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Arith.v
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Require Import FJ_tactics.
Require Import List.
Require Import Functors.
Require Import Names.
Require Import FunctionalExtensionality.
Section Arith.
(* ============================================== *)
(* TYPES *)
(* ============================================== *)
(* The Arithmetic Type. *)
Inductive AType (A : Set) : Set :=
| TNat : AType A.
Definition AType_fmap : forall (A B : Set) (f : A -> B),
AType A -> AType B := fun A B _ _ => TNat _.
Global Instance AType_Functor : Functor AType :=
{| fmap := AType_fmap |}.
destruct a; reflexivity.
(* fmap id *)
destruct a; reflexivity.
Defined.
Variable D : Set -> Set.
Context {Fun_D : Functor D}.
Definition DType := DType D.
Context {Sub_AType_D : AType :<: D}.
(* Constructor + Universal Property. *)
Context {WF_Sub_AType_D : WF_Functor _ _ Sub_AType_D _ _}.
Definition tnat' : DType := inject' (TNat _).
Definition tnat : Fix D := proj1_sig tnat'.
Global Instance UP'_tnat :
Universal_Property'_fold tnat := proj2_sig tnat'.
(* Induction Principle for Nat Types. *)
Definition ind_alg_AType
(P : forall d : Fix D, Universal_Property'_fold d -> Prop)
(H : UP'_P P tnat)
(d : AType (sig (UP'_P P))) : sig (UP'_P P) :=
match d with
| TNat => exist _ _ H
end.
Lemma WF_ind_alg_AType (Name : Set)
(P : forall e : Fix D, Universal_Property'_fold e -> Prop)
(H : UP'_P P tnat)
{Sub_AType_D' : AType :<: D} :
(forall a, inj (Sub_Functor := Sub_AType_D) a =
inj (A := Fix D) (Sub_Functor := Sub_AType_D') a) ->
WF_Ind (Name := Name) {| p_algebra := fun H0 => ind_alg_AType P H H0|}.
constructor; intros.
simpl; unfold ind_alg_AType; destruct e; simpl.
unfold tnat; simpl; rewrite wf_functor; simpl; apply f_equal; eauto.
Defined.
(* Type Equality Section. *)
Definition isTNat : Fix D -> bool :=
fun typ =>
match project typ with
| Some TNat => true
| None => false
end.
Definition AType_eq_DType (R : Set) (rec : R -> eq_DTypeR D)
(e : AType R) : eq_DTypeR D :=
match e with
| TNat => fun t => isTNat (proj1_sig t)
end.
Global Instance MAlgebra_eq_DType_Arith T:
FAlgebra eq_DTypeName T (eq_DTypeR D) AType :=
{| f_algebra := AType_eq_DType T|}.
Context {eq_DType_DT : forall T, FAlgebra eq_DTypeName T (eq_DTypeR D) D}.
Context {WF_DType_eq_DT : forall T, @WF_FAlgebra eq_DTypeName T _ _ _
Sub_AType_D (MAlgebra_eq_DType_Arith T) (eq_DType_DT _)}.
Lemma AType_eq_DType_eq_H : UP'_P (eq_DType_eq_P D) tnat.
unfold UP'_P; econstructor.
unfold eq_DType_eq_P; intros.
unfold eq_DType, mfold, tnat, tnat', inject' in H; simpl in H;
repeat rewrite wf_functor in H; simpl in H; unfold in_t in H.
rewrite (wf_algebra (WF_FAlgebra := WF_DType_eq_DT _)) in H; simpl in H.
unfold isTNat in H.
caseEq (project (proj1_sig d2)); rewrite H0 in H;
try discriminate; destruct a.
unfold project in H0.
apply inj_prj in H0.
unfold tnat, tnat'; simpl; rewrite wf_functor; simpl.
unfold AType_fmap.
generalize (f_equal (in_t_UP' _ _ ) H0); intros.
apply (f_equal (@proj1_sig _ _)) in H1;
rewrite in_out_UP'_inverse in H1.
rewrite H1; simpl; rewrite wf_functor; simpl; unfold AType_fmap;
reflexivity.
exact (proj2_sig d2).
Qed.
Global Instance PAlgebra_eq_DType_eq_AType :
PAlgebra eq_DType_eqName (sig (UP'_P (eq_DType_eq_P D))) AType.
Proof.
constructor; unfold Algebra; intros.
eapply (ind_alg_AType (eq_DType_eq_P D) AType_eq_DType_eq_H H).
Defined.
(* ============================================== *)
(* EXPRESSIONS *)
(* ============================================== *)
Inductive Arith (a : Set) : Set :=
| Lit : nat -> Arith a
| Add : a -> a -> Arith a.
Definition Arith_fmap (B C : Set) (f : B -> C) (Aa : Arith B) : Arith C :=
match Aa with
| Lit n => Lit _ n
| Add a b => Add _ (f a) (f b)
end.
Global Instance Arith_Functor : Functor Arith :=
{| fmap := Arith_fmap |}.
destruct a; reflexivity.
(* fmap id *)
destruct a; reflexivity.
Defined.
Variable F : Set -> Set.
Context {Fun_F : Functor F}.
Definition Exp := Exp F.
Context {Sub_Arith_F : Arith :<: F}.
(* Constructor + Universal Property. *)
Context {WF_Sub_Arith_F : WF_Functor _ _ Sub_Arith_F _ _}.
Definition lit' (n : nat) : Exp := inject' (Lit _ n).
Definition lit (n : nat) : Fix F := proj1_sig (lit' n).
Global Instance UP'_lit {n : nat} :
Universal_Property'_fold (lit n) := proj2_sig (lit' n).
