Stacks annotated with types, variables no longer annotated.

theories/assoc.v

@@ -13,17 +13,15 @@ Require Export fin_maps.
 [lexico], it automatically guarantees that no duplicates exist. *)
 Definition assoc (K : Type) `{Lexico K, !TrichotomyT lexico,
     !StrictOrder lexico} (A : Type) : Type :=
-  dsig (λ l : list (K * A), StronglySorted lexico (fst <$> l)).
+  dsig (λ l : list (K * A), StronglySorted lexico (l.*1)).
 
 Section assoc.
 Context `{Lexico K, !StrictOrder lexico,
   ∀ x y : K, Decision (x = y), !TrichotomyT lexico}.
 
 Infix "⊂" := lexico.
-Notation assoc_before j l :=
-  (Forall (lexico j) (fst <$> l)) (only parsing).
-Notation assoc_wf l :=
-  (StronglySorted (lexico) (fst <$> l)) (only parsing).
+Notation assoc_before j l := (Forall (lexico j) (l.*1)) (only parsing).
+Notation assoc_wf l := (StronglySorted (lexico) (l.*1)) (only parsing).
 
 Lemma assoc_before_transitive {A} (l : list (K * A)) i j :
   i ⊂ j → assoc_before j l → assoc_before i l.

theories/base.v

@@ -431,6 +431,11 @@ Notation "' ( x1 , x2 , x3  , x4 , x5 , x6 ) ← y ; z" :=
   (y ≫= (λ x : _, let ' (x1,x2,x3,x4,x5,x6) := x in z))
   (at level 65, next at level 35, only parsing, right associativity) : C_scope.
 
+Notation "ps .*1" := (fmap (M:=list) fst ps)
+  (at level 10, format "ps .*1").
+Notation "ps .*2" := (fmap (M:=list) snd ps)
+  (at level 10, format "ps .*2").
+
 Class MGuard (M : Type → Type) :=
   mguard: ∀ P {dec : Decision P} {A}, (P → M A) → M A.
 Arguments mguard _ _ _ !_ _ _ /.

theories/fin_maps.v

@@ -455,7 +455,7 @@ Proof. apply map_empty; intros i. by rewrite lookup_omap, lookup_empty. Qed.
 Lemma map_to_list_unique {A} (m : M A) i x y :
   (i,x) ∈ map_to_list m → (i,y) ∈ map_to_list m → x = y.
 Proof. rewrite !elem_of_map_to_list. congruence. Qed.
-Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup (fst <$> map_to_list m).
+Lemma NoDup_fst_map_to_list {A} (m : M A) : NoDup ((map_to_list m).*1).
 Proof. eauto using NoDup_fmap_fst, map_to_list_unique, NoDup_map_to_list. Qed.
 Lemma elem_of_map_of_list_1_help {A} (l : list (K * A)) i x :
   (i,x) ∈ l → (∀ y, (i,y) ∈ l → y = x) → map_of_list l !! i = Some x.
@@ -468,11 +468,11 @@ Proof.
   * rewrite lookup_insert_ne by done; eauto.
 Qed.
 Lemma elem_of_map_of_list_1 {A} (l : list (K * A)) i x :
-  NoDup (fst <$> l) → (i,x) ∈ l → map_of_list l !! i = Some x.
+  NoDup (l.*1) → (i,x) ∈ l → map_of_list l !! i = Some x.
 Proof.
   intros ? Hx; apply elem_of_map_of_list_1_help; eauto using NoDup_fmap_fst.
   intros y; revert Hx. rewrite !elem_of_list_lookup; intros [i' Hi'] [j' Hj'].
-  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (fst <$> l) i;
+  cut (i' = j'); [naive_solver|]. apply NoDup_lookup with (l.*1) i;
     by rewrite ?list_lookup_fmap, ?Hi', ?Hj'.
 Qed.
 Lemma elem_of_map_of_list_2 {A} (l : list (K * A)) i x :
@@ -483,16 +483,16 @@ Proof.
     rewrite ?lookup_insert, ?lookup_insert_ne; intuition congruence.
 Qed.
 Lemma elem_of_map_of_list {A} (l : list (K * A)) i x :
-  NoDup (fst <$> l) → (i,x) ∈ l ↔ map_of_list l !! i = Some x.
+  NoDup (l.*1) → (i,x) ∈ l ↔ map_of_list l !! i = Some x.
 Proof. split; auto using elem_of_map_of_list_1, elem_of_map_of_list_2. Qed.
 Lemma not_elem_of_map_of_list_1 {A} (l : list (K * A)) i :
-  i ∉ fst <$> l → map_of_list l !! i = None.
+  i ∉ l.*1 → map_of_list l !! i = None.
 Proof.
   rewrite elem_of_list_fmap, eq_None_not_Some. intros Hi [x ?]; destruct Hi.
   exists (i,x); simpl; auto using elem_of_map_of_list_2.
 Qed.
 Lemma not_elem_of_map_of_list_2 {A} (l : list (K * A)) i :
-  map_of_list l !! i = None → i ∉ fst <$> l.
+  map_of_list l !! i = None → i ∉ l.*1.
 Proof.
   induction l as [|[j y] l IH]; csimpl; [rewrite elem_of_nil; tauto|].
   rewrite elem_of_cons. destruct (decide (i = j)); simplify_equality.
@@ -500,17 +500,16 @@ Proof.
   * by rewrite lookup_insert_ne; intuition.
 Qed.
 Lemma not_elem_of_map_of_list {A} (l : list (K * A)) i :
-  i ∉ fst <$> l ↔ map_of_list l !! i = None.
+  i ∉ l.*1 ↔ map_of_list l !! i = None.
 Proof. red; auto using not_elem_of_map_of_list_1,not_elem_of_map_of_list_2. Qed.
 Lemma map_of_list_proper {A} (l1 l2 : list (K * A)) :
-  NoDup (fst <$> l1) → l1 ≡ₚ l2 → map_of_list l1 = map_of_list l2.
+  NoDup (l1.*1) → l1 ≡ₚ l2 → map_of_list l1 = map_of_list l2.
 Proof.
   intros ? Hperm. apply map_eq. intros i. apply option_eq. intros x.
   by rewrite <-!elem_of_map_of_list; rewrite <-?Hperm.
 Qed.
 Lemma map_of_list_inj {A} (l1 l2 : list (K * A)) :
-  NoDup (fst <$> l1) → NoDup (fst <$> l2) →
-  map_of_list l1 = map_of_list l2 → l1 ≡ₚ l2.
+  NoDup (l1.*1) → NoDup (l2.*1) → map_of_list l1 = map_of_list l2 → l1 ≡ₚ l2.
 Proof.
   intros ?? Hl1l2. apply NoDup_Permutation; auto using (NoDup_fmap_1 fst).
   intros [i x]. by rewrite !elem_of_map_of_list, Hl1l2.
@@ -522,7 +521,7 @@ Proof.
     by auto using NoDup_fst_map_to_list.
 Qed.
 Lemma map_to_of_list {A} (l : list (K * A)) :
-  NoDup (fst <$> l) → map_to_list (map_of_list l) ≡ₚ l.
+  NoDup (l.*1) → map_to_list (map_of_list l) ≡ₚ l.
 Proof. auto using map_of_list_inj, NoDup_fst_map_to_list, map_of_to_list. Qed.
 Lemma map_to_list_inj {A} (m1 m2 : M A) :
   map_to_list m1 ≡ₚ map_to_list m2 → m1 = m2.
@@ -564,9 +563,9 @@ Lemma map_to_list_insert_inv {A} (m : M A) l i x :
   map_to_list m ≡ₚ (i,x) :: l → m = <[i:=x]>(map_of_list l).
 Proof.
   intros Hperm. apply map_to_list_inj.
-  assert (NoDup (fst <$> (i, x) :: l)) as Hnodup.
-  { rewrite <-Hperm. auto using NoDup_fst_map_to_list. }
-  csimpl in *. rewrite NoDup_cons in Hnodup. destruct Hnodup.
+  assert (i ∉ l.*1 ∧ NoDup (l.*1)) as [].
+  { rewrite <-NoDup_cons. change (NoDup (((i,x)::l).*1)). rewrite <-Hperm.
+    auto using NoDup_fst_map_to_list. }
   rewrite Hperm, map_to_list_insert, map_to_of_list;
     auto using not_elem_of_map_of_list_1.
 Qed.
@@ -597,7 +596,7 @@ Qed.
 Lemma map_ind {A} (P : M A → Prop) :
   P ∅ → (∀ i x m, m !! i = None → P m → P (<[i:=x]>m)) → ∀ m, P m.
 Proof.
-  intros ? Hins. cut (∀ l, NoDup (fst <$> l) → ∀ m, map_to_list m ≡ₚ l → P m).
+  intros ? Hins. cut (∀ l, NoDup (l.*1) → ∀ m, map_to_list m ≡ₚ l → P m).
   { intros help m.
     apply (help (map_to_list m)); auto using NoDup_fst_map_to_list. }
   induction l as [|[i x] l IH]; intros Hnodup m Hml.

