993 lines
30 KiB
Coq
993 lines
30 KiB
Coq
Require Import ssreflect.
|
||
Require Import FunInd.
|
||
Require Import Arith.Wf_nat.
|
||
Require Import Psatz.
|
||
From stdpp Require Import relations (rtc (..), rtc_once, rtc_r).
|
||
From Hammer Require Import Tactics.
|
||
Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
|
||
From Equations Require Import Equations.
|
||
|
||
|
||
(* Trying my best to not write C style module_funcname *)
|
||
Module Par.
|
||
Inductive R {n} : Tm n -> Tm n -> Prop :=
|
||
(***************** Beta ***********************)
|
||
| AppAbs a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
|
||
| AppPair a0 a1 b0 b1 c0 c1:
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R c0 c1 ->
|
||
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
|
||
| ProjAbs p a0 a1 :
|
||
R a0 a1 ->
|
||
R (Proj p (Abs a0)) (Abs (Proj p a1))
|
||
| ProjPair p a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (Proj p (Pair a0 b0)) (if p is PL then a1 else b1)
|
||
|
||
(****************** Eta ***********************)
|
||
| AppEta a0 a1 :
|
||
R a0 a1 ->
|
||
R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
|
||
| PairEta a0 a1 :
|
||
R a0 a1 ->
|
||
R a0 (Pair (Proj PL a1) (Proj PR a1))
|
||
|
||
(*************** Congruence ********************)
|
||
| Var i : R (VarTm i) (VarTm i)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (Abs a0) (Abs a1)
|
||
| AppCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (App a0 b0) (App a1 b1)
|
||
| PairCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (Pair a0 b0) (Pair a1 b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (Proj p a0) (Proj p a1)
|
||
| PiCong A0 A1 B0 B1:
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (Pi A0 B0) (Pi A1 B1)
|
||
| BotCong :
|
||
R Bot Bot.
|
||
|
||
|
||
Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
|
||
t = subst_Tm (scons b1 VarTm) a1 ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
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 AppEta' n (a0 a1 b : Tm n) :
|
||
b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
|
||
R a0 a1 ->
|
||
R a0 b.
|
||
Proof. move => ->; apply AppEta. 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 : match goal with
|
||
| [ |- context[var_zero]] => move => *; apply : AppEta'; eauto; by asimpl
|
||
| _ => qauto ctrs:R use:ProjPair'
|
||
end.
|
||
Qed.
|
||
|
||
End Par.
|
||
|
||
Module Pars.
|
||
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
|
||
rtc Par.R a b -> rtc Par.R (ren_Tm ξ a) (ren_Tm ξ b).
|
||
Proof.
|
||
induction 1; hauto lq:on ctrs:rtc use:Par.renaming.
|
||
Qed.
|
||
|
||
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
|
||
rtc Par.R a b ->
|
||
rtc Par.R (subst_Tm ρ a) (subst_Tm ρ b).
|
||
Admitted.
|
||
End Pars.
|
||
|
||
(***************** Beta rules only ***********************)
|
||
Module RPar.
|
||
Inductive R {n} : Tm n -> Tm n -> Prop :=
|
||
(***************** Beta ***********************)
|
||
| AppAbs a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (App (Abs a0) b0) (subst_Tm (scons b1 VarTm) a1)
|
||
| AppPair a0 a1 b0 b1 c0 c1:
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R c0 c1 ->
|
||
R (App (Pair a0 b0) c0) (Pair (App a1 c1) (App b1 c1))
|
||
| ProjAbs p a0 a1 :
|
||
R a0 a1 ->
|
||
R (Proj p (Abs a0)) (Abs (Proj p a1))
|
||
| ProjPair p a0 a1 b0 b1 :
|
||
R 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)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (Abs a0) (Abs a1)
|
||
| AppCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (App a0 b0) (App a1 b1)
|
||
| PairCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (Pair a0 b0) (Pair a1 b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (Proj p a0) (Proj p a1)
|
||
| PiCong A0 A1 B0 B1:
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (Pi A0 B0) (Pi A1 B1)
|
||
| BotCong :
|
||
R Bot Bot.
|
||
|
||
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
|
||
|
||
Lemma refl n (a : Tm n) : R a a.
|
||
Proof.
|
||
induction a; hauto lq:on ctrs:R.
|
||
Qed.
|
||
|
||
Lemma AppAbs' n a0 a1 (b0 b1 t : Tm n) :
|
||
t = subst_Tm (scons b1 VarTm) a1 ->
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
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 use:ProjPair'.
|
||
Qed.
|
||
|
||
Lemma morphing_ren n m p (ρ0 ρ1 : fin n -> Tm m) (ξ : fin m -> fin p) :
|
||
(forall i, R (ρ0 i) (ρ1 i)) ->
|
||
(forall i, R ((funcomp (ren_Tm ξ) ρ0) i) ((funcomp (ren_Tm ξ) ρ1) i)).
|
||
Proof. eauto using renaming. Qed.