Definition add' (m n : Exp) : Exp := inject' (Add _ m n).
Definition add (m n : Fix F)
{UP'_m : Universal_Property'_fold m}
{UP'_n : Universal_Property'_fold n}
: Fix F := proj1_sig (add' (exist _ _ UP'_m) (exist _ _ UP'_n)).
Global Instance UP'_add {m n : Fix F}
{UP'_m : Universal_Property'_fold m}
{UP'_n : Universal_Property'_fold n}
:
Universal_Property'_fold (add m n) :=
proj2_sig (add' (exist _ _ UP'_m) (exist _ _ UP'_n)).
(* Induction Principles for Arith. *)
Definition ind_alg_Arith
(P : forall e : Fix F, Universal_Property'_fold e -> Prop)
(H : forall n, UP'_P P (lit n))
(H0 : forall m n
(IHm : UP'_P P m)
(IHn : UP'_P P n),
UP'_P P (@add m n (proj1_sig IHm) (proj1_sig IHn)))
(e : Arith (sig (UP'_P P)))
:
sig (UP'_P P) :=
match e with
| Lit n => exist _ (lit n) (H n)
| Add m n => exist (UP'_P P) _
(H0 (proj1_sig m) (proj1_sig n) (proj2_sig m) (proj2_sig n))
end.
Definition ind2_alg_Arith
{E E' : Set -> Set}
{Fun_E : Functor E}
{Fun_E' : Functor E'}
{Sub_Arith_E : Arith :<: E}
{Sub_Arith_E' : Arith :<: E'}
(P : forall e : (Fix E) * (Fix E'),
Universal_Property'_fold (fst e) /\ Universal_Property'_fold (snd e) -> Prop)
(H : forall n, UP'_P2 P (inject (Lit _ n), inject (Lit _ n)))
(H0 : forall m n
(IHm : UP'_P2 P m)
(IHn : UP'_P2 P n),
UP'_P2 P (inject (Add _ (exist _ _ (proj1 (proj1_sig IHm)))
(exist _ _ (proj1 (proj1_sig IHn)))),
inject (Add _ (exist _ _ (proj2 (proj1_sig IHm)))
(exist _ _ (proj2 (proj1_sig IHn))))))
(e : Arith (sig (UP'_P2 P)))
:
sig (UP'_P2 P) :=
match e with
| Lit n => exist _ _ (H n)
| Add m n => exist (UP'_P2 P) _
(H0 (proj1_sig m) (proj1_sig n) (proj2_sig m) (proj2_sig n))
end.
(* ============================================== *)
(* TYPING *)
(* ============================================== *)
(* Typing Arithmetic Expressions. *)
Definition Arith_typeof (R : Set) (rec : R -> typeofR D)
(e : Arith R) : typeofR D :=
match e with
| Lit n => Some (inject' (TNat _))
| Add m n => match (rec m), (rec n) with
| Some T1, Some T2 =>
match isTNat (proj1_sig T1), isTNat (proj1_sig T2) with
| true, true => Some (inject' (TNat _))
| _, _ => None
end
| _, _ => None
end
end.
Global Instance MAlgebra_typeof_Arith T:
FAlgebra TypeofName T (typeofR D) Arith :=
{| f_algebra := Arith_typeof T|}.
(* ============================================== *)
(* VALUES *)
(* ============================================== *)
(* Arithmetic Values. *)
Inductive NatValue (A : Set) : Set :=
| VI : nat -> NatValue A.
Definition VI_fmap : forall (A B : Set) (f : A -> B),
NatValue A -> NatValue B :=
fun A B _ e => match e with
| VI n => VI _ n
end.
Global Instance VI_Functor : Functor NatValue :=
{| fmap := VI_fmap |}.
destruct a; reflexivity.
destruct a; reflexivity.
Defined.
Variable V : Set -> Set.
Context {Fun_V : Functor V}.
Definition Value := Value V.
Context {Sub_NatValue_V : NatValue :<: V}.
(* Constructor + Universal Property. *)
Context {WF_Sub_NatValue_F : WF_Functor _ _ Sub_NatValue_V _ _}.
Definition vi' (n : nat) : Value := inject' (VI _ n).
Definition vi (n : nat) : Fix V := proj1_sig (vi' n).
Global Instance UP'_vi {n : nat} :
Universal_Property'_fold (vi n) := proj2_sig (vi' n).
(* Constructor Testing for Arithmetic Values. *)
Definition isVI : Fix V -> option nat :=
fun exp =>
match project exp with
| Some (VI n) => Some n
| None => None
end.
Context {Sub_StuckValue_V : StuckValue :<: V}.
Definition stuck' : nat -> Value := stuck' _.
Definition stuck : nat -> Fix V := stuck _.
(* ============================================== *)
(* EVALUATION *)
(* ============================================== *)
Context {Sub_BotValue_V : BotValue :<: V}.
(* Evaluation Algebra for Arithemetic Expressions. *)
Definition Arith_eval (R : Set) (rec : R -> evalR V)
(e : Arith R) : evalR V :=
match e with
| Lit n => (fun _ => vi' n)
| Add m n => (fun env =>
let m' := (rec m env) in
let n' := (rec n env) in
match isVI (proj1_sig m'), isVI (proj1_sig n') with
| Some m', Some n' => vi' (m' + n')
| _, _ =>
if @isBot _ Fun_V Sub_BotValue_V (proj1_sig m') then @bot' _ Fun_V Sub_BotValue_V else
if @isBot _ Fun_V Sub_BotValue_V (proj1_sig n') then @bot' _ Fun_V Sub_BotValue_V else
stuck' 4
end)
end.