theories/list.v

@@ -2053,6 +2053,8 @@ Section Forall_Exists.
   Proof. rewrite Forall_lookup. eauto. Qed.
   Lemma Forall_lookup_2 l : (∀ i x, l !! i = Some x → P x) → Forall P l.
   Proof. by rewrite Forall_lookup. Qed.
+  Lemma Forall_tail l : Forall P l → Forall P (tail l).
+  Proof. destruct 1; simpl; auto. Qed.
   Lemma Forall_alter f l i :
     Forall P l → (∀ x, l!!i = Some x → P x → P (f x)) → Forall P (alter f i l).
   Proof.
@@ -2575,6 +2577,8 @@ Section fmap.
   Proof.
     induction l as [|?? IH]; csimpl; by rewrite ?reverse_cons, ?fmap_app, ?IH.
   Qed.
+  Lemma fmap_tail l : f <$> tail l = tail (f <$> l).
+  Proof. by destruct l. Qed.
   Lemma fmap_last l : last (f <$> l) = f <$> last l.
   Proof. induction l as [|? []]; simpl; auto. Qed.
   Lemma fmap_replicate n x :  f <$> replicate n x = replicate n (f x).
@@ -2673,7 +2677,7 @@ Lemma list_alter_fmap_mono {A} (f : A → A) (g : A → A) l i :
   Forall (λ x, f (g x) = g (f x)) l → f <$> alter g i l = alter g i (f <$> l).
 Proof. auto using list_alter_fmap. Qed.
 Lemma NoDup_fmap_fst {A B} (l : list (A * B)) :
-  (∀ x y1 y2, (x,y1) ∈ l → (x,y2) ∈ l → y1 = y2) → NoDup l → NoDup (fst <$> l).
+  (∀ x y1 y2, (x,y1) ∈ l → (x,y2) ∈ l → y1 = y2) → NoDup l → NoDup (l.*1).
 Proof.
   intros Hunique. induction 1 as [|[x1 y1] l Hin Hnodup IH]; csimpl; constructor.
   * rewrite elem_of_list_fmap.
@@ -3040,8 +3044,7 @@ Section zip_with.
   Proof. revert l. induction k; intros [|??] ??; f_equal'; auto with lia. Qed.
   Lemma zip_with_zip l k : zip_with f l k = curry f <$> zip l k.
   Proof. revert k. by induction l; intros [|??]; f_equal'. Qed.
-  Lemma zip_with_fst_snd lk :
-    zip_with f (fst <$> lk) (snd <$> lk) = curry f <$> lk.
+  Lemma zip_with_fst_snd lk : zip_with f (lk.*1) (lk.*2) = curry f <$> lk.
   Proof. by induction lk as [|[]]; f_equal'. Qed.
   Lemma zip_with_replicate n x y :
     zip_with f (replicate n x) (replicate n y) = replicate n (f x y).
@@ -3095,11 +3098,11 @@ Section zip.
   Context {A B : Type}.
   Implicit Types l : list A.
   Implicit Types k : list B.
-  Lemma fst_zip l k : length l ≤ length k → fst <$> zip l k = l.
+  Lemma fst_zip l k : length l ≤ length k → (zip l k).*1 = l.
   Proof. by apply fmap_zip_with_l. Qed.
-  Lemma snd_zip l k : length k ≤ length l → snd <$> zip l k = k.
+  Lemma snd_zip l k : length k ≤ length l → (zip l k).*2 = k.
   Proof. by apply fmap_zip_with_r. Qed.
-  Lemma zip_fst_snd (lk : list (A * B)) : zip (fst <$> lk) (snd <$> lk) = lk.
+  Lemma zip_fst_snd (lk : list (A * B)) : zip (lk.*1) (lk.*2) = lk.
   Proof. by induction lk as [|[]]; f_equal'. Qed.
   Lemma Forall2_fst P l1 l2 k1 k2 :
     length l2 = length k2 → Forall2 P l1 k1 →
@@ -3292,6 +3295,8 @@ Ltac simplify_list_equality ::= repeat
     rewrite take_app_alt in H by solve_length
   | H : context [drop _ (_ ++ _)] |- _ =>
     rewrite drop_app_alt in H by solve_length
+  | H : ?l !! ?i = _, H2 : context [(_ <$> ?l) !! ?i] |- _ =>
+     rewrite list_lookup_fmap, H in H2
   end.
 Ltac decompose_Forall_hyps :=
   repeat match goal with

theories/mapset.v

@@ -24,7 +24,7 @@ Instance mapset_intersection: Intersection (mapset (M unit)) := λ X1 X2,
 Instance mapset_difference: Difference (mapset (M unit)) := λ X1 X2,
   let (m1) := X1 in let (m2) := X2 in Mapset (m1 ∖ m2).
 Instance mapset_elems: Elements K (mapset (M unit)) := λ X,
-  let (m) := X in fst <$> map_to_list m.
+  let (m) := X in (map_to_list m).*1.
 
 Lemma mapset_eq (X1 X2 : mapset (M unit)) : X1 = X2 ↔ ∀ x, x ∈ X1 ↔ x ∈ X2.
 Proof.