Theory NthRoot

Up to index of Isabelle/HOL/HOL-Complex

theory NthRoot
imports Parity Deriv
begin

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

header {* Nth Roots of Real Numbers *}

theory NthRoot
imports "~~/src/HOL/Library/Parity" "../Hyperreal/Deriv"
begin

subsection {* Existence of Nth Root *}

text {* Existence follows from the Intermediate Value Theorem *}

lemma realpow_pos_nth:
  assumes n: "0 < n"
  assumes a: "0 < a"
  shows "∃r>0. r ^ n = (a::real)"
proof -
  have "∃r≥0. r ≤ (max 1 a) ∧ r ^ n = a"
  proof (rule IVT)
    show "0 ^ n ≤ a" using n a by (simp add: power_0_left)
    show "0 ≤ max 1 a" by simp
    from n have n1: "1 ≤ n" by simp
    have "a ≤ max 1 a ^ 1" by simp
    also have "max 1 a ^ 1 ≤ max 1 a ^ n"
      using n1 by (rule power_increasing, simp)
    finally show "a ≤ max 1 a ^ n" .
    show "∀r. 0 ≤ r ∧ r ≤ max 1 a --> isCont (λx. x ^ n) r"
      by (simp add: isCont_power)
  qed
  then obtain r where r: "0 ≤ r ∧ r ^ n = a" by fast
  with n a have "r ≠ 0" by (auto simp add: power_0_left)
  with r have "0 < r ∧ r ^ n = a" by simp
  thus ?thesis ..
qed

(* Used by Integration/RealRandVar.thy in AFP *)
lemma realpow_pos_nth2: "(0::real) < a ==> ∃r>0. r ^ Suc n = a"
by (blast intro: realpow_pos_nth)

text {* Uniqueness of nth positive root *}

lemma realpow_pos_nth_unique:
  "[|0 < n; 0 < a|] ==> ∃!r. 0 < r ∧ r ^ n = (a::real)"
apply (auto intro!: realpow_pos_nth)
apply (rule_tac n=n in power_eq_imp_eq_base, simp_all)
done

subsection {* Nth Root *}

text {* We define roots of negative reals such that
  @{term "root n (- x) = - root n x"}. This allows
  us to omit side conditions from many theorems. *}

definition
  root :: "[nat, real] => real" where
  "root n x = (if 0 < x then (THE u. 0 < u ∧ u ^ n = x) else
               if x < 0 then - (THE u. 0 < u ∧ u ^ n = - x) else 0)"

lemma real_root_zero [simp]: "root n 0 = 0"
unfolding root_def by simp

lemma real_root_minus: "0 < n ==> root n (- x) = - root n x"
unfolding root_def by simp

lemma real_root_gt_zero: "[|0 < n; 0 < x|] ==> 0 < root n x"
apply (simp add: root_def)
apply (drule (1) realpow_pos_nth_unique)
apply (erule theI' [THEN conjunct1])
done

lemma real_root_pow_pos: (* TODO: rename *)
  "[|0 < n; 0 < x|] ==> root n x ^ n = x"
apply (simp add: root_def)
apply (drule (1) realpow_pos_nth_unique)
apply (erule theI' [THEN conjunct2])
done

lemma real_root_pow_pos2 [simp]: (* TODO: rename *)
  "[|0 < n; 0 ≤ x|] ==> root n x ^ n = x"
by (auto simp add: order_le_less real_root_pow_pos)

lemma odd_real_root_pow: "odd n ==> root n x ^ n = x"
apply (rule_tac x=0 and y=x in linorder_le_cases)
apply (erule (1) real_root_pow_pos2 [OF odd_pos])
apply (subgoal_tac "root n (- x) ^ n = - x")
apply (simp add: real_root_minus odd_pos)
apply (simp add: odd_pos)
done

lemma real_root_ge_zero: "[|0 < n; 0 ≤ x|] ==> 0 ≤ root n x"
by (auto simp add: order_le_less real_root_gt_zero)

lemma real_root_power_cancel: "[|0 < n; 0 ≤ x|] ==> root n (x ^ n) = x"
apply (subgoal_tac "0 ≤ x ^ n")
apply (subgoal_tac "0 ≤ root n (x ^ n)")
apply (subgoal_tac "root n (x ^ n) ^ n = x ^ n")
apply (erule (3) power_eq_imp_eq_base)
apply (erule (1) real_root_pow_pos2)
apply (erule (1) real_root_ge_zero)
apply (erule zero_le_power)
done

lemma odd_real_root_power_cancel: "odd n ==> root n (x ^ n) = x"
apply (rule_tac x=0 and y=x in linorder_le_cases)
apply (erule (1) real_root_power_cancel [OF odd_pos])
apply (subgoal_tac "root n ((- x) ^ n) = - x")
apply (simp add: real_root_minus odd_pos)
apply (erule real_root_power_cancel [OF odd_pos], simp)
done

lemma real_root_pos_unique:
  "[|0 < n; 0 ≤ y; y ^ n = x|] ==> root n x = y"
by (erule subst, rule real_root_power_cancel)

