section ‹Common discrete functions›
theory Discrete
imports Main
begin
subsection ‹Discrete logarithm›
context
begin
qualified fun log :: "nat ⇒ nat"
where [simp del]: "log n = (if n < 2 then 0 else Suc (log (n div 2)))"
lemma log_induct [consumes 1, case_names one double]:
fixes n :: nat
assumes "n > 0"
assumes one: "P 1"
assumes double: "⋀n. n ≥ 2 ⟹ P (n div 2) ⟹ P n"
shows "P n"
using ‹n > 0› proof (induct n rule: log.induct)
fix n
assume "¬ n < 2 ⟹
0 < n div 2 ⟹ P (n div 2)"
then have *: "n ≥ 2 ⟹ P (n div 2)" by simp
assume "n > 0"
show "P n"
proof (cases "n = 1")
case True with one show ?thesis by simp
next
case False with ‹n > 0› have "n ≥ 2" by auto
moreover with * have "P (n div 2)" .
ultimately show ?thesis by (rule double)
qed
qed
lemma log_zero [simp]: "log 0 = 0"
by (simp add: log.simps)
lemma log_one [simp]: "log 1 = 0"
by (simp add: log.simps)
lemma log_Suc_zero [simp]: "log (Suc 0) = 0"
using log_one by simp
lemma log_rec: "n ≥ 2 ⟹ log n = Suc (log (n div 2))"
by (simp add: log.simps)
lemma log_twice [simp]: "n ≠ 0 ⟹ log (2 * n) = Suc (log n)"
by (simp add: log_rec)
lemma log_half [simp]: "log (n div 2) = log n - 1"
proof (cases "n < 2")
case True
then have "n = 0 ∨ n = 1" by arith
then show ?thesis by (auto simp del: One_nat_def)
next
case False
then show ?thesis by (simp add: log_rec)
qed
lemma log_exp [simp]: "log (2 ^ n) = n"
by (induct n) simp_all
lemma log_mono: "mono log"
proof
fix m n :: nat
assume "m ≤ n"
then show "log m ≤ log n"
proof (induct m arbitrary: n rule: log.induct)
case (1 m)
then have mn2: "m div 2 ≤ n div 2" by arith
show "log m ≤ log n"
proof (cases "m ≥ 2")
case False
then have "m = 0 ∨ m = 1" by arith
then show ?thesis by (auto simp del: One_nat_def)
next
case True then have "¬ m < 2" by simp
with mn2 have "n ≥ 2" by arith
from True have m2_0: "m div 2 ≠ 0" by arith
with mn2 have n2_0: "n div 2 ≠ 0" by arith
from ‹¬ m < 2› "1.hyps" mn2 have "log (m div 2) ≤ log (n div 2)" by blast
with m2_0 n2_0 have "log (2 * (m div 2)) ≤ log (2 * (n div 2))" by simp
with m2_0 n2_0 ‹m ≥ 2› ‹n ≥ 2› show ?thesis by (simp only: log_rec [of m] log_rec [of n]) simp
qed
qed
qed
lemma log_exp2_le:
assumes "n > 0"
shows "2 ^ log n ≤ n"
using assms proof (induct n rule: log_induct)
show "2 ^ log 1 ≤ (1 :: nat)" by simp
next
fix n :: nat
assume "n ≥ 2"
with log_mono have "log n ≥ Suc 0"
by (simp add: log.simps)
assume "2 ^ log (n div 2) ≤ n div 2"
with ‹n ≥ 2› have "2 ^ (log n - Suc 0) ≤ n div 2" by simp
then have "2 ^ (log n - Suc 0) * 2 ^ 1 ≤ n div 2 * 2" by simp
with ‹log n ≥ Suc 0› have "2 ^ log n ≤ n div 2 * 2"
unfolding power_add [symmetric] by simp
also have "n div 2 * 2 ≤ n" by (cases "even n") simp_all
finally show "2 ^ log n ≤ n" .
qed
subsection ‹Discrete square root›
qualified definition sqrt :: "nat ⇒ nat"
where "sqrt n = Max {m. m⇧2 ≤ n}"
lemma sqrt_aux:
fixes n :: nat
shows "finite {m. m⇧2 ≤ n}" and "{m. m⇧2 ≤ n} ≠ {}"
proof -
{ fix m
assume "m⇧2 ≤ n"
then have "m ≤ n"
by (cases m) (simp_all add: power2_eq_square)
} note ** = this
then have "{m. m⇧2 ≤ n} ⊆ {m. m ≤ n}" by auto
then show "finite {m. m⇧2 ≤ n}" by (rule finite_subset) rule
have "0⇧2 ≤ n" by simp
then show *: "{m. m⇧2 ≤ n} ≠ {}" by blast
qed
lemma [code]: "sqrt n = Max (Set.filter (λm. m⇧2 ≤ n) {0..n})"
proof -
from power2_nat_le_imp_le [of _ n] have "{m. m ≤ n ∧ m⇧2 ≤ n} = {m. m⇧2 ≤ n}" by auto
then show ?thesis by (simp add: sqrt_def Set.filter_def)
qed
lemma sqrt_inverse_power2 [simp]: "sqrt (n⇧2) = n"
proof -
have "{m. m ≤ n} ≠ {}" by auto
then have "Max {m. m ≤ n} ≤ n" by auto
then show ?thesis
by (auto simp add: sqrt_def power2_nat_le_eq_le intro: antisym)
qed
lemma sqrt_zero [simp]: "sqrt 0 = 0"
using sqrt_inverse_power2 [of 0] by simp
lemma sqrt_one [simp]: "sqrt 1 = 1"
using sqrt_inverse_power2 [of 1] by simp
lemma mono_sqrt: "mono sqrt"
proof
fix m n :: nat
have *: "0 * 0 ≤ m" by simp
assume "m ≤ n"
then show "sqrt m ≤ sqrt n"
by (auto intro!: Max_mono ‹0 * 0 ≤ m› finite_less_ub simp add: power2_eq_square sqrt_def)
qed
lemma sqrt_greater_zero_iff [simp]: "sqrt n > 0 ⟷ n > 0"
proof -
have *: "0 < Max {m. m⇧2 ≤ n} ⟷ (∃a∈{m. m⇧2 ≤ n}. 0 < a)"
by (rule Max_gr_iff) (fact sqrt_aux)+
show ?thesis
proof
assume "0 < sqrt n"
then have "0 < Max {m. m⇧2 ≤ n}" by (simp add: sqrt_def)
with * show "0 < n" by (auto dest: power2_nat_le_imp_le)
next
assume "0 < n"
then have "1⇧2 ≤ n ∧ 0 < (1::nat)" by simp
then have "∃q. q⇧2 ≤ n ∧ 0 < q" ..
with * have "0 < Max {m. m⇧2 ≤ n}" by blast
then show "0 < sqrt n" by (simp add: sqrt_def)
qed
qed
lemma sqrt_power2_le [simp]: "(sqrt n)⇧2 ≤ n"
proof (cases "n > 0")
case False then show ?thesis by simp
next
case True then have "sqrt n > 0" by simp
then have "mono (times (Max {m. m⇧2 ≤ n}))" by (auto intro: mono_times_nat simp add: sqrt_def)
then have *: "Max {m. m⇧2 ≤ n} * Max {m. m⇧2 ≤ n} = Max (times (Max {m. m⇧2 ≤ n}) ` {m. m⇧2 ≤ n})"
using sqrt_aux [of n] by (rule mono_Max_commute)
have "Max (op * (Max {m. m * m ≤ n}) ` {m. m * m ≤ n}) ≤ n"
apply (subst Max_le_iff)
apply (metis (mono_tags) finite_imageI finite_less_ub le_square)
apply simp
apply (metis le0 mult_0_right)
apply auto
proof -
fix q
assume "q * q ≤ n"
show "Max {m. m * m ≤ n} * q ≤ n"
proof (cases "q > 0")
case False then show ?thesis by simp
next
case True then have "mono (times q)" by (rule mono_times_nat)
then have "q * Max {m. m * m ≤ n} = Max (times q ` {m. m * m ≤ n})"
using sqrt_aux [of n] by (auto simp add: power2_eq_square intro: mono_Max_commute)
then have "Max {m. m * m ≤ n} * q = Max (times q ` {m. m * m ≤ n})" by (simp add: ac_simps)
then show ?thesis
apply simp
apply (subst Max_le_iff)
apply auto
apply (metis (mono_tags) finite_imageI finite_less_ub le_square)
apply (metis ‹q * q ≤ n›)
apply (metis ‹q * q ≤ n› le_cases mult_le_mono1 mult_le_mono2 order_trans)
done
qed
qed
with * show ?thesis by (simp add: sqrt_def power2_eq_square)
qed
lemma sqrt_le: "sqrt n ≤ n"
using sqrt_aux [of n] by (auto simp add: sqrt_def intro: power2_nat_le_imp_le)
end
end