Global Instance MAlgebra_eval_Arith T :
FAlgebra EvalName T (evalR V) Arith :=
{| f_algebra := Arith_eval T |}.
(* ============================================== *)
(* PRETTY PRINTING *)
(* ============================================== *)
(* Pretty Printing Functions*)
Require Import Ascii.
Require Import String.
Global Instance MAlgebra_DTypePrint_AType T:
FAlgebra DTypePrintName T DTypePrintR AType :=
{| f_algebra := fun rec e => append "tnat" "" |}.
Global Instance MAlgebra_ExpPrint_Arith T :
FAlgebra ExpPrintName T (ExpPrintR) Arith :=
{| f_algebra := fun rec e => match e with
| Lit n => fun i => String (ascii_of_nat (n + 48)) EmptyString
| Add m n => fun i => append "(" ((rec m i) ++ " + " ++ (rec n i) ++ ")")
end |}.
Global Instance MAlgebra_ValuePrint_AType T :
FAlgebra ValuePrintName T ValuePrintR NatValue :=
{| f_algebra := fun rec e =>
match e with
| VI n => String (ascii_of_nat (n + 48)) EmptyString
end |}.
(* ============================================== *)
(* TYPE SOUNDNESS *)
(* ============================================== *)
Context {eval_F : FAlgebra EvalName Exp (evalR V) F}.
Context {WF_eval_F : @WF_FAlgebra EvalName _ _ Arith F
Sub_Arith_F (MAlgebra_eval_Arith _) eval_F}.
(* Continuity of Evaluation. *)
Context {WF_SubBotValue_V : WF_Functor BotValue V Sub_BotValue_V Bot_Functor Fun_V}.
Context {SV : (SubValue_i V -> Prop) -> SubValue_i V -> Prop}.
Context {Sub_SV_refl_SV : Sub_iFunctor (SubValue_refl V) SV}.
(* Lit case. *)
Ltac WF_FAlg_rewrite := repeat rewrite wf_functor; simpl;
repeat rewrite out_in_fmap; simpl;
repeat rewrite wf_functor; simpl;
repeat rewrite wf_algebra; simpl.
Lemma eval_continuous_Exp_H : forall n,
UP'_P (eval_continuous_Exp_P V F SV) (lit n).
unfold eval_continuous_Exp_P; intros; econstructor; intros.
unfold beval, mfold, lit; simpl; unfold inject.
WF_FAlg_rewrite.
apply inject_i.
constructor.
reflexivity.
Qed.
(* Add case. *)
Context {Dis_VI_Bot : Distinct_Sub_Functor _ Sub_NatValue_V Sub_BotValue_V}.
(* Inversion principles for natural number SubValues. *)
Definition SV_invertVI_P (i : SubValue_i V) :=
forall n, proj1_sig (sv_a _ i) = vi n ->
proj1_sig (sv_b _ i) = vi n.
Inductive SV_invertVI_Name := ece_invertvi_name.
Context {SV_invertVI_SV :
iPAlgebra SV_invertVI_Name SV_invertVI_P SV}.
Global Instance SV_invertVI_refl :
iPAlgebra SV_invertVI_Name SV_invertVI_P (SubValue_refl V).
econstructor; intros.
unfold iAlgebra; intros; unfold SV_invertVI_P.
inversion H; subst; simpl; congruence.
Defined.
Lemma SV_invertVI_default : forall V'
(Fun_V' : Functor V')
(SV' : (SubValue_i V -> Prop) -> SubValue_i V -> Prop)
(sub_V'_V : V' :<: V)
(WF_V' : WF_Functor V' V sub_V'_V Fun_V' Fun_V),
(forall (i : SubValue_i V) (H : SV' SV_invertVI_P i),
exists v', proj1_sig (sv_a _ i) = inject v') ->
Distinct_Sub_Functor _ Sub_NatValue_V sub_V'_V ->
iPAlgebra SV_invertVI_Name SV_invertVI_P SV'.
econstructor; intros.
unfold iAlgebra; intros; unfold SV_invertVI_P.
destruct (H _ H1) as [v' eq_v'].
intros; rewrite eq_v' in H2.
elimtype False.
unfold vi, inject, vi', inject' in H2; simpl in H2.
apply sym_eq in H2.
apply (inject_discriminate H0 _ _ H2).
Defined.
Global Instance SV_invertVI_Bot :
iPAlgebra SV_invertVI_Name SV_invertVI_P (SubValue_Bot V).
econstructor; intros.
unfold iAlgebra; intros; unfold SV_invertVI_P.
inversion H; subst; simpl; intros.
elimtype False.
rewrite H0 in H1.
unfold vi, inject, vi', inject' in H1; simpl in H1.
repeat rewrite out_in_inverse, wf_functor in H1; simpl in H1.
eapply (inject_discriminate Dis_VI_Bot); unfold inject; simpl; eauto.
Defined.
Context {iFun_F : iFunctor SV}.
Definition SV_invertVI := ifold_ SV _ (ip_algebra (iPAlgebra := SV_invertVI_SV)).
Definition SV_invertVI'_P (i : SubValue_i V) :=
forall n, proj1_sig (sv_b _ i) = vi n ->
proj1_sig (sv_a _ i) = vi n \/ proj1_sig (sv_a _ i) = bot _.
Inductive SV_invertVI'_Name := ece_invertvi'_name.
Context {SV_invertVI'_SV :
iPAlgebra SV_invertVI'_Name SV_invertVI'_P SV}.
Global Instance SV_invertVI'_refl :
iPAlgebra SV_invertVI'_Name SV_invertVI'_P (SubValue_refl V).
econstructor; intros.
unfold iAlgebra; intros; unfold SV_invertVI'_P.
inversion H; subst; simpl; eauto.
intros.
left; congruence.
Defined.
Global Instance SV_invertVI'_Bot :
iPAlgebra SV_invertVI'_Name SV_invertVI'_P (SubValue_Bot V).
econstructor; intros.
unfold iAlgebra; intros; unfold SV_invertVI'_P.
inversion H; subst; simpl; eauto.
Defined.
Definition SV_invertVI' := ifold_ SV _ (ip_algebra (iPAlgebra := SV_invertVI'_SV)).
(* End Inversion principles for SubValue.*)
Context {SV_invertBot_SV :
iPAlgebra SV_invertBot_Name (SV_invertBot_P V) SV}.
Context {Sub_SV_Bot_SV : Sub_iFunctor (SubValue_Bot V) SV}.
Lemma project_bot_vi : forall n,
project (F := V) (G := BotValue) (vi n) = None.
Proof.
intros; unfold project, vi; simpl; rewrite out_in_fmap.
repeat rewrite wf_functor; simpl; unfold VI_fmap.
caseEq (prj (sub_F := BotValue) (inj (sub_G := V) (VI (sig (@Universal_Property'_fold V _)) n))).
apply inj_prj in H; elimtype False; eapply (inject_discriminate Dis_VI_Bot);
unfold inject; repeat apply f_equal; apply H.
auto.
Qed.
Lemma project_vi_bot : project (F := V) (G := NatValue) (bot _) = None.
Proof.
intros; unfold project, bot; simpl; rewrite out_in_fmap.
repeat rewrite wf_functor; simpl; unfold Bot_fmap.
caseEq (prj (sub_F := NatValue) (inj (sub_G := V) (Bot (sig (@Universal_Property'_fold V _))))).
apply inj_prj in H; elimtype False; eapply (inject_discriminate Dis_VI_Bot);
unfold inject; repeat apply f_equal; apply sym_eq in H; apply H.
auto.
Qed.
Lemma project_vi_vi : forall n,
project (F := V) (G := NatValue) (vi n) = Some (VI _ n).
Proof.
intros; unfold project, vi, inject; simpl; rewrite out_in_fmap.
repeat rewrite wf_functor; simpl; unfold VI_fmap.
rewrite prj_inj; reflexivity.
Qed.
Lemma eval_continuous_Exp_H0 : forall
(m n : Fix F)
(IHm : UP'_P (eval_continuous_Exp_P V F SV) m)
(IHn : UP'_P (eval_continuous_Exp_P V F SV) n),
UP'_P (eval_continuous_Exp_P V F SV) (@add m n (proj1_sig IHm) (proj1_sig IHn)).
unfold eval_continuous_Exp_P; intros.
destruct IHm as [m_UP' IHm].
destruct IHn as [n_UP' IHn].
econstructor; intros; eauto with typeclass_instances.
unfold beval, mfold, add; simpl.
unfold inject; simpl; repeat rewrite out_in_fmap; simpl;
repeat rewrite wf_functor; simpl.
repeat rewrite (wf_algebra (WF_FAlgebra := WF_eval_F)); simpl.
repeat erewrite bF_UP_in_out.
caseEq (project (G := NatValue)
(proj1_sig (boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma))).
unfold isVI at 1, evalR, Names.Exp; rewrite H2.
destruct n1.
generalize (H (exist _ m m_UP') _ _ _ H0 H1); simpl; intros.
generalize (inj_prj _ _ H2); rename H2 into H2'; intros.
assert (proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma) = vi n1) as Eval_m.
unfold vi, vi', inject'; rewrite <- H2; rewrite in_out_UP'_inverse; eauto.
exact (proj2_sig _).
clear H2; rename H3 into SubV_m.
unfold isVI; unfold eval, mfold in SubV_m.
caseEq (project (G := NatValue) (proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma))).
destruct n2.
generalize (H (exist _ n n_UP') _ _ _ H0 H1); simpl; intros.
generalize (inj_prj _ _ H2); rename H2 into H3'; intros.
assert (proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma) = vi n2) as Eval_n.
unfold vi, vi', inject'; rewrite <- H2; rewrite in_out_UP'_inverse; eauto.
exact (proj2_sig _).
clear H2; rename H3 into SubV_n.
unfold isVI; unfold eval, mfold in SubV_n.
generalize (SV_invertVI _ SubV_m _ Eval_m).
generalize (SV_invertVI _ SubV_n _ Eval_n).
simpl; unfold beval at 1; unfold beval at 1; unfold evalR, Names.Exp; intros.
rewrite H3, H2.
unfold project, vi, vi'; simpl; repeat rewrite out_in_fmap;
repeat rewrite wf_functor; repeat rewrite prj_inj;
repeat rewrite wf_functor; simpl.
apply (inject_i (subGF := Sub_SV_refl_SV)); constructor; eauto.
unfold isBot; rewrite Eval_m.
caseEq (project (G := BotValue) (vi n1)).
destruct b; generalize (inj_prj _ _ H3); intro.
assert (vi n1 = bot _) by
(unfold vi, vi', bot, bot', inject' at -1; rewrite <- H4;
rewrite in_out_UP'_inverse; eauto with typeclass_instances).
unfold vi, bot in H5.
elimtype False; eapply (inject_discriminate Dis_VI_Bot _ _ H5).
