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Simplify the syntax by combining proj1 and proj2

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This commit is contained in:
Yiyun Liu 2024-12-22 10:38:58 -05:00
parent e8ec23a3e7
commit ccbb9a1395
3 changed files with 109 additions and 145 deletions
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7
syntax.sig
View file

@ -1,8 +1,9 @@
Tm(VarTm) : Type
-- nat : Type
PTag : Type
PL : PTag
PR : PTag
Abs : (bind Tm in Tm) -> Tm
App : Tm -> Tm -> Tm
Pair : Tm -> Tm -> Tm
Proj1 : Tm -> Tm
Proj2 : Tm -> Tm
Proj : PTag -> Tm -> Tm

80
theories/Autosubst2/syntax.v
View file

@ -5,13 +5,26 @@ Require Import Setoid Morphisms Relation_Definitions.
Module Core.
Inductive PTag : Type :=
| PL : PTag
| PR : PTag.
Lemma congr_PL : PL = PL.
Proof.
exact (eq_refl).
Qed.
Lemma congr_PR : PR = PR.
Proof.
exact (eq_refl).
Qed.
Inductive Tm (n_Tm : nat) : Type :=
| VarTm : fin n_Tm -> Tm n_Tm
| Abs : Tm (S n_Tm) -> Tm n_Tm
| App : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Pair : Tm n_Tm -> Tm n_Tm -> Tm n_Tm
| Proj1 : Tm n_Tm -> Tm n_Tm
| Proj2 : Tm n_Tm -> Tm n_Tm.
| Proj : PTag -> Tm n_Tm -> Tm n_Tm.
Lemma congr_Abs {m_Tm : nat} {s0 : Tm (S m_Tm)} {t0 : Tm (S m_Tm)}
(H0 : s0 = t0) : Abs m_Tm s0 = Abs m_Tm t0.
@ -35,16 +48,12 @@ exact (eq_trans (eq_trans eq_refl (ap (fun x => Pair m_Tm x s1) H0))
(ap (fun x => Pair m_Tm t0 x) H1)).
Qed.
Lemma congr_Proj1 {m_Tm : nat} {s0 : Tm m_Tm} {t0 : Tm m_Tm} (H0 : s0 = t0) :
Proj1 m_Tm s0 = Proj1 m_Tm t0.
Lemma congr_Proj {m_Tm : nat} {s0 : PTag} {s1 : Tm m_Tm} {t0 : PTag}
{t1 : Tm m_Tm} (H0 : s0 = t0) (H1 : s1 = t1) :
Proj m_Tm s0 s1 = Proj m_Tm t0 t1.
Proof.
exact (eq_trans eq_refl (ap (fun x => Proj1 m_Tm x) H0)).
Qed.
Lemma congr_Proj2 {m_Tm : nat} {s0 : Tm m_Tm} {t0 : Tm m_Tm} (H0 : s0 = t0) :
Proj2 m_Tm s0 = Proj2 m_Tm t0.
Proof.
exact (eq_trans eq_refl (ap (fun x => Proj2 m_Tm x) H0)).
exact (eq_trans (eq_trans eq_refl (ap (fun x => Proj m_Tm x s1) H0))
(ap (fun x => Proj m_Tm t0 x) H1)).
Qed.
Lemma upRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n) :
@ -66,8 +75,7 @@ Fixpoint ren_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
| Abs _ s0 => Abs n_Tm (ren_Tm (upRen_Tm_Tm xi_Tm) s0)
| App _ s0 s1 => App n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
| Pair _ s0 s1 => Pair n_Tm (ren_Tm xi_Tm s0) (ren_Tm xi_Tm s1)
| Proj1 _ s0 => Proj1 n_Tm (ren_Tm xi_Tm s0)
| Proj2 _ s0 => Proj2 n_Tm (ren_Tm xi_Tm s0)
| Proj _ s0 s1 => Proj n_Tm s0 (ren_Tm xi_Tm s1)
end.
