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Commit c8613be0 authored by Adam Chlipala's avatar Adam Chlipala
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Switch DepList to inductive, not recursive, types

parent b87eea7b
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src/DataStruct.v
View file @ c8613be0
......@@ -841,7 +841,7 @@ Qed.
(** remove printing * *)
(** Some of the type family definitions from this chapter are duplicated in the [DepList] module of the book source. Only the recursive versions of length-indexed and heterogeneous lists are included, and they are renamed without the [f] prefixes, e.g., [ilist] in place of [filist].
(** Some of the type family definitions and associated functions from this chapter are duplicated in the [DepList] module of the book source. Some of their names have been changed to be more sensible in a general context.
%\begin{enumerate}%#<ol>#
......
src/DepList.v
View file @ c8613be0
(* Copyright (c) 2008, Adam Chlipala
(* Copyright (c) 2008-2009, Adam Chlipala
*
* This work is licensed under a
* Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
......@@ -17,56 +17,68 @@ Set Implicit Arguments.
Section ilist.
Variable A : Type.
Fixpoint ilist (n : nat) : Type :=
match n with
| O => unit
| S n' => A * ilist n'
end%type.
Definition inil : ilist O := tt.
Definition icons n x (ls : ilist n) : ilist (S n) := (x, ls).
Definition hd n (ls : ilist (S n)) : A := fst ls.
Definition tl n (ls : ilist (S n)) : ilist n := snd ls.
Inductive ilist : nat -> Type :=
| INil : ilist O
| ICons : forall n, A -> ilist n -> ilist (S n).
Implicit Arguments icons [n].
Fixpoint fin (n : nat) : Type :=
match n with
| O => Empty_set
| S n' => option (fin n')
Definition hd n (ls : ilist (S n)) :=
match ls in ilist n' return match n' with
| O => unit
| S _ => A
end with
| INil => tt
| ICons _ x _ => x
end.
Definition tl n (ls : ilist (S n)) :=
match ls in ilist n' return match n' with
| O => unit
| S n => ilist n
end with
| INil => tt
| ICons _ _ ls' => ls'
end.
Fixpoint get (n : nat) : ilist n -> fin n -> A :=
match n return ilist n -> fin n -> A with
| O => fun _ idx => match idx with end
| S n' => fun ls idx =>
match idx with
| None => fst ls
| Some idx' => get n' (snd ls) idx'
Inductive fin : nat -> Set :=
| First : forall n, fin (S n)
| Next : forall n, fin n -> fin (S n).
Fixpoint get n (ls : ilist n) : fin n -> A :=
match ls with
| INil => fun idx =>
match idx in fin n' return (match n' with
| O => A
| S _ => unit
end) with
| First _ => tt
| Next _ _ => tt
end
| ICons _ x ls' => fun idx =>
match idx in fin n' return (fin (pred n') -> A) -> A with
| First _ => fun _ => x
| Next _ idx' => fun get_ls' => get_ls' idx'
end (get ls')
end.
Section everywhere.
Variable x : A.
Fixpoint everywhere (n : nat) : ilist n :=
match n return ilist n with
| O => inil
| S n' => icons x (everywhere n')
match n with
| O => INil
| S n' => ICons x (everywhere n')
end.
End everywhere.
Section singleton.
Variables x default : A.
Fixpoint singleton (n m : nat) {struct n} : ilist n :=
match n return ilist n with
| O => inil
Fixpoint singleton (n m : nat) : ilist n :=
match n with
| O => INil
| S n' =>
match m with
| O => icons x (everywhere default n')
| S m' => icons default (singleton n' m')
| O => ICons x (everywhere default n')
| S m' => ICons default (singleton n' m')
end
end.
End singleton.
......@@ -74,10 +86,10 @@ Section ilist.
Section map2.
Variable f : A -> A -> A.
Fixpoint map2 (n : nat) : ilist n -> ilist n -> ilist n :=
match n return ilist n -> ilist n -> ilist n with
| O => fun _ _ => inil
| S n' => fun ls1 ls2 => icons (f (hd ls1) (hd ls2)) (map2 _ (tl ls1) (tl ls2))
Fixpoint map2 n (il1 : ilist n) : ilist n -> ilist n :=
match il1 in ilist n return ilist n -> ilist n with
| INil => fun _ => INil
| ICons _ x il1' => fun il2 => ICons (f x (hd il2)) (map2 il1' (tl il2))
end.