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma))).
destruct b.
apply inject_i; constructor; reflexivity.
generalize (H (exist _ n n_UP') _ _ _ H0 H1) as SubV_n; simpl; intros.
caseEq (project (G := NatValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
destruct n2.
caseEq (project (G := NatValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma'))).
destruct n3.
generalize (inj_prj _ _ H5); rename H5 into H5'; intros.
assert (proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma') = vi n2) as Eval_m' by
(unfold vi, vi', inject'; rewrite <- H5;
rewrite in_out_UP'_inverse; unfold eval, mfold; eauto;
exact (proj2_sig _)).
unfold beval in SubV_m.
generalize (inj_prj _ _ H6); rename H6 into H6'; intros.
assert (proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma') = vi n3) as Eval_n'.
unfold vi, vi', inject'.
(unfold vi, vi', inject'; rewrite <- H6;
rewrite in_out_UP'_inverse; unfold eval, mfold; eauto;
exact (proj2_sig _)).
destruct (SV_invertVI' _ SubV_n _ Eval_n') as [n_eq_vi | n_eq_bot];
simpl in *|-.
unfold beval, mfold, evalR, Names.Exp in n_eq_vi; rewrite n_eq_vi in H2.
unfold vi, project, inject in H2; simpl in H2; rewrite
out_in_fmap in H2.
rewrite fmap_fusion in H2; rewrite wf_functor in H2; simpl in H2.
rewrite (prj_inj _ ) in H2; discriminate.
unfold beval, mfold, evalR, Names.Exp in n_eq_bot; rewrite n_eq_bot in H4.
unfold bot, project, inject in H4; simpl in H4; rewrite out_in_fmap in H4.
rewrite fmap_fusion, wf_functor in H4; simpl in H4.
rewrite (prj_inj _ ) in H4; discriminate.
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
destruct b.
generalize (inj_prj _ _ H7); rename H7 into H7'; intros.
assert (proj1_sig
(beval _ _ n0 (exist Universal_Property'_fold m m_UP') gamma') = bot _ ) as Eval_m' by
(apply (f_equal (in_t_UP' _ _)) in H7; apply (f_equal (@proj1_sig _ _)) in H7;
rewrite in_out_UP'_inverse in H7; [apply H7 | exact (proj2_sig _)]).
generalize (SV_invertBot _ SV _ _ SubV_m Eval_m'); simpl; intro H8;
unfold beval, mfold, evalR, Names.Exp in H8; rewrite H8 in Eval_m.
elimtype False; eapply (inject_discriminate Dis_VI_Bot (VI _ n1)); eauto.
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma'))).
destruct b.
generalize (inj_prj _ _ H8); rename H8 into H8'; intros.
assert (proj1_sig
(beval _ _ n0 (exist Universal_Property'_fold n n_UP') gamma') = bot _ ) as Eval_n' by
(apply (f_equal (in_t_UP' _ _)) in H8; apply (f_equal (@proj1_sig _ _)) in H8;
rewrite in_out_UP'_inverse in H8; [apply H8 | exact (proj2_sig _)]).
generalize (SV_invertBot _ SV _ _ SubV_n Eval_n'); simpl; intro H9;
unfold beval, mfold, evalR, Names.Exp in H9. rewrite H9 in H4.
unfold project, bot, bot' in H4; simpl in H4; rewrite out_in_fmap in H4;
simpl in H4; repeat rewrite wf_functor in H4; simpl in H4;
rewrite prj_inj in H4; discriminate.
apply (inject_i (subGF := Sub_SV_refl_SV)); constructor; eauto.
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
destruct b.
unfold project in H6.
apply inj_prj in H6; apply (f_equal (in_t_UP' _ _)) in H6;
apply (f_equal (@proj1_sig _ _)) in H6.
rewrite in_out_UP'_inverse in H6; simpl.
generalize (SV_invertBot _ SV _ _ SubV_m H6); simpl; intro.
unfold beval, evalR, Names.Exp in H7; rewrite H7 in Eval_m.
elimtype False; eapply (inject_discriminate Dis_VI_Bot (VI _ n1)); eauto.
exact (proj2_sig _).
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma'))).
destruct b.
unfold project in H7.
apply inj_prj in H7; apply (f_equal (in_t_UP' _ _)) in H7;
apply (f_equal (@proj1_sig _ _)) in H7.
rewrite in_out_UP'_inverse in H7; simpl.
generalize (SV_invertBot _ SV _ _ SubV_n H7); simpl; intro.
unfold beval, evalR, Names.Exp in H8; rewrite H8 in H4.
unfold project, bot, bot' in H4; simpl in H4; rewrite out_in_fmap in H4;
simpl in H4; repeat rewrite wf_functor in H4; simpl in H4;
rewrite prj_inj in H4; discriminate.
exact (proj2_sig _).
apply (inject_i (subGF := Sub_SV_refl_SV)); constructor; eauto.
unfold isVI, evalR, Names.Exp; rewrite H2.
unfold isBot.
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma))).
destruct b.
apply inj_prj in H3; apply (f_equal (in_t_UP' _ _)) in H3;
apply (f_equal (@proj1_sig _ _)) in H3.
assert (proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma) = bot _) as Eval_m.
unfold bot, bot', inject'; rewrite <- H3; rewrite in_out_UP'_inverse; eauto.
exact (proj2_sig _).
apply (inject_i (subGF := Sub_SV_Bot_SV)); constructor; eauto.
