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(* Title : Transcendental.thy
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Author : Jacques D. Fleuriot
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Copyright : 1998,1999 University of Cambridge
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1999,2001 University of Edinburgh
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Conversion to Isar and new proofs by Lawrence C Paulson, 2004
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*)
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header{*Power Series, Transcendental Functions etc.*}
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theory Transcendental
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import NthRoot Fact HSeries EvenOdd Lim
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begin
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constdefs
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root :: "[nat,real] => real"
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"root n x == (@u. ((0::real) < x --> 0 < u) & (u ^ n = x))"
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sqrt :: "real => real"
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"sqrt x == root 2 x"
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exp :: "real => real"
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"exp x == suminf(%n. inverse(real (fact n)) * (x ^ n))"
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sin :: "real => real"
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"sin x == suminf(%n. (if even(n) then 0 else
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((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
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diffs :: "(nat => real) => nat => real"
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"diffs c == (%n. real (Suc n) * c(Suc n))"
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cos :: "real => real"
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"cos x == suminf(%n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n))
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else 0) * x ^ n)"
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ln :: "real => real"
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"ln x == (@u. exp u = x)"
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pi :: "real"
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"pi == 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"
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tan :: "real => real"
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"tan x == (sin x)/(cos x)"
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arcsin :: "real => real"
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"arcsin y == (@x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"
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arcos :: "real => real"
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"arcos y == (@x. 0 \<le> x & x \<le> pi & cos x = y)"
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arctan :: "real => real"
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"arctan y == (@x. -(pi/2) < x & x < pi/2 & tan x = y)"
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lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
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apply (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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paulson@15077
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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>)"
|
paulson@15077
|
251 |
apply (simp (no_asm) add: real_add_commute del: realpow_Suc)
|
paulson@15077
|
252 |
done
|
paulson@15077
|
253 |
|
paulson@15077
|
254 |
lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"
|
paulson@15077
|
255 |
apply (rule_tac n = 1 in realpow_increasing)
|
paulson@15077
|
256 |
apply (auto simp add: numeral_2_eq_2 [symmetric] real_sqrt_ge_zero simp
|
paulson@15077
|
257 |
del: realpow_Suc)
|
paulson@15077
|
258 |
done
|
paulson@15077
|
259 |
|
paulson@15077
|
260 |
|
paulson@15077
|
261 |
subsection{*Exponential Function*}
|
paulson@15077
|
262 |
|
paulson@15077
|
263 |
lemma summable_exp: "summable (%n. inverse (real (fact n)) * x ^ n)"
|
paulson@15077
|
264 |
apply (cut_tac 'a = real in zero_less_one [THEN dense], safe)
|
paulson@15077
|
265 |
apply (cut_tac x = r in reals_Archimedean3, auto)
|
paulson@15077
|
266 |
apply (drule_tac x = "\<bar>x\<bar>" in spec, safe)
|
paulson@15077
|
267 |
apply (rule_tac N = n and c = r in ratio_test)
|
paulson@15077
|
268 |
apply (auto simp add: abs_mult mult_assoc [symmetric] simp del: fact_Suc)
|
paulson@15077
|
269 |
apply (rule mult_right_mono)
|
paulson@15077
|
270 |
apply (rule_tac b1 = "\<bar>x\<bar>" in mult_commute [THEN ssubst])
|
paulson@15077
|
271 |
apply (subst fact_Suc)
|
paulson@15077
|
272 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
273 |
apply (auto simp add: abs_mult inverse_mult_distrib)
|
paulson@15077
|
274 |
apply (auto simp add: mult_assoc [symmetric] abs_eqI2 positive_imp_inverse_positive)
|
paulson@15077
|
275 |
apply (rule order_less_imp_le)
|
paulson@15077
|
276 |
apply (rule_tac z1 = "real (Suc na) " in real_mult_less_iff1 [THEN iffD1])
|
paulson@15077
|
277 |
apply (auto simp add: real_not_refl2 [THEN not_sym] mult_assoc abs_inverse)
|
paulson@15077
|
278 |
apply (erule order_less_trans)
|
paulson@15077
|
279 |
apply (auto simp add: mult_less_cancel_left mult_ac)
|
paulson@15077
|
280 |
done
|
paulson@15077
|
281 |
|
paulson@15077
|
282 |
|
paulson@15077
|
283 |
lemma summable_sin:
|
paulson@15077
|
284 |
"summable (%n.
|
paulson@15077
|
285 |
(if even n then 0
|
paulson@15077
|
286 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *
|
paulson@15077
|
287 |
x ^ n)"
|
paulson@15077
|
288 |
apply (unfold real_divide_def)
|
paulson@15077
|
289 |
apply (rule_tac g = " (%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n) " in summable_comparison_test)
|
paulson@15077
|
290 |
apply (rule_tac [2] summable_exp)
|
paulson@15077
|
291 |
apply (rule_tac x = 0 in exI)
|
paulson@15077
|
292 |
apply (auto simp add: power_abs [symmetric] abs_mult zero_le_mult_iff)
|
paulson@15077
|
293 |
done
|
paulson@15077
|
294 |
|
paulson@15077
|
295 |
lemma summable_cos:
|
paulson@15077
|
296 |
"summable (%n.
|
paulson@15077
|
297 |
(if even n then
|
paulson@15077
|
298 |
(- 1) ^ (n div 2)/(real (fact n)) else 0) * x ^ n)"
|
paulson@15077
|
299 |
apply (unfold real_divide_def)
|
paulson@15077
|
300 |
apply (rule_tac g = " (%n. inverse (real (fact n)) * \<bar>x\<bar> ^ n) " in summable_comparison_test)
|
paulson@15077
|
301 |
apply (rule_tac [2] summable_exp)
|
paulson@15077
|
302 |
apply (rule_tac x = 0 in exI)
|
paulson@15077
|
303 |
apply (auto simp add: power_abs [symmetric] abs_mult zero_le_mult_iff)
|
paulson@15077
|
304 |
done
|
paulson@15077
|
305 |
|
paulson@15077
|
306 |
lemma lemma_STAR_sin [simp]: "(if even n then 0
|
paulson@15077
|
307 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * 0 ^ n = 0"
|
paulson@15077
|
308 |
apply (induct_tac "n", auto)
|
paulson@15077
|
309 |
done
|
paulson@15077
|
310 |
|
paulson@15077
|
311 |
lemma lemma_STAR_cos [simp]: "0 < n -->
|
paulson@15077
|
312 |
(- 1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
|
paulson@15077
|
313 |
apply (induct_tac "n", auto)
|
paulson@15077
|
314 |
done
|
paulson@15077
|
315 |
|
paulson@15077
|
316 |
lemma lemma_STAR_cos1 [simp]: "0 < n -->
|
paulson@15077
|
317 |
(-1) ^ (n div 2)/(real (fact n)) * 0 ^ n = 0"
|
paulson@15077
|
318 |
apply (induct_tac "n", auto)
|
paulson@15077
|
319 |
done
|
paulson@15077
|
320 |
|
paulson@15077
|
321 |
lemma lemma_STAR_cos2 [simp]: "sumr 1 n (%n. if even n
|
paulson@15077
|
322 |
then (- 1) ^ (n div 2)/(real (fact n)) *
|
paulson@15077
|
323 |
0 ^ n
|
paulson@15077
|
324 |
else 0) = 0"
|
paulson@15077
|
325 |
apply (induct_tac "n")
|
paulson@15077
|
326 |
apply (case_tac [2] "n", auto)
|
paulson@15077
|
327 |
done
|
paulson@15077
|
328 |
|
paulson@15077
|
329 |
lemma exp_converges: "(%n. inverse (real (fact n)) * x ^ n) sums exp(x)"
|
paulson@15077
|
330 |
apply (unfold exp_def)
|
paulson@15077
|
331 |
apply (rule summable_exp [THEN summable_sums])
|
paulson@15077
|
332 |
done
|
paulson@15077
|
333 |
|
paulson@15077
|
334 |
lemma sin_converges:
|
paulson@15077
|
335 |
"(%n. (if even n then 0
|
paulson@15077
|
336 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *
|
paulson@15077
|
337 |
x ^ n) sums sin(x)"
|
paulson@15077
|
338 |
apply (unfold sin_def)
|
paulson@15077
|
339 |
apply (rule summable_sin [THEN summable_sums])
|
paulson@15077
|
340 |
done
|
paulson@15077
|
341 |
|
paulson@15077
|
342 |
lemma cos_converges:
|
paulson@15077
|
343 |
"(%n. (if even n then
|
paulson@15077
|
344 |
(- 1) ^ (n div 2)/(real (fact n))
|
paulson@15077
|
345 |
else 0) * x ^ n) sums cos(x)"
|
paulson@15077
|
346 |
apply (unfold cos_def)
|
paulson@15077
|
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"
|
paulson@15077
|
351 |
apply (induct_tac "n", auto)
|
paulson@15077
|
352 |
apply (subgoal_tac "p = Suc n")
|
paulson@15077
|
353 |
apply (simp (no_asm_simp), auto)
|
paulson@15077
|
354 |
apply (drule sym)
|
paulson@15077
|
355 |
apply (simp add: Suc_diff_le mult_commute realpow_Suc [symmetric]
|
paulson@15077
|
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:
|
paulson@15077
|
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)
|
paulson@15077
|
366 |
apply (rule sumr_subst)
|
paulson@15077
|
367 |
apply (intro strip)
|
paulson@15077
|
368 |
apply (subst lemma_realpow_diff)
|
paulson@15077
|
369 |
apply (auto simp add: mult_ac)
|
paulson@15077
|
370 |
done
|
paulson@15077
|
371 |
|
paulson@15077
|
372 |
lemma lemma_realpow_diff_sumr2: "x ^ (Suc n) - y ^ (Suc n) =
|
paulson@15077
|
373 |
(x - y) * sumr 0 (Suc n) (%p. (x ^ p) * (y ^(n - p)))"
|
paulson@15077
|
374 |
apply (induct_tac "n", simp)
|
paulson@15077
|
375 |
apply (auto simp del: sumr_Suc)
|
paulson@15077
|
376 |
apply (subst sumr_Suc)
|
paulson@15077
|
377 |
apply (drule sym)
|
paulson@15077
|
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@15081
|
396 |
==> summable (%n. \<bar>f(n)\<bar> * (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@15081
|
402 |
apply (rule_tac g = "%n. K * \<bar>z ^ n\<bar> * inverse (\<bar>x ^ n\<bar>)" in summable_comparison_test)
|
paulson@15077
|
403 |
apply (rule_tac x = 0 in exI, safe)
|
paulson@15081
|
404 |
apply (subgoal_tac "0 < \<bar>x ^ n\<bar> ")
|
paulson@15081
|
405 |
apply (rule_tac c="\<bar>x ^ n\<bar>" 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@15081
|
415 |
apply (subgoal_tac "\<bar>z ^ n\<bar> * 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@15081
|
505 |
lemma lemma_termdiff3: "[| h \<noteq> 0; \<bar>z\<bar> \<le> K; \<bar>z + h\<bar> \<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@15081
|
535 |
(\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>f h\<bar> \<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@15081
|
565 |
apply (subgoal_tac "\<forall>h. 0 < \<bar>h\<bar> & \<bar>h\<bar> < k --> \<bar>suminf (g h)\<bar> \<le> suminf f * \<bar>h\<bar>")
|
paulson@15077
|
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@15081
|
576 |
apply (rule_tac j = "suminf (%n. \<bar>g h n\<bar>)" 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@15081
|
594 |
f = "%n. \<bar>c n\<bar> * 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@15081
|
612 |
apply (subgoal_tac " (%n. real n * (real (Suc n) * (\<bar>c (Suc n)\<bar> * (r ^ (n - Suc 0))))) = (%n. diffs (%m. real (m - Suc 0) * \<bar>c m\<bar> * inverse r) n * (r ^ n))")
|
paulson@15077
|
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@15085
|
667 |
apply (rule sums_unique)
|
paulson@15079
|
668 |
apply (simp add: diff_def divide_inverse 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 (simp add: suminf_diff [OF sums_summable sums_summable] add_ac mult_ac)
|
paulson@15077
|
694 |
apply (rule_tac f = suminf in arg_cong)
|
paulson@15077
|
695 |
apply (rule ext)
|
paulson@15077
|
696 |
apply (simp add: diff_def right_distrib add_ac mult_ac)
|
paulson@15077
|
697 |
done
|
paulson@15077
|
698 |
|
paulson@15077
|
699 |
|
paulson@15077
|
700 |
subsection{*Formal Derivatives of Exp, Sin, and Cos Series*}
|
paulson@15077
|
701 |
|
paulson@15077
|
702 |
lemma exp_fdiffs:
|
paulson@15077
|
703 |
"diffs (%n. inverse(real (fact n))) = (%n. inverse(real (fact n)))"
|
paulson@15077
|
704 |
apply (unfold diffs_def)
|
paulson@15077
|
705 |
apply (rule ext)
|
paulson@15077
|
706 |
apply (subst fact_Suc)
|
paulson@15077
|
707 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
708 |
apply (subst inverse_mult_distrib)
|
paulson@15077
|
709 |
apply (auto simp add: mult_assoc [symmetric])
|
paulson@15077
|
710 |
done
|
paulson@15077
|
711 |
|
paulson@15077
|
712 |
lemma sin_fdiffs:
|
paulson@15077
|
713 |
"diffs(%n. if even n then 0
|
paulson@15077
|
714 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n)))
|
paulson@15077
|
715 |
= (%n. if even n then
|
paulson@15077
|
716 |
(- 1) ^ (n div 2)/(real (fact n))
|
paulson@15077
|
717 |
else 0)"
|
paulson@15077
|
718 |
apply (unfold diffs_def real_divide_def)
|
paulson@15077
|
719 |
apply (rule ext)
|
paulson@15077
|
720 |
apply (subst fact_Suc)
|
paulson@15077
|
721 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
722 |
apply (subst even_nat_Suc)
|
paulson@15077
|
723 |
apply (subst inverse_mult_distrib, auto)
|
paulson@15077
|
724 |
done
|
paulson@15077
|
725 |
|
paulson@15077
|
726 |
lemma sin_fdiffs2:
|
paulson@15077
|
727 |
"diffs(%n. if even n then 0
|
paulson@15077
|
728 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) n
|
paulson@15077
|
729 |
= (if even n then
|
paulson@15077
|
730 |
(- 1) ^ (n div 2)/(real (fact n))
|
paulson@15077
|
731 |
else 0)"
|
paulson@15077
|
732 |
apply (unfold diffs_def real_divide_def)
|
paulson@15077
|
733 |
apply (subst fact_Suc)
|
paulson@15077
|
734 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
735 |
apply (subst even_nat_Suc)
|
paulson@15077
|
736 |
apply (subst inverse_mult_distrib, auto)
|
paulson@15077
|
737 |
done
|
paulson@15077
|
738 |
|
paulson@15077
|
739 |
lemma cos_fdiffs:
|
paulson@15077
|
740 |
"diffs(%n. if even n then
|
paulson@15077
|
741 |
(- 1) ^ (n div 2)/(real (fact n)) else 0)
|
paulson@15077
|
742 |
= (%n. - (if even n then 0
|
paulson@15077
|
743 |
else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n))))"
|
paulson@15077
|
744 |
apply (unfold diffs_def real_divide_def)
|
paulson@15077
|
745 |
apply (rule ext)
|
paulson@15077
|
746 |
apply (subst fact_Suc)
|
paulson@15077
|
747 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
748 |
apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)
|
paulson@15077
|
749 |
done
|
paulson@15077
|
750 |
|
paulson@15077
|
751 |
|
paulson@15077
|
752 |
lemma cos_fdiffs2:
|
paulson@15077
|
753 |
"diffs(%n. if even n then
|
paulson@15077
|
754 |
(- 1) ^ (n div 2)/(real (fact n)) else 0) n
|
paulson@15077
|
755 |
= - (if even n then 0
|
paulson@15077
|
756 |
else (- 1) ^ ((n - Suc 0)div 2)/(real (fact n)))"
|
paulson@15077
|
757 |
apply (unfold diffs_def real_divide_def)
|
paulson@15077
|
758 |
apply (subst fact_Suc)
|
paulson@15077
|
759 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
760 |
apply (auto simp add: mult_assoc odd_Suc_mult_two_ex)
|
paulson@15077
|
761 |
done
|
paulson@15077
|
762 |
|
paulson@15077
|
763 |
text{*Now at last we can get the derivatives of exp, sin and cos*}
|
paulson@15077
|
764 |
|
paulson@15077
|
765 |
lemma lemma_sin_minus:
|
paulson@15077
|
766 |
"- sin x =
|
paulson@15077
|
767 |
suminf(%n. - ((if even n then 0
|
paulson@15077
|
768 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) * x ^ n))"
|
paulson@15077
|
769 |
by (auto intro!: sums_unique sums_minus sin_converges)
|
paulson@15077
|
770 |
|
paulson@15077
|
771 |
lemma lemma_exp_ext: "exp = (%x. suminf (%n. inverse (real (fact n)) * x ^ n))"
|
paulson@15077
|
772 |
by (auto intro!: ext simp add: exp_def)
|
paulson@15077
|
773 |
|
paulson@15077
|
774 |
lemma DERIV_exp [simp]: "DERIV exp x :> exp(x)"
|
paulson@15077
|
775 |
apply (unfold exp_def)
|
paulson@15077
|
776 |
apply (subst lemma_exp_ext)
|
paulson@15077
|
777 |
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
|
778 |
apply (rule_tac [2] K = "1 + \<bar>x\<bar> " in termdiffs)
|
paulson@15077
|
779 |
apply (auto intro: exp_converges [THEN sums_summable] simp add: exp_fdiffs, arith)
|
paulson@15077
|
780 |
done
|
paulson@15077
|
781 |
|
paulson@15077
|
782 |
lemma lemma_sin_ext:
|
paulson@15077
|
783 |
"sin = (%x. suminf(%n.
|
paulson@15077
|
784 |
(if even n then 0
|
paulson@15077
|
785 |
else (- 1) ^ ((n - Suc 0) div 2)/(real (fact n))) *
|
paulson@15077
|
786 |
x ^ n))"
|
paulson@15077
|
787 |
by (auto intro!: ext simp add: sin_def)
|
paulson@15077
|
788 |
|
paulson@15077
|
789 |
lemma lemma_cos_ext:
|
paulson@15077
|
790 |
"cos = (%x. suminf(%n.
|
paulson@15077
|
791 |
(if even n then (- 1) ^ (n div 2)/(real (fact n)) else 0) *
|
paulson@15077
|
792 |
x ^ n))"
|
paulson@15077
|
793 |
by (auto intro!: ext simp add: cos_def)
|
paulson@15077
|
794 |
|
paulson@15077
|
795 |
lemma DERIV_sin [simp]: "DERIV sin x :> cos(x)"
|
paulson@15077
|
796 |
apply (unfold cos_def)
|
paulson@15077
|
797 |
apply (subst lemma_sin_ext)
|
paulson@15077
|
798 |
apply (auto simp add: sin_fdiffs2 [symmetric])
|
paulson@15077
|
799 |
apply (rule_tac K = "1 + \<bar>x\<bar> " in termdiffs)
|
paulson@15077
|
800 |
apply (auto intro: sin_converges cos_converges sums_summable intro!: sums_minus [THEN sums_summable] simp add: cos_fdiffs sin_fdiffs, arith)
|
paulson@15077
|
801 |
done
|
paulson@15077
|
802 |
|
paulson@15077
|
803 |
lemma DERIV_cos [simp]: "DERIV cos x :> -sin(x)"
|
paulson@15077
|
804 |
apply (subst lemma_cos_ext)
|
paulson@15077
|
805 |
apply (auto simp add: lemma_sin_minus cos_fdiffs2 [symmetric] minus_mult_left)
|
paulson@15077
|
806 |
apply (rule_tac K = "1 + \<bar>x\<bar> " in termdiffs)
|
paulson@15077
|
807 |
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
|
808 |
done
|
paulson@15077
|
809 |
|
paulson@15077
|
810 |
|
paulson@15077
|
811 |
subsection{*Properties of the Exponential Function*}
|
paulson@15077
|
812 |
|
paulson@15077
|
813 |
lemma exp_zero [simp]: "exp 0 = 1"
|
paulson@15077
|
814 |
proof -
|
paulson@15077
|
815 |
have "(\<Sum>n = 0..<1. inverse (real (fact n)) * 0 ^ n) =
|
paulson@15077
|
816 |
suminf (\<lambda>n. inverse (real (fact n)) * 0 ^ n)"
|
paulson@15077
|
817 |
by (rule series_zero [rule_format, THEN sums_unique],
|
paulson@15077
|
818 |
case_tac "m", auto)
|
paulson@15077
|
819 |
thus ?thesis by (simp add: exp_def)
|
paulson@15077
|
820 |
qed
|
paulson@15077
|
821 |
|
paulson@15077
|
822 |
lemma exp_ge_add_one_self [simp]: "0 \<le> x ==> (1 + x) \<le> exp(x)"
|
paulson@15077
|
823 |
apply (drule real_le_imp_less_or_eq, auto)
|
paulson@15077
|
824 |
apply (unfold exp_def)
|
paulson@15077
|
825 |
apply (rule real_le_trans)
|
paulson@15077
|
826 |
apply (rule_tac [2] n = 2 and f = " (%n. inverse (real (fact n)) * x ^ n) " in series_pos_le)
|
paulson@15077
|
827 |
apply (auto intro: summable_exp simp add: numeral_2_eq_2 zero_le_power zero_le_mult_iff)
|
paulson@15077
|
828 |
done
|
paulson@15077
|
829 |
|
paulson@15077
|
830 |
lemma exp_gt_one [simp]: "0 < x ==> 1 < exp x"
|
paulson@15077
|
831 |
apply (rule order_less_le_trans)
|
paulson@15077
|
832 |
apply (rule_tac [2] exp_ge_add_one_self, auto)
|
paulson@15077
|
833 |
done
|
paulson@15077
|
834 |
|
paulson@15077
|
835 |
lemma DERIV_exp_add_const: "DERIV (%x. exp (x + y)) x :> exp(x + y)"
|
paulson@15077
|
836 |
proof -
|
paulson@15077
|
837 |
have "DERIV (exp \<circ> (\<lambda>x. x + y)) x :> exp (x + y) * (1+0)"
|
paulson@15077
|
838 |
by (fast intro: DERIV_chain DERIV_add DERIV_exp DERIV_Id DERIV_const)
|
paulson@15077
|
839 |
thus ?thesis by (simp add: o_def)
|
paulson@15077
|
840 |
qed
|
paulson@15077
|
841 |
|
paulson@15077
|
842 |
lemma DERIV_exp_minus [simp]: "DERIV (%x. exp (-x)) x :> - exp(-x)"
|
paulson@15077
|
843 |
proof -
|
paulson@15077
|
844 |
have "DERIV (exp \<circ> uminus) x :> exp (- x) * - 1"
|
paulson@15077
|
845 |
by (fast intro: DERIV_chain DERIV_minus DERIV_exp DERIV_Id)
|
paulson@15077
|
846 |
thus ?thesis by (simp add: o_def)
|
paulson@15077
|
847 |
qed
|
paulson@15077
|
848 |
|
paulson@15077
|
849 |
lemma DERIV_exp_exp_zero [simp]: "DERIV (%x. exp (x + y) * exp (- x)) x :> 0"
|
paulson@15077
|
850 |
proof -
|
paulson@15077
|
851 |
have "DERIV (\<lambda>x. exp (x + y) * exp (- x)) x
|
paulson@15077
|
852 |
:> exp (x + y) * exp (- x) + - exp (- x) * exp (x + y)"
|
paulson@15077
|
853 |
by (fast intro: DERIV_exp_add_const DERIV_exp_minus DERIV_mult)
|
paulson@15077
|
854 |
thus ?thesis by simp
|
paulson@15077
|
855 |
qed
|
paulson@15077
|
856 |
|
paulson@15077
|
857 |
lemma exp_add_mult_minus [simp]: "exp(x + y)*exp(-x) = exp(y)"
|
paulson@15077
|
858 |
proof -
|
paulson@15077
|
859 |
have "\<forall>x. DERIV (%x. exp (x + y) * exp (- x)) x :> 0" by simp
|
paulson@15077
|
860 |
hence "exp (x + y) * exp (- x) = exp (0 + y) * exp (- 0)"
|
paulson@15077
|
861 |
by (rule DERIV_isconst_all)
|
paulson@15077
|
862 |
thus ?thesis by simp
|
paulson@15077
|
863 |
qed
|
paulson@15077
|
864 |
|
paulson@15077
|
865 |
lemma exp_mult_minus [simp]: "exp x * exp(-x) = 1"
|
paulson@15077
|
866 |
proof -
|
paulson@15077
|
867 |
have "exp (x + 0) * exp (- x) = exp 0" by (rule exp_add_mult_minus)
|
paulson@15077
|
868 |
thus ?thesis by simp
|
paulson@15077
|
869 |
qed
|
paulson@15077
|
870 |
|
paulson@15077
|
871 |
lemma exp_mult_minus2 [simp]: "exp(-x)*exp(x) = 1"
|
paulson@15077
|
872 |
by (simp add: mult_commute)
|
paulson@15077
|
873 |
|
paulson@15077
|
874 |
|
paulson@15077
|
875 |
lemma exp_minus: "exp(-x) = inverse(exp(x))"
|
paulson@15077
|
876 |
by (auto intro: inverse_unique [symmetric])
|
paulson@15077
|
877 |
|
paulson@15077
|
878 |
lemma exp_add: "exp(x + y) = exp(x) * exp(y)"
|
paulson@15077
|
879 |
proof -
|
paulson@15077
|
880 |
have "exp x * exp y = exp x * (exp (x + y) * exp (- x))" by simp
|
paulson@15077
|
881 |
thus ?thesis by (simp (no_asm_simp) add: mult_ac)
|
paulson@15077
|
882 |
qed
|
paulson@15077
|
883 |
|
paulson@15077
|
884 |
text{*Proof: because every exponential can be seen as a square.*}
|
paulson@15077
|
885 |
lemma exp_ge_zero [simp]: "0 \<le> exp x"
|
paulson@15077
|
886 |
apply (rule_tac t = x in real_sum_of_halves [THEN subst])
|
paulson@15077
|
887 |
apply (subst exp_add, auto)
|
paulson@15077
|
888 |
done
|
paulson@15077
|
889 |
|
paulson@15077
|
890 |
lemma exp_not_eq_zero [simp]: "exp x \<noteq> 0"
|
paulson@15077
|
891 |
apply (cut_tac x = x in exp_mult_minus2)
|
paulson@15077
|
892 |
apply (auto simp del: exp_mult_minus2)
|
paulson@15077
|
893 |
done
|
paulson@15077
|
894 |
|
paulson@15077
|
895 |
lemma exp_gt_zero [simp]: "0 < exp x"
|
paulson@15077
|
896 |
by (simp add: order_less_le)
|
paulson@15077
|
897 |
|
paulson@15077
|
898 |
lemma inv_exp_gt_zero [simp]: "0 < inverse(exp x)"
|
paulson@15077
|
899 |
by (auto intro: positive_imp_inverse_positive)
|
paulson@15077
|
900 |
|
paulson@15081
|
901 |
lemma abs_exp_cancel [simp]: "\<bar>exp x\<bar> = exp x"
|
paulson@15077
|
902 |
by (auto simp add: abs_eqI2)
|
paulson@15077
|
903 |
|
paulson@15077
|
904 |
lemma exp_real_of_nat_mult: "exp(real n * x) = exp(x) ^ n"
|
paulson@15077
|
905 |
apply (induct_tac "n")
|
paulson@15077
|
906 |
apply (auto simp add: real_of_nat_Suc right_distrib exp_add mult_commute)
|
paulson@15077
|
907 |
done
|
paulson@15077
|
908 |
|
paulson@15077
|
909 |
lemma exp_diff: "exp(x - y) = exp(x)/(exp y)"
|
paulson@15077
|
910 |
apply (unfold real_diff_def real_divide_def)
|
paulson@15077
|
911 |
apply (simp (no_asm) add: exp_add exp_minus)
|
paulson@15077
|
912 |
done
|
paulson@15077
|
913 |
|
paulson@15077
|
914 |
|
paulson@15077
|
915 |
lemma exp_less_mono:
|
paulson@15077
|
916 |
assumes xy: "x < y" shows "exp x < exp y"
|
paulson@15077
|
917 |
proof -
|
paulson@15077
|
918 |
have "1 < exp (y + - x)"
|
paulson@15077
|
919 |
by (rule real_less_sum_gt_zero [THEN exp_gt_one])
|
paulson@15077
|
920 |
hence "exp x * inverse (exp x) < exp y * inverse (exp x)"
|
paulson@15077
|
921 |
by (auto simp add: exp_add exp_minus)
|
paulson@15077
|
922 |
thus ?thesis
|
paulson@15077
|
923 |
by (simp add: divide_inverse [symmetric] pos_less_divide_eq)
|
paulson@15077
|
924 |
qed
|
paulson@15077
|
925 |
|
paulson@15077
|
926 |
lemma exp_less_cancel: "exp x < exp y ==> x < y"
|
paulson@15077
|
927 |
apply (rule ccontr)
|
paulson@15077
|
928 |
apply (simp add: linorder_not_less order_le_less)
|
paulson@15077
|
929 |
apply (auto dest: exp_less_mono)
|
paulson@15077
|
930 |
done
|
paulson@15077
|
931 |
|
paulson@15077
|
932 |
lemma exp_less_cancel_iff [iff]: "(exp(x) < exp(y)) = (x < y)"
|
paulson@15077
|
933 |
by (auto intro: exp_less_mono exp_less_cancel)
|
paulson@15077
|
934 |
|
paulson@15077
|
935 |
lemma exp_le_cancel_iff [iff]: "(exp(x) \<le> exp(y)) = (x \<le> y)"
|
paulson@15077
|
936 |
by (auto simp add: linorder_not_less [symmetric])
|
paulson@15077
|
937 |
|
paulson@15077
|
938 |
lemma exp_inj_iff [iff]: "(exp x = exp y) = (x = y)"
|
paulson@15077
|
939 |
by (simp add: order_eq_iff)
|
paulson@15077
|
940 |
|
paulson@15077
|
941 |
lemma lemma_exp_total: "1 \<le> y ==> \<exists>x. 0 \<le> x & x \<le> y - 1 & exp(x) = y"
|
paulson@15077
|
942 |
apply (rule IVT)
|
paulson@15077
|
943 |
apply (auto intro: DERIV_exp [THEN DERIV_isCont] simp add: le_diff_eq)
|
paulson@15077
|
944 |
apply (subgoal_tac "1 + (y - 1) \<le> exp (y - 1)")
|
paulson@15077
|
945 |
apply simp
|
paulson@15077
|
946 |
apply (rule exp_ge_add_one_self, simp)
|
paulson@15077
|
947 |
done
|
paulson@15077
|
948 |
|
paulson@15077
|
949 |
lemma exp_total: "0 < y ==> \<exists>x. exp x = y"
|
paulson@15077
|
950 |
apply (rule_tac x = 1 and y = y in linorder_cases)
|
paulson@15077
|
951 |
apply (drule order_less_imp_le [THEN lemma_exp_total])
|
paulson@15077
|
952 |
apply (rule_tac [2] x = 0 in exI)
|
paulson@15077
|
953 |
apply (frule_tac [3] real_inverse_gt_one)
|
paulson@15077
|
954 |
apply (drule_tac [4] order_less_imp_le [THEN lemma_exp_total], auto)
|
paulson@15077
|
955 |
apply (rule_tac x = "-x" in exI)
|
paulson@15077
|
956 |
apply (simp add: exp_minus)
|
paulson@15077
|
957 |
done
|
paulson@15077
|
958 |
|
paulson@15077
|
959 |
|
paulson@15077
|
960 |
subsection{*Properties of the Logarithmic Function*}
|
paulson@15077
|
961 |
|
paulson@15077
|
962 |
lemma ln_exp[simp]: "ln(exp x) = x"
|
paulson@15077
|
963 |
by (simp add: ln_def)
|
paulson@15077
|
964 |
|
paulson@15077
|
965 |
lemma exp_ln_iff[simp]: "(exp(ln x) = x) = (0 < x)"
|
paulson@15077
|
966 |
apply (auto dest: exp_total)
|
paulson@15077
|
967 |
apply (erule subst, simp)
|
paulson@15077
|
968 |
done
|
paulson@15077
|
969 |
|
paulson@15077
|
970 |
lemma ln_mult: "[| 0 < x; 0 < y |] ==> ln(x * y) = ln(x) + ln(y)"
|
paulson@15077
|
971 |
apply (rule exp_inj_iff [THEN iffD1])
|
paulson@15077
|
972 |
apply (frule real_mult_order)
|
paulson@15077
|
973 |
apply (auto simp add: exp_add exp_ln_iff [symmetric] simp del: exp_inj_iff exp_ln_iff)
|
paulson@15077
|
974 |
done
|
paulson@15077
|
975 |
|
paulson@15077
|
976 |
lemma ln_inj_iff[simp]: "[| 0 < x; 0 < y |] ==> (ln x = ln y) = (x = y)"
|
paulson@15077
|
977 |
apply (simp only: exp_ln_iff [symmetric])
|
paulson@15077
|
978 |
apply (erule subst)+
|
paulson@15077
|
979 |
apply simp
|
paulson@15077
|
980 |
done
|
paulson@15077
|
981 |
|
paulson@15077
|
982 |
lemma ln_one[simp]: "ln 1 = 0"
|
paulson@15077
|
983 |
by (rule exp_inj_iff [THEN iffD1], auto)
|
paulson@15077
|
984 |
|
paulson@15077
|
985 |
lemma ln_inverse: "0 < x ==> ln(inverse x) = - ln x"
|
paulson@15077
|
986 |
apply (rule_tac a1 = "ln x" in add_left_cancel [THEN iffD1])
|
paulson@15077
|
987 |
apply (auto simp add: positive_imp_inverse_positive ln_mult [symmetric])
|
paulson@15077
|
988 |
done
|
paulson@15077
|
989 |
|
paulson@15077
|
990 |
lemma ln_div:
|
paulson@15077
|
991 |
"[|0 < x; 0 < y|] ==> ln(x/y) = ln x - ln y"
|
paulson@15077
|
992 |
apply (unfold real_divide_def)
|
paulson@15077
|
993 |
apply (auto simp add: positive_imp_inverse_positive ln_mult ln_inverse)
|
paulson@15077
|
994 |
done
|
paulson@15077
|
995 |
|
paulson@15077
|
996 |
lemma ln_less_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x < ln y) = (x < y)"
|
paulson@15077
|
997 |
apply (simp only: exp_ln_iff [symmetric])
|
paulson@15077
|
998 |
apply (erule subst)+
|
paulson@15077
|
999 |
apply simp
|
paulson@15077
|
1000 |
done
|
paulson@15077
|
1001 |
|
paulson@15077
|
1002 |
lemma ln_le_cancel_iff[simp]: "[| 0 < x; 0 < y|] ==> (ln x \<le> ln y) = (x \<le> y)"
|
paulson@15077
|
1003 |
by (auto simp add: linorder_not_less [symmetric])
|
paulson@15077
|
1004 |
|
paulson@15077
|
1005 |
lemma ln_realpow: "0 < x ==> ln(x ^ n) = real n * ln(x)"
|
paulson@15077
|
1006 |
by (auto dest!: exp_total simp add: exp_real_of_nat_mult [symmetric])
|
paulson@15077
|
1007 |
|
paulson@15077
|
1008 |
lemma ln_add_one_self_le_self [simp]: "0 \<le> x ==> ln(1 + x) \<le> x"
|
paulson@15077
|
1009 |
apply (rule ln_exp [THEN subst])
|
paulson@15077
|
1010 |
apply (rule ln_le_cancel_iff [THEN iffD2], auto)
|
paulson@15077
|
1011 |
done
|
paulson@15077
|
1012 |
|
paulson@15077
|
1013 |
lemma ln_less_self [simp]: "0 < x ==> ln x < x"
|
paulson@15077
|
1014 |
apply (rule order_less_le_trans)
|
paulson@15077
|
1015 |
apply (rule_tac [2] ln_add_one_self_le_self)
|
paulson@15077
|
1016 |
apply (rule ln_less_cancel_iff [THEN iffD2], auto)
|
paulson@15077
|
1017 |
done
|
paulson@15077
|
1018 |
|
paulson@15077
|
1019 |
lemma ln_ge_zero:
|
paulson@15077
|
1020 |
assumes x: "1 \<le> x" shows "0 \<le> ln x"
|
paulson@15077
|
1021 |
proof -
|
paulson@15077
|
1022 |
have "0 < x" using x by arith
|
paulson@15077
|
1023 |
hence "exp 0 \<le> exp (ln x)"
|
paulson@15077
|
1024 |
by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
|
paulson@15077
|
1025 |
thus ?thesis by (simp only: exp_le_cancel_iff)
|
paulson@15077
|
1026 |
qed
|
paulson@15077
|
1027 |
|
paulson@15077
|
1028 |
lemma ln_ge_zero_imp_ge_one:
|
paulson@15077
|
1029 |
assumes ln: "0 \<le> ln x"
|
paulson@15077
|
1030 |
and x: "0 < x"
|
paulson@15077
|
1031 |
shows "1 \<le> x"
|
paulson@15077
|
1032 |
proof -
|
paulson@15077
|
1033 |
from ln have "ln 1 \<le> ln x" by simp
|
paulson@15077
|
1034 |
thus ?thesis by (simp add: x del: ln_one)
|
paulson@15077
|
1035 |
qed
|
paulson@15077
|
1036 |
|
paulson@15077
|
1037 |
lemma ln_ge_zero_iff [simp]: "0 < x ==> (0 \<le> ln x) = (1 \<le> x)"
|
paulson@15077
|
1038 |
by (blast intro: ln_ge_zero ln_ge_zero_imp_ge_one)
|
paulson@15077
|
1039 |
|
paulson@15077
|
1040 |
lemma ln_gt_zero:
|
paulson@15077
|
1041 |
assumes x: "1 < x" shows "0 < ln x"
|
paulson@15077
|
1042 |
proof -
|
paulson@15077
|
1043 |
have "0 < x" using x by arith
|
paulson@15077
|
1044 |
hence "exp 0 < exp (ln x)"
|
paulson@15077
|
1045 |
by (simp add: x exp_ln_iff [symmetric] del: exp_ln_iff)
|
paulson@15077
|
1046 |
thus ?thesis by (simp only: exp_less_cancel_iff)
|
paulson@15077
|
1047 |
qed
|
paulson@15077
|
1048 |
|
paulson@15077
|
1049 |
lemma ln_gt_zero_imp_gt_one:
|
paulson@15077
|
1050 |
assumes ln: "0 < ln x"
|
paulson@15077
|
1051 |
and x: "0 < x"
|
paulson@15077
|
1052 |
shows "1 < x"
|
paulson@15077
|
1053 |
proof -
|
paulson@15077
|
1054 |
from ln have "ln 1 < ln x" by simp
|
paulson@15077
|
1055 |
thus ?thesis by (simp add: x del: ln_one)
|
paulson@15077
|
1056 |
qed
|
paulson@15077
|
1057 |
|
paulson@15077
|
1058 |
lemma ln_gt_zero_iff [simp]: "0 < x ==> (0 < ln x) = (1 < x)"
|
paulson@15077
|
1059 |
by (blast intro: ln_gt_zero ln_gt_zero_imp_gt_one)
|
paulson@15077
|
1060 |
|
paulson@15077
|
1061 |
lemma ln_not_eq_zero [simp]: "[| 0 < x; x \<noteq> 1 |] ==> ln x \<noteq> 0"
|
paulson@15077
|
1062 |
apply safe
|
paulson@15077
|
1063 |
apply (drule exp_inj_iff [THEN iffD2])
|
paulson@15077
|
1064 |
apply (drule exp_ln_iff [THEN iffD2], auto)
|
paulson@15077
|
1065 |
done
|
paulson@15077
|
1066 |
|
paulson@15077
|
1067 |
lemma ln_less_zero: "[| 0 < x; x < 1 |] ==> ln x < 0"
|
paulson@15077
|
1068 |
apply (rule exp_less_cancel_iff [THEN iffD1])
|
paulson@15077
|
1069 |
apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
|
paulson@15077
|
1070 |
done
|
paulson@15077
|
1071 |
|
paulson@15077
|
1072 |
lemma exp_ln_eq: "exp u = x ==> ln x = u"
|
paulson@15077
|
1073 |
by auto
|
paulson@15077
|
1074 |
|
paulson@15077
|
1075 |
|
paulson@15077
|
1076 |
subsection{*Basic Properties of the Trigonometric Functions*}
|
paulson@15077
|
1077 |
|
paulson@15077
|
1078 |
lemma sin_zero [simp]: "sin 0 = 0"
|
paulson@15077
|
1079 |
by (auto intro!: sums_unique [symmetric] LIMSEQ_const
|
paulson@15077
|
1080 |
simp add: sin_def sums_def simp del: power_0_left)
|
paulson@15077
|
1081 |
|
paulson@15077
|
1082 |
lemma lemma_series_zero2: "(\<forall>m. n \<le> m --> f m = 0) --> f sums sumr 0 n f"
|
paulson@15077
|
1083 |
by (auto intro: series_zero)
|
paulson@15077
|
1084 |
|
paulson@15077
|
1085 |
lemma cos_zero [simp]: "cos 0 = 1"
|
paulson@15077
|
1086 |
apply (unfold cos_def)
|
paulson@15077
|
1087 |
apply (rule sums_unique [symmetric])
|
paulson@15077
|
1088 |
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
|
1089 |
apply auto
|
paulson@15077
|
1090 |
done
|
paulson@15077
|
1091 |
|
paulson@15077
|
1092 |
lemma DERIV_sin_sin_mult [simp]:
|
paulson@15077
|
1093 |
"DERIV (%x. sin(x)*sin(x)) x :> cos(x) * sin(x) + cos(x) * sin(x)"
|
paulson@15077
|
1094 |
by (rule DERIV_mult, auto)
|
paulson@15077
|
1095 |
|
paulson@15077
|
1096 |
lemma DERIV_sin_sin_mult2 [simp]:
|
paulson@15077
|
1097 |
"DERIV (%x. sin(x)*sin(x)) x :> 2 * cos(x) * sin(x)"
|
paulson@15077
|
1098 |
apply (cut_tac x = x in DERIV_sin_sin_mult)
|
paulson@15077
|
1099 |
apply (auto simp add: mult_assoc)
|
paulson@15077
|
1100 |
done
|
paulson@15077
|
1101 |
|
paulson@15077
|
1102 |
lemma DERIV_sin_realpow2 [simp]:
|
paulson@15077
|
1103 |
"DERIV (%x. (sin x)\<twosuperior>) x :> cos(x) * sin(x) + cos(x) * sin(x)"
|
paulson@15077
|
1104 |
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
|
paulson@15077
|
1105 |
|
paulson@15077
|
1106 |
lemma DERIV_sin_realpow2a [simp]:
|
paulson@15077
|
1107 |
"DERIV (%x. (sin x)\<twosuperior>) x :> 2 * cos(x) * sin(x)"
|
paulson@15077
|
1108 |
by (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1109 |
|
paulson@15077
|
1110 |
lemma DERIV_cos_cos_mult [simp]:
|
paulson@15077
|
1111 |
"DERIV (%x. cos(x)*cos(x)) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
|
paulson@15077
|
1112 |
by (rule DERIV_mult, auto)
|
paulson@15077
|
1113 |
|
paulson@15077
|
1114 |
lemma DERIV_cos_cos_mult2 [simp]:
|
paulson@15077
|
1115 |
"DERIV (%x. cos(x)*cos(x)) x :> -2 * cos(x) * sin(x)"
|
paulson@15077
|
1116 |
apply (cut_tac x = x in DERIV_cos_cos_mult)
|
paulson@15077
|
1117 |
apply (auto simp add: mult_ac)
|
paulson@15077
|
1118 |
done
|
paulson@15077
|
1119 |
|
paulson@15077
|
1120 |
lemma DERIV_cos_realpow2 [simp]:
|
paulson@15077
|
1121 |
"DERIV (%x. (cos x)\<twosuperior>) x :> -sin(x) * cos(x) + -sin(x) * cos(x)"
|
paulson@15077
|
1122 |
by (auto simp add: numeral_2_eq_2 real_mult_assoc [symmetric])
|
paulson@15077
|
1123 |
|
paulson@15077
|
1124 |
lemma DERIV_cos_realpow2a [simp]:
|
paulson@15077
|
1125 |
"DERIV (%x. (cos x)\<twosuperior>) x :> -2 * cos(x) * sin(x)"
|
paulson@15077
|
1126 |
by (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1127 |
|
paulson@15077
|
1128 |
lemma lemma_DERIV_subst: "[| DERIV f x :> D; D = E |] ==> DERIV f x :> E"
|
paulson@15077
|
1129 |
by auto
|
paulson@15077
|
1130 |
|
paulson@15077
|
1131 |
lemma DERIV_cos_realpow2b: "DERIV (%x. (cos x)\<twosuperior>) x :> -(2 * cos(x) * sin(x))"
|
paulson@15077
|
1132 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1133 |
apply (rule DERIV_cos_realpow2a, auto)
|
paulson@15077
|
1134 |
done
|
paulson@15077
|
1135 |
|
paulson@15077
|
1136 |
(* most useful *)
|
paulson@15077
|
1137 |
lemma DERIV_cos_cos_mult3 [simp]: "DERIV (%x. cos(x)*cos(x)) x :> -(2 * cos(x) * sin(x))"
|
paulson@15077
|
1138 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1139 |
apply (rule DERIV_cos_cos_mult2, auto)
|
paulson@15077
|
1140 |
done
|
paulson@15077
|
1141 |
|
paulson@15077
|
1142 |
lemma DERIV_sin_circle_all:
|
paulson@15077
|
1143 |
"\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :>
|
paulson@15077
|
1144 |
(2*cos(x)*sin(x) - 2*cos(x)*sin(x))"
|
paulson@15077
|
1145 |
apply (unfold real_diff_def, safe)
|
paulson@15077
|
1146 |
apply (rule DERIV_add)
|
paulson@15077
|
1147 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1148 |
done
|
paulson@15077
|
1149 |
|
paulson@15077
|
1150 |
lemma DERIV_sin_circle_all_zero [simp]: "\<forall>x. DERIV (%x. (sin x)\<twosuperior> + (cos x)\<twosuperior>) x :> 0"
|
paulson@15077
|
1151 |
by (cut_tac DERIV_sin_circle_all, auto)
|
paulson@15077
|
1152 |
|
paulson@15077
|
1153 |
lemma sin_cos_squared_add [simp]: "((sin x)\<twosuperior>) + ((cos x)\<twosuperior>) = 1"
|
paulson@15077
|
1154 |
apply (cut_tac x = x and y = 0 in DERIV_sin_circle_all_zero [THEN DERIV_isconst_all])
|
paulson@15077
|
1155 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1156 |
done
|
paulson@15077
|
1157 |
|
paulson@15077
|
1158 |
lemma sin_cos_squared_add2 [simp]: "((cos x)\<twosuperior>) + ((sin x)\<twosuperior>) = 1"
|
paulson@15077
|
1159 |
apply (subst real_add_commute)
|
paulson@15077
|
1160 |
apply (simp (no_asm) del: realpow_Suc)
|
paulson@15077
|
1161 |
done
|
paulson@15077
|
1162 |
|
paulson@15077
|
1163 |
lemma sin_cos_squared_add3 [simp]: "cos x * cos x + sin x * sin x = 1"
|
paulson@15077
|
1164 |
apply (cut_tac x = x in sin_cos_squared_add2)
|
paulson@15077
|
1165 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1166 |
done
|
paulson@15077
|
1167 |
|
paulson@15077
|
1168 |
lemma sin_squared_eq: "(sin x)\<twosuperior> = 1 - (cos x)\<twosuperior>"
|
paulson@15077
|
1169 |
apply (rule_tac a1 = "(cos x)\<twosuperior> " in add_right_cancel [THEN iffD1])
|
paulson@15077
|
1170 |
apply (simp del: realpow_Suc)
|
paulson@15077
|
1171 |
done
|
paulson@15077
|
1172 |
|
paulson@15077
|
1173 |
lemma cos_squared_eq: "(cos x)\<twosuperior> = 1 - (sin x)\<twosuperior>"
|
paulson@15077
|
1174 |
apply (rule_tac a1 = "(sin x)\<twosuperior>" in add_right_cancel [THEN iffD1])
|
paulson@15077
|
1175 |
apply (simp del: realpow_Suc)
|
paulson@15077
|
1176 |
done
|
paulson@15077
|
1177 |
|
paulson@15077
|
1178 |
lemma real_gt_one_ge_zero_add_less: "[| 1 < x; 0 \<le> y |] ==> 1 < x + (y::real)"
|
paulson@15077
|
1179 |
by arith
|
paulson@15077
|
1180 |
|
paulson@15081
|
1181 |
lemma abs_sin_le_one [simp]: "\<bar>sin x\<bar> \<le> 1"
|
paulson@15077
|
1182 |
apply (auto simp add: linorder_not_less [symmetric])
|
paulson@15077
|
1183 |
apply (drule_tac n = "Suc 0" in power_gt1)
|
paulson@15077
|
1184 |
apply (auto simp del: realpow_Suc)
|
paulson@15077
|
1185 |
apply (drule_tac r1 = "cos x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
|
paulson@15077
|
1186 |
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
|
paulson@15077
|
1187 |
done
|
paulson@15077
|
1188 |
|
paulson@15077
|
1189 |
lemma sin_ge_minus_one [simp]: "-1 \<le> sin x"
|
paulson@15077
|
1190 |
apply (insert abs_sin_le_one [of x])
|
paulson@15077
|
1191 |
apply (simp add: abs_le_interval_iff del: abs_sin_le_one)
|
paulson@15077
|
1192 |
done
|
paulson@15077
|
1193 |
|
paulson@15077
|
1194 |
lemma sin_le_one [simp]: "sin x \<le> 1"
|
paulson@15077
|
1195 |
apply (insert abs_sin_le_one [of x])
|
paulson@15077
|
1196 |
apply (simp add: abs_le_interval_iff del: abs_sin_le_one)
|
paulson@15077
|
1197 |
done
|
paulson@15077
|
1198 |
|
paulson@15081
|
1199 |
lemma abs_cos_le_one [simp]: "\<bar>cos x\<bar> \<le> 1"
|
paulson@15077
|
1200 |
apply (auto simp add: linorder_not_less [symmetric])
|
paulson@15077
|
1201 |
apply (drule_tac n = "Suc 0" in power_gt1)
|
paulson@15077
|
1202 |
apply (auto simp del: realpow_Suc)
|
paulson@15077
|
1203 |
apply (drule_tac r1 = "sin x" in realpow_two_le [THEN [2] real_gt_one_ge_zero_add_less])
|
paulson@15077
|
1204 |
apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
|
paulson@15077
|
1205 |
done
|
paulson@15077
|
1206 |
|
paulson@15077
|
1207 |
lemma cos_ge_minus_one [simp]: "-1 \<le> cos x"
|
paulson@15077
|
1208 |
apply (insert abs_cos_le_one [of x])
|
paulson@15077
|
1209 |
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
|
paulson@15077
|
1210 |
done
|
paulson@15077
|
1211 |
|
paulson@15077
|
1212 |
lemma cos_le_one [simp]: "cos x \<le> 1"
|
paulson@15077
|
1213 |
apply (insert abs_cos_le_one [of x])
|
paulson@15077
|
1214 |
apply (simp add: abs_le_interval_iff del: abs_cos_le_one)
|
paulson@15077
|
1215 |
done
|
paulson@15077
|
1216 |
|
paulson@15077
|
1217 |
lemma DERIV_fun_pow: "DERIV g x :> m ==>
|
paulson@15077
|
1218 |
DERIV (%x. (g x) ^ n) x :> real n * (g x) ^ (n - 1) * m"
|
paulson@15077
|
1219 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1220 |
apply (rule_tac f = " (%x. x ^ n) " in DERIV_chain2)
|
paulson@15077
|
1221 |
apply (rule DERIV_pow, auto)
|
paulson@15077
|
1222 |
done
|
paulson@15077
|
1223 |
|
paulson@15077
|
1224 |
lemma DERIV_fun_exp: "DERIV g x :> m ==> DERIV (%x. exp(g x)) x :> exp(g x) * m"
|
paulson@15077
|
1225 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1226 |
apply (rule_tac f = exp in DERIV_chain2)
|
paulson@15077
|
1227 |
apply (rule DERIV_exp, auto)
|
paulson@15077
|
1228 |
done
|
paulson@15077
|
1229 |
|
paulson@15077
|
1230 |
lemma DERIV_fun_sin: "DERIV g x :> m ==> DERIV (%x. sin(g x)) x :> cos(g x) * m"
|
paulson@15077
|
1231 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1232 |
apply (rule_tac f = sin in DERIV_chain2)
|
paulson@15077
|
1233 |
apply (rule DERIV_sin, auto)
|
paulson@15077
|
1234 |
done
|
paulson@15077
|
1235 |
|
paulson@15077
|
1236 |
lemma DERIV_fun_cos: "DERIV g x :> m ==> DERIV (%x. cos(g x)) x :> -sin(g x) * m"
|
paulson@15077
|
1237 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1238 |
apply (rule_tac f = cos in DERIV_chain2)
|
paulson@15077
|
1239 |
apply (rule DERIV_cos, auto)
|
paulson@15077
|
1240 |
done
|
paulson@15077
|
1241 |
|
paulson@15077
|
1242 |
lemmas DERIV_intros = DERIV_Id DERIV_const DERIV_cos DERIV_cmult
|
paulson@15077
|
1243 |
DERIV_sin DERIV_exp DERIV_inverse DERIV_pow
|
paulson@15077
|
1244 |
DERIV_add DERIV_diff DERIV_mult DERIV_minus
|
paulson@15077
|
1245 |
DERIV_inverse_fun DERIV_quotient DERIV_fun_pow
|
paulson@15077
|
1246 |
DERIV_fun_exp DERIV_fun_sin DERIV_fun_cos
|
paulson@15077
|
1247 |
DERIV_Id DERIV_const DERIV_cos
|
paulson@15077
|
1248 |
|
paulson@15077
|
1249 |
(* lemma *)
|
paulson@15077
|
1250 |
lemma lemma_DERIV_sin_cos_add: "\<forall>x.