End map2.
......@@ -86,107 +98,112 @@ Section ilist.
Variable f : A -> B -> B.
Variable i : B.
Fixpoint foldr (n : nat) : ilist n -> B :=
match n return ilist n -> B with
| O => fun _ => i
| S n' => fun ils => f (hd ils) (foldr n' (tl ils))
Fixpoint foldr n (il : ilist n) : B :=
match il with
| INil => i
| ICons _ x il' => f x (foldr il')
end.
End fold.
End ilist.
Implicit Arguments inil [A].
Implicit Arguments icons [A n].
Implicit Arguments icons [A n].
Implicit Arguments get [A n].
Implicit Arguments map2 [A n].
Implicit Arguments foldr [A B n].
Implicit Arguments INil [A].
Implicit Arguments First [n].
Section imap.
Variables A B : Type.
Variable f : A -> B.
Fixpoint imap (n : nat) : ilist A n -> ilist B n :=
match n return ilist A n -> ilist B n with
| O => fun _ => inil
| S n' => fun ls => icons (f (hd ls)) (imap _ (tl ls))
Fixpoint imap n (il : ilist A n) : ilist B n :=
match il with
| INil => INil
| ICons _ x il' => ICons (f x) (imap il')
end.
End imap.
Implicit Arguments imap [A B n].
Section hlist.
Variable A : Type.
Variable B : A -> Type.
Fixpoint hlist (ls : list A) : Type :=
match ls with
Inductive hlist : list A -> Type :=
| HNil : hlist nil
| HCons : forall (x : A) (ls : list A), B x -> hlist ls -> hlist (x :: ls).
Definition hhd ls (hl : hlist ls) :=
match hl in hlist ls return match ls with
| nil => unit
| x :: ls' => B x * hlist ls'
end%type.
| x :: _ => B x
end with
| HNil => tt
| HCons _ _ v _ => v
end.
Definition hnil : hlist nil := tt.
Definition hcons (x : A) (ls : list A) (v : B x) (hls : hlist ls) : hlist (x :: ls) :=
(v, hls).
Definition htl ls (hl : hlist ls) :=
match hl in hlist ls return match ls with
| nil => unit
| _ :: ls' => hlist ls'
end with
| HNil => tt
| HCons _ _ _ hl' => hl'
end.
Variable elm : A.
Fixpoint member (ls : list A) : Type :=
match ls with
| nil => Empty_set
| x :: ls' => (x = elm) + member ls'
end%type.
Definition hfirst (x : A) (ls : list A) (pf : x = elm) : member (x :: ls) :=
inl _ pf.
Definition hnext (x : A) (ls : list A) (m : member ls) : member (x :: ls) :=
inr _ m.
Fixpoint hget (ls : list A) : hlist ls -> member ls -> B elm :=
match ls return hlist ls -> member ls -> B elm with
| nil => fun _ idx => match idx with end
| _ :: ls' => fun mls idx =>
match idx with
| inl pf => match pf with
| refl_equal => fst mls
end
| inr idx' => hget ls' (snd mls) idx'
Inductive member : list A -> Type :=
| HFirst : forall ls, member (elm :: ls)
| HNext : forall x ls, member ls -> member (x :: ls).
Fixpoint hget ls (mls : hlist ls) : member ls -> B elm :=
match mls with
| HNil => fun mem =>
match mem in member ls' return (match ls' with
| nil => B elm
| _ :: _ => unit
end) with
| HFirst _ => tt
| HNext _ _ _ => tt
end
| HCons _ _ x mls' => fun mem =>
match mem in member ls' return (match ls' with
| nil => Empty_set
| x' :: ls'' =>
B x' -> (member ls'' -> B elm) -> B elm
end) with
| HFirst _ => fun x _ => x
| HNext _ _ mem' => fun _ get_mls' => get_mls' mem'
end x (hget mls')
end.
