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Commit e1ab379e authored by Adam Chlipala's avatar Adam Chlipala
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Parts I want to keep compile with 8.2

parent 7e33bd55
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.hgignore
View file @ e1ab379e
......@@ -18,3 +18,6 @@ templates/*.v
staging/html/.dir
cpdt.tgz
*.glob
*.v.d
Makefile
View file @ e1ab379e
MODULES_NODOC := Axioms AxiomsImpred Tactics MoreSpecif DepList
MODULES_NODOC := Axioms Tactics MoreSpecif DepList
MODULES_PROSE := Intro
MODULES_CODE := StackMachine InductiveTypes Predicates Coinductive Subset \
MoreDep DataStruct Equality Match Reflection Firstorder Hoas Interps \
......@@ -17,7 +17,7 @@ coq: Makefile.coq
Makefile.coq: Makefile $(VS)
coq_makefile $(VS) \
COQC = "coqc -impredicative-set -I src -dump-glob $(GLOBALS)" \
COQC = "coqc -I src -dump-glob $(GLOBALS)" \
COQDEP = "coqdep -I src" \
-o Makefile.coq
......
src/DataStruct.v
View file @ e1ab379e
......@@ -699,7 +699,7 @@ Section cfoldCond.
match n return (findex n -> exp' Bool) -> (findex n -> exp' t) -> exp' t with
| O => fun _ _ => default
| S n' => fun tests bodies =>
match tests None with
match tests None return _ with
| BConst true => bodies None
| BConst false => cfoldCond n'
(fun idx => tests (Some idx))
......@@ -743,14 +743,14 @@ Fixpoint cfold t (e : exp' t) {struct e} : exp' t :=
| Plus e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| NConst n1, NConst n2 => NConst (n1 + n2)
| _, _ => Plus e1' e2'
end
| Eq e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
| _, _ => Eq e1' e2'
end
......
src/Extensional.v
View file @ e1ab379e
......@@ -963,6 +963,7 @@ Module PatMatch.
| [ H : forall env, Some _ = Some env -> _ |- _ ] =>
destruct (H _ (refl_equal _)); clear H; intuition
| [ H : _ |- _ ] => rewrite H; intuition
| [ |- context[match ?v with inl _ => _ | inr _ => _ end] ] => destruct v; auto
end.
Qed.
......
src/Impure.v
View file @ e1ab379e
......@@ -7,232 +7,8 @@
* http://creativecommons.org/licenses/by-nc-nd/3.0/
*)
(* begin hide *)
Require Import Arith List Omega.
Require Import Axioms Tactics.
Set Implicit Arguments.
(* end hide *)
(** %\chapter{Modeling Impure Languages}% *)
(** TODO: Prose for this chapter *)
Section var.
Variable var : Type.
Inductive term : Type :=
| Var : var -> term
| App : term -> term -> term
| Abs : (var -> term) -> term
| Unit : term.
End var.
Implicit Arguments Unit [var].
Notation "# v" := (Var v) (at level 70).
Notation "()" := Unit.
Infix "@" := App (left associativity, at level 72).
Notation "\ x , e" := (Abs (fun x => e)) (at level 73).
Notation "\ ? , e" := (Abs (fun _ => e)) (at level 73).
Definition Term := forall var, term var.
Definition ident : Term := fun _ => \x, #x.
Definition unite : Term := fun _ => ().
Definition ident_self : Term := fun _ => ident _ @ ident _.
Definition ident_unit : Term := fun _ => ident _ @ unite _.
Module impredicative.
Inductive dynamic : Set :=
| Dyn : forall (dynTy : Type), dynTy -> dynamic.
Inductive computation (T : Type) : Set :=
| Return : T -> computation T
| Bind : forall (T' : Type),
computation T' -> (T' -> computation T) -> computation T
| Unpack : dynamic -> computation T.
Inductive eval : forall T, computation T -> T -> Prop :=
| EvalReturn : forall T (v : T),
eval (Return v) v
| EvalUnpack : forall T (v : T),
eval (Unpack T (Dyn v)) v
| EvalBind : forall T1 T2 (c1 : computation T1) (c2 : T1 -> computation T2) v1 v2,
eval c1 v1
-> eval (c2 v1) v2
-> eval (Bind c1 c2) v2.
(* begin thide *)
Fixpoint termDenote (e : term dynamic) : computation dynamic :=
match e with
| Var v => Return v
| App e1 e2 => Bind (termDenote e1) (fun f =>
Bind (termDenote e2) (fun x =>
Bind (Unpack (dynamic -> computation dynamic) f) (fun f' =>
f' x)))
| Abs e' => Return (Dyn (fun x => termDenote (e' x)))
| Unit => Return (Dyn tt)
end.
(* end thide *)
Definition TermDenote (E : Term) := termDenote (E _).
Eval compute in TermDenote ident.
Eval compute in TermDenote unite.
Eval compute in TermDenote ident_self.
Eval compute in TermDenote ident_unit.
Theorem eval_ident_unit : eval (TermDenote ident_unit) (Dyn tt).
(* begin thide *)
compute.
repeat econstructor.
simpl.
constructor.
Qed.
(* end thide *)
Theorem invert_ident : forall (E : Term) d,
eval (TermDenote (fun _ => ident _ @ E _)) d
-> eval (TermDenote E) d.
(* begin thide *)
inversion 1.
crush.
Focus 3.
crush.
unfold TermDenote in H0.
simpl in H0.
(** [injection H0.] *)
Abort.
(* end thide *)
End impredicative.
Module predicative.
Inductive val : Type :=
| Func : nat -> val
| VUnit.
Inductive computation : Type :=
| Return : val -> computation
| Bind : computation -> (val -> computation) -> computation
| CAbs : (val -> computation) -> computation
| CApp : val -> val -> computation.
Definition func := val -> computation.
Fixpoint get (n : nat) (ls : list func) {struct ls} : option func :=
match ls with
| nil => None
| x :: ls' =>
if eq_nat_dec n (length ls')
then Some x
else get n ls'
end.
Inductive eval : list func -> computation -> list func -> val -> Prop :=
| EvalReturn : forall ds d,
eval ds (Return d) ds d
| EvalBind : forall ds c1 c2 ds' d1 ds'' d2,
eval ds c1 ds' d1
-> eval ds' (c2 d1) ds'' d2
-> eval ds (Bind c1 c2) ds'' d2
| EvalCAbs : forall ds f,
eval ds (CAbs f) (f :: ds) (Func (length ds))
| EvalCApp : forall ds i d2 f ds' d3,
get i ds = Some f
-> eval ds (f d2) ds' d3
-> eval ds (CApp (Func i) d2) ds' d3.
(* begin thide *)
Fixpoint termDenote (e : term val) : computation :=
match e with
| Var v => Return v
| App e1 e2 => Bind (termDenote e1) (fun f =>
Bind (termDenote e2) (fun x =>
CApp f x))
| Abs e' => CAbs (fun x => termDenote (e' x))
| Unit => Return VUnit
end.
(* end thide *)
Definition TermDenote (E : Term) := termDenote (E _).
Eval compute in TermDenote ident.
Eval compute in TermDenote unite.
Eval compute in TermDenote ident_self.
Eval compute in TermDenote ident_unit.
Theorem eval_ident_unit : exists ds, eval nil (TermDenote ident_unit) ds VUnit.
(* begin thide *)
compute.
repeat econstructor.
simpl.
rewrite (eta Return).
reflexivity.
Qed.
Hint Constructors eval.
Lemma app_nil_start : forall A (ls : list A),
ls = nil ++ ls.
reflexivity.
Qed.
Lemma app_cons : forall A (x : A) (ls : list A),
x :: ls = (x :: nil) ++ ls.
reflexivity.
Qed.
Theorem eval_monotone : forall ds c ds' d,
eval ds c ds' d
-> exists ds'', ds' = ds'' ++ ds.
Hint Resolve app_nil_start app_ass app_cons.
induction 1; firstorder; subst; eauto.
Qed.
Lemma length_app : forall A (ds2 ds1 : list A),
length (ds1 ++ ds2) = length ds1 + length ds2.
induction ds1; simpl; intuition.
Qed.
Lemma get_app : forall ds2 d ds1,
get (length ds2) (ds1 ++ d :: ds2) = Some d.
Hint Rewrite length_app : cpdt.
induction ds1; crush;
match goal with
| [ |- context[if ?E then _ else _] ] => destruct E
end; crush.
Qed.
(* end thide *)
Theorem invert_ident : forall (E : Term) ds ds' d,
eval ds (TermDenote (fun _ => ident _ @ E _)) ds' d
-> eval ((fun x => Return x) :: ds) (TermDenote E) ds' d.
(* begin thide *)
inversion 1; subst.
clear H.
inversion H3; clear H3; subst.
inversion H6; clear H6; subst.
generalize (eval_monotone H2); crush.
inversion H5; clear H5; subst.
rewrite get_app in H3.
inversion H3; clear H3; subst.
inversion H7; clear H7; subst.
assumption.
Qed.
(* end thide *)
End predicative.
(** TODO: This chapter! (Old version was too impredicative) *)
src/Intensional.v
View file @ e1ab379e
......@@ -7,1078 +7,7 @@
* http://creativecommons.org/licenses/by-nc-nd/3.0/
*)
(* begin hide *)
Require Import Arith Bool String List Eqdep JMeq.
Require Import Axioms Tactics DepList.
Set Implicit Arguments.
Infix "==" := JMeq (at level 70, no associativity).
(* end hide *)
(** %\chapter{Intensional Transformations}% *)
(** TODO: Prose for this chapter *)
(** * Closure Conversion *)
Module Source.
Inductive type : Type :=
| TNat : type
| Arrow : type -> type -> type.
Notation "'Nat'" := TNat : source_scope.
Infix "-->" := Arrow (right associativity, at level 60) : source_scope.
Open Scope source_scope.
Bind Scope source_scope with type.
Delimit Scope source_scope with source.
Section vars.
Variable var : type -> Type.
Inductive exp : type -> Type :=
| Var : forall t,
var t
-> exp t
| Const : nat -> exp Nat
| Plus : exp Nat -> exp Nat -> exp Nat
| App : forall t1 t2,
exp (t1 --> t2)
-> exp t1
-> exp t2
| Abs : forall t1 t2,
(var t1 -> exp t2)
-> exp (t1 --> t2).
End vars.
Definition Exp t := forall var, exp var t.
Implicit Arguments Var [var t].
Implicit Arguments Const [var].
Implicit Arguments Plus [var].
Implicit Arguments App [var t1 t2].
Implicit Arguments Abs [var t1 t2].
Notation "# v" := (Var v) (at level 70) : source_scope.
Notation "^ n" := (Const n) (at level 70) : source_scope.
Infix "+^" := Plus (left associativity, at level 79) : source_scope.
Infix "@" := App (left associativity, at level 77) : source_scope.
Notation "\ x , e" := (Abs (fun x => e)) (at level 78) : source_scope.
Notation "\ ! , e" := (Abs (fun _ => e)) (at level 78) : source_scope.
Bind Scope source_scope with exp.
Definition zero : Exp Nat := fun _ => ^0.
Definition one : Exp Nat := fun _ => ^1.
Definition zpo : Exp Nat := fun _ => zero _ +^ one _.
Definition ident : Exp (Nat --> Nat) := fun _ => \x, #x.
Definition app_ident : Exp Nat := fun _ => ident _ @ zpo _.
Definition app : Exp ((Nat --> Nat) --> Nat --> Nat) := fun _ =>
\f, \x, #f @ #x.
Definition app_ident' : Exp Nat := fun _ => app _ @ ident _ @ zpo _.
Fixpoint typeDenote (t : type) : Set :=
match t with
| Nat => nat
| t1 --> t2 => typeDenote t1 -> typeDenote t2
end.
Fixpoint expDenote t (e : exp typeDenote t) {struct e} : typeDenote t :=
match e in (exp _ t) return (typeDenote t) with
| Var _ v => v
| Const n => n
| Plus e1 e2 => expDenote e1 + expDenote e2
| App _ _ e1 e2 => (expDenote e1) (expDenote e2)
| Abs _ _ e' => fun x => expDenote (e' x)
end.
Definition ExpDenote t (e : Exp t) := expDenote (e _).
Section exp_equiv.
Variables var1 var2 : type -> Type.
Inductive exp_equiv : list { t : type & var1 t * var2 t }%type -> forall t, exp var1 t -> exp var2 t -> Prop :=
| EqVar : forall G t (v1 : var1 t) v2,
In (existT _ t (v1, v2)) G
-> exp_equiv G (#v1) (#v2)
| EqConst : forall G n,
exp_equiv G (^n) (^n)
| EqPlus : forall G x1 y1 x2 y2,
exp_equiv G x1 x2
-> exp_equiv G y1 y2
-> exp_equiv G (x1 +^ y1) (x2 +^ y2)
| EqApp : forall G t1 t2 (f1 : exp _ (t1 --> t2)) (x1 : exp _ t1) f2 x2,
exp_equiv G f1 f2
-> exp_equiv G x1 x2
-> exp_equiv G (f1 @ x1) (f2 @ x2)
| EqAbs : forall G t1 t2 (f1 : var1 t1 -> exp var1 t2) f2,
(forall v1 v2, exp_equiv (existT _ t1 (v1, v2) :: G) (f1 v1) (f2 v2))
-> exp_equiv G (Abs f1) (Abs f2).
End exp_equiv.
Axiom Exp_equiv : forall t (E : Exp t) var1 var2,
exp_equiv nil (E var1) (E var2).
End Source.
Module Closed.
Inductive type : Type :=
| TNat : type
| Arrow : type -> type -> type
| Code : type -> type -> type -> type
| Prod : type -> type -> type
| TUnit : type.
Notation "'Nat'" := TNat : cc_scope.
Notation "'Unit'" := TUnit : cc_scope.
Infix "-->" := Arrow (right associativity, at level 60) : cc_scope.
Infix "**" := Prod (right associativity, at level 59) : cc_scope.
Notation "env @@ dom ---> ran" := (Code env dom ran) (no associativity, at level 62, dom at next level) : cc_scope.
Bind Scope cc_scope with type.
Delimit Scope cc_scope with cc.
Open Local Scope cc_scope.
Section vars.
Variable var : type -> Set.
Inductive exp : type -> Type :=
| Var : forall t,
var t
-> exp t
| Const : nat -> exp Nat
| Plus : exp Nat -> exp Nat -> exp Nat
| App : forall dom ran,
exp (dom --> ran)
-> exp dom
-> exp ran
| Pack : forall env dom ran,
exp (env @@ dom ---> ran)
-> exp env
-> exp (dom --> ran)
| EUnit : exp Unit
| Pair : forall t1 t2,
exp t1
-> exp t2
-> exp (t1 ** t2)
| 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.
Inductive funcs : Type :=
| Main : T -> funcs
| Abs : forall env dom ran,
(var env -> var dom -> exp ran)
-> (var (env @@ dom ---> ran) -> funcs)
-> funcs.
End funcs.
Definition prog t := funcs (exp t).
End vars.
Implicit Arguments Var [var t].
Implicit Arguments Const [var].
Implicit Arguments EUnit [var].
Implicit Arguments Fst [var t1 t2].
Implicit Arguments Snd [var t1 t2].
Implicit Arguments Main [var T].
Implicit Arguments Abs [var T env dom ran].
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 (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" :=
(Abs (fun x y => e) (fun f => fs))
(right associativity, at level 80, e at next level) : cc_scope.
Notation "f <== \\ ! , y , e ; fs" :=
(Abs (fun _ y => e) (fun f => fs))
(right associativity, at level 80, e at next level) : cc_scope.
Notation "f <== \\ x , ! , e ; fs" :=
(Abs (fun x _ => e) (fun f => fs))
(right associativity, at level 80, e at next level) : cc_scope.
Notation "f <== \\ ! , ! , e ; fs" :=
(Abs (fun _ _ => 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
| Nat => nat
| Unit => unit
| dom --> ran => typeDenote dom -> typeDenote ran
| t1 ** t2 => typeDenote t1 * typeDenote t2
| env @@ dom ---> ran => typeDenote env -> typeDenote dom -> typeDenote ran
end%type.
Fixpoint expDenote t (e : exp typeDenote t) {struct e} : typeDenote t :=
match e in (exp _ t) return (typeDenote t) with
| Var _ v => v
| Const n => n
| Plus e1 e2 => expDenote e1 + expDenote e2
| App _ _ f x => (expDenote f) (expDenote x)
| Pack _ _ _ f env => (expDenote f) (expDenote env)
| 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
| Abs _ _ _ e fs =>
funcsDenote (fs (fun env arg => expDenote (e env arg)))
end.
Definition progDenote t (p : prog typeDenote t) : typeDenote t :=
expDenote (funcsDenote p).
Definition Exp t := forall var, exp var t.
Definition Prog t := forall var, prog var t.
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.
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; crush.
Defined.
Hint Resolve lookup_bound_contra.
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.
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].
Import Source.
Open Local Scope cc_scope.
Definition proj1 A B (pf : A /\ B) : A :=
let (x, _) := pf in x.
Definition proj2 A B (pf : A /\ B) : B :=
let (_, y) := pf in y.
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 T1 T2 (f : T1 -> T2) (fs : funcs var T1) {struct fs}
: funcs var T2 :=
match fs with
| Main v => Main (f v)
| Abs _ _ _ e fs' => Abs e (fun x => map_funcs f (fs' x))
end.
Definition CcExp' t (E : Source.Exp t) (Hwf : wfExp (envT := nil) tt (E _)) : Prog (ccType t) :=
fun _ => map_funcs (fun f => f tt) (ccExp (E _) (envT := nil) tt Hwf).
(** ** Examples *)
Open Local Scope source_scope.
Definition ident : Source.Exp (Nat --> Nat) := fun _ => \x, #x.
Theorem ident_ok : wfExp (envT := nil) tt (ident _).
crush.
Defined.
Eval compute in CcExp' ident ident_ok.
Eval compute in ProgDenote (CcExp' ident ident_ok).
Definition app_ident : Source.Exp Nat := fun _ => ident _ @ ^0.
Theorem app_ident_ok : wfExp (envT := nil) tt (app_ident _).
crush.
Defined.
Eval compute in CcExp' app_ident app_ident_ok.
Eval compute in ProgDenote (CcExp' app_ident app_ident_ok).
Definition first : Source.Exp (Nat --> Nat --> Nat) := fun _ =>
\x, \y, #x.
Theorem first_ok : wfExp (envT := nil) tt (first _).
crush.
Defined.
Eval compute in CcExp' first first_ok.
Eval compute in ProgDenote (CcExp' first first_ok).
Definition app_first : Source.Exp Nat := fun _ => first _ @ ^1 @ ^0.
Theorem app_first_ok : wfExp (envT := nil) tt (app_first _).
crush.
Defined.
Eval compute in CcExp' app_first app_first_ok.
Eval compute in ProgDenote (CcExp' app_first app_first_ok).
(** ** Correctness *)
Section spliceFuncs_correct.
Variables T1 T2 : Type.
Variable f : T1 -> funcs typeDenote T2.
Theorem spliceFuncs_correct : forall fs,
funcsDenote (spliceFuncs fs f)
= funcsDenote (f (funcsDenote fs)).
induction fs; crush.
Qed.
End spliceFuncs_correct.
Notation "var <| to" := (match to return Set with
| None => unit
| Some t => var (ccType t)
end) (no associativity, at level 70).
Section packing_correct.
Fixpoint makeEnv (envT : list Source.type) : forall (fvs : isfree envT),
Closed.typeDenote (envType fvs)
-> envOf Closed.typeDenote fvs :=
match envT return (forall (fvs : isfree envT),
Closed.typeDenote (envType fvs)
-> envOf Closed.typeDenote fvs) with
| nil => fun _ _ => tt
| first :: rest => fun fvs =>
match fvs return (Closed.typeDenote (envType (envT := first :: rest) fvs)
-> envOf (envT := first :: rest) Closed.typeDenote fvs) with
| (false, fvs') => fun env => makeEnv rest fvs' env
| (true, fvs') => fun env => (fst env, makeEnv rest fvs' (snd env))
end
end.
Implicit Arguments makeEnv [envT fvs].
Theorem unpackExp_correct : forall t (envT : list Source.type) (fvs : isfree envT)
(e1 : Closed.exp Closed.typeDenote (envType fvs))
(e2 : envOf Closed.typeDenote fvs -> Closed.exp Closed.typeDenote t),
Closed.expDenote (unpackExp e1 e2)
= Closed.expDenote (e2 (makeEnv (Closed.expDenote e1))).
induction envT; my_crush.
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.
my_crush.
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.
my_crush.
Qed.
Hint Resolve lookup_bound_contra.
Lemma lookup_bound_contra_eq : forall t envT (fvs : isfree envT) v,
lookup_type v fvs = Some t
-> v = length envT
-> False.
my_crush; elimtype False; eauto.
Qed.
Lemma lookup_type_inner : forall t t' envT v t'' (fvs : isfree envT) e,
wfExp (envT := t' :: envT) (true, fvs) e
-> lookup_type v (snd (fvsExp (t := t) e (t' :: envT))) = Some t''
-> lookup_type v fvs = Some t''.
Hint Resolve lookup_bound_contra_eq fvsExp_minimal
lookup_type_more lookup_type_less.
Hint Extern 2 (Some _ = Some _) => elimtype False.
eauto 6.
Qed.
Lemma cast_irrel : forall T1 T2 x (H1 H2 : T1 = T2),
match H1 in _ = T return T with
| refl_equal => x
end
= match H2 in _ = T return T with
| refl_equal => x
end.
intros; generalize H1; crush;
repeat match goal with
| [ |- context[match ?pf with refl_equal => _ end] ] =>
rewrite (UIP_refl _ _ pf)
end;
reflexivity.
Qed.
Hint Immediate cast_irrel.
Hint Extern 3 (_ == _) =>
match goal with
| [ |- context[False_rect _ ?H] ] =>
apply False_rect; exact H
end.
Theorem packExp_correct : forall v t envT (fvs1 fvs2 : isfree envT)
Hincl env,
lookup_type v fvs1 = Some t
-> lookup Closed.typeDenote v env
== lookup Closed.typeDenote v
(makeEnv (Closed.expDenote
(packExp fvs1 fvs2 Hincl env))).
induction envT; my_crush.
Qed.
End packing_correct.
Implicit Arguments packExp_correct [v envT fvs1].
Implicit Arguments lookup_type_inner [t t' envT v t'' fvs e].
Implicit Arguments inclusion [t t' envT fvs e].
Lemma typeDenote_same : forall t,
Source.typeDenote t = Closed.typeDenote (ccType t).
induction t; crush.
Qed.
Hint Resolve typeDenote_same.
Fixpoint lr (t : Source.type) : Source.typeDenote t -> Closed.typeDenote (ccType t) -> Prop :=
match t return Source.typeDenote t -> Closed.typeDenote (ccType t) -> Prop with
| Nat => @eq nat
| dom --> ran => fun f1 f2 =>
forall x1 x2, lr dom x1 x2
-> lr ran (f1 x1) (f2 x2)
end.
Theorem ccExp_correct : forall t G
(e1 : Source.exp Source.typeDenote t)
(e2 : Source.exp natvar t),
exp_equiv G e1 e2
-> forall (envT : list Source.type) (fvs : isfree envT)
(env : envOf Closed.typeDenote fvs) (wf : wfExp fvs e2),
(forall t (v1 : Source.typeDenote t) (v2 : natvar t),
In (existT _ _ (v1, v2)) G
-> v2 < length envT)
-> (forall t (v1 : Source.typeDenote t) (v2 : natvar t),
In (existT _ _ (v1, v2)) G
-> forall pf,
lr t v1 (match lok pf in _ = T return T with
| refl_equal => lookup Closed.typeDenote v2 env
end))
-> lr t (Source.expDenote e1) (Closed.expDenote (funcsDenote (ccExp e2 fvs wf) env)).
Hint Rewrite spliceFuncs_correct unpackExp_correct : cpdt.
Hint Resolve packExp_correct lookup_type_inner.
induction 1; crush;
match goal with
| [ IH : _, Hlr : lr ?T ?X1 ?X2, ENV : list Source.type, F2 : natvar _ -> _ |- _ ] =>
apply (IH X1 (length ENV) (T :: ENV) (true, snd (fvsExp (F2 (length ENV)) (T :: ENV))))
end; crush;
match goal with
| [ Hlt : forall t v1 v2, _ -> _ < _, Hin : In _ _ |- _ ] =>
solve [ generalize (Hlt _ _ _ Hin); crush ]
| [ |- context[match ?pf with refl_equal => _ end] ] => generalize pf
end; simpl;
match goal with
| [ |- context[if ?E then _ else _] ] => destruct E
end; intuition; subst;
match goal with
| [ |- context[match ?pf with refl_equal => _ end] ] => rewrite (UIP_refl _ _ pf); assumption
| [ Hlt : forall t v1 v2, _ -> _ < _, Hin : In (existT _ _ (_, length _)) _ |- _ ] =>
generalize (Hlt _ _ _ Hin); crush
| [ HG : _, Hin : In _ _, wf : wfExp _ _, pf : _ = Some _,
fvs : isfree _, env : envOf _ _ |- _ ] =>
generalize (HG _ _ _ Hin (lookup_type_inner wf pf)); clear_all;
repeat match goal with
| [ |- context[match ?pf with refl_equal => _ end] ] => generalize pf
end; simpl;
generalize (packExp_correct _ fvs (inclusion wf) env pf); simpl;
match goal with
| [ |- ?X == ?Y -> _ ] =>
generalize X Y
end;
rewrite pf; rewrite (lookup_type_inner wf pf);
intros lhs rhs Heq; intros;
repeat match goal with
| [ H : _ = _ |- _ ] => rewrite (UIP_refl _ _ H) in *
end;
rewrite <- Heq; assumption
end.
Qed.
(** * Parametric version *)
Section wf.
Lemma Exp_wf' : forall G t (e1 e2 : Source.exp natvar t),
exp_equiv G e1 e2
-> forall envT (fvs : isfree envT),
(forall t (v1 v2 : natvar t), In (existT _ _ (v1, v2)) G
-> lookup_type v1 fvs = Some t)
-> wfExp fvs e1.
Hint Extern 3 (Some _ = Some _) => elimtype False; eapply lookup_bound_contra; eauto.
induction 1; crush; eauto;
match goal with
| [ H : _, envT : list Source.type |- _ ] =>
apply H with (length envT); my_crush; eauto
end.
Qed.
Theorem Exp_wf : forall t (E : Source.Exp t),
wfExp (envT := nil) tt (E _).
Hint Resolve Exp_equiv.
intros; eapply Exp_wf'; crush.
Qed.
End wf.
Definition CcExp t (E : Source.Exp t) : Prog (ccType t) :=
CcExp' E (Exp_wf E).
Lemma map_funcs_correct : forall T1 T2 (f : T1 -> T2) (fs : funcs Closed.typeDenote T1),
funcsDenote (map_funcs f fs) = f (funcsDenote fs).
induction fs; crush.
Qed.
Theorem CcExp_correct : forall (E : Source.Exp Nat),
Source.ExpDenote E
= ProgDenote (CcExp E).
Hint Rewrite map_funcs_correct : cpdt.
unfold Source.ExpDenote, ProgDenote, CcExp, CcExp', progDenote; crush;
apply (ccExp_correct
(G := nil)
(e1 := E _)
(e2 := E _)
(Exp_equiv _ _ _)
nil
tt
tt); crush.
Qed.
(** TODO: This chapter! (Old version was too complicated) *)
src/Interps.v
View file @ e1ab379e
......@@ -10,7 +10,7 @@
(* begin hide *)
Require Import String List.
Require Import AxiomsImpred Tactics.
Require Import Axioms Tactics.
Set Implicit Arguments.
(* end hide *)
......@@ -121,7 +121,7 @@ Module STLC.
| Plus e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| Const n1, Const n2 => ^(n1 + n2)
| _, _ => e1' +^ e2'
end
......@@ -301,7 +301,7 @@ Module PSLC.
Variable var : type -> Type.
Definition pairOutType t :=
match t with
match t return Type with
| t1 ** t2 => option (exp var t1 * exp var t2)
| _ => unit
end.
......@@ -326,7 +326,7 @@ Module PSLC.
| Plus e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| Const n1, Const n2 => ^(n1 + n2)
| _, _ => e1' +^ e2'
end
......@@ -392,196 +392,3 @@ Module PSLC.
Qed.
(* end thide *)
End PSLC.
(** * Type Variables *)
Module SysF.
(* EX: Follow a similar progression for System F. *)
(* begin thide *)
Section vars.
Variable tvar : Type.
Inductive type : Type :=
| Nat : type
| Arrow : type -> type -> type
| TVar : tvar -> type
| All : (tvar -> type) -> type.
Notation "## v" := (TVar v) (at level 40).
Infix "-->" := Arrow (right associativity, at level 60).
Section Subst.
Variable t : type.
Inductive Subst : (tvar -> type) -> type -> Prop :=
| SNat : Subst (fun _ => Nat) Nat
| SArrow : forall dom ran dom' ran',
Subst dom dom'
-> Subst ran ran'
-> Subst (fun v => dom v --> ran v) (dom' --> ran')
| SVarEq : Subst TVar t
| SVarNe : forall v, Subst (fun _ => ##v) (##v)
| SAll : forall ran ran',
(forall v', Subst (fun v => ran v v') (ran' v'))
-> Subst (fun v => All (ran v)) (All ran').
End Subst.
Variable var : type -> Type.
Inductive exp : type -> Type :=
| Var : forall t,
var t
-> exp t
| Const : nat -> exp Nat
| Plus : exp Nat -> exp Nat -> exp Nat
| App : forall t1 t2,
exp (t1 --> t2)
-> exp t1
-> exp t2
| Abs : forall t1 t2,
(var t1 -> exp t2)
-> exp (t1 --> t2)
| TApp : forall tf,
exp (All tf)
-> forall t tf', Subst t tf tf'
-> exp tf'
| TAbs : forall tf,
(forall v, exp (tf v))
-> exp (All tf).
End vars.
Definition Typ := forall tvar, type tvar.
Definition Exp (T : Typ) := forall tvar (var : type tvar -> Type), exp var (T _).
(* end thide *)
Implicit Arguments Nat [tvar].
Notation "## v" := (TVar v) (at level 40).
Infix "-->" := Arrow (right associativity, at level 60).
Notation "\\\ x , t" := (All (fun x => t)) (at level 65).
Implicit Arguments Var [tvar var t].
Implicit Arguments Const [tvar var].
Implicit Arguments Plus [tvar var].
Implicit Arguments App [tvar var t1 t2].
Implicit Arguments Abs [tvar var t1 t2].
Implicit Arguments TAbs [tvar var tf].
Notation "# v" := (Var v) (at level 70).
Notation "^ n" := (Const n) (at level 70).
Infix "+^" := Plus (left associativity, at level 79).
Infix "@" := App (left associativity, at level 77).
Notation "\ x , e" := (Abs (fun x => e)) (at level 78).
Notation "\ ! , e" := (Abs (fun _ => e)) (at level 78).
Notation "e @@ t" := (TApp e (t := t) _) (left associativity, at level 77).
Notation "\\ x , e" := (TAbs (fun x => e)) (at level 78).
Notation "\\ ! , e" := (TAbs (fun _ => e)) (at level 78).
Definition zero : Exp (fun _ => Nat) := fun _ _ =>
^0.
Definition ident : Exp (fun _ => \\\T, ##T --> ##T) := fun _ _ =>
\\T, \x, #x.
Definition ident_zero : Exp (fun _ => Nat).
do 2 intro; refine (ident _ @@ _ @ zero _);
repeat constructor.
Defined.
Definition ident_ident : Exp (fun _ => \\\T, ##T --> ##T).
do 2 intro; refine (ident _ @@ _ @ ident _);
repeat constructor.
Defined.
Definition ident5 : Exp (fun _ => \\\T, ##T --> ##T).
do 2 intro; refine (ident_ident _ @@ _ @ ident_ident _ @@ _ @ ident _);
repeat constructor.
Defined.
(* begin thide *)
Fixpoint typeDenote (t : type Set) : Set :=
match t with
| Nat => nat
| t1 --> t2 => typeDenote t1 -> typeDenote t2
| ##v => v
| All tf => forall T, typeDenote (tf T)
end.
Lemma Subst_typeDenote : forall t tf tf',
Subst t tf tf'
-> typeDenote (tf (typeDenote t)) = typeDenote tf'.
induction 1; crush; ext_eq; crush.
Defined.
Fixpoint expDenote t (e : exp typeDenote t) {struct e} : typeDenote t :=
match e in (exp _ t) return (typeDenote t) with
| Var _ v => v
| Const n => n
| Plus e1 e2 => expDenote e1 + expDenote e2
| App _ _ e1 e2 => (expDenote e1) (expDenote e2)
| Abs _ _ e' => fun x => expDenote (e' x)
| TApp _ e' t' _ pf => match Subst_typeDenote pf in _ = T return T with
| refl_equal => (expDenote e') (typeDenote t')
end
| TAbs _ e' => fun T => expDenote (e' T)
end.
Definition ExpDenote T (E : Exp T) := expDenote (E _ _).
(* end thide *)
Eval compute in ExpDenote zero.
Eval compute in ExpDenote ident.
Eval compute in ExpDenote ident_zero.
Eval compute in ExpDenote ident_ident.
Eval compute in ExpDenote ident5.
(* begin thide *)
Section cfold.
Variable tvar : Type.
Variable var : type tvar -> Type.
Fixpoint cfold t (e : exp var t) {struct e} : exp var t :=
match e in exp _ t return exp _ t with
| Var _ v => #v
| Const n => ^n
| Plus e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
| Const n1, Const n2 => ^(n1 + n2)
| _, _ => e1' +^ e2'
end
| App _ _ e1 e2 => cfold e1 @ cfold e2
| Abs _ _ e' => Abs (fun x => cfold (e' x))
| TApp _ e' _ _ pf => TApp (cfold e') pf
| TAbs _ e' => \\T, cfold (e' T)
end.
End cfold.
Definition Cfold T (E : Exp T) : Exp T := fun _ _ => cfold (E _ _).
Lemma cfold_correct : forall t (e : exp _ t),
expDenote (cfold e) = expDenote e.
induction e; crush; try (ext_eq; crush);
repeat (match goal with
| [ |- context[cfold ?E] ] => dep_destruct (cfold E)
end; crush).
Qed.
Theorem Cfold_correct : forall t (E : Exp t),
ExpDenote (Cfold E) = ExpDenote E.
unfold ExpDenote, Cfold; intros; apply cfold_correct.
Qed.
(* end thide *)
End SysF.
src/Match.v
View file @ e1ab379e
......@@ -784,10 +784,10 @@ Qed.
Ltac matcher :=
intros;
repeat search_prem ltac:(apply False_prem || (apply ex_prem; intro));
repeat search_conc ltac:(apply True_conc || eapply ex_conc
|| search_prem ltac:(apply Match));
try apply imp_True.
repeat search_prem ltac:(simple apply False_prem || (simple apply ex_prem; intro));
repeat search_conc ltac:(simple apply True_conc || simple eapply ex_conc
|| search_prem ltac:(simple apply Match));
try simple apply imp_True.
(* end thide *)
(** Our tactic succeeds at proving a simple example. *)
......
src/MoreDep.v
View file @ e1ab379e
......@@ -247,20 +247,20 @@ Definition pairOut t (e : exp t) :=
With [pairOut] available, we can write [cfold] in a straightforward way. There are really no surprises beyond that Coq verifies that this code has such an expressive type, given the small annotation burden. *)
Fixpoint cfold t (e : exp t) {struct e} : exp t :=
match e in (exp t) return (exp t) with
Fixpoint cfold t (e : exp t) : exp t :=
match e with
| NConst n => NConst n
| Plus e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| NConst n1, NConst n2 => NConst (n1 + n2)
| _, _ => Plus e1' e2'
end
| Eq e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| NConst n1, NConst n2 => BConst (if eq_nat_dec n1 n2 then true else false)
| _, _ => Eq e1' e2'
end
......@@ -269,7 +269,7 @@ Fixpoint cfold t (e : exp t) {struct e} : exp t :=
| And e1 e2 =>
let e1' := cfold e1 in
let e2' := cfold e2 in
match e1', e2' with
match e1', e2' return _ with
| BConst b1, BConst b2 => BConst (b1 && b2)
| _, _ => And e1' e2'
end
......@@ -1028,7 +1028,7 @@ Section dec_star.
(** Finally, we have [dec_star]. It has a straightforward implementation. We introduce a spurious match on [s] so that [simpl] will know to reduce calls to [dec_star]. The heuristic that [simpl] uses is only to unfold identifier definitions when doing so would simplify some [match] expression. *)
Definition dec_star : {star P s} + { ~star P s}.
refine (match s with
refine (match s return _ with
| "" => Reduce (dec_star' (n := length s) 0 _)
| _ => Reduce (dec_star' (n := length s) 0 _)
end); crush.
......
src/Subset.v
View file @ e1ab379e
......@@ -376,7 +376,7 @@ let rec eq_nat_dec' n m0 =
We can build "smart" versions of the usual boolean operators and put them to good use in certified programming. For instance, here is a [sumbool] version of boolean "or." *)
(* begin thide *)
Notation "x || y" := (if x then Yes else Reduce y) (at level 50).
Notation "x || y" := (if x then Yes else Reduce y).
(** Let us use it for building a function that decides list membership. We need to assume the existence of an equality decision procedure for the type of list elements. *)
......
src/Tactics.v
View file @ e1ab379e
......@@ -48,15 +48,21 @@ Ltac simplHyp invOne :=
match goal with
| [ H : ex _ |- _ ] => destruct H
| [ H : ?F ?X = ?F ?Y |- _ ] => injection H;
match goal with
| [ |- F X = F Y -> _ ] => fail 1
| [ |- _ = _ -> _ ] => try clear H; intros; try subst
end
| [ H : ?F _ _ = ?F _ _ |- _ ] => injection H;
match goal with
| [ |- _ = _ -> _ = _ -> _ ] => try clear H; intros; try subst
end
| [ H : ?F ?X = ?F ?Y |- ?G ] =>
(assert (X = Y); [ assumption | fail 1 ])
|| (injection H;
match goal with
| [ |- X = Y -> G ] =>
try clear H; intros; try subst
end)
| [ H : ?F ?X ?U = ?F ?Y ?V |- ?G ] =>
(assert (X = Y); [ assumption
| assert (U = V); [ assumption | fail 1 ] ])
|| (injection H;
match goal with
| [ |- U = V -> X = Y -> G ] =>
try clear H; intros; try subst
end)
| [ H : ?F _ |- _ ] => invert H F
| [ H : ?F _ _ |- _ ] => invert H F
......
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