Add Abs EPar If
This commit is contained in:
parent
764606cf2d
commit
f68efaf938
1 changed files with 188 additions and 32 deletions
|
|
@ -1070,6 +1070,144 @@ Module CRedRRed.
|
|||
Qed.
|
||||
End CRedRRed.
|
||||
|
||||
Module CRRed.
|
||||
Inductive R {n} : Tm n -> Tm n -> Prop :=
|
||||
(****************** Beta ***********************)
|
||||
| AppAbs a b :
|
||||
R (App (Abs a) b) (subst_Tm (scons b VarTm) a)
|
||||
| AppPair a b c:
|
||||
R (App (Pair a b) c) (Pair (App a c) (App b c))
|
||||
| ProjAbs p a :
|
||||
R (Proj p (Abs a)) (Abs (Proj p a))
|
||||
| ProjPair p a b :
|
||||
R (Proj p (Pair a b)) (if p is PL then a else b)
|
||||
| IfAbs (a : Tm (S n)) b c :
|
||||
R (If (Abs a) b c) (Abs (If a (ren_Tm shift b) (ren_Tm shift c)))
|
||||
| IfPair a b c d :
|
||||
R (If (Pair a b) c d) (Pair (If a c d) (If b c d))
|
||||
| IfBool a b c :
|
||||
R (If (BVal a) b c) (if a then b else c)
|
||||
| IfApp a b c d :
|
||||
R (If (App a b) c d) (App (If a c d) b)
|
||||
| IfProj p a b c :
|
||||
R (If (Proj p a) b c) (Proj p (If a b c))
|
||||
|
||||
(*************** Congruence ********************)
|
||||
| AbsCong a0 a1 :
|
||||
R a0 a1 ->
|
||||
R (Abs a0) (Abs a1)
|
||||
| AppCong0 a0 a1 b :
|
||||
R a0 a1 ->
|
||||
R (App a0 b) (App a1 b)
|
||||
| AppCong1 a b0 b1 :
|
||||
R b0 b1 ->
|
||||
R (App a b0) (App a b1)
|
||||
| PairCong0 a0 a1 b :
|
||||
R a0 a1 ->
|
||||
R (Pair a0 b) (Pair a1 b)
|
||||
| PairCong1 a b0 b1 :
|
||||
R b0 b1 ->
|
||||
R (Pair a b0) (Pair a b1)
|
||||
| ProjCong p a0 a1 :
|
||||
R a0 a1 ->
|
||||
R (Proj p a0) (Proj p a1)
|
||||
| BindCong0 p A0 A1 B:
|
||||
R A0 A1 ->
|
||||
R (TBind p A0 B) (TBind p A1 B)
|
||||
| BindCong1 p A B0 B1:
|
||||
R B0 B1 ->
|
||||
R (TBind p A B0) (TBind p A B1)
|
||||
| IfCong0 a0 a1 b c :
|
||||
R a0 a1 ->
|
||||
R (If a0 b c) (If a1 b c)
|
||||
| IfCong1 a b0 b1 c :
|
||||
R b0 b1 ->
|
||||
R (If a b0 c) (If a b1 c)
|
||||
| IfCong2 a b c0 c1 :
|
||||
R c0 c1 ->
|
||||
R (If a b c0) (If a b c1).
|
||||
|
||||
Lemma AppAbs' n a (b t : Tm n) :
|
||||
t = subst_Tm (scons b VarTm) a ->
|
||||
R (App (Abs a) b) t.
|
||||
Proof. move => ->. apply AppAbs. Qed.
|
||||
|
||||
Lemma IfAbs' n (a : Tm (S n)) (b c : Tm n) u :
|
||||
u = (Abs (If a (ren_Tm shift b) (ren_Tm shift c))) ->
|
||||
R (If (Abs a) b c) u.
|
||||
Proof. move => ->. apply IfAbs. Qed.
|
||||
|
||||
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
|
||||
R a b -> R (ren_Tm ξ a) (ren_Tm ξ b).
|
||||
Proof.
|
||||
move => h. move : m ξ.
|
||||
elim : n a b /h => n;
|
||||
lazymatch goal with
|
||||
| [|- context[App (Abs _) _]] => move => * /=; apply AppAbs'; by asimpl
|
||||
| [|- context[If (BVal _) _]] => hauto l:on use:IfBool
|
||||
| [|- context[Proj _ (Pair _ _)]] => hauto l:on use:ProjPair
|
||||
| [|- context[If (Abs _) _]] => move => * /=; apply IfAbs'; by asimpl
|
||||
| _ => qauto ctrs:R
|
||||
end.
|
||||
Qed.
|
||||
|
||||
End CRRed.
|
||||
|
||||
|
||||
Module CRReds.
|
||||
|
||||
#[local]Ltac solve_s_rec :=
|
||||
move => *; eapply rtc_l; eauto;
|
||||
hauto lq:on ctrs:CRRed.R.
