countable.v 9.64 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2015, Robbert Krebbers. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
2
(* This file is distributed under the terms of the BSD license. *)
3
Require Export prelude.list.
4 5 6 7 8 9 10
Local Open Scope positive.

Class Countable A `{ x y : A, Decision (x = y)} := {
  encode : A  positive;
  decode : positive  option A;
  decode_encode x : decode (encode x) = Some x
}.
Robbert Krebbers's avatar
Robbert Krebbers committed
11 12
Arguments encode : simpl never.
Arguments decode : simpl never.
13 14 15 16 17

Definition encode_nat `{Countable A} (x : A) : nat :=
  pred (Pos.to_nat (encode x)).
Definition decode_nat `{Countable A} (i : nat) : option A :=
  decode (Pos.of_nat (S i)).
18 19 20 21 22
Instance encode_injective `{Countable A} : Injective (=) (=) encode.
Proof.
  intros x y Hxy; apply (injective Some).
  by rewrite <-(decode_encode x), Hxy, decode_encode.
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
23 24
Instance encode_nat_injective `{Countable A} : Injective (=) (=) encode_nat.
Proof. unfold encode_nat; intros x y Hxy; apply (injective encode); lia. Qed.
25 26 27 28 29 30 31
Lemma decode_encode_nat `{Countable A} x : decode_nat (encode_nat x) = Some x.
Proof.
  pose proof (Pos2Nat.is_pos (encode x)).
  unfold decode_nat, encode_nat. rewrite Nat.succ_pred by lia.
  by rewrite Pos2Nat.id, decode_encode.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
32
(** * Choice principles *)
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55
Section choice.
  Context `{Countable A} (P : A  Prop) `{ x, Decision (P x)}.

  Inductive choose_step: relation positive :=
    | choose_step_None {p} : decode p = None  choose_step (Psucc p) p
    | choose_step_Some {p x} :
       decode p = Some x  ¬P x  choose_step (Psucc p) p.
  Lemma choose_step_acc : ( x, P x)  Acc choose_step 1%positive.
  Proof.
    intros [x Hx]. cut ( i p,
      i  encode x  1 + encode x = p + i  Acc choose_step p).
    { intros help. by apply (help (encode x)). }
    induction i as [|i IH] using Pos.peano_ind; intros p ??.
    { constructor. intros j. assert (p = encode x) by lia; subst.
      inversion 1 as [? Hd|?? Hd]; subst;
        rewrite decode_encode in Hd; congruence. }
    constructor. intros j.
    inversion 1 as [? Hd|? y Hd]; subst; auto with lia.
  Qed.
  Fixpoint choose_go {i} (acc : Acc choose_step i) : A :=
    match Some_dec (decode i) with
    | inleft (xHx) =>
      match decide (P x) with
Robbert Krebbers's avatar
Robbert Krebbers committed
56
      | left _ => x | right H => choose_go (Acc_inv acc (choose_step_Some Hx H))
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76
      end
    | inright H => choose_go (Acc_inv acc (choose_step_None H))
    end.
  Fixpoint choose_go_correct {i} (acc : Acc choose_step i) : P (choose_go acc).
  Proof. destruct acc; simpl. repeat case_match; auto. Qed.
  Fixpoint choose_go_pi {i} (acc1 acc2 : Acc choose_step i) :
    choose_go acc1 = choose_go acc2.
  Proof. destruct acc1, acc2; simpl; repeat case_match; auto. Qed.

  Definition choose (H:  x, P x) : A := choose_go (choose_step_acc H).
  Definition choose_correct (H:  x, P x) : P (choose H) := choose_go_correct _.
  Definition choose_pi (H1 H2 :  x, P x) :
    choose H1 = choose H2 := choose_go_pi _ _.
  Definition choice (HA :  x, P x) : { x | P x } := _choose_correct HA.
End choice.

