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ILTranslation.v
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ILTranslation.v
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(* Theorems about translation (compilation) from
the Contract DSL to payoff expression language *)
Require Import Tactics.
Require Import ILSyntax.
Require Import ILSemantics.
Require Import Denotational.
Require Import Syntax.
Require Import Utils.
Require Import Horizon.
Require Import Misc.
Require Import Reduction.
Require Import Typing.
Require Import Templates.
Require Import List.
Require Import ZArith.
Require Import Arith.
Require Import FunctionalExtensionality.
Require Import Coq.Program.Equality.
Import ListNotations.
Local Open Scope Z.
Definition tsmartPlus e1 e2 :=
match e1 with
| ILTexpr (Tnum 0) => e2
| _ => match e2 with
| ILTexpr (Tnum 0) => e1
| _ => ILTplus e1 e2
end
end.
Definition tsmartPlus' e1 e2 :=
match e1, e2 with
| ILTexpr (Tnum n1), ILTexpr (Tnum n2) => ILTexpr (Tnum (n1 + n2))
| _,_ => ILTplus e1 e2
end.
Lemma tsmartPlus_sound e1 e2 tenv:
ILTexprSem (tsmartPlus e1 e2) tenv = ILTexprSem (ILTplus e1 e2) tenv.
Proof.
destruct e1; destruct e2; simpl; try destruct e; try destruct n; try destruct e0; try destruct n0; try destruct n;
simpl; try rewrite plus_0_r; try reflexivity.
Qed.
Lemma tsmartPlus_sound' e1 e2 tenv:
ILTexprSem (tsmartPlus' e1 e2) tenv = ILTexprSem (ILTplus e1 e2) tenv.
Proof.
destruct e1; destruct e2; simpl; try destruct e; try destruct e0; try destruct n0;
simpl; try rewrite plus_0_r; try reflexivity.
Qed.
Fixpoint fromExp (t0 : ILTExprZ) (e : Exp) :=
match e with
| OpE Syntax.Add [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILAdd v1 v2)))
| OpE Sub [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILSub v1 v2)))
| OpE Mult [a1;a2] => (fromExp t0 a1) >>= fun v1 =>
(fromExp t0 a2) >>= fun v2 => pure (ILBinExpr ILMult v1 v2)
| OpE Div [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILDiv v1 v2)))
| OpE And [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILAnd v1 v2)))
| OpE Or [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILOr v1 v2)))
| OpE Less [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILLess v1 v2)))
| OpE Leq [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILLeq v1 v2)))
| OpE Equal [a1;a2] => (fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILBinExpr ILEqual v1 v2)))
| OpE Not [a] => fromExp t0 a >>=
fun v1 => pure (ILUnExpr ILNot v1)
| OpE Neg [a] => fromExp t0 a >>=
fun v1 => pure (ILUnExpr ILNeg v1)
| OpE Cond [b;a1;a2] => (fromExp t0 b) >>= (fun b_il =>
(fromExp t0 a1) >>= (fun v1 =>
(fromExp t0 a2) >>= (fun v2 =>
pure (ILIf b_il v1 v2))))
| OpE (RLit v) [] => Some (ILFloat v)
| OpE (BLit v) [] => Some (ILBool v)
| OpE _ _ => None
| Obs lab t => Some (ILModel lab (ILTplusZ (ILTnumZ t) t0))
| VarE n => None
| Acc _ _ _ => None
end.
Fixpoint fromContr (c : Contr) (t0 : ILTExpr) : option ILExpr:=
match c with
| Scale e c => fromExp (ILTexprZ t0) e
>>= fun v => (fromContr c t0)
>>= fun v' => pure (ILBinExpr ILMult v v')
| Translate t' c => fromContr c (tsmartPlus' (ILTexpr t') t0)
| If e t' c1 c2 => (fromContr c1 t0)
>>= fun v1 => (fromContr c2 t0)
>>= fun v2 => fromExp (ILTexprZ t0) e
>>= fun e => Some (ILLoopIf e v1 v2 t')
| Zero => Some (ILFloat 0)
| Let e c => None
| Transfer p1 p2 _ => Some (if (Party.eqb p1 p2) then (ILFloat 0)
else (ILPayoff t0 p1 p2))
| Both c1 c2 => (fromContr c1 t0)
>>= fun v1 => (fromContr c2 t0)
>>= fun v2 => pure (ILBinExpr ILAdd v1 v2)
end.
Lemma all_to_forall1 {A} P (x : A) :
all P [x] -> P x.
Proof.
intros. inversion H. subst. assumption.
Qed.
Lemma all_to_forall2 {A} P (x1 : A) (x2 : A) :
all P [x1;x2] -> P x1 /\ P x2.
Proof.
intros. inversion H. subst. split.
- assumption.
- apply all_to_forall1 in H3. assumption.
Qed.
Lemma all_to_forall3 {A} P (x1 x2 x3 : A) :
all P [x1;x2;x3] -> P x1 /\ P x2 /\ P x3.
Proof.
intros. inversion H. subst. split.
- assumption.
- apply all_to_forall2 in H3. assumption.
Qed.
(* Next lemmas look wierd, but I found them helpful for the proof automation*)
Lemma fromVal_RVal_f_eq f x1 x2 y1 y2:
fromVal (RVal x1) = ILRVal y1 ->
fromVal (RVal x2) = ILRVal y2 ->
fromVal (RVal (f x1 x2)) = (ILRVal (f y1 y2)).
Proof.
intros. simpl in *. inversion H. inversion H0. subst. reflexivity.
Qed.
Lemma ILRVal_zero_mult_l x1 x2:
ILRVal x1 = ILRVal 0 -> ILRVal (x1 * x2) = ILRVal 0.
Proof.
intros. inversion H. rewrite Rmult_0_l. reflexivity.
Qed.
Lemma ILRVal_zero_mult_r x1 x2:
ILRVal x2 = ILRVal 0 -> ILRVal (x1 * x2) = ILRVal 0.
Proof.
intros. inversion H. rewrite Rmult_0_r. reflexivity.
Qed.
Lemma fromVal_BVal_f_eq f x1 x2 y1 y2:
fromVal (BVal x1) = ILBVal y1 ->
fromVal (BVal x2) = ILBVal y2 ->
fromVal (BVal (f x1 x2)) = (ILBVal (f y1 y2)).
Proof.
intros. simpl in *. inversion H. inversion H0. subst. reflexivity.
Qed.
Lemma fromVal_BVal_f_eq' f x1 x2 y1 y2:
fromVal (RVal x1) = ILRVal y1 ->
fromVal (RVal x2) = ILRVal y2 ->
fromVal (BVal (f x1 x2)) = (ILBVal (f y1 y2)).
Proof.
intros. simpl in *. inversion H. inversion H0. subst. reflexivity.
Qed.
Lemma fromVal_BVal_f_eq_un f x1 y1:
fromVal (BVal x1) = ILBVal y1 ->
fromVal (BVal (f x1)) = ILBVal (f y1).
Proof.
intros. simpl in *. inversion H. reflexivity.
Qed.
Lemma fromVal_RVal_un_minus_eq x1 y1:
fromVal (RVal x1) = ILRVal y1 ->
fromVal (RVal ( - x1)) = ILRVal ( - y1).
Proof.
intros. simpl in *. inversion H. reflexivity.
Qed.
Hint Resolve of_nat_succ fromVal_RVal_f_eq fromVal_BVal_f_eq fromVal_BVal_f_eq'.
Hint Resolve fromVal_BVal_f_eq_un fromVal_RVal_un_minus_eq.
Hint Unfold liftM compose bind pure.
Hint Rewrite Nat2Z.inj_succ Rminus_0_l.
Ltac destruct_vals := repeat (match goal with
| [x : Val |- _] => destruct x; tryfalse
| [x : ILVal |- _] => destruct x; tryfalse
end).
Ltac some_inv := repeat (unfold pure in *; match goal with
| [H: Some _ = Some _ |- _] => inversion H; clear H
end).
Notation "'IL[|' e '|]'" := (ILsem e).
Ltac destruct_ILsem := repeat (match goal with
| [_ : match IL[|?e|] ?env ?ext ?d ?p1 ?p2 : _ with _ => _ end = _ |- _] =>
let newH := fresh "H" in destruct (IL[|e|] env ext d p1 p2) eqn: newH; tryfalse
end).
Hint Unfold ILTexprSem.
(* --------------------------------------------- *)
(* Soundness of contract expressions compilation *)
(* --------------------------------------------- *)
Theorem Exp_translation_sound : forall (e : Exp) (il_e : ILExpr) (envC : Env' Val) (extC : ExtEnv' Val)
(extIL : ExtEnv' ILVal) (envT : TEnv)
(v : Val) (v' : ILVal) p1 p2 t0 t0' disc t_now,
fromExp t0 e = Some il_e ->
(forall l t, fromVal (extC l t) = extIL l t) ->
E[|e|] envC (adv_ext (ILTexprSemZ t0 envT + Z.of_nat t0') extC) = Some v ->
IL[|il_e|] extIL envT t0' t_now disc p1 p2 = Some v'->
fromVal v = v'.
Proof.
intros. generalize dependent envC. generalize dependent extC.
generalize dependent extIL. generalize dependent envT.
generalize dependent il_e. generalize dependent v. generalize dependent v'.
induction e using Exp_ind'.
+ (* OpE *)
intros v' v il_e Htrans extIL envT Hilsem extC Heqext envC Hesem. destruct op;
(* Binary operations *)
try (simpl in *; do 3 try (destruct args; tryfalse); tryfalse; simpl in *;
option_inv_auto; subst; simpl in *; try (some_inv); subst; simpl in *; subst;
option_inv_auto; subst; simpl in *; destruct_vals;
subst; option_inv_auto;
apply all_to_forall2 in H; inversion_clear H;
eauto);
(* Unary operations*)
try (simpl in *; do 2 try (destruct args; tryfalse); tryfalse; simpl in *;
option_inv_auto; subst; simpl in *; some_inv; subst; simpl in *; subst;
option_inv_auto; subst; simpl in *; destruct_vals; some_inv; subst; simpl in H6; some_inv; subst;
apply all_to_forall1 in H; some_inv; autorewrite with core; eauto);
(* Lit *)
simpl in *; destruct args; tryfalse; simpl in *; option_inv_auto; tryfalse; subst; simpl in *; some_inv; subst; try reflexivity.
(* cond *)
simpl in *. do 4 try (destruct args; tryfalse); tryfalse; simpl in *;
eapply all_to_forall3 in H; inversion_clear H as [IHe1 IHe']; inversion_clear IHe' as [IHe2 IHe3];
option_inv_auto; subst; simpl in *; try (some_inv); subst.
simpl in *; subst; option_inv_auto; subst; simpl in *. destruct x0; tryfalse.
destruct x;tryfalse.
(* Bool *)
* destruct x1; tryfalse. some_inv; destruct b.
- (* condition is true*)
replace x3 with (fromVal (BVal true)) in H8; simpl in H8; eauto.
- (* condition is false *)replace x3 with (fromVal (BVal false)) in H8; simpl in H8; eauto.
(* Real *)
* destruct x1; tryfalse. some_inv; destruct b.
- (* condition is true*)
replace x3 with (fromVal (BVal true)) in H8; simpl in H8; eauto.
- (* condition is false *)replace x3 with (fromVal (BVal false)) in H8; simpl in H8; eauto.
+ (* Obs *)
intros. simpl in *. some_inv. subst. simpl in *. some_inv. subst.
replace (Z.of_nat t0' + (i + ILTexprSemZ t0 envT)) with (ILTexprSemZ t0 envT + Z.of_nat t0' + i). eapply H0. ring.
+ (* Var *)
intros. inversion H.
+ (* Acc *)
intros. inversion H.
Qed.
Lemma delay_scale_trace tr t n :
scale_trace n (delay_trace t tr) = delay_trace t (scale_trace n tr).
Proof.
unfold scale_trace, delay_trace, compose. apply functional_extensionality. intros. destruct (Compare_dec.leb t x).
- reflexivity.
- apply scale_empty_trans.
Qed.
Local Open Scope nat.
Lemma delay_trace_n_m n m tr :
n <= m ->
(delay_trace n tr) m = tr (m-n).
Proof.
intros. unfold delay_trace.
apply leb_iff in H. rewrite H. reflexivity.
Qed.
Lemma delay_trace_empty_before_n n tr t:
t < n ->
(delay_trace n tr) t = empty_trans.
Proof.
intros. unfold delay_trace. apply leb_correct_conv in H. rewrite H. reflexivity.
Qed.
Lemma sum_of_map_empty_trace : forall n n0 p1 p2 curr trace f,
trace = empty_trace ->
sum_list (map (fun t => (f t * trace t p1 p2 curr)%R) (seq n0 n)) = 0%R.
Proof.
induction n.
+ intros. reflexivity.
+ intros. simpl. rewrite IHn. rewrite H. unfold empty_trace, const_trace, empty_trans. ring. assumption.
Qed.
Lemma delay_trace_zero_before_n : forall n tr p1 p2 curr t0 f,
sum_list (map (fun t1 : nat => (f t1 * (delay_trace (n + t0) tr t1 p1 p2 curr))%R) (seq t0 n)) = 0%R.
Proof.
induction n.
+ intros. reflexivity.
+ intros. simpl. rewrite delay_trace_empty_before_n. unfold empty_trans. rewrite Rmult_0_r.
rewrite Rplus_0_l. replace (S (n + t0)) with (n + S t0).
apply IHn. omega. rewrite <- plus_Sn_m. omega.
Qed.
(* ------------------------------------------------------------------- *)
(* Various lemmas about moving around initial index of the trace's sum *)
(* ------------------------------------------------------------------- *)
Lemma sum_delay_plus n (trace0 : Trace) p1 p2 disc (a : Asset) : forall m,
sum_list (map (fun t0 : nat => (disc (1 + t0)%nat * trace0 (1 + t0)%nat p1 p2 a))%R (seq m n)%nat)%R =
sum_list (map (fun t0 : nat => (disc t0 * trace0 t0 p1 p2 a)%R) (seq (1+m) n)).
Proof.
induction n.
- reflexivity.
- intros m. simpl in *. rewrite IHn with (m:= S m).
reflexivity.
Qed.
Lemma sum_delay_plus' n (trace0 : Trace) p1 p2 disc (a : Asset) : forall m t,
sum_list (map (fun t0 : nat => (disc (t + t0)%nat * trace0 (t + t0)%nat p1 p2 a))%R (seq m n)%nat)%R =
sum_list (map (fun t0 : nat => (disc t0 * trace0 t0 p1 p2 a)%R) (seq (t+m) n)).
Proof.
induction n.
- reflexivity.
- intros m t. simpl in *. rewrite IHn with (m:= S m) (t:=t).
replace (t + S m) with (S (t + m)) by ring. reflexivity.
Qed.
Lemma sum_delay t0 t n tr p1 p2 curr f:
sum_list (map (fun t => (f t * (delay_trace (n + t0) tr) t p1 p2 curr)%R) (seq t0 (n + t))) =
sum_list (map (fun t => (f t * (delay_trace (n + t0) tr) t p1 p2 curr)%R) (seq (n + t0) t)).
Proof.
rewrite seq_split. rewrite map_app. rewrite sum_split. rewrite delay_trace_zero_before_n.
rewrite Rplus_0_l. rewrite plus_comm. reflexivity.
Qed.
Lemma sum_delay' n tr p1 p2 curr f t0 n1:
sum_list (map (fun t : nat =>(f t * delay_trace (1 + n) tr t p1 p2 curr)%R) (seq (t0 + 1) n1)) =
sum_list (map (fun t : nat => (f (S t)%nat * delay_trace n tr t p1 p2 curr)%R) (seq t0 n1)).
Proof.
generalize dependent t0. generalize dependent n1.
induction n.
- intros. simpl. generalize dependent t0. induction n1; intros.
+ reflexivity.
+ simpl. f_equal.
* rewrite plus_comm. simpl. rewrite delay_trace_S. reflexivity.
* rewrite plus_comm. simpl. rewrite <- IHn1. rewrite plus_comm. reflexivity.
- generalize dependent n. induction n1; intros.
+ reflexivity.
+ rewrite plus_comm. simpl. f_equal.
* rewrite plus_comm. simpl. rewrite delay_trace_S. rewrite plus_comm. reflexivity.
* rewrite <- IHn1. simpl. rewrite plus_comm. simpl. reflexivity.
Qed.
Lemma sum_delay'' n tr p1 p2 curr f t0 n1 n0:
sum_list (map (fun t : nat =>(f (n0 + t)%nat * delay_trace (1 + n) tr t p1 p2 curr)%R) (seq (t0 + 1) n1)) =
sum_list (map (fun t : nat => (f (S (n0 + t))%nat * delay_trace n tr t p1 p2 curr)%R) (seq t0 n1)).
Proof.
generalize dependent t0. generalize dependent n1.
induction n.
- intros. simpl. generalize dependent t0. induction n1; intros.
+ reflexivity.
+ simpl. f_equal.
* rewrite plus_comm. simpl. replace (t0 + 1 + n0) with (S (n0 + t0)) by omega. replace (t0 + 1) with (S t0) by omega.
rewrite delay_trace_S. reflexivity.
* rewrite plus_comm. simpl. rewrite <- IHn1. rewrite plus_comm. reflexivity.
- generalize dependent n. induction n1; intros.
+ reflexivity.
+ rewrite plus_comm. simpl. f_equal.
* rewrite plus_comm. simpl. replace (t0 + 1 + n0) with (S (n0 + t0)) by omega. replace (t0 + 1) with (S t0) by omega.
rewrite delay_trace_S. replace (n+1) with (S n) by omega. reflexivity.
* rewrite <- IHn1. simpl. rewrite plus_comm. simpl. reflexivity.
Qed.
(* -------------------------------------------------- *)
(* Lemmas about traces of contracts with zero horizon *)
(* -------------------------------------------------- *)
Lemma zero_horizon_empty_trans c t trace env ext tenv:
horizon c tenv = 0 ->
C[|c|] env ext tenv = Some trace ->
trace t = empty_trans.
Proof.
intros. eapply horizon_sound with (c := c) (tenv := tenv). rewrite H. omega. eassumption.
Qed.
Lemma zero_horizon_empty_trace c trace env ext tenv:
horizon c tenv = 0 ->
C[|c|] env ext tenv= Some trace ->
trace = empty_trace.
Proof.
intros. apply functional_extensionality. intros.
eapply horizon_sound with (c := c) (tenv := tenv). rewrite H. omega. eassumption.
Qed.
Lemma zero_horizon_delay_trace c m trace env ext tenv:
horizon c tenv= 0 ->
C[|c|] env ext tenv = Some trace ->
delay_trace m trace = trace.
Proof.
intros. apply functional_extensionality. intros. unfold delay_trace. destruct (Compare_dec.leb m x).
- erewrite zero_horizon_empty_trans. erewrite zero_horizon_empty_trans. reflexivity.
eassumption. eassumption. eassumption. eassumption.
- erewrite zero_horizon_empty_trans. reflexivity. eassumption. eassumption.
Qed.
Lemma sum_list_zero_horizon : forall n c t env ext trace p1 p2 curr f tenv,
horizon c tenv= 0 ->
C[|c|] env ext tenv = Some trace ->
sum_list (map (fun t => (f t * trace t p1 p2 curr)%R) (seq t n)) = 0%R.
Proof.
induction n.
+ intros. reflexivity.
+ intros. simpl. erewrite IHn. erewrite zero_horizon_empty_trans. unfold empty_trans. ring.
eassumption. eassumption. eassumption. eassumption.
Qed.
(* ------------------------------------------------------------------ *)
(* Various lemmas about summing up values in traces. Quite often we *)
(* use the fact that the traces geneated by contracts are finitely *)
(* supported. The notion of horizon allows us to delimit where *)
(* non-zero part of the trace ends. *)
(* ------------------------------------------------------------------ *)
Lemma sum_after_horizon_zero : forall n t0 c trace p1 p2 curr ext env tenv,
C[|c|] env ext tenv = Some trace ->
sum_list (map (fun t => trace t p1 p2 curr) (seq (t0 + horizon c tenv) n)) = 0%R.
Proof.
intros n.
induction n.
+ intros. reflexivity.
+ intros. simpl. erewrite horizon_sound. unfold empty_trans.
rewrite <- plus_Sn_m. erewrite IHn. ring.
eassumption. instantiate (1:=tenv). instantiate (1:=c). omega. eassumption.
Qed.
Lemma sum_delay_after_horizon_zero : forall n m t0 t1 c trace p1 p2 curr ext env f tenv,
m <= t0 ->
C[|c|] env ext tenv= Some trace ->
sum_list (map (fun t => (f t * delay_trace m trace (t1 + t)%nat p1 p2 curr)%R) (seq (t0 + horizon c tenv) n)) = 0%R.
Proof.
intros n.
induction n.
+ intros. reflexivity.
+ intros. simpl. rewrite delay_trace_n_m. erewrite horizon_sound. unfold empty_trans.
rewrite <- plus_Sn_m. erewrite IHn. ring. omega. eassumption.
instantiate (1:=tenv). instantiate (1:=c). omega. eassumption. omega.
Qed.
Lemma sum_before_after_horizon t0 t1 t c trace p1 p2 curr ext env f tenv:
C[|c|] env ext tenv = Some trace ->
sum_list (map (fun t => (f t * (delay_trace t0 trace) (t1 + t)%nat p1 p2 curr))%R (seq t0 (horizon c tenv + t))) =
sum_list (map (fun t => (f t * (delay_trace t0 trace) (t1 + t)%nat p1 p2 curr))%R (seq t0 (horizon c tenv))).
Proof.
intros. rewrite seq_split. rewrite map_app. rewrite sum_split. erewrite sum_delay_after_horizon_zero. ring.
omega. eassumption.
Qed.
Lemma sum_before_after_horizon_St t0 t1 t c trace p1 p2 curr ext env f tenv:
C[|c|] env ext tenv = Some trace ->
0 < horizon c tenv ->
sum_list (map (fun t => (f t * (delay_trace t0 trace) (t1 + t)%nat p1 p2 curr))%R (seq (S t0) (horizon c tenv - 1 + t))) =
sum_list (map (fun t => (f t * (delay_trace t0 trace) (t1 + t)%nat p1 p2 curr))%R (seq (S t0) (horizon c tenv - 1))).
Proof.
intros. rewrite seq_split. rewrite map_app. rewrite sum_split.
replace (S t0 + (horizon c tenv - 1)) with (t0 + horizon c tenv) by omega.
erewrite sum_delay_after_horizon_zero. ring.
omega. eassumption.
Qed.
Lemma sum_after_horizon_zero' : forall n c f trace p1 p2 curr ext env tenv m,
C[|c|] env ext tenv = Some trace ->
horizon c tenv <= m ->
sum_list (map (fun t => f t * trace t p1 p2 curr)%R (seq m n)) = 0%R.
Proof.
intros n.
induction n.
+ intros. reflexivity.
+ intros. simpl. erewrite horizon_sound. unfold empty_trans.
erewrite IHn;eauto. ring. eauto. eauto.
Qed.
Lemma sum_before_after_horizon' t0 c trace p1 p2 curr ext env f tenv m n:
C[|c|] env ext tenv = Some trace ->
m+t0 >= horizon c tenv ->
sum_list (map (fun t => (f t * trace t p1 p2 curr))%R (seq t0 m)) =
sum_list (map (fun t => (f t * trace t p1 p2 curr))%R (seq t0 (m+n))).
Proof.
intros. rewrite seq_split. rewrite map_app. rewrite sum_split.
erewrite sum_after_horizon_zero' with (m:=t0+m) (n:=n). ring. eauto. omega.
Qed.
