cofe_solver.v 9.94 KB
Newer Older
1
Require Export iris.cofe.
Robbert Krebbers's avatar
Robbert Krebbers committed
2 3 4

Section solver.
Context (F : cofeT  cofeT  cofeT).
5
Context `{Finhab : Inhabited (F (CofeT unit) (CofeT unit))}.
Robbert Krebbers's avatar
Robbert Krebbers committed
6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Context (map :  {A1 A2 B1 B2 : cofeT},
  ((A2 -n> A1) * (B1 -n> B2))  (F A1 B1 -n> F A2 B2)).
Arguments map {_ _ _ _} _.
Instance: Params (@map) 4.
Context (map_id :  {A B : cofeT} (x : F A B), map (cid, cid) x  x).
Context (map_comp :  {A1 A2 A3 B1 B2 B3 : cofeT}
    (f : A2 -n> A1) (g : A3 -n> A2) (f' : B1 -n> B2) (g' : B2 -n> B3) x,
  map (f  g, g'  f') x  map (g,g') (map (f,f') x)).
Context (map_contractive :  {A1 A2 B1 B2}, Contractive (@map A1 A2 B1 B2)).

Lemma map_ext {A1 A2 B1 B2 : cofeT}
  (f : A2 -n> A1) (f' : A2 -n> A1) (g : B1 -n> B2) (g' : B1 -n> B2) x x' :
  ( x, f x  f' x)  ( y, g y  g' y)  x  x' 
  map (f,g) x  map (f',g') x'.
Proof. by rewrite <-!cofe_mor_ext; intros Hf Hg Hx; rewrite Hf, Hg, Hx. Qed.

Fixpoint A (k : nat) : cofeT :=
  match k with 0 => CofeT unit | S k => F (A k) (A k) end.
Fixpoint f {k} : A k -n> A (S k) :=
25
  match k with 0 => CofeMor (λ _, inhabitant) | S k => map (g,f) end
Robbert Krebbers's avatar
Robbert Krebbers committed
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
with g {k} : A (S k) -n> A k :=
  match k with 0 => CofeMor (λ _, () : CofeT ()) | S k => map (f,g) end.
Definition f_S k (x : A (S k)) : f x = map (g,f) x := eq_refl.
Definition g_S k (x : A (S (S k))) : g x = map (f,g) x := eq_refl.
Lemma gf {k} (x : A k) : g (f x)  x.
Proof.
  induction k as [|k IH]; simpl in *; [by destruct x|].
  rewrite <-map_comp; rewrite <-(map_id _ _ x) at 2; by apply map_ext.
Qed.
Lemma fg {n k} (x : A (S k)) : n  k  f (g x) ={n}= x.
Proof.
  intros Hnk; apply dist_le with k; auto; clear Hnk.
  induction k as [|k IH]; simpl; [apply dist_0|].
  rewrite <-(map_id _ _ x) at 2; rewrite <-map_comp; by apply map_contractive.
Qed.
Arguments A _ : simpl never.
Arguments f {_} : simpl never.
Arguments g {_} : simpl never.

Record tower := {
  tower_car k :> A k;
  g_tower k : g (tower_car (S k))  tower_car k
}.
Instance tower_equiv : Equiv tower := λ X Y,  k, X k  Y k.
Instance tower_dist : Dist tower := λ n X Y,  k, X k ={n}= Y k.
Program Definition tower_chain (c : chain tower) (k : nat) : chain (A k) :=
  {| chain_car i := c i k |}.
Next Obligation. intros c k n i ?; apply (chain_cauchy c n); lia. Qed.
Program Instance tower_compl : Compl tower := λ c,
  {| tower_car n := compl (tower_chain c n) |}.
Next Obligation.
  intros c k; apply equiv_dist; intros n.
  rewrite (conv_compl (tower_chain c k) n).
  rewrite (conv_compl (tower_chain c (S k)) n); simpl.
  by rewrite (g_tower (c n) k).
Qed.
Instance tower_cofe : Cofe tower.
Proof.
  split.
  * intros X Y; split; [by intros HXY n k; apply equiv_dist|].
    intros HXY k; apply equiv_dist; intros n; apply HXY.
  * intros k; split.
    + by intros X n.
    + by intros X Y ? n.
    + by intros X Y Z ?? n; transitivity (Y n).
  * intros k X Y HXY n; apply dist_S.
    by rewrite <-(g_tower X), (HXY (S n)), g_tower.
  * intros X Y k; apply dist_0.
  * intros c n k; simpl; rewrite (conv_compl (tower_chain c k) n).
    apply (chain_cauchy c); lia.
Qed.
Definition T : cofeT := CofeT tower.