|
||||
|
||||
#[local]Ltac solve_s :=
|
||||
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
|
||||
|
||||
Lemma AbsCong n (a b : Tm (S n)) :
|
||||
rtc CRRed.R a b ->
|
||||
rtc CRRed.R (Abs a) (Abs b).
|
||||
Proof. solve_s. Qed.
|
||||
|
||||
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
|
||||
rtc CRRed.R a0 a1 ->
|
||||
rtc CRRed.R b0 b1 ->
|
||||
rtc CRRed.R (App a0 b0) (App a1 b1).
|
||||
Proof. solve_s. Qed.
|
||||
|
||||
Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
|
||||
rtc CRRed.R a0 a1 ->
|
||||
rtc CRRed.R b0 b1 ->
|
||||
rtc CRRed.R (TBind p a0 b0) (TBind p a1 b1).
|
||||
Proof. solve_s. Qed.
|
||||
|
||||
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
|
||||
rtc CRRed.R a0 a1 ->
|
||||
rtc CRRed.R b0 b1 ->
|
||||
rtc CRRed.R (Pair a0 b0) (Pair a1 b1).
|
||||
Proof. solve_s. Qed.
|
||||
|
||||
Lemma ProjCong n p (a0 a1 : Tm n) :
|
||||
rtc CRRed.R a0 a1 ->
|
||||
rtc CRRed.R (Proj p a0) (Proj p a1).
|
||||
Proof. solve_s. Qed.
|
||||
|
||||
Lemma IfCong n (a0 a1 b0 b1 c0 c1 : Tm n) :
|
||||
rtc CRRed.R a0 a1 ->
|
||||
rtc CRRed.R b0 b1 ->
|
||||
rtc CRRed.R c0 c1 ->
|
||||
rtc CRRed.R (If a0 b0 c0) (If a1 b1 c1).
|
||||
Proof. solve_s. Qed.
|
||||
|
||||
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
|
||||
rtc CRRed.R a b -> rtc CRRed.R (ren_Tm ξ a) (ren_Tm ξ b).
|
||||
Proof.
|
||||
move => h. move : m ξ.
|
||||
induction h; hauto lq:on ctrs:rtc use:CRRed.renaming.
|
||||
Qed.
|
||||
|
||||
End CRReds.
|
||||
|
||||
|
||||
(* (***************** Beta rules only ***********************) *)
|
||||
(* Module RPar'. *)
|
||||
(* Inductive R {n} : Tm n -> Tm n -> Prop := *)
|
||||
|
|
@ -1733,10 +1871,10 @@ End RPars.
|
|||
Lemma Abs_EPar n a (b : Tm n) :
|
||||
EPar.R (Abs a) b ->
|
||||
(exists d, EPar.R a d /\
|
||||
rtc RPar.R (App (ren_Tm shift b) (VarTm var_zero)) d) /\
|
||||
rtc CRRed.R (App (ren_Tm shift b) (VarTm var_zero)) d) /\
|
||||
(exists d,
|
||||
EPar.R a d /\ forall p,
|
||||
rtc RPar.R (Proj p b) (Abs (Proj p d))).
|
||||
rtc CRRed.R (Proj p b) (Abs (Proj p d))).
|
||||
Proof.
|
||||
move E : (Abs a) => u h.
|
||||
move : a E.
|
||||
|
|
@ -1747,65 +1885,83 @@ Proof.
|
|||
split; exists d.
|
||||
+ split => //.
|
||||
apply : rtc_l.
|
||||
apply RPar.AppAbs; eauto => //=.
|
||||
apply RPar.refl.
|
||||
by apply RPar.refl.
|
||||
apply CRRed.AppAbs; eauto => //=.
|
||||
move :ih1; substify; by asimpl.
|
||||
+ split => // p.
|
||||
apply : rtc_l.
|
||||
apply : RPar.ProjAbs.
|
||||
by apply RPar.refl.
|
||||
eauto using RPars.ProjCong, RPars.AbsCong.
|
||||
apply : CRRed.ProjAbs.
|
||||
eauto using CRReds.ProjCong, CRReds.AbsCong.
|
||||
- move => n ? a1 ha iha a0 ?. subst. specialize iha with (1 := eq_refl).
|
||||
move : iha => [_ [d [ih0 ih1]]].
|
||||
split.
|
||||
+ exists (Pair (Proj PL d) (Proj PR d)).
|
||||
split; first by apply EPar.PairEta.
|
||||
apply : rtc_l.
|
||||
apply RPar.AppPair; eauto using RPar.refl.
|
||||
suff h : forall p, rtc RPar.R (App (Proj p (ren_Tm shift a1)) (VarTm var_zero)) (Proj p d) by
|
||||
sfirstorder use:RPars.PairCong.
|
||||
move => p. move /(_ p) /RPars.weakening in ih1.
|
||||
apply CRRed.AppPair.
|
||||
suff h : forall p, rtc CRRed.R (App (Proj p (ren_Tm shift a1)) (VarTm var_zero)) (Proj p d) by
|
||||
sfirstorder use:CRReds.PairCong.
|
||||
move => p. move /(_ p) /CRReds.renaming in ih1.
|
||||
apply relations.rtc_transitive with (y := App (ren_Tm shift (Abs (Proj p d))) (VarTm var_zero)).
|
||||
by eauto using RPars.AppCong, rtc_refl.
|
||||
by eauto using CRReds.AppCong, rtc_refl.
|
||||
apply relations.rtc_once => /=.
|
||||
apply : RPar.AppAbs'; eauto using RPar.refl.
|
||||
by asimpl.
|
||||
apply : CRRed.AppAbs'. by asimpl.
|
||||
+ exists d. repeat split => //. move => p.
|
||||
apply : rtc_l; eauto.
|
||||
hauto q:on use:RPar.ProjPair', RPar.refl.
|
||||
case : p; sfirstorder use:CRRed.ProjPair.
|
||||
- move => n a0 a1 ha _ ? [*]. subst.
|
||||
split.
|
||||
+ exists a1. split => //.
|
||||
apply rtc_once. apply : RPar.AppAbs'; eauto using RPar.refl. by asimpl.
|
||||
apply rtc_once. apply : CRRed.AppAbs'. by asimpl.
|
||||
+ exists a1. split => // p.
|
||||
apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl.
|
||||
apply rtc_once. apply : CRRed.ProjAbs.
|
||||
Qed.
|
||||
|
||||
Lemma Abs_EPar_If n (a : Tm (S n)) q :
|
||||
EPar.R (Abs a) q ->
|
||||
exists d, EPar.R a d /\
|
||||
forall b c, rtc RPar.R (If q b c) (Abs (If d (ren_Tm shift b) (ren_Tm shift c))).
|
||||
forall b c, exists e, rtc CRRed.R (If q b c) e /\ rtc OExp.R (Abs (If d (ren_Tm shift b) (ren_Tm shift c))) e.
|
||||
Proof.
|
||||
move E : (Abs a) => u h.
|
||||
move : a E.
|
||||
elim : n u q /h => //= n.
|
||||
- move => a0 a1 ha _ b ?. subst.
|
||||
move /Abs_EPar : ha.
|
||||
move => [[d [h0 h1]] _].
|
||||
- move => a0 a1 ha iha b ?. subst.
|
||||
(* move /Abs_EPar : ha. *)
|
||||
spec_refl.
|
||||
move : iha => [d [hd h]].
|
||||
exists d.
|
||||
split => // b0 c.
|
||||
apply : rtc_l.
|
||||
apply RPar.IfAbs; auto using RPar.refl.
|
||||
apply RPars.AbsCong.
|
||||
apply RPars.IfCong; auto using rtc_refl.
|
||||
- move => a0 a1 ha _ a ?. subst.
|
||||
move /Abs_EPar : ha => [_ [d [h0 h1]]].
|
||||
split => //.
|
||||
move => b0 c0.
|
||||
move /(_ b0 c0) : h.
|
||||
move => [e [h0 h1]].
|
||||
exists (Abs (App (ren_Tm shift e) (VarTm var_zero))).
|
||||
split.
|
||||
apply : rtc_l. apply CRRed.IfAbs.
|
||||
apply : rtc_l. apply CRRed.AbsCong.
|
||||
apply CRRed.IfApp.
|
||||
apply CRReds.AbsCong. apply CRReds.AppCong; eauto using rtc_refl.
|
||||
change (If (ren_Tm shift a1) (ren_Tm shift b0) (ren_Tm shift c0)) with (ren_Tm shift (If a1 b0 c0)).
|
||||
hauto lq:on use:CRReds.renaming.
|
||||
apply : rtc_r; eauto.
|
||||
apply OExp.AppEta.
|
||||
- move => a0 a1 ha iha a ?. subst.
|
||||
spec_refl.
|
||||
move : iha => [d [h0 h1]].
|
||||
exists d. split => //.
|
||||
move => b c.
|
||||
move => b c. move/(_ b c ): h1 => [e [h1 h2]].
|
||||
eexists; split; cycle 1.
|
||||
apply : rtc_r; eauto.
|
||||
apply OExp.PairEta.
|
||||
apply : rtc_l.
|
||||
apply RPar.IfPair; auto using RPar.refl.
|
||||
Admitted.
|
||||
apply CRRed.IfPair.
|
||||
apply : rtc_l.
|
||||
apply CRRed.PairCong0.
|
||||
apply CRRed.IfProj.
|
||||
apply : rtc_l.
|
||||
apply CRRed.PairCong1.
|
||||
apply CRRed.IfProj.
|
||||
by apply CRReds.PairCong; apply CRReds.ProjCong.
|
||||
- sauto lq: on.
|
||||
Qed.
|
||||
|
||||
|
||||
Lemma Pair_EPar n (a b c : Tm n) :
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue