(* Title: HOL/Nat_Int_Bij.thy
ID: $Id$
Author: Stefan Richter, Tobias Nipkow
*)
header{* Bijections $\mathbb{N}\to\mathbb{N}^2$ and $\mathbb{N}\to\mathbb{Z}$*}
theory Nat_Int_Bij
imports Hilbert_Choice Presburger
begin
subsection{* A bijection between @{text "\<nat>"} and @{text "\<nat>\<twosuperior>"} *}
text{* Definition and proofs are from \cite[page 85]{Oberschelp:1993}. *}
definition nat2_to_nat:: "(nat * nat) \<Rightarrow> nat" where
"nat2_to_nat pair = (let (n,m) = pair in (n+m) * Suc (n+m) div 2 + n)"
definition nat_to_nat2:: "nat \<Rightarrow> (nat * nat)" where
"nat_to_nat2 = inv nat2_to_nat"
lemma dvd2_a_x_suc_a: "2 dvd a * (Suc a)"
proof (cases "2 dvd a")
case True
then show ?thesis by (rule dvd_mult2)
next
case False
then have "Suc (a mod 2) = 2" by (simp add: dvd_eq_mod_eq_0)
then have "Suc a mod 2 = 0" by (simp add: mod_Suc)
then have "2 dvd Suc a" by (simp only:dvd_eq_mod_eq_0)
then show ?thesis by (rule dvd_mult)
qed
lemma
assumes eq: "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
shows nat2_to_nat_help: "u+v \<le> x+y"
proof (rule classical)
assume "\<not> ?thesis"
then have contrapos: "x+y < u+v"
by simp
have "nat2_to_nat (x,y) < (x+y) * Suc (x+y) div 2 + Suc (x + y)"
by (unfold nat2_to_nat_def) (simp add: Let_def)
also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2"
by (simp only: div_mult_self1_is_m)
also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2
+ ((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2"
proof -
have "2 dvd (x+y)*Suc(x+y)"
by (rule dvd2_a_x_suc_a)
then have "(x+y)*Suc(x+y) mod 2 = 0"
by (simp only: dvd_eq_mod_eq_0)
also
have "2 * Suc(x+y) mod 2 = 0"
by (rule mod_mult_self1_is_0)
ultimately have
"((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2 = 0"
by simp
then show ?thesis
by simp
qed
also have "\<dots> = ((x+y)*Suc(x+y) + 2*Suc(x+y)) div 2"
by (rule div_add1_eq [symmetric])
also have "\<dots> = ((x+y+2)*Suc(x+y)) div 2"
by (simp only: add_mult_distrib [symmetric])
also from contrapos have "\<dots> \<le> ((Suc(u+v))*(u+v)) div 2"
by (simp only: mult_le_mono div_le_mono)
also have "\<dots> \<le> nat2_to_nat (u,v)"
by (unfold nat2_to_nat_def) (simp add: Let_def)
finally show ?thesis
by (simp only: eq)
qed
theorem nat2_to_nat_inj: "inj nat2_to_nat"
proof -
{
fix u v x y
assume eq1: "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
then have "u+v \<le> x+y" by (rule nat2_to_nat_help)
also from eq1 [symmetric] have "x+y \<le> u+v"
by (rule nat2_to_nat_help)
finally have eq2: "u+v = x+y" .
with eq1 have ux: "u=x"
by (simp add: nat2_to_nat_def Let_def)
with eq2 have vy: "v=y" by simp
with ux have "(u,v) = (x,y)" by simp
}
then have "\<And>x y. nat2_to_nat x = nat2_to_nat y \<Longrightarrow> x=y" by fast
then show ?thesis unfolding inj_on_def by simp
qed
lemma nat_to_nat2_surj: "surj nat_to_nat2"
by (simp only: nat_to_nat2_def nat2_to_nat_inj inj_imp_surj_inv)
lemma gauss_sum_nat_upto: "2 * (\<Sum>i\<le>n::nat. i) = n * (n + 1)"
using gauss_sum[where 'a = nat]
by (simp add:atLeast0AtMost setsum_shift_lb_Suc0_0 numeral_2_eq_2)
lemma nat2_to_nat_surj: "surj nat2_to_nat"
proof (unfold surj_def)
{
fix z::nat
def r \<equiv> "Max {r. (\<Sum>i\<le>r. i) \<le> z}"
def x \<equiv> "z - (\<Sum>i\<le>r. i)"
hence "finite {r. (\<Sum>i\<le>r. i) \<le> z}"
by (simp add: lessThan_Suc_atMost[symmetric] lessThan_Suc finite_less_ub)
also have "0 \<in> {r. (\<Sum>i\<le>r. i) \<le> z}" by simp
hence "{r::nat. (\<Sum>i\<le>r. i) \<le> z} \<noteq> {}" by fast
ultimately have a: "r \<in> {r. (\<Sum>i\<le>r. i) \<le> z} \<and> (\<forall>s \<in> {r. (\<Sum>i\<le>r. i) \<le> z}. s \<le> r)"
by (simp add: r_def del:mem_Collect_eq)
{
assume "r<x"
hence "r+1\<le>x" by simp
hence "(\<Sum>i\<le>r. i)+(r+1)\<le>z" using x_def by arith
hence "(r+1) \<in> {r. (\<Sum>i\<le>r. i) \<le> z}" by simp
with a have "(r+1)\<le>r" by simp
}
hence b: "x\<le>r" by force
def y \<equiv> "r-x"
have "2*z=2*(\<Sum>i\<le>r. i)+2*x" using x_def a by simp arith
also have "\<dots> = r * (r+1) + 2*x" using gauss_sum_nat_upto by simp
also have "\<dots> = (x+y)*(x+y+1)+2*x" using y_def b by simp
also { have "2 dvd ((x+y)*(x+y+1))" using dvd2_a_x_suc_a by simp }
hence "\<dots> = 2 * nat2_to_nat(x,y)"
using nat2_to_nat_def by (simp add: Let_def dvd_mult_div_cancel)
finally have "z=nat2_to_nat (x, y)" by simp
}
thus "\<forall>y. \<exists>x. y = nat2_to_nat x" by fast
qed
subsection{* A bijection between @{text "\<nat>"} and @{text "\<int>"} *}
definition nat_to_int_bij :: "nat \<Rightarrow> int" where
"nat_to_int_bij n = (if 2 dvd n then int(n div 2) else -int(Suc n div 2))"
definition int_to_nat_bij :: "int \<Rightarrow> nat" where
"int_to_nat_bij i = (if 0<=i then 2*nat(i) else 2*nat(-i) - 1)"
lemma i2n_n2i_id: "int_to_nat_bij (nat_to_int_bij n) = n"
by (simp add: int_to_nat_bij_def nat_to_int_bij_def) presburger
lemma n2i_i2n_id: "nat_to_int_bij(int_to_nat_bij i) = i"
proof -
have "ALL m n::nat. m>0 \<longrightarrow> 2 * m - Suc 0 \<noteq> 2 * n" by presburger
thus ?thesis
by(simp add: nat_to_int_bij_def int_to_nat_bij_def, simp add:dvd_def)
qed
lemma inv_nat_to_int_bij: "inv nat_to_int_bij = int_to_nat_bij"
by (simp add: i2n_n2i_id inv_equality n2i_i2n_id)
lemma inv_int_to_nat_bij: "inv int_to_nat_bij = nat_to_int_bij"
by (simp add: i2n_n2i_id inv_equality n2i_i2n_id)
lemma surj_nat_to_int_bij: "surj nat_to_int_bij"
by (blast intro: n2i_i2n_id surjI)
lemma surj_int_to_nat_bij: "surj int_to_nat_bij"
by (blast intro: i2n_n2i_id surjI)
lemma inj_nat_to_int_bij: "inj nat_to_int_bij"
by(simp add:inv_int_to_nat_bij[symmetric] surj_int_to_nat_bij surj_imp_inj_inv)
lemma inj_int_to_nat_bij: "inj int_to_nat_bij"
by(simp add:inv_nat_to_int_bij[symmetric] surj_nat_to_int_bij surj_imp_inj_inv)
end