Fixpoint ff {k} (i : nat) : A k -n> A (i + k) :=
  match i with 0 => cid | S i => f  ff i end.
Fixpoint gg {k} (i : nat) : A (i + k) -n> A k :=
  match i with 0 => cid | S i => gg i  g end.
Lemma ggff {k i} (x : A k) : gg i (ff i x)  x.
Proof. induction i as [|i IH]; simpl; [done|by rewrite (gf (ff i x)),IH]. Qed.
Lemma f_tower {n k} (X : tower) : n  k  f (X k) ={n}= X (S k).
Proof. intros. by rewrite <-(fg (X (S k))), <-(g_tower X). Qed.
Lemma ff_tower {n} k i (X : tower) : n  k  ff i (X k) ={n}= X (i + k).
Proof.
  intros; induction i as [|i IH]; simpl; [done|].
  by rewrite IH, (f_tower X) by lia.
Qed.
Lemma gg_tower k i (X : tower) : gg i (X (i + k))  X k.
Proof. by induction i as [|i IH]; simpl; [done|rewrite g_tower, IH]. Qed.

Instance tower_car_ne n k : Proper (dist n ==> dist n) (λ X, tower_car X k).
Proof. by intros X Y HX. Qed.
Definition project (k : nat) : T -n> A k := CofeMor (λ X : T, tower_car X k).

Definition coerce {i j} (H : i = j) : A i -n> A j :=
  eq_rect _ (λ i', A i -n> A i') cid _ H.
Lemma coerce_id {i} (H : i = i) (x : A i) : coerce H x = x.
Proof. unfold coerce. by rewrite (proof_irrel H (eq_refl i)). Qed.
Lemma coerce_proper {i j} (x y : A i) (H1 H2 : i = j) :
  x = y  coerce H1 x = coerce H2 y.
Proof. by destruct H1; rewrite !coerce_id. Qed.
Lemma g_coerce {k j} (H : S k = S j) (x : A (S k)) :
  g (coerce H x) = coerce (Nat.succ_inj _ _ H) (g x).
Proof. by assert (k = j) by lia; subst; rewrite !coerce_id. Qed.
Lemma coerce_f {k j} (H : S k = S j) (x : A k) :
  coerce H (f x) = f (coerce (Nat.succ_inj _ _ H) x).
Proof. by assert (k = j) by lia; subst; rewrite !coerce_id. Qed.
Lemma gg_gg {k i i1 i2 j} (H1 : k = i + j) (H2 : k = i2 + (i1 + j)) (x : A k) :
  gg i (coerce H1 x) = gg i1 (gg i2 (coerce H2 x)).
Proof.
  assert (i = i2 + i1) by lia; simplify_equality'. revert j x H1.
  induction i2 as [|i2 IH]; intros j X H1; simplify_equality';
    [by rewrite coerce_id|by rewrite g_coerce, IH].
Qed.
Lemma ff_ff {k i i1 i2 j} (H1 : i + k = j) (H2 : i1 + (i2 + k) = j) (x : A k) :
  coerce H1 (ff i x) = coerce H2 (ff i1 (ff i2 x)).
Proof.
  assert (i = i1 + i2) by lia; simplify_equality'.
  induction i1 as [|i1 IH]; simplify_equality';
    [by rewrite coerce_id|by rewrite coerce_f, IH].
Qed.

Definition embed' {k} (i : nat) : A k -n> A i :=
  match le_lt_dec i k with
  | left H => gg (k-i)  coerce (eq_sym (Nat.sub_add _ _ H))
  | right H => coerce (Nat.sub_add k i (Nat.lt_le_incl _ _ H))  ff (i-k)
  end.
Lemma g_embed' {k i} (x : A k) : g (embed' (S i) x)  embed' i x.
Proof.
  unfold embed'; destruct (le_lt_dec (S i) k), (le_lt_dec i k); simpl.
  * symmetry; by erewrite (@gg_gg _ _ 1 (k - S i)); simpl.
  * exfalso; lia.
  * assert (i = k) by lia; subst.
    rewrite (ff_ff _ (eq_refl (1 + (0 + k)))); simpl; rewrite gf.
    by rewrite (gg_gg _ (eq_refl (0 + (0 + k)))).
  * assert (H : 1 + ((i - k) + k) = S i) by lia; rewrite (ff_ff _ H); simpl.
    rewrite <-(gf (ff (i - k) x)) at 2; rewrite g_coerce.
    by erewrite coerce_proper by done.
Qed.
Program Definition embed_inf (k : nat) (x : A k) : T :=
  {| tower_car n := embed' n x |}.
Next Obligation. intros k x i. apply g_embed'. Qed.
Instance embed_inf_ne k n : Proper (dist n ==> dist n) (embed_inf k).
Proof. by intros x y Hxy i; simpl; rewrite Hxy. Qed.
Definition embed (k : nat) : A k -n> T := CofeMor (embed_inf k).
Lemma embed_f k (x : A k) : embed (S k) (f x)  embed k x.
Proof.
  rewrite equiv_dist; intros n i.
  unfold embed_inf, embed; simpl; unfold embed'.
  destruct (le_lt_dec i (S k)), (le_lt_dec i k); simpl.
  * assert (H : S k = S (k - i) + (0 + i)) by lia; rewrite (gg_gg _ H); simpl.
    by erewrite g_coerce, gf, coerce_proper by done.
  * assert (S k = 0 + (0 + i)) as H by lia.
    rewrite (gg_gg _ H); simplify_equality'.
    by rewrite (ff_ff _ (eq_refl (1 + (0 + k)))).
  * exfalso; lia.
  * assert (H : (i - S k) + (1 + k) = i) by lia; rewrite (ff_ff _ H); simpl.
    by erewrite coerce_proper by done.
Qed.
Lemma embed_tower j n (X : T) : n  j  embed j (X j) ={n}= X.
Proof.
  intros Hn i; simpl; unfold embed'; destruct (le_lt_dec i j) as [H|H]; simpl.
  * rewrite <-(gg_tower i (j - i) X).
    apply (_ : Proper (_ ==> _) (gg _)); by destruct (eq_sym _).
  * rewrite (ff_tower j (i-j) X) by lia; by destruct (Nat.sub_add _ _ _).
Qed.

