERIV_subst)
paulson@15077
  1239
apply (rule_tac f = cos in DERIV_chain2)
paulson@15077
  1240
apply (rule DERIV_cos, auto)
paulson@15077
  1241
done
paulson@15077
  1242
paulson@15077
  1243
lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult 
paulson@15077
  1244
                    DERIV_sin  DERIV_exp  DERIV_inverse DERIV_pow 
paulson@15077
  1245
                    DERIV_add  DERIV_diff  DERIV_mult  DERIV_minus 
paulson@15077
  1246
                    DERIV_inverse_fun DERIV_quotient DERIV_fun_pow 
paulson@15077
  1247
                    DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos 
paulson@15077
  1248
                    DERIV_Id DERIV_const DERIV_cos
paulson@15077
  1249
paulson@15077
  1250
(* lemma *)
paulson@15077
  1251
lemma lemma_DERIV_sin_cos_add: "\<forall>x.  
paulson@15077
  1252
         DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1253
               (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
paulson@15077
  1254
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1255
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1256
  --{*replaces the old @{text DERIV_tac}*}
paulson@15077
  1257
apply (auto simp add: real_diff_def left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1258
done
paulson@15077
  1259
paulson@15077
  1260
lemma sin_cos_add [simp]:
paulson@15077
  1261
     "(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +  
paulson@15077
  1262
      (cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
paulson@15077
  1263
apply (cut_tac y = 0 and x = x and y7 = y 
paulson@15077
  1264
       in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
paulson@15077
  1265
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1266
done
paulson@15077
  1267
paulson@15077
  1268
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
paulson@15077
  1269
apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077
  1270
apply (auto dest!: real_sum_squares_cancel_a 
paulson@15077
  1271
            simp add: numeral_2_eq_2 simp del: sin_cos_add)
paulson@15077
  1272
done
paulson@15077
  1273
paulson@15077
  1274
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
paulson@15077
  1275
apply (cut_tac x = x and y = y in sin_cos_add)
paulson@15077
  1276
apply (auto dest!: real_sum_squares_cancel_a
paulson@15077
  1277
            simp add: numeral_2_eq_2 simp del: sin_cos_add)
paulson@15077
  1278
done
paulson@15077
  1279
paulson@15077
  1280
lemma lemma_DERIV_sin_cos_minus: "\<forall>x.  
paulson@15077
  1281
          DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
paulson@15077
  1282
apply (safe, rule lemma_DERIV_subst)
paulson@15077
  1283
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1284
apply (auto simp add: real_diff_def left_distrib right_distrib mult_ac add_ac)
paulson@15077
  1285
done
paulson@15077
  1286
paulson@15077
  1287
lemma sin_cos_minus [simp]: "(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
paulson@15077
  1288
apply (cut_tac y = 0 and x = x in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
paulson@15077
  1289
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1290
done
paulson@15077
  1291
paulson@15077
  1292
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
paulson@15077
  1293
apply (cut_tac x = x in sin_cos_minus)
paulson@15077
  1294
apply (auto dest!: real_sum_squares_cancel_a simp add: numeral_2_eq_2 simp del: sin_cos_minus)
paulson@15077
  1295
done
paulson@15077
  1296
paulson@15077
  1297
lemma cos_minus [simp]: "cos (-x) = cos(x)"
paulson@15077
  1298
apply (cut_tac x = x in sin_cos_minus)
paulson@15077
  1299
apply (auto dest!: real_sum_squares_cancel_a simp add: numeral_2_eq_2 simp del: sin_cos_minus)
paulson@15077
  1300
done
paulson@15077
  1301
paulson@15077
  1302
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
paulson@15077
  1303
apply (unfold real_diff_def)
paulson@15077
  1304
apply (simp (no_asm) add: sin_add)
paulson@15077
  1305
done
paulson@15077
  1306
paulson@15077
  1307
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
paulson@15077
  1308
by (simp add: sin_diff mult_commute)
paulson@15077
  1309
paulson@15077
  1310
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
paulson@15077
  1311
apply (unfold real_diff_def)
paulson@15077
  1312
apply (simp (no_asm) add: cos_add)
paulson@15077
  1313
done
paulson@15077
  1314
paulson@15077
  1315
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
paulson@15077
  1316
by (simp add: cos_diff mult_commute)
paulson@15077
  1317
paulson@15077
  1318
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
paulson@15077
  1319
by (cut_tac x = x and y = x in sin_add, auto)
paulson@15077
  1320
paulson@15077
  1321
paulson@15077
  1322
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
paulson@15077
  1323
apply (cut_tac x = x and y = x in cos_add)
paulson@15077
  1324
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1325
done
paulson@15077
  1326
paulson@15077
  1327
paulson@15077
  1328
subsection{*The Constant Pi*}
paulson@15077
  1329
paulson@15077
  1330
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"}; 
paulson@15077
  1331
   hence define pi.*}
paulson@15077
  1332
paulson@15077
  1333
lemma sin_paired:
paulson@15077
  1334
     "(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1)) 
paulson@15077
  1335
      sums  sin x"
paulson@15077
  1336
proof -
paulson@15077
  1337
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1338
            (if even k then 0
paulson@15077
  1339
             else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
paulson@15077
  1340
            x ^ k) 
paulson@15077
  1341
	sums
paulson@15077
  1342
	suminf (\<lambda>n. (if even n then 0
paulson@15077
  1343
		     else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
paulson@15077
  1344
	            x ^ n)" 
paulson@15077
  1345
    by (rule sin_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1346
  thus ?thesis by (simp add: mult_ac sin_def)
paulson@15077
  1347
qed
paulson@15077
  1348
paulson@15077
  1349
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1350
apply (subgoal_tac 
paulson@15077
  1351
       "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1352
              (- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1)) 
paulson@15077
  1353
     sums suminf (\<lambda>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
paulson@15077
  1354
 prefer 2
paulson@15077
  1355
 apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1356
apply (rotate_tac 2)
paulson@15077
  1357
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
paulson@15077
  1358
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1359
apply (frule sums_unique)
paulson@15077
  1360
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1361
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
paulson@15077
  1362
apply (auto simp del: fact_Suc realpow_Suc)
paulson@15077
  1363
apply (erule sums_summable)
paulson@15077
  1364
apply (case_tac "m=0")
paulson@15077
  1365
apply (simp (no_asm_simp))
paulson@15077
  1366
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
paulson@15077
  1367
apply (simp only: mult_less_cancel_left, simp)
paulson@15077
  1368
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
paulson@15077
  1369
apply (subgoal_tac "x*x < 2*3", simp) 
paulson@15077
  1370
apply (rule mult_strict_mono)
paulson@15077
  1371
apply (auto simp add: real_of_nat_Suc simp del: fact_Suc)
paulson@15077
  1372
apply (subst fact_Suc)
paulson@15077
  1373
apply (subst fact_Suc)
paulson@15077
  1374
apply (subst fact_Suc)
paulson@15077
  1375
apply (subst fact_Suc)
paulson@15077
  1376
apply (subst real_of_nat_mult)
paulson@15077
  1377
apply (subst real_of_nat_mult)
paulson@15077
  1378
apply (subst real_of_nat_mult)
paulson@15077
  1379
apply (subst real_of_nat_mult)
paulson@15077
  1380
apply (simp (no_asm) add: real_divide_def inverse_mult_distrib del: fact_Suc)
paulson@15077
  1381
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
paulson@15077
  1382
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right) 
paulson@15077
  1383
apply (auto simp add: mult_assoc simp del: fact_Suc)
paulson@15077
  1384
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right) 
paulson@15077
  1385
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
paulson@15077
  1386
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)") 
paulson@15077
  1387
apply (erule ssubst)+
paulson@15077
  1388
apply (auto simp del: fact_Suc)
paulson@15077
  1389
apply (subgoal_tac "0 < x ^ (4 * m) ")
paulson@15077
  1390
 prefer 2 apply (simp only: zero_less_power) 
paulson@15077
  1391
apply (simp (no_asm_simp) add: mult_less_cancel_left)
paulson@15077
  1392
apply (rule mult_strict_mono)
paulson@15077
  1393
apply (simp_all (no_asm_simp))
paulson@15077
  1394
done
paulson@15077
  1395
paulson@15077
  1396
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
paulson@15077
  1397
by (auto intro: sin_gt_zero)
paulson@15077
  1398
paulson@15077
  1399
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
paulson@15077
  1400
apply (cut_tac x = x in sin_gt_zero1)
paulson@15077
  1401
apply (auto simp add: cos_squared_eq cos_double)
paulson@15077
  1402
done
paulson@15077
  1403
paulson@15077
  1404
lemma cos_paired:
paulson@15077
  1405
     "(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
paulson@15077
  1406
proof -
paulson@15077
  1407
  have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
paulson@15077
  1408
            (if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
paulson@15077
  1409
            x ^ k) 
paulson@15077
  1410
        sums
paulson@15077
  1411
	suminf
paulson@15077
  1412
	 (\<lambda>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
paulson@15077
  1413
	      x ^ n)" 
paulson@15077
  1414
    by (rule cos_converges [THEN sums_summable, THEN sums_group], simp) 
paulson@15077
  1415
  thus ?thesis by (simp add: mult_ac cos_def)
paulson@15077
  1416
qed
paulson@15077
  1417
paulson@15077
  1418
declare zero_less_power [simp]
paulson@15077
  1419
paulson@15077
  1420
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
paulson@15077
  1421
by simp
paulson@15077
  1422
paulson@15077
  1423
lemma cos_two_less_zero: "cos (2) < 0"
paulson@15077
  1424
apply (cut_tac x = 2 in cos_paired)
paulson@15077
  1425
apply (drule sums_minus)
paulson@15077
  1426
apply (rule neg_less_iff_less [THEN iffD1]) 
paulson@15077
  1427
apply (frule sums_unique, auto)
paulson@15077
  1428
apply (rule_tac y = "sumr 0 (Suc (Suc (Suc 0))) (%n. - ((- 1) ^ n / (real (fact (2 * n))) * 2 ^ (2 * n))) " in order_less_trans)
paulson@15077
  1429
apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
paulson@15077
  1430
apply (simp (no_asm) add: mult_assoc del: sumr_Suc)
paulson@15077
  1431
apply (rule sumr_pos_lt_pair)
paulson@15077
  1432
apply (erule sums_summable, safe)
paulson@15077
  1433
apply (simp (no_asm) add: real_divide_def mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1434
apply (rule real_mult_inverse_cancel2)
paulson@15077
  1435
apply (rule real_of_nat_fact_gt_zero)+
paulson@15077
  1436
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
paulson@15077
  1437
apply (subst fact_lemma) 
paulson@15077
  1438
apply (subst fact_Suc)
paulson@15077
  1439
apply (subst real_of_nat_mult)
paulson@15077
  1440
apply (erule ssubst, subst real_of_nat_mult)
paulson@15077
  1441
apply (rule real_mult_less_mono, force)
paulson@15077
  1442
prefer 2 apply force
paulson@15077
  1443
apply (rule_tac [2] real_of_nat_fact_gt_zero)
paulson@15077
  1444
apply (rule real_of_nat_less_iff [THEN iffD2])
paulson@15077
  1445
apply (rule fact_less_mono, auto)
paulson@15077
  1446
done
paulson@15077
  1447
declare cos_two_less_zero [simp]
paulson@15077
  1448
declare cos_two_less_zero [THEN real_not_refl2, simp]
paulson@15077
  1449
declare cos_two_less_zero [THEN order_less_imp_le, simp]
paulson@15077
  1450
paulson@15077
  1451
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
paulson@15077
  1452
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
paulson@15077
  1453
apply (rule_tac [2] IVT2)
paulson@15077
  1454
apply (auto intro: DERIV_isCont DERIV_cos)
paulson@15077
  1455
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1456
apply (rule ccontr)
paulson@15077
  1457
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
paulson@15077
  1458
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
paulson@15077
  1459
apply (drule_tac f = cos in Rolle)
paulson@15077
  1460
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1461
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
paulson@15077
  1462
apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
paulson@15077
  1463
apply (assumption, rule_tac y=y in order_less_le_trans, simp_all) 
paulson@15077
  1464
apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all) 
paulson@15077
  1465
done
paulson@15077
  1466
    
paulson@15077
  1467
lemma pi_half: "pi/2 = (@x. 0 \<le> x & x \<le> 2 & cos x = 0)"
paulson@15077
  1468
by (simp add: pi_def)
paulson@15077
  1469
paulson@15077
  1470
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
paulson@15077
  1471
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1472
apply (auto intro!: someI2 simp add: pi_half)
paulson@15077
  1473
done
paulson@15077
  1474
paulson@15077
  1475
lemma pi_half_gt_zero: "0 < pi / 2"
paulson@15077
  1476
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1477
apply (auto simp add: pi_half)
paulson@15077
  1478
apply (rule someI2, blast, safe)
paulson@15077
  1479
apply (drule_tac y = xa in real_le_imp_less_or_eq)
paulson@15077
  1480
apply (safe, simp)
paulson@15077
  1481
done
paulson@15077
  1482
declare pi_half_gt_zero [simp]
paulson@15077
  1483
declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
paulson@15077
  1484
declare pi_half_gt_zero [THEN order_less_imp_le, simp]
paulson@15077
  1485
paulson@15077
  1486
lemma pi_half_less_two: "pi / 2 < 2"
paulson@15077
  1487
apply (rule cos_is_zero [THEN ex1E])
paulson@15077
  1488
apply (auto simp add: pi_half)
paulson@15077
  1489
apply (rule someI2, blast, safe)
paulson@15077
  1490
apply (drule_tac x = xa in order_le_imp_less_or_eq)
paulson@15077
  1491
apply (safe, simp)
paulson@15077
  1492
done
paulson@15077
  1493
declare pi_half_less_two [simp]
paulson@15077
  1494
declare pi_half_less_two [THEN real_not_refl2, simp]
paulson@15077
  1495
declare pi_half_less_two [THEN order_less_imp_le, simp]
paulson@15077
  1496
paulson@15077
  1497
lemma pi_gt_zero [simp]: "0 < pi"
paulson@15077
  1498
apply (rule_tac c="inverse 2" in mult_less_imp_less_right) 
paulson@15077
  1499
apply (cut_tac pi_half_gt_zero, simp_all)
paulson@15077
  1500
done
paulson@15077
  1501
paulson@15077
  1502
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
paulson@15077
  1503
by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])
paulson@15077
  1504
paulson@15077
  1505
lemma pi_not_less_zero [simp]: "~ (pi < 0)"
paulson@15077
  1506
apply (insert pi_gt_zero)
paulson@15077
  1507
apply (blast elim: order_less_asym) 
paulson@15077
  1508
done
paulson@15077
  1509
paulson@15077
  1510
lemma pi_ge_zero [simp]: "0 \<le> pi"
paulson@15077
  1511
by (auto intro: order_less_imp_le)
paulson@15077
  1512
paulson@15077
  1513
lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
paulson@15077
  1514
by auto
paulson@15077
  1515
paulson@15077
  1516
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
paulson@15077
  1517
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
paulson@15077
  1518
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
paulson@15077
  1519
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1520
done
paulson@15077
  1521
paulson@15077
  1522
lemma cos_pi [simp]: "cos pi = -1"
paulson@15077
  1523
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
paulson@15077
  1524
paulson@15077
  1525
lemma sin_pi [simp]: "sin pi = 0"
paulson@15077
  1526
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
paulson@15077
  1527
paulson@15077
  1528
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
paulson@15077
  1529
apply (unfold real_diff_def)
paulson@15077
  1530
apply (simp (no_asm) add: cos_add)
paulson@15077
  1531
done
paulson@15077
  1532
paulson@15077
  1533
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
paulson@15077
  1534
apply (simp (no_asm) add: cos_add)
paulson@15077
  1535
done
paulson@15077
  1536
declare minus_sin_cos_eq [symmetric, simp]
paulson@15077
  1537
paulson@15077
  1538
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
paulson@15077
  1539
apply (unfold real_diff_def)
paulson@15077
  1540
apply (simp (no_asm) add: sin_add)
paulson@15077
  1541
done
paulson@15077
  1542
declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]
paulson@15077
  1543
paulson@15077
  1544
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
paulson@15077
  1545
apply (simp (no_asm) add: sin_add)
paulson@15077
  1546
done
paulson@15077
  1547
paulson@15077
  1548
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
paulson@15077
  1549
apply (simp (no_asm) add: sin_add)
paulson@15077
  1550
done
paulson@15077
  1551
paulson@15077
  1552
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
paulson@15077
  1553
apply (simp (no_asm) add: cos_add)
paulson@15077
  1554
done
paulson@15077
  1555
paulson@15077
  1556
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
paulson@15077
  1557
by (simp add: sin_add cos_double)
paulson@15077
  1558
paulson@15077
  1559
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
paulson@15077
  1560
by (simp add: cos_add cos_double)
paulson@15077
  1561
paulson@15077
  1562
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
paulson@15077
  1563
apply (induct_tac "n")
paulson@15077
  1564
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1565
done
paulson@15077
  1566
paulson@15077
  1567
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
paulson@15077
  1568
apply (induct_tac "n")
paulson@15077
  1569
apply (auto simp add: real_of_nat_Suc left_distrib)
paulson@15077
  1570
done
paulson@15077
  1571
paulson@15077
  1572
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
paulson@15077
  1573
apply (cut_tac n = n in sin_npi)
paulson@15077
  1574
apply (auto simp add: mult_commute simp del: sin_npi)
paulson@15077
  1575
done
paulson@15077
  1576
paulson@15077
  1577
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
paulson@15077
  1578
by (simp add: cos_double)
paulson@15077
  1579
paulson@15077
  1580
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
paulson@15077
  1581
apply (simp (no_asm))
paulson@15077
  1582
done
paulson@15077
  1583
paulson@15077
  1584
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
paulson@15077
  1585
apply (rule sin_gt_zero, assumption)
paulson@15077
  1586
apply (rule order_less_trans, assumption)
paulson@15077
  1587
apply (rule pi_half_less_two)
paulson@15077
  1588
done
paulson@15077
  1589
paulson@15077
  1590
lemma sin_less_zero: 
paulson@15077
  1591
  assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
paulson@15077
  1592
proof -
paulson@15077
  1593
  have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2) 
paulson@15077
  1594
  thus ?thesis by simp
paulson@15077
  1595
qed
paulson@15077
  1596
paulson@15077
  1597
lemma pi_less_4: "pi < 4"
paulson@15077
  1598
by (cut_tac pi_half_less_two, auto)
paulson@15077
  1599
paulson@15077
  1600
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1601
apply (cut_tac pi_less_4)
paulson@15077
  1602
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
paulson@15077
  1603
apply (force intro: DERIV_isCont DERIV_cos)
paulson@15077
  1604
apply (cut_tac cos_is_zero, safe)
paulson@15077
  1605
apply (rename_tac y z)
paulson@15077
  1606
apply (drule_tac x = y in spec)
paulson@15077
  1607
apply (drule_tac x = "pi/2" in spec, simp) 
paulson@15077
  1608
done
paulson@15077
  1609
paulson@15077
  1610
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
paulson@15077
  1611
apply (rule_tac x = x and y = 0 in linorder_cases)
paulson@15077
  1612
apply (rule cos_minus [THEN subst])
paulson@15077
  1613
apply (rule cos_gt_zero)
paulson@15077
  1614
apply (auto intro: cos_gt_zero)
paulson@15077
  1615
done
paulson@15077
  1616
 
paulson@15077
  1617
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
paulson@15077
  1618
apply (auto simp add: order_le_less cos_gt_zero_pi)
paulson@15077
  1619
apply (subgoal_tac "x = pi/2", auto) 
paulson@15077
  1620
done
paulson@15077
  1621
paulson@15077
  1622
lemma sin_gt_zero_pi: "[| 0 < x; x < pi  |] ==> 0 < sin x"
paulson@15077
  1623
apply (subst sin_cos_eq)
paulson@15077
  1624
apply (rotate_tac 1)
paulson@15077
  1625
apply (drule real_sum_of_halves [THEN ssubst])
paulson@15077
  1626
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
paulson@15077
  1627
done
paulson@15077
  1628
paulson@15077
  1629
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
paulson@15077
  1630
by (auto simp add: order_le_less sin_gt_zero_pi)
paulson@15077
  1631
paulson@15077
  1632
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
paulson@15077
  1633
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
paulson@15077
  1634
apply (rule_tac [2] IVT2)
paulson@15077
  1635
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
paulson@15077
  1636
apply (cut_tac x = xa and y = y in linorder_less_linear)
paulson@15077
  1637
apply (rule ccontr, auto)
paulson@15077
  1638
apply (drule_tac f = cos in Rolle)
paulson@15077
  1639
apply (drule_tac [5] f = cos in Rolle)
paulson@15077
  1640
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
paulson@15077
  1641
            dest!: DERIV_cos [THEN DERIV_unique] 
paulson@15077
  1642
            simp add: differentiable_def)
paulson@15077
  1643
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
paulson@15077
  1644
done
paulson@15077
  1645
paulson@15077
  1646
lemma sin_total:
paulson@15077
  1647
     "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
paulson@15077
  1648
apply (rule ccontr)
paulson@15077
  1649
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
  1650
apply (erule swap)
paulson@15077
  1651
apply (simp del: minus_sin_cos_eq [symmetric])
paulson@15077
  1652
apply (cut_tac y="-y" in cos_total, simp) apply simp 
paulson@15077
  1653
apply (erule ex1E)
paulson@15077
  1654
apply (rule_tac a = "x - (pi/2) " in ex1I)
paulson@15077
  1655
apply (simp (no_asm) add: real_add_assoc)
paulson@15077
  1656
apply (rotate_tac 3)
paulson@15077
  1657
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all) 
paulson@15077
  1658
done
paulson@15077
  1659
paulson@15077
  1660
lemma reals_Archimedean4:
paulson@15077
  1661
     "[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
paulson@15077
  1662
apply (auto dest!: reals_Archimedean3)
paulson@15077
  1663
apply (drule_tac x = x in spec, clarify) 
paulson@15077
  1664
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
paulson@15077
  1665
 prefer 2 apply (erule LeastI) 
paulson@15077
  1666
apply (case_tac "LEAST m::nat. x < real m * y", simp) 
paulson@15077
  1667
apply (subgoal_tac "~ x < real nat * y")
paulson@15077
  1668
 prefer 2 apply (rule not_less_Least, simp, force)  
paulson@15077
  1669
done
paulson@15077
  1670
paulson@15077
  1671
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic 
paulson@15077
  1672
   now causes some unwanted re-arrangements of literals!   *)
paulson@15077
  1673
lemma cos_zero_lemma: "[| 0 \<le> x; cos x = 0 |] ==>  
paulson@15077
  1674
      \<exists>n::nat. ~even n & x = real n * (pi/2)"
paulson@15077
  1675
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
paulson@15077
  1676
apply (subgoal_tac "0 \<le> x - real n * pi & (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
paulson@15077
  1677
apply safe
paulson@15077
  1678
  prefer 3 apply (simp add: cos_diff) 
paulson@15077
  1679
 prefer 2 apply (simp add: real_of_nat_Suc left_distrib) 
paulson@15077
  1680
apply (simp add: cos_diff)
paulson@15077
  1681
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
paulson@15077
  1682
apply (rule_tac [2] cos_total, safe)
paulson@15077
  1683
apply (drule_tac x = "x - real n * pi" in spec)
paulson@15077
  1684
apply (drule_tac x = "pi/2" in spec)
paulson@15077
  1685
apply (simp add: cos_diff)
paulson@15077
  1686
apply (rule_tac x = "Suc (2 * n) " in exI)
paulson@15077
  1687
apply (simp add: real_of_nat_Suc left_distrib, auto)
paulson@15077
  1688
done
paulson@15077
  1689
paulson@15077
  1690
lemma sin_zero_lemma: "[| 0 \<le> x; sin x = 0 |] ==>  
paulson@15077
  1691
      \<exists>n::nat. even n & x = real n * (pi/2)"
paulson@15077
  1692
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
paulson@15077
  1693
 apply (clarify, rule_tac x = "n - 1" in exI)
paulson@15077
  1694
 apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
paulson@15077
  1695
apply (rule cos_zero_lemma, clarify) 
paulson@15077
  1696
apply (rule minus_le_iff [THEN iffD1])  
paulson@15077
  1697
apply (rule_tac y=0 in order_trans, auto)
paulson@15077
  1698
done
paulson@15077
  1699
paulson@15077
  1700
paulson@15077
  1701
(* also spoilt by numeral arithmetic *)
paulson@15077
  1702
lemma cos_zero_iff: "(cos x = 0) =  
paulson@15077
  1703
      ((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |    
paulson@15077
  1704
       (\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1705
apply (rule iffI)
paulson@15077
  1706
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1707
apply (drule cos_zero_lemma, assumption+)
paulson@15077
  1708
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp) 
paulson@15077
  1709
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1710
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib) 
paulson@15077
  1711
apply (auto simp add: cos_add)
paulson@15077
  1712
done
paulson@15077
  1713
paulson@15077
  1714
(* ditto: but to a lesser extent *)
paulson@15077
  1715
lemma sin_zero_iff: "(sin x = 0) =  
paulson@15077
  1716
      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |    
paulson@15077
  1717
       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
paulson@15077
  1718
apply (rule iffI)
paulson@15077
  1719
apply (cut_tac linorder_linear [of 0 x], safe)
paulson@15077
  1720
apply (drule sin_zero_lemma, assumption+)
paulson@15077
  1721
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
paulson@15077
  1722
apply (force simp add: minus_equation_iff [of x]) 
paulson@15077
  1723
apply (auto simp add: even_mult_two_ex)
paulson@15077
  1724
done
paulson@15077
  1725
paulson@15077
  1726
paulson@15077
  1727
subsection{*Tangent*}
paulson@15077
  1728
paulson@15077
  1729
lemma tan_zero [simp]: "tan 0 = 0"
paulson@15077
  1730
by (simp add: tan_def)
paulson@15077
  1731
paulson@15077
  1732
lemma tan_pi [simp]: "tan pi = 0"
paulson@15077
  1733
by (simp add: tan_def)
paulson@15077
  1734
paulson@15077
  1735
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
paulson@15077
  1736
by (simp add: tan_def)
paulson@15077
  1737
paulson@15077
  1738
lemma tan_minus [simp]: "tan (-x) = - tan x"
paulson@15077
  1739
by (simp add: tan_def minus_mult_left)
paulson@15077
  1740
paulson@15077
  1741
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
paulson@15077
  1742
by (simp add: tan_def)
paulson@15077
  1743
paulson@15077
  1744
lemma lemma_tan_add1: 
paulson@15077
  1745
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1746
        ==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
paulson@15077
  1747
apply (unfold tan_def real_divide_def)
paulson@15077
  1748
apply (auto simp del: inverse_mult_distrib simp add: inverse_mult_distrib [symmetric] mult_ac)
paulson@15077
  1749
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  1750
apply (auto simp del: inverse_mult_distrib simp add: mult_assoc left_diff_distrib cos_add)
paulson@15077
  1751
done
paulson@15077
  1752
paulson@15077
  1753
lemma add_tan_eq: 
paulson@15077
  1754
      "[| cos x \<noteq> 0; cos y \<noteq> 0 |]  
paulson@15077
  1755
       ==> tan x + tan y = sin(x + y)/(cos x * cos y)"
paulson@15077
  1756
apply (unfold tan_def)
paulson@15077
  1757
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
paulson@15077
  1758
apply (auto simp add: mult_assoc left_distrib)
paulson@15077
  1759
apply (simp (no_asm) add: sin_add)
paulson@15077
  1760
done
paulson@15077
  1761
paulson@15077
  1762
lemma tan_add: "[| 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@15077
  1768
lemma tan_double: "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]  
paulson@15077
  1769
      ==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
paulson@15077
  1770
apply (insert tan_add [of x x]) 
paulson@15077
  1771
apply (simp add: mult_2 [symmetric])  
paulson@15077
  1772
apply (auto simp add: numeral_2_eq_2)
paulson@15077
  1773
done
paulson@15077
  1774
paulson@15077
  1775
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
paulson@15077
  1776
apply (unfold tan_def real_divide_def)
paulson@15077
  1777
apply (auto intro!: sin_gt_zero2 cos_gt_zero_pi intro!: real_mult_order positive_imp_inverse_positive)
paulson@15077
  1778
done
paulson@15077
  1779
paulson@15077
  1780
lemma tan_less_zero: 
paulson@15077
  1781
  assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
paulson@15077
  1782
proof -
paulson@15077
  1783
  have "0 < tan (- x)" using prems by (simp only: tan_gt_zero) 
paulson@15077
  1784
  thus ?thesis by simp
paulson@15077
  1785
qed
paulson@15077
  1786
paulson@15077
  1787
lemma lemma_DERIV_tan:
paulson@15077
  1788
     "cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1789
apply (rule lemma_DERIV_subst)
paulson@15077
  1790
apply (best intro!: DERIV_intros intro: DERIV_chain2) 
paulson@15077
  1791
apply (auto simp add: real_divide_def numeral_2_eq_2)
paulson@15077
  1792
done
paulson@15077
  1793
paulson@15077
  1794
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
paulson@15077
  1795
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
paulson@15077
  1796
paulson@15077
  1797
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
paulson@15077
  1798
apply (unfold real_divide_def)
paulson@15077
  1799
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
paulson@15077
  1800
apply simp 
paulson@15077
  1801
apply (rule LIM_mult2)
paulson@15077
  1802
apply (rule_tac [2] inverse_1 [THEN subst])
paulson@15077
  1803
apply (rule_tac [2] LIM_inverse)
paulson@15077
  1804
apply (simp_all add: divide_inverse [symmetric]) 
paulson@15077
  1805
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric]) 
paulson@15077
  1806
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
paulson@15077
  1807
done
paulson@15077
  1808
paulson@15077
  1809
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
paulson@15077
  1810
apply (cut_tac LIM_cos_div_sin)
paulson@15077
  1811
apply (simp only: LIM_def)
paulson@15077
  1812
apply (drule_tac x = "inverse y" in spec, safe, force)
paulson@15077
  1813
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
paulson@15077
  1814
apply (rule_tac x = " (pi/2) - e" in exI)
paulson@15077
  1815
apply (simp (no_asm_simp))
paulson@15077
  1816
apply (drule_tac x = " (pi/2) - e" in spec)
paulson@15077
  1817
apply (auto simp add: abs_eqI2 tan_def)
paulson@15077
  1818
apply (rule inverse_less_iff_less [THEN iffD1])
paulson@15077
  1819
apply (auto simp add: real_divide_def)
paulson@15077
  1820
apply (rule real_mult_order)
paulson@15077
  1821
apply (subgoal_tac [3] "0 < sin e")
paulson@15077
  1822
apply (subgoal_tac [3] "0 < cos e")
paulson@15077
  1823
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: inverse_mult_distrib abs_mult)
paulson@15077
  1824
apply (drule_tac a = "cos e" in positive_imp_inverse_positive)
paulson@15077
  1825
apply (drule_tac x = "inverse (cos e) " in abs_eqI2)
paulson@15077
  1826
apply (auto dest!: abs_eqI2 simp add: mult_ac)
paulson@15077
  1827
done
paulson@15077
  1828
paulson@15077
  1829
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
paulson@15077
  1830
apply (frule real_le_imp_less_or_eq, safe)
paulson@15077
  1831
 prefer 2 apply force
paulson@15077
  1832
apply (drule lemma_tan_total, safe)
paulson@15077
  1833
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
paulson@15077
  1834
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
paulson@15077
  1835
apply (drule_tac y = xa in order_le_imp_less_or_eq)
paulson@15077
  1836
apply (auto dest: cos_gt_zero)
paulson@15077
  1837
done
paulson@15077
  1838
paulson@15077
  1839
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1840
apply (cut_tac linorder_linear [of 0 y], safe)
paulson@15077
  1841
apply (drule tan_total_pos)
paulson@15077
  1842
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
paulson@15077
  1843
apply (rule_tac [3] x = "-x" in exI)
paulson@15077
  1844
apply (auto intro!: exI)
paulson@15077
  1845
done
paulson@15077
  1846
paulson@15077
  1847
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
paulson@15077
  1848
apply (cut_tac y = y in lemma_tan_total1, auto)
paulson@15077
  1849
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
paulson@15077
  1850
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
paulson@15077
  1851
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
paulson@15077
  1852
apply (rule_tac [4] Rolle)
paulson@15077
  1853
apply (rule_tac [2] Rolle)
paulson@15077
  1854
apply (auto intro!: DERIV_tan DERIV_isCont exI 
paulson@15077
  1855
            simp add: differentiable_def)
paulson@15077
  1856
txt{*Now, simulate TRYALL*}
paulson@15077
  1857
apply (rule_tac [!] DERIV_tan asm_rl)
paulson@15077
  1858
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
paulson@15077
  1859
	    simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym]) 
paulson@15077
  1860
done
paulson@15077
  1861
paulson@15077
  1862
lemma arcsin_pi: "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1863
      ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
paulson@15077
  1864
apply (drule sin_total, assumption)
paulson@15077
  1865
apply (erule ex1E)
paulson@15077
  1866
apply (unfold arcsin_def)
paulson@15077
  1867
apply (rule someI2, blast) 
paulson@15077
  1868
apply (force intro: order_trans) 
paulson@15077
  1869
done
paulson@15077
  1870
paulson@15077
  1871
lemma arcsin: "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1872
      ==> -(pi/2) \<le> arcsin y &  
paulson@15077
  1873
           arcsin y \<le> pi/2 & sin(arcsin y) = y"
paulson@15077
  1874
apply (unfold arcsin_def)
paulson@15077
  1875
apply (drule sin_total, assumption)
paulson@15077
  1876
apply (fast intro: someI2)
paulson@15077
  1877
done
paulson@15077
  1878
paulson@15077
  1879
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
paulson@15077
  1880
by (blast dest: arcsin)
paulson@15077
  1881
      
paulson@15077
  1882
lemma arcsin_bounded:
paulson@15077
  1883
     "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
paulson@15077
  1884
by (blast dest: arcsin)
paulson@15077
  1885
paulson@15077
  1886
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
paulson@15077
  1887
by (blast dest: arcsin)
paulson@15077
  1888
paulson@15077
  1889
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
paulson@15077
  1890
by (blast dest: arcsin)
paulson@15077
  1891
paulson@15077
  1892
lemma arcsin_lt_bounded:
paulson@15077
  1893
     "[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
paulson@15077
  1894
apply (frule order_less_imp_le)
paulson@15077
  1895
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1896
apply (frule arcsin_bounded)
paulson@15077
  1897
apply (safe, simp)
paulson@15077
  1898
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
paulson@15077
  1899
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
paulson@15077
  1900
apply (drule_tac [!] f = sin in arg_cong, auto)
paulson@15077
  1901
done
paulson@15077
  1902
paulson@15077
  1903
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
paulson@15077
  1904
apply (unfold arcsin_def)
paulson@15077
  1905
apply (rule some1_equality)
paulson@15077
  1906
apply (rule sin_total, auto)
paulson@15077
  1907
done
paulson@15077
  1908
paulson@15077
  1909
lemma arcos: "[| -1 \<le> y; y \<le> 1 |]  
paulson@15077
  1910
      ==> 0 \<le> arcos y & arcos y \<le> pi & cos(arcos y) = y"
paulson@15077
  1911
apply (unfold arcos_def)
paulson@15077
  1912
apply (drule cos_total, assumption)
paulson@15077
  1913
apply (fast intro: someI2)
paulson@15077
  1914
done
paulson@15077
  1915
paulson@15077
  1916
lemma cos_arcos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arcos y) = y"
paulson@15077
  1917
by (blast dest: arcos)
paulson@15077
  1918
      
paulson@15077
  1919
lemma arcos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y & arcos y \<le> pi"
paulson@15077
  1920
by (blast dest: arcos)
paulson@15077
  1921
paulson@15077
  1922
lemma arcos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y"
paulson@15077
  1923
by (blast dest: arcos)
paulson@15077
  1924
paulson@15077
  1925
lemma arcos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcos y \<le> pi"
paulson@15077
  1926
by (blast dest: arcos)
paulson@15077
  1927
paulson@15077
  1928
lemma arcos_lt_bounded: "[| -1 < y; y < 1 |]  
paulson@15077
  1929
      ==> 0 < arcos y & arcos y < pi"
paulson@15077
  1930
apply (frule order_less_imp_le)
paulson@15077
  1931
apply (frule_tac y = y in order_less_imp_le)
paulson@15077
  1932
apply (frule arcos_bounded, auto)
paulson@15077
  1933
apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
paulson@15077
  1934
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
paulson@15077
  1935
apply (drule_tac [!] f = cos in arg_cong, auto)
paulson@15077
  1936
done
paulson@15077
  1937
paulson@15077
  1938
lemma arcos_cos: "[|0 \<le> x; x \<le> pi |] ==> arcos(cos x) = x"
paulson@15077
  1939
apply (unfold arcos_def)
paulson@15077
  1940
apply (auto intro!: some1_equality cos_total)
paulson@15077
  1941
done
paulson@15077
  1942
paulson@15077
  1943
lemma arcos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arcos(cos x) = -x"
paulson@15077
  1944
apply (unfold arcos_def)
paulson@15077
  1945
apply (auto intro!: some1_equality cos_total)
paulson@15077
  1946
done
paulson@15077
  1947
paulson@15077
  1948
lemma arctan [simp]:
paulson@15077
  1949
     "- (pi/2) < arctan y  & arctan y < pi/2 & tan (arctan y) = y"
paulson@15077
  1950
apply (cut_tac y = y in tan_total)
paulson@15077
  1951
apply (unfold arctan_def)
paulson@15077
  1952
apply (fast intro: someI2)
paulson@15077
  1953
done
paulson@15077
  1954
paulson@15077
  1955
lemma tan_arctan: "tan(arctan y) = y"
paulson@15077
  1956
by auto
paulson@15077
  1957
paulson@15077
  1958
lemma arctan_bounded: "- (pi/2) < arctan y  & arctan y < pi/2"
paulson@15077
  1959
by (auto simp only: arctan)
paulson@15077
  1960
paulson@15077
  1961
lemma arctan_lbound: "- (pi/2) < arctan y"
paulson@15077
  1962
by auto
paulson@15077
  1963
paulson@15077
  1964
lemma arctan_ubound: "arctan y < pi/2"
paulson@15077
  1965
by (auto simp only: arctan)
paulson@15077
  1966
paulson@15077
  1967
lemma arctan_tan: 
paulson@15077
  1968
      "[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
paulson@15077
  1969
apply (unfold arctan_def)
paulson@15077
  1970
apply (rule some1_equality)
paulson@15077
  1971
apply (rule tan_total, auto)
paulson@15077
  1972
done
paulson@15077
  1973
paulson@15077
  1974
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
paulson@15077
  1975
by (insert arctan_tan [of 0], simp)
paulson@15077
  1976
paulson@15077
  1977
lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
paulson@15077
  1978
apply (auto simp add: cos_zero_iff)
paulson@15077
  1979
apply (case_tac "n")
paulson@15077
  1980
apply (case_tac [3] "n")
paulson@15077
  1981
apply (cut_tac [2] y = x in arctan_ubound)
paulson@15077
  1982
apply (cut_tac [4] y = x in arctan_lbound) 
paulson@15077
  1983
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
paulson@15077
  1984
done
paulson@15077
  1985
paulson@15077
  1986
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
paulson@15077
  1987
apply (rule power_inverse [THEN subst])
paulson@15077
  1988
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
paulson@15077
  1989
apply (auto dest: realpow_not_zero 
paulson@15077
  1990
        simp add: power_mult_distrib left_distrib realpow_divide tan_def 
paulson@15077
  1991
                  mult_assoc power_inverse [symmetric] 
paulson@15077
  1992
        simp del: realpow_Suc)
paulson@15077
  1993
done
paulson@15077
  1994
paulson@15077
  1995
lemma lemma_sin_cos_eq [simp]: "sin (xa + 1 / 2 * real (Suc m) * pi) =  
paulson@15077
  1996
      cos (xa + 1 / 2 * real  (m) * pi)"
paulson@15077
  1997
apply (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  1998
done
paulson@15077
  1999
paulson@15077
  2000
lemma lemma_sin_cos_eq2 [simp]: "sin (xa + real (Suc m) * pi / 2) =  
paulson@15077
  2001
      cos (xa + real (m) * pi / 2)"
paulson@15077
  2002
apply (simp only: cos_add sin_add real_divide_def real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  2003
done
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@15077
  2012
(* which further simplifies to (- 1 ^ m) !! *)
paulson@15077
  2013
lemma sin_cos_npi [simp]: "sin ((real m + 1/2) * pi) = cos (real m * pi)"
paulson@15077
  2014
by (auto simp add: right_distrib sin_add left_distrib mult_ac)
paulson@15077
  2015
paulson@15077
  2016
lemma sin_cos_npi2 [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
paulson@15077
  2017
apply (cut_tac m = n in sin_cos_npi)
paulson@15077
  2018
apply (simp only: real_of_nat_Suc left_distrib real_divide_def, auto)
paulson@15077
  2019
done
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@15077
  2040
lemma [simp]: "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
paulson@15077
  2041
apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
paulson@15077
  2042
done
paulson@15077
  2043
paulson@15077
  2044
(*NEEDED??*)
paulson@15077
  2045
lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
paulson@15077
  2046
apply (simp only: cos_add sin_add real_divide_def real_of_nat_Suc left_distrib right_distrib, auto)
paulson@15077
  2047
done
paulson@15077
  2048
paulson@15077
  2049
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
paulson@15077
  2050
by (simp only: cos_add sin_add real_divide_def real_of_nat_Suc left_distrib right_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@15077
  2068
lemma sin_zero_abs_cos_one: "sin x = 0 ==> abs(cos x) = 1"
paulson@15077
  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@15077
  2080
lemma real_root_less_mono: "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
paulson@15077
  2081
apply (frule order_le_less_trans, assumption)
paulson@15077
  2082
apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
paulson@15077
  2083
apply (rotate_tac 1, assumption)
paulson@15077
  2084
apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
paulson@15077
  2085
apply (rotate_tac 3, assumption)
paulson@15077
  2086
apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
paulson@15077
  2087
apply (frule_tac n = n in real_root_pos_pos_le)
paulson@15077
  2088
apply (frule_tac n = n in real_root_pos_pos)
paulson@15077
  2089
apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
paulson@15077
  2090
apply (assumption, assumption)
paulson@15077
  2091
apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
paulson@15077
  2092
apply auto
paulson@15077
  2093
apply (drule_tac f = "%x. x ^ (Suc n) " in arg_cong)
paulson@15077
  2094
apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
paulson@15077
  2095
done
paulson@15077
  2096
paulson@15077
  2097
lemma real_root_le_mono: "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
paulson@15077
  2098
apply (drule_tac y = y in order_le_imp_less_or_eq)
paulson@15077
  2099
apply (auto dest: real_root_less_mono intro: order_less_imp_le)
paulson@15077
  2100
done
paulson@15077
  2101
paulson@15077
  2102
lemma real_root_less_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
paulson@15077
  2103
apply (auto intro: real_root_less_mono)
paulson@15077
  2104
apply (rule ccontr, drule linorder_not_less [THEN iffD1])
paulson@15077
  2105
apply (drule_tac x = y and n = n in real_root_le_mono, auto)
paulson@15077
  2106
done
paulson@15077
  2107
paulson@15077
  2108
lemma real_root_le_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x \<le> root(Suc n) y) = (x \<le> y)"
paulson@15077
  2109
apply (auto intro: real_root_le_mono)
paulson@15077
  2110
apply (simp (no_asm) add: linorder_not_less [symmetric])
paulson@15077
  2111
apply auto
paulson@15077
  2112
apply (drule_tac x = y and n = n in real_root_less_mono, auto)
paulson@15077
  2113
done
paulson@15077
  2114
paulson@15077
  2115
lemma real_root_eq_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
paulson@15077
  2116
apply (auto intro!: order_antisym)
paulson@15077
  2117
apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
paulson@15077
  2118
apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
paulson@15077
  2119
done
paulson@15077
  2120
paulson@15077
  2121
lemma real_root_pos_unique: "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
paulson@15077
  2122
by (auto dest: real_root_pos2 simp del: realpow_Suc)
paulson@15077
  2123
paulson@15077
  2124
lemma real_root_mult: "[| 0 \<le> x; 0 \<le> y |] 
paulson@15077
  2125
      ==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
paulson@15077
  2126
apply (rule real_root_pos_unique)
paulson@15077
  2127
apply (auto intro!: real_root_pos_pos_le simp add: power_mult_distrib zero_le_mult_iff real_root_pow_pos2 simp del: realpow_Suc)
paulson@15077
  2128
done
paulson@15077
  2129
paulson@15077
  2130
lemma real_root_inverse: "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
paulson@15077
  2131
apply (rule real_root_pos_unique)
paulson@15077
  2132
apply (auto intro: real_root_pos_pos_le simp add: power_inverse [symmetric] real_root_pow_pos2 simp del: realpow_Suc)
paulson@15077
  2133
done
paulson@15077
  2134
paulson@15077
  2135
lemma real_root_divide: 
paulson@15077
  2136
     "[| 0 \<le> x; 0 \<le> y |]  
paulson@15077
  2137
      ==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
paulson@15077
  2138
apply (unfold real_divide_def)
paulson@15077
  2139
apply (auto simp add: real_root_mult real_root_inverse)
paulson@15077
  2140
done
paulson@15077
  2141
paulson@15077
  2142
lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
paulson@15077
  2143
apply (unfold sqrt_def)
paulson@15077
  2144
apply (auto intro: real_root_less_mono)
paulson@15077
  2145
done
paulson@15077
  2146
paulson@15077
  2147
lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
paulson@15077
  2148
apply (unfold sqrt_def)
paulson@15077
  2149
apply (auto intro: real_root_le_mono)
paulson@15077
  2150
done
paulson@15077
  2151
paulson@15077
  2152
lemma real_sqrt_less_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
paulson@15077
  2153
by (unfold sqrt_def, auto)
paulson@15077
  2154
paulson@15077
  2155
lemma real_sqrt_le_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
paulson@15077
  2156
by (unfold sqrt_def, auto)
paulson@15077
  2157
paulson@15077
  2158
lemma real_sqrt_eq_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
paulson@15077
  2159
by (unfold sqrt_def, auto)
paulson@15077
  2160
paulson@15077
  2161
lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
paulson@15077
  2162
apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077
  2163
apply (rule_tac [2] real_sqrt_less_mono)
paulson@15077
  2164
apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
paulson@15077
  2165
done
paulson@15077
  2166
paulson@15077
  2167
lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
paulson@15077
  2168
apply (rule real_sqrt_one [THEN subst], safe)
paulson@15077
  2169
apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
paulson@15077
  2170
done
paulson@15077
  2171
paulson@15077
  2172
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
paulson@15077
  2173
apply (unfold real_divide_def)
paulson@15077
  2174
apply (case_tac "r=0")
paulson@15077
  2175
apply (auto simp add: inverse_mult_distrib mult_ac)
paulson@15077
  2176
done
paulson@15077
  2177
paulson@15077
  2178
paulson@15077
  2179
subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}
paulson@15077
  2180
paulson@15077
  2181
lemma lemma_real_divide_sqrt: 
paulson@15077
  2182
    "0 < x ==> 0 \<le> x/(sqrt (x * x + y * y))"
paulson@15077
  2183
apply (unfold real_divide_def)
paulson@15077
  2184
apply (rule real_mult_order [THEN order_less_imp_le], assumption)
paulson@15077
  2185
apply (subgoal_tac "0 < inverse (sqrt (x\<twosuperior> + y\<twosuperior>))") 
paulson@15077
  2186
 apply (simp add: numeral_2_eq_2)
paulson@15077
  2187
apply (simp add: real_sqrt_sum_squares_ge1 [THEN [2] order_less_le_trans]) 
paulson@15077
  2188
done
paulson@15077
  2189
paulson@15077
  2190
lemma lemma_real_divide_sqrt_ge_minus_one:
paulson@15077
  2191
     "0 < x ==> -1 \<le> x/(sqrt (x * x + y * y))"
paulson@15077
  2192
apply (rule real_le_trans)
paulson@15077
  2193
apply (rule_tac [2] lemma_real_divide_sqrt, auto)
paulson@15077
  2194
done
paulson@15077
  2195
paulson@15077
  2196
lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
paulson@15077
  2197
apply (rule real_sqrt_gt_zero)
paulson@15077
  2198
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077
  2199
apply (auto simp add: zero_less_mult_iff)
paulson@15077
  2200
done
paulson@15077
  2201
paulson@15077
  2202
lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
paulson@15077
  2203
apply (rule real_sqrt_gt_zero)
paulson@15077
  2204
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith) 
paulson@15077
  2205
apply (auto simp add: zero_less_mult_iff)
paulson@15077
  2206
done
paulson@15077
  2207
paulson@15077
  2208
lemma real_sqrt_sum_squares_gt_zero3: "x \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2209
apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077
  2210
apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
paulson@15077
  2211
done
paulson@15077
  2212
paulson@15077
  2213
lemma real_sqrt_sum_squares_gt_zero3a: "y \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2214
apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
paulson@15077
  2215
apply (auto simp add: real_add_commute)
paulson@15077
  2216
done
paulson@15077
  2217
paulson@15077
  2218
lemma real_sqrt_sum_squares_eq_cancel [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = x ==> y = 0"
paulson@15077
  2219
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, auto)
paulson@15077
  2220
paulson@15077
  2221
lemma real_sqrt_sum_squares_eq_cancel2 [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = y ==> x = 0"
paulson@15077
  2222
apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
paulson@15077
  2223
apply (simp add: real_add_commute)
paulson@15077
  2224
done
paulson@15077
  2225
paulson@15077
  2226
lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077
  2227
by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)
paulson@15077
  2228
paulson@15077
  2229
lemma lemma_real_divide_sqrt_ge_minus_one2:
paulson@15077
  2230
     "x < 0 ==> -1 \<le> x/(sqrt (x * x + y * y))"
paulson@15077
  2231
apply (case_tac "y = 0", auto)
paulson@15077
  2232
apply (frule abs_minus_eqI2)
paulson@15077
  2233
apply (auto simp add: inverse_minus_eq)
paulson@15077
  2234
apply (rule order_less_imp_le)
paulson@15077
  2235
apply (rule_tac z1 = "sqrt (x * x + y * y) " in real_mult_less_iff1 [THEN iffD1])
paulson@15077
  2236
apply (frule_tac [2] y2 = y in
paulson@15077
  2237
       real_sqrt_sum_squares_gt_zero1 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2238
apply (auto intro: real_sqrt_sum_squares_gt_zero1 simp add: mult_ac)
paulson@15077
  2239
apply (cut_tac x = "-x" and y = y in real_sqrt_sum_squares_ge1)
paulson@15077
  2240
apply (drule order_le_less [THEN iffD1], safe) 
paulson@15077
  2241
apply (simp add: numeral_2_eq_2)
paulson@15077
  2242
apply (drule sym [THEN real_sqrt_sum_squares_eq_cancel], simp)
paulson@15077
  2243
done
paulson@15077
  2244
paulson@15077
  2245
lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) \<le> 1"
paulson@15077
  2246
by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)
paulson@15077
  2247
paulson@15077
  2248
paulson@15077
  2249
lemma cos_x_y_ge_minus_one: "-1 \<le> x / sqrt (x * x + y * y)"
paulson@15077
  2250
apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
paulson@15077
  2251
apply (rule lemma_real_divide_sqrt_ge_minus_one2)
paulson@15077
  2252
apply (rule_tac [3] lemma_real_divide_sqrt_ge_minus_one, auto)
paulson@15077
  2253
done
paulson@15077
  2254
paulson@15077
  2255
lemma cos_x_y_ge_minus_one1a [simp]: "-1 \<le> y / sqrt (x * x + y * y)"
paulson@15077
  2256
apply (cut_tac x = y and y = x in cos_x_y_ge_minus_one)
paulson@15077
  2257
apply (simp add: real_add_commute)
paulson@15077
  2258
done
paulson@15077
  2259
paulson@15077
  2260
lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) \<le> 1"
paulson@15077
  2261
apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
paulson@15077
  2262
apply (rule lemma_real_divide_sqrt_le_one)
paulson@15077
  2263
apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
paulson@15077
  2264
done
paulson@15077
  2265
paulson@15077
  2266
lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) \<le> 1"
paulson@15077
  2267
apply (cut_tac x = y and y = x in cos_x_y_le_one)
paulson@15077
  2268
apply (simp add: real_add_commute)
paulson@15077
  2269
done
paulson@15077
  2270
paulson@15077
  2271
declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2272
declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2273
paulson@15077
  2274
declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077
  2275
declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp] 
paulson@15077
  2276
paulson@15077
  2277
lemma cos_abs_x_y_ge_minus_one [simp]:
paulson@15077
  2278
     "-1 \<le> \<bar>x\<bar> / sqrt (x * x + y * y)"
paulson@15077
  2279
apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077
  2280
apply (auto simp add: abs_minus_eqI2 abs_eqI2)
paulson@15077
  2281
apply (drule lemma_real_divide_sqrt_ge_minus_one, force)
paulson@15077
  2282
done
paulson@15077
  2283
paulson@15077
  2284
lemma cos_abs_x_y_le_one [simp]: "\<bar>x\<bar> / sqrt (x * x + y * y) \<le> 1"
paulson@15077
  2285
apply (cut_tac x = x and y = 0 in linorder_less_linear)
paulson@15077
  2286
apply (auto simp add: abs_minus_eqI2 abs_eqI2)
paulson@15077
  2287
apply (drule lemma_real_divide_sqrt_ge_minus_one2, force)
paulson@15077
  2288
done
paulson@15077
  2289
paulson@15077
  2290
declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077
  2291
declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp] 
paulson@15077
  2292
paulson@15077
  2293
lemma minus_pi_less_zero: "-pi < 0"
paulson@15077
  2294
apply (simp (no_asm))
paulson@15077
  2295
done
paulson@15077
  2296
declare minus_pi_less_zero [simp]
paulson@15077
  2297
declare minus_pi_less_zero [THEN order_less_imp_le, simp]
paulson@15077
  2298
paulson@15077
  2299
lemma arcos_ge_minus_pi: "[| -1 \<le> y; y \<le> 1 |] ==> -pi \<le> arcos y"
paulson@15077
  2300
apply (rule real_le_trans)
paulson@15077
  2301
apply (rule_tac [2] arcos_lbound, auto)
paulson@15077
  2302
done
paulson@15077
  2303
paulson@15077
  2304
declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp] 
paulson@15077
  2305
paulson@15077
  2306
(* How tedious! *)
paulson@15077
  2307
lemma lemma_divide_rearrange:
paulson@15077
  2308
     "[| x + (y::real) \<noteq> 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
paulson@15077
  2309
apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
paulson@15077
  2310
apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
paulson@15077
  2311
prefer 2 apply assumption
paulson@15077
  2312
apply (rotate_tac [2] 2)
paulson@15077
  2313
apply (drule_tac [2] mult_assoc [THEN subst])
paulson@15077
  2314
apply (rotate_tac [2] 2)
paulson@15077
  2315
apply (frule_tac [2] left_inverse [THEN subst])
paulson@15077
  2316
prefer 2 apply assumption