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma))).
destruct b.
apply inj_prj in H4; apply (f_equal (in_t_UP' _ _)) in H4;
apply (f_equal (@proj1_sig _ _)) in H4.
assert (proj1_sig
(boundedFix_UP m0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma) = bot _) as Eval_n.
unfold bot, bot', inject'; rewrite <- H4; rewrite in_out_UP'_inverse; eauto.
exact (proj2_sig _).
apply (inject_i (subGF := Sub_SV_Bot_SV)); constructor; eauto.
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma))).
rename H5 into Eval_n.
unfold isVI; unfold eval, mfold in Eval_n.
apply inj_prj in Eval_n; apply (f_equal (in_t_UP' _ _)) in Eval_n;
apply (f_equal (@proj1_sig _ _)) in Eval_n.
rewrite in_out_UP'_inverse in Eval_n.
caseEq (project (G := NatValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
generalize (H (exist _ m m_UP') _ _ _ H0 H1) as SubV_m; intros.
destruct n1.
apply inj_prj in H5; apply (f_equal (in_t_UP' _ _)) in H5;
apply (f_equal (@proj1_sig _ _)) in H5;
rewrite in_out_UP'_inverse in H5;
unfold beval, evalR, Names.Exp in SubV_m, H5.
destruct (SV_invertVI' _ SubV_m _ H5); simpl in H6.
rewrite H6 in H2; unfold project, vi, vi' in H2; simpl in H2.
rewrite out_in_fmap in H2; repeat rewrite wf_functor in H2.
rewrite prj_inj in H2; discriminate.
rewrite H6 in H3; unfold project, bot, bot' in H3; simpl in H3.
rewrite out_in_fmap in H3; repeat rewrite wf_functor in H3.
rewrite prj_inj in H3; discriminate.
exact (proj2_sig _).
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
unfold evalR, Names.Exp in Eval_n.
destruct b0.
apply inj_prj in H6; apply (f_equal (in_t_UP' _ _)) in H6;
apply (f_equal (@proj1_sig _ _)) in H6;
rewrite in_out_UP'_inverse in H6.
generalize (H (exist _ m m_UP') _ _ _ H0 H1) as SubV_m; intros.
unfold beval, evalR, Names.Exp in SubV_m, H6.
generalize (SV_invertBot _ _ _ _ SubV_m H6); simpl;
intros; rewrite H7 in H3.
unfold project, bot, bot' in H3; simpl in H3.
rewrite out_in_fmap in H3; repeat rewrite wf_functor in H3.
rewrite prj_inj in H3; discriminate.
exact (proj2_sig _).
caseEq (project (G := BotValue) (proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma'))).
destruct b0.
apply inj_prj in H7; apply (f_equal (in_t_UP' _ _)) in H7;
apply (f_equal (@proj1_sig _ _)) in H7;
rewrite in_out_UP'_inverse in H7.
generalize (H (exist _ n n_UP') _ _ _ H0 H1) as SubV_n; intros.
unfold beval, evalR, Names.Exp in SubV_n, H7.
generalize (SV_invertBot _ _ _ _ SubV_n H7); simpl;
intros; rewrite H8 in H4.
unfold project, bot, bot' in H4; simpl in H4.
rewrite out_in_fmap in H4; repeat rewrite wf_functor in H4.
rewrite prj_inj in H4; discriminate.
exact (proj2_sig _).
apply (inject_i (subGF := Sub_SV_refl_SV)); constructor; eauto.
exact (proj2_sig _).
caseEq (project (G := NatValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
destruct n1.
generalize (H (exist _ m m_UP') _ _ _ H0 H1) as SubV_m; intros.
apply inj_prj in H6; apply (f_equal (in_t_UP' _ _)) in H6;
apply (f_equal (@proj1_sig _ _)) in H6;
rewrite in_out_UP'_inverse in H6;
unfold beval, evalR, Names.Exp in SubV_m, H6.
destruct (SV_invertVI' _ SubV_m _ H6); simpl in H7.
rewrite H7 in H2; unfold project, vi, vi' in H2; simpl in H2.
rewrite out_in_fmap in H2; repeat rewrite wf_functor in H2.
rewrite prj_inj in H2; discriminate.
rewrite H7 in H3; unfold project, bot, bot' in H3; simpl in H3.
rewrite out_in_fmap in H3; repeat rewrite wf_functor in H3.
rewrite prj_inj in H3; discriminate.
exact (proj2_sig _).
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold m m_UP') gamma'))).
destruct b.
generalize (H (exist _ m m_UP') _ _ _ H0 H1) as SubV_m; intros.
apply inj_prj in H7; apply (f_equal (in_t_UP' _ _)) in H7;
apply (f_equal (@proj1_sig _ _)) in H7;
rewrite in_out_UP'_inverse in H7;
unfold beval, evalR, Names.Exp in SubV_m, H7.
generalize (SV_invertBot _ _ _ _ SubV_m H7); simpl;
intros.
rewrite H8 in H3; unfold project, bot, bot' in H3; simpl in H3.
rewrite out_in_fmap in H3; repeat rewrite wf_functor in H3.
rewrite prj_inj in H3; discriminate.
exact (proj2_sig _).
caseEq (project (G := BotValue)
(proj1_sig
(boundedFix_UP n0 f_algebra
(fun _ : Env (Names.Value V) => bot' V)
(exist Universal_Property'_fold n n_UP') gamma'))).
destruct b.
generalize (H (exist _ n n_UP') _ _ _ H0 H1) as SubV_n; intros.
apply inj_prj in H8; apply (f_equal (in_t_UP' _ _)) in H8;
apply (f_equal (@proj1_sig _ _)) in H8;
rewrite in_out_UP'_inverse in H8;
unfold beval, evalR, Names.Exp in SubV_n, H8.
generalize (SV_invertBot _ _ _ _ SubV_n H8); simpl;
intros.
rewrite H9 in H4; unfold project, bot, bot' in H4; simpl in H4.
rewrite out_in_fmap in H4; repeat rewrite wf_functor in H4.
rewrite prj_inj in H4; discriminate.
exact (proj2_sig _).
apply (inject_i (subGF := Sub_SV_refl_SV)); constructor; eauto.
