(*  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 SEQ Parity

begin

subsection {* Existence of Nth Root *}

text {*

  Various lemmas needed for this result. We follow the proof given by

  John Lindsay Orr (\texttt{jorr@math.unl.edu}) in his Analysis

  Webnotes available at \url{http://www.math.unl.edu/~webnotes}.

  Lemmas about sequences of reals are used to reach the result.

*}

lemma lemma_nth_realpow_non_empty:

     "[| (0::real) < a; 0 < n |] ==> \<exists>s. s : {x. x ^ n <= a & 0 < x}"

apply (case_tac "1 <= a")

apply (rule_tac x = 1 in exI)

apply (drule_tac [2] linorder_not_le [THEN iffD1])

apply (drule_tac [2] less_not_refl2 [THEN not0_implies_Suc], simp) 

apply (force intro!: realpow_Suc_le_self simp del: realpow_Suc)

done

text{*Used only just below*}

lemma realpow_ge_self2: "[| (1::real) \<le> r; 0 < n |] ==> r \<le> r ^ n"

by (insert power_increasing [of 1 n r], simp)

lemma lemma_nth_realpow_isUb_ex:

     "[| (0::real) < a; 0 < n |]  

      ==> \<exists>u. isUb (UNIV::real set) {x. x ^ n <= a & 0 < x} u"

apply (case_tac "1 <= a")

apply (rule_tac x = a in exI)

apply (drule_tac [2] linorder_not_le [THEN iffD1])

apply (rule_tac [2] x = 1 in exI)

apply (rule_tac [!] setleI [THEN isUbI], safe)

apply (simp_all (no_asm))

apply (rule_tac [!] ccontr)

apply (drule_tac [!] linorder_not_le [THEN iffD1])

apply (drule realpow_ge_self2, assumption)

apply (drule_tac n = n in realpow_less)

apply (assumption+)

apply (drule real_le_trans, assumption)

apply (drule_tac y = "y ^ n" in order_less_le_trans, assumption, simp) 

apply (drule_tac n = n in zero_less_one [THEN realpow_less], auto)

done

lemma nth_realpow_isLub_ex:

     "[| (0::real) < a; 0 < n |]  

      ==> \<exists>u. isLub (UNIV::real set) {x. x ^ n <= a & 0 < x} u"

by (blast intro: lemma_nth_realpow_isUb_ex lemma_nth_realpow_non_empty reals_complete)

subsubsection {* First Half -- Lemmas First *}

lemma lemma_nth_realpow_seq:

     "isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u  

           ==> u + inverse(real (Suc k)) ~: {x. x ^ n <= a & 0 < x}"

apply (safe, drule isLubD2, blast)

apply (simp add: linorder_not_less [symmetric])

done

lemma lemma_nth_realpow_isLub_gt_zero:

     "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  

         0 < a; 0 < n |] ==> 0 < u"

apply (drule lemma_nth_realpow_non_empty, auto)

apply (drule_tac y = s in isLub_isUb [THEN isUbD])

apply (auto intro: order_less_le_trans)

done

lemma lemma_nth_realpow_isLub_ge:

     "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  

         0 < a; 0 < n |] ==> ALL k. a <= (u + inverse(real (Suc k))) ^ n"

apply safe

apply (frule lemma_nth_realpow_seq, safe)

apply (auto elim: order_less_asym simp add: linorder_not_less [symmetric]

            iff: real_0_less_add_iff) --{*legacy iff rule!*}

apply (simp add: linorder_not_less)

apply (rule order_less_trans [of _ 0])

apply (auto intro: lemma_nth_realpow_isLub_gt_zero)

done

text{*First result we want*}

lemma realpow_nth_ge:

     "[| (0::real) < a; 0 < n;  

     isLub (UNIV::real set)  

     {x. x ^ n <= a & 0 < x} u |] ==> a <= u ^ n"

apply (frule lemma_nth_realpow_isLub_ge, safe)

apply (rule LIMSEQ_inverse_real_of_nat_add [THEN LIMSEQ_pow, THEN LIMSEQ_le_const])

apply (auto simp add: real_of_nat_def)

done

subsubsection {* Second Half *}

lemma less_isLub_not_isUb:

     "[| isLub (UNIV::real set) S u; x < u |]  

           ==> ~ isUb (UNIV::real set) S x"

apply safe

apply (drule isLub_le_isUb, assumption)

apply (drule order_less_le_trans, auto)

done

lemma not_isUb_less_ex:

     "~ isUb (UNIV::real set) S u ==> \<exists>x \<in> S. u < x"

apply (rule ccontr, erule contrapos_np)

apply (rule setleI [THEN isUbI])

apply (auto simp add: linorder_not_less [symmetric])

done

lemma real_mult_less_self: "0 < r ==> r * (1 + -inverse(real (Suc n))) < r"

apply (simp (no_asm) add: right_distrib)

apply (rule add_less_cancel_left [of "-r", THEN iffD1])

apply (auto intro: mult_pos_pos

            simp add: add_assoc [symmetric] neg_less_0_iff_less)

done

lemma real_of_nat_inverse_le_iff:

  "(inverse (real(Suc n)) \<le> r) = (1 \<le> r * real(Suc n))"

by (simp add: inverse_eq_divide pos_divide_le_eq)

lemma real_mult_add_one_minus_ge_zero:

     "0 < r ==>  0 <= r*(1 + -inverse(real (Suc n)))"

by (simp add: zero_le_mult_iff real_of_nat_inverse_le_iff real_0_le_add_iff)

lemma lemma_nth_realpow_isLub_le:

     "[| isLub (UNIV::real set) {x. x ^ n <= a & (0::real) < x} u;  

       0 < a; 0 < n |] ==> ALL k. (u*(1 + -inverse(real (Suc k)))) ^ n <= a"

apply safe

apply (frule less_isLub_not_isUb [THEN not_isUb_less_ex])

apply (rule_tac n = k in real_mult_less_self)

apply (blast intro: lemma_nth_realpow_isLub_gt_zero, safe)

apply (drule_tac n = k in

        lemma_nth_realpow_isLub_gt_zero [THEN real_mult_add_one_minus_ge_zero], assumption+)

apply (blast intro: order_trans order_less_imp_le power_mono) 

done

text{*Second result we want*}

lemma realpow_nth_le:

     "[| (0::real) < a; 0 < n;  

     isLub (UNIV::real set)  

     {x. x ^ n <= a & 0 < x} u |] ==> u ^ n <= a"

apply (frule lemma_nth_realpow_isLub_le, safe)

apply (rule LIMSEQ_inverse_real_of_nat_add_minus_mult

                [THEN LIMSEQ_pow, THEN LIMSEQ_le_const2])

apply (auto simp add: real_of_nat_def)

done

text{*The theorem at last!*}

lemma realpow_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. r ^ n = a"

apply (frule nth_realpow_isLub_ex, auto)

apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym)

done

text {* positive only *}

lemma realpow_pos_nth: "[| (0::real) < a; 0 < n |] ==> \<exists>r. 0 < r & r ^ n = a"

apply (frule nth_realpow_isLub_ex, auto)

apply (auto intro: realpow_nth_le realpow_nth_ge order_antisym lemma_nth_realpow_isLub_gt_zero)

done

lemma realpow_pos_nth2: "(0::real) < a  ==> \<exists>r. 0 < r & r ^ Suc n = a"

by (blast intro: realpow_pos_nth)

text {* uniqueness of nth positive root *}

lemma realpow_pos_nth_unique:

     "[| (0::real) < a; 0 < n |] ==> EX! r. 0 < r & r ^ n = a"

apply (auto intro!: realpow_pos_nth)

apply (cut_tac x = r and y = y in linorder_less_linear, auto)

apply (drule_tac x = r in realpow_less)

apply (drule_tac [4] x = y in realpow_less, auto)