Lemma delay_add_trace n tr1 tr2:
add_trace (delay_trace n tr1) (delay_trace n tr2) = delay_trace n (add_trace tr1 tr2).
Proof.
unfold add_trace, delay_trace, compose. apply functional_extensionality. intros. destruct (Compare_dec.leb n x).
- reflexivity.
- rewrite add_empty_trans_l. reflexivity.
Qed.
Lemma map_delay_trace : forall n m t0 x0 p1 p2 curr,
m <= t0 ->
map (fun t : nat => delay_trace m x0 t p1 p2 curr) (seq t0 n) =
map (fun t : nat => x0 (t-m) p1 p2 curr) (seq t0 n).
Proof.
induction n.
- reflexivity.
- intros. simpl. rewrite delay_trace_n_m. apply f_equal. apply IHn. omega. omega.
Qed.
Ltac destruct_option_vals :=
repeat (match goal with
| [x : match ?X : option Val with _ => _ end = _ |- _] => destruct X; tryfalse
| [x : match ?X : option ILVal with _ => _ end = _ |- _] => destruct X; tryfalse
end).
Ltac rewrite_option :=
repeat (match goal with
| [H : _ : option ?X |- match _ : option ?X with _ => _ end = _] => rewrite H
end).
(* Definition of the contract compilation soundness *)
Definition translation_sound c envC extC t0 t0' :=
forall (il_e : ILExpr)
(extIL : ExtEnv' ILVal)
(v' : ILVal) p1 p2 curr v trace (disc : nat -> R ) tenv t_now,
(forall a a', Asset.eqb a a' = true) ->
(forall l t, fromVal (extC l t) = extIL l t) ->
fromContr c t0 = Some il_e ->
C[|Translate (Tnum (ILTexprSem t0 tenv + t0')) c|] envC extC tenv = Some trace ->
sum_list (map (fun t => (disc t * trace t p1 p2 curr)%R)
(seq (ILTexprSem t0 tenv + t0') (horizon c tenv))) = v ->
(IL[|il_e|] extIL tenv) t0' t_now disc p1 p2 = Some v'->
fromVal (RVal v) = v'.
Ltac destruct_ilexpr := repeat (match goal with
| [x : match ?X : option ILExpr with _ => _ end = _ |- _] =>
let newH := fresh "H" in destruct X eqn: newH; tryfalse
end).
Hint Rewrite Z2Nat.inj_add Nat2Z.id Nat2Z.inj_add : ILDB.
Hint Resolve Zle_0_nat.
Lemma delay_trace_at d t : delay_trace d t d = t O.
Proof.
unfold delay_trace.
assert (Compare_dec.leb d d = true) as E by (apply leb_correct; auto).
rewrite E. rewrite minus_diag. reflexivity.
Qed.
Lemma plus0_0_n : forall n : nat,
plus0 0 n = n.
Proof.
intros.
destruct n as [| n']; reflexivity.
Qed.
Lemma fromVal_RVal_eq v : fromVal (RVal v) = ILRVal v.
Proof. reflexivity. Qed.
Lemma tsmartPlus_0_r t : ILTexprSem (tsmartPlus t (ILTexpr (Tnum 0))) = ILTexprSem t.
Proof.
unfold tsmartPlus. destruct t; try destruct e; try destruct n; try reflexivity.
Qed.
Lemma fold_unfold_ILTexprSem t t0 tenv:
ILTexprSem (match t with
| Tvar _ => match t0 with
| ILTexpr (Tnum 0) => ILTexpr t
| _ => ILTplus (ILTexpr t) t0
end
| Tnum 0 => t0
| Tnum (S _) => match t0 with
| ILTexpr (Tnum 0) => ILTexpr t
| _ => ILTplus (ILTexpr t) t0
end
end) tenv = ILTexprSem (tsmartPlus (ILTexpr t) t0) tenv.
Proof. reflexivity. Qed.
Lemma fold_unfold_ILTexprSem' t t0 tenv:
ILTexprSem (match t with
| Tvar _ => ILTplus (ILTexpr t) t0
| Tnum n1 =>
match t0 with
| ILTplus _ _ => ILTplus (ILTexpr t) t0
| ILTexpr (Tvar _) => ILTplus (ILTexpr t) t0
| ILTexpr (Tnum n2) => ILTexpr (Tnum (n1 + n2))
end
end) tenv = ILTexprSem (tsmartPlus' (ILTexpr t) t0) tenv.
Proof. reflexivity. Qed.
(* ------------------------------------------------------ *)
(* Soundness of compilation from the Contact DSL to the *)
(* payoff expression language wrt. denotational semantics *)
(* of the contract language. *)
(* ------------------------------------------------------ *)
Theorem Contr_translation_sound: forall c envC extC t0 t0',
translation_sound c envC extC t0 t0'.
Proof.
intro c. induction c; unfold translation_sound.
+ (* Zero *)
intros. simpl in *. some_inv. subst. simpl in *. some_inv.
reflexivity.
+ (* Let *)
intros. tryfalse.
+ (* Transfer *)
intros. simpl in *. some_inv. subst. rewrite Rplus_0_r. unfold compose in H2. some_inv.
rewrite delay_trace_at. unfold singleton_trace, singleton_trans.
destruct (Party.eqb p p0).
- simpl in *. rewrite Rmult_0_r. some_inv. subst. reflexivity.
- simpl in *. some_inv. unfold eval_payoff.
destruct (Party.eqb p p1); destruct (Party.eqb p0 p2);
try (rewrite H; rewrite Rmult_1_r; reflexivity);
simpl; destruct (Party.eqb p p2); destruct (Party.eqb p0 p1); simpl;
try (rewrite H); apply f_equal;
try (rewrite plus_comm); try ring; try assumption; reflexivity.
+ (* Scale *)
intros. simpl in *. option_inv_auto. some_inv. subst. simpl in *. option_inv_auto.
destruct_vals. some_inv. subst. rewrite <- delay_scale_trace.
unfold scale_trace, compose, scale_trans.
rewrite summ_list_common_factor. rewrite <- fromVal_RVal_eq. apply fromVal_RVal_f_eq.