Qed.
Lemma project_bot_bot : project (F := V) (G := BotValue) (bot _) = Some (Bot _).
Proof.
intros; unfold project, bot; simpl; rewrite out_in_fmap.
repeat rewrite wf_functor; simpl; unfold Bot_fmap.
rewrite prj_inj; reflexivity.
Qed.
Global Instance Arith_eval_continuous_Exp :
PAlgebra EC_ExpName (sig (UP'_P (eval_continuous_Exp_P V F SV))) Arith.
Proof.
constructor; unfold Algebra; intros.
eapply ind_alg_Arith.
apply eval_continuous_Exp_H.
apply eval_continuous_Exp_H0.
assumption.
Defined.
Lemma WF_ind_alg_Arith (Name : Set)
(P : forall e : Fix F, Universal_Property'_fold e -> Prop)
(H : forall n, UP'_P P (lit n))
(H0 : forall m n
(IHm : UP'_P P m)
(IHn : UP'_P P n),
UP'_P P (@add m n (proj1_sig IHm) (proj1_sig IHn)))
{Sub_Arith_F' : Arith :<: F} :
(forall a, inj (Sub_Functor := Sub_Arith_F) a =
inj (A := (Fix F)) (Sub_Functor := Sub_Arith_F') a) ->
WF_Ind (Name := Name) {| p_algebra := ind_alg_Arith P H H0|}.
constructor; intros.
simpl; unfold ind_alg_Arith; destruct e; simpl.
unfold lit; simpl; rewrite wf_functor; simpl; apply f_equal; eauto.
unfold add; simpl; rewrite wf_functor; simpl; apply f_equal; eauto.
Defined.
Context {eval_continuous_Exp_E : PAlgebra EC_ExpName
(sig (UP'_P (eval_continuous_Exp_P V F SV))) F}.
Context {WF_Ind_EC_Exp : WF_Ind eval_continuous_Exp_E}.
(* ============================================== *)
(* WELL-FORMED NAT VALUES *)
(* ============================================== *)
Variable WFV : (WFValue_i D V -> Prop) -> WFValue_i D V -> Prop.
Variable funWFV : iFunctor WFV.
(** Natrual Numbers are well-formed **)
Inductive WFValue_VI (WFV : WFValue_i D V -> Prop) : WFValue_i D V -> Prop :=
| WFV_VI : forall n v T,
proj1_sig v = vi n ->
proj1_sig T = tnat ->
WFValue_VI WFV (mk_WFValue_i D V v T).
Definition ind_alg_WFV_VI (P : WFValue_i D V -> Prop)
(H : forall n v T veq Teq, P (mk_WFValue_i _ _ v T))
i (e : WFValue_VI P i) : P i :=
match e in WFValue_VI _ i return P i with
| WFV_VI n v T veq Teq => H n v T veq Teq
end.
Definition WFV_VI_ifmap (A B : WFValue_i D V -> Prop) i (f : forall i, A i -> B i)
(WFV_a : WFValue_VI A i) : WFValue_VI B i :=
match WFV_a in (WFValue_VI _ s) return (WFValue_VI B s)
with
| WFV_VI n v T veq Teq => WFV_VI B n v T veq Teq
end.
Global Instance iFun_WFV_VI : iFunctor WFValue_VI.
constructor 1 with (ifmap := WFV_VI_ifmap).
destruct a; simpl; intros; reflexivity.
destruct a; simpl; intros; reflexivity.
Defined.
Variable Sub_WFV_VI_WFV : Sub_iFunctor WFValue_VI WFV.
Global Instance WFV_proj1_a_VI :
iPAlgebra WFV_proj1_a_Name (WFV_proj1_a_P D V WFV) WFValue_VI.
econstructor; intros.
unfold iAlgebra; intros; unfold WFV_proj1_a_P.
inversion H; subst; simpl; intros.
apply (inject_i (subGF := Sub_WFV_VI_WFV)); econstructor; simpl; eauto.
rewrite H3; eauto.
Defined.
Global Instance WFV_proj1_b_VI :
iPAlgebra WFV_proj1_b_Name (WFV_proj1_b_P D V WFV) WFValue_VI.
econstructor; intros.
unfold iAlgebra; intros; unfold WFV_proj1_b_P.
inversion H; subst; simpl; intros.
apply (inject_i (subGF := Sub_WFV_VI_WFV)); econstructor; simpl; eauto.
rewrite H3; eauto.
Defined.
(* Inversion principles for Well-formed natural numbers. *)
Definition WF_invertVI_P (i : WFValue_i D V) :=
proj1_sig (wfv_b _ _ i) = tnat ->
WFValue_VI (iFix WFV) i \/ (proj1_sig (wfv_a D V i) = bot V).
Inductive WF_invertVI_Name := wfv_invertvi_name.
Context {WF_invertVI_WFV :
iPAlgebra WF_invertVI_Name WF_invertVI_P WFV}.
Global Instance WF_invertVI_VI :
iPAlgebra WF_invertVI_Name WF_invertVI_P WFValue_VI.
econstructor; intros.
unfold iAlgebra; intros; unfold WF_invertVI_P.
inversion H; subst; simpl; intros.
left; econstructor; eassumption.
