From Ltac2 Require Ltac2.
Import Ltac2.Notations.
Import Ltac2.Control.
Require Import ssreflect ssrbool.
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.
Ltac2 spec_refl () :=
List.iter
(fun a => match a with
| (i, _, _) =>
let h := Control.hyp i in
try (specialize $h with (1 := eq_refl))
end) (Control.hyps ()).
Ltac spec_refl := ltac2:(spec_refl ()).
(* 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)
| BindCong p A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i)
| BotCong :
R Bot Bot.
Lemma refl n (a : Tm n) : R a a.
elim : n /a; hauto 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 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.
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 => n a0 a1 b0 b1 ha iha hb ihb m ρ0 ρ1 hρ /=.
eapply AppAbs' with (a1 := subst_Tm (up_Tm_Tm ρ1) a1); eauto.
by asimpl.
hauto l:on use:renaming inv:option.
- hauto lq:on rew:off ctrs:R.
- hauto l:on inv:option use:renaming ctrs:R.
- hauto lq:on use:ProjPair'.
- move => n a0 a1 ha iha m ρ0 ρ1 hρ /=.
apply : AppEta'; eauto. by asimpl.
- hauto lq:on ctrs:R.
- sfirstorder.
- hauto l:on inv:option ctrs:R use:renaming.
- hauto q:on ctrs:R.
- qauto l:on ctrs:R.
- qauto l:on ctrs:R.
- hauto l:on inv:option ctrs:R use:renaming.
- sfirstorder.
- sfirstorder.
- sfirstorder.
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 antirenaming n m (a : Tm n) (b : Tm m) (ξ : fin n -> fin m) :
R (ren_Tm ξ a) b -> exists b0, R a b0 /\ ren_Tm ξ b0 = b.
Proof.
move E : (ren_Tm ξ a) => u h.
move : n ξ a E. elim : m u b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//=.
move => c c0 [+ ?]. subst.
case : c => //=.
move => c [?]. subst.
spec_refl.
move : iha => [c1][ih0]?. subst.
move : ihb => [c2][ih1]?. subst.
eexists. split.
apply AppAbs; eauto.
by asimpl.
- move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ξ []//=.
move => []//= t t0 t1 [*]. subst.
spec_refl.
move : iha => [? [*]].
move : ihb => [? [*]].
move : ihc => [? [*]].
eexists. split.
apply AppPair; hauto. subst.
by asimpl.
- move => n p a0 a1 ha iha m ξ []//= p0 []//= t [*]. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjAbs; eauto. by asimpl.
- move => n p a0 a1 b0 b1 ha iha hb ihb m ξ []//= p0 []//= t t0[*].
subst. spec_refl.
move : iha => [b0 [? ?]].
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
- move => n a0 a1 ha iha m ξ a ?. subst.
spec_refl. move : iha => [a0 [? ?]]. subst.
eexists. split. apply AppEta; eauto.
by asimpl.
- move => n a0 a1 ha iha m ξ a ?. subst.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply PairEta; eauto.
by asimpl.
- move => n i m ξ []//=.
hauto l:on.
- move => n a0 a1 ha iha m ξ []//= t [*]. subst.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
by asimpl.
- move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0 [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply AppCong; eauto.
done.
- move => n a0 a1 b0 b1 ha iha hb ihb m ξ []//= t t0[*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply PairCong; eauto.
by asimpl.
- move => n p a0 a1 ha iha m ξ []//= p0 t [*]. subst.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. by apply ProjCong; eauto.
by asimpl.
- move => n p A0 A1 B0 B1 ha iha hB ihB m ξ []//= ? t t0 [*]. subst.
spec_refl.
move : iha => [b0 [? ?]].
move : ihB => [c0 [? ?]]. subst.
eexists. split. by apply BindCong; eauto.
by asimpl.
- move => n k m ξ []//=. hauto l:on.
- move => n i n0 ξ []//=. hauto l:on.
- hauto q:on inv:Tm ctrs:R.
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).
induction 1; hauto l:on ctrs:rtc use:Par.substing.
Qed.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ξ : fin n -> fin m) :
rtc Par.R (ren_Tm ξ a) b -> exists b0, rtc Par.R a b0 /\ ren_Tm ξ b0 = b.
Proof.
move E :(ren_Tm ξ a) => u h.
move : a E.
elim : u b /h.
- sfirstorder.