|
||
|
||
Lemma morphing_ext n m (ρ0 ρ1 : fin n -> Tm m) a b :
|
||
R a b ->
|
||
(forall i, R (ρ0 i) (ρ1 i)) ->
|
||
(forall i, R ((scons a ρ0) i) ((scons b ρ1) i)).
|
||
Proof. hauto q:on inv:option. Qed.
|
||
|
||
Lemma morphing_up n m (ρ0 ρ1 : fin n -> Tm m) :
|
||
(forall i, R (ρ0 i) (ρ1 i)) ->
|
||
(forall i, R (up_Tm_Tm ρ0 i) (up_Tm_Tm ρ1 i)).
|
||
Proof. hauto l:on ctrs:R use:morphing_ext, morphing_ren unfold:up_Tm_Tm. Qed.
|
||
|
||
Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
|
||
(forall i, R (ρ0 i) (ρ1 i)) ->
|
||
R a b -> R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
|
||
Proof.
|
||
move => + h. move : m ρ0 ρ1.
|
||
elim : n a b /h.
|
||
- move => *.
|
||
apply : AppAbs'; eauto using morphing_up.
|
||
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.
|
||
- hauto lq:on ctrs:R.
|
||
- hauto lq:on ctrs:R.
|
||
- hauto lq:on ctrs:R use:morphing_up.
|
||
- hauto lq:on ctrs:R.
|
||
Qed.
|
||
|
||
Lemma substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
|
||
R a b ->
|
||
R (subst_Tm ρ a) (subst_Tm ρ b).
|
||
Proof. hauto l:on use:morphing, refl. Qed.
|
||
|
||
Lemma cong n (a b : Tm (S n)) c d :
|
||
R a b ->
|
||
R c d ->
|
||
R (subst_Tm (scons c VarTm) a) (subst_Tm (scons d VarTm) b).
|
||
Proof.
|
||
move => h0 h1. apply morphing => //=.
|
||
qauto l:on ctrs:R inv:option.
|
||
Qed.
|
||
End RPar.
|
||
|
||
Module EPar.
|
||
Inductive R {n} : Tm n -> Tm n -> Prop :=
|
||
(****************** Eta ***********************)
|
||
| AppEta a0 a1 :
|
||
R a0 a1 ->
|
||
R a0 (Abs (App (ren_Tm shift a1) (VarTm var_zero)))
|
||
| PairEta a0 a1 :
|
||
R a0 a1 ->
|
||
R a0 (Pair (Proj PL a1) (Proj PR a1))
|
||
|
||
(*************** Congruence ********************)
|
||
| Var i : R (VarTm i) (VarTm i)
|
||
| AbsCong a0 a1 :
|
||
R a0 a1 ->
|
||
R (Abs a0) (Abs a1)
|
||
| AppCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (App a0 b0) (App a1 b1)
|
||
| PairCong a0 a1 b0 b1 :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (Pair a0 b0) (Pair a1 b1)
|
||
| ProjCong p a0 a1 :
|
||
R a0 a1 ->
|
||
R (Proj p a0) (Proj p a1)
|
||
| PiCong A0 A1 B0 B1:
|
||
R A0 A1 ->
|
||
R B0 B1 ->
|
||
R (Pi A0 B0) (Pi A1 B1)
|
||
| BotCong :
|
||
R Bot Bot.
|
||
|
||
Lemma refl n (a : Tm n) : EPar.R a a.
|
||
Proof.
|
||
induction a; hauto lq:on ctrs:EPar.R.
|
||
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 => n a0 a1 ha iha m ξ /=.
|
||
move /(_ _ ξ) /AppEta : iha.
|
||
by asimpl.
|
||
|
||
all : qauto ctrs:R.
|
||
Qed.
|
||
|
||
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
|
||
|
||
Lemma AppEta' n (a0 a1 b : Tm n) :
|
||
b = (Abs (App (ren_Tm shift a1) (VarTm var_zero))) ->
|
||
R a0 a1 ->
|
||
R a0 b.
|
||
Proof. move => ->; apply AppEta. Qed.
|
||
|
||
Lemma morphing n m (a b : Tm n) (ρ0 ρ1 : fin n -> Tm m) :
|
||
R a b ->
|
||
(forall i, R (ρ0 i) (ρ1 i)) ->
|
||
R (subst_Tm ρ0 a) (subst_Tm ρ1 b).
|
||
Proof.
|
||
move => h. move : m ρ0 ρ1. elim : n a b / h => n.
|
||
- move => a0 a1 ha iha m ρ0 ρ1 hρ /=.
|
||
apply : AppEta'; eauto. by asimpl.
|
||
- hauto lq:on ctrs:R.
|
||
- hauto lq:on ctrs:R.
|
||
- hauto l:on ctrs:R use:renaming inv:option.
|
||
- hauto q:on ctrs:R.
|
||
- hauto q:on ctrs:R.
|
||
- hauto q:on ctrs:R.
|
||
- hauto l:on ctrs:R use:renaming inv:option.
|
||
- hauto lq:on ctrs:R.
|
||
Qed.
|
||
|
||
Lemma substing n a0 a1 (b0 b1 : Tm n) :
|
||
R a0 a1 ->
|
||
R b0 b1 ->
|
||
R (subst_Tm (scons b0 VarTm) a0) (subst_Tm (scons b1 VarTm) a1).
|
||
Proof.
|
||
move => h0 h1. apply morphing => //.
|
||
hauto lq:on ctrs:R inv:option.
|
||
Qed.
|
||
|
||
End EPar.
|
||
|
||
|
||
Module OExp.
|
||
Inductive R {n} : Tm n -> Tm n -> Prop :=
|
||
(****************** Eta ***********************)
|
||
| AppEta a :
|
||
R a (Abs (App (ren_Tm shift a) (VarTm var_zero)))
|
||
| PairEta a :
|
||
R a (Pair (Proj PL a) (Proj PR a)).
|
||
|
||
Lemma merge n (t a b : Tm n) :
|
||
rtc R a b ->
|
||
EPar.R t a ->
|
||
EPar.R t b.
|
||
Proof.
|
||
move => h. move : t. elim : a b /h.
|
||
- eauto using EPar.refl.
|
||
- hauto q:on ctrs:EPar.R inv:R.
|
||
Qed.
|
||
|
||
Lemma commutativity n (a b c : Tm n) :
|
||
EPar.R a b -> R a c -> exists d, R b d /\ EPar.R c d.
|
||
Proof.
|
||
move => h.
|
||
inversion 1; subst.
|
||
- hauto q:on ctrs:EPar.R, R use:EPar.renaming, EPar.refl.
|
||
- hauto lq:on ctrs:EPar.R, R.
|
||
Qed.
|
||
|
||
Lemma commutativity0 n (a b c : Tm n) :
|
||
EPar.R a b -> rtc R a c -> exists d, rtc R b d /\ EPar.R c d.
|
||
Proof.
|
||
move => + h. move : b.
|
||
elim : a c / h.
|
||
- sfirstorder.
|
||
- hauto lq:on rew:off ctrs:rtc use:commutativity.
|
||
Qed.
|
||
|
||
End OExp.
|
||
|
||
|
||
Local Ltac com_helper :=
|
||
split; [hauto lq:on ctrs:RPar.R use: RPar.refl, RPar.renaming
|
||
|hauto lq:on ctrs:EPar.R use:EPar.refl, EPar.renaming].
|
||
|
||
Module RPars.
|
||
|
||
#[local]Ltac solve_s_rec :=
|
||
move => *; eapply rtc_l; eauto;
|
||
hauto lq:on ctrs:RPar.R use:RPar.refl.
|
||
|
||
#[local]Ltac solve_s :=
|
||
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
|
||
|
||
Lemma AbsCong n (a b : Tm (S n)) :
|
||
rtc RPar.R a b ->
|
||
rtc RPar.R (Abs a) (Abs b).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
|
||
rtc RPar.R a0 a1 ->
|
||
rtc RPar.R b0 b1 ->
|
||
rtc RPar.R (App a0 b0) (App a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma PiCong n (a0 a1 : Tm n) b0 b1 :
|
||
rtc RPar.R a0 a1 ->
|
||
rtc RPar.R b0 b1 ->
|
||
rtc RPar.R (Pi a0 b0) (Pi a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
|
||
rtc RPar.R a0 a1 ->
|
||
rtc RPar.R b0 b1 ->
|
||
rtc RPar.R (Pair a0 b0) (Pair a1 b1).
|
||
Proof. solve_s. Qed.
|
||
|
||
Lemma ProjCong n p (a0 a1 : Tm n) :
|
||
rtc RPar.R a0 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) :
|
||
rtc RPar.R a0 a1 ->
|
||
rtc RPar.R (ren_Tm ξ a0) (ren_Tm ξ a1).
|
||
Proof.
|
||
induction 1.
|
||
- apply rtc_refl.
|
||
- eauto using RPar.renaming, rtc_l.
|
||
Qed.
|
||
|
||
Lemma weakening n (a0 a1 : Tm n) :
|
||
rtc RPar.R a0 a1 ->
|
||
rtc RPar.R (ren_Tm shift a0) (ren_Tm shift a1).
|
||
Proof. apply renaming. Qed.
|
||
|
||
Lemma Abs_inv n (a : Tm (S n)) b :
|
||
rtc RPar.R (Abs a) b -> exists a', b = Abs a' /\ rtc RPar.R a a'.
|
||
Proof.
|
||
move E : (Abs a) => b0 h. move : a E.
|
||
elim : b0 b / h.
|
||
- hauto lq:on ctrs:rtc.
|
||
- hauto lq:on ctrs:rtc inv:RPar.R, rtc.
|
||
Qed.
|
||
|
||
Lemma morphing n m (a b : Tm n) (ρ : fin n -> Tm m) :
|
||
rtc RPar.R a b ->
|
||
rtc RPar.R (subst_Tm ρ a) (subst_Tm ρ b).
|
||
Proof. induction 1; qauto l:on ctrs:rtc use:RPar.substing. Qed.