|
paulson@15077
|
1251 |
DERIV (%x. (sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
|
paulson@15077
|
1252 |
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2) x :> 0"
|
paulson@15077
|
1253 |
apply (safe, rule lemma_DERIV_subst)
|
paulson@15077
|
1254 |
apply (best intro!: DERIV_intros intro: DERIV_chain2)
|
paulson@15077
|
1255 |
--{*replaces the old @{text DERIV_tac}*}
|
paulson@15077
|
1256 |
apply (auto simp add: real_diff_def left_distrib right_distrib mult_ac add_ac)
|
paulson@15077
|
1257 |
done
|
paulson@15077
|
1258 |
|
paulson@15077
|
1259 |
lemma sin_cos_add [simp]:
|
paulson@15077
|
1260 |
"(sin (x + y) - (sin x * cos y + cos x * sin y)) ^ 2 +
|
paulson@15077
|
1261 |
(cos (x + y) - (cos x * cos y - sin x * sin y)) ^ 2 = 0"
|
paulson@15077
|
1262 |
apply (cut_tac y = 0 and x = x and y7 = y
|
paulson@15077
|
1263 |
in lemma_DERIV_sin_cos_add [THEN DERIV_isconst_all])
|
paulson@15077
|
1264 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1265 |
done
|
paulson@15077
|
1266 |
|
paulson@15077
|
1267 |
lemma sin_add: "sin (x + y) = sin x * cos y + cos x * sin y"
|
paulson@15077
|
1268 |
apply (cut_tac x = x and y = y in sin_cos_add)
|
paulson@15077
|
1269 |
apply (auto dest!: real_sum_squares_cancel_a
|
paulson@15085
|
1270 |
simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
|
paulson@15077
|
1271 |
done
|
paulson@15077
|
1272 |
|
paulson@15077
|
1273 |
lemma cos_add: "cos (x + y) = cos x * cos y - sin x * sin y"
|
paulson@15077
|
1274 |
apply (cut_tac x = x and y = y in sin_cos_add)
|
paulson@15077
|
1275 |
apply (auto dest!: real_sum_squares_cancel_a
|
paulson@15085
|
1276 |
simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_add)
|
paulson@15077
|
1277 |
done
|
paulson@15077
|
1278 |
|
paulson@15085
|
1279 |
lemma lemma_DERIV_sin_cos_minus:
|
paulson@15085
|
1280 |
"\<forall>x. DERIV (%x. (sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2) x :> 0"
|
paulson@15077
|
1281 |
apply (safe, rule lemma_DERIV_subst)
|
paulson@15077
|
1282 |
apply (best intro!: DERIV_intros intro: DERIV_chain2)
|
paulson@15077
|
1283 |
apply (auto simp add: real_diff_def left_distrib right_distrib mult_ac add_ac)
|
paulson@15077
|
1284 |
done
|
paulson@15077
|
1285 |
|
paulson@15085
|
1286 |
lemma sin_cos_minus [simp]:
|
paulson@15085
|
1287 |
"(sin(-x) + (sin x)) ^ 2 + (cos(-x) - (cos x)) ^ 2 = 0"
|
paulson@15085
|
1288 |
apply (cut_tac y = 0 and x = x
|
paulson@15085
|
1289 |
in lemma_DERIV_sin_cos_minus [THEN DERIV_isconst_all])
|
paulson@15077
|
1290 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1291 |
done
|
paulson@15077
|
1292 |
|
paulson@15077
|
1293 |
lemma sin_minus [simp]: "sin (-x) = -sin(x)"
|
paulson@15077
|
1294 |
apply (cut_tac x = x in sin_cos_minus)
|
paulson@15085
|
1295 |
apply (auto dest!: real_sum_squares_cancel_a
|
paulson@15085
|
1296 |
simp add: numeral_2_eq_2 real_add_eq_0_iff simp del: sin_cos_minus)
|
paulson@15077
|
1297 |
done
|
paulson@15077
|
1298 |
|
paulson@15077
|
1299 |
lemma cos_minus [simp]: "cos (-x) = cos(x)"
|
paulson@15077
|
1300 |
apply (cut_tac x = x in sin_cos_minus)
|
paulson@15085
|
1301 |
apply (auto dest!: real_sum_squares_cancel_a
|
paulson@15085
|
1302 |
simp add: numeral_2_eq_2 simp del: sin_cos_minus)
|
paulson@15077
|
1303 |
done
|
paulson@15077
|
1304 |
|
paulson@15077
|
1305 |
lemma sin_diff: "sin (x - y) = sin x * cos y - cos x * sin y"
|
paulson@15077
|
1306 |
apply (unfold real_diff_def)
|
paulson@15077
|
1307 |
apply (simp (no_asm) add: sin_add)
|
paulson@15077
|
1308 |
done
|
paulson@15077
|
1309 |
|
paulson@15077
|
1310 |
lemma sin_diff2: "sin (x - y) = cos y * sin x - sin y * cos x"
|
paulson@15077
|
1311 |
by (simp add: sin_diff mult_commute)
|
paulson@15077
|
1312 |
|
paulson@15077
|
1313 |
lemma cos_diff: "cos (x - y) = cos x * cos y + sin x * sin y"
|
paulson@15077
|
1314 |
apply (unfold real_diff_def)
|
paulson@15077
|
1315 |
apply (simp (no_asm) add: cos_add)
|
paulson@15077
|
1316 |
done
|
paulson@15077
|
1317 |
|
paulson@15077
|
1318 |
lemma cos_diff2: "cos (x - y) = cos y * cos x + sin y * sin x"
|
paulson@15077
|
1319 |
by (simp add: cos_diff mult_commute)
|
paulson@15077
|
1320 |
|
paulson@15077
|
1321 |
lemma sin_double [simp]: "sin(2 * x) = 2* sin x * cos x"
|
paulson@15077
|
1322 |
by (cut_tac x = x and y = x in sin_add, auto)
|
paulson@15077
|
1323 |
|
paulson@15077
|
1324 |
|
paulson@15077
|
1325 |
lemma cos_double: "cos(2* x) = ((cos x)\<twosuperior>) - ((sin x)\<twosuperior>)"
|
paulson@15077
|
1326 |
apply (cut_tac x = x and y = x in cos_add)
|
paulson@15077
|
1327 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1328 |
done
|
paulson@15077
|
1329 |
|
paulson@15077
|
1330 |
|
paulson@15077
|
1331 |
subsection{*The Constant Pi*}
|
paulson@15077
|
1332 |
|
paulson@15077
|
1333 |
text{*Show that there's a least positive @{term x} with @{term "cos(x) = 0"};
|
paulson@15077
|
1334 |
hence define pi.*}
|
paulson@15077
|
1335 |
|
paulson@15077
|
1336 |
lemma sin_paired:
|
paulson@15077
|
1337 |
"(%n. (- 1) ^ n /(real (fact (2 * n + 1))) * x ^ (2 * n + 1))
|
paulson@15077
|
1338 |
sums sin x"
|
paulson@15077
|
1339 |
proof -
|
paulson@15077
|
1340 |
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
|
paulson@15077
|
1341 |
(if even k then 0
|
paulson@15077
|
1342 |
else (- 1) ^ ((k - Suc 0) div 2) / real (fact k)) *
|
paulson@15077
|
1343 |
x ^ k)
|
paulson@15077
|
1344 |
sums
|
paulson@15077
|
1345 |
suminf (\<lambda>n. (if even n then 0
|
paulson@15077
|
1346 |
else (- 1) ^ ((n - Suc 0) div 2) / real (fact n)) *
|
paulson@15077
|
1347 |
x ^ n)"
|
paulson@15077
|
1348 |
by (rule sin_converges [THEN sums_summable, THEN sums_group], simp)
|
paulson@15077
|
1349 |
thus ?thesis by (simp add: mult_ac sin_def)
|
paulson@15077
|
1350 |
qed
|
paulson@15077
|
1351 |
|
paulson@15077
|
1352 |
lemma sin_gt_zero: "[|0 < x; x < 2 |] ==> 0 < sin x"
|
paulson@15077
|
1353 |
apply (subgoal_tac
|
paulson@15077
|
1354 |
"(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
|
paulson@15077
|
1355 |
(- 1) ^ k / real (fact (2 * k + 1)) * x ^ (2 * k + 1))
|
paulson@15077
|
1356 |
sums suminf (\<lambda>n. (- 1) ^ n / real (fact (2 * n + 1)) * x ^ (2 * n + 1))")
|
paulson@15077
|
1357 |
prefer 2
|
paulson@15077
|
1358 |
apply (rule sin_paired [THEN sums_summable, THEN sums_group], simp)
|
paulson@15077
|
1359 |
apply (rotate_tac 2)
|
paulson@15077
|
1360 |
apply (drule sin_paired [THEN sums_unique, THEN ssubst])
|
paulson@15077
|
1361 |
apply (auto simp del: fact_Suc realpow_Suc)
|
paulson@15077
|
1362 |
apply (frule sums_unique)
|
paulson@15077
|
1363 |
apply (auto simp del: fact_Suc realpow_Suc)
|
paulson@15077
|
1364 |
apply (rule_tac n1 = 0 in series_pos_less [THEN [2] order_le_less_trans])
|
paulson@15077
|
1365 |
apply (auto simp del: fact_Suc realpow_Suc)
|
paulson@15077
|
1366 |
apply (erule sums_summable)
|
paulson@15077
|
1367 |
apply (case_tac "m=0")
|
paulson@15077
|
1368 |
apply (simp (no_asm_simp))
|
paulson@15077
|
1369 |
apply (subgoal_tac "6 * (x * (x * x) / real (Suc (Suc (Suc (Suc (Suc (Suc 0))))))) < 6 * x")
|
paulson@15077
|
1370 |
apply (simp only: mult_less_cancel_left, simp)
|
paulson@15077
|
1371 |
apply (simp (no_asm_simp) add: numeral_2_eq_2 [symmetric] mult_assoc [symmetric])
|
paulson@15077
|
1372 |
apply (subgoal_tac "x*x < 2*3", simp)
|
paulson@15077
|
1373 |
apply (rule mult_strict_mono)
|
paulson@15085
|
1374 |
apply (auto simp add: real_0_less_add_iff real_of_nat_Suc simp del: fact_Suc)
|
paulson@15077
|
1375 |
apply (subst fact_Suc)
|
paulson@15077
|
1376 |
apply (subst fact_Suc)
|
paulson@15077
|
1377 |
apply (subst fact_Suc)
|
paulson@15077
|
1378 |
apply (subst fact_Suc)
|
paulson@15077
|
1379 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
1380 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
1381 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
1382 |
apply (subst real_of_nat_mult)
|
paulson@15079
|
1383 |
apply (simp (no_asm) add: divide_inverse inverse_mult_distrib del: fact_Suc)
|
paulson@15077
|
1384 |
apply (auto simp add: mult_assoc [symmetric] simp del: fact_Suc)
|
paulson@15077
|
1385 |
apply (rule_tac c="real (Suc (Suc (4*m)))" in mult_less_imp_less_right)
|
paulson@15077
|
1386 |
apply (auto simp add: mult_assoc simp del: fact_Suc)
|
paulson@15077
|
1387 |
apply (rule_tac c="real (Suc (Suc (Suc (4*m))))" in mult_less_imp_less_right)
|
paulson@15077
|
1388 |
apply (auto simp add: mult_assoc mult_less_cancel_left simp del: fact_Suc)
|
paulson@15077
|
1389 |
apply (subgoal_tac "x * (x * x ^ (4*m)) = (x ^ (4*m)) * (x * x)")
|
paulson@15077
|
1390 |
apply (erule ssubst)+
|
paulson@15077
|
1391 |
apply (auto simp del: fact_Suc)
|
paulson@15077
|
1392 |
apply (subgoal_tac "0 < x ^ (4 * m) ")
|
paulson@15077
|
1393 |
prefer 2 apply (simp only: zero_less_power)
|
paulson@15077
|
1394 |
apply (simp (no_asm_simp) add: mult_less_cancel_left)
|
paulson@15077
|
1395 |
apply (rule mult_strict_mono)
|
paulson@15077
|
1396 |
apply (simp_all (no_asm_simp))
|
paulson@15077
|
1397 |
done
|
paulson@15077
|
1398 |
|
paulson@15077
|
1399 |
lemma sin_gt_zero1: "[|0 < x; x < 2 |] ==> 0 < sin x"
|
paulson@15077
|
1400 |
by (auto intro: sin_gt_zero)
|
paulson@15077
|
1401 |
|
paulson@15077
|
1402 |
lemma cos_double_less_one: "[| 0 < x; x < 2 |] ==> cos (2 * x) < 1"
|
paulson@15077
|
1403 |
apply (cut_tac x = x in sin_gt_zero1)
|
paulson@15077
|
1404 |
apply (auto simp add: cos_squared_eq cos_double)
|
paulson@15077
|
1405 |
done
|
paulson@15077
|
1406 |
|
paulson@15077
|
1407 |
lemma cos_paired:
|
paulson@15077
|
1408 |
"(%n. (- 1) ^ n /(real (fact (2 * n))) * x ^ (2 * n)) sums cos x"
|
paulson@15077
|
1409 |
proof -
|
paulson@15077
|
1410 |
have "(\<lambda>n. \<Sum>k = n * 2..<n * 2 + 2.
|
paulson@15077
|
1411 |
(if even k then (- 1) ^ (k div 2) / real (fact k) else 0) *
|
paulson@15077
|
1412 |
x ^ k)
|
paulson@15077
|
1413 |
sums
|
paulson@15077
|
1414 |
suminf
|
paulson@15077
|
1415 |
(\<lambda>n. (if even n then (- 1) ^ (n div 2) / real (fact n) else 0) *
|
paulson@15077
|
1416 |
x ^ n)"
|
paulson@15077
|
1417 |
by (rule cos_converges [THEN sums_summable, THEN sums_group], simp)
|
paulson@15077
|
1418 |
thus ?thesis by (simp add: mult_ac cos_def)
|
paulson@15077
|
1419 |
qed
|
paulson@15077
|
1420 |
|
paulson@15077
|
1421 |
declare zero_less_power [simp]
|
paulson@15077
|
1422 |
|
paulson@15077
|
1423 |
lemma fact_lemma: "real (n::nat) * 4 = real (4 * n)"
|
paulson@15077
|
1424 |
by simp
|
paulson@15077
|
1425 |
|
paulson@15077
|
1426 |
lemma cos_two_less_zero: "cos (2) < 0"
|
paulson@15077
|
1427 |
apply (cut_tac x = 2 in cos_paired)
|
paulson@15077
|
1428 |
apply (drule sums_minus)
|
paulson@15077
|
1429 |
apply (rule neg_less_iff_less [THEN iffD1])
|
paulson@15077
|
1430 |
apply (frule sums_unique, auto)
|
paulson@15077
|
1431 |
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
|
1432 |
apply (simp (no_asm) add: fact_num_eq_if realpow_num_eq_if del: fact_Suc realpow_Suc)
|
paulson@15077
|
1433 |
apply (simp (no_asm) add: mult_assoc del: sumr_Suc)
|
paulson@15077
|
1434 |
apply (rule sumr_pos_lt_pair)
|
paulson@15077
|
1435 |
apply (erule sums_summable, safe)
|
paulson@15085
|
1436 |
apply (simp (no_asm) add: divide_inverse real_0_less_add_iff mult_assoc [symmetric]
|
paulson@15085
|
1437 |
del: fact_Suc)
|
paulson@15077
|
1438 |
apply (rule real_mult_inverse_cancel2)
|
paulson@15077
|
1439 |
apply (rule real_of_nat_fact_gt_zero)+
|
paulson@15077
|
1440 |
apply (simp (no_asm) add: mult_assoc [symmetric] del: fact_Suc)
|
paulson@15077
|
1441 |
apply (subst fact_lemma)
|
paulson@15077
|
1442 |
apply (subst fact_Suc)
|
paulson@15077
|
1443 |
apply (subst real_of_nat_mult)
|
paulson@15077
|
1444 |
apply (erule ssubst, subst real_of_nat_mult)
|
paulson@15077
|
1445 |
apply (rule real_mult_less_mono, force)
|
paulson@15077
|
1446 |
prefer 2 apply force
|
paulson@15077
|
1447 |
apply (rule_tac [2] real_of_nat_fact_gt_zero)
|
paulson@15077
|
1448 |
apply (rule real_of_nat_less_iff [THEN iffD2])
|
paulson@15077
|
1449 |
apply (rule fact_less_mono, auto)
|
paulson@15077
|
1450 |
done
|
paulson@15077
|
1451 |
declare cos_two_less_zero [simp]
|
paulson@15077
|
1452 |
declare cos_two_less_zero [THEN real_not_refl2, simp]
|
paulson@15077
|
1453 |
declare cos_two_less_zero [THEN order_less_imp_le, simp]
|
paulson@15077
|
1454 |
|
paulson@15077
|
1455 |
lemma cos_is_zero: "EX! x. 0 \<le> x & x \<le> 2 & cos x = 0"
|
paulson@15077
|
1456 |
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> 2 & cos x = 0")
|
paulson@15077
|
1457 |
apply (rule_tac [2] IVT2)
|
paulson@15077
|
1458 |
apply (auto intro: DERIV_isCont DERIV_cos)
|
paulson@15077
|
1459 |
apply (cut_tac x = xa and y = y in linorder_less_linear)
|
paulson@15077
|
1460 |
apply (rule ccontr)
|
paulson@15077
|
1461 |
apply (subgoal_tac " (\<forall>x. cos differentiable x) & (\<forall>x. isCont cos x) ")
|
paulson@15077
|
1462 |
apply (auto intro: DERIV_cos DERIV_isCont simp add: differentiable_def)
|
paulson@15077
|
1463 |
apply (drule_tac f = cos in Rolle)
|
paulson@15077
|
1464 |
apply (drule_tac [5] f = cos in Rolle)
|
paulson@15077
|
1465 |
apply (auto dest!: DERIV_cos [THEN DERIV_unique] simp add: differentiable_def)
|
paulson@15077
|
1466 |
apply (drule_tac y1 = xa in order_le_less_trans [THEN sin_gt_zero])
|
paulson@15077
|
1467 |
apply (assumption, rule_tac y=y in order_less_le_trans, simp_all)
|
paulson@15077
|
1468 |
apply (drule_tac y1 = y in order_le_less_trans [THEN sin_gt_zero], assumption, simp_all)
|
paulson@15077
|
1469 |
done
|
paulson@15077
|
1470 |
|
paulson@15077
|
1471 |
lemma pi_half: "pi/2 = (@x. 0 \<le> x & x \<le> 2 & cos x = 0)"
|
paulson@15077
|
1472 |
by (simp add: pi_def)
|
paulson@15077
|
1473 |
|
paulson@15077
|
1474 |
lemma cos_pi_half [simp]: "cos (pi / 2) = 0"
|
paulson@15077
|
1475 |
apply (rule cos_is_zero [THEN ex1E])
|
paulson@15077
|
1476 |
apply (auto intro!: someI2 simp add: pi_half)
|
paulson@15077
|
1477 |
done
|
paulson@15077
|
1478 |
|
paulson@15077
|
1479 |
lemma pi_half_gt_zero: "0 < pi / 2"
|
paulson@15077
|
1480 |
apply (rule cos_is_zero [THEN ex1E])
|
paulson@15077
|
1481 |
apply (auto simp add: pi_half)
|
paulson@15077
|
1482 |
apply (rule someI2, blast, safe)
|
paulson@15077
|
1483 |
apply (drule_tac y = xa in real_le_imp_less_or_eq)
|
paulson@15077
|
1484 |
apply (safe, simp)
|
paulson@15077
|
1485 |
done
|
paulson@15077
|
1486 |
declare pi_half_gt_zero [simp]
|
paulson@15077
|
1487 |
declare pi_half_gt_zero [THEN real_not_refl2, THEN not_sym, simp]
|
paulson@15077
|
1488 |
declare pi_half_gt_zero [THEN order_less_imp_le, simp]
|
paulson@15077
|
1489 |
|
paulson@15077
|
1490 |
lemma pi_half_less_two: "pi / 2 < 2"
|
paulson@15077
|
1491 |
apply (rule cos_is_zero [THEN ex1E])
|
paulson@15077
|
1492 |
apply (auto simp add: pi_half)
|
paulson@15077
|
1493 |
apply (rule someI2, blast, safe)
|
paulson@15077
|
1494 |
apply (drule_tac x = xa in order_le_imp_less_or_eq)
|
paulson@15077
|
1495 |
apply (safe, simp)
|
paulson@15077
|
1496 |
done
|
paulson@15077
|
1497 |
declare pi_half_less_two [simp]
|
paulson@15077
|
1498 |
declare pi_half_less_two [THEN real_not_refl2, simp]
|
paulson@15077
|
1499 |
declare pi_half_less_two [THEN order_less_imp_le, simp]
|
paulson@15077
|
1500 |
|
paulson@15077
|
1501 |
lemma pi_gt_zero [simp]: "0 < pi"
|
paulson@15077
|
1502 |
apply (rule_tac c="inverse 2" in mult_less_imp_less_right)
|
paulson@15077
|
1503 |
apply (cut_tac pi_half_gt_zero, simp_all)
|
paulson@15077
|
1504 |
done
|
paulson@15077
|
1505 |
|
paulson@15077
|
1506 |
lemma pi_neq_zero [simp]: "pi \<noteq> 0"
|
paulson@15077
|
1507 |
by (rule pi_gt_zero [THEN real_not_refl2, THEN not_sym])
|
paulson@15077
|
1508 |
|
paulson@15077
|
1509 |
lemma pi_not_less_zero [simp]: "~ (pi < 0)"
|
paulson@15077
|
1510 |
apply (insert pi_gt_zero)
|
paulson@15077
|
1511 |
apply (blast elim: order_less_asym)
|
paulson@15077
|
1512 |
done
|
paulson@15077
|
1513 |
|
paulson@15077
|
1514 |
lemma pi_ge_zero [simp]: "0 \<le> pi"
|
paulson@15077
|
1515 |
by (auto intro: order_less_imp_le)
|
paulson@15077
|
1516 |
|
paulson@15077
|
1517 |
lemma minus_pi_half_less_zero [simp]: "-(pi/2) < 0"
|
paulson@15077
|
1518 |
by auto
|
paulson@15077
|
1519 |
|
paulson@15077
|
1520 |
lemma sin_pi_half [simp]: "sin(pi/2) = 1"
|
paulson@15077
|
1521 |
apply (cut_tac x = "pi/2" in sin_cos_squared_add2)
|
paulson@15077
|
1522 |
apply (cut_tac sin_gt_zero [OF pi_half_gt_zero pi_half_less_two])
|
paulson@15077
|
1523 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1524 |
done
|
paulson@15077
|
1525 |
|
paulson@15077
|
1526 |
lemma cos_pi [simp]: "cos pi = -1"
|
paulson@15077
|
1527 |
by (cut_tac x = "pi/2" and y = "pi/2" in cos_add, simp)
|
paulson@15077
|
1528 |
|
paulson@15077
|
1529 |
lemma sin_pi [simp]: "sin pi = 0"
|
paulson@15077
|
1530 |
by (cut_tac x = "pi/2" and y = "pi/2" in sin_add, simp)
|
paulson@15077
|
1531 |
|
paulson@15077
|
1532 |
lemma sin_cos_eq: "sin x = cos (pi/2 - x)"
|
paulson@15077
|
1533 |
apply (unfold real_diff_def)
|
paulson@15077
|
1534 |
apply (simp (no_asm) add: cos_add)
|
paulson@15077
|
1535 |
done
|
paulson@15077
|
1536 |
|
paulson@15077
|
1537 |
lemma minus_sin_cos_eq: "-sin x = cos (x + pi/2)"
|
paulson@15077
|
1538 |
apply (simp (no_asm) add: cos_add)
|
paulson@15077
|
1539 |
done
|
paulson@15077
|
1540 |
declare minus_sin_cos_eq [symmetric, simp]
|
paulson@15077
|
1541 |
|
paulson@15077
|
1542 |
lemma cos_sin_eq: "cos x = sin (pi/2 - x)"
|
paulson@15077
|
1543 |
apply (unfold real_diff_def)
|
paulson@15077
|
1544 |
apply (simp (no_asm) add: sin_add)
|
paulson@15077
|
1545 |
done
|
paulson@15077
|
1546 |
declare sin_cos_eq [symmetric, simp] cos_sin_eq [symmetric, simp]
|
paulson@15077
|
1547 |
|
paulson@15077
|
1548 |
lemma sin_periodic_pi [simp]: "sin (x + pi) = - sin x"
|
paulson@15077
|
1549 |
apply (simp (no_asm) add: sin_add)
|
paulson@15077
|
1550 |
done
|
paulson@15077
|
1551 |
|
paulson@15077
|
1552 |
lemma sin_periodic_pi2 [simp]: "sin (pi + x) = - sin x"
|
paulson@15077
|
1553 |
apply (simp (no_asm) add: sin_add)
|
paulson@15077
|
1554 |
done
|
paulson@15077
|
1555 |
|
paulson@15077
|
1556 |
lemma cos_periodic_pi [simp]: "cos (x + pi) = - cos x"
|
paulson@15077
|
1557 |
apply (simp (no_asm) add: cos_add)
|
paulson@15077
|
1558 |
done
|
paulson@15077
|
1559 |
|
paulson@15077
|
1560 |
lemma sin_periodic [simp]: "sin (x + 2*pi) = sin x"
|
paulson@15077
|
1561 |
by (simp add: sin_add cos_double)
|
paulson@15077
|
1562 |
|
paulson@15077
|
1563 |
lemma cos_periodic [simp]: "cos (x + 2*pi) = cos x"
|
paulson@15077
|
1564 |
by (simp add: cos_add cos_double)
|
paulson@15077
|
1565 |
|
paulson@15077
|
1566 |
lemma cos_npi [simp]: "cos (real n * pi) = -1 ^ n"
|
paulson@15077
|
1567 |
apply (induct_tac "n")
|
paulson@15077
|
1568 |
apply (auto simp add: real_of_nat_Suc left_distrib)
|
paulson@15077
|
1569 |
done
|
paulson@15077
|
1570 |
|
paulson@15077
|
1571 |
lemma sin_npi [simp]: "sin (real (n::nat) * pi) = 0"
|
paulson@15077
|
1572 |
apply (induct_tac "n")
|
paulson@15077
|
1573 |
apply (auto simp add: real_of_nat_Suc left_distrib)
|
paulson@15077
|
1574 |
done
|
paulson@15077
|
1575 |
|
paulson@15077
|
1576 |
lemma sin_npi2 [simp]: "sin (pi * real (n::nat)) = 0"
|
paulson@15077
|
1577 |
apply (cut_tac n = n in sin_npi)
|
paulson@15077
|
1578 |
apply (auto simp add: mult_commute simp del: sin_npi)
|
paulson@15077
|
1579 |
done
|
paulson@15077
|
1580 |
|
paulson@15077
|
1581 |
lemma cos_two_pi [simp]: "cos (2 * pi) = 1"
|
paulson@15077
|
1582 |
by (simp add: cos_double)
|
paulson@15077
|
1583 |
|
paulson@15077
|
1584 |
lemma sin_two_pi [simp]: "sin (2 * pi) = 0"
|
paulson@15077
|
1585 |
apply (simp (no_asm))
|
paulson@15077
|
1586 |
done
|
paulson@15077
|
1587 |
|
paulson@15077
|
1588 |
lemma sin_gt_zero2: "[| 0 < x; x < pi/2 |] ==> 0 < sin x"
|
paulson@15077
|
1589 |
apply (rule sin_gt_zero, assumption)
|
paulson@15077
|
1590 |
apply (rule order_less_trans, assumption)
|
paulson@15077
|
1591 |
apply (rule pi_half_less_two)
|
paulson@15077
|
1592 |
done
|
paulson@15077
|
1593 |
|
paulson@15077
|
1594 |
lemma sin_less_zero:
|
paulson@15077
|
1595 |
assumes lb: "- pi/2 < x" and "x < 0" shows "sin x < 0"
|
paulson@15077
|
1596 |
proof -
|
paulson@15077
|
1597 |
have "0 < sin (- x)" using prems by (simp only: sin_gt_zero2)
|
paulson@15077
|
1598 |
thus ?thesis by simp
|
paulson@15077
|
1599 |
qed
|
paulson@15077
|
1600 |
|
paulson@15077
|
1601 |
lemma pi_less_4: "pi < 4"
|
paulson@15077
|
1602 |
by (cut_tac pi_half_less_two, auto)
|
paulson@15077
|
1603 |
|
paulson@15077
|
1604 |
lemma cos_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < cos x"
|
paulson@15077
|
1605 |
apply (cut_tac pi_less_4)
|
paulson@15077
|
1606 |
apply (cut_tac f = cos and a = 0 and b = x and y = 0 in IVT2_objl, safe, simp_all)
|
paulson@15077
|
1607 |
apply (force intro: DERIV_isCont DERIV_cos)
|
paulson@15077
|
1608 |
apply (cut_tac cos_is_zero, safe)
|
paulson@15077
|
1609 |
apply (rename_tac y z)
|
paulson@15077
|
1610 |
apply (drule_tac x = y in spec)
|
paulson@15077
|
1611 |
apply (drule_tac x = "pi/2" in spec, simp)
|
paulson@15077
|
1612 |
done
|
paulson@15077
|
1613 |
|
paulson@15077
|
1614 |
lemma cos_gt_zero_pi: "[| -(pi/2) < x; x < pi/2 |] ==> 0 < cos x"
|
paulson@15077
|
1615 |
apply (rule_tac x = x and y = 0 in linorder_cases)
|
paulson@15077
|
1616 |
apply (rule cos_minus [THEN subst])
|
paulson@15077
|
1617 |
apply (rule cos_gt_zero)
|
paulson@15077
|
1618 |
apply (auto intro: cos_gt_zero)
|
paulson@15077
|
1619 |
done
|
paulson@15077
|
1620 |
|
paulson@15077
|
1621 |
lemma cos_ge_zero: "[| -(pi/2) \<le> x; x \<le> pi/2 |] ==> 0 \<le> cos x"
|
paulson@15077
|
1622 |
apply (auto simp add: order_le_less cos_gt_zero_pi)
|
paulson@15077
|
1623 |
apply (subgoal_tac "x = pi/2", auto)
|
paulson@15077
|
1624 |
done
|
paulson@15077
|
1625 |
|
paulson@15077
|
1626 |
lemma sin_gt_zero_pi: "[| 0 < x; x < pi |] ==> 0 < sin x"
|
paulson@15077
|
1627 |
apply (subst sin_cos_eq)
|
paulson@15077
|
1628 |
apply (rotate_tac 1)
|
paulson@15077
|
1629 |
apply (drule real_sum_of_halves [THEN ssubst])
|
paulson@15077
|
1630 |
apply (auto intro!: cos_gt_zero_pi simp del: sin_cos_eq [symmetric])
|
paulson@15077
|
1631 |
done
|
paulson@15077
|
1632 |
|
paulson@15077
|
1633 |
lemma sin_ge_zero: "[| 0 \<le> x; x \<le> pi |] ==> 0 \<le> sin x"
|
paulson@15077
|
1634 |
by (auto simp add: order_le_less sin_gt_zero_pi)
|
paulson@15077
|
1635 |
|
paulson@15077
|
1636 |
lemma cos_total: "[| -1 \<le> y; y \<le> 1 |] ==> EX! x. 0 \<le> x & x \<le> pi & (cos x = y)"
|
paulson@15077
|
1637 |
apply (subgoal_tac "\<exists>x. 0 \<le> x & x \<le> pi & cos x = y")
|
paulson@15077
|
1638 |
apply (rule_tac [2] IVT2)
|
paulson@15077
|
1639 |
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos)
|
paulson@15077
|
1640 |
apply (cut_tac x = xa and y = y in linorder_less_linear)
|
paulson@15077
|
1641 |
apply (rule ccontr, auto)
|
paulson@15077
|
1642 |
apply (drule_tac f = cos in Rolle)
|
paulson@15077
|
1643 |
apply (drule_tac [5] f = cos in Rolle)
|
paulson@15077
|
1644 |
apply (auto intro: order_less_imp_le DERIV_isCont DERIV_cos
|
paulson@15077
|
1645 |
dest!: DERIV_cos [THEN DERIV_unique]
|
paulson@15077
|
1646 |
simp add: differentiable_def)
|
paulson@15077
|
1647 |
apply (auto dest: sin_gt_zero_pi [OF order_le_less_trans order_less_le_trans])
|
paulson@15077
|
1648 |
done
|
paulson@15077
|
1649 |
|
paulson@15077
|
1650 |
lemma sin_total:
|
paulson@15077
|
1651 |
"[| -1 \<le> y; y \<le> 1 |] ==> EX! x. -(pi/2) \<le> x & x \<le> pi/2 & (sin x = y)"
|
paulson@15077
|
1652 |
apply (rule ccontr)
|
paulson@15077
|
1653 |
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
|
1654 |
apply (erule swap)
|
paulson@15077
|
1655 |
apply (simp del: minus_sin_cos_eq [symmetric])
|
paulson@15077
|
1656 |
apply (cut_tac y="-y" in cos_total, simp) apply simp
|
paulson@15077
|
1657 |
apply (erule ex1E)
|
paulson@15077
|
1658 |
apply (rule_tac a = "x - (pi/2) " in ex1I)
|
paulson@15077
|
1659 |
apply (simp (no_asm) add: real_add_assoc)
|
paulson@15077
|
1660 |
apply (rotate_tac 3)
|
paulson@15077
|
1661 |
apply (drule_tac x = "xa + pi/2" in spec, safe, simp_all)
|
paulson@15077
|
1662 |
done
|
paulson@15077
|
1663 |
|
paulson@15077
|
1664 |
lemma reals_Archimedean4:
|
paulson@15077
|
1665 |
"[| 0 < y; 0 \<le> x |] ==> \<exists>n. real n * y \<le> x & x < real (Suc n) * y"
|
paulson@15077
|
1666 |
apply (auto dest!: reals_Archimedean3)
|
paulson@15077
|
1667 |
apply (drule_tac x = x in spec, clarify)
|
paulson@15077
|
1668 |
apply (subgoal_tac "x < real(LEAST m::nat. x < real m * y) * y")
|
paulson@15077
|
1669 |
prefer 2 apply (erule LeastI)
|
paulson@15077
|
1670 |
apply (case_tac "LEAST m::nat. x < real m * y", simp)
|
paulson@15077
|
1671 |
apply (subgoal_tac "~ x < real nat * y")
|
paulson@15077
|
1672 |
prefer 2 apply (rule not_less_Least, simp, force)
|
paulson@15077
|
1673 |
done
|
paulson@15077
|
1674 |
|
paulson@15077
|
1675 |
(* Pre Isabelle99-2 proof was simpler- numerals arithmetic
|
paulson@15077
|
1676 |
now causes some unwanted re-arrangements of literals! *)
|
paulson@15077
|
1677 |
lemma cos_zero_lemma: "[| 0 \<le> x; cos x = 0 |] ==>
|
paulson@15077
|
1678 |
\<exists>n::nat. ~even n & x = real n * (pi/2)"
|
paulson@15077
|
1679 |
apply (drule pi_gt_zero [THEN reals_Archimedean4], safe)
|
paulson@15086
|
1680 |
apply (subgoal_tac "0 \<le> x - real n * pi &
|
paulson@15086
|
1681 |
(x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
|
paulson@15086
|
1682 |
apply (auto simp add: compare_rls)
|
paulson@15077
|
1683 |
prefer 3 apply (simp add: cos_diff)
|
paulson@15077
|
1684 |
prefer 2 apply (simp add: real_of_nat_Suc left_distrib)
|
paulson@15077
|
1685 |
apply (simp add: cos_diff)
|
paulson@15077
|
1686 |
apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
|
paulson@15077
|
1687 |
apply (rule_tac [2] cos_total, safe)
|
paulson@15077
|
1688 |
apply (drule_tac x = "x - real n * pi" in spec)
|
paulson@15077
|
1689 |
apply (drule_tac x = "pi/2" in spec)
|
paulson@15077
|
1690 |
apply (simp add: cos_diff)
|
paulson@15077
|
1691 |
apply (rule_tac x = "Suc (2 * n) " in exI)
|
paulson@15077
|
1692 |
apply (simp add: real_of_nat_Suc left_distrib, auto)
|
paulson@15077
|
1693 |
done
|
paulson@15077
|
1694 |
|
paulson@15077
|
1695 |
lemma sin_zero_lemma: "[| 0 \<le> x; sin x = 0 |] ==>
|
paulson@15077
|
1696 |
\<exists>n::nat. even n & x = real n * (pi/2)"
|
paulson@15077
|
1697 |
apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
|
paulson@15077
|
1698 |
apply (clarify, rule_tac x = "n - 1" in exI)
|
paulson@15077
|
1699 |
apply (force simp add: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
|
paulson@15085
|
1700 |
apply (rule cos_zero_lemma)
|
paulson@15085
|
1701 |
apply (simp_all add: add_increasing)
|
paulson@15077
|
1702 |
done
|
paulson@15077
|
1703 |
|
paulson@15077
|
1704 |
|
paulson@15077
|
1705 |
lemma cos_zero_iff: "(cos x = 0) =
|
paulson@15077
|
1706 |
((\<exists>n::nat. ~even n & (x = real n * (pi/2))) |
|
paulson@15077
|
1707 |
(\<exists>n::nat. ~even n & (x = -(real n * (pi/2)))))"
|
paulson@15077
|
1708 |
apply (rule iffI)
|
paulson@15077
|
1709 |
apply (cut_tac linorder_linear [of 0 x], safe)
|
paulson@15077
|
1710 |
apply (drule cos_zero_lemma, assumption+)
|
paulson@15077
|
1711 |
apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
|
paulson@15077
|
1712 |
apply (force simp add: minus_equation_iff [of x])
|
paulson@15077
|
1713 |
apply (auto simp only: odd_Suc_mult_two_ex real_of_nat_Suc left_distrib)
|
paulson@15077
|
1714 |
apply (auto simp add: cos_add)
|
paulson@15077
|
1715 |
done
|
paulson@15077
|
1716 |
|
paulson@15077
|
1717 |
(* ditto: but to a lesser extent *)
|
paulson@15077
|
1718 |
lemma sin_zero_iff: "(sin x = 0) =
|
paulson@15077
|
1719 |
((\<exists>n::nat. even n & (x = real n * (pi/2))) |
|
paulson@15077
|
1720 |
(\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
|
paulson@15077
|
1721 |
apply (rule iffI)
|
paulson@15077
|
1722 |
apply (cut_tac linorder_linear [of 0 x], safe)
|
paulson@15077
|
1723 |
apply (drule sin_zero_lemma, assumption+)
|
paulson@15077
|
1724 |
apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
|
paulson@15077
|
1725 |
apply (force simp add: minus_equation_iff [of x])
|
paulson@15077
|
1726 |
apply (auto simp add: even_mult_two_ex)
|
paulson@15077
|
1727 |
done
|
paulson@15077
|
1728 |
|
paulson@15077
|
1729 |
|
paulson@15077
|
1730 |
subsection{*Tangent*}
|
paulson@15077
|
1731 |
|
paulson@15077
|
1732 |
lemma tan_zero [simp]: "tan 0 = 0"
|
paulson@15077
|
1733 |
by (simp add: tan_def)
|
paulson@15077
|
1734 |
|
paulson@15077
|
1735 |
lemma tan_pi [simp]: "tan pi = 0"
|
paulson@15077
|
1736 |
by (simp add: tan_def)
|
paulson@15077
|
1737 |
|
paulson@15077
|
1738 |
lemma tan_npi [simp]: "tan (real (n::nat) * pi) = 0"
|
paulson@15077
|
1739 |
by (simp add: tan_def)
|
paulson@15077
|
1740 |
|
paulson@15077
|
1741 |
lemma tan_minus [simp]: "tan (-x) = - tan x"
|
paulson@15077
|
1742 |
by (simp add: tan_def minus_mult_left)
|
paulson@15077
|
1743 |
|
paulson@15077
|
1744 |
lemma tan_periodic [simp]: "tan (x + 2*pi) = tan x"
|
paulson@15077
|
1745 |
by (simp add: tan_def)
|
paulson@15077
|
1746 |
|
paulson@15077
|
1747 |
lemma lemma_tan_add1:
|
paulson@15077
|
1748 |
"[| cos x \<noteq> 0; cos y \<noteq> 0 |]
|
paulson@15077
|
1749 |
==> 1 - tan(x)*tan(y) = cos (x + y)/(cos x * cos y)"
|
paulson@15077
|
1750 |
apply (unfold tan_def real_divide_def)
|
paulson@15077
|
1751 |
apply (auto simp del: inverse_mult_distrib simp add: inverse_mult_distrib [symmetric] mult_ac)
|
paulson@15077
|
1752 |
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
|
paulson@15077
|
1753 |
apply (auto simp del: inverse_mult_distrib simp add: mult_assoc left_diff_distrib cos_add)
|
paulson@15077
|
1754 |
done
|
paulson@15077
|
1755 |
|
paulson@15077
|
1756 |
lemma add_tan_eq:
|
paulson@15077
|
1757 |
"[| cos x \<noteq> 0; cos y \<noteq> 0 |]
|
paulson@15077
|
1758 |
==> tan x + tan y = sin(x + y)/(cos x * cos y)"
|
paulson@15077
|
1759 |
apply (unfold tan_def)
|
paulson@15077
|
1760 |
apply (rule_tac c1 = "cos x * cos y" in real_mult_right_cancel [THEN subst])
|
paulson@15077
|
1761 |
apply (auto simp add: mult_assoc left_distrib)
|
paulson@15077
|
1762 |
apply (simp (no_asm) add: sin_add)
|
paulson@15077
|
1763 |
done
|
paulson@15077
|
1764 |
|
paulson@15077
|
1765 |
lemma tan_add: "[| cos x \<noteq> 0; cos y \<noteq> 0; cos (x + y) \<noteq> 0 |]
|
paulson@15077
|
1766 |
==> tan(x + y) = (tan(x) + tan(y))/(1 - tan(x) * tan(y))"
|
paulson@15077
|
1767 |
apply (simp (no_asm_simp) add: add_tan_eq lemma_tan_add1)
|
paulson@15077
|
1768 |
apply (simp add: tan_def)
|
paulson@15077
|
1769 |
done
|
paulson@15077
|
1770 |
|
paulson@15077
|
1771 |
lemma tan_double: "[| cos x \<noteq> 0; cos (2 * x) \<noteq> 0 |]
|
paulson@15077
|
1772 |
==> tan (2 * x) = (2 * tan x)/(1 - (tan(x) ^ 2))"
|
paulson@15077
|
1773 |
apply (insert tan_add [of x x])
|
paulson@15077
|
1774 |
apply (simp add: mult_2 [symmetric])
|
paulson@15077
|
1775 |
apply (auto simp add: numeral_2_eq_2)
|
paulson@15077
|
1776 |
done
|
paulson@15077
|
1777 |
|
paulson@15077
|
1778 |
lemma tan_gt_zero: "[| 0 < x; x < pi/2 |] ==> 0 < tan x"
|
paulson@15077
|
1779 |
apply (unfold tan_def real_divide_def)
|
paulson@15077
|
1780 |
apply (auto intro!: sin_gt_zero2 cos_gt_zero_pi intro!: real_mult_order positive_imp_inverse_positive)
|
paulson@15077
|
1781 |
done
|
paulson@15077
|
1782 |
|
paulson@15077
|
1783 |
lemma tan_less_zero:
|
paulson@15077
|
1784 |
assumes lb: "- pi/2 < x" and "x < 0" shows "tan x < 0"
|
paulson@15077
|
1785 |
proof -
|
paulson@15077
|
1786 |
have "0 < tan (- x)" using prems by (simp only: tan_gt_zero)
|
paulson@15077
|
1787 |
thus ?thesis by simp
|
paulson@15077
|
1788 |
qed
|
paulson@15077
|
1789 |
|
paulson@15077
|
1790 |
lemma lemma_DERIV_tan:
|
paulson@15077
|
1791 |
"cos x \<noteq> 0 ==> DERIV (%x. sin(x)/cos(x)) x :> inverse((cos x)\<twosuperior>)"
|
paulson@15077
|
1792 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
1793 |
apply (best intro!: DERIV_intros intro: DERIV_chain2)
|
paulson@15079
|
1794 |
apply (auto simp add: divide_inverse numeral_2_eq_2)
|
paulson@15077
|
1795 |
done
|
paulson@15077
|
1796 |
|
paulson@15077
|
1797 |
lemma DERIV_tan [simp]: "cos x \<noteq> 0 ==> DERIV tan x :> inverse((cos x)\<twosuperior>)"
|
paulson@15077
|
1798 |
by (auto dest: lemma_DERIV_tan simp add: tan_def [symmetric])
|
paulson@15077
|
1799 |
|
paulson@15077
|
1800 |
lemma LIM_cos_div_sin [simp]: "(%x. cos(x)/sin(x)) -- pi/2 --> 0"
|
paulson@15077
|
1801 |
apply (unfold real_divide_def)
|
paulson@15077
|
1802 |
apply (subgoal_tac "(\<lambda>x. cos x * inverse (sin x)) -- pi * inverse 2 --> 0*1")
|
paulson@15077
|
1803 |
apply simp
|
paulson@15077
|
1804 |
apply (rule LIM_mult2)
|
paulson@15077
|
1805 |
apply (rule_tac [2] inverse_1 [THEN subst])
|
paulson@15077
|
1806 |
apply (rule_tac [2] LIM_inverse)
|
paulson@15077
|
1807 |
apply (simp_all add: divide_inverse [symmetric])
|
paulson@15077
|
1808 |
apply (simp_all only: isCont_def [symmetric] cos_pi_half [symmetric] sin_pi_half [symmetric])
|
paulson@15077
|
1809 |
apply (blast intro!: DERIV_isCont DERIV_sin DERIV_cos)+
|
paulson@15077
|
1810 |
done
|
paulson@15077
|
1811 |
|
paulson@15077
|
1812 |
lemma lemma_tan_total: "0 < y ==> \<exists>x. 0 < x & x < pi/2 & y < tan x"
|
paulson@15077
|
1813 |
apply (cut_tac LIM_cos_div_sin)
|
paulson@15077
|
1814 |
apply (simp only: LIM_def)
|
paulson@15077
|
1815 |
apply (drule_tac x = "inverse y" in spec, safe, force)
|
paulson@15077
|
1816 |
apply (drule_tac ?d1.0 = s in pi_half_gt_zero [THEN [2] real_lbound_gt_zero], safe)
|
paulson@15077
|
1817 |
apply (rule_tac x = " (pi/2) - e" in exI)
|
paulson@15077
|
1818 |
apply (simp (no_asm_simp))
|
paulson@15077
|
1819 |
apply (drule_tac x = " (pi/2) - e" in spec)
|
paulson@15077
|
1820 |
apply (auto simp add: abs_eqI2 tan_def)
|
paulson@15077
|
1821 |
apply (rule inverse_less_iff_less [THEN iffD1])
|
paulson@15079
|
1822 |
apply (auto simp add: divide_inverse)
|
paulson@15077
|
1823 |
apply (rule real_mult_order)
|
paulson@15077
|
1824 |
apply (subgoal_tac [3] "0 < sin e")
|
paulson@15077
|
1825 |
apply (subgoal_tac [3] "0 < cos e")
|
paulson@15077
|
1826 |
apply (auto intro: cos_gt_zero sin_gt_zero2 simp add: inverse_mult_distrib abs_mult)
|
paulson@15077
|
1827 |
apply (drule_tac a = "cos e" in positive_imp_inverse_positive)
|
paulson@15077
|
1828 |
apply (drule_tac x = "inverse (cos e) " in abs_eqI2)
|
paulson@15077
|
1829 |
apply (auto dest!: abs_eqI2 simp add: mult_ac)
|
paulson@15077
|
1830 |
done
|
paulson@15077
|
1831 |
|
paulson@15077
|
1832 |
lemma tan_total_pos: "0 \<le> y ==> \<exists>x. 0 \<le> x & x < pi/2 & tan x = y"
|
paulson@15077
|
1833 |
apply (frule real_le_imp_less_or_eq, safe)
|
paulson@15077
|
1834 |
prefer 2 apply force
|
paulson@15077
|
1835 |
apply (drule lemma_tan_total, safe)
|
paulson@15077
|
1836 |
apply (cut_tac f = tan and a = 0 and b = x and y = y in IVT_objl)
|
paulson@15077
|
1837 |
apply (auto intro!: DERIV_tan [THEN DERIV_isCont])
|
paulson@15077
|
1838 |
apply (drule_tac y = xa in order_le_imp_less_or_eq)
|
paulson@15077
|
1839 |
apply (auto dest: cos_gt_zero)
|
paulson@15077
|
1840 |
done
|
paulson@15077
|
1841 |
|
paulson@15077
|
1842 |
lemma lemma_tan_total1: "\<exists>x. -(pi/2) < x & x < (pi/2) & tan x = y"
|
paulson@15077
|
1843 |
apply (cut_tac linorder_linear [of 0 y], safe)
|
paulson@15077
|
1844 |
apply (drule tan_total_pos)
|
paulson@15077
|
1845 |
apply (cut_tac [2] y="-y" in tan_total_pos, safe)
|
paulson@15077
|
1846 |
apply (rule_tac [3] x = "-x" in exI)
|
paulson@15077
|
1847 |
apply (auto intro!: exI)
|
paulson@15077
|
1848 |
done
|
paulson@15077
|
1849 |
|
paulson@15077
|
1850 |
lemma tan_total: "EX! x. -(pi/2) < x & x < (pi/2) & tan x = y"
|
paulson@15077
|
1851 |
apply (cut_tac y = y in lemma_tan_total1, auto)
|
paulson@15077
|
1852 |
apply (cut_tac x = xa and y = y in linorder_less_linear, auto)
|
paulson@15077
|
1853 |
apply (subgoal_tac [2] "\<exists>z. y < z & z < xa & DERIV tan z :> 0")
|
paulson@15077
|
1854 |
apply (subgoal_tac "\<exists>z. xa < z & z < y & DERIV tan z :> 0")
|
paulson@15077
|
1855 |
apply (rule_tac [4] Rolle)
|
paulson@15077
|
1856 |
apply (rule_tac [2] Rolle)
|
paulson@15077
|
1857 |
apply (auto intro!: DERIV_tan DERIV_isCont exI
|
paulson@15077
|
1858 |
simp add: differentiable_def)
|
paulson@15077
|
1859 |
txt{*Now, simulate TRYALL*}
|
paulson@15077
|
1860 |
apply (rule_tac [!] DERIV_tan asm_rl)
|
paulson@15077
|
1861 |
apply (auto dest!: DERIV_unique [OF _ DERIV_tan]
|
paulson@15077
|
1862 |
simp add: cos_gt_zero_pi [THEN real_not_refl2, THEN not_sym])
|
paulson@15077
|
1863 |
done
|
paulson@15077
|
1864 |
|
paulson@15077
|
1865 |
lemma arcsin_pi: "[| -1 \<le> y; y \<le> 1 |]
|
paulson@15077
|
1866 |
==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi & sin(arcsin y) = y"
|
paulson@15077
|
1867 |
apply (drule sin_total, assumption)
|
paulson@15077
|
1868 |
apply (erule ex1E)
|
paulson@15077
|
1869 |
apply (unfold arcsin_def)
|
paulson@15077
|
1870 |
apply (rule someI2, blast)
|
paulson@15077
|
1871 |
apply (force intro: order_trans)
|
paulson@15077
|
1872 |
done
|
paulson@15077
|
1873 |
|
paulson@15077
|
1874 |
lemma arcsin: "[| -1 \<le> y; y \<le> 1 |]
|
paulson@15077
|
1875 |
==> -(pi/2) \<le> arcsin y &
|
paulson@15077
|
1876 |
arcsin y \<le> pi/2 & sin(arcsin y) = y"
|
paulson@15077
|
1877 |
apply (unfold arcsin_def)
|
paulson@15077
|
1878 |
apply (drule sin_total, assumption)
|
paulson@15077
|
1879 |
apply (fast intro: someI2)
|
paulson@15077
|
1880 |
done
|
paulson@15077
|
1881 |
|
paulson@15077
|
1882 |
lemma sin_arcsin [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> sin(arcsin y) = y"
|
paulson@15077
|
1883 |
by (blast dest: arcsin)
|
paulson@15077
|
1884 |
|
paulson@15077
|
1885 |
lemma arcsin_bounded:
|
paulson@15077
|
1886 |
"[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y & arcsin y \<le> pi/2"
|
paulson@15077
|
1887 |
by (blast dest: arcsin)
|
paulson@15077
|
1888 |
|
paulson@15077
|
1889 |
lemma arcsin_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> -(pi/2) \<le> arcsin y"
|
paulson@15077
|
1890 |
by (blast dest: arcsin)
|
paulson@15077
|
1891 |
|
paulson@15077
|
1892 |
lemma arcsin_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcsin y \<le> pi/2"
|
paulson@15077
|
1893 |
by (blast dest: arcsin)
|
paulson@15077
|
1894 |
|
paulson@15077
|
1895 |
lemma arcsin_lt_bounded:
|
paulson@15077
|
1896 |
"[| -1 < y; y < 1 |] ==> -(pi/2) < arcsin y & arcsin y < pi/2"
|
paulson@15077
|
1897 |
apply (frule order_less_imp_le)
|
paulson@15077
|
1898 |
apply (frule_tac y = y in order_less_imp_le)
|
paulson@15077
|
1899 |
apply (frule arcsin_bounded)
|
paulson@15077
|
1900 |
apply (safe, simp)
|
paulson@15077
|
1901 |
apply (drule_tac y = "arcsin y" in order_le_imp_less_or_eq)
|
paulson@15077
|
1902 |
apply (drule_tac [2] y = "pi/2" in order_le_imp_less_or_eq, safe)
|
paulson@15077
|
1903 |
apply (drule_tac [!] f = sin in arg_cong, auto)
|
paulson@15077
|
1904 |
done
|
paulson@15077
|
1905 |
|
paulson@15077
|
1906 |
lemma arcsin_sin: "[|-(pi/2) \<le> x; x \<le> pi/2 |] ==> arcsin(sin x) = x"
|
paulson@15077
|
1907 |
apply (unfold arcsin_def)
|
paulson@15077
|
1908 |
apply (rule some1_equality)
|
paulson@15077
|
1909 |
apply (rule sin_total, auto)
|
paulson@15077
|
1910 |
done
|
paulson@15077
|
1911 |
|
paulson@15077
|
1912 |
lemma arcos: "[| -1 \<le> y; y \<le> 1 |]
|
paulson@15077
|
1913 |
==> 0 \<le> arcos y & arcos y \<le> pi & cos(arcos y) = y"
|
paulson@15077
|
1914 |
apply (unfold arcos_def)
|
paulson@15077
|
1915 |
apply (drule cos_total, assumption)
|
paulson@15077
|
1916 |
apply (fast intro: someI2)
|
paulson@15077
|
1917 |
done
|
paulson@15077
|
1918 |
|
paulson@15077
|
1919 |
lemma cos_arcos [simp]: "[| -1 \<le> y; y \<le> 1 |] ==> cos(arcos y) = y"
|
paulson@15077
|
1920 |
by (blast dest: arcos)
|
paulson@15077
|
1921 |
|
paulson@15077
|
1922 |
lemma arcos_bounded: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y & arcos y \<le> pi"
|
paulson@15077
|
1923 |
by (blast dest: arcos)
|
paulson@15077
|
1924 |
|
paulson@15077
|
1925 |
lemma arcos_lbound: "[| -1 \<le> y; y \<le> 1 |] ==> 0 \<le> arcos y"
|
paulson@15077
|
1926 |
by (blast dest: arcos)
|
paulson@15077
|
1927 |
|
paulson@15077
|
1928 |
lemma arcos_ubound: "[| -1 \<le> y; y \<le> 1 |] ==> arcos y \<le> pi"
|
paulson@15077
|
1929 |
by (blast dest: arcos)
|
paulson@15077
|
1930 |
|
paulson@15077
|
1931 |
lemma arcos_lt_bounded: "[| -1 < y; y < 1 |]
|
paulson@15077
|
1932 |
==> 0 < arcos y & arcos y < pi"
|
paulson@15077
|
1933 |
apply (frule order_less_imp_le)
|
paulson@15077
|
1934 |
apply (frule_tac y = y in order_less_imp_le)
|
paulson@15077
|
1935 |
apply (frule arcos_bounded, auto)
|
paulson@15077
|
1936 |
apply (drule_tac y = "arcos y" in order_le_imp_less_or_eq)
|
paulson@15077
|
1937 |
apply (drule_tac [2] y = pi in order_le_imp_less_or_eq, auto)
|
paulson@15077
|
1938 |
apply (drule_tac [!] f = cos in arg_cong, auto)
|
paulson@15077
|
1939 |
done
|
paulson@15077
|
1940 |
|
paulson@15077
|
1941 |
lemma arcos_cos: "[|0 \<le> x; x \<le> pi |] ==> arcos(cos x) = x"
|
paulson@15077
|
1942 |
apply (unfold arcos_def)
|
paulson@15077
|
1943 |
apply (auto intro!: some1_equality cos_total)
|
paulson@15077
|
1944 |
done
|
paulson@15077
|
1945 |
|
paulson@15077
|
1946 |
lemma arcos_cos2: "[|x \<le> 0; -pi \<le> x |] ==> arcos(cos x) = -x"
|
paulson@15077
|
1947 |
apply (unfold arcos_def)
|
paulson@15077
|
1948 |
apply (auto intro!: some1_equality cos_total)
|
paulson@15077
|
1949 |
done
|
paulson@15077
|
1950 |
|
paulson@15077
|
1951 |
lemma arctan [simp]:
|
paulson@15077
|
1952 |
"- (pi/2) < arctan y & arctan y < pi/2 & tan (arctan y) = y"
|
paulson@15077
|
1953 |
apply (cut_tac y = y in tan_total)
|
paulson@15077
|
1954 |
apply (unfold arctan_def)
|
paulson@15077
|
1955 |
apply (fast intro: someI2)
|
paulson@15077
|
1956 |
done
|
paulson@15077
|
1957 |
|
paulson@15077
|
1958 |
lemma tan_arctan: "tan(arctan y) = y"
|
paulson@15077
|
1959 |
by auto
|
paulson@15077
|
1960 |
|
paulson@15077
|
1961 |
lemma arctan_bounded: "- (pi/2) < arctan y & arctan y < pi/2"
|
paulson@15077
|
1962 |
by (auto simp only: arctan)
|
paulson@15077
|
1963 |
|
paulson@15077
|
1964 |
lemma arctan_lbound: "- (pi/2) < arctan y"
|
paulson@15077
|
1965 |
by auto
|
paulson@15077
|
1966 |
|
paulson@15077
|
1967 |
lemma arctan_ubound: "arctan y < pi/2"
|
paulson@15077
|
1968 |
by (auto simp only: arctan)
|
paulson@15077
|
1969 |
|
paulson@15077
|
1970 |
lemma arctan_tan:
|
paulson@15077
|
1971 |
"[|-(pi/2) < x; x < pi/2 |] ==> arctan(tan x) = x"
|
paulson@15077
|
1972 |
apply (unfold arctan_def)
|
paulson@15077
|
1973 |
apply (rule some1_equality)
|
paulson@15077
|
1974 |
apply (rule tan_total, auto)
|
paulson@15077
|
1975 |
done
|
paulson@15077
|
1976 |
|
paulson@15077
|
1977 |
lemma arctan_zero_zero [simp]: "arctan 0 = 0"
|
paulson@15077
|
1978 |
by (insert arctan_tan [of 0], simp)
|
paulson@15077
|
1979 |
|
paulson@15077
|
1980 |
lemma cos_arctan_not_zero [simp]: "cos(arctan x) \<noteq> 0"
|
paulson@15077
|
1981 |
apply (auto simp add: cos_zero_iff)
|
paulson@15077
|
1982 |
apply (case_tac "n")
|
paulson@15077
|
1983 |
apply (case_tac [3] "n")
|
paulson@15077
|
1984 |
apply (cut_tac [2] y = x in arctan_ubound)
|
paulson@15077
|
1985 |
apply (cut_tac [4] y = x in arctan_lbound)
|
paulson@15077
|
1986 |
apply (auto simp add: real_of_nat_Suc left_distrib mult_less_0_iff)
|
paulson@15077
|
1987 |
done
|
paulson@15077
|
1988 |
|
paulson@15077
|
1989 |
lemma tan_sec: "cos x \<noteq> 0 ==> 1 + tan(x) ^ 2 = inverse(cos x) ^ 2"
|
paulson@15077
|
1990 |
apply (rule power_inverse [THEN subst])
|
paulson@15077
|
1991 |
apply (rule_tac c1 = "(cos x)\<twosuperior>" in real_mult_right_cancel [THEN iffD1])
|
paulson@15077
|
1992 |
apply (auto dest: realpow_not_zero
|
paulson@15077
|
1993 |
simp add: power_mult_distrib left_distrib realpow_divide tan_def
|
paulson@15077
|
1994 |
mult_assoc power_inverse [symmetric]
|
paulson@15077
|
1995 |
simp del: realpow_Suc)
|
paulson@15077
|
1996 |
done
|
paulson@15077
|
1997 |
|
paulson@15085
|
1998 |
text{*NEEDED??*}
|
paulson@15085
|
1999 |
lemma [simp]: "sin (xa + 1 / 2 * real (Suc m) * pi) =
|
paulson@15077
|
2000 |
cos (xa + 1 / 2 * real (m) * pi)"
|
paulson@15077
|
2001 |
apply (simp only: cos_add sin_add real_of_nat_Suc left_distrib right_distrib, auto)
|
paulson@15077
|
2002 |
done
|
paulson@15077
|
2003 |
|
paulson@15085
|
2004 |
text{*NEEDED??*}
|
paulson@15085
|
2005 |
lemma [simp]: "sin (xa + real (Suc m) * pi / 2) =
|
paulson@15077
|
2006 |
cos (xa + real (m) * pi / 2)"
|
paulson@15079
|
2007 |
apply (simp only: cos_add sin_add divide_inverse real_of_nat_Suc left_distrib right_distrib, auto)
|
paulson@15077
|
2008 |
done
|
paulson@15077
|
2009 |
|
paulson@15077
|
2010 |
lemma DERIV_sin_add [simp]: "DERIV (%x. sin (x + k)) xa :> cos (xa + k)"
|
paulson@15077
|
2011 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
2012 |
apply (rule_tac f = sin and g = "%x. x + k" in DERIV_chain2)
|
paulson@15077
|
2013 |
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
|
paulson@15077
|
2014 |
apply (simp (no_asm))
|
paulson@15077
|
2015 |
done
|
paulson@15077
|
2016 |
|
paulson@15077
|
2017 |
(* which further simplifies to (- 1 ^ m) !! *)
|
paulson@15077
|
2018 |
lemma sin_cos_npi [simp]: "sin ((real m + 1/2) * pi) = cos (real m * pi)"
|
paulson@15077
|
2019 |
by (auto simp add: right_distrib sin_add left_distrib mult_ac)
|
paulson@15077
|
2020 |
|
paulson@15077
|
2021 |
lemma sin_cos_npi2 [simp]: "sin (real (Suc (2 * n)) * pi / 2) = (-1) ^ n"
|
paulson@15077
|
2022 |
apply (cut_tac m = n in sin_cos_npi)
|
paulson@15079
|
2023 |
apply (simp only: real_of_nat_Suc left_distrib divide_inverse, auto)
|
paulson@15077
|
2024 |
done
|
paulson@15077
|
2025 |
|
paulson@15077
|
2026 |
lemma cos_2npi [simp]: "cos (2 * real (n::nat) * pi) = 1"
|
paulson@15077
|
2027 |
by (simp add: cos_double mult_assoc power_add [symmetric] numeral_2_eq_2)
|
paulson@15077
|
2028 |
|
paulson@15077
|
2029 |
lemma cos_3over2_pi [simp]: "cos (3 / 2 * pi) = 0"
|
paulson@15077
|
2030 |
apply (subgoal_tac "3/2 = (1+1 / 2::real)")
|
paulson@15077
|
2031 |
apply (simp only: left_distrib)
|
paulson@15077
|
2032 |
apply (auto simp add: cos_add mult_ac)
|
paulson@15077
|
2033 |
done
|
paulson@15077
|
2034 |
|
paulson@15077
|
2035 |
lemma sin_2npi [simp]: "sin (2 * real (n::nat) * pi) = 0"
|
paulson@15077
|
2036 |
by (auto simp add: mult_assoc)
|
paulson@15077
|
2037 |
|
paulson@15077
|
2038 |
lemma sin_3over2_pi [simp]: "sin (3 / 2 * pi) = - 1"
|
paulson@15077
|
2039 |
apply (subgoal_tac "3/2 = (1+1 / 2::real)")
|
paulson@15077
|
2040 |
apply (simp only: left_distrib)
|
paulson@15077
|
2041 |
apply (auto simp add: sin_add mult_ac)
|
paulson@15077
|
2042 |
done
|
paulson@15077
|
2043 |
|
paulson@15077
|
2044 |
(*NEEDED??*)
|
paulson@15077
|
2045 |
lemma [simp]: "cos(x + 1 / 2 * real(Suc m) * pi) = -sin (x + 1 / 2 * real m * pi)"
|
paulson@15077
|
2046 |
apply (simp only: cos_add sin_add real_of_nat_Suc right_distrib left_distrib minus_mult_right, auto)
|
paulson@15077
|
2047 |
done
|
paulson@15077
|
2048 |
|
paulson@15077
|
2049 |
(*NEEDED??*)
|
paulson@15077
|
2050 |
lemma [simp]: "cos (x + real(Suc m) * pi / 2) = -sin (x + real m * pi / 2)"
|
paulson@15079
|
2051 |
apply (simp only: cos_add sin_add divide_inverse real_of_nat_Suc left_distrib right_distrib, auto)
|
paulson@15077
|
2052 |
done
|
paulson@15077
|
2053 |
|
paulson@15077
|
2054 |
lemma cos_pi_eq_zero [simp]: "cos (pi * real (Suc (2 * m)) / 2) = 0"
|
paulson@15079
|
2055 |
by (simp only: cos_add sin_add divide_inverse real_of_nat_Suc left_distrib right_distrib, auto)
|
paulson@15077
|
2056 |
|
paulson@15077
|
2057 |
lemma DERIV_cos_add [simp]: "DERIV (%x. cos (x + k)) xa :> - sin (xa + k)"
|
paulson@15077
|
2058 |
apply (rule lemma_DERIV_subst)
|
paulson@15077
|
2059 |
apply (rule_tac f = cos and g = "%x. x + k" in DERIV_chain2)
|
paulson@15077
|
2060 |
apply (best intro!: DERIV_intros intro: DERIV_chain2)+
|
paulson@15077
|
2061 |
apply (simp (no_asm))
|
paulson@15077
|
2062 |
done
|
paulson@15077
|
2063 |
|
paulson@15077
|
2064 |
lemma isCont_cos [simp]: "isCont cos x"
|
paulson@15077
|
2065 |
by (rule DERIV_cos [THEN DERIV_isCont])
|
paulson@15077
|
2066 |
|
paulson@15077
|
2067 |
lemma isCont_sin [simp]: "isCont sin x"
|
paulson@15077
|
2068 |
by (rule DERIV_sin [THEN DERIV_isCont])
|
paulson@15077
|
2069 |
|
paulson@15077
|
2070 |
lemma isCont_exp [simp]: "isCont exp x"
|
paulson@15077
|
2071 |
by (rule DERIV_exp [THEN DERIV_isCont])
|
paulson@15077
|
2072 |
|
paulson@15081
|
2073 |
lemma sin_zero_abs_cos_one: "sin x = 0 ==> \<bar>cos x\<bar> = 1"
|
paulson@15077
|
2074 |
by (auto simp add: sin_zero_iff even_mult_two_ex)
|
paulson@15077
|
2075 |
|
paulson@15077
|
2076 |
lemma exp_eq_one_iff [simp]: "(exp x = 1) = (x = 0)"
|
paulson@15077
|
2077 |
apply auto
|
paulson@15077
|
2078 |
apply (drule_tac f = ln in arg_cong, auto)
|
paulson@15077
|
2079 |
done
|
paulson@15077
|
2080 |
|
paulson@15077
|
2081 |
lemma cos_one_sin_zero: "cos x = 1 ==> sin x = 0"
|
paulson@15077
|
2082 |
by (cut_tac x = x in sin_cos_squared_add3, auto)
|
paulson@15077
|
2083 |
|
paulson@15077
|
2084 |
|
paulson@15077
|
2085 |
lemma real_root_less_mono: "[| 0 \<le> x; x < y |] ==> root(Suc n) x < root(Suc n) y"
|
paulson@15077
|
2086 |
apply (frule order_le_less_trans, assumption)
|
paulson@15077
|
2087 |
apply (frule_tac n1 = n in real_root_pow_pos2 [THEN ssubst])
|
paulson@15077
|
2088 |
apply (rotate_tac 1, assumption)
|
paulson@15077
|
2089 |
apply (frule_tac n1 = n in real_root_pow_pos [THEN ssubst])
|
paulson@15077
|
2090 |
apply (rotate_tac 3, assumption)
|
paulson@15077
|
2091 |
apply (drule_tac y = "root (Suc n) y ^ Suc n" in order_less_imp_le)
|
paulson@15077
|
2092 |
apply (frule_tac n = n in real_root_pos_pos_le)
|
paulson@15077
|
2093 |
apply (frule_tac n = n in real_root_pos_pos)
|
paulson@15077
|
2094 |
apply (drule_tac x = "root (Suc n) x" and y = "root (Suc n) y" in realpow_increasing)
|
paulson@15077
|
2095 |
apply (assumption, assumption)
|
paulson@15077
|
2096 |
apply (drule_tac x = "root (Suc n) x" in order_le_imp_less_or_eq)
|
paulson@15077
|
2097 |
apply auto
|
paulson@15077
|
2098 |
apply (drule_tac f = "%x. x ^ (Suc n) " in arg_cong)
|
paulson@15077
|
2099 |
apply (auto simp add: real_root_pow_pos2 simp del: realpow_Suc)
|
paulson@15077
|
2100 |
done
|
paulson@15077
|
2101 |
|
paulson@15077
|
2102 |
lemma real_root_le_mono: "[| 0 \<le> x; x \<le> y |] ==> root(Suc n) x \<le> root(Suc n) y"
|
paulson@15077
|
2103 |
apply (drule_tac y = y in order_le_imp_less_or_eq)
|
paulson@15077
|
2104 |
apply (auto dest: real_root_less_mono intro: order_less_imp_le)
|
paulson@15077
|
2105 |
done
|
paulson@15077
|
2106 |
|
paulson@15077
|
2107 |
lemma real_root_less_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x < root(Suc n) y) = (x < y)"
|
paulson@15077
|
2108 |
apply (auto intro: real_root_less_mono)
|
paulson@15077
|
2109 |
apply (rule ccontr, drule linorder_not_less [THEN iffD1])
|
paulson@15077
|
2110 |
apply (drule_tac x = y and n = n in real_root_le_mono, auto)
|
paulson@15077
|
2111 |
done
|
paulson@15077
|
2112 |
|
paulson@15077
|
2113 |
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
|
2114 |
apply (auto intro: real_root_le_mono)
|
paulson@15077
|
2115 |
apply (simp (no_asm) add: linorder_not_less [symmetric])
|
paulson@15077
|
2116 |
apply auto
|
paulson@15077
|
2117 |
apply (drule_tac x = y and n = n in real_root_less_mono, auto)
|
paulson@15077
|
2118 |
done
|
paulson@15077
|
2119 |
|
paulson@15077
|
2120 |
lemma real_root_eq_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (root(Suc n) x = root(Suc n) y) = (x = y)"
|
paulson@15077
|
2121 |
apply (auto intro!: order_antisym)
|
paulson@15077
|
2122 |
apply (rule_tac n1 = n in real_root_le_iff [THEN iffD1])
|
paulson@15077
|
2123 |
apply (rule_tac [4] n1 = n in real_root_le_iff [THEN iffD1], auto)
|
paulson@15077
|
2124 |
done
|
paulson@15077
|
2125 |
|
paulson@15077
|
2126 |
lemma real_root_pos_unique: "[| 0 \<le> x; 0 \<le> y; y ^ (Suc n) = x |] ==> root (Suc n) x = y"
|
paulson@15077
|
2127 |
by (auto dest: real_root_pos2 simp del: realpow_Suc)
|
paulson@15077
|
2128 |
|
paulson@15077
|
2129 |
lemma real_root_mult: "[| 0 \<le> x; 0 \<le> y |]
|
paulson@15077
|
2130 |
==> root(Suc n) (x * y) = root(Suc n) x * root(Suc n) y"
|
paulson@15077
|
2131 |
apply (rule real_root_pos_unique)
|
paulson@15077
|
2132 |
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
|
2133 |
done
|
paulson@15077
|
2134 |
|
paulson@15077
|
2135 |
lemma real_root_inverse: "0 \<le> x ==> (root(Suc n) (inverse x) = inverse(root(Suc n) x))"
|
paulson@15077
|
2136 |
apply (rule real_root_pos_unique)
|
paulson@15077
|
2137 |
apply (auto intro: real_root_pos_pos_le simp add: power_inverse [symmetric] real_root_pow_pos2 simp del: realpow_Suc)
|
paulson@15077
|
2138 |
done
|
paulson@15077
|
2139 |
|
paulson@15077
|
2140 |
lemma real_root_divide:
|
paulson@15077
|
2141 |
"[| 0 \<le> x; 0 \<le> y |]
|
paulson@15077
|
2142 |
==> (root(Suc n) (x / y) = root(Suc n) x / root(Suc n) y)"
|
paulson@15077
|
2143 |
apply (unfold real_divide_def)
|
paulson@15077
|
2144 |
apply (auto simp add: real_root_mult real_root_inverse)
|
paulson@15077
|
2145 |
done
|
paulson@15077
|
2146 |
|
paulson@15077
|
2147 |
lemma real_sqrt_less_mono: "[| 0 \<le> x; x < y |] ==> sqrt(x) < sqrt(y)"
|
paulson@15077
|
2148 |
apply (unfold sqrt_def)
|
paulson@15077
|
2149 |
apply (auto intro: real_root_less_mono)
|
paulson@15077
|
2150 |
done
|
paulson@15077
|
2151 |
|
paulson@15077
|
2152 |
lemma real_sqrt_le_mono: "[| 0 \<le> x; x \<le> y |] ==> sqrt(x) \<le> sqrt(y)"
|
paulson@15077
|
2153 |
apply (unfold sqrt_def)
|
paulson@15077
|
2154 |
apply (auto intro: real_root_le_mono)
|
paulson@15077
|
2155 |
done
|
paulson@15077
|
2156 |
|
paulson@15077
|
2157 |
lemma real_sqrt_less_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) < sqrt(y)) = (x < y)"
|
paulson@15077
|
2158 |
by (unfold sqrt_def, auto)
|
paulson@15077
|
2159 |
|
paulson@15077
|
2160 |
lemma real_sqrt_le_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) \<le> sqrt(y)) = (x \<le> y)"
|
paulson@15077
|
2161 |
by (unfold sqrt_def, auto)
|
paulson@15077
|
2162 |
|
paulson@15077
|
2163 |
lemma real_sqrt_eq_iff [simp]: "[| 0 \<le> x; 0 \<le> y |] ==> (sqrt(x) = sqrt(y)) = (x = y)"
|
paulson@15077
|
2164 |
by (unfold sqrt_def, auto)
|
paulson@15077
|
2165 |
|
paulson@15077
|
2166 |
lemma real_sqrt_sos_less_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"
|
paulson@15077
|
2167 |
apply (rule real_sqrt_one [THEN subst], safe)
|
paulson@15077
|
2168 |
apply (rule_tac [2] real_sqrt_less_mono)
|
paulson@15077
|
2169 |
apply (drule real_sqrt_less_iff [THEN [2] rev_iffD1], auto)
|
paulson@15077
|
2170 |
done
|
paulson@15077
|
2171 |
|
paulson@15077
|
2172 |
lemma real_sqrt_sos_eq_one_iff [simp]: "(sqrt(x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"
|
paulson@15077
|
2173 |
apply (rule real_sqrt_one [THEN subst], safe)
|
paulson@15077
|
2174 |
apply (drule real_sqrt_eq_iff [THEN [2] rev_iffD1], auto)
|
paulson@15077
|
2175 |
done
|
paulson@15077
|
2176 |
|
paulson@15077
|
2177 |
lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
|
paulson@15077
|
2178 |
apply (unfold real_divide_def)
|
paulson@15077
|
2179 |
apply (case_tac "r=0")
|
paulson@15077
|
2180 |
apply (auto simp add: inverse_mult_distrib mult_ac)
|
paulson@15077
|
2181 |
done
|
paulson@15077
|
2182 |
|
paulson@15077
|
2183 |
|
paulson@15077
|
2184 |
subsection{*Theorems About Sqrt, Transcendental Functions for Complex*}
|
paulson@15077
|
2185 |
|
paulson@15077
|
2186 |
lemma lemma_real_divide_sqrt:
|
paulson@15077
|
2187 |
"0 < x ==> 0 \<le> x/(sqrt (x * x + y * y))"
|
paulson@15077
|
2188 |
apply (unfold real_divide_def)
|
paulson@15077
|
2189 |
apply (rule real_mult_order [THEN order_less_imp_le], assumption)
|
paulson@15077
|
2190 |
apply (subgoal_tac "0 < inverse (sqrt (x\<twosuperior> + y\<twosuperior>))")
|
paulson@15077
|
2191 |
apply (simp add: numeral_2_eq_2)
|
paulson@15077
|
2192 |
apply (simp add: real_sqrt_sum_squares_ge1 [THEN [2] order_less_le_trans])
|
paulson@15077
|
2193 |
done
|
paulson@15077
|
2194 |
|
paulson@15077
|
2195 |
lemma lemma_real_divide_sqrt_ge_minus_one:
|
paulson@15077
|
2196 |
"0 < x ==> -1 \<le> x/(sqrt (x * x + y * y))"
|
paulson@15077
|
2197 |
apply (rule real_le_trans)
|
paulson@15077
|
2198 |
apply (rule_tac [2] lemma_real_divide_sqrt, auto)
|
paulson@15077
|
2199 |
done
|
paulson@15077
|
2200 |
|
paulson@15077
|
2201 |
lemma real_sqrt_sum_squares_gt_zero1: "x < 0 ==> 0 < sqrt (x * x + y * y)"
|
paulson@15077
|
2202 |
apply (rule real_sqrt_gt_zero)
|
paulson@15077
|
2203 |
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith)
|
paulson@15077
|
2204 |
apply (auto simp add: zero_less_mult_iff)
|
paulson@15077
|
2205 |
done
|
paulson@15077
|
2206 |
|
paulson@15077
|
2207 |
lemma real_sqrt_sum_squares_gt_zero2: "0 < x ==> 0 < sqrt (x * x + y * y)"
|
paulson@15077
|
2208 |
apply (rule real_sqrt_gt_zero)
|
paulson@15077
|
2209 |
apply (subgoal_tac "0 < x*x & 0 \<le> y*y", arith)
|
paulson@15077
|
2210 |
apply (auto simp add: zero_less_mult_iff)
|
paulson@15077
|
2211 |
done
|
paulson@15077
|
2212 |
|
paulson@15077
|
2213 |
lemma real_sqrt_sum_squares_gt_zero3: "x \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
|
paulson@15077
|
2214 |
apply (cut_tac x = x and y = 0 in linorder_less_linear)
|
paulson@15077
|
2215 |
apply (auto intro: real_sqrt_sum_squares_gt_zero2 real_sqrt_sum_squares_gt_zero1 simp add: numeral_2_eq_2)
|
paulson@15077
|
2216 |
done
|
paulson@15077
|
2217 |
|
paulson@15077
|
2218 |
lemma real_sqrt_sum_squares_gt_zero3a: "y \<noteq> 0 ==> 0 < sqrt(x\<twosuperior> + y\<twosuperior>)"
|
paulson@15077
|
2219 |
apply (drule_tac y = x in real_sqrt_sum_squares_gt_zero3)
|
paulson@15077
|
2220 |
apply (auto simp add: real_add_commute)
|
paulson@15077
|
2221 |
done
|
paulson@15077
|
2222 |
|
paulson@15077
|
2223 |
lemma real_sqrt_sum_squares_eq_cancel [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = x ==> y = 0"
|
paulson@15077
|
2224 |
by (drule_tac f = "%x. x\<twosuperior>" in arg_cong, auto)
|
paulson@15077
|
2225 |
|
paulson@15077
|
2226 |
lemma real_sqrt_sum_squares_eq_cancel2 [simp]: "sqrt(x\<twosuperior> + y\<twosuperior>) = y ==> x = 0"
|
paulson@15077
|
2227 |
apply (rule_tac x = y in real_sqrt_sum_squares_eq_cancel)
|
paulson@15077
|
2228 |
apply (simp add: real_add_commute)
|
paulson@15077
|
2229 |
done
|
paulson@15077
|
2230 |
|
paulson@15077
|
2231 |
lemma lemma_real_divide_sqrt_le_one: "x < 0 ==> x/(sqrt (x * x + y * y)) \<le> 1"
|
paulson@15077
|
2232 |
by (insert lemma_real_divide_sqrt_ge_minus_one [of "-x" y], simp)
|
paulson@15077
|
2233 |
|
paulson@15077
|
2234 |
lemma lemma_real_divide_sqrt_ge_minus_one2:
|
paulson@15077
|
2235 |
"x < 0 ==> -1 \<le> x/(sqrt (x * x + y * y))"
|
paulson@15077
|
2236 |
apply (case_tac "y = 0", auto)
|
paulson@15077
|
2237 |
apply (frule abs_minus_eqI2)
|
paulson@15077
|
2238 |
apply (auto simp add: inverse_minus_eq)
|
paulson@15077
|
2239 |
apply (rule order_less_imp_le)
|
paulson@15077
|
2240 |
apply (rule_tac z1 = "sqrt (x * x + y * y) " in real_mult_less_iff1 [THEN iffD1])
|
paulson@15077
|
2241 |
apply (frule_tac [2] y2 = y in
|
paulson@15077
|
2242 |
real_sqrt_sum_squares_gt_zero1 [THEN real_not_refl2, THEN not_sym])
|
paulson@15077
|
2243 |
apply (auto intro: real_sqrt_sum_squares_gt_zero1 simp add: mult_ac)
|
paulson@15077
|
2244 |
apply (cut_tac x = "-x" and y = y in real_sqrt_sum_squares_ge1)
|
paulson@15077
|
2245 |
apply (drule order_le_less [THEN iffD1], safe)
|
paulson@15077
|
2246 |
apply (simp add: numeral_2_eq_2)
|
paulson@15077
|
2247 |
apply (drule sym [THEN real_sqrt_sum_squares_eq_cancel], simp)
|
paulson@15077
|
2248 |
done
|
paulson@15077
|
2249 |
|
paulson@15077
|
2250 |
lemma lemma_real_divide_sqrt_le_one2: "0 < x ==> x/(sqrt (x * x + y * y)) \<le> 1"
|
paulson@15077
|
2251 |
by (cut_tac x = "-x" and y = y in lemma_real_divide_sqrt_ge_minus_one2, auto)
|
paulson@15077
|
2252 |
|
paulson@15077
|
2253 |
|
paulson@15077
|
2254 |
lemma cos_x_y_ge_minus_one: "-1 \<le> x / sqrt (x * x + y * y)"
|
paulson@15077
|
2255 |
apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
|
paulson@15077
|
2256 |
apply (rule lemma_real_divide_sqrt_ge_minus_one2)
|
paulson@15077
|
2257 |
apply (rule_tac [3] lemma_real_divide_sqrt_ge_minus_one, auto)
|
paulson@15077
|
2258 |
done
|
paulson@15077
|
2259 |
|
paulson@15077
|
2260 |
lemma cos_x_y_ge_minus_one1a [simp]: "-1 \<le> y / sqrt (x * x + y * y)"
|
paulson@15077
|
2261 |
apply (cut_tac x = y and y = x in cos_x_y_ge_minus_one)
|
paulson@15077
|
2262 |
apply (simp add: real_add_commute)
|
paulson@15077
|
2263 |
done
|
paulson@15077
|
2264 |
|
paulson@15077
|
2265 |
lemma cos_x_y_le_one [simp]: "x / sqrt (x * x + y * y) \<le> 1"
|
paulson@15077
|
2266 |
apply (cut_tac x = x and y = 0 in linorder_less_linear, safe)
|
paulson@15077
|
2267 |
apply (rule lemma_real_divide_sqrt_le_one)
|
paulson@15077
|
2268 |
apply (rule_tac [3] lemma_real_divide_sqrt_le_one2, auto)
|
paulson@15077
|
2269 |
done
|
paulson@15077
|
2270 |
|
paulson@15077
|
2271 |
lemma cos_x_y_le_one2 [simp]: "y / sqrt (x * x + y * y) \<le> 1"
|
paulson@15077
|
2272 |
apply (cut_tac x = y and y = x in cos_x_y_le_one)
|
paulson@15077
|
2273 |
apply (simp add: real_add_commute)
|
paulson@15077
|
2274 |
done
|
paulson@15077
|
2275 |
|
paulson@15077
|
2276 |
declare cos_arcos [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp]
|
paulson@15077
|
2277 |
declare arcos_bounded [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp]
|
paulson@15077
|
2278 |
|
paulson@15077
|
2279 |
declare cos_arcos [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp]
|
paulson@15077
|
2280 |
declare arcos_bounded [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2, simp]
|
paulson@15077
|
2281 |
|
paulson@15077
|
2282 |
lemma cos_abs_x_y_ge_minus_one [simp]:
|
paulson@15077
|
2283 |
"-1 \<le> \<bar>x\<bar> / sqrt (x * x + y * y)"
|
paulson@15077
|
2284 |
apply (cut_tac x = x and y = 0 in linorder_less_linear)
|
paulson@15077
|
2285 |
apply (auto simp add: abs_minus_eqI2 abs_eqI2)
|
paulson@15077
|
2286 |
apply (drule lemma_real_divide_sqrt_ge_minus_one, force)
|
paulson@15077
|
2287 |
done
|
paulson@15077
|
2288 |
|
paulson@15077
|
2289 |
lemma cos_abs_x_y_le_one [simp]: "\<bar>x\<bar> / sqrt (x * x + y * y) \<le> 1"
|
paulson@15077
|
2290 |
apply (cut_tac x = x and y = 0 in linorder_less_linear)
|
paulson@15077
|
2291 |
apply (auto simp add: abs_minus_eqI2 abs_eqI2)
|
paulson@15077
|
2292 |
apply (drule lemma_real_divide_sqrt_ge_minus_one2, force)
|
paulson@15077
|
2293 |
done
|
paulson@15077
|
2294 |
|
paulson@15077
|
2295 |
declare cos_arcos [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp]
|
paulson@15077
|
2296 |
declare arcos_bounded [OF cos_abs_x_y_ge_minus_one cos_abs_x_y_le_one, simp]
|
paulson@15077
|
2297 |
|
paulson@15077
|
2298 |
lemma minus_pi_less_zero: "-pi < 0"
|
paulson@15077
|
2299 |
apply (simp (no_asm))
|
paulson@15077
|
2300 |
done
|
paulson@15077
|
2301 |
declare minus_pi_less_zero [simp]
|
paulson@15077
|
2302 |
declare minus_pi_less_zero [THEN order_less_imp_le, simp]
|
paulson@15077
|
2303 |
|
paulson@15077
|
2304 |
lemma arcos_ge_minus_pi: "[| -1 \<le> y; y \<le> 1 |] ==> -pi \<le> arcos y"
|
paulson@15077
|
2305 |
apply (rule real_le_trans)
|
paulson@15077
|
2306 |
apply (rule_tac [2] arcos_lbound, auto)
|
paulson@15077
|
2307 |
done
|
paulson@15077
|
2308 |
|
paulson@15077
|
2309 |
declare arcos_ge_minus_pi [OF cos_x_y_ge_minus_one cos_x_y_le_one, simp]
|
paulson@15077
|
2310 |
|
paulson@15077
|
2311 |
(* How tedious! *)
|
paulson@15077
|
2312 |
lemma lemma_divide_rearrange:
|
paulson@15077
|
2313 |
"[| x + (y::real) \<noteq> 0; 1 - z = x/(x + y) |] ==> z = y/(x + y)"
|
paulson@15077
|
2314 |
apply (rule_tac c1 = "x + y" in real_mult_right_cancel [THEN iffD1])
|
paulson@15077
|
2315 |
apply (frule_tac [2] c1 = "x + y" in real_mult_right_cancel [THEN iffD2])
|
paulson@15077
|
2316 |
prefer 2 apply assumption
|
paulson@15077
|
2317 |
apply (rotate_tac [2] 2)
|
paulson@15077
|
2318 |
apply (drule_tac [2] mult_assoc [THEN subst])
|
paulson@15077
|
2319 |
apply (rotate_tac [2] 2)
|
paulson@15077
|
2320 |
apply (frule_tac [2] left_inverse [THEN subst])
|
paulson@15077
|
2321 |
prefer 2 apply assumption
|
paulson@15077
|
2322 |
apply (erule_tac [2] V = " (1 - z) * (x + y) = x / (x + y) * (x + y) " in thin_rl)
|
paulson@15077
|
2323 |
apply (erule_tac [2] V = "1 - z = x / (x + y) " in thin_rl)
|
paulson@15077
|
2324 |
apply (auto simp add: mult_assoc)
|
paulson@15077
|
2325 |
apply (auto simp add: right_distrib left_diff_distrib)
|
paulson@15077
|
2326 |
done
|
paulson@15077
|
2327 |
|
paulson@15077
|
2328 |
lemma lemma_cos_sin_eq:
|
paulson@15077
|
2329 |
"[| 0 < x * x + y * y;
|
paulson@15077
|
2330 |
1 - (sin xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2 |]
|
paulson@15077
|
2331 |
==> (sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2"
|
paulson@15077
|
2332 |
by (auto intro: lemma_divide_rearrange
|
paulson@15077
|
2333 |
simp add: realpow_divide power2_eq_square [symmetric])
|
paulson@15077
|
2334 |
|
paulson@15077
|
2335 |
|
paulson@15077
|
2336 |
lemma lemma_sin_cos_eq:
|
paulson@15077
|
2337 |
"[| 0 < x * x + y * y;
|
paulson@15077
|
2338 |
1 - (cos xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2 |]
|
paulson@15077
|
2339 |
==> (cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2"
|
paulson@15077
|
2340 |
apply (auto simp add: realpow_divide power2_eq_square [symmetric])
|
paulson@15085
|
2341 |
apply (subst add_commute)
|
paulson@15085
|
2342 |
apply (rule lemma_divide_rearrange, simp add: real_add_eq_0_iff)
|
paulson@15077
|
2343 |
apply (simp add: add_commute)
|
paulson@15077
|
2344 |
done
|
paulson@15077
|
2345 |
|
paulson@15077
|
2346 |
lemma sin_x_y_disj:
|
paulson@15077
|
2347 |
"[| x \<noteq> 0;
|
paulson@15077
|
2348 |
cos xa = x / sqrt (x * x + y * y)
|
paulson@15077
|
2349 |
|] ==> sin xa = y / sqrt (x * x + y * y) |
|
paulson@15077
|
2350 |
sin xa = - y / sqrt (x * x + y * y)"
|
paulson@15077
|
2351 |
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
|
paulson@15077
|
2352 |
apply (frule_tac y = y in real_sum_square_gt_zero)
|
paulson@15077
|
2353 |
apply (simp add: cos_squared_eq)
|
paulson@15077
|
2354 |
apply (subgoal_tac "(sin xa)\<twosuperior> = (y / sqrt (x * x + y * y)) ^ 2")
|
paulson@15077
|
2355 |
apply (rule_tac [2] lemma_cos_sin_eq)
|
paulson@15077
|
2356 |
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
|
paulson@15077
|
2357 |
done
|
paulson@15077
|
2358 |
|
paulson@15077
|
2359 |
lemma lemma_cos_not_eq_zero: "x \<noteq> 0 ==> x / sqrt (x * x + y * y) \<noteq> 0"
|
paulson@15077
|
2360 |
apply (unfold real_divide_def)
|
paulson@15077
|
2361 |
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
|
2362 |
apply (auto simp add: power2_eq_square)
|
paulson@15077
|
2363 |
done
|
paulson@15077
|
2364 |
|
paulson@15077
|
2365 |
lemma cos_x_y_disj: "[| x \<noteq> 0;
|
paulson@15077
|
2366 |
sin xa = y / sqrt (x * x + y * y)
|
paulson@15077
|
2367 |
|] ==> cos xa = x / sqrt (x * x + y * y) |
|
paulson@15077
|
2368 |
cos xa = - x / sqrt (x * x + y * y)"
|
paulson@15077
|
2369 |
apply (drule_tac f = "%x. x\<twosuperior>" in arg_cong)
|
paulson@15077
|
2370 |
apply (frule_tac y = y in real_sum_square_gt_zero)
|
paulson@15077
|
2371 |
apply (simp add: sin_squared_eq del: realpow_Suc)
|
paulson@15077
|
2372 |
apply (subgoal_tac "(cos xa)\<twosuperior> = (x / sqrt (x * x + y * y)) ^ 2")
|
paulson@15077
|
2373 |
apply (rule_tac [2] lemma_sin_cos_eq)
|
paulson@15077
|
2374 |
apply (auto simp add: realpow_two_disj numeral_2_eq_2 simp del: realpow_Suc)
|
paulson@15077
|
2375 |
done
|
paulson@15077
|
2376 |
|
paulson@15077
|
2377 |
lemma real_sqrt_divide_less_zero: "0 < y ==> - y / sqrt (x * x + y * y) < 0"
|
paulson@15077
|
2378 |
apply (case_tac "x = 0")
|
paulson@15077
|
2379 |
apply (auto simp add: abs_eqI2)
|
paulson@15077
|
2380 |
apply (drule_tac y = y in real_sqrt_sum_squares_gt_zero3)
|
paulson@15079
|
2381 |
apply (auto simp add: zero_less_mult_iff divide_inverse power2_eq_square)
|
paulson@15077
|
2382 |
done
|
paulson@15077
|
2383 |
|
paulson@15077
|
2384 |
lemma polar_ex1: "[| x \<noteq> 0; 0 < y |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
|
paulson@15077
|
2385 |
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>) " in exI)
|
paulson@15077
|
2386 |
apply (rule_tac x = "arcos (x / sqrt (x * x + y * y))" in exI)
|
paulson@15077
|
2387 |
apply auto
|
paulson@15077
|
2388 |
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
|
paulson@15077
|
2389 |
apply (auto simp add: power2_eq_square)
|
paulson@15077
|
2390 |
apply (unfold arcos_def)
|
paulson@15077
|
2391 |
apply (cut_tac x1 = x and y1 = y
|
paulson@15077
|
2392 |
in cos_total [OF cos_x_y_ge_minus_one cos_x_y_le_one])
|
paulson@15077
|
2393 |
apply (rule someI2_ex, blast)
|
paulson@15077
|
2394 |
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
|
2395 |
apply (frule sin_x_y_disj, blast)
|
paulson@15077
|
2396 |
apply (drule_tac y2 = y in real_sqrt_sum_squares_gt_zero3 [THEN real_not_refl2, THEN not_sym])
|
paulson@15077
|
2397 |
apply (auto simp add: power2_eq_square)
|
paulson@15077
|
2398 |
apply (drule sin_ge_zero, assumption)
|
paulson@15077
|
2399 |
apply (drule_tac x = x in real_sqrt_divide_less_zero, auto)
|
paulson@15077
|
2400 |
done
|
paulson@15077
|
2401 |
|
paulson@15077
|
2402 |
lemma real_sum_squares_cancel2a: "x * x = -(y * y) ==> y = (0::real)"
|
paulson@15085
|
2403 |
by (auto intro: real_sum_squares_cancel iff: real_add_eq_0_iff)
|
paulson@15077
|
2404 |
|
paulson@15077
|
2405 |
lemma polar_ex2: "[| x \<noteq> 0; y < 0 |] ==> \<exists>r a. x = r * cos a & y = r * sin a"
|
paulson@15077
|
2406 |
apply (cut_tac x = 0 and y = x in linorder_less_linear, auto)
|
paulson@15077
|
2407 |
apply (rule_tac x = "sqrt (x\<twosuperior> + y\<twosuperior>) " in exI)
|
paulson@15077
|
2408 |
apply (rule_tac x = "arcsin (y / sqrt (x * x + y * y))" in exI)
|
paulson@15085
|
2409 |
apply (auto dest: real_sum_squares_cancel2a
|
paulson@15085
|
2410 |
simp add: power2_eq_square real_0_le_add_iff real_add_eq_0_iff)
|
paulson@15077
|
2411 |
apply (unfold arcsin_def)
|
paulson@15077
|
2412 |
apply (cut_tac x1 = x and y1 = y
|
paulson@15077
|
2413 |
in sin_total [OF cos_x_y_ge_minus_one1a cos_x_y_le_one2])
|
paulson@15077
|
2414 |
apply (rule someI2_ex, blast)
|
paulson@15077
|
2415 |
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@15085
|
2416 |
apply (cut_tac x=x and y=y in cos_x_y_disj, simp, blast)
|
paulson@15085
|
2417 |
apply (auto simp add: real_0_le_add_iff real_add_eq_0_iff)
|
paulson@15077
|
2418 |
apply (drule cos_ge_zero, force)
|
paulson@15077
|
2419 |
apply (drule_tac x = y in real_sqrt_divide_less_zero)
|
paulson@15085
|
2420 |
apply (auto simp add: add_commute)
|
paulson@15077
|
2421 |
apply (insert polar_ex1 [of x "-y"], simp, clarify)
|
paulson@15077
|
2422 |
apply (rule_tac x = r in exI)
|
paulson@15077
|
2423 |
apply (rule_tac x = "-a" in exI, simp)
|
paulson@15077
|
2424 |
done
|
paulson@15077
|
2425 |
|
paulson@15077
|
2426 |
lemma polar_Ex: "\<exists>r a. x = r * cos a & y = r * sin a"
|
paulson@15077
|
2427 |
apply (case_tac "x = 0", auto)
|
paulson@15077
|
2428 |
apply (rule_tac x = y in exI)
|
paulson@15077
|
2429 |
apply (rule_tac x = "pi/2" in exI, auto)
|
paulson@15077
|
2430 |
apply (cut_tac x = 0 and y = y in linorder_less_linear, auto)
|
paulson@15077
|
2431 |
apply (rule_tac [2] x = x in exI)
|
paulson@15077
|
2432 |
apply (rule_tac [2] x = 0 in exI, auto)
|
paulson@15077
|
2433 |
apply (blast intro: polar_ex1 polar_ex2)+
|
paulson@15077
|
2434 |
done
|
paulson@15077
|
2435 |
|
paulson@15077
|
2436 |
lemma real_sqrt_ge_abs1 [simp]: "\<bar>x\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
|
paulson@15077
|
2437 |
apply (rule_tac n = 1 in realpow_increasing)
|
paulson@15077
|
2438 |
apply (auto simp add: numeral_2_eq_2 [symmetric])
|
paulson@15077
|
2439 |
done
|
paulson@15077
|
2440 |
|
paulson@15077
|
2441 |
lemma real_sqrt_ge_abs2 [simp]: "\<bar>y\<bar> \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
|
paulson@15077
|
2442 |
apply (rule real_add_commute [THEN subst])
|
paulson@15077
|
2443 |
apply (rule real_sqrt_ge_abs1)
|
paulson@15077
|
2444 |
done
|
paulson@15077
|
2445 |
declare real_sqrt_ge_abs1 [simp] real_sqrt_ge_abs2 [simp]
|
paulson@15077
|
2446 |
|
paulson@15077
|
2447 |
lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
|
paulson@15077
|
2448 |
by (auto intro: real_sqrt_gt_zero)
|
paulson@15077
|
2449 |
|
paulson@15077
|
2450 |
lemma real_sqrt_two_ge_zero [simp]: "0 \<le> sqrt 2"
|
paulson@15077
|
2451 |
by (auto intro: real_sqrt_ge_zero)
|
paulson@15077
|
2452 |
|
paulson@15077
|
2453 |
lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
|
paulson@15077
|
2454 |
apply (rule order_less_le_trans [of _ "7/5"], simp)
|
paulson@15077
|
2455 |
apply (rule_tac n = 1 in realpow_increasing)
|
paulson@15077
|
2456 |
prefer 3 apply (simp add: numeral_2_eq_2 [symmetric] del: realpow_Suc)
|
paulson@15077
|
2457 |
apply (simp_all add: numeral_2_eq_2)
|
paulson@15077
|
2458 |
done
|
paulson@15077
|
2459 |
|
paulson@15077
|
2460 |
lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
|
paulson@15077
|
2461 |
apply (rule_tac z1 = "inverse u" in real_mult_less_iff1 [THEN iffD1], auto)
|
paulson@15077
|
2462 |
apply (rule_tac z1 = "sqrt 2" in real_mult_less_iff1 [THEN iffD1], auto)
|
paulson@15077
|
2463 |
done
|
paulson@15077
|
2464 |
|
paulson@15077
|
2465 |
lemma four_x_squared:
|
paulson@15077
|
2466 |
fixes x::real
|
paulson@15077
|
2467 |
shows "4 * x\<twosuperior> = (2 * x)\<twosuperior>"
|
paulson@15077
|
2468 |
by (simp add: power2_eq_square)
|
paulson@15077
|
2469 |
|
paulson@15077
|
2470 |
|
paulson@15077
|
2471 |
text{*Needed for the infinitely close relation over the nonstandard
|
paulson@15077
|
2472 |
complex numbers*}
|
paulson@15077
|
2473 |
lemma lemma_sqrt_hcomplex_capprox:
|
paulson@15077
|
2474 |
"[| 0 < u; x < u/2; y < u/2; 0 \<le> x; 0 \<le> y |] ==> sqrt (x\<twosuperior> + y\<twosuperior>) < u"
|
paulson@15077
|
2475 |
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
|
paulson@15077
|
2476 |
apply (erule_tac [2] lemma_real_divide_sqrt_less)
|
paulson@15077
|
2477 |
apply (rule_tac n = 1 in realpow_increasing)
|
paulson@15077
|
2478 |
apply (auto simp add: real_0_le_divide_iff realpow_divide numeral_2_eq_2 [symmetric]
|
paulson@15077
|
2479 |
simp del: realpow_Suc)
|
paulson@15077
|
2480 |
apply (rule_tac t = "u\<twosuperior>" in real_sum_of_halves [THEN subst])
|
paulson@15077
|
2481 |
apply (rule add_mono)
|
paulson@15077
|
2482 |
apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono)
|
paulson@15077
|
2483 |
done
|
paulson@15077
|
2484 |
|
paulson@15077
|
2485 |
declare real_sqrt_sum_squares_ge_zero [THEN abs_eqI1, simp]
|
paulson@15077
|
2486 |
|
paulson@15077
|
2487 |
|
paulson@15077
|
2488 |
subsection{*A Few Theorems Involving Ln, Derivatives, etc.*}
|
paulson@15077
|
2489 |
|
paulson@15077
|
2490 |
lemma lemma_DERIV_ln:
|
paulson@15077
|
2491 |
"DERIV ln z :> l ==> DERIV (%x. exp (ln x)) z :> exp (ln z) * l"
|
paulson@15077
|
2492 |
by (erule DERIV_fun_exp)
|
paulson@15077
|
2493 |
|
paulson@15077
|
2494 |
lemma STAR_exp_ln: "0 < z ==> ( *f* (%x. exp (ln x))) z = z"
|
paulson@15077
|
2495 |
apply (rule_tac z = z in eq_Abs_hypreal)
|
paulson@15077
|
2496 |
apply (auto simp add: starfun hypreal_zero_def hypreal_less)
|
paulson@15077
|
2497 |
done
|
paulson@15077
|
2498 |
|
paulson@15077
|
2499 |
lemma hypreal_add_Infinitesimal_gt_zero: "[|e : Infinitesimal; 0 < x |] ==> 0 < hypreal_of_real x + e"
|
paulson@15077
|
2500 |
apply (rule_tac c1 = "-e" in add_less_cancel_right [THEN iffD1])
|
paulson@15077
|
2501 |
apply (auto intro: Infinitesimal_less_SReal)
|
paulson@15077
|
2502 |
done
|
paulson@15077
|
2503 |
|
paulson@15077
|
2504 |
lemma NSDERIV_exp_ln_one: "0 < z ==> NSDERIV (%x. exp (ln x)) z :> 1"
|
paulson@15077
|
2505 |
apply (unfold nsderiv_def NSLIM_def, auto)
|
paulson@15077
|
2506 |
apply (rule ccontr)
|
paulson@15077
|
2507 |
apply (subgoal_tac "0 < hypreal_of_real z + h")
|
paulson@15077
|
2508 |
apply (drule STAR_exp_ln)
|
paulson@15077
|
2509 |
apply (rule_tac [2] hypreal_add_Infinitesimal_gt_zero)
|
paulson@15077
|
2510 |
apply (subgoal_tac "h/h = 1")
|
paulson@15077
|
2511 |
apply (auto simp add: exp_ln_iff [symmetric] simp del: exp_ln_iff)
|
paulson@15077
|
2512 |
done
|
paulson@15077
|
2513 |
|
paulson@15077
|
2514 |
lemma DERIV_exp_ln_one: "0 < z ==> DERIV (%x. exp (ln x)) z :> 1"
|
paulson@15077
|
2515 |
by (auto intro: NSDERIV_exp_ln_one simp add: NSDERIV_DERIV_iff [symmetric])
|
paulson@15077
|
2516 |
|
paulson@15077
|
2517 |
lemma lemma_DERIV_ln2: "[| 0 < z; DERIV ln z :> l |] ==> exp (ln z) * l = 1"
|
paulson@15077
|
2518 |
apply (rule DERIV_unique)
|
paulson@15077
|
2519 |
apply (rule lemma_DERIV_ln)
|
paulson@15077
|
2520 |
apply (rule_tac [2] DERIV_exp_ln_one, auto)
|
paulson@15077
|
2521 |
done
|
paulson@15077
|
2522 |
|
paulson@15077
|
2523 |
lemma lemma_DERIV_ln3: "[| 0 < z; DERIV ln z :> l |] ==> l = 1/(exp (ln z))"
|
paulson@15077
|
2524 |
apply (rule_tac c1 = "exp (ln z) " in real_mult_left_cancel [THEN iffD1])
|
paulson@15077
|
2525 |
apply (auto intro: lemma_DERIV_ln2)
|
paulson@15077
|
2526 |
done
|
paulson@15077
|
2527 |
|
paulson@15077
|
2528 |
lemma lemma_DERIV_ln4: "[| 0 < z; DERIV ln z :> l |] ==> l = 1/z"
|
paulson@15077
|
2529 |
apply (rule_tac t = z in exp_ln_iff [THEN iffD2, THEN subst])
|
paulson@15077
|
2530 |
apply (auto intro: lemma_DERIV_ln3)
|
paulson@15077
|
2531 |
done
|
paulson@15077
|
2532 |
|
paulson@15077
|
2533 |
(* need to rename second isCont_inverse *)
|
paulson@15077
|
2534 |
|
paulson@15077
|
2535 |
lemma isCont_inv_fun: "[| 0 < d; \<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
|
paulson@15077
|
2536 |
\<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
|
paulson@15077
|
2537 |
==> isCont g (f x)"
|
paulson@15077
|
2538 |
apply (simp (no_asm) add: isCont_iff LIM_def)
|
paulson@15077
|
2539 |
apply safe
|
paulson@15077
|
2540 |
apply (drule_tac ?d1.0 = r in real_lbound_gt_zero)
|
paulson@15077
|
2541 |
apply (assumption, safe)
|
paulson@15077
|
2542 |
apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> (g (f z) = z) ")
|
paulson@15077
|
2543 |
prefer 2 apply force
|
paulson@15077
|
2544 |
apply (subgoal_tac "\<forall>z. \<bar>z - x\<bar> \<le> e --> isCont f z")
|
paulson@15077
|
2545 |
prefer 2 apply force
|
paulson@15077
|
2546 |
apply (drule_tac d = e in isCont_inj_range)
|
paulson@15077
|
2547 |
prefer 2 apply (assumption, assumption, safe)
|
paulson@15077
|
2548 |
apply (rule_tac x = ea in exI, auto)
|
paulson@15085
|
2549 |
apply (drule_tac x = "f (x) + xa" and P = "%y. \<bar>y - f x\<bar> \<le> ea \<longrightarrow> (\<exists>z. \<bar>z - x\<bar> \<le> e \<and> f z = y)" in spec)
|
paulson@15077
|
2550 |
apply auto
|
paulson@15077
|
2551 |
apply (drule sym, auto, arith)
|
paulson@15077
|
2552 |
done
|
paulson@15077
|
2553 |
|
paulson@15077
|
2554 |
lemma isCont_inv_fun_inv:
|
paulson@15077
|
2555 |
"[| 0 < d;
|
paulson@15077
|
2556 |
\<forall>z. \<bar>z - x\<bar> \<le> d --> g(f(z)) = z;
|
paulson@15077
|
2557 |
\<forall>z. \<bar>z - x\<bar> \<le> d --> isCont f z |]
|
paulson@15077
|
2558 |
==> \<exists>e. 0 < e &
|
paulson@15081
|
2559 |
(\<forall>y. 0 < \<bar>y - f(x)\<bar> & \<bar>y - f(x)\<bar> < e --> f(g(y)) = y)"
|
paulson@15077
|
2560 |
apply (drule isCont_inj_range)
|
paulson@15077
|
2561 |
prefer 2 apply (assumption, assumption, auto)
|
paulson@15077
|
2562 |
apply (rule_tac x = e in exI, auto)
|
paulson@15077
|
2563 |
apply (rotate_tac 2)
|
paulson@15077
|
2564 |
apply (drule_tac x = y in spec, auto)
|
paulson@15077
|
2565 |
done
|
paulson@15077
|
2566 |
|
paulson@15077
|
2567 |
|
paulson@15077
|
2568 |
text{*Bartle/Sherbert: Introduction to Real Analysis, Theorem 4.2.9, p. 110*}
|
paulson@15077
|
2569 |
lemma LIM_fun_gt_zero: "[| f -- c --> l; 0 < l |]
|
paulson@15077
|
2570 |
==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> 0 < f x)"
|
paulson@15077
|
2571 |
apply (auto simp add: LIM_def)
|
paulson@15077
|
2572 |
apply (drule_tac x = "l/2" in spec, safe, force)
|
paulson@15077
|
2573 |
apply (rule_tac x = s in exI)
|
paulson@15077
|
2574 |
apply (auto simp only: abs_interval_iff)
|
paulson@15077
|
2575 |
done
|
paulson@15077
|
2576 |
|
paulson@15077
|
2577 |
lemma LIM_fun_less_zero: "[| f -- c --> l; l < 0 |]
|
paulson@15077
|
2578 |
==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x < 0)"
|
paulson@15077
|
2579 |
apply (auto simp add: LIM_def)
|
paulson@15077
|
2580 |
apply (drule_tac x = "-l/2" in spec, safe, force)
|
paulson@15077
|
2581 |
apply (rule_tac x = s in exI)
|
paulson@15077
|
2582 |
apply (auto simp only: abs_interval_iff)
|
paulson@15077
|
2583 |
done
|
paulson@15077
|
2584 |
|
paulson@15077
|
2585 |
|
paulson@15077
|
2586 |
lemma LIM_fun_not_zero:
|
paulson@15077
|
2587 |
"[| f -- c --> l; l \<noteq> 0 |]
|
paulson@15077
|
2588 |
==> \<exists>r. 0 < r & (\<forall>x. x \<noteq> c & \<bar>c - x\<bar> < r --> f x \<noteq> 0)"
|
paulson@15077
|
2589 |
apply (cut_tac x = l and y = 0 in linorder_less_linear, auto)
|
paulson@15077
|
2590 |
apply (drule LIM_fun_less_zero)
|
paulson@15077
|
2591 |
apply (drule_tac [3] LIM_fun_gt_zero, auto)
|
paulson@15077
|
2592 |
apply (rule_tac [!] x = r in exI, auto)
|
paulson@15077
|
2593 |
done
|
paulson@15077
|
2594 |
|
paulson@15077
|
2595 |
ML
|
paulson@15077
|
2596 |
{*
|
paulson@15077
|
2597 |
val inverse_unique = thm "inverse_unique";
|
paulson@15077
|
2598 |
val real_root_zero = thm "real_root_zero";
|
paulson@15077
|
2599 |
val real_root_pow_pos = thm "real_root_pow_pos";
|
paulson@15077
|
2600 |
val real_root_pow_pos2 = thm "real_root_pow_pos2";
|
paulson@15077
|
2601 |
val real_root_pos = thm "real_root_pos";
|
paulson@15077
|
2602 |
val real_root_pos2 = thm "real_root_pos2";
|
paulson@15077
|
2603 |
val real_root_pos_pos = thm "real_root_pos_pos";
|
paulson@15077
|
2604 |
val real_root_pos_pos_le = thm "real_root_pos_pos_le";
|
paulson@15077
|
2605 |
val real_root_one = thm "real_root_one";
|
paulson@15077
|
2606 |
val root_2_eq = thm "root_2_eq";
|
paulson@15077
|
2607 |
val real_sqrt_zero = thm "real_sqrt_zero";
|
paulson@15077
|
2608 |
val real_sqrt_one = thm "real_sqrt_one";
|
paulson@15077
|
2609 |
val real_sqrt_pow2_iff = thm "real_sqrt_pow2_iff";
|
paulson@15077
|
2610 |
val real_sqrt_pow2 = thm "real_sqrt_pow2";
|
paulson@15077
|
2611 |
val real_sqrt_abs_abs = thm "real_sqrt_abs_abs";
|
paulson@15077
|
2612 |
val real_pow_sqrt_eq_sqrt_pow = thm "real_pow_sqrt_eq_sqrt_pow";
|
paulson@15077
|
2613 |
val real_pow_sqrt_eq_sqrt_abs_pow2 = thm "real_pow_sqrt_eq_sqrt_abs_pow2";
|
paulson@15077
|
2614 |
val real_sqrt_pow_abs = thm "real_sqrt_pow_abs";
|
paulson@15077
|
2615 |
val not_real_square_gt_zero = thm "not_real_square_gt_zero";
|
paulson@15077
|
2616 |
val real_mult_self_eq_zero_iff = thm "real_mult_self_eq_zero_iff";
|
paulson@15077
|
2617 |
val real_sqrt_gt_zero = thm "real_sqrt_gt_zero";
|
paulson@15077
|
2618 |
val real_sqrt_ge_zero = thm "real_sqrt_ge_zero";
|
paulson@15077
|
2619 |
val sqrt_eqI = thm "sqrt_eqI";
|
paulson@15077
|
2620 |
val real_sqrt_mult_distrib = thm "real_sqrt_mult_distrib";
|
paulson@15077
|
2621 |
val real_sqrt_mult_distrib2 = thm "real_sqrt_mult_distrib2";
|
paulson@15077
|
2622 |
val real_sqrt_sum_squares_ge_zero = thm "real_sqrt_sum_squares_ge_zero";
