Theory Pell_Algorithm
subsection ‹Executable code›
theory Pell_Algorithm
imports
Pell
Efficient_Discrete_Sqrt
"HOL-Library.Discrete"
"HOL-Library.While_Combinator"
"HOL-Library.Stream"
begin
subsubsection ‹Efficient computation of powers by squaring›
text ‹
The following is a tail-recursive implementation of exponentiation by squaring.
It works for any binary operation ‹f› that fulfils ‹f x (f x z) = f (f x x) z›, i.\,e.\
some weak form of associativity.
›
context
fixes f :: "'a ⇒ 'a ⇒ 'a"
begin
function efficient_power :: "'a ⇒ 'a ⇒ nat ⇒ 'a" where
"efficient_power y x 0 = y"
| "efficient_power y x (Suc 0) = f x y"
| "n ≠ 0 ⟹ even n ⟹ efficient_power y x n = efficient_power y (f x x) (n div 2)"
| "n ≠ 1 ⟹ odd n ⟹ efficient_power y x n = efficient_power (f x y) (f x x) (n div 2)"
by force+
termination by (relation "measure (snd ∘ snd)") (auto elim: oddE)
lemma efficient_power_code [code]:
"efficient_power y x n =
(if n = 0 then y
else if n = 1 then f x y
else if even n then efficient_power y (f x x) (n div 2)
else efficient_power (f x y) (f x x) (n div 2))"
by (induction y x n rule: efficient_power.induct) auto
lemma efficient_power_correct:
assumes "⋀x z. f x (f x z) = f (f x x) z"
shows "efficient_power y x n = (f x ^^ n) y"
proof -
have [simp]: "f ^^ 2 = (λx. f (f x))" for f :: "'a ⇒ 'a"
by (simp add: eval_nat_numeral o_def)
show ?thesis
by (induction y x n rule: efficient_power.induct)
(auto elim!: evenE oddE simp: funpow_mult [symmetric] funpow_Suc_right assms
simp del: funpow.simps(2))
qed
end
subsubsection ‹Multiplication and powers of solutions›
text ‹
We define versions of Pell solution multiplication and exponentiation specialised
to natural numbers, both for efficiency reasons and to circumvent the problem of
generating code for definitions made inside locales.
›
fun pell_mul_nat :: "nat ⇒ nat × nat ⇒ _" where
"pell_mul_nat D (a, b) (x, y) = (a * x + D * b * y, a * y + b * x)"
lemma (in pell) pell_mul_nat_correct [simp]: "pell_mul_nat D = pell.pell_mul D"
by (auto simp add: pell_mul_def fun_eq_iff)
definition efficient_pell_power :: "nat ⇒ nat × nat ⇒ nat ⇒ nat × nat" where
"efficient_pell_power D z n = efficient_power (pell_mul_nat D) (1, 0) z n"
lemma efficient_pell_power_correct [simp]:
"efficient_pell_power D z n = (pell_mul_nat D z ^^ n) (1, 0)"
unfolding efficient_pell_power_def
by (intro efficient_power_correct) (auto simp: algebra_simps)
subsubsection ‹Finding the fundamental solution›
text ‹
In the following, we set up a very simple algorithm for computing the fundamental
solution ‹(x, y)›. We try inreasing values for ‹y› until $1 + Dy^2$ is a perfect
square, which we check using an efficient square-detection algorithm. This is efficient
enough to work on some interesting small examples.
Much better algorithms (typically based on the continued fraction expansion of $\sqrt{D}$)
are available, but they are also considerably more complicated.
›
lemma Discrete_sqrt_square_is_square:
assumes "is_square n"
shows "Discrete.sqrt n ^ 2 = n"
using assms unfolding is_nth_power_def by force
definition find_fund_sol_step :: "nat ⇒ nat × nat + nat × nat ⇒ _" where
"find_fund_sol_step D = (λInl (y, y') ⇒
(case get_nat_sqrt y' of
Some x ⇒ Inr (x, y)
| None ⇒ Inl (y + 1, y' + D * (2 * y + 1))))"
definition find_fund_sol where
"find_fund_sol D =
(if square_test D then
(0, 0)
else
sum.projr (while sum.isl (find_fund_sol_step D) (Inl (1, 1 + D))))"
lemma fund_sol_code:
assumes "¬is_square (D :: nat)"
shows "pell.fund_sol D = sum.projr (while isl (find_fund_sol_step D) (Inl (Suc 0, Suc D)))"
proof -
from assms interpret pell D by unfold_locales
note [simp] = find_fund_sol_step_def
define f where "f = find_fund_sol_step D"
define P :: "nat ⇒ bool" where "P = (λy. y > 0 ∧ is_square (y^2 * D + 1))"
define Q :: "nat × nat ⇒ bool" where
"Q = (λ(x,y). P y ∧ (∀y'∈{0<..<y}. ¬P y') ∧ x = Discrete.sqrt (y^2 * D + 1))"
define R :: "nat × nat + nat × nat ⇒ bool"
where "R = (λs. case s of
Inl (m, m') ⇒ m > 0 ∧ (m' = m^2 * D + 1) ∧ (∀y∈{0<..<m}. ¬is_square (y^2 * D + 1))
| Inr x ⇒ Q x)"
define rel :: "((nat × nat + nat × nat) × (nat × nat + nat × nat)) set"
where "rel = {(A,B). (case (A, B) of
(Inl (m, _), Inl (m', _)) ⇒ m' > 0 ∧ m > m' ∧ m ≤ snd fund_sol
| (Inr _, Inl (m', _)) ⇒ m' ≤ snd fund_sol
| _ ⇒ False) ∧ A = f B}"
obtain x y where xy: "sum.projr (while isl f (Inl (Suc 0, Suc D))) = (x, y)"
by (cases "sum.projr (while isl f (Inl (Suc 0, Suc D)))")
have neq_fund_solI: "y ≠ snd fund_sol" if "¬ is_square (Suc (y⇧2 * D))" for y
proof
assume "y = snd fund_sol"
with fund_sol_is_nontriv_solution have "Suc (y⇧2 * D) = fst fund_sol ^ 2"
by (simp add: nontriv_solution_def case_prod_unfold)
hence "is_square (Suc (y⇧2 * D))" by simp
with that show False by contradiction
qed
have "case_sum (λ_. False) Q (while sum.isl f (Inl (m, m^2 * D + 1)))"
if "∀y∈{0<..<m}. ¬is_square (y^2 * D + 1)" "m > 0" for m
proof (rule while_rule[where b = sum.isl])
show "R (Inl (m, m⇧2 * D + 1))"
using that by (auto simp: R_def)
next
fix s assume "R s" "isl s"
thus "R (f s)"
by (auto simp: not_less_less_Suc_eq Q_def P_def R_def f_def get_nat_sqrt_def
power2_eq_square algebra_simps split: sum.splits prod.splits)
next
fix s assume "R s" "¬isl s"
thus "case s of Inl _ ⇒ False | Inr x ⇒ Q x"
by (auto simp: R_def split: sum.splits)
next
fix s assume s: "R s" "isl s"
show "(f s, s) ∈ rel"
proof (cases s)
case [simp]: (Inl s')
obtain a b where [simp]: "s' = (a, b)" by (cases s')
from s have *: "a > 0" "b = Suc (a⇧2 * D)" "⋀y. y ∈ {0<..<a} ⟹ ¬ is_square (Suc (y⇧2 * D))"
by (auto simp: R_def)
have "a < snd fund_sol" if **: "¬ is_square (Suc (a⇧2 * D))"
proof -
from neq_fund_solI have "y' ≠ snd fund_sol" if "y' ∈ {0<..<Suc a}" for y'
using * ** that by (cases "y' = a") auto
moreover have "snd fund_sol ≠ 0" using fund_sol_is_nontriv_solution
by (intro notI, cases fund_sol) (auto simp: nontriv_solution_altdef)
ultimately have "∀y'≤a. y' ≠ snd fund_sol" by (auto simp: less_Suc_eq_le)
thus "snd fund_sol > a" by (cases "a < snd fund_sol") (auto simp: not_less)
qed
moreover have "a ≤ snd fund_sol"
proof -
have "∀y'∈{0<..<a}. y' ≠ snd fund_sol" using neq_fund_solI *
by (auto simp: less_Suc_eq_le)
moreover have "snd fund_sol ≠ 0" using fund_sol_is_nontriv_solution
by (intro notI, cases fund_sol) (auto simp: nontriv_solution_altdef)
ultimately have "∀y'<a. y' ≠ snd fund_sol" by (auto simp: less_Suc_eq_le)
thus "snd fund_sol ≥ a" by (cases "a ≤ snd fund_sol") (auto simp: not_less)
qed
ultimately show ?thesis using *
by (auto simp: f_def get_nat_sqrt_def rel_def)
qed (insert s, auto)
next
define rel'
where "rel' = {(y, x). (case x of Inl (m, _) ⇒ m ≤ snd fund_sol | Inr _ ⇒ False) ∧ y = f x}"
have "wf rel'" unfolding rel'_def
by (rule wf_if_measure[where f = "λz. case z of Inl (m, _) ⇒ Suc (snd fund_sol) - m | _ ⇒ 0"])
(auto split: prod.splits sum.splits simp: f_def get_nat_sqrt_def)
moreover have "rel ⊆ rel'"
proof safe
fix w z assume "(w, z) ∈ rel"
thus "(w, z) ∈ rel'" by (cases w; cases z) (auto simp: rel_def rel'_def)
qed
ultimately show "wf rel" by (rule wf_subset)
qed
from this[of 1] and xy have *: "Q (x, y)"
by (auto split: sum.splits)
from * have "is_square (Suc (y⇧2 * D))" by (simp add: Q_def P_def)
with * have "x⇧2 = Suc (y⇧2 * D)" "y > 0"
by (auto simp: Q_def P_def Discrete_sqrt_square_is_square)
hence "nontriv_solution (x, y)"
by (auto simp: nontriv_solution_def)
from this have "snd fund_sol ≤ snd (x, y)"
by (rule fund_sol_minimal'')
moreover have "snd fund_sol ≥ y"
proof -
from * have "(∀y'∈{0<..<y}. ¬ is_square (Suc (y'⇧2 * D)))"
by (simp add: Q_def P_def)
with neq_fund_solI have "(∀y'∈{0<..<y}. y' ≠ snd fund_sol)"
by auto
moreover have "snd fund_sol ≠ 0"
using fund_sol_is_nontriv_solution
by (cases fund_sol) (auto intro!: Nat.gr0I simp: nontriv_solution_altdef)
ultimately have "(∀y'<y. y' ≠ snd fund_sol)" by auto
thus "snd fund_sol ≥ y" by (cases "snd fund_sol ≥ y") (auto simp: not_less)
qed
ultimately have "snd fund_sol = y" by simp
with solutions_linorder_strict[of x y "fst fund_sol" "snd fund_sol"]
fund_sol_is_nontriv_solution ‹nontriv_solution (x, y)›
have "fst fund_sol = x" by (cases fund_sol) (auto simp: nontriv_solution_altdef)
with ‹snd fund_sol = y› have "fund_sol = (x, y)"
by (cases fund_sol) simp
with xy show ?thesis by (simp add: f_def)
qed
lemma find_fund_sol_correct: "find_fund_sol D = (if is_square D then (0, 0) else pell.fund_sol D)"
by (simp add: find_fund_sol_def fund_sol_code square_test_correct)
subsubsection ‹The infinite list of all solutions›
definition pell_solutions :: "nat ⇒ (nat × nat) stream" where
"pell_solutions D = (let z = find_fund_sol D in siterate (pell_mul_nat D z) (1, 0))"
lemma (in pell) "snth (pell_solutions D) n = nth_solution n"
by (simp add: pell_solutions_def Let_def find_fund_sol_correct nonsquare_D nth_solution_def
pell_power_def pell_mul_commutes[of _ fund_sol])
subsubsection ‹Computing the $n$-th solution›
definition find_nth_solution :: "nat ⇒ nat ⇒ nat × nat" where
"find_nth_solution D n =
(if is_square D then (0, 0) else
let z = sum.projr (while isl (find_fund_sol_step D) (Inl (Suc 0, Suc D)))
in efficient_pell_power D z n)"
lemma (in pell) find_nth_solution_correct: "find_nth_solution D n = nth_solution n"
by (simp add: find_nth_solution_def nonsquare_D nth_solution_def fund_sol_code
pell_power_def pell_mul_commutes[of _ "projr _"])
end