|
||
|
||
Lemma substing n (a b : Tm (S n)) c :
|
||
rtc RPar.R a b ->
|
||
rtc RPar.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
|
||
Proof. hauto lq:on use:morphing inv:option. Qed.
|
||
|
||
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) /\
|
||
(exists 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.
|
||
elim : n u b /h => //=.
|
||
- move => n a0 a1 ha iha b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
move : iha => [[d [ih0 ih1]] _].
|
||
split; exists d.
|
||
+ split => //.
|
||
apply : rtc_l.
|
||
apply RPar.AppAbs; eauto => //=.
|
||
apply RPar.refl.
|
||
by apply RPar.refl.
|
||
move :ih1; substify; by asimpl.
|
||
+ split => // p.
|
||
apply : rtc_l.
|
||
apply : RPar.ProjAbs.
|
||
by apply RPar.refl.
|
||
eauto using RPars.ProjCong, RPars.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 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.
|
||
+ 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. split => // p.
|
||
apply rtc_once. apply : RPar.ProjAbs; eauto using RPar.refl.
|
||
Qed.
|
||
|
||
Lemma Pair_EPar n (a b c : Tm n) :
|
||
EPar.R (Pair a b) c ->
|
||
(forall p, exists d, rtc RPar.R (Proj p c) d /\ EPar.R (if p is PL then a else b) d) /\
|
||
(exists d0 d1, rtc RPar.R (App (ren_Tm shift c) (VarTm var_zero))
|
||
(Pair (App (ren_Tm shift d0) (VarTm var_zero))(App (ren_Tm shift d1) (VarTm var_zero))) /\
|
||
EPar.R a d0 /\ EPar.R b d1).
|
||
Proof.
|
||
move E : (Pair a b) => u h. move : a b E.
|
||
elim : n u c /h => //=.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
move : iha => [_ [d0 [d1 [ih0 [ih1 ih2]]]]].
|
||
split.
|
||
+ move => p.
|
||
exists (Abs (App (ren_Tm shift (if p is PL then d0 else d1)) (VarTm var_zero))).
|
||
split.
|
||
* apply : relations.rtc_transitive.
|
||
** apply RPars.ProjCong. apply RPars.AbsCong. eassumption.
|
||
** apply : rtc_l. apply RPar.ProjAbs; eauto using RPar.refl. apply RPars.AbsCong.
|
||
apply : rtc_l. apply RPar.ProjPair; eauto using RPar.refl.
|
||
hauto l:on.
|
||
* hauto lq:on use:EPar.AppEta'.
|
||
+ exists d0, d1.
|
||
repeat split => //.
|
||
apply : rtc_l. apply : RPar.AppAbs'; eauto using RPar.refl => //=.
|
||
by asimpl; renamify.
|
||
- move => n a0 a1 ha iha a b ?. subst. specialize iha with (1 := eq_refl).
|
||
split => [p|].
|
||
+ move : iha => [/(_ p) [d [ih0 ih1]] _].
|
||
exists d. split=>//.
|
||
apply : rtc_l. apply RPar.ProjPair; eauto using RPar.refl.
|
||
set q := (X in rtc RPar.R X d).
|
||
by have -> : q = Proj p a1 by hauto lq:on.
|
||
+ move :iha => [iha _].
|
||
move : (iha PL) => [d0 [ih0 ih0']].
|
||
move : (iha PR) => [d1 [ih1 ih1']] {iha}.
|
||
exists d0, d1.
|
||
apply RPars.weakening in ih0, ih1.
|
||
repeat split => //=.
|
||
apply : rtc_l. apply RPar.AppPair; eauto using RPar.refl.
|
||
apply RPars.PairCong; apply RPars.AppCong; eauto using rtc_refl.
|
||
- move => n a0 a1 b0 b1 ha _ hb _ a b [*]. subst.
|
||
split.
|
||
+ move => p.
|
||
exists (if p is PL then a1 else b1).
|
||
split.
|
||
* apply rtc_once. apply : RPar.ProjPair'; eauto using RPar.refl.
|
||
* hauto lq:on rew:off.
|
||
+ exists a1, b1.
|
||
split. apply rtc_once. apply RPar.AppPair; eauto using RPar.refl.
|
||
split => //.
|
||
Qed.
|
||
|
||
Lemma commutativity0 n (a b0 b1 : Tm n) :
|
||
EPar.R a b0 -> RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
|
||
Proof.
|
||
move => h. move : b1.
|
||
elim : n a b0 / h.
|
||
- move => n a b0 ha iha b1 hb.
|
||
move : iha (hb) => /[apply].
|
||
move => [c [ih0 ih1]].
|
||
exists (Abs (App (ren_Tm shift c) (VarTm var_zero))).
|
||
split.
|
||
+ hauto lq:on ctrs:rtc use:RPars.AbsCong, RPars.AppCong, RPars.renaming.
|
||
+ 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 (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.
|
||
elim /RPar.inv => //= _.
|
||
move => a2 a3 ? [*]. subst.
|
||
hauto lq:on ctrs:rtc, RPar.R, EPar.R use:RPars.AbsCong.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb b2.
|
||
elim /RPar.inv => //= _.
|
||
+ move => a2 a3 b3 b4 h0 h1 [*]. subst.
|
||
move /(_ _ ltac:(by eauto)) : ihb => [b [ihb0 ihb1]].
|
||
have {}/iha : RPar.R (Abs a2) (Abs a3) by hauto lq:on ctrs:RPar.R.
|
||
move => [c [ih0 /Abs_EPar [[d [ih1 ih2]] _]]].
|
||
exists (subst_Tm (scons b VarTm) d).
|
||
split.
|
||
(* By substitution *)
|
||
* move /RPars.substing : ih2.
|
||
move /(_ b).
|
||
asimpl.
|
||
eauto using relations.rtc_transitive, RPars.AppCong.
|
||
(* By EPar morphing *)
|
||
* by apply EPar.substing.
|
||
+ move => a2 a3 b3 b4 c0 c1 h0 h1 h2 [*]. subst.
|
||
move /(_ _ ltac:(by eauto using RPar.PairCong)) : iha
|
||
=> [c [ihc0 ihc1]].
|
||
move /(_ _ ltac:(by eauto)) : ihb => [d [ihd0 ihd1]].
|
||
move /Pair_EPar : ihc1 => [_ [d0 [d1 [ih0 [ih1 ih2]]]]].
|
||
move /RPars.substing : ih0. move /(_ d).
|
||
asimpl => h.
|
||
exists (Pair (App d0 d) (App d1 d)).
|
||
split.
|
||
hauto lq:on use:relations.rtc_transitive, RPars.AppCong.
|
||
apply EPar.PairCong; by apply EPar.AppCong.
|
||
+ hauto lq:on ctrs:EPar.R use:RPars.AppCong.
|
||
- hauto lq:on ctrs:EPar.R inv:RPar.R use:RPars.PairCong.
|
||
- move => n p a b0 h0 ih0 b1.
|
||
elim /RPar.inv => //= _.
|
||
+ 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]].
|
||
move /Pair_EPar : ih1 => [/(_ p)[d [ihd ihd']] _].
|
||
exists d. split => //.
|
||
hauto lq:on use:RPars.ProjCong, relations.rtc_transitive.
|
||
+ hauto lq:on ctrs:EPar.R use:RPars.ProjCong.
|
||
- hauto lq:on inv:RPar.R ctrs:EPar.R, rtc use:RPars.PiCong.
|
||
- hauto l:on ctrs:EPar.R inv:RPar.R.
|
||
Qed.
|
||
|
||
Lemma commutativity1 n (a b0 b1 : Tm n) :
|
||
EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ EPar.R b1 c.
|
||
Proof.
|
||
move => + h. move : b0.
|
||
elim : a b1 / h.
|
||
- sfirstorder.
|
||
- qauto l:on use:relations.rtc_transitive, commutativity0.
|
||
Qed.
|
||
|
||
Lemma commutativity n (a b0 b1 : Tm n) :
|
||
rtc EPar.R a b0 -> rtc RPar.R a b1 -> exists c, rtc RPar.R b0 c /\ rtc EPar.R b1 c.
|
||
move => h. move : b1. elim : a b0 /h.
|
||
- sfirstorder.
|
||
- move => a0 a1 a2 + ha1 ih b1 +.
|
||
move : commutativity1; repeat move/[apply].
|
||
hauto q:on ctrs:rtc.
|
||
Qed.
|
||
|
||
Lemma Abs_EPar' n a (b : Tm n) :
|
||
EPar.R (Abs a) b ->
|
||
(exists d, EPar.R a d /\
|
||
rtc OExp.R (Abs d) b).
|
||
Proof.
|
||
move E : (Abs a) => u h.
|
||
move : a E.
|
||
elim : n u b /h => //=.
|
||
- move => n a0 a1 ha iha a ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- move => n a0 a1 ha iha a ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- hauto l:on ctrs:OExp.R.
|
||
Qed.
|
||
|
||
Lemma Proj_EPar' n p a (b : Tm n) :
|
||
EPar.R (Proj p a) b ->
|
||
(exists d, EPar.R a d /\
|
||
rtc OExp.R (Proj p d) b).
|
||
Proof.
|
||
move E : (Proj p a) => u h.
|
||
move : p a E.
|
||
elim : n u b /h => //=.
|
||
- move => n a0 a1 ha iha a p ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- move => n a0 a1 ha iha a p ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- hauto l:on ctrs:OExp.R.
|
||
Qed.
|
||
|
||
Lemma App_EPar' n (a b u : Tm n) :
|
||
EPar.R (App a b) u ->
|
||
(exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (App a0 b0) u).
|
||
Proof.
|
||
move E : (App a b) => t h.
|
||
move : a b E. elim : n t u /h => //=.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- hauto l:on ctrs:OExp.R.
|
||
Qed.
|
||
|
||
Lemma Pi_EPar' n (a : Tm n) b u :
|
||
EPar.R (Pi a b) u ->
|
||
(exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (Pi a0 b0) u).
|
||
Proof.
|
||
move E : (Pi a b) => t h.
|
||
move : a b E. elim : n t u /h => //=.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- hauto l:on ctrs:OExp.R.
|
||
Qed.
|
||
|
||
Lemma Pair_EPar' n (a b u : Tm n) :
|
||
EPar.R (Pair a b) u ->
|
||
exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (Pair a0 b0) u.
|
||
Proof.
|
||
move E : (Pair a b) => t h.
|
||
move : a b E. elim : n t u /h => //=.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- move => n a0 a1 ha iha a b ?. subst.
|
||
specialize iha with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- hauto l:on ctrs:OExp.R.
|
||
Qed.
|
||
|
||
Lemma Bot_EPar' n (u : Tm n) :
|
||
EPar.R Bot u ->
|
||
rtc OExp.R Bot u.
|
||
move E : Bot => t h.
|
||
move : E. elim : n t u /h => //=.
|
||
- move => n a0 a1 h ih ?. subst.
|
||
specialize ih with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- move => n a0 a1 h ih ?. subst.
|
||
specialize ih with (1 := eq_refl).
|
||
hauto lq:on ctrs:OExp.R use:rtc_r.
|
||
- hauto l:on ctrs:OExp.R.
|
||
Qed.
|
||
|
||
Lemma EPar_diamond n (c a1 b1 : Tm n) :
|
||
EPar.R c a1 ->
|
||
EPar.R c b1 ->
|
||
exists d2, EPar.R a1 d2 /\ EPar.R b1 d2.
|
||
Proof.
|
||
move => h. move : b1. elim : n c a1 / h.
|
||
- move => n c a1 ha iha b1 /iha [d2 [hd0 hd1]].
|
||
exists(Abs (App (ren_Tm shift d2) (VarTm var_zero))).
|
||
hauto lq:on ctrs:EPar.R use:EPar.renaming.
|
||
- hauto lq:on rew:off ctrs:EPar.R.
|
||
- hauto lq:on use:EPar.refl.
|
||
- move => n a0 a1 ha iha a2.
|
||
move /Abs_EPar' => [d [hd0 hd1]].
|
||
move : iha hd0; repeat move/[apply].
|
||
move => [d2 [h0 h1]].
|
||
have : EPar.R (Abs d) (Abs d2) by eauto using EPar.AbsCong.
|
||
move : OExp.commutativity0 hd1; repeat move/[apply].
|
||
move => [d1 [hd1 hd2]].
|
||
exists d1. hauto lq:on ctrs:EPar.R use:OExp.merge.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb c.
|
||
move /App_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
|
||
have : EPar.R (App a2 b2)(App a3 b3)
|
||
by hauto l:on use:EPar.AppCong.
|
||
move : OExp.commutativity0 h2; repeat move/[apply].
|
||
move => [d h].
|
||
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb c.
|
||
move /Pair_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
|
||
have : EPar.R (Pair a2 b2)(Pair a3 b3)
|
||
by hauto l:on use:EPar.PairCong.
|
||
move : OExp.commutativity0 h2; repeat move/[apply].
|
||
move => [d h].
|
||
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
|
||
- move => n p a0 a1 ha iha b.
|
||
move /Proj_EPar' => [d [/iha [d2 h] h1]] {iha}.
|
||
have : EPar.R (Proj p d) (Proj p d2)
|
||
by hauto l:on use:EPar.ProjCong.
|
||
move : OExp.commutativity0 h1; repeat move/[apply].
|
||
move => [d1 h1].
|
||
exists d1. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb c.
|
||
move /Pi_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
|
||
have : EPar.R (Pi a2 b2)(Pi a3 b3)
|
||
by hauto l:on use:EPar.PiCong.
|
||
move : OExp.commutativity0 h2; repeat move/[apply].
|
||
move => [d h].
|
||
exists d. hauto lq:on rew:off ctrs:EPar.R use:OExp.merge.
|
||
- qauto use:Bot_EPar', EPar.refl.
|
||
Qed.
|
||
|
||
Function tstar {n} (a : Tm n) :=
|
||
match a with
|
||
| VarTm i => a
|
||
| Abs a => Abs (tstar a)
|
||
| App (Abs a) b => subst_Tm (scons (tstar b) VarTm) (tstar a)
|
||
| App (Pair a b) c =>
|
||
Pair (App (tstar a) (tstar c)) (App (tstar b) (tstar c))
|
||
| App a b => App (tstar a) (tstar b)
|
||
| Pair a b => Pair (tstar a) (tstar b)
|
||
| Proj p (Pair a b) => if p is PL then (tstar a) else (tstar b)
|
||
| Proj p (Abs a) => (Abs (Proj p (tstar a)))
|
||
| Proj p a => Proj p (tstar a)
|
||
| Pi a b => Pi (tstar a) (tstar b)
|
||
| Bot => Bot
|
||
end.
|
||
|
||
Lemma RPar_triangle n (a : Tm n) : forall b, RPar.R a b -> RPar.R b (tstar a).
|
||
Proof.
|
||
apply tstar_ind => {n a}.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R.