lemma odd_real_root_unique:
  "[|odd n; y ^ n = x|] ==> root n x = y"
by (erule subst, rule odd_real_root_power_cancel)

lemma real_root_one [simp]: "0 < n ==> root n 1 = 1"
by (simp add: real_root_pos_unique)

text {* Root function is strictly monotonic, hence injective *}

lemma real_root_less_mono_lemma:
  "[|0 < n; 0 ≤ x; x < y|] ==> root n x < root n y"
apply (subgoal_tac "0 ≤ y")
apply (subgoal_tac "root n x ^ n < root n y ^ n")
apply (erule power_less_imp_less_base)
apply (erule (1) real_root_ge_zero)
apply simp
apply simp
done

lemma real_root_less_mono: "[|0 < n; x < y|] ==> root n x < root n y"
apply (cases "0 ≤ x")
apply (erule (2) real_root_less_mono_lemma)
apply (cases "0 ≤ y")
apply (rule_tac y=0 in order_less_le_trans)
apply (subgoal_tac "0 < root n (- x)")
apply (simp add: real_root_minus)
apply (simp add: real_root_gt_zero)
apply (simp add: real_root_ge_zero)
apply (subgoal_tac "root n (- y) < root n (- x)")
apply (simp add: real_root_minus)
apply (simp add: real_root_less_mono_lemma)
done

lemma real_root_le_mono: "[|0 < n; x ≤ y|] ==> root n x ≤ root n y"
by (auto simp add: order_le_less real_root_less_mono)

lemma real_root_less_iff [simp]:
  "0 < n ==> (root n x < root n y) = (x < y)"
apply (cases "x < y")
apply (simp add: real_root_less_mono)
apply (simp add: linorder_not_less real_root_le_mono)
done

lemma real_root_le_iff [simp]:
  "0 < n ==> (root n x ≤ root n y) = (x ≤ y)"
apply (cases "x ≤ y")
apply (simp add: real_root_le_mono)
apply (simp add: linorder_not_le real_root_less_mono)
done

lemma real_root_eq_iff [simp]:
  "0 < n ==> (root n x = root n y) = (x = y)"
by (simp add: order_eq_iff)

lemmas real_root_gt_0_iff [simp] = real_root_less_iff [where x=0, simplified]
lemmas real_root_lt_0_iff [simp] = real_root_less_iff [where y=0, simplified]
lemmas real_root_ge_0_iff [simp] = real_root_le_iff [where x=0, simplified]
lemmas real_root_le_0_iff [simp] = real_root_le_iff [where y=0, simplified]
lemmas real_root_eq_0_iff [simp] = real_root_eq_iff [where y=0, simplified]

lemma real_root_gt_1_iff [simp]: "0 < n ==> (1 < root n y) = (1 < y)"
by (insert real_root_less_iff [where x=1], simp)

lemma real_root_lt_1_iff [simp]: "0 < n ==> (root n x < 1) = (x < 1)"
by (insert real_root_less_iff [where y=1], simp)

lemma real_root_ge_1_iff [simp]: "0 < n ==> (1 ≤ root n y) = (1 ≤ y)"
by (insert real_root_le_iff [where x=1], simp)

lemma real_root_le_1_iff [simp]: "0 < n ==> (root n x ≤ 1) = (x ≤ 1)"
by (insert real_root_le_iff [where y=1], simp)

lemma real_root_eq_1_iff [simp]: "0 < n ==> (root n x = 1) = (x = 1)"
by (insert real_root_eq_iff [where y=1], simp)

text {* Roots of roots *}

lemma real_root_Suc_0 [simp]: "root (Suc 0) x = x"
by (simp add: odd_real_root_unique)

lemma real_root_pos_mult_exp:
  "[|0 < m; 0 < n; 0 < x|] ==> root (m * n) x = root m (root n x)"
by (rule real_root_pos_unique, simp_all add: power_mult)

lemma real_root_mult_exp:
  "[|0 < m; 0 < n|] ==> root (m * n) x = root m (root n x)"
apply (rule linorder_cases [where x=x and y=0])
apply (subgoal_tac "root (m * n) (- x) = root m (root n (- x))")
apply (simp add: real_root_minus)
apply (simp_all add: real_root_pos_mult_exp)
done

lemma real_root_commute:
  "[|0 < m; 0 < n|] ==> root m (root n x) = root n (root m x)"
by (simp add: real_root_mult_exp [symmetric] mult_commute)

text {* Monotonicity in first argument *}

lemma real_root_strict_decreasing:
  "[|0 < n; n < N; 1 < x|] ==> root N x < root n x"
apply (subgoal_tac "root n (root N x) ^ n < root N (root n x) ^ N", simp)
apply (simp add: real_root_commute power_strict_increasing
            del: real_root_pow_pos2)
done

lemma real_root_strict_increasing:
  "[|0 < n; n < N; 0 < x; x < 1|] ==> root n x < root N x"
apply (subgoal_tac "root N (root n x) ^ N < root n (root N x) ^ n", simp)
apply (simp add: real_root_commute power_strict_decreasing
            del: real_root_pow_pos2)
done

lemma real_root_decreasing:
  "[|0 < n; n < N; 1 ≤ x|] ==> root N x ≤ root n x"
by (auto simp add: order_le_less real_root_strict_decreasing)

lemma real_root_increasing:
  "[|0 < n; n < N; 0 ≤ x; x ≤ 1|] ==> root n x ≤ root N x"
by (auto simp add: order_le_less real_root_strict_increasing)

text {* Roots of multiplication and division *}

lemma real_root_mult_lemma:
  "[|0 < n; 0 ≤ x; 0 ≤ y|] ==> root n (x * y) = root n x * root n y"
by (simp add: real_root_pos_unique mult_nonneg_nonneg power_mult_distrib)

lemma real_root_inverse_lemma:
  "[|0 < n; 0 ≤ x|] ==> root n (inverse x) = inverse (root n x)"
by (simp add: real_root_pos_unique power_inverse [symmetric])

lemma real_root_mult:
  assumes n: "0 < n"
  shows "root n (x * y) = root n x * root n y"
proof (rule linorder_le_cases, rule_tac [!] linorder_le_cases)
  assume "0 ≤ x" and "0 ≤ y"
  thus ?thesis by (rule real_root_mult_lemma [OF n])
next
  assume "0 ≤ x" and "y ≤ 0"
  hence "0 ≤ x" and "0 ≤ - y" by simp_all
  hence "root n (x * - y) = root n x * root n (- y)"
    by (rule real_root_mult_lemma [OF n])
  thus ?thesis by (simp add: real_root_minus [OF n])
next
  assume "x ≤ 0" and "0 ≤ y"
  hence "0 ≤ - x" and "0 ≤ y" by simp_all
  hence "root n (- x * y) = root n (- x) * root n y"
    by (rule real_root_mult_lemma [OF n])
  thus ?thesis by (simp add: real_root_minus [OF n])
next
  assume "x ≤ 0" and "y ≤ 0"
  hence "0 ≤ - x" and "0 ≤ - y" by simp_all
  hence "root n (- x * - y) = root n (- x) * root n (- y)"
    by (rule real_root_mult_lemma [OF n])
  thus ?thesis by (simp add: real_root_minus [OF n])
qed

lemma real_root_inverse:
  assumes n: "0 < n"
  shows "root n (inverse x) = inverse (root n x)"
proof (rule linorder_le_cases)
  assume "0 ≤ x"
  thus ?thesis by (rule real_root_inverse_lemma [OF n])
next
  assume "x ≤ 0"
  hence "0 ≤ - x" by simp
  hence "root n (inverse (- x)) = inverse (root n (- x))"
    by (rule real_root_inverse_lemma [OF n])
  thus ?thesis by (simp add: real_root_minus [OF n])
qed

lemma real_root_divide:
  "0 < n ==> root n (x / y) = root n x / root n y"
by (simp add: divide_inverse real_root_mult real_root_inverse)

lemma real_root_power:
  "0 < n ==> root n (x ^ k) = root n x ^ k"
by (induct k, simp_all add: real_root_mult)

lemma real_root_abs: "0 < n ==> root n ¦x¦ = ¦root n x¦"
by (simp add: abs_if real_root_minus)

text {* Continuity and derivatives *}

lemma isCont_root_pos:
  assumes n: "0 < n"
  assumes x: "0 < x"
  shows "isCont (root n) x"
proof -
  have "isCont (root n) (root n x ^ n)"
  proof (rule isCont_inverse_function [where f="λa. a ^ n"])
    show "0 < root n x" using n x by simp
    show "∀z. ¦z - root n x¦ ≤ root n x --> root n (z ^ n) = z"
      by (simp add: abs_le_iff real_root_power_cancel n)
    show "∀z. ¦z - root n x¦ ≤ root n x --> isCont (λa. a ^ n) z"
      by (simp add: isCont_power)
  qed
  thus ?thesis using n x by simp
qed

lemma isCont_root_neg:
  "[|0 < n; x < 0|] ==> isCont (root n) x"
apply (subgoal_tac "isCont (λx. - root n (- x)) x")
apply (simp add: real_root_minus)
apply (rule isCont_o2 [OF isCont_minus [OF isCont_ident]])
apply (simp add: isCont_minus isCont_root_pos)
done

lemma isCont_root_zero:
  "0 < n ==> isCont (root n) 0"
unfolding isCont_def
apply (rule LIM_I)
apply (rule_tac x="r ^ n" in exI, safe)
apply (simp)
apply (simp add: real_root_abs [symmetric])
apply (rule_tac n="n" in power_less_imp_less_base, simp_all)
done

lemma isCont_real_root: "0 < n ==> isCont (root n) x"
apply (rule_tac x=x and y=0 in linorder_cases)
apply (simp_all add: isCont_root_pos isCont_root_neg isCont_root_zero)
done

lemma DERIV_real_root:
  assumes n: "0 < n"
  assumes x: "0 < x"
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
proof (rule DERIV_inverse_function)
  show "0 < x" using x .
  show "x < x + 1" by simp
  show "∀y. 0 < y ∧ y < x + 1 --> root n y ^ n = y"
    using n by simp
  show "DERIV (λx. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
    by (rule DERIV_pow)
  show "real n * root n x ^ (n - Suc 0) ≠ 0"
    using n x by simp
  show "isCont (root n) x"
    using n by (rule isCont_real_root)
qed