Lemma up_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm) :
@ -90,8 +98,7 @@ Fixpoint subst_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
| Abs _ s0 => Abs n_Tm (subst_Tm (up_Tm_Tm sigma_Tm) s0)
| App _ s0 s1 => App n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
| Pair _ s0 s1 => Pair n_Tm (subst_Tm sigma_Tm s0) (subst_Tm sigma_Tm s1)
| Proj1 _ s0 => Proj1 n_Tm (subst_Tm sigma_Tm s0)
| Proj2 _ s0 => Proj2 n_Tm (subst_Tm sigma_Tm s0)
| Proj _ s0 s1 => Proj n_Tm s0 (subst_Tm sigma_Tm s1)
end.
Lemma upId_Tm_Tm {m_Tm : nat} (sigma : fin m_Tm -> Tm m_Tm)
@ -126,8 +133,7 @@ subst_Tm sigma_Tm s = s :=
| Pair _ s0 s1 =>
congr_Pair (idSubst_Tm sigma_Tm Eq_Tm s0)
(idSubst_Tm sigma_Tm Eq_Tm s1)
| Proj1 _ s0 => congr_Proj1 (idSubst_Tm sigma_Tm Eq_Tm s0)
| Proj2 _ s0 => congr_Proj2 (idSubst_Tm sigma_Tm Eq_Tm s0)
| Proj _ s0 s1 => congr_Proj (eq_refl s0) (idSubst_Tm sigma_Tm Eq_Tm s1)
end.
Lemma upExtRen_Tm_Tm {m : nat} {n : nat} (xi : fin m -> fin n)
@ -164,8 +170,8 @@ Fixpoint extRen_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
| Pair _ s0 s1 =>
congr_Pair (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
(extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
| Proj1 _ s0 => congr_Proj1 (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
| Proj2 _ s0 => congr_Proj2 (extRen_Tm xi_Tm zeta_Tm Eq_Tm s0)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (extRen_Tm xi_Tm zeta_Tm Eq_Tm s1)
end.
Lemma upExt_Tm_Tm {m : nat} {n_Tm : nat} (sigma : fin m -> Tm n_Tm)
@ -204,8 +210,7 @@ Fixpoint ext_Tm {m_Tm : nat} {n_Tm : nat} (sigma_Tm : fin m_Tm -> Tm n_Tm)
| Pair _ s0 s1 =>
congr_Pair (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
(ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
| Proj1 _ s0 => congr_Proj1 (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
| Proj2 _ s0 => congr_Proj2 (ext_Tm sigma_Tm tau_Tm Eq_Tm s0)
| Proj _ s0 s1 => congr_Proj (eq_refl s0) (ext_Tm sigma_Tm tau_Tm Eq_Tm s1)
end.
Lemma up_ren_ren_Tm_Tm {k : nat} {l : nat} {m : nat} (xi : fin k -> fin l)
@ -243,8 +248,8 @@ Fixpoint compRenRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
| Pair _ s0 s1 =>
congr_Pair (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
(compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
| Proj1 _ s0 => congr_Proj1 (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
| Proj2 _ s0 => congr_Proj2 (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s0)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (compRenRen_Tm xi_Tm zeta_Tm rho_Tm Eq_Tm s1)
end.
Lemma up_ren_subst_Tm_Tm {k : nat} {l : nat} {m_Tm : nat}
@ -291,10 +296,9 @@ Fixpoint compRenSubst_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
| Pair _ s0 s1 =>
congr_Pair (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
| Proj1 _ s0 =>
congr_Proj1 (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
| Proj2 _ s0 =>
congr_Proj2 (compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s0)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compRenSubst_Tm xi_Tm tau_Tm theta_Tm Eq_Tm s1)
end.
Lemma up_subst_ren_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -362,10 +366,9 @@ ren_Tm zeta_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
| Pair _ s0 s1 =>
congr_Pair (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
| Proj1 _ s0 =>
congr_Proj1 (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
| Proj2 _ s0 =>
congr_Proj2 (compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s0)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstRen_Tm sigma_Tm zeta_Tm theta_Tm Eq_Tm s1)
end.
Lemma up_subst_subst_Tm_Tm {k : nat} {l_Tm : nat} {m_Tm : nat}
@ -434,10 +437,9 @@ subst_Tm tau_Tm (subst_Tm sigma_Tm s) = subst_Tm theta_Tm s :=
| Pair _ s0 s1 =>
congr_Pair (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
| Proj1 _ s0 =>
congr_Proj1 (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
| Proj2 _ s0 =>
congr_Proj2 (compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s0)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0)
(compSubstSubst_Tm sigma_Tm tau_Tm theta_Tm Eq_Tm s1)
end.