|
||
- hauto lq:on use:RPar.cong, RPar.refl ctrs:RPar.R inv:RPar.R.
|
||
- hauto lq:on rew:off ctrs:RPar.R inv:RPar.R.
|
||
- hauto lq:on rew:off inv:RPar.R ctrs:RPar.R.
|
||
- hauto lq:on rew:off inv:RPar.R ctrs:RPar.R.
|
||
- hauto drew:off inv:RPar.R use:RPar.refl, RPar.ProjPair'.
|
||
- hauto drew:off inv:RPar.R use:RPar.refl, RPar.ProjPair'.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R.
|
||
- hauto lq:on inv:RPar.R ctrs:RPar.R.
|
||
Qed.
|
||
|
||
Lemma RPar_diamond n (c a1 b1 : Tm n) :
|
||
RPar.R c a1 ->
|
||
RPar.R c b1 ->
|
||
exists d2, RPar.R a1 d2 /\ RPar.R b1 d2.
|
||
Proof. hauto l:on use:RPar_triangle. Qed.
|
||
|
||
Lemma RPar_confluent n (c a1 b1 : Tm n) :
|
||
rtc RPar.R c a1 ->
|
||
rtc RPar.R c b1 ->
|
||
exists d2, rtc RPar.R a1 d2 /\ rtc RPar.R b1 d2.
|
||
Proof.
|
||
sfirstorder use:relations.diamond_confluent, RPar_diamond.
|
||
Qed.
|
||
|
||
Lemma EPar_confluent n (c a1 b1 : Tm n) :
|
||
rtc EPar.R c a1 ->
|
||
rtc EPar.R c b1 ->
|
||
exists d2, rtc EPar.R a1 d2 /\ rtc EPar.R b1 d2.
|
||
Proof.
|
||
sfirstorder use:relations.diamond_confluent, EPar_diamond.
|
||
Qed.
|
||
|
||
Fixpoint depth_tm {n} (a : Tm n) :=
|
||
match a with
|
||
| VarTm _ => 1
|
||
| Pi A B => 1 + max (depth_tm A) (depth_tm B)
|
||
| Abs a => 1 + depth_tm a
|
||
| App a b => 1 + max (depth_tm a) (depth_tm b)
|
||
| Proj p a => 1 + depth_tm a
|
||
| Pair a b => 1 + max (depth_tm a) (depth_tm b)
|
||
| Bot => 1
|
||
end.
|
||
|
||
Lemma depth_ren n m (ξ: fin n -> fin m) a :
|
||
depth_tm a = depth_tm (ren_Tm ξ a).
|
||
Proof.
|
||
move : m ξ. elim : n / a; scongruence.
|
||
Qed.
|
||
|
||
Local Ltac prov_tac := sfirstorder use:depth_ren.
|
||
|
||
#[tactic="prov_tac"]Equations prov {n} (A : Tm n) (B : Tm (S n)) (a : Tm n) : Prop by wf (depth_tm a) lt :=
|
||
prov A B (Pi A0 B0) := rtc Par.R A A0 /\ rtc Par.R B B0;
|
||
prov A B (Abs a) := prov (ren_Tm shift A) (ren_Tm (upRen_Tm_Tm shift) B) a;
|
||
prov A B (App a b) := prov A B a;
|
||
prov A B (Pair a b) := prov A B a /\ prov A B b;
|
||
prov A B (Proj p a) := prov A B a;
|
||
prov A B Bot := False;
|
||
prov A B (VarTm _) := False.
|
||
|
||
Equations extract {n} (a : Tm n) : Tm n by wf (depth_tm a) lt :=
|
||
extract (Pi A B) := Pi A B;
|
||
extract (Abs a) := extract (subst_Tm (scons Bot VarTm) a);
|
||
extract (App a b) := extract a;
|
||
extract (Pair a b) := extract a;
|
||
extract (Proj p a) := extract a;
|
||
extract Bot := Bot;
|
||
extract (VarTm _) := Bot.
|
||
Next Obligation.
|
||
Admitted.
|
||
|
||
Next Obligation.
|
||
sfirstorder.
|
||
Qed.
|
||
|
||
Next Obligation.
|
||
sfirstorder.
|
||
Qed.
|
||
|
||
Lemma prov_extract n A B (a : Tm n) :
|
||
prov A B a -> exists A0 B0,
|
||
extract a = Pi A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0.
|
||
Proof.
|
||
move : A B. elim : n / a => //=.
|
||
- move => n a ih A B.
|
||
simp prov.
|
||
move /ih.
|
||
simp extract.
|
||
move => [A0][B0][h0][h1]h2.
|
||
(* anti renaming for par *)
|
||
have : exists A1, rtc Par.R A A1 /\ ren_Tm shift A1 = A0 by admit.
|
||
move => [A1 [h3 h4]].
|
||
have : exists B1, rtc Par.R B B1 /\ ren_Tm (upRen_Tm_Tm shift) B1 = B0
|
||
by admit.
|
||
move => [B1 [h5 h6]].
|
||
subst.