- eapply Exp_translation_sound; try (simpl in *; some_inv; subst);
try eassumption. simpl. rewrite <- Nat2Z.inj_add. eassumption.
- eapply IHc with (envC := envC) (curr := curr); try eassumption. autounfold in *. simpl.
rewrite H3. reflexivity. reflexivity.
+ (*Translate *)
intros. simpl in *.
option_inv_auto. rewrite adv_ext_iter in H3. rewrite delay_trace_iter.
eapply IHc with (envC := envC); try eassumption. simpl.
rewrite Nat2Z.inj_add. rewrite fold_unfold_ILTexprSem'. rewrite tsmartPlus_sound'. simpl.
rewrite Nat2Z.inj_add. rewrite Nat2Z.inj_add in H3.
rewrite Zplus_comm in H3. rewrite Zplus_assoc in H3. rewrite H3.
reflexivity. simpl. rewrite fold_unfold_ILTexprSem'. rewrite tsmartPlus_sound'. simpl.
rewrite <- plus_assoc. rewrite <- sum_delay. destruct (horizon c tenv) eqn:Hhor.
* simpl.
eapply sum_list_zero_horizon with (c:=c) (tenv:=tenv)
(ext:=(adv_ext (Z.of_nat (ILTexprSem t0 tenv + t0') +
Z.of_nat (TexprSem t tenv)) extC)) (env:=envC); eauto.
erewrite zero_horizon_delay_trace;eauto.
* unfold plus0. reflexivity.
+ (* Both *)
intros. simpl in *. option_inv_auto. some_inv. subst. simpl in *.
option_inv_auto. destruct_vals. some_inv.
rewrite <- delay_add_trace. unfold add_trace, add_trans.
rewrite summ_list_plus. rewrite <- fromVal_RVal_eq.
apply fromVal_RVal_f_eq.
- eapply IHc1 with (envC:=envC); try eassumption.
autounfold in *. simpl. rewrite H2. reflexivity.
destruct (Max.max_dec (horizon c1 tenv) (horizon c2 tenv)) as [Hmax_h1 | Hmax_h2].
* rewrite Hmax_h1. reflexivity.
* rewrite Hmax_h2. apply Nat.max_r_iff in Hmax_h2.
assert (Hh2eq: horizon c2 tenv = horizon c1 tenv + (horizon c2 tenv - horizon c1 tenv))
by omega.
rewrite Hh2eq. erewrite sum_before_after_horizon with (t1:=0). reflexivity. eassumption.
- eapply IHc2 with (envC:=envC); try eassumption. autounfold in *. simpl. rewrite H3. reflexivity.
destruct (Max.max_dec (horizon c1 tenv) (horizon c2 tenv)) as [Hmax_h1 | Hmax_h2].
* rewrite Hmax_h1. apply Nat.max_l_iff in Hmax_h1.
assert (Hh2eq: horizon c1 tenv = horizon c2 tenv + (horizon c1 tenv - horizon c2 tenv))
by omega.
rewrite Hh2eq. erewrite sum_before_after_horizon with (t1:=0). reflexivity. eassumption.
* rewrite Hmax_h2. reflexivity.
+ (* If *)
intros. simpl in *. option_inv_auto. simpl in *. unfold loop_if_sem in H5.
generalize dependent t0. generalize dependent v'.
generalize dependent e. generalize dependent x.
generalize dependent t0'.
remember (TexprSem t tenv) eqn:Ht.
revert t Ht.
induction n.
- (* n = 0 *)
intros.
simpl in *. destruct (fromExp (ILTexprZ t0) e) eqn:Hfromexp; tryfalse.
option_inv_auto. simpl in *. subst.
destruct (E[|e|] envC (adv_ext (Z.of_nat (ILTexprSem t0 tenv + t0')) extC)) eqn: Hexp; tryfalse.
destruct v; tryfalse. destruct b.
* (* Condition evaluates to true *)
replace x3 with (fromVal (BVal true)) in H8. simpl in H8.
eapply IHc1 with (envC:=envC); try eassumption.
** simpl. rewrite H6. reflexivity.
** rewrite plus0_0_n.
destruct (Max.max_dec (horizon c1 tenv) (horizon c2 tenv)) as [Hmax_h1 | Hmax_h2].
*** rewrite Hmax_h1. reflexivity.
*** rewrite Hmax_h2. apply Nat.max_r_iff in Hmax_h2.
assert (Hh2eq: horizon c2 tenv = horizon c1 tenv + (horizon c2 tenv - horizon c1 tenv))
by omega.
rewrite Hh2eq. erewrite sum_before_after_horizon with (t1:=0). reflexivity.
eassumption.
** eapply Exp_translation_sound; rewrite Nat2Z.inj_add in Hexp; try eassumption.
* (* Condition evaluates to false *)
replace x3 with (fromVal (BVal false)) in H8. simpl in H8.
eapply IHc2 with (envC:=envC); try eassumption. simpl. rewrite H6.
reflexivity. rewrite plus0_0_n.
destruct (Max.max_dec (horizon c1 tenv) (horizon c2 tenv)) as [Hmax_h1 | Hmax_h2].
rewrite Hmax_h1. apply Nat.max_l_iff in Hmax_h1.
assert (Hh2eq: horizon c1 tenv = horizon c2 tenv+ (horizon c1 tenv - horizon c2 tenv))
by omega.
rewrite Hh2eq. simpl. erewrite sum_before_after_horizon with (t1:=0).
reflexivity. eassumption.
rewrite Hmax_h2. simpl. reflexivity.
eapply Exp_translation_sound; rewrite Nat2Z.inj_add in Hexp; try eassumption.
- (* n = S n' *)
intros. simpl in *. option_inv_auto.
destruct (E[|e|] envC (adv_ext (Z.of_nat (ILTexprSem t0 tenv + t0')) extC)) eqn: Hexp; tryfalse.
destruct v; tryfalse. destruct b.
* (* Condition evaluates to true *)
replace x3 with (fromVal (BVal true)) in H8. simpl in H8.
eapply IHc1 with (envC:=envC) (tenv:=tenv); try eassumption. simpl. rewrite H6. reflexivity.
destruct (Max.max_dec (horizon c1 tenv) (horizon c2 tenv)) as [Hmax_h1 | Hmax_h2].
(* max = horizon c1 *)
rewrite Hmax_h1. destruct (horizon c1 tenv) eqn:Hhor1. reflexivity.
unfold plus0. rewrite <- Hhor1. replace (S n + horizon c1 tenv) with (horizon c1 tenv + S n).
erewrite sum_before_after_horizon with (t1:=0); try reflexivity; try eassumption. ring.
(* max = horizon c2 *)
rewrite Hmax_h2. apply Nat.max_r_iff in Hmax_h2.
destruct (horizon c2 tenv) eqn:Hhor2.
simpl. eapply sum_list_zero_horizon with (c:=c1) (tenv:=tenv).
omega. erewrite zero_horizon_delay_trace with (c:=c1) (tenv:=tenv).
eassumption. omega. eassumption.
unfold plus0. rewrite <- Hhor2.
assert (Hh2eq: S n + horizon c2 tenv =
horizon c1 tenv + ((horizon c2 tenv - horizon c1 tenv)) + S n) by omega.
rewrite Hh2eq. rewrite <- plus_assoc.
erewrite sum_before_after_horizon with (t1:=0). reflexivity. eassumption.
eapply Exp_translation_sound with (envC:=envC) (envT:=tenv);
rewrite Nat2Z.inj_add in Hexp; autounfold; try eassumption.
* (* Condition evaluates to false *)
destruct (max (horizon c1 tenv) (horizon c2 tenv)) eqn:Hmax. simpl in *.
Focus 2.
unfold plus0 in *. rewrite <- Hmax. rewrite <- Hmax in IHn.
destruct (Max.max_dec (horizon c1 tenv) (horizon c2 tenv)) as [Hmax_h1 | Hmax_h2]. simpl.
(* max = horizon c1 *)
rewrite Hmax_h1. rewrite Hmax_h1 in IHn.
replace x3 with (fromVal (BVal false)) in H8. simpl in H8. rewrite adv_ext_iter in H6.
option_inv_auto. rewrite delay_trace_at. rewrite delay_trace_empty_before_n.
unfold empty_trans. rewrite Rmult_0_r. rewrite Rplus_0_l.
rewrite delay_trace_iter. simpl. rewrite plus_n_Sm.
eapply IHn with (t:=Tnum n); try eassumption. reflexivity. rewrite <- plus_n_Sm.
replace (Z.of_nat (S (ILTexprSem t0 tenv + t0')))
with (Z.of_nat (ILTexprSem t0 tenv + t0') + 1)%Z.
apply H4. rewrite <- of_nat_succ. apply f_equal. omega. omega.
eapply Exp_translation_sound; try eassumption. rewrite Nat2Z.inj_add in Hexp; eassumption.
(* max = horizon c2 *)
rewrite Hmax_h2. rewrite Hmax_h2 in IHn. simpl.
replace x3 with (fromVal (BVal false)) in H8. simpl in H8. rewrite adv_ext_iter in H6.
option_inv_auto. rewrite delay_trace_at. rewrite delay_trace_empty_before_n.
unfold empty_trans. rewrite Rmult_0_r. rewrite Rplus_0_l.
rewrite delay_trace_iter. simpl. rewrite plus_n_Sm.
eapply IHn with (t:=Tnum n); try eassumption. reflexivity.
replace (Z.of_nat (ILTexprSem t0 tenv + S t0'))
with (Z.of_nat (ILTexprSem t0 tenv + t0') + 1)%Z.
apply H4. rewrite <- of_nat_succ. apply f_equal. omega. omega.
eapply Exp_translation_sound; try eassumption. rewrite Nat2Z.inj_add in Hexp; eassumption.
(* max = 0*)
replace x3 with (fromVal (BVal false)) in H8. simpl in H8. rewrite adv_ext_iter in H6.
option_inv_auto. eapply IHn with (t:=Tnum n); try eassumption; try reflexivity.
replace (Z.of_nat (ILTexprSem t0 tenv + S t0'))
with (Z.of_nat (ILTexprSem t0 tenv + t0') + 1)%Z.
apply H4. rewrite <- of_nat_succ. apply f_equal. ring.
eapply Exp_translation_sound; try eassumption. rewrite Nat2Z.inj_add in Hexp; eassumption.
Qed.
(* Lemma real_exp_compile_sem_total e il ext env tenv p1 p2 t0 G d: *)
(* G |-E e ∶ REAL -> fromExp t0 e = Some il -> *)
(* exists v, IL[|il|]env ext tenv 0 d p1 p2 = Some (ILRVal v). *)
(* Proof. *)
(* intros T C. *)
(* generalize dependent il. *)
(* induction T using TypeExp_ind';intros;tryfalse. *)
(* + inversion H;subst; simpl in *;tryfalse. *)
(* (* Literals *) *)
(* * destruct es; tryfalse; *)
(* destruct il; tryfalse; some_inv; subst; simpl in *; eauto. *)
(* (* Unary *) *)
(* * do 2 try (destruct es; tryfalse); tryfalse. option_inv_auto. *)
(* some_inv. subst. inversion_clear H1. destruct (H2 eq_refl x) as [v Hv];auto. *)
(* exists (-v)%R. simpl. autounfold. rewrite Hv. reflexivity. *)
(* * do 4 try (destruct es; tryfalse); tryfalse. option_inv_auto. *)
(* some_inv. subst. inversion_clear H1. inversion_clear H6. *)
(* inversion_clear H7. *)
(* destruct (H2 eq_refl x) as [v Hv];auto. *)
(* exists (-v)%R. simpl. autounfold. rewrite Hv. reflexivity. *)
Check ex.
Lemma adv_ext_neg_adv t0 t1 (ext : ExtEnv' ILVal) obs: adv_ext (-t0) ext obs (t1 + t0)%Z = ext obs t1.
Proof.
unfold adv_ext. replace (- t0 + (t1 + t0))%Z with t1 by ring.
reflexivity.
Qed.
