src/HOL/Hyperreal/Transcendental.thy

made proofs more robust

(*  Title       : Transcendental.thy
    Author      : Jacques D. Fleuriot
    Copyright   : 1998,1999 University of Cambridge
                  1999,2001 University of Edinburgh
    Conversion to Isar and new proofs by Lawrence C Paulson, 2004
*)

header{*Power Series, Transcendental Functions etc.*}

theory Transcendental
imports NthRoot Fact HSeries EvenOdd Lim
begin

constdefs
    root :: "[nat,real] => real"
    "root n x == (@u. ((0::real) < x --> 0 < u) & (u ^ n = x))"

    sqrt :: "real => real"
    "sqrt x == root 2 x"

    exp :: "real => real"
    "exp x == suminf(%n. inverse(real (fact n)) * (x ^ n))"

    sin :: "real => real"
    "sin x == suminf(%n. (if even(n) then 0 else
             ((- 1) ^ ((n - Suc 0) div 2))/(real (fact n))) * x ^ n)"
 
    diffs :: "(nat => real) => nat => real"
    "diffs c == (%n. real (Suc n) * c(Suc n))"

    cos :: "real => real"
    "cos x == suminf(%n. (if even(n) then ((- 1) ^ (n div 2))/(real (fact n)) 
                          else 0) * x ^ n)"
  
    ln :: "real => real"
    "ln x == (@u. exp u = x)"

    pi :: "real"
    "pi == 2 * (@x. 0 \<le> (x::real) & x \<le> 2 & cos x = 0)"

    tan :: "real => real"
    "tan x == (sin x)/(cos x)"

    arcsin :: "real => real"
    "arcsin y == (@x. -(pi/2) \<le> x & x \<le> pi/2 & sin x = y)"

    arcos :: "real => real"
    "arcos y == (@x. 0 \<le> x & x \<le> pi & cos x = y)"
     
    arctan :: "real => real"
    "arctan y == (@x. -(pi/2) < x & x < pi/2 & tan x = y)"


lemma real_root_zero [simp]: "root (Suc n) 0 = 0"
apply (simp add: root_def)
apply (safe intro!: some_equality power_0_Suc elim!: realpow_zero_zero)
done

lemma real_root_pow_pos: 
     "0 < x ==> (root(Suc n) x) ^ (Suc n) = x"
apply (simp add: root_def)
apply (drule_tac n = n in realpow_pos_nth2)
apply (auto intro: someI2)
done

lemma real_root_pow_pos2: "0 \<le> x ==> (root(Suc n) x) ^ (Suc n) = x"
by (auto dest!: real_le_imp_less_or_eq dest: real_root_pow_pos)

lemma real_root_pos: 
     "0 < x ==> root(Suc n) (x ^ (Suc n)) = x"
apply (simp add: root_def)
apply (rule some_equality)
apply (frule_tac [2] n = n in zero_less_power)
apply (auto simp add: zero_less_mult_iff)
apply (rule_tac x = u and y = x in linorder_cases)
apply (drule_tac n1 = n and x = u in zero_less_Suc [THEN [3] realpow_less])
apply (drule_tac [4] n1 = n and x = x in zero_less_Suc [THEN [3] realpow_less])
apply (auto simp add: order_less_irrefl)
done

lemma real_root_pos2: "0 \<le> x ==> root(Suc n) (x ^ (Suc n)) = x"
by (auto dest!: real_le_imp_less_or_eq real_root_pos)

lemma real_root_pos_pos: 
     "0 < x ==> 0 \<le> root(Suc n) x"
apply (simp add: root_def)
apply (drule_tac n = n in realpow_pos_nth2)
apply (safe, rule someI2)
apply (auto intro!: order_less_imp_le dest: zero_less_power 
            simp add: zero_less_mult_iff)
done

lemma real_root_pos_pos_le: "0 \<le> x ==> 0 \<le> root(Suc n) x"
by (auto dest!: real_le_imp_less_or_eq dest: real_root_pos_pos)

lemma real_root_one [simp]: "root (Suc n) 1 = 1"
apply (simp add: root_def)
apply (rule some_equality, auto)
apply (rule ccontr)
apply (rule_tac x = u and y = 1 in linorder_cases)
apply (drule_tac n = n in realpow_Suc_less_one)
apply (drule_tac [4] n = n in power_gt1_lemma)
apply (auto simp add: order_less_irrefl)
done


subsection{*Square Root*}

text{*needed because 2 is a binary numeral!*}
lemma root_2_eq [simp]: "root 2 = root (Suc (Suc 0))"
by (simp del: nat_numeral_0_eq_0 nat_numeral_1_eq_1 
         add: nat_numeral_0_eq_0 [symmetric])

lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
by (simp add: sqrt_def)

lemma real_sqrt_one [simp]: "sqrt 1 = 1"
by (simp add: sqrt_def)

lemma real_sqrt_pow2_iff [simp]: "((sqrt x)\<twosuperior> = x) = (0 \<le> x)"
apply (simp add: sqrt_def)
apply (rule iffI) 
 apply (cut_tac r = "root 2 x" in realpow_two_le)
 apply (simp add: numeral_2_eq_2)
apply (subst numeral_2_eq_2)
apply (erule real_root_pow_pos2)
done

