countable.v 8.4 KB
 Robbert Krebbers committed Feb 08, 2015 1 ``````(* Copyright (c) 2012-2015, Robbert Krebbers. *) `````` Robbert Krebbers committed May 02, 2014 2 ``````(* This file is distributed under the terms of the BSD license. *) `````` Robbert Krebbers committed Nov 16, 2015 3 ``````Require Export prelude.list. `````` Robbert Krebbers committed Jun 17, 2013 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 ``````Local Obligation Tactic := idtac. 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 }. 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)). 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. 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 (x↾Hx) => match decide (P x) with | left _ => x | right H => choose_go (Acc_inv acc (choose_step_Some Hx H)) 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 committed Feb 01, 2017 71 `````` intros y. by rewrite (choose_correct (λ x, f x = y) (surjective f y)). `````` Robbert Krebbers committed Jun 17, 2013 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 ``````Qed. (** ** Instances *) Program Instance option_countable `{Countable A} : Countable (option A) := {| 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) |}. Next Obligation. intros ??? [x|]; simpl; repeat case_decide; auto with lia. by rewrite Pos.pred_succ, decode_encode. Qed. 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. 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 := {| encode xy := let (x,y) := xy in prod_encode (encode x) (encode y); decode p := x ← prod_decode_fst p ≫= decode; y ← prod_decode_snd p ≫= decode; Some (x, y) |}. Next Obligation. intros ?????? [x y]; simpl. rewrite prod_decode_encode_fst, prod_decode_encode_snd. `````` Robbert Krebbers committed Jun 16, 2014 168 `````` csimpl. by rewrite !decode_encode. `````` Robbert Krebbers committed Jun 17, 2013 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 ``````Qed. Fixpoint list_encode_ (l : list positive) : positive := match l with [] => 1 | x :: l => prod_encode x (list_encode_ l) end. Definition list_encode (l : list positive) : positive := prod_encode (Pos.of_nat (S (length l))) (list_encode_ l). Fixpoint list_decode_ (n : nat) (p : positive) : option (list positive) := match n with | O => guard (p = 1); Some [] | S n => x ← prod_decode_fst p; pl ← prod_decode_snd p; l ← list_decode_ n pl; Some (x :: l) end. Definition list_decode (p : positive) : option (list positive) := pn ← prod_decode_fst p; pl ← prod_decode_snd p; list_decode_ (pred (Pos.to_nat pn)) pl. Lemma list_decode_encode l : list_decode (list_encode l) = Some l. Proof. cut (list_decode_ (length l) (list_encode_ l) = Some l). { intros help. unfold list_decode, list_encode. `````` Robbert Krebbers committed Jun 16, 2014 191 `````` rewrite prod_decode_encode_fst, prod_decode_encode_snd; csimpl. `````` Robbert Krebbers committed Jun 17, 2013 192 193 194 195 196 197 198 199 200 201 202 `````` by rewrite Nat2Pos.id by done; simpl. } induction l; simpl; auto. by rewrite prod_decode_encode_fst, prod_decode_encode_snd; simplify_option_equality. Qed. Program Instance list_countable `{Countable A} : Countable (list A) := {| encode l := list_encode (encode <\$> l); decode p := list_decode p ≫= mapM decode |}. Next Obligation. `````` Robbert Krebbers committed Feb 01, 2017 203 `````` intros ??? l; simpl; rewrite list_decode_encode; simpl. `````` Robbert Krebbers committed Jun 17, 2013 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 `````` apply mapM_fmap_Some; auto using decode_encode. Qed. Program Instance pos_countable : Countable positive := {| encode := id; decode := Some; decode_encode x := eq_refl |}. 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. intros [|p]; simpl; repeat case_decide; auto with lia. by rewrite Pos.pred_succ. Qed. Program Instance Z_countable : Countable Z := {| 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 |}. Next Obligation. by intros [|p|p]. Qed. Program Instance nat_countable : Countable nat := {| encode x := encode (N.of_nat x); decode p := N.to_nat <\$> decode p |}. Next Obligation. `````` Robbert Krebbers committed Feb 01, 2017 230 `````` intros x; lazy beta; rewrite decode_encode; csimpl. by rewrite Nat2N.id. `````` Robbert Krebbers committed Jun 17, 2013 231 ``Qed.``