|
||
have {}h0 : extract a = ren_Tm shift (Pi A1 B1) by done.
|
||
have : exists a1, extract a1 = Pi A1 B1 /\ ren_Tm shift a1 = a by admit.
|
||
move => [a1 [h6 ?]]. subst.
|
||
asimpl. exists A1, B1.
|
||
repeat split => //=.
|
||
- hauto l:on rew:db:prov, extract.
|
||
- hauto l:on rew:db:prov, extract.
|
||
- hauto l:on rew:db:prov, extract.
|
||
- qauto l:on rew:db:prov, extract.
|
||
Admitted.
|
||
|
||
Lemma tm_depth_ind (P : forall n, Tm n -> Prop) :
|
||
(forall n (a : Tm n), (forall m (b : Tm m), depth_tm b < depth_tm a -> P m b) -> P n a) -> forall n a, P n a.
|
||
move => ih.
|
||
suff : forall m n (a : Tm n), depth_tm a <= m -> P n a by sfirstorder.
|
||
elim.
|
||
- move => n a h.
|
||
apply ih. lia.
|
||
- move => n ih0 m a h.
|
||
apply : ih.
|
||
move => m0 b h0.
|
||
apply : ih0.
|
||
lia.
|
||
Qed.
|
||
|
||
Lemma prov_ren n m (ξ : fin n -> fin m) A B a :
|
||
prov A B a -> prov (ren_Tm ξ A) (ren_Tm (upRen_Tm_Tm ξ) B) (ren_Tm ξ a).
|
||
Proof.
|
||
move : m ξ A B. elim : n / a.
|
||
- sfirstorder rew:db:prov.
|
||
- move => n a ih m ξ A B.
|
||
simp prov. simpl.
|
||
move /ih => {ih}.
|
||
move /(_ _ (upRen_Tm_Tm ξ)).
|
||
simp prov. by asimpl.
|
||
- hauto q:on rew:db:prov.
|
||
- qauto l:on rew:db:prov.
|
||
- hauto lq:on rew:db:prov.
|
||
- move => n A0 ih B0 h0 m ξ A B. simpl.
|
||
simp prov.
|
||
hauto l:on use:Pars.renaming.
|
||
- sfirstorder.
|
||
Qed.
|
||
|
||
Lemma prov_morph n m (ρ : fin n -> Tm m) A B a :
|
||
prov A B a ->
|
||
prov (subst_Tm ρ A) (subst_Tm (up_Tm_Tm ρ) B) (subst_Tm ρ a).
|
||
Proof.
|
||
move : m ρ A B. elim : n / a.
|
||
- hauto q:on rew:db:prov.
|
||
- move => n a ih m ρ A B.
|
||
simp prov => /=.
|
||
move /ih => {ih}.
|
||
move /(_ _ (up_Tm_Tm ρ)). asimpl.
|
||
simp prov. by asimpl.
|
||
- hauto q:on rew:db:prov.
|
||
- hauto q:on rew:db:prov.
|
||
- hauto lq:on rew:db:prov.
|
||
- hauto l:on use:Pars.substing rew:db:prov.
|
||
- qauto rew:db:prov.
|
||
Qed.
|
||
|
||
Lemma EPar_Par n (a b : Tm n) : EPar.R a b -> Par.R a b.
|
||
Proof.
|
||
move => h. elim : n a b /h; qauto ctrs:Par.R.
|
||
Qed.
|
||
|
||
Lemma prov_par n (A : Tm n) B a b : prov A B a -> Par.R a b -> prov A B b.
|
||
Proof.
|
||
move => + h. move : A B.
|
||
elim : n a b /h.
|
||
- move => n a0 a1 b0 b1 ha iha hb ihb A B /=.
|
||
simp prov => h.
|
||
have : prov (ren_Tm shift A) (ren_Tm (upRen_Tm_Tm shift) B) a1 by admit.
|
||
move /(prov_morph _ _ (scons b1 VarTm)).
|
||
by asimpl.
|
||
- hauto lq:on rew:db:prov.
|
||
- hauto lq:on rew:db:prov.
|
||
- hauto lq:on rew:db:prov.
|
||
- move => n a0 a1 ha iha A B. simp prov. move /iha.
|
||
hauto l:on use:prov_ren.
|
||
- hauto l:on rew:db:prov.
|
||
- simp prov.
|
||
- move => n a0 a1 ha iha A B. simp prov.
|
||
- hauto l:on rew:db:prov.
|
||
- hauto l:on rew:db:prov.
|
||
- hauto lq:on rew:db:prov.
|
||
- move => n A0 A1 B0 B1 hA ihA hB ihB A B. simp prov.
|
||
move => [hA0 hA1].
|
||
eauto using rtc_r.
|
||
- sfirstorder.
|
||
Admitted.
|
||
|
||
Lemma Par_confluent n (c a1 b1 : Tm n) :
|
||
rtc Par.R c a1 ->
|
||
rtc Par.R c b1 ->
|
||
exists d2, rtc Par.R a1 d2 /\ rtc Par.R b1 d2.
|
||
Proof.
|
||
Admitted.
|