lemma [simp]: "(sqrt(u2\<twosuperior> + v2\<twosuperior>))\<twosuperior> = u2\<twosuperior> + v2\<twosuperior>"
by (rule realpow_two_le_add_order [THEN real_sqrt_pow2_iff [THEN iffD2]])

lemma real_sqrt_pow2 [simp]: "0 \<le> x ==> (sqrt x)\<twosuperior> = x"
by (simp add: real_sqrt_pow2_iff)

lemma real_sqrt_abs_abs [simp]: "sqrt\<bar>x\<bar> ^ 2 = \<bar>x\<bar>"
by (rule real_sqrt_pow2_iff [THEN iffD2], arith)

lemma real_pow_sqrt_eq_sqrt_pow: 
      "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(x\<twosuperior>)"
apply (simp add: sqrt_def)
apply (subst numeral_2_eq_2)
apply (auto intro: real_root_pow_pos2 [THEN ssubst] real_root_pos2 [THEN ssubst] simp del: realpow_Suc)
done

lemma real_pow_sqrt_eq_sqrt_abs_pow2:
     "0 \<le> x ==> (sqrt x)\<twosuperior> = sqrt(\<bar>x\<bar> ^ 2)" 
by (simp add: real_pow_sqrt_eq_sqrt_pow [symmetric])

lemma real_sqrt_pow_abs: "0 \<le> x ==> (sqrt x)\<twosuperior> = \<bar>x\<bar>"
apply (rule real_sqrt_abs_abs [THEN subst])
apply (rule_tac x1 = x in real_pow_sqrt_eq_sqrt_abs_pow2 [THEN ssubst])
apply (rule_tac [2] real_pow_sqrt_eq_sqrt_pow [symmetric])
apply (assumption, arith)
done

lemma not_real_square_gt_zero [simp]: "(~ (0::real) < x*x) = (x = 0)"
apply auto
apply (cut_tac x = x and y = 0 in linorder_less_linear)
apply (simp add: zero_less_mult_iff)
done

lemma real_mult_self_eq_zero_iff [simp]: "(r * r = 0) = (r = (0::real))"
by auto

lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt(x)"
apply (simp add: sqrt_def root_def)
apply (drule realpow_pos_nth2 [where n=1], safe)
apply (rule someI2, auto)
done

lemma real_sqrt_ge_zero: "0 \<le> x ==> 0 \<le> sqrt(x)"
by (auto intro: real_sqrt_gt_zero simp add: order_le_less)

lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 \<le> sqrt(x*x + y*y)"
by (rule real_sqrt_ge_zero [OF real_mult_self_sum_ge_zero]) 


(*we need to prove something like this:
lemma "[|r ^ n = a; 0<n; 0 < a \<longrightarrow> 0 < r|] ==> root n a = r"
apply (case_tac n, simp) 
apply (simp add: root_def) 
apply (rule someI2 [of _ r], safe)
apply (auto simp del: realpow_Suc dest: power_inject_base)
*)

lemma sqrt_eqI: "[|r\<twosuperior> = a; 0 \<le> r|] ==> sqrt a = r"
apply (unfold sqrt_def root_def) 
apply (rule someI2 [of _ r], auto) 
apply (auto simp add: numeral_2_eq_2 simp del: realpow_Suc 
            dest: power_inject_base) 
done

lemma real_sqrt_mult_distrib: 
     "[| 0 \<le> x; 0 \<le> y |] ==> sqrt(x*y) = sqrt(x) * sqrt(y)"
apply (rule sqrt_eqI)
apply (simp add: power_mult_distrib)  
apply (simp add: zero_le_mult_iff real_sqrt_ge_zero) 
done

lemma real_sqrt_mult_distrib2:
     "[|0\<le>x; 0\<le>y |] ==> sqrt(x*y) =  sqrt(x) * sqrt(y)"
by (auto intro: real_sqrt_mult_distrib simp add: order_le_less)

lemma real_sqrt_sum_squares_ge_zero [simp]: "0 \<le> sqrt (x\<twosuperior> + y\<twosuperior>)"
by (auto intro!: real_sqrt_ge_zero)

lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
     "0 \<le> sqrt ((x\<twosuperior> + y\<twosuperior>)*(xa\<twosuperior> + ya\<twosuperior>))"
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)

lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
     "sqrt ((x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)) ^ 2 = (x\<twosuperior> + y\<twosuperior>) * (xa\<twosuperior> + ya\<twosuperior>)"
by (auto simp add: real_sqrt_pow2_iff zero_le_mult_iff simp del: realpow_Suc)

lemma real_sqrt_abs [simp]: "sqrt(x\<twosuperior>) = \<bar>x\<bar>"
apply (rule abs_realpow_two [THEN subst])
apply (rule real_sqrt_abs_abs [THEN subst])
apply (subst real_pow_sqrt_eq_sqrt_pow)
apply (auto simp add: numeral_2_eq_2 abs_mult)
done

lemma real_sqrt_abs2 [simp]: "sqrt(x*x) = \<bar>x\<bar>"
apply (rule realpow_two [THEN subst])
apply (subst numeral_2_eq_2 [symmetric])
apply (rule real_sqrt_abs)
done

lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)\<twosuperior>"