Lemma renRen_Tm {k_Tm : nat} {l_Tm : nat} {m_Tm : nat}
@ -546,8 +548,8 @@ Fixpoint rinst_inst_Tm {m_Tm : nat} {n_Tm : nat}
| Pair _ s0 s1 =>
congr_Pair (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
(rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
| Proj1 _ s0 => congr_Proj1 (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
| Proj2 _ s0 => congr_Proj2 (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s0)
| Proj _ s0 s1 =>
congr_Proj (eq_refl s0) (rinst_inst_Tm xi_Tm sigma_Tm Eq_Tm s1)
end.
Lemma rinstInst'_Tm {m_Tm : nat} {n_Tm : nat} (xi_Tm : fin m_Tm -> fin n_Tm)
@ -746,9 +748,7 @@ Core.
Arguments VarTm {n_Tm}.
Arguments Proj2 {n_Tm}.
Arguments Proj1 {n_Tm}.
Arguments Proj {n_Tm}.
Arguments Pair {n_Tm}.

157
theories/fp_red.v
View file

@ -17,18 +17,13 @@ Module Par.
R b0 b1 ->
R c0 c1 ->
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
| Proj1Abs a0 a1 :
| ProjAbs p a0 a1 :
R a0 a1 ->
R (Proj1 (Abs a0)) (Abs (Proj1 a0))
| Proj1Pair a0 a1 b :
R (Proj p (Abs a0)) (Abs (Proj p a1))
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R (Proj1 (Pair a0 b)) a1
| Proj2Abs a0 a1 :
R a0 a1 ->
R (Proj2 (Abs a0)) (Abs (Proj2 a0))
| Proj2Pair a0 a1 b :
R a0 a1 ->
R (Proj2 (Pair b a0)) a1
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
(****************** Eta ***********************)
| AppEta a0 a1 :
@ -36,7 +31,7 @@ Module Par.
R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
| PairEta a0 a1 :
R a0 a1 ->
R a0 (Pair (Proj1 a1) (Proj2 a1))
R a0 (Pair (Proj PL a1) (Proj PR a1))
(*************** Congruence ********************)
| Var i : R (VarTm i) (VarTm i)
@ -51,12 +46,9 @@ Module Par.
R a0 a1 ->
R b0 b1 ->
R (Pair a0 b0) (Pair a1 b1)
| Proj1Cong a0 a1 :
| ProjCong p a0 a1 :
R a0 a1 ->
R (Proj1 a0) (Proj1 a1)
| Proj2Cong a0 a1 :
R a0 a1 ->
R (Proj2 a0) (Proj2 a1).
R (Proj p a0) (Proj p a1).
End Par.
(***************** Beta rules only ***********************)
@ -72,18 +64,14 @@ Module RPar.
R b0 b1 ->
R c0 c1 ->
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
| Proj1Abs a0 a1 :
| ProjAbs p a0 a1 :
R a0 a1 ->
R (Proj1 (Abs a0)) (Abs (Proj1 a1))
| Proj1Pair a0 a1 b :
R (Proj p (Abs a0)) (Abs (Proj p a1))
| ProjPair p a0 a1 b0 b1 :
R a0 a1 ->
R (Proj1 (Pair a0 b)) a1
| Proj2Abs a0 a1 :
R a0 a1 ->
R (Proj2 (Abs a0)) (Abs (Proj2 a1))
| Proj2Pair a0 a1 b :
R a0 a1 ->
R (Proj2 (Pair b a0)) a1
R b0 b1 ->
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
(*************** Congruence ********************)
| Var i : R (VarTm i) (VarTm i)
@ -98,12 +86,9 @@ Module RPar.
R a0 a1 ->
R b0 b1 ->
R (Pair a0 b0) (Pair a1 b1)
| Proj1Cong a0 a1 :
| ProjCong p a0 a1 :
R a0 a1 ->
R (Proj1 a0) (Proj1 a1)
| Proj2Cong a0 a1 :
R a0 a1 ->
R (Proj2 a0) (Proj2 a1).