paulson@15077
  2317
apply (erule_tac [2] V = " (1 - z) * (x + y) = x / (x + y) * (x + y) " in thin_rl)
paulson@15077
  2318
apply (erule_tac [2] V = "1 - z = x / (x + y) " in thin_rl)
paulson@15077
  2319
apply (auto simp add: mult_assoc)
paulson@15077
  2320
apply (auto simp add: right_distrib left_diff_distrib)
paulson@15077
  2321
done
paulson@15077
  2322
paulson@15077
  2323
lemma lemma_cos_sin_eq:
paulson@15077
  2324
     "[| 0 < x * x + y * y;  
paulson@15077
  2325
         1 - (sin xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2 |] 
paulson@15077
  2326
      ==> (sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2"
paulson@15077
  2327
by (auto intro: lemma_divide_rearrange
paulson@15077
  2328
         simp add: realpow_divide power2_eq_square [symmetric])
paulson@15077
  2329
paulson@15077
  2330
paulson@15077
  2331
lemma lemma_sin_cos_eq:
paulson@15077
  2332
     "[| 0 < x * x + y * y;  
paulson@15077
  2333
         1 - (cos xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2 |]
paulson@15077
  2334
      ==> (cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2"
paulson@15077
  2335
apply (auto simp add: realpow_divide power2_eq_square [symmetric])
paulson@15077
  2336
apply (rule add_commute [THEN subst])
paulson@15077
  2337
apply (rule lemma_divide_rearrange, simp)
paulson@15077
  2338
apply (simp add: add_commute)
paulson@15077
  2339
done
paulson@15077
  2340
paulson@15077
  2341
lemma sin_x_y_disj:
paulson@15077
  2342
     "[| x \<noteq> 0;  
paulson@15077
  2343
         cos xa = x / sqrt (x * x + y * y)  
paulson@15077
  2344
      |] ==>  sin xa = y / sqrt (x * x + y * y) |  
paulson@15077
  2345
              sin xa = - y / sqrt (x * x + y * y)"
paulson@15077
  2346
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077
  2347
apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077
  2348
apply (simp add: cos_squared_eq)
paulson@15077
  2349
apply (subgoal_tac "(sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2")
paulson@15077
  2350
apply (rule_tac [2] lemma_cos_sin_eq)
paulson@15077
  2351
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077
  2352
done
paulson@15077
  2353
paulson@15077
  2354
lemma lemma_cos_not_eq_zero: "x \<noteq> 0 ==> x / sqrt (x * x + y * y) \<noteq> 0"
paulson@15077
  2355
apply (unfold real_divide_def)
paulson@15077
  2356
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
  2357
apply (auto simp add: power2_eq_square)
paulson@15077
  2358
done
paulson@15077
  2359
paulson@15077
  2360
lemma cos_x_y_disj: "[| x \<noteq> 0;  
paulson@15077
  2361
         sin xa = y / sqrt (x * x + y * y)  
paulson@15077
  2362
      |] ==>  cos xa = x / sqrt (x * x + y * y) |  
paulson@15077
  2363
              cos xa = - x / sqrt (x * x + y * y)"
paulson@15077
  2364
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
paulson@15077
  2365
apply (frule_tac y = y in real_sum_square_gt_zero)
paulson@15077
  2366
apply (simp add: sin_squared_eq del: realpow_Suc)
paulson@15077
  2367
apply (subgoal_tac "(cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2")
paulson@15077
  2368
apply (rule_tac [2] lemma_sin_cos_eq)
paulson@15077
  2369
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
paulson@15077
  2370
done
paulson@15077
  2371
paulson@15077
  2372
lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
paulson@15077
  2373
apply (case_tac "x = 0")
paulson@15077
  2374
apply (auto simp add: abs_eqI2)
paulson@15077
  2375
apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
paulson@15077
  2376
apply (auto simp add: zero_less_mult_iff real_divide_def power2_eq_square)
paulson@15077
  2377
done
paulson@15077
  2378
paulson@15077
  2379
lemma polar_ex1: "[| x \<noteq> 0; 0 < y |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2380
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>) " in exI)
paulson@15077
  2381
apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
paulson@15077
  2382
apply auto
paulson@15077
  2383
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2384
apply (auto simp add: power2_eq_square)
paulson@15077
  2385
apply (unfold arcos_def)
paulson@15077
  2386
apply (cut_tac x1 = x and y1 = y 
paulson@15077
  2387
       in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
paulson@15077
  2388
apply (rule someI2_ex, blast)
paulson@15077
  2389
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
  2390
apply (frule sin_x_y_disj, blast)
paulson@15077
  2391
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
paulson@15077
  2392
apply (auto simp add: power2_eq_square)
paulson@15077
  2393
apply (drule sin_ge_zero, assumption)
paulson@15077
  2394
apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
paulson@15077
  2395
done
paulson@15077
  2396
paulson@15077
  2397
lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
paulson@15077
  2398
by (auto intro: real_sum_squares_cancel)
paulson@15077
  2399
paulson@15077
  2400
lemma polar_ex2: "[| x \<noteq> 0; y < 0 |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2401
apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
paulson@15077
  2402
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>) " in exI)
paulson@15077
  2403
apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI)
paulson@15077
  2404
apply (auto dest: real_sum_squares_cancel2a simp add: power2_eq_square)
paulson@15077
  2405
apply (unfold arcsin_def)
paulson@15077
  2406
apply (cut_tac x1 = x and y1 = y 
paulson@15077
  2407
       in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
paulson@15077
  2408
apply (rule someI2_ex, blast)
paulson@15077
  2409
apply (erule_tac V = "EX! xa. - (pi/2) \<le> xa & xa \<le> pi/2 & sin xa = y / sqrt (x * x + y * y) " in thin_rl)
paulson@15077
  2410
apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast, auto)
paulson@15077
  2411
apply (drule cos_ge_zero, force)
paulson@15077
  2412
apply (drule_tac x = y in real_sqrt_divide_less_zero)
paulson@15077
  2413
apply (auto simp add: real_add_commute)
paulson@15077
  2414
apply (insert polar_ex1 [of x "-y"], simp, clarify) 
paulson@15077
  2415
apply (rule_tac x = r in exI)
paulson@15077
  2416
apply (rule_tac x = "-a" in exI, simp) 
paulson@15077
  2417
done
paulson@15077
  2418
paulson@15077
  2419
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
paulson@15077
  2420
apply (case_tac "x = 0", auto)
paulson@15077
  2421
apply (rule_tac x = y in exI)
paulson@15077
  2422
apply (rule_tac x = "pi/2" in exI, auto)
paulson@15077
  2423
apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
paulson@15077
  2424
apply (rule_tac [2] x = x in exI)
paulson@15077
  2425
apply (rule_tac [2] x = 0 in exI, auto)
paulson@15077
  2426
apply (blast intro: polar_ex1 polar_ex2)+
paulson@15077
  2427
done
paulson@15077
  2428
paulson@15077
  2429
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2430
apply (rule_tac n = 1 in realpow_increasing)
paulson@15077
  2431
apply (auto simp add: numeral_2_eq_2 [symmetric])
paulson@15077
  2432
done
paulson@15077
  2433
paulson@15077
  2434
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
paulson@15077
  2435
apply (rule real_add_commute [THEN subst])
paulson@15077
  2436
apply (rule real_sqrt_ge_abs1)
paulson@15077
  2437
done
paulson@15077
  2438
declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]
paulson@15077
  2439
paulson@15077
  2440
lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
paulson@15077
  2441
by (auto intro: real_sqrt_gt_zero)
paulson@15077
  2442
paulson@15077
  2443
lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
paulson@15077
  2444
by (auto intro: real_sqrt_ge_zero)
paulson@15077
  2445
paulson@15077
  2446
lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
paulson@15077
  2447
apply (rule order_less_le_trans [of _ "7/5"], simp) 
paulson@15077
  2448
apply (rule_tac n = 1 in realpow_increasing)
paulson@15077
  2449
  prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
paulson@15077
  2450
apply (simp_all add: numeral_2_eq_2)
paulson@15077
  2451
done
paulson@15077
  2452