lemma DERIV_odd_real_root:
  assumes n: "odd n"
  assumes x: "x ≠ 0"
  shows "DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))"
proof (rule DERIV_inverse_function)
  show "x - 1 < x" by simp
  show "x < x + 1" by simp
  show "∀y. x - 1 < y ∧ y < x + 1 --> root n y ^ n = y"
    using n by (simp add: odd_real_root_pow)
  show "DERIV (λx. x ^ n) (root n x) :> real n * root n x ^ (n - Suc 0)"
    by (rule DERIV_pow)
  show "real n * root n x ^ (n - Suc 0) ≠ 0"
    using odd_pos [OF n] x by simp
  show "isCont (root n) x"
    using odd_pos [OF n] by (rule isCont_real_root)
qed

subsection {* Square Root *}

definition
  sqrt :: "real => real" where
  "sqrt = root 2"

lemma pos2: "0 < (2::nat)" by simp

lemma real_sqrt_unique: "[|y² = x; 0 ≤ y|] ==> sqrt x = y"
unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])

lemma real_sqrt_abs [simp]: "sqrt (x²) = ¦x¦"
apply (rule real_sqrt_unique)
apply (rule power2_abs)
apply (rule abs_ge_zero)
done

lemma real_sqrt_pow2 [simp]: "0 ≤ x ==> (sqrt x)² = x"
unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])

lemma real_sqrt_pow2_iff [simp]: "((sqrt x)² = x) = (0 ≤ x)"
apply (rule iffI)
apply (erule subst)
apply (rule zero_le_power2)
apply (erule real_sqrt_pow2)
done

lemma real_sqrt_zero [simp]: "sqrt 0 = 0"
unfolding sqrt_def by (rule real_root_zero)

lemma real_sqrt_one [simp]: "sqrt 1 = 1"
unfolding sqrt_def by (rule real_root_one [OF pos2])

lemma real_sqrt_minus: "sqrt (- x) = - sqrt x"
unfolding sqrt_def by (rule real_root_minus [OF pos2])

lemma real_sqrt_mult: "sqrt (x * y) = sqrt x * sqrt y"
unfolding sqrt_def by (rule real_root_mult [OF pos2])

lemma real_sqrt_inverse: "sqrt (inverse x) = inverse (sqrt x)"
unfolding sqrt_def by (rule real_root_inverse [OF pos2])

lemma real_sqrt_divide: "sqrt (x / y) = sqrt x / sqrt y"
unfolding sqrt_def by (rule real_root_divide [OF pos2])

lemma real_sqrt_power: "sqrt (x ^ k) = sqrt x ^ k"
unfolding sqrt_def by (rule real_root_power [OF pos2])

lemma real_sqrt_gt_zero: "0 < x ==> 0 < sqrt x"
unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])

lemma real_sqrt_ge_zero: "0 ≤ x ==> 0 ≤ sqrt x"
unfolding sqrt_def by (rule real_root_ge_zero [OF pos2])

lemma real_sqrt_less_mono: "x < y ==> sqrt x < sqrt y"
unfolding sqrt_def by (rule real_root_less_mono [OF pos2])

lemma real_sqrt_le_mono: "x ≤ y ==> sqrt x ≤ sqrt y"
unfolding sqrt_def by (rule real_root_le_mono [OF pos2])

lemma real_sqrt_less_iff [simp]: "(sqrt x < sqrt y) = (x < y)"
unfolding sqrt_def by (rule real_root_less_iff [OF pos2])

lemma real_sqrt_le_iff [simp]: "(sqrt x ≤ sqrt y) = (x ≤ y)"
unfolding sqrt_def by (rule real_root_le_iff [OF pos2])

lemma real_sqrt_eq_iff [simp]: "(sqrt x = sqrt y) = (x = y)"
unfolding sqrt_def by (rule real_root_eq_iff [OF pos2])

lemmas real_sqrt_gt_0_iff [simp] = real_sqrt_less_iff [where x=0, simplified]
lemmas real_sqrt_lt_0_iff [simp] = real_sqrt_less_iff [where y=0, simplified]
lemmas real_sqrt_ge_0_iff [simp] = real_sqrt_le_iff [where x=0, simplified]
lemmas real_sqrt_le_0_iff [simp] = real_sqrt_le_iff [where y=0, simplified]
lemmas real_sqrt_eq_0_iff [simp] = real_sqrt_eq_iff [where y=0, simplified]

lemmas real_sqrt_gt_1_iff [simp] = real_sqrt_less_iff [where x=1, simplified]
lemmas real_sqrt_lt_1_iff [simp] = real_sqrt_less_iff [where y=1, simplified]
lemmas real_sqrt_ge_1_iff [simp] = real_sqrt_le_iff [where x=1, simplified]
lemmas real_sqrt_le_1_iff [simp] = real_sqrt_le_iff [where y=1, simplified]
lemmas real_sqrt_eq_1_iff [simp] = real_sqrt_eq_iff [where y=1, simplified]

lemma isCont_real_sqrt: "isCont sqrt x"
unfolding sqrt_def by (rule isCont_real_root [OF pos2])

lemma DERIV_real_sqrt:
  "0 < x ==> DERIV sqrt x :> inverse (sqrt x) / 2"
unfolding sqrt_def by (rule DERIV_real_root [OF pos2, simplified])

