# Theory Sqrt_Babylonian_Auxiliary

(* Title: Computing Square Roots using the Babylonian Method Author: René Thiemann <rene.thiemann@uibk.ac.at> Maintainer: René Thiemann License: LGPL *) (* Copyright 2009-2014 René Thiemann This file is part of IsaFoR/CeTA. IsaFoR/CeTA is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. IsaFoR/CeTA is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with IsaFoR/CeTA. If not, see <http://www.gnu.org/licenses/>. *) section ‹Auxiliary lemmas which might be moved into the Isabelle distribution.› theory Sqrt_Babylonian_Auxiliary imports Complex_Main begin lemma mod_div_equality_int: "(n :: int) div x * x = n - n mod x" using div_mult_mod_eq[of n x] by arith lemma div_is_floor_divide_rat: "n div y = ⌊rat_of_int n / rat_of_int y⌋" unfolding Fract_of_int_quotient[symmetric] floor_Fract by simp lemma div_is_floor_divide_real: "n div y = ⌊real_of_int n / of_int y⌋" unfolding div_is_floor_divide_rat[of n y] by (metis Ratreal_def of_rat_divide of_rat_of_int_eq real_floor_code) lemma floor_div_pos_int: fixes r :: "'a :: floor_ceiling" assumes n: "n > 0" shows "⌊r / of_int n⌋ = ⌊r⌋ div n" (is "?l = ?r") proof - let ?of_int = "of_int :: int ⇒ 'a" define rhs where "rhs = ⌊r⌋ div n" let ?n = "?of_int n" define m where "m = ⌊r⌋ mod n" let ?m = "?of_int m" from div_mult_mod_eq[of "floor r" n] have dm: "rhs * n + m = ⌊r⌋" unfolding rhs_def m_def by simp have mn: "m < n" and m0: "m ≥ 0" using n m_def by auto define e where "e = r - ?of_int ⌊r⌋" have e0: "e ≥ 0" unfolding e_def by (metis diff_self eq_iff floor_diff_of_int zero_le_floor) have e1: "e < 1" unfolding e_def by (metis diff_self dual_order.refl floor_diff_of_int floor_le_zero) have "r = ?of_int ⌊r⌋ + e" unfolding e_def by simp also have "⌊r⌋ = rhs * n + m" using dm by simp finally have "r = ?of_int (rhs * n + m) + e" . hence "r / ?n = ?of_int (rhs * n) / ?n + (e + ?m) / ?n" using n by (simp add: field_simps) also have "?of_int (rhs * n) / ?n = ?of_int rhs" using n by auto finally have *: "r / ?of_int n = (e + ?of_int m) / ?of_int n + ?of_int rhs" by simp have "?l = rhs + floor ((e + ?m) / ?n)" unfolding * by simp also have "floor ((e + ?m) / ?n) = 0" proof (rule floor_unique) show "?of_int 0 ≤ (e + ?m) / ?n" using e0 m0 n by (metis add_increasing2 divide_nonneg_pos of_int_0 of_int_0_le_iff of_int_0_less_iff) show "(e + ?m) / ?n < ?of_int 0 + 1" proof (rule ccontr) from n have n': "?n > 0" "?n ≥ 0" by simp_all assume "¬ ?thesis" hence "(e + ?m) / ?n ≥ 1" by auto from mult_right_mono[OF this n'(2)] have "?n ≤ e + ?m" using n'(1) by simp also have "?m ≤ ?n - 1" using mn by (metis of_int_1 of_int_diff of_int_le_iff zle_diff1_eq) finally have "?n ≤ e + ?n - 1" by auto with e1 show False by arith qed qed finally show ?thesis unfolding rhs_def by simp qed lemma floor_div_neg_int: fixes r :: "'a :: floor_ceiling" assumes n: "n < 0" shows "⌊r / of_int n⌋ = ⌈r⌉ div n" proof - from n have n': "- n > 0" by auto have "⌊r / of_int n⌋ = ⌊ - r / of_int (- n)⌋" using n by (metis floor_of_int floor_zero less_int_code(1) minus_divide_left minus_minus nonzero_minus_divide_right of_int_minus) also have "… = ⌊ - r ⌋ div (- n)" by (rule floor_div_pos_int[OF n']) also have "… = ⌈ r ⌉ div n" using n by (metis ceiling_def div_minus_right) finally show ?thesis . qed lemma divide_less_floor1: "n / y < of_int (floor (n / y)) + 1" by (metis floor_less_iff less_add_one of_int_1 of_int_add) context linordered_idom begin lemma sgn_int_pow_if [simp]: "sgn x ^ p = (if even p then 1 else sgn x)" if "x ≠ 0" using that by (induct p) simp_all lemma compare_pow_le_iff: "p > 0 ⟹ (x :: 'a) ≥ 0 ⟹ y ≥ 0 ⟹ (x ^ p ≤ y ^ p) = (x ≤ y)" by (rule power_mono_iff) lemma compare_pow_less_iff: "p > 0 ⟹ (x :: 'a) ≥ 0 ⟹ y ≥ 0 ⟹ (x ^ p < y ^ p) = (x < y)" using compare_pow_le_iff [of p x y] using local.dual_order.order_iff_strict local.power_strict_mono by blast end lemma quotient_of_int[simp]: "quotient_of (of_int i) = (i,1)" by (metis Rat.of_int_def quotient_of_int) lemma quotient_of_nat[simp]: "quotient_of (of_nat i) = (int i,1)" by (metis Rat.of_int_def Rat.quotient_of_int of_int_of_nat_eq) lemma square_lesseq_square: "⋀ x y. 0 ≤ (x :: 'a :: linordered_field) ⟹ 0 ≤ y ⟹ (x * x ≤ y * y) = (x ≤ y)" by (metis mult_mono mult_strict_mono' not_less) lemma square_less_square: "⋀ x y. 0 ≤ (x :: 'a :: linordered_field) ⟹ 0 ≤ y ⟹ (x * x < y * y) = (x < y)" by (metis mult_mono mult_strict_mono' not_less) lemma sqrt_sqrt[simp]: "x ≥ 0 ⟹ sqrt x * sqrt x = x" by (metis real_sqrt_pow2 power2_eq_square) lemma abs_lesseq_square: "abs (x :: real) ≤ abs y ⟷ x * x ≤ y * y" using square_lesseq_square[of "abs x" "abs y"] by auto end

# Theory Log_Impl

(* Title: Computing Square Roots using the Babylonian Method Author: René Thiemann <rene.thiemann@uibk.ac.at> Maintainer: René Thiemann License: LGPL *) section ‹A Fast Logarithm Algorithm› theory Log_Impl imports Sqrt_Babylonian_Auxiliary begin text ‹We implement the discrete logarithm function in a manner similar to a repeated squaring exponentiation algorithm.› text ‹In order to prove termination of the algorithm without intermediate checks we need to ensure that we only use proper bases, i.e., values of at least 2. This will be encoded into a separate type.› typedef proper_base = "{x :: int. x ≥ 2}" by auto setup_lifting type_definition_proper_base lift_definition get_base :: "proper_base ⇒ int" is "λ x. x" . lift_definition square_base :: "proper_base ⇒ proper_base" is "λ x. x * x" proof - fix i :: int assume i: "2 ≤ i" have "2 * 2 ≤ i * i" by (rule mult_mono[OF i i], insert i, auto) thus "2 ≤ i * i" by auto qed lift_definition into_base :: "int ⇒ proper_base" is "λ x. if x ≥ 2 then x else 2" by auto lemma square_base: "get_base (square_base b) = get_base b * get_base b" by (transfer, auto) lemma get_base_2: "get_base b ≥ 2" by (transfer, auto) lemma b_less_square_base_b: "get_base b < get_base (square_base b)" unfolding square_base using get_base_2[of b] by simp lemma b_less_div_base_b: assumes xb: "¬ x < get_base b" shows "x div get_base b < x" proof - from get_base_2[of b] have b: "get_base b ≥ 2" . with xb have x2: "x ≥ 2" by auto with b int_div_less_self[of x "(get_base b)"] show ?thesis by auto qed text ‹We now state the main algorithm.› function log_main :: "proper_base ⇒ int ⇒ nat × int" where "log_main b x = (if x < get_base b then (0,1) else case log_main (square_base b) x of (z, bz) ⇒ let l = 2 * z; bz1 = bz * get_base b in if x < bz1 then (l,bz) else (Suc l,bz1))" by pat_completeness auto termination by (relation "measure (λ (b,x). nat (1 + x - get_base b))", insert b_less_square_base_b, auto) lemma log_main: "x > 0 ⟹ log_main b x = (y,by) ⟹ by = (get_base b)^y ∧ (get_base b)^y ≤ x ∧ x < (get_base b)^(Suc y)" proof (induct b x arbitrary: y "by" rule: log_main.induct) case (1 b x y "by") note x = 1(2) note y = 1(3) note IH = 1(1) let ?b = "get_base b" show ?case proof (cases "x < ?b") case True with x y show ?thesis by auto next case False obtain z bz where zz: "log_main (square_base b) x = (z,bz)" by (cases "log_main (square_base b) x", auto) have id: "get_base (square_base b) ^ k = ?b ^ (2 * k)" for k unfolding square_base by (simp add: power_mult semiring_normalization_rules(29)) from IH[OF False x zz, unfolded id] have z: "?b ^ (2 * z) ≤ x" "x < ?b ^ (2 * Suc z)" and bz: "bz = get_base b ^ (2 * z)" by auto from y[unfolded log_main.simps[of b x] Let_def zz split] bz False have yy: "(if x < bz * ?b then (2 * z, bz) else (Suc (2 * z), bz * ?b)) = (y, by)" by auto show ?thesis proof (cases "x < bz * ?b") case True with yy have yz: "y = 2 * z" "by = bz" by auto from True z(1) bz show ?thesis unfolding yz by (auto simp: ac_simps) next case False with yy have yz: "y = Suc (2 * z)" "by = ?b * bz" by auto from False have "?b ^ Suc (2 * z) ≤ x" by (auto simp: bz ac_simps) with z(2) bz show ?thesis unfolding yz by auto qed qed qed text ‹We then derive the floor- and ceiling-log functions.› definition log_floor :: "int ⇒ int ⇒ nat" where "log_floor b x = fst (log_main (into_base b) x)" definition log_ceiling :: "int ⇒ int ⇒ nat" where "log_ceiling b x = (case log_main (into_base b) x of (y,by) ⇒ if x = by then y else Suc y)" lemma log_floor_sound: assumes "b > 1" "x > 0" "log_floor b x = y" shows "b^y ≤ x" "x < b^(Suc y)" proof - from assms(1,3) have id: "get_base (into_base b) = b" by transfer auto obtain yy bb where log: "log_main (into_base b) x = (yy,bb)" by (cases "log_main (into_base b) x", auto) from log_main[OF assms(2) log] assms(3)[unfolded log_floor_def log] id show "b^y ≤ x" "x < b^(Suc y)" by auto qed lemma log_ceiling_sound: assumes "b > 1" "x > 0" "log_ceiling b x = y" shows "x ≤ b^y" "y ≠ 0 ⟹ b^(y - 1) < x" proof - from assms(1,3) have id: "get_base (into_base b) = b" by transfer auto obtain yy bb where log: "log_main (into_base b) x = (yy,bb)" by (cases "log_main (into_base b) x", auto) from log_main[OF assms(2) log, unfolded id] assms(3)[unfolded log_ceiling_def log split] have bnd: "b ^ yy ≤ x" "x < b ^ Suc yy" and y: "y = (if x = b ^ yy then yy else Suc yy)" by auto have "x ≤ b^y ∧ (y ≠ 0 ⟶ b^(y - 1) < x)" proof (cases "x = b ^ yy") case True with y bnd assms(1) show ?thesis by (cases yy, auto) next case False with y bnd show ?thesis by auto qed thus "x ≤ b^y" "y ≠ 0 ⟹ b^(y - 1) < x" by auto qed text ‹Finally, we connect it to the @{const log} function working on real numbers.› lemma log_floor[simp]: assumes b: "b > 1" and x: "x > 0" shows "log_floor b x = ⌊log b x⌋" proof - obtain y where y: "log_floor b x = y" by auto note main = log_floor_sound[OF assms y] from b x have *: "1 < real_of_int b" "0 < real_of_int (b ^ y)" "0 < real_of_int x" and **: "1 < real_of_int b" "0 < real_of_int x" "0 < real_of_int (b ^ Suc y)" by auto show ?thesis unfolding y proof (rule sym, rule floor_unique) show "real_of_int (int y) ≤ log (real_of_int b) (real_of_int x)" using main(1)[folded log_le_cancel_iff[OF *, unfolded of_int_le_iff]] using log_pow_cancel[of b y] b by auto show "log (real_of_int b) (real_of_int x) < real_of_int (int y) + 1" using main(2)[folded log_less_cancel_iff[OF **, unfolded of_int_less_iff]] using log_pow_cancel[of b "Suc y"] b by auto qed qed lemma log_ceiling[simp]: assumes b: "b > 1" and x: "x > 0" shows "log_ceiling b x = ⌈log b x⌉" proof - obtain y where y: "log_ceiling b x = y" by auto note main = log_ceiling_sound[OF assms y] from b x have *: "1 < real_of_int b" "0 < real_of_int (b ^ (y - 1))" "0 < real_of_int x" and **: "1 < real_of_int b" "0 < real_of_int x" "0 < real_of_int (b ^ y)" by auto show ?thesis unfolding y proof (rule sym, rule ceiling_unique) show "log (real_of_int b) (real_of_int x) ≤ real_of_int (int y)" using main(1)[folded log_le_cancel_iff[OF **, unfolded of_int_le_iff]] using log_pow_cancel[of b y] b by auto from x have x: "x ≥ 1" by auto show "real_of_int (int y) - 1 < log (real_of_int b) (real_of_int x)" proof (cases "y = 0") case False thus ?thesis using main(2)[folded log_less_cancel_iff[OF *, unfolded of_int_less_iff]] using log_pow_cancel[of b "y - 1"] b x by auto next case True have "real_of_int (int y) - 1 = log b (1/b)" using True b by (subst log_divide, auto) also have "… < log b 1" by (subst log_less_cancel_iff, insert b, auto) also have "… ≤ log b x" by (subst log_le_cancel_iff, insert b x, auto) finally show "real_of_int (int y) - 1 < log (real_of_int b) (real_of_int x)" . qed qed qed end

# Theory NthRoot_Impl

(* Title: Computing Square Roots using the Babylonian Method Author: René Thiemann <rene.thiemann@uibk.ac.at> Maintainer: René Thiemann License: LGPL *) (* Copyright 2009-2014 René Thiemann This file is part of IsaFoR/CeTA. IsaFoR/CeTA is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. IsaFoR/CeTA is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with IsaFoR/CeTA. If not, see <http://www.gnu.org/licenses/>. *) section ‹Executable algorithms for $p$-th roots› theory NthRoot_Impl imports Log_Impl Cauchy.CauchysMeanTheorem begin text ‹ We implemented algorithms to decide $\sqrt[p]{n} \in \rats$ and to compute $\lfloor \sqrt[p]{n} \rfloor$. To this end, we use a variant of Newton iteration which works with integer division instead of floating point or rational division. To get suitable starting values for the Newton iteration, we also implemented a function to approximate logarithms. › subsection ‹Logarithm› text ‹For computing the $p$-th root of a number $n$, we must choose a starting value in the iteration. Here, we use @{term "2 ^ (nat ⌈of_int ⌈log 2 n⌉ / p⌉)"}. › text ‹We use a partial efficient algorithm, which does not terminate on corner-cases, like $b = 0$ or $p = 1$, and invoke it properly afterwards. Then there is a second algorithm which terminates on these corner-cases by additional guards and on which we can perform induction. › subsection ‹Computing the $p$-th root of an integer number› text ‹Using the logarithm, we can define an executable version of the intended starting value. Its main property is the inequality @{term "(start_value x p) ^ p ≥ x"}, i.e., the start value is larger than the p-th root. This property is essential, since our algorithm will abort as soon as we fall below the p-th root.› definition start_value :: "int ⇒ nat ⇒ int" where "start_value n p = 2 ^ (nat ⌈of_nat (log_ceiling 2 n) / rat_of_nat p⌉)" lemma start_value_main: assumes x: "x ≥ 0" and p: "p > 0" shows "x ≤ (start_value x p)^p ∧ start_value x p ≥ 0" proof (cases "x = 0") case True with p show ?thesis unfolding start_value_def True by simp next case False with x have x: "x > 0" by auto define l2x where "l2x = ⌈log 2 x⌉" define pow where "pow = nat ⌈rat_of_int l2x / of_nat p⌉" have "root p x = x powr (1 / p)" by (rule root_powr_inverse, insert x p, auto) also have "… = (2 powr (log 2 x)) powr (1 / p)" using powr_log_cancel[of 2 x] x by auto also have "… = 2 powr (log 2 x * (1 / p))" by (rule powr_powr) also have "log 2 x * (1 / p) = log 2 x / p" using p by auto finally have r: "root p x = 2 powr (log 2 x / p)" . have lp: "log 2 x ≥ 0" using x by auto hence l2pos: "l2x ≥ 0" by (auto simp: l2x_def) have "log 2 x / p ≤ l2x / p" using x p unfolding l2x_def by (metis divide_right_mono le_of_int_ceiling of_nat_0_le_iff) also have "… ≤ ⌈l2x / (p :: real)⌉" by (simp add: ceiling_correct) also have "l2x / real p = l2x / real_of_rat (of_nat p)" by (metis of_rat_of_nat_eq) also have "of_int l2x = real_of_rat (of_int l2x)" by (metis of_rat_of_int_eq) also have "real_of_rat (of_int l2x) / real_of_rat (of_nat p) = real_of_rat (rat_of_int l2x / of_nat p)" by (metis of_rat_divide) also have "⌈real_of_rat (rat_of_int l2x / rat_of_nat p)⌉ = ⌈rat_of_int l2x / of_nat p⌉" by simp also have "⌈rat_of_int l2x / of_nat p⌉ ≤ real pow" unfolding pow_def by auto finally have le: "log 2 x / p ≤ pow" . from powr_mono[OF le, of 2, folded r] have "root p x ≤ 2 powr pow" by auto also have "… = 2 ^ pow" by (rule powr_realpow, auto) also have "… = of_int ((2 :: int) ^ pow)" by simp also have "pow = (nat ⌈of_int (log_ceiling 2 x) / rat_of_nat p⌉)" unfolding pow_def l2x_def using x by simp also have "real_of_int ((2 :: int) ^ … ) = start_value x p" unfolding start_value_def by simp finally have less: "root p x ≤ start_value x p" . have "0 ≤ root p x" using p x by auto also have "… ≤ start_value x p" by (rule less) finally have start: "0 ≤ start_value x p" by simp from power_mono[OF less, of p] have "root p (of_int x) ^ p ≤ of_int (start_value x p) ^ p" using p x by auto also have "… = start_value x p ^ p" by simp also have "root p (of_int x) ^ p = x" using p x by force finally have "x ≤ (start_value x p) ^ p" by presburger with start show ?thesis by auto qed lemma start_value: assumes x: "x ≥ 0" and p: "p > 0" shows "x ≤ (start_value x p) ^ p" "start_value x p ≥ 0" using start_value_main[OF x p] by auto text ‹We now define the Newton iteration to compute the $p$-th root. We are working on the integers, where every @{term "(/)"} is replaced by @{term "(div)"}. We are proving several things within a locale which ensures that $p > 0$, and where $pm = p - 1$. › locale fixed_root = fixes p pm :: nat assumes p: "p = Suc pm" begin function root_newton_int_main :: "int ⇒ int ⇒ int × bool" where "root_newton_int_main x n = (if (x < 0 ∨ n < 0) then (0,False) else (if x ^ p ≤ n then (x, x ^ p = n) else root_newton_int_main ((n div (x ^ pm) + x * int pm) div (int p)) n))" by pat_completeness auto end text ‹For the executable algorithm we omit the guard and use a let-construction› partial_function (tailrec) root_int_main' :: "nat ⇒ int ⇒ int ⇒ int ⇒ int ⇒ int × bool" where [code]: "root_int_main' pm ipm ip x n = (let xpm = x^pm; xp = xpm * x in if xp ≤ n then (x, xp = n) else root_int_main' pm ipm ip ((n div xpm + x * ipm) div ip) n)" text ‹In the following algorithm, we start the iteration. It will compute @{term "⌊root p n⌋"} and a boolean to indicate whether the root is exact.› definition root_int_main :: "nat ⇒ int ⇒ int × bool" where "root_int_main p n ≡ if p = 0 then (1,n = 1) else let pm = p - 1 in root_int_main' pm (int pm) (int p) (start_value n p) n" text ‹Once we have proven soundness of @{const fixed_root.root_newton_int_main} and equivalence to @{const root_int_main}, it is easy to assemble the following algorithm which computes all roots for arbitrary integers.› definition root_int :: "nat ⇒ int ⇒ int list" where "root_int p x ≡ if p = 0 then [] else if x = 0 then [0] else let e = even p; s = sgn x; x' = abs x in if x < 0 ∧ e then [] else case root_int_main p x' of (y,True) ⇒ if e then [y,-y] else [s * y] | _ ⇒ []" text ‹We start with proving termination of @{const fixed_root.root_newton_int_main}.› context fixed_root begin lemma iteration_mono_eq: assumes xn: "x ^ p = (n :: int)" shows "(n div x ^ pm + x * int pm) div int p = x" proof - have [simp]: "⋀ n. (x + x * n) = x * (1 + n)" by (auto simp: field_simps) show ?thesis unfolding xn[symmetric] p by simp qed lemma p0: "p ≠ 0" unfolding p by auto text ‹The following property is the essential property for proving termination of @{const "root_newton_int_main"}. › lemma iteration_mono_less: assumes x: "x ≥ 0" and n: "n ≥ 0" and xn: "x ^ p > (n :: int)" shows "(n div x ^ pm + x * int pm) div int p < x" proof - let ?sx = "(n div x ^ pm + x * int pm) div int p" from xn have xn_le: "x ^ p ≥ n" by auto from xn x n have x0: "x > 0" using not_le p by fastforce from p have xp: "x ^ p = x * x ^ pm" by auto from x n have "n div x ^ pm * x ^ pm ≤ n" by (auto simp add: minus_mod_eq_div_mult [symmetric] mod_int_pos_iff not_less power_le_zero_eq) also have "… ≤ x ^ p" using xn by auto finally have le: "n div x ^ pm ≤ x" using x x0 unfolding xp by simp have "?sx ≤ (x^p div x ^ pm + x * int pm) div int p" by (rule zdiv_mono1, insert le p0, unfold xp, auto) also have "x^p div x ^ pm = x" unfolding xp by auto also have "x + x * int pm = x * int p" unfolding p by (auto simp: field_simps) also have "x * int p div int p = x" using p by force finally have le: "?sx ≤ x" . { assume "?sx = x" from arg_cong[OF this, of "λ x. x * int p"] have "x * int p ≤ (n div x ^ pm + x * int pm) div (int p) * int p" using p0 by simp also have "… ≤ n div x ^ pm + x * int pm" unfolding mod_div_equality_int using p by auto finally have "n div x^pm ≥ x" by (auto simp: p field_simps) from mult_right_mono[OF this, of "x ^ pm"] have ge: "n div x^pm * x^pm ≥ x^p" unfolding xp using x by auto from div_mult_mod_eq[of n "x^pm"] have "n div x^pm * x^pm = n - n mod x^pm" by arith from ge[unfolded this] have le: "x^p ≤ n - n mod x^pm" . from x n have ge: "n mod x ^ pm ≥ 0" by (auto simp add: mod_int_pos_iff not_less power_le_zero_eq) from le ge have "n ≥ x^p" by auto with xn have False by auto } with le show ?thesis unfolding p by fastforce qed lemma iteration_mono_lesseq: assumes x: "x ≥ 0" and n: "n ≥ 0" and xn: "x ^ p ≥ (n :: int)" shows "(n div x ^ pm + x * int pm) div int p ≤ x" proof (cases "x ^ p = n") case True from iteration_mono_eq[OF this] show ?thesis by simp next case False with assms have "x ^ p > n" by auto from iteration_mono_less[OF x n this] show ?thesis by simp qed termination (* of root_newton_int_main *) proof - let ?mm = "λ x n :: int. nat x" let ?m1 = "λ (x,n). ?mm x n" let ?m = "measures [?m1]" show ?thesis proof (relation ?m) fix x n :: int assume "¬ x ^ p ≤ n" hence x: "x ^ p > n" by auto assume "¬ (x < 0 ∨ n < 0)" hence x_n: "x ≥ 0" "n ≥ 0" by auto from x x_n have x0: "x > 0" using p by (cases "x = 0", auto) from iteration_mono_less[OF x_n x] x0 show "(((n div x ^ pm + x * int pm) div int p, n), x, n) ∈ ?m" by auto qed auto qed text ‹We next prove that @{const root_int_main'} is a correct implementation of @{const root_newton_int_main}. We additionally prove that the result is always positive, a lower bound, and that the returned boolean indicates whether the result has a root or not. We prove all these results in one go, so that we can share the inductive proof. › abbreviation root_main' where "root_main' ≡ root_int_main' pm (int pm) (int p)" lemmas root_main'_simps = root_int_main'.simps[of pm "int pm" "int p"] lemma root_main'_newton_pos: "x ≥ 0 ⟹ n ≥ 0 ⟹ root_main' x n = root_newton_int_main x n ∧ (root_main' x n = (y,b) ⟶ y ≥ 0 ∧ y^p ≤ n ∧ b = (y^p = n))" proof (induct x n rule: root_newton_int_main.induct) case (1 x n) have pm_x[simp]: "x ^ pm * x = x ^ p" unfolding p by simp from 1 have id: "(x < 0 ∨ n < 0) = False" by auto note d = root_main'_simps[of x n] root_newton_int_main.simps[of x n] id if_False Let_def show ?case proof (cases "x ^ p ≤ n") case True thus ?thesis unfolding d using 1(2) by auto next case False hence id: "(x ^ p ≤ n) = False" by simp from 1(3) 1(2) have not: "¬ (x < 0 ∨ n < 0)" by auto then have x: "x > 0 ∨ x = 0" by auto with ‹0 ≤ n› have "0 ≤ (n div x ^ pm + x * int pm) div int p" by (auto simp add: p algebra_simps pos_imp_zdiv_nonneg_iff power_0_left) then show ?thesis unfolding d id pm_x by (rule 1(1)[OF not False _ 1(3)]) qed qed lemma root_main': "x ≥ 0 ⟹ n ≥ 0 ⟹ root_main' x n = root_newton_int_main x n" using root_main'_newton_pos by blast lemma root_main'_pos: "x ≥ 0 ⟹ n ≥ 0 ⟹ root_main' x n = (y,b) ⟹ y ≥ 0" using root_main'_newton_pos by blast lemma root_main'_sound: "x ≥ 0 ⟹ n ≥ 0 ⟹ root_main' x n = (y,b) ⟹ b = (y ^ p = n)" using root_main'_newton_pos by blast text ‹In order to prove completeness of the algorithms, we provide sharp upper and lower bounds for @{const root_main'}. For the upper bounds, we use Cauchy's mean theorem where we added the non-strict variant to Porter's formalization of this theorem.› lemma root_main'_lower: "x ≥ 0 ⟹ n ≥ 0 ⟹ root_main' x n = (y,b) ⟹ y ^ p ≤ n" using root_main'_newton_pos by blast lemma root_newton_int_main_upper: shows "y ^ p ≥ n ⟹ y ≥ 0 ⟹ n ≥ 0 ⟹ root_newton_int_main y n = (x,b) ⟹ n < (x + 1) ^ p" proof (induct y n rule: root_newton_int_main.induct) case (1 y n) from 1(3) have y0: "y ≥ 0" . then have "y > 0 ∨ y = 0" by auto from 1(4) have n0: "n ≥ 0" . define y' where "y' = (n div (y ^ pm) + y * int pm) div (int p)" from ‹y > 0 ∨ y = 0› ‹n ≥ 0› have y'0: "y' ≥ 0" by (auto simp add: y'_def p algebra_simps pos_imp_zdiv_nonneg_iff power_0_left) let ?rt = "root_newton_int_main" from 1(5) have rt: "?rt y n = (x,b)" by auto from y0 n0 have not: "¬ (y < 0 ∨ n < 0)" "(y < 0 ∨ n < 0) = False" by auto note rt = rt[unfolded root_newton_int_main.simps[of y n] not(2) if_False, folded y'_def] note IH = 1(1)[folded y'_def, OF not(1) _ _ y'0 n0] show ?case proof (cases "y ^ p ≤ n") case False note yyn = this with rt have rt: "?rt y' n = (x,b)" by simp show ?thesis proof (cases "n ≤ y' ^ p") case True show ?thesis by (rule IH[OF False True rt]) next case False with rt have x: "x = y'" unfolding root_newton_int_main.simps[of y' n] using n0 y'0 by simp from yyn have yyn: "y^p > n" by simp from False have yyn': "n > y' ^ p" by auto { assume pm: "pm = 0" have y': "y' = n" unfolding y'_def p pm by simp with yyn' have False unfolding p pm by auto } hence pm0: "pm > 0" by auto show ?thesis proof (cases "n = 0") case True thus ?thesis unfolding p by (metis False y'0 zero_le_power) next case False note n00 = this let ?y = "of_int y :: real" let ?n = "of_int n :: real" from yyn n0 have y00: "y ≠ 0" unfolding p by auto from y00 y0 have y0: "?y > 0" by auto from n0 False have n0: "?n > 0" by auto define Y where "Y = ?y * of_int pm" define NY where "NY = ?n / ?y ^ pm" note pos_intro = divide_nonneg_pos add_nonneg_nonneg mult_nonneg_nonneg have NY0: "NY > 0" unfolding NY_def using y0 n0 by (metis NY_def zero_less_divide_iff zero_less_power) let ?ls = "NY # replicate pm ?y" have prod: "∏:replicate pm ?y = ?y ^ pm " by (induct pm, auto) have sum: "∑:replicate pm ?y = Y" unfolding Y_def by (induct pm, auto simp: field_simps) have pos: "pos ?ls" unfolding pos_def using NY0 y0 by auto have "root p ?n = gmean ?ls" unfolding gmean_def using y0 by (auto simp: p NY_def prod) also have "… < mean ?ls" proof (rule CauchysMeanTheorem_Less[OF pos het_gt_0I]) show "NY ∈ set ?ls" by simp from pm0 show "?y ∈ set ?ls" by simp have "NY < ?y" proof - from yyn have less: "?n < ?y ^ Suc pm" unfolding p by (metis of_int_less_iff of_int_power) have "NY < ?y ^ Suc pm / ?y ^ pm" unfolding NY_def by (rule divide_strict_right_mono[OF less], insert y0, auto) thus ?thesis using y0 by auto qed thus "NY ≠ ?y" by blast qed also have "… = (NY + Y) / real p" by (simp add: mean_def sum p) finally have *: "root p ?n < (NY + Y) / real p" . have "?n = (root p ?n)^p" using n0 by (metis neq0_conv p0 real_root_pow_pos) also have "… < ((NY + Y) / real p)^p" by (rule power_strict_mono[OF *], insert n0 p, auto) finally have ineq1: "?n < ((NY + Y) / real p)^p" by auto { define s where "s = n div y ^ pm + y * int pm" define S where "S = NY + Y" have Y0: "Y ≥ 0" using y0 unfolding Y_def by (metis "1.prems"(2) mult_nonneg_nonneg of_int_0_le_iff of_nat_0_le_iff) have S0: "S > 0" using NY0 Y0 unfolding S_def by auto from p have p0: "p > 0" by auto have "?n / ?y ^ pm < of_int (floor (?n / ?y^pm)) + 1" by (rule divide_less_floor1) also have "floor (?n / ?y ^ pm) = n div y^pm" unfolding div_is_floor_divide_real by (metis of_int_power) finally have "NY < of_int (n div y ^ pm) + 1" unfolding NY_def by simp hence less: "S < of_int s + 1" unfolding Y_def s_def S_def by simp { (* by sledgehammer *) have f1: "∀x⇩_{0}. rat_of_int ⌊rat_of_nat x⇩_{0}⌋ = rat_of_nat x⇩_{0}" using of_int_of_nat_eq by simp have f2: "∀x⇩_{0}. real_of_int ⌊rat_of_nat x⇩_{0}⌋ = real x⇩_{0}" using of_int_of_nat_eq by auto have f3: "∀x⇩_{0}x⇩_{1}. ⌊rat_of_int x⇩_{0}/ rat_of_int x⇩_{1}⌋ = ⌊real_of_int x⇩_{0}/ real_of_int x⇩_{1}⌋" using div_is_floor_divide_rat div_is_floor_divide_real by simp have f4: "0 < ⌊rat_of_nat p⌋" using p by simp have "⌊S⌋ ≤ s" using less floor_le_iff by auto hence "⌊rat_of_int ⌊S⌋ / rat_of_nat p⌋ ≤ ⌊rat_of_int s / rat_of_nat p⌋" using f1 f3 f4 by (metis div_is_floor_divide_real zdiv_mono1) hence "⌊S / real p⌋ ≤ ⌊rat_of_int s / rat_of_nat p⌋" using f1 f2 f3 f4 by (metis div_is_floor_divide_real floor_div_pos_int) hence "S / real p ≤ real_of_int (s div int p) + 1" using f1 f3 by (metis div_is_floor_divide_real floor_le_iff floor_of_nat less_eq_real_def) } hence "S / real p ≤ of_int(s div p) + 1" . note this[unfolded S_def s_def] } hence ge: "of_int y' + 1 ≥ (NY + Y) / p" unfolding y'_def by simp have pos1: "(NY + Y) / p ≥ 0" unfolding Y_def NY_def by (intro divide_nonneg_pos add_nonneg_nonneg mult_nonneg_nonneg, insert y0 n0 p0, auto) have pos2: "of_int y' + (1 :: rat) ≥ 0" using y'0 by auto have ineq2: "(of_int y' + 1) ^ p ≥ ((NY + Y) / p) ^ p" by (rule power_mono[OF ge pos1]) from order.strict_trans2[OF ineq1 ineq2] have "?n < of_int ((x + 1) ^ p)" unfolding x by (metis of_int_1 of_int_add of_int_power) thus "n < (x + 1) ^ p" using of_int_less_iff by blast qed qed next case True with rt have x: "x = y" by simp with 1(2) True have n: "n = y ^ p" by auto show ?thesis unfolding n x using y0 unfolding p by (metis add_le_less_mono add_less_cancel_left lessI less_add_one add.right_neutral le_iff_add power_strict_mono) qed qed lemma root_main'_upper: "x ^ p ≥ n ⟹ x ≥ 0 ⟹ n ≥ 0 ⟹ root_main' x n = (y,b) ⟹ n < (y + 1) ^ p" using root_newton_int_main_upper[of n x y b] root_main'[of x n] by auto end text ‹Now we can prove all the nice properties of @{const root_int_main}.› lemma root_int_main_all: assumes n: "n ≥ 0" and rm: "root_int_main p n = (y,b)" shows "y ≥ 0 ∧ b = (y ^ p = n) ∧ (p > 0 ⟶ y ^ p ≤ n ∧ n < (y + 1)^p) ∧ (p > 0 ⟶ x ≥ 0 ⟶ x ^ p = n ⟶ y = x ∧ b)" proof (cases "p = 0") case True with rm[unfolded root_int_main_def] have y: "y = 1" and b: "b = (n = 1)" by auto show ?thesis unfolding True y b using n by auto next case False from False have p_0: "p > 0" by auto from False have "(p = 0) = False" by simp from rm[unfolded root_int_main_def this Let_def] have rm: "root_int_main' (p - 1) (int (p - 1)) (int p) (start_value n p) n = (y,b)" by simp from start_value[OF n p_0] have start: "n ≤ (start_value n p)^p" "0 ≤ start_value n p" by auto interpret fixed_root p "p - 1" by (unfold_locales, insert False, auto) from root_main'_pos[OF start(2) n rm] have y: "y ≥ 0" . from root_main'_sound[OF start(2) n rm] have b: "b = (y ^ p = n)" . from root_main'_lower[OF start(2) n rm] have low: "y ^ p ≤ n" . from root_main'_upper[OF start n rm] have up: "n < (y + 1) ^ p" . { assume n: "x ^ p = n" and x: "x ≥ 0" with low up have low: "y ^ p ≤ x ^ p" and up: "x ^ p < (y+1) ^ p" by auto from power_strict_mono[of x y, OF _ x p_0] low have x: "x ≥ y" by arith from power_mono[of "(y + 1)" x p] y up have y: "y ≥ x" by arith from x y have "x = y" by auto with b n have "y = x ∧ b" by auto } thus ?thesis using b low up y by auto qed lemma root_int_main: assumes n: "n ≥ 0" and rm: "root_int_main p n = (y,b)" shows "y ≥ 0" "b = (y ^ p = n)" "p > 0 ⟹ y ^ p ≤ n" "p > 0 ⟹ n < (y + 1)^p" "p > 0 ⟹ x ≥ 0 ⟹ x ^ p = n ⟹ y = x ∧ b" using root_int_main_all[OF n rm, of x] by blast+ lemma root_int[simp]: assumes p: "p ≠ 0 ∨ x ≠ 1" shows "set (root_int p x) = {y . y ^ p = x}" proof (cases "p = 0") case True with p have "x ≠ 1" by auto thus ?thesis unfolding root_int_def True by auto next case False hence p: "(p = 0) = False" and p0: "p > 0" by auto note d = root_int_def p if_False Let_def show ?thesis proof (cases "x = 0") case True thus ?thesis unfolding d using p0 by auto next case False hence x: "(x = 0) = False" by auto show ?thesis proof (cases "x < 0 ∧ even p") case True hence left: "set (root_int p x) = {}" unfolding d by auto { fix y assume x: "y ^ p = x" with True have "y ^ p < 0 ∧ even p" by auto hence False by presburger } with left show ?thesis by auto next case False with x p have cond: "(x = 0) = False" "(x < 0 ∧ even p) = False" by auto obtain y b where rt: "root_int_main p ¦x¦ = (y,b)" by force have "abs x ≥ 0" by auto note rm = root_int_main[OF this rt] have "?thesis = (set (case root_int_main p ¦x¦ of (y, True) ⇒ if even p then [y, - y] else [sgn x * y] | (y, False) ⇒ []) = {y. y ^ p = x})" unfolding d cond by blast also have "(case root_int_main p ¦x¦ of (y, True) ⇒ if even p then [y, - y] else [sgn x * y] | (y, False) ⇒ []) = (if b then if even p then [y, - y] else [sgn x * y] else [])" (is "_ = ?lhs") unfolding rt by auto also have "set ?lhs = {y. y ^ p = x}" (is "_ = ?rhs") proof - { fix z assume idx: "z ^ p = x" hence eq: "(abs z) ^ p = abs x" by (metis power_abs) from idx x p0 have z: "z ≠ 0" unfolding p by auto have "(y, b) = (¦z¦, True)" using rm(5)[OF p0 _ eq] by auto hence id: "y = abs z" "b = True" by auto have "z ∈ set ?lhs" unfolding id using z by (auto simp: idx[symmetric], cases "z < 0", auto) } moreover { fix z assume z: "z ∈ set ?lhs" hence b: "b = True" by (cases b, auto) note z = z[unfolded b if_True] from rm(2) b have yx: "y ^ p = ¦x¦" by auto from rm(1) have y: "y ≥ 0" . from False have "odd p ∨ even p ∧ x ≥ 0" by auto hence "z ∈ ?rhs" proof assume odd: "odd p" with z have "z = sgn x * y" by auto hence "z ^ p = (sgn x * y) ^ p" by auto also have "… = sgn x ^ p * y ^ p" unfolding power_mult_distrib by auto also have "… = sgn x ^ p * abs x" unfolding yx by simp also have "sgn x ^ p = sgn x" using x odd by auto also have "sgn x * abs x = x" by (rule mult_sgn_abs) finally show "z ∈ ?rhs" by auto next assume even: "even p ∧ x ≥ 0" from z even have "z = y ∨ z = -y" by auto hence id: "abs z = y" using y by auto with yx x even have z: "z ≠ 0" using p0 by (cases "y = 0", auto) have "z ^ p = (sgn z * abs z) ^ p" by (simp add: mult_sgn_abs) also have "… = (sgn z * y) ^ p" using id by auto also have "… = (sgn z)^p * y ^ p" unfolding power_mult_distrib by simp also have "… = sgn z ^ p * x" unfolding yx using even by auto also have "sgn z ^ p = 1" using even z by (auto) finally show "z ∈ ?rhs" by auto qed } ultimately show ?thesis by blast qed finally show ?thesis by auto qed qed qed lemma root_int_pos: assumes x: "x ≥ 0" and ri: "root_int p x = y # ys" shows "y ≥ 0" proof - from x have abs: "abs x = x" by auto note ri = ri[unfolded root_int_def Let_def abs] from ri have p: "(p = 0) = False" by (cases p, auto) note ri = ri[unfolded p if_False] show ?thesis proof (cases "x = 0") case True with ri show ?thesis by auto next case False hence "(x = 0) = False" "(x < 0 ∧ even p) = False" using x by auto note ri = ri[unfolded this if_False] obtain y' b' where r: "root_int_main p x = (y',b')" by force note ri = ri[unfolded this] hence y: "y = (if even p then y' else sgn x * y')" by (cases b', auto) from root_int_main(1)[OF x r] have y': "0 ≤ y'" . thus ?thesis unfolding y using x False by auto qed qed subsection ‹Floor and ceiling of roots› text ‹Using the bounds for @{const root_int_main} we can easily design algorithms which compute @{term "floor (root p x)"} and @{term "ceiling (root p x)"}. To this end, we first develop algorithms for non-negative @{term x}, and later on these are used for the general case.› definition "root_int_floor_pos p x = (if p = 0 then 0 else fst (root_int_main p x))" definition "root_int_ceiling_pos p x = (if p = 0 then 0 else (case root_int_main p x of (y,b) ⇒ if b then y else y + 1))" lemma root_int_floor_pos_lower: assumes p0: "p ≠ 0" and x: "x ≥ 0" shows "root_int_floor_pos p x ^ p ≤ x" using root_int_main(3)[OF x, of p] p0 unfolding root_int_floor_pos_def by (cases "root_int_main p x", auto) lemma root_int_floor_pos_pos: assumes x: "x ≥ 0" shows "root_int_floor_pos p x ≥ 0" using root_int_main(1)[OF x, of p] unfolding root_int_floor_pos_def by (cases "root_int_main p x", auto) lemma root_int_floor_pos_upper: assumes p0: "p ≠ 0" and x: "x ≥ 0" shows "(root_int_floor_pos p x + 1) ^ p > x" using root_int_main(4)[OF x, of p] p0 unfolding root_int_floor_pos_def by (cases "root_int_main p x", auto) lemma root_int_floor_pos: assumes x: "x ≥ 0" shows "root_int_floor_pos p x = floor (root p (of_int x))" proof (cases "p = 0") case True thus ?thesis by (simp add: root_int_floor_pos_def) next case False hence p: "p > 0" by auto let ?s1 = "real_of_int (root_int_floor_pos p x)" let ?s2 = "root p (of_int x)" from x have s1: "?s1 ≥ 0" by (metis of_int_0_le_iff root_int_floor_pos_pos) from x have s2: "?s2 ≥ 0" by (metis of_int_0_le_iff real_root_pos_pos_le) from s1 have s11: "?s1 + 1 ≥ 0" by auto have id: "?s2 ^ p = of_int x" using x by (metis p of_int_0_le_iff real_root_pow_pos2) show ?thesis proof (rule floor_unique[symmetric]) show "?s1 ≤ ?s2" unfolding compare_pow_le_iff[OF p s1 s2, symmetric] unfolding id using root_int_floor_pos_lower[OF False x] by (metis of_int_le_iff of_int_power) show "?s2 < ?s1 + 1" unfolding compare_pow_less_iff[OF p s2 s11, symmetric] unfolding id using root_int_floor_pos_upper[OF False x] by (metis of_int_add of_int_less_iff of_int_power of_int_1) qed qed lemma root_int_ceiling_pos: assumes x: "x ≥ 0" shows "root_int_ceiling_pos p x = ceiling (root p (of_int x))" proof (cases "p = 0") case True thus ?thesis by (simp add: root_int_ceiling_pos_def) next case False hence p: "p > 0" by auto obtain y b where s: "root_int_main p x = (y,b)" by force note rm = root_int_main[OF x s] note rm = rm(1-2) rm(3-5)[OF p] from rm(1) have y: "y ≥ 0" by simp let ?s = "root_int_ceiling_pos p x" let ?sx = "root p (of_int x)" note d = root_int_ceiling_pos_def show ?thesis proof (cases b) case True hence id: "?s = y" unfolding s d using p by auto from rm(2) True have xy: "x = y ^ p" by auto show ?thesis unfolding id unfolding xy using y by (simp add: p real_root_power_cancel) next case False hence id: "?s = root_int_floor_pos p x + 1" unfolding d root_int_floor_pos_def using s p by simp from False have x0: "x ≠ 0" using rm(5)[of 0] using s unfolding root_int_main_def Let_def using p by (cases "x = 0", auto) show ?thesis unfolding id root_int_floor_pos[OF x] proof (rule ceiling_unique[symmetric]) show "?sx ≤ real_of_int (⌊root p (of_int x)⌋ + 1)" by (metis of_int_add real_of_int_floor_add_one_ge of_int_1) let ?l = "real_of_int (⌊root p (of_int x)⌋ + 1) - 1" let ?m = "real_of_int ⌊root p (of_int x)⌋" have "?l = ?m" by simp also have "… < ?sx" proof - have le: "?m ≤ ?sx" by (rule of_int_floor_le) have neq: "?m ≠ ?sx" proof assume "?m = ?sx" hence "?m ^ p = ?sx ^ p" by auto also have "… = of_int x" using x False by (metis p real_root_ge_0_iff real_root_pow_pos2 root_int_floor_pos root_int_floor_pos_pos zero_le_floor zero_less_Suc) finally have xs: "x = ⌊root p (of_int x)⌋ ^ p" by (metis floor_power floor_of_int) hence "⌊root p (of_int x)⌋ ∈ set (root_int p x)" using p by simp hence "root_int p x ≠ []" by force with s False ‹p ≠ 0› x x0 show False unfolding root_int_def by (cases p, auto) qed from le neq show ?thesis by arith qed finally show "?l < ?sx" . qed qed qed definition "root_int_floor p x = (if x ≥ 0 then root_int_floor_pos p x else - root_int_ceiling_pos p (- x))" definition "root_int_ceiling p x = (if x ≥ 0 then root_int_ceiling_pos p x else - root_int_floor_pos p (- x))" lemma root_int_floor[simp]: "root_int_floor p x = floor (root p (of_int x))" proof - note d = root_int_floor_def show ?thesis proof (cases "x ≥ 0") case True with root_int_floor_pos[OF True, of p] show ?thesis unfolding d by simp next case False hence "- x ≥ 0" by auto from False root_int_ceiling_pos[OF this] show ?thesis unfolding d by (simp add: real_root_minus ceiling_minus) qed qed lemma root_int_ceiling[simp]: "root_int_ceiling p x = ceiling (root p (of_int x))" proof - note d = root_int_ceiling_def show ?thesis proof (cases "x ≥ 0") case True with root_int_ceiling_pos[OF True] show ?thesis unfolding d by simp next case False hence "- x ≥ 0" by auto from False root_int_floor_pos[OF this, of p] show ?thesis unfolding d by (simp add: real_root_minus floor_minus) qed qed subsection ‹Downgrading algorithms to the naturals› definition root_nat_floor :: "nat ⇒ nat ⇒ int" where "root_nat_floor p x = root_int_floor_pos p (int x)" definition root_nat_ceiling :: "nat ⇒ nat ⇒ int" where "root_nat_ceiling p x = root_int_ceiling_pos p (int x)" definition root_nat :: "nat ⇒ nat ⇒ nat list" where "root_nat p x = map nat (take 1 (root_int p x))" lemma root_nat_floor [simp]: "root_nat_floor p x = floor (root p (real x))" unfolding root_nat_floor_def using root_int_floor_pos[of "int x" p] by auto lemma root_nat_floor_lower: assumes p0: "p ≠ 0" shows "root_nat_floor p x ^ p ≤ x" using root_int_floor_pos_lower[OF p0, of x] unfolding root_nat_floor_def by auto lemma root_nat_floor_upper: assumes p0: "p ≠ 0" shows "(root_nat_floor p x + 1) ^ p > x" using root_int_floor_pos_upper[OF p0, of x] unfolding root_nat_floor_def by auto lemma root_nat_ceiling [simp]: "root_nat_ceiling p x = ceiling (root p x)" unfolding root_nat_ceiling_def using root_int_ceiling_pos[of x p] by auto lemma root_nat: assumes p0: "p ≠ 0 ∨ x ≠ 1" shows "set (root_nat p x) = { y. y ^ p = x}" proof - { fix y assume "y ∈ set (root_nat p x)" note y = this[unfolded root_nat_def] then obtain yi ys where ri: "root_int p x = yi # ys" by (cases "root_int p x", auto) with y have y: "y = nat yi" by auto from root_int_pos[OF _ ri] have yi: "0 ≤ yi" by auto from root_int[of p "int x"] p0 ri have "yi ^ p = x" by auto from arg_cong[OF this, of nat] yi have "nat yi ^ p = x" by (metis nat_int nat_power_eq) hence "y ∈ {y. y ^ p = x}" using y by auto } moreover { fix y assume yx: "y ^ p = x" hence y: "int y ^ p = int x" by (metis of_nat_power) hence "set (root_int p (int x)) ≠ {}" using root_int[of p "int x"] p0 by (metis (mono_tags) One_nat_def ‹y ^ p = x› empty_Collect_eq nat_power_eq_Suc_0_iff) then obtain yi ys where ri: "root_int p (int x) = yi # ys" by (cases "root_int p (int x)", auto) from root_int_pos[OF _ this] have yip: "yi ≥ 0" by auto from root_int[of p "int x", unfolded ri] p0 have yi: "yi ^ p = int x" by auto with y have "int y ^ p = yi ^ p" by auto from arg_cong[OF this, of nat] have id: "y ^ p = nat yi ^ p" by (metis ‹y ^ p = x› nat_int nat_power_eq yi yip) { assume p: "p ≠ 0" hence p0: "p > 0" by auto obtain yy b where rm: "root_int_main p (int x) = (yy,b)" by force from root_int_main(5)[OF _ rm p0 _ y] have "yy = int y" and "b = True" by auto note rm = rm[unfolded this] hence "y ∈ set (root_nat p x)" unfolding root_nat_def p root_int_def using p0 p yx by auto } moreover { assume p: "p = 0" with p0 have "x ≠ 1" by auto with y p have False by auto } ultimately have "y ∈ set (root_nat p x)" by auto } ultimately show ?thesis by blast qed subsection ‹Upgrading algorithms to the rationals› text ‹The main observation to lift everything from the integers to the rationals is the fact, that one can reformulate $\frac{a}{b}^{1/p}$ as $\frac{(ab^{p-1})^{1/p}}b$.› definition root_rat_floor :: "nat ⇒ rat ⇒ int" where "root_rat_floor p x ≡ case quotient_of x of (a,b) ⇒ root_int_floor p (a * b^(p - 1)) div b" definition root_rat_ceiling :: "nat ⇒ rat ⇒ int" where "root_rat_ceiling p x ≡ - (root_rat_floor p (-x))" definition root_rat :: "nat ⇒ rat ⇒ rat list" where "root_rat p x ≡ case quotient_of x of (a,b) ⇒ concat (map (λ rb. map (λ ra. of_int ra / rat_of_int rb) (root_int p a)) (take 1 (root_int p b)))" lemma root_rat_reform: assumes q: "quotient_of x = (a,b)" shows "root p (real_of_rat x) = root p (of_int (a * b ^ (p - 1))) / of_int b" proof (cases "p = 0") case False from quotient_of_denom_pos[OF q] have b: "0 < b" by auto hence b: "0 < real_of_int b" by auto from quotient_of_div[OF q] have x: "root p (real_of_rat x) = root p (a / b)" by (metis of_rat_divide of_rat_of_int_eq) also have "a / b = a * real_of_int b ^ (p - 1) / of_int b ^ p" using b False by (cases p, auto simp: field_simps) also have "root p … = root p (a * real_of_int b ^ (p - 1)) / root p (of_int b ^ p)" by (rule real_root_divide) also have "root p (of_int b ^ p) = of_int b" using False b by (metis neq0_conv real_root_pow_pos real_root_power) also have "a * real_of_int b ^ (p - 1) = of_int (a * b ^ (p - 1))" by (metis of_int_mult of_int_power) finally show ?thesis . qed auto lemma root_rat_floor [simp]: "root_rat_floor p x = floor (root p (of_rat x))" proof - obtain a b where q: "quotient_of x = (a,b)" by force from quotient_of_denom_pos[OF q] have b: "b > 0" . show ?thesis unfolding root_rat_floor_def q split root_int_floor unfolding root_rat_reform[OF q] floor_div_pos_int[OF b] .. qed lemma root_rat_ceiling [simp]: "root_rat_ceiling p x = ceiling (root p (of_rat x))" unfolding root_rat_ceiling_def ceiling_def real_root_minus root_rat_floor of_rat_minus .. lemma root_rat[simp]: assumes p: "p ≠ 0 ∨ x ≠ 1" shows "set (root_rat p x) = { y. y ^ p = x}" proof (cases "p = 0") case False note p = this obtain a b where q: "quotient_of x = (a,b)" by force note x = quotient_of_div[OF q] have b: "b > 0" by (rule quotient_of_denom_pos[OF q]) note d = root_rat_def q split set_concat set_map { fix q assume "q ∈ set (root_rat p x)" note mem = this[unfolded d] from mem obtain rb xs where rb: "root_int p b = Cons rb xs" by (cases "root_int p b", auto) note mem = mem[unfolded this] from mem obtain ra where ra: "ra ∈ set (root_int p a)" and q: "q = of_int ra / of_int rb" by (cases "root_int p a", auto) from rb have "rb ∈ set (root_int p b)" by auto with ra p have rb: "b = rb ^ p" and ra: "a = ra ^ p" by auto have "q ∈ {y. y ^ p = x}" unfolding q x ra rb by (auto simp: power_divide) } moreover { fix q assume "q ∈ {y. y ^ p = x}" hence "q ^ p = of_int a / of_int b" unfolding x by auto hence eq: "of_int b * q ^ p = of_int a" using b by auto obtain z n where quo: "quotient_of q = (z,n)" by force note qzn = quotient_of_div[OF quo] have n: "n > 0" using quotient_of_denom_pos[OF quo] . from eq[unfolded qzn] have "rat_of_int b * of_int z^p / of_int n^p = of_int a" unfolding power_divide by simp from arg_cong[OF this, of "λ x. x * of_int n^p"] n have "rat_of_int b * of_int z^p = of_int a * of_int n ^ p" by auto also have "rat_of_int b * of_int z^p = rat_of_int (b * z^p)" unfolding of_int_mult of_int_power .. also have "of_int a * rat_of_int n ^ p = of_int (a * n ^ p)" unfolding of_int_mult of_int_power .. finally have id: "a * n ^ p = b * z ^ p" by linarith from quotient_of_coprime[OF quo] have cop: "coprime (z ^ p) (n ^ p)" by simp from coprime_crossproduct_int[OF quotient_of_coprime[OF q] this] arg_cong[OF id, of abs] have "¦n ^ p¦ = ¦b¦" by (simp add: field_simps abs_mult) with n b have bnp: "b = n ^ p" by auto hence rn: "n ∈ set (root_int p b)" using p by auto then obtain rb rs where rb: "root_int p b = Cons rb rs" by (cases "root_int p b", auto) from id[folded bnp] b have "a = z ^ p" by auto hence a: "z ∈ set (root_int p a)" using p by auto from root_int_pos[OF _ rb] b have rb0: "rb ≥ 0" by auto from root_int[OF disjI1[OF p], of b] rb have "rb ^ p = b" by auto with bnp have id: "rb ^ p = n ^ p" by auto have "rb = n" by (rule power_eq_imp_eq_base[OF id], insert n rb0 p, auto) with rb have b: "n ∈ set (take 1 (root_int p b))" by auto have "q ∈ set (root_rat p x)" unfolding d qzn using b a by auto } ultimately show ?thesis by blast next case True with p have x: "x ≠ 1" by auto obtain a b where q: "quotient_of x = (a,b)" by force show ?thesis unfolding True root_rat_def q split root_int_def using x by auto qed end

# Theory Sqrt_Babylonian

(* Title: Computing Square Roots using the Babylonian Method Author: René Thiemann <rene.thiemann@uibk.ac.at> Maintainer: René Thiemann License: LGPL *) (* Copyright 2009-2014 René Thiemann This file is part of IsaFoR/CeTA. IsaFoR/CeTA is free software: you can redistribute it and/or modify it under the terms of the GNU Lesser General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. IsaFoR/CeTA is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. You should have received a copy of the GNU Lesser General Public License along with IsaFoR/CeTA. If not, see <http://www.gnu.org/licenses/>. *) theory Sqrt_Babylonian imports Sqrt_Babylonian_Auxiliary NthRoot_Impl begin section ‹Executable algorithms for square roots› text ‹ This theory provides executable algorithms for computing square-roots of numbers which are all based on the Babylonian method (which is also known as Heron's method or Newton's method). For integers / naturals / rationals precise algorithms are given, i.e., here $sqrt\ x$ delivers a list of all integers / naturals / rationals $y$ where $y^2 = x$. To this end, the Babylonian method has been adapted by using integer-divisions. In addition to the precise algorithms, we also provide approximation algorithms. One works for arbitrary linear ordered fields, where some number $y$ is computed such that @{term "abs(y^2 - x) < ε"}. Moreover, for the naturals, integers, and rationals we provide algorithms to compute @{term "floor (sqrt x)"} and @{term "ceiling (sqrt x)"} which are all based on the underlying algorithm that is used to compute the precise square-roots on integers, if these exist. The major motivation for developing the precise algorithms was given by \ceta{} \cite{CeTA}, a tool for certifiying termination proofs. Here, non-linear equations of the form $(a_1x_1 + \dots a_nx_n)^2 = p$ had to be solved over the integers, where $p$ is a concrete polynomial. For example, for the equation $(ax + by)^2 = 4x^2 - 12xy + 9y^2$ one easily figures out that $a^2 = 4, b^2 = 9$, and $ab = -6$, which results in a possible solution $a = \sqrt 4 = 2, b = - \sqrt 9 = -3$. › subsection ‹The Babylonian method› text ‹ The Babylonian method for computing $\sqrt n$ iteratively computes \[ x_{i+1} = \frac{\frac n{x_i} + x_i}2 \] until $x_i^2 \approx n$. Note that if $x_0^2 \geq n$, then for all $i$ we have both $x_i^2 \geq n$ and $x_i \geq x_{i+1}$. › subsection ‹The Babylonian method using integer division› text ‹ First, the algorithm is developed for the non-negative integers. Here, the division operation $\frac xy$ is replaced by @{term "x div y = ⌊of_int x / of_int y⌋"}. Note that replacing @{term "⌊of_int x / of_int y⌋"} by @{term "⌈of_int x / of_int y⌉"} would lead to non-termination in the following algorithm. We explicititly develop the algorithm on the integers and not on the naturals, as the calculations on the integers have been much easier. For example, $y - x + x = y$ on the integers, which would require the side-condition $y \geq x$ for the naturals. These conditions will make the reasoning much more tedious---as we have experienced in an earlier state of this development where everything was based on naturals. Since the elements $x_0, x_1, x_2,\dots$ are monotone decreasing, in the main algorithm we abort as soon as $x_i^2 \leq n$.› text ‹\textbf{Since in the meantime, all of these algorithms have been generalized to arbitrary $p$-th roots in @{theory Sqrt_Babylonian.NthRoot_Impl}, we just instantiate the general algorithms by $p = 2$ and then provide specialized code equations which are more efficient than the general purpose algorithms.}› definition sqrt_int_main' :: "int ⇒ int ⇒ int × bool" where [simp]: "sqrt_int_main' x n = root_int_main' 1 1 2 x n" lemma sqrt_int_main'_code[code]: "sqrt_int_main' x n = (let x2 = x * x in if x2 ≤ n then (x, x2 = n) else sqrt_int_main' ((n div x + x) div 2) n)" using root_int_main'.simps[of 1 1 2 x n] unfolding Let_def by auto definition sqrt_int_main :: "int ⇒ int × bool" where [simp]: "sqrt_int_main x = root_int_main 2 x" lemma sqrt_int_main_code[code]: "sqrt_int_main x = sqrt_int_main' (start_value x 2) x" by (simp add: root_int_main_def Let_def) definition sqrt_int :: "int ⇒ int list" where "sqrt_int x = root_int 2 x" lemma sqrt_int_code[code]: "sqrt_int x = (if x < 0 then [] else case sqrt_int_main x of (y,True) ⇒ if y = 0 then [0] else [y,-y] | _ ⇒ [])" proof - interpret fixed_root 2 1 by (unfold_locales, auto) obtain b y where res: "root_int_main 2 x = (b,y)" by force show ?thesis unfolding sqrt_int_def root_int_def Let_def using root_int_main[OF _ res] using res by simp qed lemma sqrt_int[simp]: "set (sqrt_int x) = {y. y * y = x}" unfolding sqrt_int_def by (simp add: power2_eq_square) lemma sqrt_int_pos: assumes res: "sqrt_int x = Cons s ms" shows "s ≥ 0" proof - note res = res[unfolded sqrt_int_code Let_def, simplified] from res have x0: "x ≥ 0" by (cases ?thesis, auto) obtain ss b where call: "sqrt_int_main x = (ss,b)" by force from res[unfolded call] x0 have "ss = s" by (cases b, cases "ss = 0", auto) from root_int_main(1)[OF x0 call[unfolded this sqrt_int_main_def]] show ?thesis . qed definition [simp]: "sqrt_int_floor_pos x = root_int_floor_pos 2 x" lemma sqrt_int_floor_pos_code[code]: "sqrt_int_floor_pos x = fst (sqrt_int_main x)" by (simp add: root_int_floor_pos_def) lemma sqrt_int_floor_pos: assumes x: "x ≥ 0" shows "sqrt_int_floor_pos x = ⌊ sqrt (of_int x) ⌋" using root_int_floor_pos[OF x, of 2] by (simp add: sqrt_def) definition [simp]: "sqrt_int_ceiling_pos x = root_int_ceiling_pos 2 x" lemma sqrt_int_ceiling_pos_code[code]: "sqrt_int_ceiling_pos x = (case sqrt_int_main x of (y,b) ⇒ if b then y else y + 1)" by (simp add: root_int_ceiling_pos_def) lemma sqrt_int_ceiling_pos: assumes x: "x ≥ 0" shows "sqrt_int_ceiling_pos x = ⌈ sqrt (of_int x) ⌉" using root_int_ceiling_pos[OF x, of 2] by (simp add: sqrt_def) definition "sqrt_int_floor x = root_int_floor 2 x" lemma sqrt_int_floor_code[code]: "sqrt_int_floor x = (if x ≥ 0 then sqrt_int_floor_pos x else - sqrt_int_ceiling_pos (- x))" unfolding sqrt_int_floor_def root_int_floor_def by simp lemma sqrt_int_floor[simp]: "sqrt_int_floor x = ⌊ sqrt (of_int x) ⌋" by (simp add: sqrt_int_floor_def sqrt_def) definition "sqrt_int_ceiling x = root_int_ceiling 2 x" lemma sqrt_int_ceiling_code[code]: "sqrt_int_ceiling x = (if x ≥ 0 then sqrt_int_ceiling_pos x else - sqrt_int_floor_pos (- x))" unfolding sqrt_int_ceiling_def root_int_ceiling_def by simp lemma sqrt_int_ceiling[simp]: "sqrt_int_ceiling x = ⌈ sqrt (of_int x) ⌉" by (simp add: sqrt_int_ceiling_def sqrt_def) lemma sqrt_int_ceiling_bound: "0 ≤ x ⟹ x ≤ (sqrt_int_ceiling x)^2" unfolding sqrt_int_ceiling using le_of_int_ceiling sqrt_le_D by (metis of_int_power_le_of_int_cancel_iff) subsection ‹Square roots for the naturals› definition sqrt_nat :: "nat ⇒ nat list" where "sqrt_nat x = root_nat 2 x" lemma sqrt_nat_code[code]: "sqrt_nat x ≡ map nat (take 1 (sqrt_int (int x)))" unfolding sqrt_nat_def root_nat_def sqrt_int_def by simp lemma sqrt_nat[simp]: "set (sqrt_nat x) = { y. y * y = x}" unfolding sqrt_nat_def using root_nat[of 2 x] by (simp add: power2_eq_square) definition sqrt_nat_floor :: "nat ⇒ int" where "sqrt_nat_floor x = root_nat_floor 2 x" lemma sqrt_nat_floor_code[code]: "sqrt_nat_floor x = sqrt_int_floor_pos (int x)" unfolding sqrt_nat_floor_def root_nat_floor_def by simp lemma sqrt_nat_floor[simp]: "sqrt_nat_floor x = ⌊ sqrt (real x) ⌋" unfolding sqrt_nat_floor_def by (simp add: sqrt_def) definition sqrt_nat_ceiling :: "nat ⇒ int" where "sqrt_nat_ceiling x = root_nat_ceiling 2 x" lemma sqrt_nat_ceiling_code[code]: "sqrt_nat_ceiling x = sqrt_int_ceiling_pos (int x)" unfolding sqrt_nat_ceiling_def root_nat_ceiling_def by simp lemma sqrt_nat_ceiling[simp]: "sqrt_nat_ceiling x = ⌈ sqrt (real x) ⌉" unfolding sqrt_nat_ceiling_def by (simp add: sqrt_def) subsection ‹Square roots for the rationals› definition sqrt_rat :: "rat ⇒ rat list" where "sqrt_rat x = root_rat 2 x" lemma sqrt_rat_code[code]: "sqrt_rat x = (case quotient_of x of (z,n) ⇒ (case sqrt_int n of [] ⇒ [] | sn # xs ⇒ map (λ sz. of_int sz / of_int sn) (sqrt_int z)))" proof - obtain z n where q: "quotient_of x = (z,n)" by force show ?thesis unfolding sqrt_rat_def root_rat_def q split sqrt_int_def by (cases "root_int 2 n", auto) qed lemma sqrt_rat[simp]: "set (sqrt_rat x) = { y. y * y = x}" unfolding sqrt_rat_def using root_rat[of 2 x] by (simp add: power2_eq_square) lemma sqrt_rat_pos: assumes sqrt: "sqrt_rat x = Cons s ms" shows "s ≥ 0" proof - obtain z n where q: "quotient_of x = (z,n)" by force note sqrt = sqrt[unfolded sqrt_rat_code q, simplified] let ?sz = "sqrt_int z" let ?sn = "sqrt_int n" from q have n: "n > 0" by (rule quotient_of_denom_pos) from sqrt obtain sz mz where sz: "?sz = sz # mz" by (cases ?sn, auto) from sqrt obtain sn mn where sn: "?sn = sn # mn" by (cases ?sn, auto) from sqrt_int_pos[OF sz] sqrt_int_pos[OF sn] have pos: "0 ≤ sz" "0 ≤ sn" by auto from sqrt sz sn have s: "s = of_int sz / of_int sn" by auto show ?thesis unfolding s using pos by (metis of_int_0_le_iff zero_le_divide_iff) qed definition sqrt_rat_floor :: "rat ⇒ int" where "sqrt_rat_floor x = root_rat_floor 2 x" lemma sqrt_rat_floor_code[code]: "sqrt_rat_floor x = (case quotient_of x of (a,b) ⇒ sqrt_int_floor (a * b) div b)" unfolding sqrt_rat_floor_def root_rat_floor_def by (simp add: sqrt_def) lemma sqrt_rat_floor[simp]: "sqrt_rat_floor x = ⌊ sqrt (of_rat x) ⌋" unfolding sqrt_rat_floor_def by (simp add: sqrt_def) definition sqrt_rat_ceiling :: "rat ⇒ int" where "sqrt_rat_ceiling x = root_rat_ceiling 2 x" lemma sqrt_rat_ceiling_code[code]: "sqrt_rat_ceiling x = - (sqrt_rat_floor (-x))" unfolding sqrt_rat_ceiling_def sqrt_rat_floor_def root_rat_ceiling_def by simp lemma sqrt_rat_ceiling: "sqrt_rat_ceiling x = ⌈ sqrt (of_rat x) ⌉" unfolding sqrt_rat_ceiling_def by (simp add: sqrt_def) lemma sqr_rat_of_int: assumes x: "x * x = rat_of_int i" shows "∃ j :: int. j * j = i" proof - from x have mem: "x ∈ set (sqrt_rat (rat_of_int i))" by simp from x have "rat_of_int i ≥ 0" by (metis zero_le_square) hence *: "quotient_of (rat_of_int i) = (i,1)" by (metis quotient_of_int) have 1: "sqrt_int 1 = [1,-1]" by code_simp from mem sqrt_rat_code * split 1 have x: "x ∈ rat_of_int ` {y. y * y = i}" by auto thus ?thesis by auto qed subsection ‹Approximating square roots› text ‹ The difference to the previous algorithms is that now we abort, once the distance is below $\epsilon$. Moreover, here we use standard division and not integer division. This part is not yet generalized by @{theory Sqrt_Babylonian.NthRoot_Impl}. We first provide the executable version without guard @{term "x > 0"} as partial function, and afterwards prove termination and soundness for a similar algorithm that is defined within the upcoming locale. › partial_function (tailrec) sqrt_approx_main_impl :: "'a :: linordered_field ⇒ 'a ⇒ 'a ⇒ 'a" where [code]: "sqrt_approx_main_impl ε n x = (if x * x - n < ε then x else sqrt_approx_main_impl ε n ((n / x + x) / 2))" text ‹We setup a locale where we ensure that we have standard assumptions: positive $\epsilon$ and positive $n$. We require sort @{term floor_ceiling}, since @{term "⌊ x ⌋"} is used for the termination argument.› locale sqrt_approximation = fixes ε :: "'a :: {linordered_field,floor_ceiling}" and n :: 'a assumes ε : "ε > 0" and n: "n > 0" begin function sqrt_approx_main :: "'a ⇒ 'a" where "sqrt_approx_main x = (if x > 0 then (if x * x - n < ε then x else sqrt_approx_main ((n / x + x) / 2)) else 0)" by pat_completeness auto text ‹Termination essentially is a proof of convergence. Here, one complication is the fact that the limit is not always defined. E.g., if @{typ "'a"} is @{typ rat} then there is no square root of 2. Therefore, the error-rate $\frac x{\sqrt n} - 1$ is not expressible. Instead we use the expression $\frac{x^2}n - 1$ as error-rate which does not require any square-root operation.› termination proof - define er where "er x = (x * x / n - 1)" for x define c where "c = 2 * n / ε" define m where "m x = nat ⌊ c * er x ⌋" for x have c: "c > 0" unfolding c_def using n ε by auto show ?thesis proof show "wf (measures [m])" by simp next fix x assume x: "0 < x" and xe: "¬ x * x - n < ε" define y where "y = (n / x + x) / 2" show "((n / x + x) / 2,x) ∈ measures [m]" unfolding y_def[symmetric] proof (rule measures_less) from n have inv_n: "1 / n > 0" by auto from xe have "x * x - n ≥ ε" by simp from this[unfolded mult_le_cancel_left_pos[OF inv_n, of ε, symmetric]] have erxen: "er x ≥ ε / n" unfolding er_def using n by (simp add: field_simps) have en: "ε / n > 0" and ne: "n / ε > 0" using ε n by auto from en erxen have erx: "er x > 0" by linarith have pos: "er x * 4 + er x * (er x * 4) > 0" using erx by (auto intro: add_pos_nonneg) have "er y = 1 / 4 * (n / (x * x) - 2 + x * x / n)" unfolding er_def y_def using x n by (simp add: field_simps) also have "… = 1 / 4 * er x * er x / (1 + er x)" unfolding er_def using x n by (simp add: field_simps) finally have "er y = 1 / 4 * er x * er x / (1 + er x)" . also have "… < 1 / 4 * (1 + er x) * er x / (1 + er x)" using erx erx pos by (auto simp: field_simps) also have "… = er x / 4" using erx by (simp add: field_simps) finally have er_y_x: "er y ≤ er x / 4" by linarith from erxen have "c * er x ≥ 2" unfolding c_def mult_le_cancel_left_pos[OF ne, of _ "er x", symmetric] using n ε by (auto simp: field_simps) hence pos: "⌊c * er x⌋ > 0" "⌊c * er x⌋ ≥ 2" by auto show "m y < m x" unfolding m_def nat_mono_iff[OF pos(1)] proof - have "⌊c * er y⌋ ≤ ⌊c * (er x / 4)⌋" by (rule floor_mono, unfold mult_le_cancel_left_pos[OF c], rule er_y_x) also have "… < ⌊c * er x / 4 + 1⌋" by auto also have "… ≤ ⌊c * er x⌋" by (rule floor_mono, insert pos(2), simp add: field_simps) finally show "⌊c * er y⌋ < ⌊c * er x⌋" . qed qed qed qed text ‹Once termination is proven, it is easy to show equivalence of @{const sqrt_approx_main_impl} and @{const sqrt_approx_main}.› lemma sqrt_approx_main_impl: "x > 0 ⟹ sqrt_approx_main_impl ε n x = sqrt_approx_main x" proof (induct x rule: sqrt_approx_main.induct) case (1 x) hence x: "x > 0" by auto hence nx: "0 < (n / x + x) / 2" using n by (auto intro: pos_add_strict) note simps = sqrt_approx_main_impl.simps[of _ _ x] sqrt_approx_main.simps[of x] show ?case proof (cases "x * x - n < ε") case True thus ?thesis unfolding simps using x by auto next case False show ?thesis using 1(1)[OF x False nx] unfolding simps using x False by auto qed qed text ‹Also soundness is not complicated.› lemma sqrt_approx_main_sound: assumes x: "x > 0" and xx: "x * x > n" shows "sqrt_approx_main x * sqrt_approx_main x > n ∧ sqrt_approx_main x * sqrt_approx_main x - n < ε" using assms proof (induct x rule: sqrt_approx_main.induct) case (1 x) from 1 have x: "x > 0" "(x > 0) = True" by auto note simp = sqrt_approx_main.simps[of x, unfolded x if_True] show ?case proof (cases "x * x - n < ε") case True with 1 show ?thesis unfolding simp by simp next case False let ?y = "(n / x + x) / 2" from False simp have simp: "sqrt_approx_main x = sqrt_approx_main ?y" by simp from n x have y: "?y > 0" by (auto intro: pos_add_strict) note IH = 1(1)[OF x(1) False y] from x have x4: "4 * x * x > 0" by (auto intro: mult_sign_intros) show ?thesis unfolding simp proof (rule IH) show "n < ?y * ?y" unfolding mult_less_cancel_left_pos[OF x4, of n, symmetric] proof - have id: "4 * x * x * (?y * ?y) = 4 * x * x * n + (n - x * x) * (n - x * x)" using x(1) by (simp add: field_simps) from 1(3) have "x * x - n > 0" by auto from mult_pos_pos[OF this this] show "4 * x * x * n < 4 * x * x * (?y * ?y)" unfolding id by (simp add: field_simps) qed qed qed qed end text ‹It remains to assemble everything into one algorithm.› definition sqrt_approx :: "'a :: {linordered_field,floor_ceiling} ⇒ 'a ⇒ 'a" where "sqrt_approx ε x ≡ if ε > 0 then (if x = 0 then 0 else let xpos = abs x in sqrt_approx_main_impl ε xpos (xpos + 1)) else 0" lemma sqrt_approx: assumes ε: "ε > 0" shows "¦sqrt_approx ε x * sqrt_approx ε x - ¦x¦¦ < ε" proof (cases "x = 0") case True with ε show ?thesis unfolding sqrt_approx_def by auto next case False let ?x = "¦x¦" let ?sqrti = "sqrt_approx_main_impl ε ?x (?x + 1)" let ?sqrt = "sqrt_approximation.sqrt_approx_main ε ?x (?x + 1)" define sqrt where "sqrt = ?sqrt" from False have x: "?x > 0" "?x + 1 > 0" by auto interpret sqrt_approximation ε ?x by (unfold_locales, insert x ε, auto) from False ε have "sqrt_approx ε x = ?sqrti" unfolding sqrt_approx_def by (simp add: Let_def) also have "?sqrti = ?sqrt" by (rule sqrt_approx_main_impl, auto) finally have id: "sqrt_approx ε x = sqrt" unfolding sqrt_def . have sqrt: "sqrt * sqrt > ?x ∧ sqrt * sqrt - ?x < ε" unfolding sqrt_def by (rule sqrt_approx_main_sound[OF x(2)], insert x mult_pos_pos[OF x(1) x(1)], auto simp: field_simps) show ?thesis unfolding id using sqrt by auto qed subsection ‹Some tests› text ‹Testing executabity and show that sqrt 2 is irrational› lemma "¬ (∃ i :: rat. i * i = 2)" proof - have "set (sqrt_rat 2) = {}" by eval thus ?thesis by simp qed text ‹Testing speed› lemma "¬ (∃ i :: int. i * i = 1234567890123456789012345678901234567890)" proof - have "set (sqrt_int 1234567890123456789012345678901234567890) = {}" by eval thus ?thesis by simp qed text ‹The following test› value "let ε = 1 / 100000000 :: rat; s = sqrt_approx ε 2 in (s, s * s - 2, ¦s * s - 2¦ < ε)" text ‹results in (1.4142135623731116, 4.738200762148612e-14, True).› end