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] \<Rightarrow> real" where

  "root n x = (if 0 < x then (THE u. 0 < u \<and> u ^ n = x) else

               if x < 0 then - (THE u. 0 < u \<and> 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 \<Longrightarrow> root n (- x) = - root n x"

unfolding root_def by simp

lemma real_root_gt_zero: "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 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 *)

  "\<lbrakk>0 < n; 0 < x\<rbrakk> \<Longrightarrow> 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 *)

  "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n x ^ n = x"

by (auto simp add: order_le_less real_root_pow_pos)

lemma real_root_ge_zero: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 0 \<le> root n x"

by (auto simp add: order_le_less real_root_gt_zero)

lemma real_root_power_cancel: "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> root n (x ^ n) = x"

apply (subgoal_tac "0 \<le> x ^ n")

apply (subgoal_tac "0 \<le> 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 real_root_pos_unique:

  "\<lbrakk>0 < n; 0 \<le> y; y ^ n = x\<rbrakk> \<Longrightarrow> root n x = y"

by (erule subst, rule real_root_power_cancel)

lemma real_root_one [simp]: "0 < n \<Longrightarrow> 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:

  "\<lbrakk>0 < n; 0 \<le> x; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"

apply (subgoal_tac "0 \<le> 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: "\<lbrakk>0 < n; x < y\<rbrakk> \<Longrightarrow> root n x < root n y"

apply (cases "0 \<le> x")

apply (erule (2) real_root_less_mono_lemma)

apply (cases "0 \<le> 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: "\<lbrakk>0 < n; x \<le> y\<rbrakk> \<Longrightarrow> root n x \<le> root n y"

by (auto simp add: order_le_less real_root_less_mono)

lemma real_root_less_iff [simp]:

  "0 < n \<Longrightarrow> (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 \<Longrightarrow> (root n x \<le> root n y) = (x \<le> y)"

apply (cases "x \<le> 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 \<Longrightarrow> (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]

text {* Roots of multiplication and division *}

lemma real_root_mult_lemma:

  "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> 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:

  "\<lbrakk>0 < n; 0 \<le> x\<rbrakk> \<Longrightarrow> 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 \<le> x" and "0 \<le> y"

  thus ?thesis by (rule real_root_mult_lemma [OF n])

next

  assume "0 \<le> x" and "y \<le> 0"

  hence "0 \<le> x" and "0 \<le> - 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 \<le> 0" and "0 \<le> y"

  hence "0 \<le> - x" and "0 \<le> 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 \<le> 0" and "y \<le> 0"

  hence "0 \<le> - x" and "0 \<le> - 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 \<le> x"

  thus ?thesis by (rule real_root_inverse_lemma [OF n])

next

  assume "x \<le> 0"

  hence "0 \<le> - 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 \<Longrightarrow> 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 \<Longrightarrow> root n (x ^ k) = root n x ^ k"

by (induct k, simp_all add: real_root_mult)

subsection {* Square Root *}

definition

  sqrt :: "real \<Rightarrow> real" where

  "sqrt = root 2"

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

lemma real_sqrt_unique: "\<lbrakk>y\<twosuperior> = x; 0 \<le> y\<rbrakk> \<Longrightarrow> sqrt x = y"

unfolding sqrt_def by (rule real_root_pos_unique [OF pos2])

lemma real_sqrt_abs [simp]: "sqrt (x\<twosuperior>) = \<bar>x\<bar>"

apply (rule real_sqrt_unique)

apply (rule power2_abs)

apply (rule abs_ge_zero)

done

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

unfolding sqrt_def by (rule real_root_pow_pos2 [OF pos2])