|
paulson@15077
|
2623 |
val real_sqrt_sum_squares_mult_ge_zero = thm "real_sqrt_sum_squares_mult_ge_zero";
|
paulson@15077
|
2624 |
val real_sqrt_sum_squares_mult_squared_eq = thm "real_sqrt_sum_squares_mult_squared_eq";
|
paulson@15077
|
2625 |
val real_sqrt_abs = thm "real_sqrt_abs";
|
paulson@15077
|
2626 |
val real_sqrt_abs2 = thm "real_sqrt_abs2";
|
paulson@15077
|
2627 |
val real_sqrt_pow2_gt_zero = thm "real_sqrt_pow2_gt_zero";
|
paulson@15077
|
2628 |
val real_sqrt_not_eq_zero = thm "real_sqrt_not_eq_zero";
|
paulson@15077
|
2629 |
val real_inv_sqrt_pow2 = thm "real_inv_sqrt_pow2";
|
paulson@15077
|
2630 |
val real_sqrt_eq_zero_cancel = thm "real_sqrt_eq_zero_cancel";
|
paulson@15077
|
2631 |
val real_sqrt_eq_zero_cancel_iff = thm "real_sqrt_eq_zero_cancel_iff";
|
paulson@15077
|
2632 |
val real_sqrt_sum_squares_ge1 = thm "real_sqrt_sum_squares_ge1";
|
paulson@15077
|
2633 |
val real_sqrt_sum_squares_ge2 = thm "real_sqrt_sum_squares_ge2";
|
paulson@15077
|
2634 |
val real_sqrt_ge_one = thm "real_sqrt_ge_one";
|
paulson@15077
|
2635 |
val summable_exp = thm "summable_exp";
|
paulson@15077
|
2636 |
val summable_sin = thm "summable_sin";
|
paulson@15077
|
2637 |
val summable_cos = thm "summable_cos";
|
paulson@15077
|
2638 |
val exp_converges = thm "exp_converges";
|
paulson@15077
|
2639 |
val sin_converges = thm "sin_converges";
|
paulson@15077
|
2640 |
val cos_converges = thm "cos_converges";
|
paulson@15077
|
2641 |
val powser_insidea = thm "powser_insidea";
|
paulson@15077
|
2642 |
val powser_inside = thm "powser_inside";
|
paulson@15077
|
2643 |
val diffs_minus = thm "diffs_minus";
|
paulson@15077
|
2644 |
val diffs_equiv = thm "diffs_equiv";
|
paulson@15077
|
2645 |
val less_add_one = thm "less_add_one";
|
paulson@15077
|
2646 |
val termdiffs_aux = thm "termdiffs_aux";
|
paulson@15077
|
2647 |
val termdiffs = thm "termdiffs";
|
paulson@15077
|
2648 |
val exp_fdiffs = thm "exp_fdiffs";
|
paulson@15077
|
2649 |
val sin_fdiffs = thm "sin_fdiffs";
|
paulson@15077
|
2650 |
val sin_fdiffs2 = thm "sin_fdiffs2";
|
paulson@15077
|
2651 |
val cos_fdiffs = thm "cos_fdiffs";
|
paulson@15077
|
2652 |
val cos_fdiffs2 = thm "cos_fdiffs2";
|
paulson@15077
|
2653 |
val DERIV_exp = thm "DERIV_exp";
|
paulson@15077
|
2654 |
val DERIV_sin = thm "DERIV_sin";
|
paulson@15077
|
2655 |
val DERIV_cos = thm "DERIV_cos";
|
paulson@15077
|
2656 |
val exp_zero = thm "exp_zero";
|
paulson@15077
|
2657 |
val exp_ge_add_one_self = thm "exp_ge_add_one_self";
|
paulson@15077
|
2658 |
val exp_gt_one = thm "exp_gt_one";
|
paulson@15077
|
2659 |
val DERIV_exp_add_const = thm "DERIV_exp_add_const";
|
paulson@15077
|
2660 |
val DERIV_exp_minus = thm "DERIV_exp_minus";
|
paulson@15077
|
2661 |
val DERIV_exp_exp_zero = thm "DERIV_exp_exp_zero";
|
paulson@15077
|
2662 |
val exp_add_mult_minus = thm "exp_add_mult_minus";
|
paulson@15077
|
2663 |
val exp_mult_minus = thm "exp_mult_minus";
|
paulson@15077
|
2664 |
val exp_mult_minus2 = thm "exp_mult_minus2";
|
paulson@15077
|
2665 |
val exp_minus = thm "exp_minus";
|
paulson@15077
|
2666 |
val exp_add = thm "exp_add";
|
paulson@15077
|
2667 |
val exp_ge_zero = thm "exp_ge_zero";
|
paulson@15077
|
2668 |
val exp_not_eq_zero = thm "exp_not_eq_zero";
|
paulson@15077
|
2669 |
val exp_gt_zero = thm "exp_gt_zero";
|
paulson@15077
|
2670 |
val inv_exp_gt_zero = thm "inv_exp_gt_zero";
|
paulson@15077
|
2671 |
val abs_exp_cancel = thm "abs_exp_cancel";
|
paulson@15077
|
2672 |
val exp_real_of_nat_mult = thm "exp_real_of_nat_mult";
|
paulson@15077
|
2673 |
val exp_diff = thm "exp_diff";
|
paulson@15077
|
2674 |
val exp_less_mono = thm "exp_less_mono";
|
paulson@15077
|
2675 |
val exp_less_cancel = thm "exp_less_cancel";
|
paulson@15077
|
2676 |
val exp_less_cancel_iff = thm "exp_less_cancel_iff";
|
paulson@15077
|
2677 |
val exp_le_cancel_iff = thm "exp_le_cancel_iff";
|
paulson@15077
|
2678 |
val exp_inj_iff = thm "exp_inj_iff";
|
paulson@15077
|
2679 |
val exp_total = thm "exp_total";
|
paulson@15077
|
2680 |
val ln_exp = thm "ln_exp";
|
paulson@15077
|
2681 |
val exp_ln_iff = thm "exp_ln_iff";
|
paulson@15077
|
2682 |
val ln_mult = thm "ln_mult";
|
paulson@15077
|
2683 |
val ln_inj_iff = thm "ln_inj_iff";
|
paulson@15077
|
2684 |
val ln_one = thm "ln_one";
|
paulson@15077
|
2685 |
val ln_inverse = thm "ln_inverse";
|
paulson@15077
|
2686 |
val ln_div = thm "ln_div";
|
paulson@15077
|
2687 |
val ln_less_cancel_iff = thm "ln_less_cancel_iff";
|
paulson@15077
|
2688 |
val ln_le_cancel_iff = thm "ln_le_cancel_iff";
|
paulson@15077
|
2689 |
val ln_realpow = thm "ln_realpow";
|
paulson@15077
|
2690 |
val ln_add_one_self_le_self = thm "ln_add_one_self_le_self";
|
paulson@15077
|
2691 |
val ln_less_self = thm "ln_less_self";
|
paulson@15077
|
2692 |
val ln_ge_zero = thm "ln_ge_zero";
|
paulson@15077
|
2693 |
val ln_gt_zero = thm "ln_gt_zero";
|
paulson@15077
|
2694 |
val ln_not_eq_zero = thm "ln_not_eq_zero";
|
paulson@15077
|
2695 |
val ln_less_zero = thm "ln_less_zero";
|
paulson@15077
|
2696 |
val exp_ln_eq = thm "exp_ln_eq";
|
paulson@15077
|
2697 |
val sin_zero = thm "sin_zero";
|
paulson@15077
|
2698 |
val cos_zero = thm "cos_zero";
|
paulson@15077
|
2699 |
val DERIV_sin_sin_mult = thm "DERIV_sin_sin_mult";
|
paulson@15077
|
2700 |
val DERIV_sin_sin_mult2 = thm "DERIV_sin_sin_mult2";
|
paulson@15077
|
2701 |
val DERIV_sin_realpow2 = thm "DERIV_sin_realpow2";
|
paulson@15077
|
2702 |
val DERIV_sin_realpow2a = thm "DERIV_sin_realpow2a";
|
paulson@15077
|
2703 |
val DERIV_cos_cos_mult = thm "DERIV_cos_cos_mult";
|
paulson@15077
|
2704 |
val DERIV_cos_cos_mult2 = thm "DERIV_cos_cos_mult2";
|
paulson@15077
|
2705 |
val DERIV_cos_realpow2 = thm "DERIV_cos_realpow2";
|
paulson@15077
|
2706 |
val DERIV_cos_realpow2a = thm "DERIV_cos_realpow2a";
|
paulson@15077
|
2707 |
val DERIV_cos_realpow2b = thm "DERIV_cos_realpow2b";
|
paulson@15077
|
2708 |
val DERIV_cos_cos_mult3 = thm "DERIV_cos_cos_mult3";
|
paulson@15077
|
2709 |
val DERIV_sin_circle_all = thm "DERIV_sin_circle_all";
|
paulson@15077
|
2710 |
val DERIV_sin_circle_all_zero = thm "DERIV_sin_circle_all_zero";
|
paulson@15077
|
2711 |
val sin_cos_squared_add = thm "sin_cos_squared_add";
|
paulson@15077
|
2712 |
val sin_cos_squared_add2 = thm "sin_cos_squared_add2";
|
paulson@15077
|
2713 |
val sin_cos_squared_add3 = thm "sin_cos_squared_add3";
|
paulson@15077
|
2714 |
val sin_squared_eq = thm "sin_squared_eq";
|
paulson@15077
|
2715 |
val cos_squared_eq = thm "cos_squared_eq";
|
paulson@15077
|
2716 |
val real_gt_one_ge_zero_add_less = thm "real_gt_one_ge_zero_add_less";
|
paulson@15077
|
2717 |
val abs_sin_le_one = thm "abs_sin_le_one";
|
paulson@15077
|
2718 |
val sin_ge_minus_one = thm "sin_ge_minus_one";
|
paulson@15077
|
2719 |
val sin_le_one = thm "sin_le_one";
|
paulson@15077
|
2720 |
val abs_cos_le_one = thm "abs_cos_le_one";
|
paulson@15077
|
2721 |
val cos_ge_minus_one = thm "cos_ge_minus_one";
|
paulson@15077
|
2722 |
val cos_le_one = thm "cos_le_one";
|
paulson@15077
|
2723 |
val DERIV_fun_pow = thm "DERIV_fun_pow";
|
paulson@15077
|
2724 |
val DERIV_fun_exp = thm "DERIV_fun_exp";
|
paulson@15077
|
2725 |
val DERIV_fun_sin = thm "DERIV_fun_sin";
|
paulson@15077
|
2726 |
val DERIV_fun_cos = thm "DERIV_fun_cos";
|
paulson@15077
|
2727 |
val DERIV_intros = thms "DERIV_intros";
|
paulson@15077
|
2728 |
val sin_cos_add = thm "sin_cos_add";
|
paulson@15077
|
2729 |
val sin_add = thm "sin_add";
|
paulson@15077
|
2730 |
val cos_add = thm "cos_add";
|
paulson@15077
|
2731 |
val sin_cos_minus = thm "sin_cos_minus";
|
paulson@15077
|
2732 |
val sin_minus = thm "sin_minus";
|
paulson@15077
|
2733 |
val cos_minus = thm "cos_minus";
|
paulson@15077
|
2734 |
val sin_diff = thm "sin_diff";
|
paulson@15077
|
2735 |
val sin_diff2 = thm "sin_diff2";
|
paulson@15077
|
2736 |
val cos_diff = thm "cos_diff";
|
paulson@15077
|
2737 |
val cos_diff2 = thm "cos_diff2";
|
paulson@15077
|
2738 |
val sin_double = thm "sin_double";
|
paulson@15077
|
2739 |
val cos_double = thm "cos_double";
|
paulson@15077
|
2740 |
val sin_paired = thm "sin_paired";
|
paulson@15077
|
2741 |
val sin_gt_zero = thm "sin_gt_zero";
|
paulson@15077
|
2742 |
val sin_gt_zero1 = thm "sin_gt_zero1";
|
paulson@15077
|
2743 |
val cos_double_less_one = thm "cos_double_less_one";
|
paulson@15077
|
2744 |
val cos_paired = thm "cos_paired";
|
paulson@15077
|
2745 |
val cos_two_less_zero = thm "cos_two_less_zero";
|
paulson@15077
|
2746 |
val cos_is_zero = thm "cos_is_zero";
|
paulson@15077
|
2747 |
val pi_half = thm "pi_half";
|
paulson@15077
|
2748 |
val cos_pi_half = thm "cos_pi_half";
|
paulson@15077
|
2749 |
val pi_half_gt_zero = thm "pi_half_gt_zero";
|
paulson@15077
|
2750 |
val pi_half_less_two = thm "pi_half_less_two";
|
paulson@15077
|
2751 |
val pi_gt_zero = thm "pi_gt_zero";
|
paulson@15077
|
2752 |
val pi_neq_zero = thm "pi_neq_zero";
|
paulson@15077
|
2753 |
val pi_not_less_zero = thm "pi_not_less_zero";
|
paulson@15077
|
2754 |
val pi_ge_zero = thm "pi_ge_zero";
|
paulson@15077
|
2755 |
val minus_pi_half_less_zero = thm "minus_pi_half_less_zero";
|
paulson@15077
|
2756 |
val sin_pi_half = thm "sin_pi_half";
|
paulson@15077
|
2757 |
val cos_pi = thm "cos_pi";
|
paulson@15077
|
2758 |
val sin_pi = thm "sin_pi";
|
paulson@15077
|
2759 |
val sin_cos_eq = thm "sin_cos_eq";
|
paulson@15077
|
2760 |
val minus_sin_cos_eq = thm "minus_sin_cos_eq";
|
paulson@15077
|
2761 |
val cos_sin_eq = thm "cos_sin_eq";
|
paulson@15077
|
2762 |
val sin_periodic_pi = thm "sin_periodic_pi";
|
paulson@15077
|
2763 |
val sin_periodic_pi2 = thm "sin_periodic_pi2";
|
paulson@15077
|
2764 |
val cos_periodic_pi = thm "cos_periodic_pi";
|
paulson@15077
|
2765 |
val sin_periodic = thm "sin_periodic";
|
paulson@15077
|
2766 |
val cos_periodic = thm "cos_periodic";
|
paulson@15077
|
2767 |
val cos_npi = thm "cos_npi";
|
paulson@15077
|
2768 |
val sin_npi = thm "sin_npi";
|
paulson@15077
|
2769 |
val sin_npi2 = thm "sin_npi2";
|
paulson@15077
|
2770 |
val cos_two_pi = thm "cos_two_pi";
|
paulson@15077
|
2771 |
val sin_two_pi = thm "sin_two_pi";
|
paulson@15077
|
2772 |
val sin_gt_zero2 = thm "sin_gt_zero2";
|
paulson@15077
|
2773 |
val sin_less_zero = thm "sin_less_zero";
|
paulson@15077
|
2774 |
val pi_less_4 = thm "pi_less_4";
|
paulson@15077
|
2775 |
val cos_gt_zero = thm "cos_gt_zero";
|
paulson@15077
|
2776 |
val cos_gt_zero_pi = thm "cos_gt_zero_pi";
|
paulson@15077
|
2777 |
val cos_ge_zero = thm "cos_ge_zero";
|
paulson@15077
|
2778 |
val sin_gt_zero_pi = thm "sin_gt_zero_pi";
|
paulson@15077
|
2779 |
val sin_ge_zero = thm "sin_ge_zero";
|
paulson@15077
|
2780 |
val cos_total = thm "cos_total";
|
paulson@15077
|
2781 |
val sin_total = thm "sin_total";
|
paulson@15077
|
2782 |
val reals_Archimedean4 = thm "reals_Archimedean4";
|
paulson@15077
|
2783 |
val cos_zero_lemma = thm "cos_zero_lemma";
|
paulson@15077
|
2784 |
val sin_zero_lemma = thm "sin_zero_lemma";
|
paulson@15077
|
2785 |
val cos_zero_iff = thm "cos_zero_iff";
|
paulson@15077
|
2786 |
val sin_zero_iff = thm "sin_zero_iff";
|
paulson@15077
|
2787 |
val tan_zero = thm "tan_zero";
|
paulson@15077
|
2788 |
val tan_pi = thm "tan_pi";
|
paulson@15077
|
2789 |
val tan_npi = thm "tan_npi";
|
paulson@15077
|
2790 |
val tan_minus = thm "tan_minus";
|
paulson@15077
|
2791 |
val tan_periodic = thm "tan_periodic";
|
paulson@15077
|
2792 |
val add_tan_eq = thm "add_tan_eq";
|
paulson@15077
|
2793 |
val tan_add = thm "tan_add";
|
paulson@15077
|
2794 |
val tan_double = thm "tan_double";
|
paulson@15077
|
2795 |
val tan_gt_zero = thm "tan_gt_zero";
|
paulson@15077
|
2796 |
val tan_less_zero = thm "tan_less_zero";
|
paulson@15077
|
2797 |
val DERIV_tan = thm "DERIV_tan";
|
paulson@15077
|
2798 |
val LIM_cos_div_sin = thm "LIM_cos_div_sin";
|
paulson@15077
|
2799 |
val tan_total_pos = thm "tan_total_pos";
|
paulson@15077
|
2800 |
val tan_total = thm "tan_total";
|
paulson@15077
|
2801 |
val arcsin_pi = thm "arcsin_pi";
|
paulson@15077
|
2802 |
val arcsin = thm "arcsin";
|
paulson@15077
|
2803 |
val sin_arcsin = thm "sin_arcsin";
|
paulson@15077
|
2804 |
val arcsin_bounded = thm "arcsin_bounded";
|
paulson@15077
|
2805 |
val arcsin_lbound = thm "arcsin_lbound";
|
paulson@15077
|
2806 |
val arcsin_ubound = thm "arcsin_ubound";
|
paulson@15077
|
2807 |
val arcsin_lt_bounded = thm "arcsin_lt_bounded";
|
paulson@15077
|
2808 |
val arcsin_sin = thm "arcsin_sin";
|
paulson@15077
|
2809 |
val arcos = thm "arcos";
|
paulson@15077
|
2810 |
val cos_arcos = thm "cos_arcos";
|
paulson@15077
|
2811 |
val arcos_bounded = thm "arcos_bounded";
|
paulson@15077
|
2812 |
val arcos_lbound = thm "arcos_lbound";
|
paulson@15077
|
2813 |
val arcos_ubound = thm "arcos_ubound";
|
paulson@15077
|
2814 |
val arcos_lt_bounded = thm "arcos_lt_bounded";
|
paulson@15077
|
2815 |
val arcos_cos = thm "arcos_cos";
|
paulson@15077
|
2816 |
val arcos_cos2 = thm "arcos_cos2";
|
paulson@15077
|
2817 |
val arctan = thm "arctan";
|
paulson@15077
|
2818 |
val tan_arctan = thm "tan_arctan";
|
paulson@15077
|
2819 |
val arctan_bounded = thm "arctan_bounded";
|
paulson@15077
|
2820 |
val arctan_lbound = thm "arctan_lbound";
|
paulson@15077
|
2821 |
val arctan_ubound = thm "arctan_ubound";
|
paulson@15077
|
2822 |
val arctan_tan = thm "arctan_tan";
|
paulson@15077
|
2823 |
val arctan_zero_zero = thm "arctan_zero_zero";
|
paulson@15077
|
2824 |
val cos_arctan_not_zero = thm "cos_arctan_not_zero";
|
paulson@15077
|
2825 |
val tan_sec = thm "tan_sec";
|
paulson@15077
|
2826 |
val DERIV_sin_add = thm "DERIV_sin_add";
|
paulson@15077
|
2827 |
val sin_cos_npi = thm "sin_cos_npi";
|
paulson@15077
|
2828 |
val sin_cos_npi2 = thm "sin_cos_npi2";
|
paulson@15077
|
2829 |
val cos_2npi = thm "cos_2npi";
|
paulson@15077
|
2830 |
val cos_3over2_pi = thm "cos_3over2_pi";
|
paulson@15077
|
2831 |
val sin_2npi = thm "sin_2npi";
|
paulson@15077
|
2832 |
val sin_3over2_pi = thm "sin_3over2_pi";
|
paulson@15077
|
2833 |
val cos_pi_eq_zero = thm "cos_pi_eq_zero";
|
paulson@15077
|
2834 |
val DERIV_cos_add = thm "DERIV_cos_add";
|
paulson@15077
|
2835 |
val isCont_cos = thm "isCont_cos";
|
paulson@15077
|
2836 |
val isCont_sin = thm "isCont_sin";
|
paulson@15077
|
2837 |
val isCont_exp = thm "isCont_exp";
|
paulson@15077
|
2838 |
val sin_zero_abs_cos_one = thm "sin_zero_abs_cos_one";
|
paulson@15077
|
2839 |
val exp_eq_one_iff = thm "exp_eq_one_iff";
|
paulson@15077
|
2840 |
val cos_one_sin_zero = thm "cos_one_sin_zero";
|
paulson@15077
|
2841 |
val real_root_less_mono = thm "real_root_less_mono";
|
paulson@15077
|
2842 |
val real_root_le_mono = thm "real_root_le_mono";
|
paulson@15077
|
2843 |
val real_root_less_iff = thm "real_root_less_iff";
|
paulson@15077
|
2844 |
val real_root_le_iff = thm "real_root_le_iff";
|
paulson@15077
|
2845 |
val real_root_eq_iff = thm "real_root_eq_iff";
|
paulson@15077
|
2846 |
val real_root_pos_unique = thm "real_root_pos_unique";
|
paulson@15077
|
2847 |
val real_root_mult = thm "real_root_mult";
|
paulson@15077
|
2848 |
val real_root_inverse = thm "real_root_inverse";
|
paulson@15077
|
2849 |
val real_root_divide = thm "real_root_divide";
|
paulson@15077
|
2850 |
val real_sqrt_less_mono = thm "real_sqrt_less_mono";
|
paulson@15077
|
2851 |
val real_sqrt_le_mono = thm "real_sqrt_le_mono";
|
paulson@15077
|
2852 |
val real_sqrt_less_iff = thm "real_sqrt_less_iff";
|
paulson@15077
|
2853 |
val real_sqrt_le_iff = thm "real_sqrt_le_iff";
|
paulson@15077
|
2854 |
val real_sqrt_eq_iff = thm "real_sqrt_eq_iff";
|
paulson@15077
|
2855 |
val real_sqrt_sos_less_one_iff = thm "real_sqrt_sos_less_one_iff";
|
paulson@15077
|
2856 |
val real_sqrt_sos_eq_one_iff = thm "real_sqrt_sos_eq_one_iff";
|
paulson@15077
|
2857 |
val real_divide_square_eq = thm "real_divide_square_eq";
|
paulson@15077
|
2858 |
val real_sqrt_sum_squares_gt_zero1 = thm "real_sqrt_sum_squares_gt_zero1";
|
paulson@15077
|
2859 |
val real_sqrt_sum_squares_gt_zero2 = thm "real_sqrt_sum_squares_gt_zero2";
|
paulson@15077
|
2860 |
val real_sqrt_sum_squares_gt_zero3 = thm "real_sqrt_sum_squares_gt_zero3";
|
paulson@15077
|
2861 |
val real_sqrt_sum_squares_gt_zero3a = thm "real_sqrt_sum_squares_gt_zero3a";
|
paulson@15077
|
2862 |
val real_sqrt_sum_squares_eq_cancel = thm "real_sqrt_sum_squares_eq_cancel";
|
paulson@15077
|
2863 |
val real_sqrt_sum_squares_eq_cancel2 = thm "real_sqrt_sum_squares_eq_cancel2";
|
paulson@15077
|
2864 |
val cos_x_y_ge_minus_one = thm "cos_x_y_ge_minus_one";
|
paulson@15077
|
2865 |
val cos_x_y_ge_minus_one1a = thm "cos_x_y_ge_minus_one1a";
|
paulson@15077
|
2866 |
val cos_x_y_le_one = thm "cos_x_y_le_one";
|
paulson@15077
|
2867 |
val cos_x_y_le_one2 = thm "cos_x_y_le_one2";
|
paulson@15077
|
2868 |
val cos_abs_x_y_ge_minus_one = thm "cos_abs_x_y_ge_minus_one";
|
paulson@15077
|
2869 |
val cos_abs_x_y_le_one = thm "cos_abs_x_y_le_one";
|
paulson@15077
|
2870 |
val minus_pi_less_zero = thm "minus_pi_less_zero";
|
paulson@15077
|
2871 |
val arcos_ge_minus_pi = thm "arcos_ge_minus_pi";
|
paulson@15077
|
2872 |
val sin_x_y_disj = thm "sin_x_y_disj";
|
paulson@15077
|
2873 |
val cos_x_y_disj = thm "cos_x_y_disj";
|
paulson@15077
|
2874 |
val real_sqrt_divide_less_zero = thm "real_sqrt_divide_less_zero";
|
paulson@15077
|
2875 |
val polar_ex1 = thm "polar_ex1";
|
paulson@15077
|
2876 |
val polar_ex2 = thm "polar_ex2";
|
paulson@15077
|
2877 |
val polar_Ex = thm "polar_Ex";
|
paulson@15077
|
2878 |
val real_sqrt_ge_abs1 = thm "real_sqrt_ge_abs1";
|
paulson@15077
|
2879 |
val real_sqrt_ge_abs2 = thm "real_sqrt_ge_abs2";
|
paulson@15077
|
2880 |
val real_sqrt_two_gt_zero = thm "real_sqrt_two_gt_zero";
|
paulson@15077
|
2881 |
val real_sqrt_two_ge_zero = thm "real_sqrt_two_ge_zero";
|
paulson@15077
|
2882 |
val real_sqrt_two_gt_one = thm "real_sqrt_two_gt_one";
|
paulson@15077
|
2883 |
val STAR_exp_ln = thm "STAR_exp_ln";
|
paulson@15077
|
2884 |
val hypreal_add_Infinitesimal_gt_zero = thm "hypreal_add_Infinitesimal_gt_zero";
|
paulson@15077
|
2885 |
val NSDERIV_exp_ln_one = thm "NSDERIV_exp_ln_one";
|
paulson@15077
|
2886 |
val DERIV_exp_ln_one = thm "DERIV_exp_ln_one";
|
paulson@15077
|
2887 |
val isCont_inv_fun = thm "isCont_inv_fun";
|
paulson@15077
|
2888 |
val isCont_inv_fun_inv = thm "isCont_inv_fun_inv";
|
paulson@15077
|
2889 |
val LIM_fun_gt_zero = thm "LIM_fun_gt_zero";
|
paulson@15077
|
2890 |
val LIM_fun_less_zero = thm "LIM_fun_less_zero";
|
paulson@15077
|
2891 |
val LIM_fun_not_zero = thm "LIM_fun_not_zero";
|
paulson@15077
|
2892 |
*}
|
paulson@12196
|
2893 |
|
paulson@12196
|
2894 |
end
|