Lemma surjective_cancel `{Countable A} `{ x y : B, Decision (x = y)}
  (f : A  B) `{!Surjective (=) f} : { g : B  A & Cancel (=) f g }.
Proof.
  exists (λ y, choose (λ x, f x = y) (surjective f y)).
Robbert Krebbers's avatar
Robbert Krebbers committed
77
  intros y. by rewrite (choose_correct (λ x, f x = y) (surjective f y)).
78 79
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
80 81
(** * Instances *)
(** ** Option *)
82
Program Instance option_countable `{Countable A} : Countable (option A) := {|
Robbert Krebbers's avatar
Robbert Krebbers committed
83 84
  encode o := match o with None => 1 | Some x => Pos.succ (encode x) end;
  decode p := if decide (p = 1) then Some None else Some <$> decode (Pos.pred p)
85 86 87 88 89 90
|}.
Next Obligation.
  intros ??? [x|]; simpl; repeat case_decide; auto with lia.
  by rewrite Pos.pred_succ, decode_encode.
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
91
(** ** Sums *)
92 93 94 95 96 97 98 99 100 101 102
Program Instance sum_countable `{Countable A} `{Countable B} :
  Countable (A + B)%type := {|
    encode xy :=
      match xy with inl x => (encode x)~0 | inr y => (encode y)~1 end;
    decode p :=
      match p with
      | 1 => None | p~0 => inl <$> decode p | p~1 => inr <$> decode p
      end
  |}.
Next Obligation. by intros ?????? [x|y]; simpl; rewrite decode_encode. Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
103
(** ** Products *)
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
Fixpoint prod_encode_fst (p : positive) : positive :=
  match p with
  | 1 => 1
  | p~0 => (prod_encode_fst p)~0~0
  | p~1 => (prod_encode_fst p)~0~1
  end.
Fixpoint prod_encode_snd (p : positive) : positive :=
  match p with
  | 1 => 1~0
  | p~0 => (prod_encode_snd p)~0~0
  | p~1 => (prod_encode_snd p)~1~0
  end.
Fixpoint prod_encode (p q : positive) : positive :=
  match p, q with
  | 1, 1 => 1~1
  | p~0, 1 => (prod_encode_fst p)~1~0
  | p~1, 1 => (prod_encode_fst p)~1~1
  | 1, q~0 => (prod_encode_snd q)~0~1
  | 1, q~1 => (prod_encode_snd q)~1~1
  | p~0, q~0 => (prod_encode p q)~0~0
  | p~0, q~1 => (prod_encode p q)~1~0
  | p~1, q~0 => (prod_encode p q)~0~1
  | p~1, q~1 => (prod_encode p q)~1~1
  end.
Fixpoint prod_decode_fst (p : positive) : option positive :=
  match p with
  | p~0~0 => (~0) <$> prod_decode_fst p
  | p~0~1 => Some match prod_decode_fst p with Some q => q~1 | _ => 1 end
  | p~1~0 => (~0) <$> prod_decode_fst p
  | p~1~1 => Some match prod_decode_fst p with Some q => q~1 | _ => 1 end
  | 1~0 => None
  | 1~1 => Some 1
  | 1 => Some 1
  end.
Fixpoint prod_decode_snd (p : positive) : option positive :=
  match p with
  | p~0~0 => (~0) <$> prod_decode_snd p
  | p~0~1 => (~0) <$> prod_decode_snd p
  | p~1~0 => Some match prod_decode_snd p with Some q => q~1 | _ => 1 end
  | p~1~1 => Some match prod_decode_snd p with Some q => q~1 | _ => 1 end
  | 1~0 => Some 1
  | 1~1 => Some 1
  | 1 => None
  end.

Lemma prod_decode_encode_fst p q : prod_decode_fst (prod_encode p q) = Some p.
Proof.
  assert ( p, prod_decode_fst (prod_encode_fst p) = Some p).
  { intros p'. by induction p'; simplify_option_equality. }
  assert ( p, prod_decode_fst (prod_encode_snd p) = None).
  { intros p'. by induction p'; simplify_option_equality. }
  revert q. by induction p; intros [?|?|]; simplify_option_equality.
Qed.
Lemma prod_decode_encode_snd p q : prod_decode_snd (prod_encode p q) = Some q.
Proof.
  assert ( p, prod_decode_snd (prod_encode_snd p) = Some p).
  { intros p'. by induction p'; simplify_option_equality. }
  assert ( p, prod_decode_snd (prod_encode_fst p) = None).
  { intros p'. by induction p'; simplify_option_equality. }
  revert q. by induction p; intros [?|?|]; simplify_option_equality.
Qed.
Program Instance prod_countable `{Countable A} `{Countable B} :
  Countable (A * B)%type := {|
Robbert Krebbers's avatar
Robbert Krebbers committed
167
    encode xy := prod_encode (encode (xy.1)) (encode (xy.2));
168 169 170 171 172 173
    decode p :=
     x  prod_decode_fst p = decode;
     y  prod_decode_snd p = decode; Some (x, y)
  |}.
Next Obligation.
  intros ?????? [x y]; simpl.
Robbert Krebbers's avatar
Robbert Krebbers committed
174 175
  rewrite prod_decode_encode_fst, prod_decode_encode_snd; simpl.
  by rewrite !decode_encode.
176 177
Qed.

Robbert Krebbers's avatar
Robbert Krebbers committed
178 179 180 181 182 183 184
(** ** Lists *)
(* Lists are encoded as 1 separated sequences of 0s corresponding to the unary
representation of the elements. *)
Fixpoint list_encode `{Countable A} (acc : positive) (l : list A) : positive :=
  match l with
  | [] => acc
  | x :: l => list_encode (Nat.iter (encode_nat x) (~0) (acc~1)) l
185
  end.
Robbert Krebbers's avatar
Robbert Krebbers committed
186 187 188 189 190 191 192 193 194 195 196
Fixpoint list_decode `{Countable A} (acc : list A)
    (n : nat) (p : positive) : option (list A) :=
  match p with
  | 1 => Some acc
  | p~0 => list_decode acc (S n) p
  | p~1 => x  decode_nat n; list_decode (x :: acc) O p
  end.
Lemma x0_iter_x1 n acc : Nat.iter n (~0) acc~1 = acc ++ Nat.iter n (~0) 3.
Proof. by induction n; f_equal'. Qed.
Lemma list_encode_app' `{Countable A} (l1 l2 : list A) acc :
  list_encode acc (l1 ++ l2) = list_encode acc l1 ++ list_encode 1 l2.
197
Proof.
Robbert Krebbers's avatar
Robbert Krebbers committed
198 199 200
  revert acc; induction l1; simpl; auto.
  induction l2 as [|x l IH]; intros acc; simpl; [by rewrite ?(left_id_L _ _)|].
  by rewrite !(IH (Nat.iter _ _ _)), (associative_L _), x0_iter_x1.
201
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
202 203
Program Instance list_countable `{Countable A} : Countable (list A) :=
  {| encode := list_encode 1; decode := list_decode [] 0 |}.
204
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
205 206 207 208 209 210 211 212 213 214
  intros A ??; simpl.
  assert ( m acc n p, list_decode acc n (Nat.iter m (~0) p)
    = list_decode acc (n + m) p) as decode_iter.
  { induction m as [|m IH]; intros acc n p; simpl; [by rewrite Nat.add_0_r|].
    by rewrite IH, Nat.add_succ_r. }
  cut ( l acc, list_decode acc 0 (list_encode 1 l) = Some (l ++ acc))%list.
  { by intros help l; rewrite help, (right_id_L _ _). }
  induction l as [|x l IH] using @rev_ind; intros acc; [done|].
  rewrite list_encode_app'; simpl; rewrite <-x0_iter_x1, decode_iter; simpl.
  by rewrite decode_encode_nat; simpl; rewrite IH, <-(associative_L _).
215
Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
216 217 218 219 220 221 222 223 224 225 226 227 228
Lemma list_encode_app `{Countable A} (l1 l2 : list A) :
  encode (l1 ++ l2)%list = encode l1 ++ encode l2.
Proof. apply list_encode_app'. Qed.
Lemma list_encode_cons `{Countable A} x (l : list A) :
  encode (x :: l) = Nat.iter (encode_nat x) (~0) 3 ++ encode l.
Proof. apply (list_encode_app' [_]). Qed.
Lemma list_encode_suffix `{Countable A} (l k : list A) :
  l `suffix_of` k   q, encode k = q ++ encode l.
Proof. intros [l' ->]; exists (encode l'); apply list_encode_app. Qed.

(** ** Numbers *)
Instance pos_countable : Countable positive :=
  {| encode := id; decode := Some; decode_encode x := eq_refl |}.
229 230 231 232 233
Program Instance N_countable : Countable N := {|
  encode x := match x with N0 => 1 | Npos p => Pos.succ p end;
  decode p := if decide (p = 1) then Some 0%N else Some (Npos (Pos.pred p))
|}.
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
234
  by intros [|p];simpl;[|rewrite decide_False,Pos.pred_succ by (by destruct p)].
235 236
Qed.
Program Instance Z_countable : Countable Z := {|
Robbert Krebbers's avatar
Robbert Krebbers committed
237 238
  encode x := match x with Z0 => 1 | Zpos p => p~0 | Zneg p => p~1 end;
  decode p := Some match p with 1 => Z0 | p~0 => Zpos p | p~1 => Zneg p end
239 240
|}.
Next Obligation. by intros [|p|p]. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
241 242
Program Instance nat_countable : Countable nat :=
  {| encode x := encode (N.of_nat x); decode p := N.to_nat <$> decode p |}.
243
Next Obligation.
Robbert Krebbers's avatar
Robbert Krebbers committed
244
  by intros x; lazy beta; rewrite decode_encode; csimpl; rewrite Nat2N.id.
245
Qed.