Certified Programming with Dependent Types: src/Intensional.v comparison

comparison src/Intensional.v @ 183:02569049397b

Closure conversion defined

author	Adam Chlipala <adamc@hcoop.net>
date	Sun, 16 Nov 2008 14:01:33 -0500
parents	24b99e025fe8
children	699fd70c04a7

comparison

equal deleted inserted replaced

-:24b99e025fe8
+:02569049397b
 * The license text is available at:
 *   http://creativecommons.org/licenses/by-nc-nd/3.0/
 *)
 (* begin hide *)
-Require Import String List.
+Require Import Arith Bool String List.
 Require Import Axioms Tactics DepList.
 Set Implicit Arguments.
 (* end hide *)
 Axiom Exp_equiv : forall t (E : Exp t) var1 var2,
 exp_equiv nil (E var1) (E var2).
 End Source.
-Section Closed.
+Module Closed.
 Inductive type : Type :=
 | TNat : type
 | Arrow : type -> type -> type
 | Code : type -> type -> type -> type
 | Prod : type -> type -> type
 | Fst : forall t1 t2,
 exp (t1 ** t2)
 -> exp t1
 | Snd : forall t1 t2,
 exp (t1 ** t2)
+-> exp t2
+| Let : forall t1 t2,
+exp t1
+-> (var t1 -> exp t2)
 -> exp t2.
 Section funcs.
 Variable T : Type.
 Notation "# v" := (Var v) (at level 70) : cc_scope.
 Notation "^ n" := (Const n) (at level 70) : cc_scope.
 Infix "+^" := Plus (left associativity, at level 79) : cc_scope.
-Infix "@@" := App (no associativity, at level 75) : cc_scope.
+Infix "@" := App (left associativity, at level 77) : cc_scope.
 Infix "##" := Pack (no associativity, at level 71) : cc_scope.
 Notation "()" := EUnit : cc_scope.
 Notation "[ x1 , x2 ]" := (Pair x1 x2) (at level 73) : cc_scope.
 Notation "#1 x" := (Fst x) (at level 72) : cc_scope.
 Notation "#2 x" := (Snd x) (at level 72) : cc_scope.
-Notation "f <= \\ x , y , e ; fs" :=
+Notation "f <== \\ x , y , e ; fs" :=
 (Abs (fun x y => e) (fun f => fs))
 (right associativity, at level 80, e at next level) : cc_scope.
+Notation "x <- e1 ; e2" := (Let e1 (fun x => e2))
+(right associativity, at level 80, e1 at next level) : cc_scope.
 Bind Scope cc_scope with exp funcs prog.
 Fixpoint typeDenote (t : type) : Set :=
 match t with
 | EUnit => tt
 | Pair _ _ e1 e2 => (expDenote e1, expDenote e2)
 | Fst _ _ e' => fst (expDenote e')
 | Snd _ _ e' => snd (expDenote e')
+| Let _ _ e1 e2 => expDenote (e2 (expDenote e1))
 end.
 Fixpoint funcsDenote T (fs : funcs typeDenote T) : T :=
 match fs with
 | Main v => v
 Definition ExpDenote t (E : Exp t) := expDenote (E _).
 Definition ProgDenote t (P : Prog t) := progDenote (P _).
 End Closed.
+Import Source Closed.
+Section splice.
+Open Local Scope cc_scope.
+Fixpoint spliceFuncs var T1 (fs : funcs var T1)
+T2 (f : T1 -> funcs var T2) {struct fs} : funcs var T2 :=
+match fs with
+| Main v => f v
+| Abs _ _ _ e fs' => Abs e (fun x => spliceFuncs (fs' x) f)
+end.
+End splice.
+Notation "x <-- e1 ; e2" := (spliceFuncs e1 (fun x => e2))
+(right associativity, at level 80, e1 at next level) : cc_scope.
+Definition natvar (_ : Source.type) := nat.
+Definition isfree := hlist (fun (_ : Source.type) => bool).
+Ltac maybe_destruct E :=
+match goal with
+| [ x : _ |- _ ] =>
+match E with
+| x => idtac
+end
+| _ =>
+match E with
+| eq_nat_dec _ _ => idtac
+end
+end; destruct E.
+Ltac my_crush :=
+crush; repeat (match goal with
+| [ x : (_ * _)%type |- _ ] => destruct x
+| [ |- context[if ?B then _ else _] ] => maybe_destruct B
+| [ _ : context[if ?B then _ else _] |- _ ] => maybe_destruct B
+end; crush).
+Section isfree.
+Import Source.
+Open Local Scope source_scope.
+Hint Extern 3 False => omega.
+Fixpoint lookup_type (envT : list Source.type) (n : nat) {struct envT}
+: isfree envT -> option Source.type :=
+match envT return (isfree envT -> _) with
+| nil => fun _ => None
+| first :: rest => fun fvs =>
+if eq_nat_dec n (length rest)
+then match fvs with
+| (true, _) => Some first
+| (false, _) => None
+end
+else lookup_type rest n (snd fvs)
+end.
+Implicit Arguments lookup_type [envT].
+Notation ok := (fun (envT : list Source.type) (fvs : isfree envT)
+(n : nat) (t : Source.type)
+=> lookup_type n fvs = Some t).
+Fixpoint wfExp (envT : list Source.type) (fvs : isfree envT)
+t (e : Source.exp natvar t) {struct e} : Prop :=
+match e with
+| Var t v => ok envT fvs v t
+| Const _ => True
+| Plus e1 e2 => wfExp envT fvs e1 /\ wfExp envT fvs e2
+| App _ _ e1 e2 => wfExp envT fvs e1 /\ wfExp envT fvs e2
+| Abs dom _ e' => wfExp (dom :: envT) (true ::: fvs) (e' (length envT))
+end.
+Implicit Arguments wfExp [envT t].
+Theorem wfExp_weaken : forall t (e : exp natvar t) envT (fvs fvs' : isfree envT),
+wfExp fvs e
+-> (forall n t, ok _ fvs n t -> ok _ fvs' n t)
+-> wfExp fvs' e.
+Hint Extern 1 (lookup_type (envT := _ :: _) _ _ = Some _) =>
+simpl in *; my_crush.
+induction e; my_crush; eauto.
+Defined.
+Fixpoint isfree_none (envT : list Source.type) : isfree envT :=
+match envT return (isfree envT) with
+| nil => tt
+| _ :: _ => (false, isfree_none _)
+end.
+Implicit Arguments isfree_none [envT].
+Fixpoint isfree_one (envT : list Source.type) (n : nat) {struct envT} : isfree envT :=
+match envT return (isfree envT) with
+| nil => tt
+| _ :: rest =>
+if eq_nat_dec n (length rest)
+then (true, isfree_none)
+else (false, isfree_one _ n)
+end.
+Implicit Arguments isfree_one [envT].
+Fixpoint isfree_merge (envT : list Source.type) : isfree envT -> isfree envT -> isfree envT :=
+match envT return (isfree envT -> isfree envT -> isfree envT) with
+| nil => fun _ _ => tt
+| _ :: _ => fun fv1 fv2 => (fst fv1 || fst fv2, isfree_merge _ (snd fv1) (snd fv2))
+end.
+Implicit Arguments isfree_merge [envT].
+Fixpoint fvsExp t (e : exp natvar t)
+(envT : list Source.type) {struct e} : isfree envT :=
+match e with
+| Var _ n => isfree_one n
+| Const _ => isfree_none
+| Plus e1 e2 => isfree_merge (fvsExp e1 envT) (fvsExp e2 envT)
+| App _ _ e1 e2 => isfree_merge (fvsExp e1 envT) (fvsExp e2 envT)
+| Abs dom _ e' => snd (fvsExp (e' (length envT)) (dom :: envT))
+end.
+Lemma isfree_one_correct : forall t (v : natvar t) envT fvs,
+ok envT fvs v t
+-> ok envT (isfree_one (envT:=envT) v) v t.
+induction envT; my_crush; eauto.
+Defined.
+Lemma isfree_merge_correct1 : forall t (v : natvar t) envT fvs1 fvs2,
+ok envT fvs1 v t
+-> ok envT (isfree_merge (envT:=envT) fvs1 fvs2) v t.
+induction envT; my_crush; eauto.
+Defined.
+Hint Rewrite orb_true_r : cpdt.
+Lemma isfree_merge_correct2 : forall t (v : natvar t) envT fvs1 fvs2,
+ok envT fvs2 v t
+-> ok envT (isfree_merge (envT:=envT) fvs1 fvs2) v t.
+induction envT; my_crush; eauto.
+Defined.
+Hint Resolve isfree_one_correct isfree_merge_correct1 isfree_merge_correct2.
+Lemma fvsExp_correct : forall t (e : exp natvar t)
+envT (fvs : isfree envT), wfExp fvs e
+-> forall (fvs' : isfree envT),
+(forall v t, ok envT (fvsExp e envT) v t -> ok envT fvs' v t)
+-> wfExp fvs' e.
+Hint Extern 1 (_ = _) =>
+match goal with
+| [ H : lookup_type _ (fvsExp ?X ?Y) = _ |- _ ] =>
+destruct (fvsExp X Y); my_crush
+end.
+induction e; my_crush; eauto.
+Defined.
+Lemma lookup_type_unique : forall v t1 t2 envT (fvs1 fvs2 : isfree envT),
+lookup_type v fvs1 = Some t1
+-> lookup_type v fvs2 = Some t2
+-> t1 = t2.
+induction envT; my_crush; eauto.