Program Definition unfold_chain (X : T) : chain (F T T) :=
  {| chain_car n := map (project n,embed n) (f (X n)) |}.
Next Obligation.
  intros X n i Hi.
  assert ( k, i = k + n) as [k ?] by (exists (i - n); lia); subst; clear Hi.
  induction k as [|k Hk]; simpl; [done|].
  rewrite Hk, (f_tower X), f_S, <-map_comp by lia.
  apply dist_S, map_contractive.
  split; intros Y; symmetry; apply equiv_dist; [apply g_tower|apply embed_f].
Qed.
Definition unfold' (X : T) : F T T := compl (unfold_chain X).
Instance unfold_ne : Proper (dist n ==> dist n) unfold'.
Proof.
  by intros n X Y HXY; unfold unfold'; apply compl_ne; simpl; rewrite (HXY n).
Qed.
Definition unfold : T -n> F T T := CofeMor unfold'.

Program Definition fold' (X : F T T) : T :=
  {| tower_car n := g (map (embed n,project n) X) |}.
Next Obligation.
  intros X k; apply (_ : Proper (() ==> ()) g).
  rewrite <-(map_comp _ _ _ _ _ _ (embed (S k)) f (project (S k)) g).
  apply map_ext; [apply embed_f|intros Y; apply g_tower|done].
Qed.
Instance fold_ne : Proper (dist n ==> dist n) fold'.
Proof. by intros n X Y HXY k; simpl; unfold fold'; simpl; rewrite HXY. Qed.
Definition fold : F T T -n> T := CofeMor fold'.

Definition fold_unfold X : fold (unfold X)  X.
Proof.
  assert (map_ff_gg :  i k (x : A (S i + k)) (H : S i + k = i + S k),
    map (ff i, gg i) x  gg i (coerce H x)).
  { intros i; induction i as [|i IH]; intros k x H; simpl.
    { by rewrite coerce_id, map_id. }
    rewrite map_comp, g_coerce; apply IH. }
  rewrite equiv_dist; intros n k; unfold unfold, fold; simpl.
  rewrite <-g_tower, <-(gg_tower _ n); apply (_ : Proper (_ ==> _) g).
  transitivity (map (ff n, gg n) (X (S (n + k)))).
  { unfold unfold'; rewrite (conv_compl (unfold_chain X) n).
    rewrite (chain_cauchy (unfold_chain X) n (n + k)) by lia; simpl.
    rewrite (f_tower X), <-map_comp by lia.
    apply dist_S. apply map_contractive; split; intros x; simpl; unfold embed'.
    * destruct (le_lt_dec _ _); simpl.
      { assert (n = 0) by lia; subst. apply dist_0. }
      by rewrite (ff_ff _ (eq_refl (n + (0 + k)))).
    * destruct (le_lt_dec _ _); [|exfalso; lia]; simpl.
      by rewrite (gg_gg _ (eq_refl (0 + (n + k)))). }
  assert (H: S n + k = n + S k) by lia; rewrite (map_ff_gg _ _ _ H).
  apply (_ : Proper (_ ==> _) (gg _)); by destruct H.
Qed.
Definition unfold_fold X : unfold (fold X)  X.
Proof.
  rewrite equiv_dist; intros n; unfold unfold; simpl.
  unfold unfold'; rewrite (conv_compl (unfold_chain (fold X)) n); simpl.
  rewrite (fg (map (embed _,project n) X)),
    <-map_comp by lia; rewrite <-(map_id _ _ X) at 2.
  apply dist_S, map_contractive; split; intros Y i; apply embed_tower; lia.
Qed.
End solver.