Defined.
Global Instance WF_invertVI_Bot :
iPAlgebra WF_invertVI_Name WF_invertVI_P (WFValue_Bot _ _).
econstructor; intros.
unfold iAlgebra; intros; unfold WF_invertVI_P.
inversion H; subst; simpl; intros.
inversion H; subst; rewrite H3; right; reflexivity.
Defined.
Definition WF_invertVI := ifold_ WFV _ (ip_algebra (iPAlgebra := WF_invertVI_WFV)).
Context {WFV_proj1_a_WFV :
iPAlgebra WFV_proj1_a_Name (WFV_proj1_a_P D V WFV) WFV}.
Context {WFV_proj1_b_WFV :
iPAlgebra WFV_proj1_b_Name (WFV_proj1_b_P D V WFV) WFV}.
Lemma Arith_eval_Soundness_H
(typeof_R eval_R : Set) typeof_rec eval_rec
{eval_F' : FAlgebra EvalName eval_R (evalR V) F}
{WF_eval_F' : @WF_FAlgebra EvalName _ _ Arith F
Sub_Arith_F (MAlgebra_eval_Arith _) (eval_F')} :
forall n : nat,
forall gamma'' : Env (Names.Value V),
forall T : Names.DType D,
Arith_typeof typeof_R typeof_rec (Lit _ n) = Some T ->
WFValueC D V WFV (Arith_eval eval_R eval_rec (Lit _ n) gamma'') T.
intros n gamma'' T H4; intros.
apply (inject_i (subGF := Sub_WFV_VI_WFV)); econstructor; eauto.
simpl.
unfold vi, vi', inject; simpl; eauto.
unfold typeof, mfold, lit in H4; simpl in H4.
injection H4; intros; subst.
reflexivity.
Defined.
Lemma Arith_eval_Soundness_H0
(typeof_R eval_R : Set) typeof_rec eval_rec
{eval_F' : FAlgebra EvalName eval_R (evalR V) F}
{WF_eval_F' : @WF_FAlgebra EvalName _ _ Arith F
Sub_Arith_F (MAlgebra_eval_Arith _) (eval_F')} :
forall (a b : typeof_R) (a' b' : eval_R),
forall gamma'' : Env (Names.Value V),
(forall T : Names.DType D,
typeof_rec a = Some T ->
WFValueC D V WFV (eval_rec a' gamma'') T) ->
(forall T : Names.DType D,
typeof_rec b = Some T ->
WFValueC D V WFV (eval_rec b' gamma'') T) ->
forall T : Names.DType D,
Arith_typeof typeof_R typeof_rec (Add _ a b) = Some T ->
WFValueC D V WFV (Arith_eval eval_R eval_rec (Add _ a' b') gamma'') T.
simpl; intros a b a' b' gamma'' IH_a IH_b T H4.
caseEq (typeof_rec a); intros; rename H into typeof_a;
unfold typeofR in typeof_a, H4; rewrite typeof_a in H4;
try discriminate.
caseEq (typeof_rec b); intros; rename H into typeof_b;
unfold typeofR in typeof_b, H4; rewrite typeof_b in H4;
try discriminate.
caseEq (isTNat (proj1_sig d)); intros; rename H into d_eq; rewrite
d_eq in H4; try discriminate.
caseEq (isTNat (proj1_sig d0)); intros; rename H into d0_eq; rewrite
d0_eq in H4; try discriminate.
injection H4; intros; subst; clear H4.
unfold isTNat in d_eq, d0_eq.
caseEq (project (proj1_sig d)); intros; rewrite H in d_eq;
try discriminate; clear d_eq; rename H into d_eq; destruct a0.
caseEq (project (proj1_sig d0)); intros; rewrite H in d0_eq;
try discriminate; clear d0_eq; rename H into d0_eq; destruct a0.
apply project_inject in d_eq; apply project_inject in d0_eq;
eauto with typeclass_instances.
generalize (IH_a _ typeof_a) as WF_a;
generalize (IH_b _ typeof_b) as WF_b; intros.
unfold WFValueC in *|-*.
destruct (WF_invertVI _ WF_a d_eq) as [beval_a' | beval_a'];
inversion beval_a'; subst.
rewrite H1; unfold isVI, project, vi, vi', inject'; simpl;
rewrite out_in_fmap; repeat rewrite wf_functor; simpl; rewrite prj_inj.
destruct (WF_invertVI _ WF_b d0_eq) as [beval_b' | beval_b'];
inversion beval_b'; subst.
rewrite H3; unfold isVI, project, vi, vi', inject'; simpl;
rewrite out_in_fmap; repeat rewrite wf_functor; simpl; rewrite prj_inj.
apply (inject_i (subGF := Sub_WFV_VI_WFV)); econstructor; eauto.
simpl; rewrite wf_functor; simpl; eauto.
unfold vi, vi', inject'; simpl; eauto;
rewrite wf_functor; simpl; reflexivity.
rewrite H0; unfold bot, isVI, project, inject, inject'; simpl;
rewrite out_in_fmap; repeat rewrite wf_functor; simpl; unfold Bot_fmap.
caseEq (prj (Sub_Functor := Sub_NatValue_V) (A:= (sig (@Universal_Property'_fold V _)))
(inj (Bot (sig Universal_Property'_fold)))).
elimtype False; eapply (inject_discriminate Dis_VI_Bot n0).
unfold inject; simpl; apply inj_prj in H; erewrite <- H; reflexivity.
caseEq (isBot V (in_t (inj (VI (Fix V) n)))).
simpl in beval_b'.