# hub.darcs.net :: rdockins -> domains -> files

A Coq library for domain theory

## root / pairing.v

 ```1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 ``` ``````(* Copyright (c) 2014, Robert Dockins *) Require Import NArith. Local Open Scope N_scope. (** * Bitwise pairing function on [N]. Here we define a "bitwise" pairing isomorphism on the nonnegative integers. This is one of many different ways to witness the isormporphism between [N] and [N×N]. We will use the pairing function to define the union of countable sets and other similar constructions. The details of the isomorphism are unimportant once the isomorphism is defined. We prove this particular isomorphism because it requires almost no facts about arithmetic, and the proofs go by simple inductions on the binary representation of positives. *) Fixpoint inflate (x:positive) : positive := match x with | xH => xO xH | xO x' => xO (xO (inflate x')) | xI x' => xO (xI (inflate x')) end. Fixpoint inflate' (x:positive) : positive := match x with | xH => xH | xO x' => xO (xO (inflate' x')) | xI x' => xI (xO (inflate' x')) end. Fixpoint deflate (x:positive) : N := match x with | xH => N0 | xO xH => Npos xH | xI xH => Npos xH | xO (xO x') => match deflate x' with N0 => N0 | Npos q => Npos (xO q) end | xI (xO x') => match deflate x' with N0 => N0 | Npos q => Npos (xO q) end | xO (xI x') => match deflate x' with N0 => Npos xH | Npos q => Npos (xI q) end | xI (xI x') => match deflate x' with N0 => Npos xH | Npos q => Npos (xI q) end end. Fixpoint deflate' (x:positive) : N := match x with | xH => Npos xH | xO xH => N0 | xI xH => Npos xH | xO (xO x') => match deflate' x' with N0 => N0 | Npos q => Npos (xO q) end | xI (xO x') => match deflate' x' with N0 => Npos xH | Npos q => Npos (xI q) end | xO (xI x') => match deflate' x' with N0 => N0 | Npos q => Npos (xO q) end | xI (xI x') => match deflate' x' with N0 => Npos xH | Npos q => Npos (xI q) end end. Lemma deflate_inflate0 : forall x, deflate (inflate x) = Npos x. Proof. induction x; simpl; intros; auto. rewrite IHx; auto. rewrite IHx; auto. Qed. Lemma deflate_inflate0' : forall y, deflate' (inflate' y) = Npos y. Proof. induction y; simpl; intros; auto. rewrite IHy; auto. rewrite IHy; auto. Qed. Lemma deflate_inflate1 : forall y, deflate (inflate' y) = 0. Proof. induction y; simpl; auto. rewrite IHy; auto. rewrite IHy; auto. Qed. Lemma deflate_inflate1' : forall x, deflate' (inflate x) = 0. Proof. induction x; simpl; auto. rewrite IHx; auto. rewrite IHx; auto. Qed. Lemma deflate_inflate : forall x y, deflate (inflate x + inflate' y) = Npos x. Proof. induction x; simpl; intros. destruct y; simpl; f_equal; auto. rewrite IHx. auto. rewrite IHx. auto. rewrite deflate_inflate0. auto. destruct y; simpl; f_equal; auto. rewrite IHx. auto. rewrite IHx. auto. rewrite deflate_inflate0. auto. destruct y; simpl; f_equal; auto. rewrite deflate_inflate1. auto. rewrite deflate_inflate1. auto. Qed. Lemma deflate_inflate' : forall y x, deflate' (inflate x + inflate' y) = Npos y. Proof. induction y; simpl; intros; auto. destruct x; simpl. rewrite IHy. auto. rewrite IHy. auto. rewrite deflate_inflate0'. auto. destruct x; simpl. rewrite IHy. auto. rewrite IHy. auto. rewrite deflate_inflate0'. auto. destruct x; simpl. rewrite deflate_inflate1'. auto. rewrite deflate_inflate1'. auto. auto. Qed. Lemma deflate00 : forall p, deflate (p~0~0) = 2*(deflate p). Proof. intros. simpl. case_eq (deflate p); auto. Qed. Lemma deflate01 : forall p, deflate (p~0~1) = 2*(deflate p). Proof. intros. simpl. case_eq (deflate p); auto. Qed. Lemma deflate10 : forall p, deflate (p~1~0) = 2*(deflate p) + 1 . Proof. intros. simpl. case_eq (deflate p); auto. Qed. Lemma deflate11 : forall p, deflate (p~1~1) = 2*(deflate p) + 1 . Proof. intros. simpl. case_eq (deflate p); auto. Qed. Lemma deflate00' : forall p, deflate' (p~0~0) = 2*(deflate' p). Proof. intros. simpl. case_eq (deflate' p); auto. Qed. Lemma deflate01' : forall p, deflate' (p~0~1) = 2*(deflate' p)+1. Proof. intros. simpl. case_eq (deflate' p); auto. Qed. Lemma deflate10' : forall p, deflate' (p~1~0) = 2*(deflate' p). Proof. intros. simpl. case_eq (deflate' p); auto. Qed. Lemma deflate11' : forall p, deflate' (p~1~1) = 2*(deflate' p) + 1 . Proof. intros. simpl. case_eq (deflate' p); auto. Qed. Definition pairing (p:N*N) : N := match p with | (N0, N0) => N0 | (N0, Npos y) => Npos (inflate' y) | (Npos x, N0) => Npos (inflate x) | (Npos x, Npos y) => Npos (inflate x + inflate' y) end. Definition unpairing (z:N) : N*N := match z with | N0 => (N0,N0) | Npos z => (deflate z, deflate' z) end. Lemma pairing00 : forall p q, pairing (2*p, 2*q) = 4*pairing (p,q). Proof. simpl; intros. destruct p; destruct q; simpl; auto. Qed. Lemma pairing10 : forall p q, pairing (2*p + 1, 2*q) = 4*pairing (p,q)+2. Proof. simpl; intros. destruct p; destruct q; simpl; auto. Qed. Lemma pairing01 : forall p q, pairing (2*p, 2*q + 1) = 4*pairing (p,q)+1. Proof. simpl; intros. destruct p; destruct q; simpl; auto. Qed. Lemma pairing11 : forall p q, pairing (2*p + 1, 2*q + 1) = 4*pairing (p,q)+3. Proof. simpl; intros. destruct p; destruct q; simpl; auto. Qed. Lemma unpairing_pairing : forall p, unpairing (pairing p) = p. Proof. intros [x y]. destruct x; destruct y; simpl; auto. rewrite deflate_inflate1. rewrite deflate_inflate0'. auto. rewrite deflate_inflate1'. rewrite deflate_inflate0. auto. rewrite deflate_inflate. rewrite deflate_inflate'. auto. Qed. Lemma pairing_unpairing : forall z, pairing (unpairing z) = z. Proof. intro z. destruct z. simpl; auto. unfold unpairing. revert p. fix 1. intro p. destruct p. destruct p. rewrite deflate11. rewrite deflate11'. rewrite pairing11. rewrite pairing_unpairing. auto. rewrite deflate01. rewrite deflate01'. rewrite pairing01. rewrite pairing_unpairing. auto. simpl. auto. destruct p. rewrite deflate10. rewrite deflate10'. rewrite pairing10. rewrite pairing_unpairing. auto. rewrite deflate00. rewrite deflate00'. rewrite pairing00. rewrite pairing_unpairing. auto. simpl. auto. simpl. auto. Qed. Global Opaque unpairing pairing.``````