Fixpoint happ (ls1 ls2 : list A) {struct ls1} : hlist ls1 -> hlist ls2 -> hlist (ls1 ++ ls2) :=
match ls1 return hlist ls1 -> hlist ls2 -> hlist (ls1 ++ ls2) with
| nil => fun _ hls2 => hls2
| _ :: _ => fun hls1 hls2 => (fst hls1, happ _ _ (snd hls1) hls2)
Fixpoint happ (ls1 : list A) (hl1 : hlist ls1) : forall ls2, hlist ls2 -> hlist (ls1 ++ ls2) :=
match hl1 in hlist ls1 return forall ls2, hlist ls2 -> hlist (ls1 ++ ls2) with
| HNil => fun _ hl2 => hl2
| HCons _ _ x hl1' => fun _ hl2 => HCons x (happ hl1' hl2)
end.
Variable f : forall x, B x.
Fixpoint hmake (ls : list A) : hlist ls :=
match ls return hlist ls with
| nil => hnil
| x :: ls' => hcons _ (f x) (hmake ls')
match ls with
| nil => HNil
| x :: ls' => HCons (f x) (hmake ls')
end.
Implicit Arguments hget [ls].
Theorem hget_hmake : forall ls (m : member ls),
hget (hmake ls) m = f elm.
induction ls; crush.
case a0; reflexivity.
induction ls; crush;
match goal with
| [ |- context[match ?E with HFirst _ => _ | HNext _ _ _ => _ end] ] => dep_destruct E
end; crush.
Qed.
End hlist.
Implicit Arguments hnil [A B].
Implicit Arguments hcons [A B x ls].
Implicit Arguments hget [A B elm ls].
Implicit Arguments happ [A B ls1 ls2].
Implicit Arguments HNil [A B].
Implicit Arguments HCons [A B x ls].
Implicit Arguments hmake [A B].
Implicit Arguments hfirst [A elm x ls].
Implicit Arguments hnext [A elm x ls].
Implicit Arguments HFirst [A elm ls].
Implicit Arguments HNext [A elm x ls].
Infix ":::" := hcons (right associativity, at level 60).
Infix ":::" := HCons (right associativity, at level 60).
Infix "+++" := happ (right associativity, at level 60).
Section hmap.
......@@ -195,23 +212,23 @@ Section hmap.
Variable f : forall x, B1 x -> B2 x.
Fixpoint hmap (ls : list A) : hlist B1 ls -> hlist B2 ls :=
match ls return hlist B1 ls -> hlist B2 ls with
| nil => fun _ => hnil
| _ :: _ => fun hl => f (fst hl) ::: hmap _ (snd hl)
Fixpoint hmap (ls : list A) (hl : hlist B1 ls) : hlist B2 ls :=
match hl with
| HNil => HNil
| HCons _ _ x hl' => f x ::: hmap hl'
end.
Implicit Arguments hmap [ls].
Theorem hmap_happ : forall ls2 (h2 : hlist B1 ls2) ls1 (h1 : hlist B1 ls1),
hmap h1 +++ hmap h2 = hmap (h1 +++ h2).
induction ls1; crush.
induction h1; crush.
Qed.
Theorem hget_hmap : forall elm ls (hls : hlist B1 ls) (m : member elm ls),
hget (hmap hls) m = f (hget hls m).
induction ls; crush.
case a1; crush.
induction hls; crush;
match goal with
| [ |- context[match ?E with HFirst _ => _ | HNext _ _ _ => _ end] ] => dep_destruct E
end; crush.
Qed.
End hmap.
......
src/Extensional.v
View file @ c8613be0
(* Copyright (c) 2008, Adam Chlipala
(* Copyright (c) 2008-2009, Adam Chlipala
*
* This work is licensed under a
* Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
......@@ -499,7 +499,7 @@ Module PatMatch.
Fixpoint patDenote t ts (p : pat t ts) {struct p} : typeDenote t -> option (hlist typeDenote ts) :=
match p in (pat t ts) return (typeDenote t -> option (hlist typeDenote ts)) with
| PVar _ => fun v => Some (v, tt)
| PVar _ => fun v => Some (v ::: HNil)
| PPair _ _ _ _ p1 p2 => fun v =>
match patDenote p1 (fst v), patDenote p2 (snd v) with
| Some tup1, Some tup2 => Some (happ tup1 tup2)
......@@ -525,18 +525,18 @@ Module PatMatch.
: (forall ts, member ts tss -> option (hlist typeDenote ts))
-> (forall ts, member ts tss -> hlist typeDenote ts -> typeDenote t2)
-> typeDenote t2 :=
match tss return (forall ts, member ts tss -> _)
-> (forall ts, member ts tss -> _)
match tss return (forall ts, member ts tss -> option (hlist typeDenote ts))
-> (forall ts, member ts tss -> hlist typeDenote ts -> typeDenote t2)
-> _ with
| nil => fun _ _ =>
default
| ts :: tss' => fun (envs : forall ts', member ts' (ts :: tss') -> option (hlist typeDenote ts'))
(bodies : forall ts', member ts' (ts :: tss') -> hlist typeDenote ts' -> typeDenote t2) =>
match envs _ (hfirst (refl_equal _)) with
| None => matchesDenote tss'
(fun _ mem => envs _ (hnext mem))
(fun _ mem => bodies _ (hnext mem))
| Some env => (bodies _ (hfirst (refl_equal _))) env
match envs _ HFirst with
| None => matchesDenote
(fun _ mem => envs _ (HNext mem))
(fun _ mem => bodies _ (HNext mem))
| Some env => (bodies _ HFirst) env
end
end.
End matchesDenote.
......@@ -735,23 +735,23 @@ Module PatMatch.
match p in (pat t1 ts) return ((hlist (exp var) ts -> result)
-> result -> choice_tree var t1 result) with
| PVar _ => fun succ fail =>
everywhere (fun disc => succ (disc, tt))
everywhere (fun disc => succ (disc ::: HNil))
| PPair _ _ _ _ p1 p2 => fun succ fail =>
elaboratePat _ p1
elaboratePat p1
(fun tup1 =>
elaboratePat _ p2
elaboratePat p2
(fun tup2 =>
succ (happ tup1 tup2))
fail)
(everywhere (fun _ => fail))
| PInl _ _ _ p' => fun succ fail =>
(elaboratePat _ p' succ fail,
(elaboratePat p' succ fail,
everywhere (fun _ => fail))
| PInr _ _ _ p' => fun succ fail =>
(everywhere (fun _ => fail),
elaboratePat _ p' succ fail)
elaboratePat p' succ fail)
end.
Implicit Arguments elaboratePat [var t1 ts result].
......@@ -760,8 +760,8 @@ Module PatMatch.
-> hlist (exp var) ts -> exp var t :=
match ts return ((hlist var ts -> exp var t)
-> hlist (exp var) ts -> exp var t) with
| nil => fun f _ => f tt
| _ :: _ => fun f tup => letify _ (fun tup' => x <- fst tup; f (x, tup')) (snd tup)
| nil => fun f _ => f HNil
| _ :: _ => fun f tup => letify (fun tup' => x <- hhd tup; f (x ::: tup')) (htl tup)
end.
Implicit Arguments letify [var t ts].
......@@ -809,14 +809,14 @@ Module PatMatch.
| ts :: tss' => fun (ps : forall ts', member ts' (ts :: tss') -> pat t1 ts')
(es : forall ts', member ts' (ts :: tss') -> hlist var ts' -> Elab.exp var t2) =>
merge (@mergeOpt _)
(elaboratePat (ps _ (hfirst (refl_equal _)))
(elaboratePat (ps _ HFirst)
(fun ts => Some (letify
(fun ts' => es _ (hfirst (refl_equal _)) ts')
(fun ts' => es _ HFirst ts')
ts))
None)
(elaborateMatches tss'
(fun _ mem => ps _ (hnext mem))
(fun _ mem => es _ (hnext mem)))
(elaborateMatches
(fun _ mem => ps _ (HNext mem))
(fun _ mem => es _ (HNext mem)))
end.
Implicit Arguments elaborateMatches [var t1 t2 tss].
......@@ -926,7 +926,7 @@ Module PatMatch.
result (succ : hlist (Elab.exp typeDenote) ts -> result)
(fail : result) v env,
patDenote p v = Some env
-> exists env', grab (elaboratePat typeDenote p succ fail) v = succ env'
-> exists env', grab (elaboratePat p succ fail) v = succ env'
/\ env = hmap Elab.expDenote env'.
Hint Resolve hmap_happ.
......@@ -935,7 +935,7 @@ Module PatMatch.
| [ |- context[grab (everywhere ?succ) ?v] ] =>
generalize (everywhere_correct succ (#v)%elab)
| [ H : forall result sudc fail, _ |- context[grab (elaboratePat _ _ ?S ?F) ?V] ] =>
| [ H : forall result succ fail, _ |- context[grab (elaboratePat _ ?S ?F) ?V] ] =>
generalize (H _ S F V); clear H
| [ H1 : context[match ?E with Some _ => _ | None => _ end],
H2 : forall env, ?E = Some env -> _ |- _ ] =>
......@@ -949,14 +949,14 @@ Module PatMatch.
result (succ : hlist (Elab.exp typeDenote) ts -> result)
(fail : result) v,
patDenote p v = None
-> grab (elaboratePat typeDenote p succ fail) v = fail.
-> grab (elaboratePat p succ fail) v = fail.
Hint Resolve everywhere_fail.
induction p; try solve [ crush ];
simpl; fold choice_tree; intuition; simpl in *;
repeat match goal with
| [ IH : forall result succ fail v, patDenote ?P v = _ -> _
|- context[grab (elaboratePat _ ?P ?S ?F) ?V] ] =>
|- context[grab (elaboratePat ?P ?S ?F) ?V] ] =>
generalize (IH _ S F V); clear IH; intro IH;
generalize (elaboratePat_correct P S F V); intros;
destruct (patDenote P V); try discriminate
......@@ -973,7 +973,7 @@ Module PatMatch.
(env : hlist (Elab.exp typeDenote) ts),
Elab.expDenote (letify f env)
= Elab.expDenote (f (hmap Elab.expDenote env)).
induction ts; crush.
induction ts; crush; dep_destruct env; crush.
Qed.
Theorem elaborate_correct : forall t (e : Source.exp typeDenote t),
......@@ -986,7 +986,7 @@ Module PatMatch.
| [ tss : list (list type) |- _ ] =>
induction tss; crush;
match goal with
| [ |- context[grab (elaboratePat _ ?P ?S ?F) ?V] ] =>
| [ |- context[grab (elaboratePat ?P ?S ?F) ?V] ] =>
case_eq (patDenote P V); [intros env Heq;
destruct (elaboratePat_correct P S F _ Heq); crush;
match goal with
......
src/Generic.v
View file @ c8613be0
(* Copyright (c) 2008, Adam Chlipala
(* Copyright (c) 2008-2009, Adam Chlipala
*
* This work is licensed under a
* Creative Commons Attribution-Noncommercial-No Derivative Works 3.0
......@@ -70,17 +70,17 @@ Notation "[ v , r # n ~> x ]" := ((fun v r => x) : constructorDenote _ (Con _ n)
(* begin thide *)
Definition Empty_set_den : datatypeDenote Empty_set Empty_set_dt :=
hnil.
HNil.
Definition unit_den : datatypeDenote unit unit_dt :=
[!, ! ~> tt] ::: hnil.
[!, ! ~> tt] ::: HNil.
Definition bool_den : datatypeDenote bool bool_dt :=
[!, ! ~> true] ::: [!, ! ~> false] ::: hnil.
[!, ! ~> true] ::: [!, ! ~> false] ::: HNil.
Definition nat_den : datatypeDenote nat nat_dt :=
[!, ! ~> O] ::: [!, r # 1 ~> S (hd r)] ::: hnil.
[!, ! ~> O] ::: [!, r # 1 ~> S (hd r)] ::: HNil.
Definition list_den (A : Type) : datatypeDenote (list A) (list_dt A) :=
[!, ! ~> nil] ::: [x, r # 1 ~> x :: hd r] ::: hnil.
[!, ! ~> nil] ::: [x, r # 1 ~> x :: hd r] ::: HNil.
Definition tree_den (A : Type) : datatypeDenote (tree A) (tree_dt A) :=
[v, ! ~> Leaf v] ::: [!, r # 2 ~> Node (hd r) (hd (tl r))] ::: hnil.
[v, ! ~> Leaf v] ::: [!, r # 2 ~> Node (hd r) (hd (tl r))] ::: HNil.
(* end thide *)
......@@ -100,36 +100,36 @@ Definition Empty_set_fix : fixDenote Empty_set Empty_set_dt :=
Eval compute in size Empty_set_fix.
Definition unit_fix : fixDenote unit unit_dt :=
fun R cases _ => (fst cases) tt inil.
fun R cases _ => (hhd cases) tt INil.
Eval compute in size unit_fix.
Definition bool_fix : fixDenote bool bool_dt :=
fun R cases b => if b
then (fst cases) tt inil
else (fst (snd cases)) tt inil.
then (hhd cases) tt INil
else (hhd (htl cases)) tt INil.
Eval compute in size bool_fix.
Definition nat_fix : fixDenote nat nat_dt :=
fun R cases => fix F (n : nat) : R :=
match n with
| O => (fst cases) tt inil
| S n' => (fst (snd cases)) tt (icons (F n') inil)
| O => (hhd cases) tt INil
| S n' => (hhd (htl cases)) tt (ICons (F n') INil)
end.
Eval cbv beta iota delta -[plus] in size nat_fix.
Definition list_fix (A : Type) : fixDenote (list A) (list_dt A) :=
fun R cases => fix F (ls : list A) : R :=
match ls with
| nil => (fst cases) tt inil
| x :: ls' => (fst (snd cases)) x (icons (F ls') inil)
| nil => (hhd cases) tt INil
| x :: ls' => (hhd (htl cases)) x (ICons (F ls') INil)
end.
Eval cbv beta iota delta -[plus] in fun A => size (@list_fix A).
Definition tree_fix (A : Type) : fixDenote (tree A) (tree_dt A) :=
fun R cases => fix F (t : tree A) : R :=
match t with
| Leaf x => (fst cases) x inil
| Node t1 t2 => (fst (snd cases)) tt (icons (F t1) (icons (F t2) inil))
| Leaf x => (hhd cases) x INil
| Node t1 t2 => (hhd (htl cases)) tt (ICons (F t1) (ICons (F t2) INil))
end.
Eval cbv beta iota delta -[plus] in fun A => size (@tree_fix A).
(* end thide *)
......@@ -157,15 +157,15 @@ Definition print T dt (pr : print_datatype dt) (fx : fixDenote T dt) : T -> stri
++ foldr (fun s acc => ", " ++ s ++ acc) ")" r) pr).
(* end thide *)
Eval compute in print hnil Empty_set_fix.
Eval compute in print (^ "tt" (fun _ => "") ::: hnil) unit_fix.
Eval compute in print HNil Empty_set_fix.
Eval compute in print (^ "tt" (fun _ => "") ::: HNil) unit_fix.
Eval compute in print (^ "true" (fun _ => "")
::: ^ "false" (fun _ => "")
::: hnil) bool_fix.
::: HNil) bool_fix.
Definition print_nat := print (^ "O" (fun _ => "")
::: ^ "S" (fun _ => "")
::: hnil) nat_fix.
::: HNil) nat_fix.
Eval cbv beta iota delta -[append] in print_nat.
Eval simpl in print_nat 0.
Eval simpl in print_nat 1.
......@@ -174,11 +174,11 @@ Eval simpl in print_nat 2.
Eval cbv beta iota delta -[append] in fun A (pr : A -> string) =>
print (^ "nil" (fun _ => "")
::: ^ "cons" pr
::: hnil) (@list_fix A).
::: HNil) (@list_fix A).
Eval cbv beta iota delta -[append] in fun A (pr : A -> string) =>
print (^ "Leaf" pr
::: ^ "Node" (fun _ => "")
::: hnil) (@tree_fix A).
::: HNil) (@tree_fix A).
(** ** Mapping *)
......@@ -225,16 +225,15 @@ Section ok.
forall (R : Type) (cases : datatypeDenote R dt)
c (m : member c dt)
(x : nonrecursive c) (r : ilist T (recursive c)),
fx R cases ((hget dd m) x r)
= (hget cases m) x (imap (fx R cases) r).
fx cases ((hget dd m) x r)
= (hget cases m) x (imap (fx cases) r).
End ok.
Implicit Arguments datatypeDenoteOk [T dt].
Lemma foldr_plus : forall n (ils : ilist nat n),
foldr plus 1 ils > 0.
induction n; crush.
generalize (IHn b); crush.
induction ils; crush.
Qed.
(* end thide *)
......@@ -265,12 +264,11 @@ Theorem map_id : forall T dt
end; crush.
induction (recursive c); crush.
destruct r; reflexivity.
destruct r; crush.
rewrite (H None).
unfold icons.
dep_destruct r; reflexivity.
dep_destruct r; crush.
rewrite (H First).
f_equal.
apply IHn; crush.
apply (H (Some i0)).
apply (H (Next i)).
Qed.
(* end thide *)
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