--- a/src/HOL/Library/NatPair.thy Mon Sep 01 19:17:04 2008 +0200
+++ b/src/HOL/Library/NatPair.thy Mon Sep 01 19:17:37 2008 +0200
@@ -7,17 +7,18 @@
header {* Pairs of Natural Numbers *}
theory NatPair
-imports Plain "~~/src/HOL/Presburger"
+imports Main
begin
text{*
- An injective function from @{text "\<nat>\<twosuperior>"} to @{text \<nat>}. Definition
+ A bijection between @{text "\<nat>\<twosuperior>"} and @{text \<nat>}. 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 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")
@@ -88,4 +89,46 @@
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
+
end