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_sqrt_abs2 [simp]: "sqrt(x*x) = ¦x¦"
apply (subst power2_eq_square [symmetric])
apply (rule real_sqrt_abs)
done

lemma real_sqrt_pow2_gt_zero: "0 < x ==> 0 < (sqrt x)²"
by simp (* TODO: delete *)

lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x ≠ 0"
by simp (* TODO: delete *)

lemma real_inv_sqrt_pow2: "0 < x ==> inverse (sqrt(x)) ^ 2 = inverse x"
by (simp add: power_inverse [symmetric])

lemma real_sqrt_eq_zero_cancel: "[| 0 ≤ x; sqrt(x) = 0|] ==> x = 0"
by simp

lemma real_sqrt_ge_one: "1 ≤ x ==> 1 ≤ sqrt x"
by simp

lemma real_sqrt_two_gt_zero [simp]: "0 < sqrt 2"
by simp

lemma real_sqrt_two_ge_zero [simp]: "0 ≤ sqrt 2"
by simp

lemma real_sqrt_two_gt_one [simp]: "1 < sqrt 2"
by simp

lemma sqrt_divide_self_eq:
  assumes nneg: "0 ≤ x"
  shows "sqrt x / x = inverse (sqrt x)"
proof cases
  assume "x=0" thus ?thesis by simp
next
  assume nz: "x≠0" 
  hence pos: "0<x" using nneg by arith
  show ?thesis
  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 
    show "sqrt x / x ≠ 0" by (simp add: divide_inverse nneg nz) 
    show "inverse (sqrt x) / (sqrt x / x) = 1"
      by (simp add: divide_inverse mult_assoc [symmetric] 
                  power2_eq_square [symmetric] real_inv_sqrt_pow2 pos nz) 
  qed
qed

lemma real_divide_square_eq [simp]: "(((r::real) * a) / (r * r)) = a / r"
apply (simp add: divide_inverse)
apply (case_tac "r=0")
apply (auto simp add: mult_ac)
done

lemma lemma_real_divide_sqrt_less: "0 < u ==> u / sqrt 2 < u"
by (simp add: divide_less_eq mult_compare_simps)

lemma four_x_squared: 
  fixes x::real
  shows "4 * x² = (2 * x)²"
by (simp add: power2_eq_square)

subsection {* Square Root of Sum of Squares *}

lemma real_sqrt_mult_self_sum_ge_zero [simp]: "0 ≤ sqrt(x*x + y*y)"
by (rule real_sqrt_ge_zero [OF sum_squares_ge_zero])

lemma real_sqrt_sum_squares_ge_zero [simp]: "0 ≤ sqrt (x² + y²)"
by simp

declare real_sqrt_sum_squares_ge_zero [THEN abs_of_nonneg, simp]

lemma real_sqrt_sum_squares_mult_ge_zero [simp]:
     "0 ≤ sqrt ((x² + y²)*(xa² + ya²))"
by (auto intro!: real_sqrt_ge_zero simp add: zero_le_mult_iff)

lemma real_sqrt_sum_squares_mult_squared_eq [simp]:
     "sqrt ((x² + y²) * (xa² + ya²)) ^ 2 = (x² + y²) * (xa² + ya²)"
by (auto simp add: zero_le_mult_iff)

lemma real_sqrt_sum_squares_eq_cancel: "sqrt (x² + y²) = x ==> y = 0"
by (drule_tac f = "%x. x²" in arg_cong, simp)

lemma real_sqrt_sum_squares_eq_cancel2: "sqrt (x² + y²) = y ==> x = 0"
by (drule_tac f = "%x. x²" in arg_cong, simp)

lemma real_sqrt_sum_squares_ge1 [simp]: "x ≤ sqrt (x² + y²)"
by (rule power2_le_imp_le, simp_all)

lemma real_sqrt_sum_squares_ge2 [simp]: "y ≤ sqrt (x² + y²)"
by (rule power2_le_imp_le, simp_all)

lemma real_sqrt_ge_abs1 [simp]: "¦x¦ ≤ sqrt (x² + y²)"
by (rule power2_le_imp_le, simp_all)

lemma real_sqrt_ge_abs2 [simp]: "¦y¦ ≤ sqrt (x² + y²)"
by (rule power2_le_imp_le, simp_all)

lemma le_real_sqrt_sumsq [simp]: "x ≤ sqrt (x * x + y * y)"
by (simp add: power2_eq_square [symmetric])

lemma power2_sum:
  fixes x y :: "'a::{number_ring,recpower}"
  shows "(x + y)² = x² + y² + 2 * x * y"
by (simp add: ring_distribs power2_eq_square)

lemma power2_diff:
  fixes x y :: "'a::{number_ring,recpower}"
  shows "(x - y)² = x² + y² - 2 * x * y"
by (simp add: ring_distribs power2_eq_square)

lemma real_sqrt_sum_squares_triangle_ineq:
  "sqrt ((a + c)² + (b + d)²) ≤ sqrt (a² + b²) + sqrt (c² + d²)"
apply (rule power2_le_imp_le, simp)
apply (simp add: power2_sum)
apply (simp only: mult_assoc right_distrib [symmetric])
apply (rule mult_left_mono)
apply (rule power2_le_imp_le)
apply (simp add: power2_sum power_mult_distrib)
apply (simp add: ring_distribs)
apply (subgoal_tac "0 ≤ b² * c² + a² * d² - 2 * (a * c) * (b * d)", simp)
apply (rule_tac b="(a * d - b * c)²" in ord_le_eq_trans)
apply (rule zero_le_power2)
apply (simp add: power2_diff power_mult_distrib)
apply (simp add: mult_nonneg_nonneg)
apply simp
apply (simp add: add_increasing)
done

lemma real_sqrt_sum_squares_less:
  "[|¦x¦ < u / sqrt 2; ¦y¦ < u / sqrt 2|] ==> sqrt (x² + y²) < u"
apply (rule power2_less_imp_less, simp)
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
apply (drule power_strict_mono [OF _ abs_ge_zero pos2])
apply (simp add: power_divide)
apply (drule order_le_less_trans [OF abs_ge_zero])
apply (simp add: zero_less_divide_iff)
done

text{*Needed for the infinitely close relation over the nonstandard
    complex numbers*}
lemma lemma_sqrt_hcomplex_capprox:
     "[| 0 < u; x < u/2; y < u/2; 0 ≤ x; 0 ≤ y |] ==> sqrt (x² + y²) < u"
apply (rule_tac y = "u/sqrt 2" in order_le_less_trans)
apply (erule_tac [2] lemma_real_divide_sqrt_less)
apply (rule power2_le_imp_le)
apply (auto simp add: real_0_le_divide_iff power_divide)
apply (rule_tac t = "u²" in real_sum_of_halves [THEN subst])
apply (rule add_mono)
apply (auto simp add: four_x_squared simp del: realpow_Suc intro: power_mono)
done

text "Legacy theorem names:"
lemmas real_root_pos2 = real_root_power_cancel
lemmas real_root_pos_pos = real_root_gt_zero [THEN order_less_imp_le]
lemmas real_root_pos_pos_le = real_root_ge_zero
lemmas real_sqrt_mult_distrib = real_sqrt_mult
lemmas real_sqrt_mult_distrib2 = real_sqrt_mult
lemmas real_sqrt_eq_zero_cancel_iff = real_sqrt_eq_0_iff

(* needed for CauchysMeanTheorem.het_base from AFP *)
lemma real_root_pos: "0 < x ==> root (Suc n) (x ^ (Suc n)) = x"
by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])

end

Existence of Nth Root

lemma realpow_pos_nth:

  [| 0 < n; 0 < a |] ==> ∃r>0. r ^ n = a

lemma realpow_pos_nth2:

  0 < a ==> ∃r>0. r ^ Suc n = a

lemma realpow_pos_nth_unique:

  [| 0 < n; 0 < a |] ==> ∃!r. 0 < rr ^ n = a

Nth Root

lemma real_root_zero:

  root n 0 = 0

lemma real_root_minus:

  0 < n ==> root n (- x) = - root n x

lemma real_root_gt_zero:

  [| 0 < n; 0 < x |] ==> 0 < root n x

lemma real_root_pow_pos:

  [| 0 < n; 0 < x |] ==> root n x ^ n = x

lemma real_root_pow_pos2:

  [| 0 < n; 0  x |] ==> root n x ^ n = x

lemma odd_real_root_pow:

  odd n ==> root n x ^ n = x

lemma real_root_ge_zero:

  [| 0 < n; 0  x |] ==> 0  root n x

lemma real_root_power_cancel:

  [| 0 < n; 0  x |] ==> root n (x ^ n) = x

lemma odd_real_root_power_cancel:

  odd n ==> root n (x ^ n) = x

lemma real_root_pos_unique:

  [| 0 < n; 0  y; y ^ n = x |] ==> root n x = y

lemma odd_real_root_unique:

  [| odd n; y ^ n = x |] ==> root n x = y

lemma real_root_one:

  0 < n ==> root n 1 = 1

lemma real_root_less_mono_lemma:

  [| 0 < n; 0  x; x < y |] ==> root n x < root n y

lemma real_root_less_mono:

  [| 0 < n; x < y |] ==> root n x < root n y

lemma real_root_le_mono:

  [| 0 < n; x  y |] ==> root n x  root n y

lemma real_root_less_iff:

  0 < n ==> (root n x < root n y) = (x < y)

lemma real_root_le_iff:

  0 < n ==> (root n x  root n y) = (x  y)

lemma real_root_eq_iff:

  0 < n ==> (root n x = root n y) = (x = y)

lemma real_root_gt_0_iff:

  0 < n ==> (0 < root n y) = (0 < y)

lemma real_root_lt_0_iff:

  0 < n ==> (root n x < 0) = (x < 0)

lemma real_root_ge_0_iff:

  0 < n ==> (0  root n y) = (0  y)

lemma real_root_le_0_iff:

  0 < n ==> (root n x  0) = (x  0)

lemma real_root_eq_0_iff:

  0 < n ==> (root n x = 0) = (x = 0)

lemma real_root_gt_1_iff:

  0 < n ==> (1 < root n y) = (1 < y)

lemma real_root_lt_1_iff:

  0 < n ==> (root n x < 1) = (x < 1)

lemma real_root_ge_1_iff:

  0 < n ==> (1  root n y) = (1  y)

lemma real_root_le_1_iff:

  0 < n ==> (root n x  1) = (x  1)

lemma real_root_eq_1_iff:

  0 < n ==> (root n x = 1) = (x = 1)

lemma real_root_Suc_0:

  root (Suc 0) x = x

lemma real_root_pos_mult_exp:

  [| 0 < m; 0 < n; 0 < x |] ==> root (m * n) x = root m (root n x)

lemma real_root_mult_exp:

  [| 0 < m; 0 < n |] ==> root (m * n) x = root m (root n x)

lemma real_root_commute:

  [| 0 < m; 0 < n |] ==> root m (root n x) = root n (root m x)

lemma real_root_strict_decreasing:

  [| 0 < n; n < N; 1 < x |] ==> root N x < root n x

lemma real_root_strict_increasing:

  [| 0 < n; n < N; 0 < x; x < 1 |] ==> root n x < root N x

lemma real_root_decreasing:

  [| 0 < n; n < N; 1  x |] ==> root N x  root n x

lemma real_root_increasing:

  [| 0 < n; n < N; 0  x; x  1 |] ==> root n x  root N x

lemma real_root_mult_lemma:

  [| 0 < n; 0  x; 0  y |] ==> root n (x * y) = root n x * root n y

lemma real_root_inverse_lemma:

  [| 0 < n; 0  x |] ==> root n (inverse x) = inverse (root n x)

lemma real_root_mult:

  0 < n ==> root n (x * y) = root n x * root n y

lemma real_root_inverse:

  0 < n ==> root n (inverse x) = inverse (root n x)

lemma real_root_divide:

  0 < n ==> root n (x / y) = root n x / root n y

lemma real_root_power:

  0 < n ==> root n (x ^ k) = root n x ^ k

lemma real_root_abs:

  0 < n ==> root n ¦x¦ = ¦root n x¦

lemma isCont_root_pos:

  [| 0 < n; 0 < x |] ==> isCont (root n) x

lemma isCont_root_neg:

  [| 0 < n; x < 0 |] ==> isCont (root n) x

lemma isCont_root_zero:

  0 < n ==> isCont (root n) 0

lemma isCont_real_root:

  0 < n ==> isCont (root n) x

lemma DERIV_real_root:

  [| 0 < n; 0 < x |]
  ==> DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))

lemma DERIV_odd_real_root:

  [| odd n; x  0 |]
  ==> DERIV (root n) x :> inverse (real n * root n x ^ (n - Suc 0))

Square Root

lemma pos2:

  0 < 2

lemma real_sqrt_unique:

  [| y ^ 2 = x; 0  y |] ==> sqrt x = y

lemma real_sqrt_abs:

  sqrt (x ^ 2) = ¦x¦

lemma real_sqrt_pow2:

  0  x ==> sqrt x ^ 2 = x

lemma real_sqrt_pow2_iff:

  (sqrt x ^ 2 = x) = (0  x)

lemma real_sqrt_zero:

  sqrt 0 = 0

lemma real_sqrt_one:

  sqrt 1 = 1

lemma real_sqrt_minus:

  sqrt (- x) = - sqrt x

lemma real_sqrt_mult:

  sqrt (x * y) = sqrt x * sqrt y

lemma real_sqrt_inverse:

  sqrt (inverse x) = inverse (sqrt x)

lemma real_sqrt_divide:

  sqrt (x / y) = sqrt x / sqrt y

lemma real_sqrt_power:

  sqrt (x ^ k) = sqrt x ^ k

lemma real_sqrt_gt_zero:

  0 < x ==> 0 < sqrt x

lemma real_sqrt_ge_zero:

  0  x ==> 0  sqrt x

lemma real_sqrt_less_mono:

  x < y ==> sqrt x < sqrt y

lemma real_sqrt_le_mono:

  x  y ==> sqrt x  sqrt y

lemma real_sqrt_less_iff:

  (sqrt x < sqrt y) = (x < y)

lemma real_sqrt_le_iff:

  (sqrt x  sqrt y) = (x  y)

lemma real_sqrt_eq_iff:

  (sqrt x = sqrt y) = (x = y)

lemma real_sqrt_gt_0_iff:

  (0 < sqrt y) = (0 < y)

lemma real_sqrt_lt_0_iff:

  (sqrt x < 0) = (x < 0)

lemma real_sqrt_ge_0_iff:

  (0  sqrt y) = (0  y)

lemma real_sqrt_le_0_iff:

  (sqrt x  0) = (x  0)

lemma real_sqrt_eq_0_iff:

  (sqrt x = 0) = (x = 0)

lemma real_sqrt_gt_1_iff:

  (1 < sqrt y) = (1 < y)

lemma real_sqrt_lt_1_iff:

  (sqrt x < 1) = (x < 1)

lemma real_sqrt_ge_1_iff:

  (1  sqrt y) = (1  y)

lemma real_sqrt_le_1_iff:

  (sqrt x  1) = (x  1)

lemma real_sqrt_eq_1_iff:

  (sqrt x = 1) = (x = 1)

lemma isCont_real_sqrt:

  isCont sqrt x

lemma DERIV_real_sqrt:

  0 < x ==> DERIV sqrt x :> inverse (sqrt x) / 2

lemma not_real_square_gt_zero:

  0 < x * x) = (x = 0)

lemma real_sqrt_abs2:

  sqrt (x * x) = ¦x¦

lemma real_sqrt_pow2_gt_zero:

  0 < x ==> 0 < sqrt x ^ 2

lemma real_sqrt_not_eq_zero:

  0 < x ==> sqrt x  0

lemma real_inv_sqrt_pow2:

  0 < x ==> inverse (sqrt x) ^ 2 = inverse x

lemma real_sqrt_eq_zero_cancel:

  [| 0  x; sqrt x = 0 |] ==> x = 0

lemma real_sqrt_ge_one:

  1  x ==> 1  sqrt x

lemma real_sqrt_two_gt_zero:

  0 < sqrt 2

lemma real_sqrt_two_ge_zero:

  0  sqrt 2

lemma real_sqrt_two_gt_one:

  1 < sqrt 2

lemma sqrt_divide_self_eq:

  0  x ==> sqrt x / x = inverse (sqrt x)

lemma real_divide_square_eq:

  r * a / (r * r) = a / r

lemma lemma_real_divide_sqrt_less:

  0 < u ==> u / sqrt 2 < u

lemma four_x_squared:

  4 * x ^ 2 = (2 * x) ^ 2

Square Root of Sum of Squares

lemma real_sqrt_mult_self_sum_ge_zero:

  0  sqrt (x * x + y * y)

lemma real_sqrt_sum_squares_ge_zero:

  0  sqrt (x ^ 2 + y ^ 2)

lemma real_sqrt_sum_squares_mult_ge_zero:

  0  sqrt ((x ^ 2 + y ^ 2) * (xa ^ 2 + ya ^ 2))

lemma real_sqrt_sum_squares_mult_squared_eq:

  sqrt ((x ^ 2 + y ^ 2) * (xa ^ 2 + ya ^ 2)) ^ 2 =
  (x ^ 2 + y ^ 2) * (xa ^ 2 + ya ^ 2)

lemma real_sqrt_sum_squares_eq_cancel:

  sqrt (x ^ 2 + y ^ 2) = x ==> y = 0

lemma real_sqrt_sum_squares_eq_cancel2:

  sqrt (x ^ 2 + y ^ 2) = y ==> x = 0

lemma real_sqrt_sum_squares_ge1:

  x  sqrt (x ^ 2 + y ^ 2)

lemma real_sqrt_sum_squares_ge2:

  y  sqrt (x ^ 2 + y ^ 2)

lemma real_sqrt_ge_abs1:

  ¦x¦  sqrt (x ^ 2 + y ^ 2)

lemma real_sqrt_ge_abs2:

  ¦y¦  sqrt (x ^ 2 + y ^ 2)

lemma le_real_sqrt_sumsq:

  x  sqrt (x * x + y * y)

lemma power2_sum:

  (x + y) ^ 2 = x ^ 2 + y ^ 2 + (2::'a) * x * y

lemma power2_diff:

  (x - y) ^ 2 = x ^ 2 + y ^ 2 - (2::'a) * x * y

lemma real_sqrt_sum_squares_triangle_ineq:

  sqrt ((a + c) ^ 2 + (b + d) ^ 2)  sqrt (a ^ 2 + b ^ 2) + sqrt (c ^ 2 + d ^ 2)

lemma real_sqrt_sum_squares_less:

  [| ¦x¦ < u / sqrt 2; ¦y¦ < u / sqrt 2 |] ==> sqrt (x ^ 2 + y ^ 2) < u

lemma lemma_sqrt_hcomplex_capprox:

  [| 0 < u; x < u / 2; y < u / 2; 0  x; 0  y |] ==> sqrt (x ^ 2 + y ^ 2) < u

lemma real_root_pos2:

  [| 0 < n; 0  x |] ==> root n (x ^ n) = x

lemma real_root_pos_pos:

  [| 0 < n1; 0 < x1 |] ==> 0  root n1 x1

lemma real_root_pos_pos_le:

  [| 0 < n; 0  x |] ==> 0  root n x

lemma real_sqrt_mult_distrib:

  sqrt (x * y) = sqrt x * sqrt y

lemma real_sqrt_mult_distrib2:

  sqrt (x * y) = sqrt x * sqrt y

lemma real_sqrt_eq_zero_cancel_iff:

  (sqrt x = 0) = (x = 0)

lemma real_root_pos:

  0 < x ==> root (Suc n) (x ^ Suc n) = x