R (Proj p a0) (Proj p a1).
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
@ -119,13 +104,20 @@ Module RPar.
R (App (Abs a0) b0) t.
Proof. move => ->. apply AppAbs. Qed.
Lemma ProjPair' n p (a0 a1 b0 b1 : Tm n) t :
t = (if p is PL then a1 else b1) ->
R a0 a1 ->
R b0 b1 ->
R (Proj p (Pair a0 b0)) t.
Proof. move => > ->. apply ProjPair. 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.
move => *; apply : AppAbs'; eauto; by asimpl.
all : qauto ctrs:R.
all : qauto ctrs:R use:ProjPair'.
Qed.
Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
@ -155,13 +147,10 @@ Module RPar.
by asimpl.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:ProjPair' use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R use:morphing_up.
- qauto.
- hauto lq:on ctrs:R use:morphing_up.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
- hauto lq:on ctrs:R.
Qed.
@ -180,7 +169,7 @@ Module EPar.
R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
| PairEta a0 a1 :
R a0 a1 ->
R a0 (Pair (Proj1 a1) (Proj2 a1))
R a0 (Pair (Proj PL a1) (Proj PR a1))
(*************** Congruence ********************)
| Var i : R (VarTm i) (VarTm i)
@ -195,12 +184,9 @@ Module EPar.
R a0 a1 ->
R b0 b1 ->
R (Pair a0 b0) (Pair a1 b1)
| Proj1Cong a0 a1 :
| ProjCong p a0 a1 :
R a0 a1 ->
R (Proj1 a0) (Proj1 a1)
| Proj2Cong a0 a1 :
R a0 a1 ->
R (Proj2 a0) (Proj2 a1).
R (Proj p a0) (Proj p a1).
Lemma refl n (a : Tm n) : EPar.R a a.
Proof.
@ -242,7 +228,6 @@ Module EPar.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
- hauto q:on ctrs:R.
Qed.
Lemma substing n a0 a1 (b0 b1 : Tm n) :
@ -287,14 +272,9 @@ Module RPars.
rtc RPar.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
Lemma Proj1Cong n (a0 a1 : Tm n) :
Lemma ProjCong n p (a0 a1 : Tm n) :
rtc RPar.R a0 a1 ->
rtc RPar.R (Proj1 a0) (Proj1 a1).
Proof. solve_s. Qed.
Lemma Proj2Cong n (a0 a1 : Tm n) :
rtc RPar.R a0 a1 ->
rtc RPar.R (Proj2 a0) (Proj2 a1).
rtc RPar.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
@ -337,9 +317,8 @@ Lemma Abs_EPar n a (b : Tm n) :
(exists d, EPar.R a d /\
rtc RPar.R (App (ren_Tm shift b) (VarTm var_zero)) d) /\
(exists d,
EPar.R a d /\
rtc RPar.R (Proj1 b) (Abs (Proj1 d)) /\
rtc RPar.R (Proj2 b) (Abs (Proj2 d))).
EPar.R a d /\ forall p,
rtc RPar.R (Proj p b) (Abs (Proj p d))).
Proof.
move E : (Abs a) => u h.
move : a E.
@ -354,47 +333,35 @@ Proof.
apply RPar.refl.
by apply RPar.refl.
move :ih1; substify; by asimpl.
+ repeat split => //.
* apply : rtc_l.
apply : RPar.Proj1Abs.
+ split => // p.
apply : rtc_l.
apply : RPar.ProjAbs.
by apply RPar.refl.
eauto using RPars.Proj1Cong, RPars.AbsCong.
* apply : rtc_l.
apply : RPar.Proj2Abs.
by apply RPar.refl.
eauto using RPars.Proj2Cong, RPars.AbsCong.
eauto using RPars.ProjCong, RPars.AbsCong.
- move => n ? a1 ha iha a0 ?. subst. specialize iha with (1 := eq_refl).
move : iha => [_ [d [ih0 [ih1 ih2]]]].
move : iha => [_ [d [ih0 ih1]]].
split.