+Defined.
+Implicit Arguments lookup_type_unique [v t1 t2 envT fvs1 fvs2].
+Hint Extern 2 (lookup_type _ _ = Some _) =>
+match goal with
+| [ H1 : lookup_type ?v _ = Some _,
+H2 : lookup_type ?v _ = Some _ |- _ ] =>
+(generalize (lookup_type_unique H1 H2); intro; subst)
+|| rewrite <- (lookup_type_unique H1 H2)
+end.
+Lemma lookup_none : forall v t envT,
+lookup_type (envT:=envT) v (isfree_none (envT:=envT)) = Some t
+-> False.
+induction envT; my_crush.
+Defined.
+Hint Extern 2 (_ = _) => elimtype False; eapply lookup_none; eassumption.
+Lemma lookup_one : forall v' v t envT,
+lookup_type (envT:=envT) v' (isfree_one (envT:=envT) v) = Some t
+-> v' = v.
+induction envT; my_crush.
+Defined.
+Implicit Arguments lookup_one [v' v t envT].
+Hint Extern 2 (lookup_type _ _ = Some _) =>
+match goal with
+| [ H : lookup_type _ _ = Some _ |- _ ] =>
+generalize (lookup_one H); intro; subst
+end.
+Lemma lookup_merge : forall v t envT (fvs1 fvs2 : isfree envT),
+lookup_type v (isfree_merge fvs1 fvs2) = Some t
+-> lookup_type v fvs1 = Some t
+\/ lookup_type v fvs2 = Some t.
+induction envT; my_crush.
+Defined.
+Implicit Arguments lookup_merge [v t envT fvs1 fvs2].
+Lemma lookup_bound : forall v t envT (fvs : isfree envT),
+lookup_type v fvs = Some t
+-> v < length envT.
+Hint Resolve lt_S.
+induction envT; my_crush; eauto.
+Defined.
+Hint Resolve lookup_bound.
+Lemma lookup_bound_contra : forall t envT (fvs : isfree envT),
+lookup_type (length envT) fvs = Some t
+-> False.
+intros; assert (length envT < length envT); eauto.
+Defined.
+Hint Resolve lookup_bound_contra.
+Hint Extern 3 (_ = _) => elimtype False; omega.
+Lemma lookup_push_drop : forall v t t' envT fvs,
+v < length envT
+-> lookup_type (envT := t :: envT) v (true, fvs) = Some t'
+-> lookup_type (envT := envT) v fvs = Some t'.
+my_crush.
+Defined.
+Lemma lookup_push_add : forall v t t' envT fvs,
+lookup_type (envT := envT) v (snd fvs) = Some t'
+-> lookup_type (envT := t :: envT) v fvs = Some t'.
+my_crush; elimtype False; eauto.
+Defined.
+Hint Resolve lookup_bound lookup_push_drop lookup_push_add.
+Theorem fvsExp_minimal : forall t (e : exp natvar t)
+envT (fvs : isfree envT), wfExp fvs e
+-> (forall v t, ok envT (fvsExp e envT) v t -> ok envT fvs v t).
+Hint Extern 1 (_ = _) =>
+match goal with
+| [ H : lookup_type _ (isfree_merge _ _) = Some _ |- _ ] =>
+destruct (lookup_merge H); clear H; eauto
+end.
+induction e; my_crush; eauto.
+Defined.
+Fixpoint ccType (t : Source.type) : Closed.type :=
+match t with
+| Nat%source => Nat
+| (dom --> ran)%source => ccType dom --> ccType ran
+end%cc.
+Open Local Scope cc_scope.
+Fixpoint envType (envT : list Source.type) : isfree envT -> Closed.type :=
+match envT return (isfree envT -> _) with
+| nil => fun _ => Unit
+| t :: _ => fun tup =>
+if fst tup
+then ccType t ** envType _ (snd tup)
+else envType _ (snd tup)
+end.
+Implicit Arguments envType [envT].
+Fixpoint envOf (var : Closed.type -> Set) (envT : list Source.type) {struct envT}
+: isfree envT -> Set :=
+match envT return (isfree envT -> _) with
+| nil => fun _ => unit
+| first :: rest => fun fvs =>
+match fvs with
+| (true, fvs') => (var (ccType first) * envOf var rest fvs')%type
+| (false, fvs') => envOf var rest fvs'
+end
+end.
+Implicit Arguments envOf [envT].
+Notation "var <| to" := (match to with
+| None => unit
+| Some t => var (ccType t)
+end) (no associativity, at level 70).