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

unfolding sqrt_def by (rule real_root_gt_zero [OF pos2])

lemma real_sqrt_ge_zero: "0 \<le> x \<Longrightarrow> 0 \<le> sqrt x"

unfolding sqrt_def by (rule real_root_ge_zero [OF pos2])

lemma real_sqrt_less_mono: "x < y \<Longrightarrow> sqrt x < sqrt y"

unfolding sqrt_def by (rule real_root_less_mono [OF pos2])

lemma real_sqrt_le_mono: "x \<le> y \<Longrightarrow> sqrt x \<le> 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 \<le> sqrt y) = (x \<le> 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 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) = \<bar>x\<bar>"

apply (subst power2_eq_square [symmetric])

apply (rule real_sqrt_abs)

done

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

by simp (* TODO: delete *)

lemma real_sqrt_not_eq_zero: "0 < x ==> sqrt x \<noteq> 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 \<le> x; sqrt(x) = 0|] ==> x = 0"

by simp

lemma real_sqrt_ge_one: "1 \<le> x ==> 1 \<le> sqrt x"

by simp

lemma sqrt_divide_self_eq:

  assumes nneg: "0 \<le> x"

  shows "sqrt x / x = inverse (sqrt x)"

proof cases

  assume "x=0" thus ?thesis by simp

next

  assume nz: "x\<noteq>0" 

  hence pos: "0<x" using nneg by arith

  show ?thesis

  proof (rule right_inverse_eq [THEN iffD1, THEN sym]) 

    show "sqrt x / x \<noteq> 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

subsection {* Square Root of Sum of Squares *}

lemma "(sqrt (x\<twosuperior> + y\<twosuperior>))\<twosuperior> = x\<twosuperior> + y\<twosuperior>"

by simp

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]) 

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: zero_le_mult_iff)

lemma real_sqrt_sum_squares_ge1 [simp]: "x \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"

by (rule power2_le_imp_le, simp_all)

lemma real_sqrt_sum_squares_ge2 [simp]: "y \<le> sqrt(x\<twosuperior> + y\<twosuperior>)"

by (rule power2_le_imp_le, simp_all)

lemma real_sqrt_sos_less_one_iff: "(sqrt (x\<twosuperior> + y\<twosuperior>) < 1) = (x\<twosuperior> + y\<twosuperior> < 1)"

by (rule real_sqrt_lt_1_iff)

lemma real_sqrt_sos_eq_one_iff: "(sqrt (x\<twosuperior> + y\<twosuperior>) = 1) = (x\<twosuperior> + y\<twosuperior> = 1)"

by (rule real_sqrt_eq_1_iff)

lemma power2_sum:

  fixes x y :: "'a::{number_ring,recpower}"

  shows "(x + y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> + 2 * x * y"

by (simp add: left_distrib right_distrib power2_eq_square)

lemma power2_diff:

  fixes x y :: "'a::{number_ring,recpower}"

  shows "(x - y)\<twosuperior> = x\<twosuperior> + y\<twosuperior> - 2 * x * y"

by (simp add: left_diff_distrib right_diff_distrib power2_eq_square)

lemma real_sqrt_sum_squares_triangle_ineq:

  "sqrt ((a + c)\<twosuperior> + (b + d)\<twosuperior>) \<le> sqrt (a\<twosuperior> + b\<twosuperior>) + sqrt (c\<twosuperior> + d\<twosuperior>)"

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_distrib)

apply (subgoal_tac "0 \<le> b\<twosuperior> * c\<twosuperior> + a\<twosuperior> * d\<twosuperior> - 2 * (a * c) * (b * d)", simp)

apply (rule_tac b="(a * d - b * c)\<twosuperior>" 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

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 \<Longrightarrow> root (Suc n) (x ^ (Suc n)) = x"

by (rule real_root_power_cancel [OF zero_less_Suc order_less_imp_le])

(* FIXME: the stronger version of real_root_less_iff

 breaks CauchysMeanTheorem.list_gmean_gt_iff from AFP. *)

declare real_root_less_iff [simp del]

lemma real_root_less_iff_nonneg [simp]:

  "\<lbrakk>0 < n; 0 \<le> x; 0 \<le> y\<rbrakk> \<Longrightarrow> (root n x < root n y) = (x < y)"

by (rule real_root_less_iff)

end