Definition consistent_ty : Ty -> ILVal -> Prop :=
fun ty v => match ty with
| REAL => exists v', v = ILRVal v'
| BOOL => exists v', v = ILBVal v'
end.
Definition toILExt (ext : ExtEnv) := (fun k t => (fromVal (ext k t)) ).
(* TODO: use more proof automation to get rid of proof code duplication *)
Theorem typed_exp_compile_sem_total e il p1 p2 ty G d ext tenv t0 t0' n:
G |-E e ∶ ty -> TypeExt ext -> fromExp t0 e = Some il ->
exists v, IL[|il|] (toILExt ext) tenv t0' n d p1 p2 = Some v
/\ consistent_ty ty v.
Proof.
intros T Text C.
generalize dependent il.
induction T using TypeExp_ind'; intros; tryfalse.
+ inversion H;subst; simpl in *.
(* Literals *)
* destruct es; tryfalse;
destruct il; tryfalse; some_inv; subst; simpl in *; eauto.
* destruct es; tryfalse;
destruct il; tryfalse; some_inv; subst; simpl in *; eauto.
(* Unary *)
* do 2 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion_clear H1 as [H2| ]. destruct (H2 _ H3) as [v Hv].
destruct Hv as [L R]. simpl in *. destruct R.
exists (ILRVal (-x0)). autounfold. rewrite L. subst. simpl.
split;eauto.
* do 2 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion_clear H1 as [H2| ]. destruct (H2 _ H3) as [v Hv].
destruct Hv as [L R]. simpl in *. destruct R.
exists (ILBVal (negb x0)). autounfold. rewrite L. subst. simpl.
split;eauto.
(* cond *)
* do 4 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion_clear H1 as [H2| ].
inversion_clear H6 as [H2'| ].
inversion_clear H7 as [H2''| ].
destruct (H2 _ H3) as [v Hv]. destruct (H6 _ H4) as [v' Hv'].
destruct (H1 _ H5) as [v'' Hv''].
intuition.
destruct t.
** simpl in *. destruct H9 as [v0 Hv0]. destruct H11 as [v1 Hv1].
destruct H13 as [v2 Hv2]. subst.
exists (ILRVal (if v0 then v2 else v1)). autounfold. rewrite H7.
destruct v0;split;eauto.
** simpl in *. destruct H9 as [v0 Hv0]. destruct H11 as [v1 Hv1].
destruct H13 as [v2 Hv2]. subst.
exists (ILBVal (if v0 then v2 else v1)). autounfold. rewrite H7.
destruct v0;split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILRVal (v0 + v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILRVal (v0 - v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILRVal (v0 * v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILRVal (v0 / v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILBVal (v0 && v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILBVal (v0 || v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILBVal (Rltb v0 v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILBVal (Rleb v0 v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
* do 3 try (destruct es; tryfalse); tryfalse. option_inv_auto.
some_inv. subst. simpl in *.
inversion H1 as [H2 | ]. subst. inversion H4 as [| ? ? ? ? H2']. subst.
destruct (H2 _ H3) as [v Hv]. destruct (H2' _ H5) as [v' Hv'].
destruct Hv as [L R]. destruct Hv' as [L' R']. simpl in *.
destruct R as [v0]. destruct R' as [v1]. subst.
exists (ILBVal (Reqb v0 v1)). autounfold. rewrite L. rewrite L'.
split;eauto.
+ simpl in *. destruct il;tryfalse. some_inv. subst. simpl. eexists.
split;eauto. unfold TypeExt in Text.
assert (H' := Text (Z.of_nat t0' + (z + ILTexprSemZ t0 tenv))%Z _ _ H).
inversion H';
subst; simpl; eexists; unfold toILExt,fromVal; rewrite <- H1; eauto.
Qed.
Lemma toILExt_adv_ext n ext : toILExt (adv_ext n ext) = (adv_ext n (toILExt ext)).
Proof.
unfold toILExt. apply functional_extensionality.
intros l. apply functional_extensionality.
intros t. reflexivity.
Qed.
Theorem typed_contr_compile_sem_total c il ext tenv p1 p2 G d t0 t0':
G |-C c ->
TypeExt ext ->
IsClosedCT c ->
fromContr c t0 = Some il ->
exists v, IL[|il|](toILExt ext) tenv t0' 0 d p1 p2 = Some (ILRVal v).
Proof.
intros T Text Cl C.
generalize dependent ext.
generalize dependent il.
generalize dependent t0.
generalize dependent t0'.
induction T; intros t0' t0 il;intros ;tryfalse.
+ simpl in *; destruct il; tryfalse; some_inv; subst; simpl; eauto.
+ simpl in *. some_inv. destruct (Party.eqb p0 p3).
* subst. simpl. eauto.
* simpl. unfold eval_payoff.
destruct (Party.eqb p0 p1),(Party.eqb p3 p2),(Party.eqb p3 p1),(Party.eqb p0 p2); simpl;eauto.
+ simpl in *. option_inv_auto. some_inv. subst. simpl in *.
inversion Cl. subst.
destruct (IHT H2 t0' t0 _ H3 ext Text) as [v Hv].
eapply typed_exp_compile_sem_total with
(t0':=t0')(tenv:=tenv) (p1:=p1) (p2:=p2) (d:=d) in H;eauto.
destruct H as [v' Htmp]. destruct Htmp as [Hv' Ht'].
simpl in *. destruct Ht' as [r Hr]. subst.
exists (r * v)%R. autounfold.
rewrite Hv. rewrite Hv'. reflexivity.
+ simpl in *. inversion Cl. subst.
destruct (IHT H0 t0' _ _ C _ Text) as [r Hr].
eauto.
+ inversion Cl. subst.
simpl in *. option_inv_auto.
edestruct IHT1 as [v Hv];eauto.
edestruct IHT2 as [v' Hv'];eauto.
some_inv. simpl.
exists (v + v')%R.
autounfold. rewrite Hv. rewrite Hv'. reflexivity.
+ simpl in *. option_inv_auto. simpl in *.
inversion Cl.
generalize dependent t0'.
generalize dependent t0. subst.
induction n;intros.
* simpl.