- move => a b c h0 h1 ih1 a0 ?. subst.
move /Par.antirenaming : h0.
move => [b0 [h2 ?]]. subst.
hauto lq:on rew:off ctrs:rtc.
Qed.
#[local]Ltac solve_s_rec :=
move => *; eapply rtc_l; eauto;
hauto lq:on ctrs:Par.R use:Par.refl.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
Lemma ProjCong n p (a0 a1 : Tm n) :
rtc Par.R a0 a1 ->
rtc Par.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
rtc Par.R a0 a1 ->
rtc Par.R b0 b1 ->
rtc Par.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
rtc Par.R a0 a1 ->
rtc Par.R b0 b1 ->
rtc Par.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
Lemma AbsCong n (a b : Tm (S n)) :
rtc Par.R a b ->
rtc Par.R (Abs a) (Abs b).
Proof. solve_s. Qed.
End Pars.
Definition var_or_const {n} (a : Tm n) :=
match a with
| VarTm _ => true
| Bot => true
| _ => false
end.
(***************** 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)
| BindCong p A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i)
| 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.
- hauto lq:on ctrs:R.
- 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.
Lemma var_or_const_imp {n} (a b : Tm n) :
var_or_const a ->
a = b -> ~~ var_or_const b -> False.
Proof.
hauto lq:on inv:Tm.
Qed.
Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
(forall i, var_or_const (up_Tm_Tm ρ i)).
Proof.
move => h /= [i|].
- asimpl.
move /(_ i) in h.
rewrite /funcomp.
move : (ρ i) h.
case => //=.
- sfirstorder.
Qed.
Local Ltac antiimp := qauto l:on use:var_or_const_imp.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E : (subst_Tm ρ a) => u hρ h.
move : n ρ hρ a E. elim : m u b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => c c0 [+ ?]. subst.
case : c => //=; first by antiimp.
move => c [?]. subst.
spec_refl.
have /var_or_const_up hρ' := hρ.
move : iha hρ' => /[apply] iha.
move : ihb hρ => /[apply] ihb.
spec_refl.
move : iha => [c1][ih0]?. subst.
move : ihb => [c2][ih1]?. subst.
eexists. split.
apply AppAbs; eauto.
by asimpl.
- move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc m ρ hρ []//=;
first by antiimp.
move => []//=; first by antiimp.
move => t t0 t1 [*]. subst.
have {}/iha := hρ => iha.
have {}/ihb := hρ => ihb.
have {}/ihc := hρ => ihc.
spec_refl.
move : iha => [? [*]].
move : ihb => [? [*]].
move : ihc => [? [*]].
eexists. split.
apply AppPair; hauto. subst.
by asimpl.
- move => n p a0 a1 ha iha m ρ hρ []//=;
first by antiimp.
move => p0 []//= t [*]; first by antiimp. subst.
have /var_or_const_up {}/iha := hρ => iha.
spec_refl. move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjAbs; eauto. by asimpl.
- move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => p0 []//=; first by antiimp. move => t t0[*].
subst.
have {}/iha := (hρ) => iha.
have {}/ihb := (hρ) => ihb.
spec_refl.
move : iha => [b0 [? ?]].
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
- move => n i m ρ hρ []//=.
hauto l:on.
- move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
move => t [*]. subst.
have /var_or_const_up {}/iha := hρ => iha.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
by asimpl.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => t t0 [*]. subst.
have {}/iha := (hρ) => iha.
have {}/ihb := (hρ) => ihb.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply AppCong; eauto.
done.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => t t0[*]. subst.
have {}/iha := (hρ) => iha.
have {}/ihb := (hρ) => ihb.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply PairCong; eauto.
by asimpl.
- move => n p a0 a1 ha iha m ρ hρ []//=;
first by antiimp.
move => p0 t [*]. subst.
have {}/iha := (hρ) => iha.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjCong; eauto. reflexivity.
- move => n p A0 A1 B0 B1 ha iha hB ihB m ρ hρ []//=;
first by antiimp.
move => ? t t0 [*]. subst.
have {}/iha := (hρ) => iha.
have /var_or_const_up {}/ihB := (hρ) => ihB.
spec_refl.
move : iha => [b0 [? ?]].
move : ihB => [c0 [? ?]]. subst.
eexists. split. by apply BindCong; eauto.
by asimpl.
- hauto q:on ctrs:R inv:Tm.
- move => n i n0 ρ hρ []//=; first by antiimp.
hauto l:on.
- move => n m ρ hρ []//=; hauto lq:on ctrs:R.
Qed.
End RPar.
(***************** 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)
| 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)
| BindCong p A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i)
| 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 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.
- hauto lq:on ctrs:R.
- 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.
Lemma var_or_const_imp {n} (a b : Tm n) :
var_or_const a ->
a = b -> ~~ var_or_const b -> False.
Proof.
hauto lq:on inv:Tm.
Qed.
Lemma var_or_const_up n m (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
(forall i, var_or_const (up_Tm_Tm ρ i)).
Proof.
move => h /= [i|].
- asimpl.
move /(_ i) in h.
rewrite /funcomp.
move : (ρ i) h.
case => //=.
- sfirstorder.
Qed.
Local Ltac antiimp := qauto l:on use:var_or_const_imp.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
R (subst_Tm ρ a) b -> exists b0, R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E : (subst_Tm ρ a) => u hρ h.
move : n ρ hρ a E. elim : m u b/h.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => c c0 [+ ?]. subst.
case : c => //=; first by antiimp.
move => c [?]. subst.
spec_refl.
have /var_or_const_up hρ' := hρ.
move : iha hρ' => /[apply] iha.
move : ihb hρ => /[apply] ihb.
spec_refl.
move : iha => [c1][ih0]?. subst.
move : ihb => [c2][ih1]?. subst.
eexists. split.
apply AppAbs; eauto.
by asimpl.
- move => n p a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => p0 []//=; first by antiimp. move => t t0[*].
subst.
have {}/iha := (hρ) => iha.
have {}/ihb := (hρ) => ihb.
spec_refl.
move : iha => [b0 [? ?]].
move : ihb => [c0 [? ?]]. subst.
eexists. split. by eauto using ProjPair.
hauto q:on.
- move => n i m ρ hρ []//=.
hauto l:on.
- move => n a0 a1 ha iha m ρ hρ []//=; first by antiimp.
move => t [*]. subst.
have /var_or_const_up {}/iha := hρ => iha.
spec_refl.
move :iha => [b0 [? ?]]. subst.
eexists. split. by apply AbsCong; eauto.
by asimpl.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => t t0 [*]. subst.
have {}/iha := (hρ) => iha.
have {}/ihb := (hρ) => ihb.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply AppCong; eauto.
done.
- move => n a0 a1 b0 b1 ha iha hb ihb m ρ hρ []//=;
first by antiimp.
move => t t0[*]. subst.
have {}/iha := (hρ) => iha.
have {}/ihb := (hρ) => ihb.
spec_refl.
move : iha => [b0 [? ?]]. subst.
move : ihb => [c0 [? ?]]. subst.
eexists. split. by apply PairCong; eauto.
by asimpl.
- move => n p a0 a1 ha iha m ρ hρ []//=;
first by antiimp.
move => p0 t [*]. subst.
have {}/iha := (hρ) => iha.
spec_refl.
move : iha => [b0 [? ?]]. subst.
eexists. split. apply ProjCong; eauto. reflexivity.
- move => n p A0 A1 B0 B1 ha iha hB ihB m ρ hρ []//=;
first by antiimp.
move => ? t t0 [*]. subst.
have {}/iha := (hρ) => iha.
have /var_or_const_up {}/ihB := (hρ) => ihB.
spec_refl.
move : iha => [b0 [? ?]].
move : ihB => [c0 [? ?]]. subst.
eexists. split. by apply BindCong; eauto.
by asimpl.
- hauto q:on ctrs:R inv:Tm.
- move => n i n0 ρ hρ []//=; first by antiimp.
hauto l:on.
- hauto q:on inv:Tm ctrs:R.
Qed.
End RPar'.
Module ERed.
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))
(*************** 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).
Derive Dependent Inversion inv with (forall n (a b : Tm n), R a b) Sort Prop.
Lemma AppEta' n a (u : Tm n) :
u = (Abs (App (ren_Tm shift a) (VarTm var_zero))) ->
R a u.
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 => n a m ξ.
apply AppEta'. by asimpl.
all : qauto ctrs:R.
Qed.
Lemma substing n m (a : Tm n) b (ρ : fin n -> Tm m) :
R a b ->
R (subst_Tm ρ a) (subst_Tm ρ b).
Proof.
move => h. move : m ρ. elim : n a b / h => n.
move => a m ρ /=.
apply : AppEta'; eauto. by asimpl.
all : hauto ctrs:R inv:option use:renaming.
Qed.
End ERed.
Module EReds.
#[local]Ltac solve_s_rec :=
move => *; eapply rtc_l; eauto;
hauto lq:on ctrs:ERed.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 ERed.R a b ->
rtc ERed.R (Abs a) (Abs b).
Proof. solve_s. Qed.
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
rtc ERed.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
Lemma BindCong n p (a0 a1 : Tm n) b0 b1 :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
rtc ERed.R (TBind p a0 b0) (TBind p a1 b1).
Proof. solve_s. Qed.
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
rtc ERed.R a0 a1 ->
rtc ERed.R b0 b1 ->
rtc ERed.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
Lemma ProjCong n p (a0 a1 : Tm n) :
rtc ERed.R a0 a1 ->
rtc ERed.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
End EReds.
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)
| BindCong p A0 A1 B0 B1:
R A0 A1 ->
R B0 B1 ->
R (TBind p A0 B0) (TBind p A1 B1)
| ConstCong k :
R (Const k) (Const k)
| UnivCong i :
R (Univ i) (Univ i)
| 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.
- hauto lq:on ctrs:R.
- 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 BindCong n p (a0 a1 : Tm n) b0 b1 :
rtc RPar.R a0 a1 ->
rtc RPar.R b0 b1 ->
rtc RPar.R (TBind p a0 b0) (TBind p 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.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
rtc RPar.R (subst_Tm ρ a) b -> exists b0, rtc RPar.R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E :(subst_Tm ρ a) => u hρ h.
move : a E.
elim : u b /h.
- sfirstorder.
- move => a b c h0 h1 ih1 a0 ?. subst.
move /RPar.antirenaming : h0.
move /(_ hρ).
move => [b0 [h2 ?]]. subst.
hauto lq:on rew:off ctrs:rtc.
Qed.
End RPars.
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 BindCong n p (a0 a1 : Tm n) b0 b1 :
rtc RPar'.R a0 a1 ->
rtc RPar'.R b0 b1 ->
rtc RPar'.R (TBind p a0 b0) (TBind p 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.
Lemma antirenaming n m (a : Tm n) (b : Tm m) (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
rtc RPar'.R (subst_Tm ρ a) b -> exists b0, rtc RPar'.R a b0 /\ subst_Tm ρ b0 = b.
Proof.
move E :(subst_Tm ρ a) => u hρ h.
move : a E.
elim : u b /h.
- sfirstorder.
- move => a b c h0 h1 ih1 a0 ?. subst.
move /RPar'.antirenaming : h0.
move /(_ hρ).
move => [b0 [h2 ?]]. subst.
hauto lq:on rew:off ctrs:rtc.
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.BindCong.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- hauto l:on ctrs:EPar.R inv:RPar.R.
- 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 Bind_EPar' n p (a : Tm n) b u :
EPar.R (TBind p a b) u ->
(exists a0 b0, EPar.R a a0 /\ EPar.R b b0 /\ rtc OExp.R (TBind p a0 b0) u).
Proof.
move E : (TBind p 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 Const_EPar' n k (u : Tm n) :
EPar.R (Const k) u ->
rtc OExp.R (Const k) u.
move E : (Const k) => t h.
move : k E. elim : n t u /h => //=.
- move => n a0 a1 h ih k ?. subst.
specialize ih with (1 := eq_refl).
hauto lq:on ctrs:OExp.R use:rtc_r.
- move => n a0 a1 h ih k ?. subst.
specialize ih 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 Univ_EPar' n i (u : Tm n) :
EPar.R (Univ i) u ->
rtc OExp.R (Univ i) u.
move E : (Univ i) => 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 p a0 a1 b0 b1 ha iha hb ihb c.
move /Bind_EPar' => [a2][b2][/iha [a3 h0]][/ihb [b3 h1]]h2 {iha ihb}.
have : EPar.R (TBind p a2 b2)(TBind p a3 b3)
by hauto l:on use:EPar.BindCong.
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:Const_EPar', EPar.refl.
- qauto use:Univ_EPar', EPar.refl.
- 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)
| TBind p a b => TBind p (tstar a) (tstar b)
| Const k => Const k
| Univ i => Univ i
| 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.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
- hauto lq:on inv:RPar.R ctrs:RPar.R.
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 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 a => Proj p (tstar' a)
| TBind p a b => TBind p (tstar' a) (tstar' b)
| Const k => