
diff -r 003dff7410c1 -r c92850d2d16c src/HOL/Library/NatPair.thy
--- a/src/HOL/Library/NatPair.thy	Tue Sep 02 22:20:27 2008 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,134 +0,0 @@
-(*  Title:      HOL/Library/NatPair.thy
-    ID:         $Id$
-    Author:     Stefan Richter
-    Copyright   2003 Technische Universitaet Muenchen
-*)
-
-header {* Pairs of Natural Numbers *}
-
-theory NatPair
-imports Main
-begin
-
-text{*
-  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 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
-
-end