+Fixpoint lookup (var : Closed.type -> Set) (envT : list Source.type) :
+forall (n : nat) (fvs : isfree envT), envOf var fvs -> var <| lookup_type n fvs :=
+match envT return (forall (n : nat) (fvs : isfree envT), envOf var fvs
+-> var <| lookup_type n fvs) with
+| nil => fun _ _ _ => tt
+| first :: rest => fun n fvs =>
+match (eq_nat_dec n (length rest)) as Heq return
+(envOf var (envT := first :: rest) fvs
+-> var <| (if Heq
+then match fvs with
+| (true, _) => Some first
+| (false, _) => None
+end
+else lookup_type n (snd fvs))) with
+| left _ =>
+match fvs return (envOf var (envT := first :: rest) fvs
+-> var <| (match fvs with
+| (true, _) => Some first
+| (false, _) => None
+end)) with
+| (true, _) => fun env => fst env
+| (false, _) => fun _ => tt
+end
+| right _ =>
+match fvs return (envOf var (envT := first :: rest) fvs
+-> var <| (lookup_type n (snd fvs))) with
+| (true, fvs') => fun env => lookup var rest n fvs' (snd env)
+| (false, fvs') => fun env => lookup var rest n fvs' env
+end
+end
+end.
+Theorem lok : forall var n t envT (fvs : isfree envT),
+lookup_type n fvs = Some t
+-> var <| lookup_type n fvs = var (ccType t).
+crush.
+Defined.
+End isfree.
+Implicit Arguments lookup_type [envT].
+Implicit Arguments lookup [envT fvs].
+Implicit Arguments wfExp [t envT].
+Implicit Arguments envType [envT].
+Implicit Arguments envOf [envT].
+Implicit Arguments lok [var n t envT fvs].
+Section lookup_hints.
+Hint Resolve lookup_bound_contra.
+(*Ltac my_chooser T k :=
+match T with
+| ptype =>
+match goal with
+| [ H : lookup _ _ = Some ?t |- _ ] => k t
+end
+| _ => default_chooser T k
+end.
+Ltac my_matching := matching equation my_chooser.*)
+Hint Resolve lookup_bound_contra.
+Lemma lookup_type_push : forall t' envT (fvs1 fvs2 : isfree envT) b1 b2,
+(forall (n : nat) (t : Source.type),
+lookup_type (envT := t' :: envT) n (b1, fvs1) = Some t ->
+lookup_type (envT := t' :: envT) n (b2, fvs2) = Some t)
+-> (forall (n : nat) (t : Source.type),
+lookup_type n fvs1 = Some t ->
+lookup_type n fvs2 = Some t).
+intros until b2; intro H; intros n t;
+generalize (H n t); my_crush; elimtype False; eauto.
+Defined.
+Lemma lookup_type_push_contra : forall t' envT (fvs1 fvs2 : isfree envT),
+(forall (n : nat) (t : Source.type),
+lookup_type (envT := t' :: envT) n (true, fvs1) = Some t ->
+lookup_type (envT := t' :: envT) n (false, fvs2) = Some t)
+-> False.
+intros until fvs2; intro H; generalize (H (length envT) t'); my_crush.
+Defined.
+End lookup_hints.
+Section packing.
+Open Local Scope cc_scope.
+Hint Resolve lookup_type_push lookup_type_push_contra.
+Definition packExp (var : Closed.type -> Set) (envT : list Source.type)
+(fvs1 fvs2 : isfree envT)
+: (forall n t, lookup_type n fvs1 = Some t -> lookup_type n fvs2 = Some t)
+-> envOf var fvs2 -> exp var (envType fvs1).
+refine (fix packExp (var : Closed.type -> Set) (envT : list Source.type) {struct envT}
+: forall fvs1 fvs2 : isfree envT,
+(forall n t, lookup_type n fvs1 = Some t -> lookup_type n fvs2 = Some t)
+-> envOf var fvs2 -> exp var (envType fvs1) :=
+match envT return (forall fvs1 fvs2 : isfree envT,
+(forall n t, lookup_type n fvs1 = Some t -> lookup_type n fvs2 = Some t)
+-> envOf var fvs2
+-> exp var (envType fvs1)) with
+| nil => fun _ _ _ _ => ()
+| first :: rest => fun fvs1 =>
+match fvs1 return (forall fvs2 : isfree (first :: rest),
+(forall n t, lookup_type (envT := first :: rest) n fvs1 = Some t
+-> lookup_type n fvs2 = Some t)
+-> envOf var fvs2
+-> exp var (envType (envT := first :: rest) fvs1)) with
+| (false, fvs1') => fun fvs2 =>
+match fvs2 return ((forall n t, lookup_type (envT := first :: rest) n (false, fvs1') = Some t
+-> lookup_type (envT := first :: rest) n fvs2 = Some t)
+-> envOf (envT := first :: rest) var fvs2
+-> exp var (envType (envT := first :: rest) (false, fvs1'))) with
+| (false, fvs2') => fun Hmin env =>
+packExp var _  fvs1' fvs2' _ env
+| (true, fvs2') => fun Hmin env =>
+packExp var _  fvs1' fvs2' _ (snd env)
+end
+| (true, fvs1') => fun fvs2 =>
+match fvs2 return ((forall n t, lookup_type (envT := first :: rest) n (true, fvs1') = Some t
+-> lookup_type (envT := first :: rest) n fvs2 = Some t)
+-> envOf (envT := first :: rest) var fvs2
+-> exp var (envType (envT := first :: rest) (true, fvs1'))) with
+| (false, fvs2') => fun Hmin env =>
+False_rect _ _
+| (true, fvs2') => fun Hmin env =>
+[#(fst env), packExp var _ fvs1' fvs2' _ (snd env)]
+end
+end
+end); eauto.
+Defined.
+Hint Resolve fvsExp_correct fvsExp_minimal.
+Hint Resolve lookup_push_drop lookup_bound lookup_push_add.
+Implicit Arguments packExp [var envT].
+Fixpoint unpackExp (var : Closed.type -> Set) t (envT : list Source.type) {struct envT}
+: forall fvs : isfree envT,
+exp var (envType fvs)
+-> (envOf var fvs -> exp var t) -> exp var t :=
+match envT return (forall fvs : isfree envT,
+exp var (envType fvs)
+-> (envOf var fvs -> exp var t) -> exp var t) with
+| nil => fun _ _ f => f tt
+| first :: rest => fun fvs =>
+match fvs return (exp var (envType (envT := first :: rest) fvs)
+-> (envOf var (envT := first :: rest) fvs -> exp var t)
+-> exp var t) with
+| (false, fvs') => fun p f =>
+unpackExp rest fvs' p f
+| (true, fvs') => fun p f =>
+x <- #1 p;
+unpackExp rest fvs' (#2 p)
+(fun env => f (x, env))
+end
+end.
+Implicit Arguments unpackExp [var t envT fvs].
+Theorem wfExp_lax : forall t t' envT (fvs : isfree envT) (e : Source.exp natvar t),
+wfExp (envT := t' :: envT) (true, fvs) e
+-> wfExp (envT := t' :: envT) (true, snd (fvsExp e (t' :: envT))) e.
+Hint Extern 1 (_ = _) =>
+match goal with
+| [ H : lookup_type _ (fvsExp ?X ?Y) = _ |- _ ] =>
+destruct (fvsExp X Y); my_crush
+end.
+eauto.
+Defined.
+Implicit Arguments wfExp_lax [t t' envT fvs e].
+Lemma inclusion : forall t t' envT fvs (e : Source.exp natvar t),
+wfExp (envT := t' :: envT) (true, fvs) e
+-> (forall n t, lookup_type n (snd (fvsExp e (t' :: envT))) = Some t
+-> lookup_type n fvs = Some t).
+eauto.
+Defined.
+Implicit Arguments inclusion [t t' envT fvs e].
+Definition env_prog var t envT (fvs : isfree envT) :=
+funcs var (envOf var fvs -> Closed.exp var t).
+Implicit Arguments env_prog [envT].
+Axiom cheat : forall T, T.
+Import Source.
+Open Local Scope cc_scope.
+Fixpoint ccExp var t (e : Source.exp natvar t)
+(envT : list Source.type) (fvs : isfree envT)
+{struct e} : wfExp fvs e -> env_prog var (ccType t) fvs :=
+match e in (Source.exp _ t) return (wfExp fvs e -> env_prog var (ccType t) fvs) with
+| Const n => fun _ => Main (fun _ => ^n)
+| Plus e1 e2 => fun wf =>
+n1 <-- ccExp var e1 _ fvs (proj1 wf);
+n2 <-- ccExp var e2 _ fvs (proj2 wf);
+Main (fun env => n1 env +^ n2 env)
+| Var _ n => fun wf =>
+Main (fun env => #(match lok wf in _ = T return T with
+| refl_equal => lookup var n env
+end))
+| App _ _ f x => fun wf =>
+f' <-- ccExp var f _ fvs (proj1 wf);
+x' <-- ccExp var x _ fvs (proj2 wf);
+Main (fun env => f' env @ x' env)
+| Abs dom _ b => fun wf =>
+b' <-- ccExp var (b (length envT)) (dom :: envT) _ (wfExp_lax wf);
+f <== \\env, arg, unpackExp (#env) (fun env => b' (arg, env));
+Main (fun env => #f ##
+packExp
+(snd (fvsExp (b (length envT)) (dom :: envT)))
+fvs (inclusion wf) env)
+end.
+End packing.
+Implicit Arguments packExp [var envT].
+Implicit Arguments unpackExp [var t envT fvs].
+Implicit Arguments ccExp [var t envT].
+Fixpoint map_funcs var result T1 T2 (f : T1 -> T2) (fs : cfuncs var result T1) {struct fs}
+: cfuncs var result T2 :=
+match fs with
+| CMain v => CMain (f v)
+| CAbs _ _ e fs' => CAbs e (fun x => map_funcs f (fs' x))
+end.
+Definition CcTerm' result (E : Pterm result) (Hwf : wfTerm (envT := nil) tt (E _)) : Cprog result :=
+fun _ => map_funcs (fun f => f tt) (ccTerm (E _) (envT := nil) tt Hwf).
+(** * Correctness *)
+Scheme pterm_equiv_mut := Minimality for pterm_equiv Sort Prop
+with pprimop_equiv_mut := Minimality for pprimop_equiv Sort Prop.
+Section splicePrim_correct.
+Variables result t t' : ptype.
+Variable ps' : ctypeDenote ([< t >]) -> cprimops ctypeDenote t'.
+Theorem splicePrim_correct : forall (ps : cprimops ctypeDenote t),
+cprimopsDenote (splicePrim ps ps')
+= cprimopsDenote (ps' (cprimopsDenote ps)).
+induction ps; equation.
+Qed.
+End splicePrim_correct.
+Section spliceTerm_correct.
+Variables result t : ptype.
+Variable e : ctypeDenote ([< t >]) -> cterm ctypeDenote result.
+Variable k : ptypeDenote result -> bool.
+Theorem spliceTerm_correct : forall (ps : cprimops ctypeDenote t),
+ctermDenote (spliceTerm ps e) k
+= ctermDenote (e (cprimopsDenote ps)) k.
+induction ps; equation.
+Qed.
+End spliceTerm_correct.
+Section spliceFuncs'_correct.
+Variable result : ptype.
+Variables T1 T2 : Type.
+Variable f : T1 -> T2.
+Variable k : ptypeDenote result -> bool.
+Theorem spliceFuncs'_correct : forall fs,
+cfuncsDenote (spliceFuncs' fs f) k
+= f (cfuncsDenote fs k).
+induction fs; equation.
+Qed.
+End spliceFuncs'_correct.
+Section spliceFuncs_correct.
+Variable result : ptype.
+Variables T1 T2 T3 : Type.
+Variable fs2 : cfuncs ctypeDenote result T2.
+Variable f : T1 -> T2 -> T3.
+Variable k : ptypeDenote result -> bool.
+Hint Rewrite spliceFuncs'_correct : ltamer.
+Theorem spliceFuncs_correct : forall fs1,
+cfuncsDenote (spliceFuncs fs1 fs2 f) k
+= f (cfuncsDenote fs1 k) (cfuncsDenote fs2 k).
+induction fs1; equation.
+Qed.
+End spliceFuncs_correct.
+Section inside_correct.
+Variable result : ptype.
+Variables T1 T2 : Type.
+Variable fs2 : T1 -> cfuncs ctypeDenote result T2.
+Variable k : ptypeDenote result -> bool.
+Theorem inside_correct : forall fs1,
+cfuncsDenote (inside fs1 fs2) k
+= cfuncsDenote (fs2 (cfuncsDenote fs1 k)) k.
+induction fs1; equation.
+Qed.
+End inside_correct.
+Notation "var <| to" := (match to with
+| None => unit
+| Some t => var ([< t >])%cc
+end) (no associativity, at level 70).
+Section packing_correct.
+Variable result : ptype.
+Hint Rewrite splicePrim_correct spliceTerm_correct : ltamer.
+Ltac my_matching := matching my_equation default_chooser.
+Fixpoint makeEnv (envT : list ptype) : forall (fvs : isfree envT),
+ptypeDenote (envType fvs)
+-> envOf ctypeDenote fvs :=
+match envT return (forall (fvs : isfree envT),
+ptypeDenote (envType fvs)
+-> envOf ctypeDenote fvs) with
+| nil => fun _ _ => tt
+| first :: rest => fun fvs =>
+match fvs return (ptypeDenote (envType (envT := first :: rest) fvs)
+-> envOf (envT := first :: rest) ctypeDenote fvs) with
+| (false, fvs') => fun env => makeEnv rest fvs' env
+| (true, fvs') => fun env => (fst env, makeEnv rest fvs' (snd env))
+end
+end.
+Theorem unpackExp_correct : forall (envT : list ptype) (fvs : isfree envT)
+(ps : cprimops ctypeDenote (envType fvs))
+(e : envOf ctypeDenote fvs -> cterm ctypeDenote result) k,
+ctermDenote (unpackExp ps e) k
+= ctermDenote (e (makeEnv _ _ (cprimopsDenote ps))) k.
+induction envT; my_matching.
+Qed.
+Lemma lookup_type_more : forall v2 envT (fvs : isfree envT) t b v,
+(v2 = length envT -> False)
+-> lookup_type v2 (envT := t :: envT) (b, fvs) = v
+-> lookup_type v2 fvs = v.
+equation.
+Qed.
+Lemma lookup_type_less : forall v2 t envT (fvs : isfree (t :: envT)) v,
+(v2 = length envT -> False)
+-> lookup_type v2 (snd fvs) = v
+-> lookup_type v2 (envT := t :: envT) fvs = v.
+equation.
+Qed.
+Lemma lookup_bound_contra_eq : forall t envT (fvs : isfree envT) v,
+lookup_type v fvs = Some t
+-> v = length envT
+-> False.
+simpler; eapply lookup_bound_contra; eauto.
+Defined.
+Lemma lookup_type_inner : forall result t envT v t' (fvs : isfree envT) e,
+wfTerm (envT := t :: envT) (true, fvs) e
+-> lookup_type v (snd (fvsTerm (result := result) e (t :: envT))) = Some t'
+-> lookup_type v fvs = Some t'.
+Hint Resolve lookup_bound_contra_eq fvsTerm_minimal
+lookup_type_more lookup_type_less.
+Hint Extern 2 (Some _ = Some _) => contradictory.
+eauto 6.
+Qed.
+Lemma cast_irrel : forall T1 T2 x (H1 H2 : T1 = T2),
+(x :? H1) = (x :? H2).
+equation.
+Qed.
+Hint Immediate cast_irrel.
+Lemma cast_irrel_unit : forall T1 T2 x y (H1 : T1 = T2) (H2 : unit = T2),
+(x :? H1) = (y :? H2).
+intros; generalize H1;
+rewrite <- H2 in H1;
+equation.
+Qed.
+Hint Immediate cast_irrel_unit.
+Hint Extern 3 (_ = _) =>
+match goal with
+| [ |- context[False_rect _ ?H] ] =>
+apply False_rect; exact H
+end.
+Theorem packExp_correct : forall v2 t envT (fvs1 fvs2 : isfree envT)
+Heq1 (Heq2 : ctypeDenote <| lookup_type v2 fvs2 = ptypeDenote t)
+Hincl env,
+(lookup ctypeDenote v2 env :? Heq2)
+= (lookup ctypeDenote v2
+(makeEnv envT fvs1
+(cprimopsDenote
+(packExp fvs1 fvs2 Hincl env))) :? Heq1).
+induction envT; my_equation.
+Qed.
+End packing_correct.
+Lemma look : forall envT n (fvs : isfree envT) t,
+lookup_type n fvs = Some t
+-> ctypeDenote <| lookup_type n fvs = ptypeDenote t.
+equation.
+Qed.
+Implicit Arguments look [envT n fvs t].
+Theorem ccTerm_correct : forall result G
+(e1 : pterm ptypeDenote result)
+(e2 : pterm natvar result),
+pterm_equiv G e1 e2
+-> forall (envT : list ptype) (fvs : isfree envT)
+(env : envOf ctypeDenote fvs) (Hwf : wfTerm fvs e2) k,
+(forall t (v1 : ptypeDenote t) (v2 : natvar t),
+In (vars (x := t) (v1, v2)) G
+-> v2 < length envT)
+-> (forall t (v1 : ptypeDenote t) (v2 : natvar t),
+In (vars (x := t) (v1, v2)) G
+-> lookup_type v2 fvs = Some t
+-> forall Heq, (lookup ctypeDenote v2 env :? Heq) = v1)
+-> ptermDenote e1 k
+= ctermDenote (cfuncsDenote (ccTerm e2 fvs Hwf) k env) k.
+Hint Rewrite splicePrim_correct spliceTerm_correct
+spliceFuncs_correct inside_correct : ltamer.
+Hint Rewrite unpackExp_correct : ltamer.
+Hint Resolve packExp_correct lookup_type_inner.
+Hint Extern 1 (_ = _) => push_vars.
+Hint Extern 1 (_ = _) =>
+match goal with
+| [ Hvars : forall t v1 v2, _,
+Hin : In _ _ |- _ ] =>
+rewrite (Hvars _ _ _ Hin)
+end.
+Hint Extern 1 (wfPrimop _ _) => tauto.
+Hint Extern 1 (_ < _) =>
+match goal with
+| [ Hvars : forall t v1 v2, _,
+Hin : In _ _ |- _ ] =>
+exact (Hvars _ _ _ Hin)
+end.
+Hint Extern 1 (lookup_type _ _ = _) => tauto.
+Hint Extern 1 (_ = _) =>
+match goal with
+| [ Hvars : forall t v1 v2, _,
+Hin : In (vars (_, length ?envT)) _ |- _ ] =>
+contradictory; assert (length envT < length envT); [auto | omega]
+end.
+Hint Extern 5 (_ = _) => symmetry.
+Hint Extern 1 (_ = _) =>
+match goal with
+| [ H : lookup_type ?v _ = Some ?t, fvs : isfree _ |- (lookup _ _ _ :? _) = _ ] =>
+let Hty := fresh "Hty" in
+(assert (Hty : lookup_type v fvs = Some t);
+[eauto
+| eapply (trans_cast (look Hty))])
+end.
+Hint Extern 3 (ptermDenote _ _ = ctermDenote _ _) =>
+match goal with
+| [ H : _ |- ptermDenote (_ ?v1) _
+= ctermDenote (cfuncsDenote (ccTerm (_ ?v2) (envT := ?envT) ?fvs _) _ _) _ ] =>
+apply (H v1 v2 envT fvs); my_simpler
+end.
+intro.
+apply (pterm_equiv_mut
+(fun G (e1 : pterm ptypeDenote result) (e2 : pterm natvar result) =>
+forall (envT : list ptype) (fvs : isfree envT)
+(env : envOf ctypeDenote fvs) (Hwf : wfTerm fvs e2) k,
+(forall t (v1 : ptypeDenote t) (v2 : natvar t),
+In (vars (x := t) (v1, v2)) G
+-> v2 < length envT)
+-> (forall t (v1 : ptypeDenote t) (v2 : natvar t),
+In (vars (x := t) (v1, v2)) G
+-> lookup_type v2 fvs = Some t
+-> forall Heq, (lookup ctypeDenote v2 env :? Heq) = v1)
+-> ptermDenote e1 k
+= ctermDenote (cfuncsDenote (ccTerm e2 fvs Hwf) k env) k)
+(fun G t (p1 : pprimop ptypeDenote result t) (p2 : pprimop natvar result t) =>
+forall (envT : list ptype) (fvs : isfree envT)
+(env : envOf ctypeDenote fvs) (Hwf : wfPrimop fvs p2) Hwf k,
+(forall t (v1 : ptypeDenote t) (v2 : natvar t),
+In (vars (x := t) (v1, v2)) G
+-> v2 < length envT)
+-> (forall t (v1 : ptypeDenote t) (v2 : natvar t),
+In (vars (x := t) (v1, v2)) G
+-> lookup_type v2 fvs = Some t
+-> forall Heq, (lookup ctypeDenote v2 env :? Heq) = v1)
+-> pprimopDenote p1 k
+= cprimopsDenote (cfuncsDenote (ccPrimop p2 fvs Hwf) k env)));
+my_simpler; eauto.
+Qed.
+(** * Parametric version *)
+Section wf.
+Variable result : ptype.
+Lemma Pterm_wf' : forall G (e1 e2 : pterm natvar result),
+pterm_equiv G e1 e2
+-> forall envT (fvs : isfree envT),
+(forall t (v1 v2 : natvar t), In (vars (v1, v2)) G
+-> lookup_type v1 fvs = Some t)
+-> wfTerm fvs e1.
+Hint Extern 3 (Some _ = Some _) => contradictory; eapply lookup_bound_contra; eauto.
+apply (pterm_equiv_mut
+(fun G (e1 e2 : pterm natvar result) =>
+forall envT (fvs : isfree envT),
+(forall t (v1 v2 : natvar t), In (vars (v1, v2)) G
+-> lookup_type v1 fvs = Some t)
+-> wfTerm (envT:=envT) fvs e1)
+(fun G t (p1 p2 : pprimop natvar result t) =>
+forall envT (fvs : isfree envT),
+(forall t (v1 v2 : natvar t), In (vars (v1, v2)) G
+-> lookup_type v1 fvs = Some t)
+-> wfPrimop (envT:=envT) fvs p1));
+simpler;
+match goal with
+| [ envT : list ptype, H : _ |- _ ] =>
+apply (H (length envT) (length envT)); simpler
+end.
+Qed.
+Theorem Pterm_wf : forall (E : Pterm result),
+wfTerm (envT := nil) tt (E _).
+intros; eapply Pterm_wf';
+[apply Pterm_equiv
+| simpler].
+Qed.
+End wf.
+Definition CcTerm result (E : Pterm result) : Cprog result :=
+CcTerm' E (Pterm_wf E).
+Lemma map_funcs_correct : forall result T1 T2 (f : T1 -> T2) (fs : cfuncs ctypeDenote result T1) k,
+cfuncsDenote (map_funcs f fs) k = f (cfuncsDenote fs k).
+induction fs; equation.
+Qed.
+Theorem CcTerm_correct : forall result (E : Pterm result) k,
+PtermDenote E k
+= CprogDenote (CcTerm E) k.
+Hint Rewrite map_funcs_correct : ltamer.
+unfold PtermDenote, CprogDenote, CcTerm, CcTerm', cprogDenote;
+simpler;
+apply (ccTerm_correct (result := result)
+(G := nil)
+(e1 := E _)
+(e2 := E _)
+(Pterm_equiv _ _ _)
+nil
+tt
+tt);
+simpler.
+Qed.

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