+ apply RPars.weakening in ih1, ih2.
exists (Pair (Proj1 d) (Proj2 d)).
split; first by by by apply EPar.PairEta.
+ 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 : rtc RPar.R (App (Proj1 (ren_Tm shift a1)) (VarTm var_zero)) (Proj1 d) /\
rtc RPar.R (App (Proj2 (ren_Tm shift a1)) (VarTm var_zero)) (Proj2 d)
by firstorder using RPars.PairCong.
split.
* apply relations.rtc_transitive with (y := App (ren_Tm shift (Abs (Proj1 d))) (VarTm var_zero)).
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 relations.rtc_transitive with (y := App (ren_Tm shift (Abs (Proj p d))) (VarTm var_zero)).
by eauto using RPars.AppCong, rtc_refl.
apply relations.rtc_once => /=.
apply : RPar.AppAbs'; eauto using RPar.refl.
by asimpl.
* apply relations.rtc_transitive with (y := App (ren_Tm shift (Abs (Proj2 d))) (VarTm var_zero)).
by eauto using RPars.AppCong, rtc_refl.
apply relations.rtc_once => /=.
apply : RPar.AppAbs'; eauto using RPar.refl.
by asimpl.
+ exists d. repeat split => //.
apply : rtc_l;eauto. apply RPar.Proj1Pair. eauto using RPar.refl.
apply : rtc_l;eauto. apply RPar.Proj2Pair. eauto using RPar.refl.
+ exists d. repeat split => //. move => p.
apply : rtc_l; eauto.
hauto q:on use:RPar.ProjPair', RPar.refl.
- move => n a0 a1 ha _ ? [*]. subst.
split.
+ exists a1. split => //.
apply rtc_once. apply : RPar.AppAbs'; eauto using RPar.refl. by asimpl.
+ exists a1. repeat split => //=.
apply rtc_once. apply : RPar.Proj1Abs; eauto using RPar.refl.
apply rtc_once. apply : RPar.Proj2Abs; eauto using RPar.refl.
+ exists a1. split => // p.
apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl.
Qed.
Lemma commutativity n (a b0 b1 : Tm n) :
@ -411,10 +378,9 @@ Proof.
+ hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming.
- move => n a b0 hb0 ihb0 b1 /[dup] hb1 {}/ihb0.
move => [c [ih0 ih1]].
exists (Pair (Proj1 c) (Proj2 c)). split.
+ apply RPars.PairCong.
by apply RPars.Proj1Cong.
by apply RPars.Proj2Cong.
exists (Pair (Proj PL c) (Proj PR c)). split.
+ apply RPars.PairCong;
by apply RPars.ProjCong.
+ hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming.
- hauto l:on ctrs:rtc inv:RPar.R.
- move => n a0 a1 h ih b1.
@ -440,20 +406,17 @@ Proof.
admit.
+ hauto lq:on ctrs:EPar.R use:RPars.AppCong.
- hauto lq:on ctrs:EPar.R inv:RPar.R use:RPars.PairCong.
- move => n a b0 h0 ih0 b1.
- move => n p a b0 h0 ih0 b1.
elim /RPar.inv => //= _.
+ move => a0 a1 h [*]. subst.
+ move => ? a0 a1 h [*]. subst.
move /(_ _ ltac:(by eauto using RPar.AbsCong)) : ih0 => [c [ih0 ih1]].
move /Abs_EPar : ih1 => [_ [d [ih1 ih2]]].
exists (Abs (Proj p d)).
qauto l:on ctrs:EPar.R use:RPars.ProjCong, @relations.rtc_transitive.
+ move => p0 a0 a1 b2 b3 h1 h2 [*]. subst.
move /(_ _ ltac:(by eauto using RPar.PairCong)) : ih0 => [c [ih0 ih1]].
admit.
+ move => a0 ? a1 h1 [*]. subst.
admit.
+ hauto lq:on ctrs:RPar.R, EPar.R.
- move => n a b0 h0 ih0 b1.
elim /RPar.inv => //= _.
+ move => a0 a1 ha [*]. subst.
admit.
+ move => a0 a1 b2 h [*]. subst.
admit.
+ hauto lq:on ctrs:RPar.R, EPar.R.
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
Admitted.
Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.