paulson@14127: (*  Title:      HOL/Library/NatPair.thy
paulson@14127:     ID:         $Id$
paulson@14127:     Author:     Christophe Tabacznyj and Lawrence C Paulson
paulson@14127:     Copyright   1996  University of Cambridge
paulson@14127: *)
paulson@14127: 
paulson@14127: header {*
paulson@14127:   \title{Pairs of Natural Numbers}
paulson@14127:   \author{Stefan Richter}
paulson@14127: *}
paulson@14127: 
paulson@14127: theory NatPair = Main:
paulson@14127: 
paulson@14127: text{*An injective function from $\mathbf{N}^2$ to
paulson@14127:   $\mathbf{N}$.  Definition and proofs are from 
paulson@14127:   Arnold Oberschelp.  Rekursionstheorie.  BI-Wissenschafts-Verlag, 1993
paulson@14127:   (page 85).  *}
paulson@14127: 
paulson@14127: constdefs 
paulson@14127:   nat2_to_nat:: "(nat * nat) \<Rightarrow> nat"
paulson@14127:   "nat2_to_nat pair \<equiv> let (n,m) = pair in (n+m) * Suc (n+m) div 2 + n"
paulson@14127: 
paulson@14127: lemma dvd2_a_x_suc_a: "2 dvd a * (Suc a)" 
paulson@14127: proof (cases "2 dvd a") 
paulson@14127:   case True
paulson@14127:   thus ?thesis by (rule dvd_mult2)
paulson@14127: next
paulson@14127:   case False 
paulson@14127:   hence "Suc (a mod 2) = 2" by (simp add: dvd_eq_mod_eq_0)
paulson@14127:   hence "Suc a mod 2 = 0" by (simp add: mod_Suc) 
paulson@14127:   hence "2 dvd Suc a" by (simp only:dvd_eq_mod_eq_0) 
paulson@14127:   thus ?thesis by (rule dvd_mult)
paulson@14127: qed
paulson@14127: 
paulson@14127: lemma assumes eq: "nat2_to_nat (u,v) = nat2_to_nat (x,y)" 
paulson@14127:   shows nat2_to_nat_help: "u+v \<le> x+y"
paulson@14127: proof (rule classical)
paulson@14127:   assume "\<not> ?thesis"
paulson@14127:   hence contrapos: "x+y < u+v" 
paulson@14127:     by simp
paulson@14127:   have "nat2_to_nat (x,y) < (x+y) * Suc (x+y) div 2 + Suc (x + y)" 
paulson@14127:     by (unfold nat2_to_nat_def) (simp add: Let_def)
paulson@14127:   also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2" 
paulson@14127:     by (simp only: div_mult_self1_is_m) 
paulson@14127:   also have "\<dots> = (x+y)*Suc(x+y) div 2 + 2 * Suc(x+y) div 2 
paulson@14127:     + ((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2" 
paulson@14127:   proof -
paulson@14127:     have "2 dvd (x+y)*Suc(x+y)" 
paulson@14127:       by (rule dvd2_a_x_suc_a)
paulson@14127:     hence "(x+y)*Suc(x+y) mod 2 = 0" 
paulson@14127:       by (simp only: dvd_eq_mod_eq_0)
paulson@14127:     also
paulson@14127:     have "2 * Suc(x+y) mod 2 = 0" 
paulson@14127:       by (rule mod_mult_self1_is_0)
paulson@14127:     ultimately have 
paulson@14127:       "((x+y)*Suc(x+y) mod 2 + 2 * Suc(x+y) mod 2) div 2 = 0" 
paulson@14127:       by simp 
paulson@14127:     thus ?thesis 
paulson@14127:       by simp
paulson@14127:   qed 
paulson@14127:   also have "\<dots> = ((x+y)*Suc(x+y) + 2*Suc(x+y)) div 2" 
paulson@14127:     by (rule div_add1_eq[THEN sym])
paulson@14127:   also have "\<dots> = ((x+y+2)*Suc(x+y)) div 2" 
paulson@14127:     by (simp only: add_mult_distrib[THEN sym])
paulson@14127:   also from contrapos have "\<dots> \<le> ((Suc(u+v))*(u+v)) div 2" 
paulson@14127:     by (simp only: mult_le_mono div_le_mono) 
paulson@14127:   also have "\<dots> \<le> nat2_to_nat (u,v)" 
paulson@14127:     by (unfold nat2_to_nat_def) (simp add: Let_def)
paulson@14127:   finally show ?thesis 
paulson@14127:     by (simp only: eq)
paulson@14127: qed
paulson@14127: 
paulson@14127: lemma nat2_to_nat_inj: "inj nat2_to_nat"
paulson@14127: proof -
paulson@14127:   {fix u v x y assume "nat2_to_nat (u,v) = nat2_to_nat (x,y)"
paulson@14127:     hence "u+v \<le> x+y" by (rule nat2_to_nat_help)
paulson@14127:     also from prems[THEN sym] have "x+y \<le> u+v" 
paulson@14127:       by (rule nat2_to_nat_help)
paulson@14127:     finally have eq: "u+v = x+y" .
paulson@14127:     with prems have ux: "u=x" 
paulson@14127:       by (simp add: nat2_to_nat_def Let_def)
paulson@14127:     with eq have vy: "v=y" 
paulson@14127:       by simp
paulson@14127:     with ux have "(u,v) = (x,y)" 
paulson@14127:       by simp
paulson@14127:   }
paulson@14127:   hence "\<And>x y. nat2_to_nat x = nat2_to_nat y \<Longrightarrow> x=y" 
paulson@14127:     by fast
paulson@14127:   thus ?thesis 
paulson@14127:     by (unfold inj_on_def) simp
paulson@14127: qed
paulson@14127: 
paulson@14127: end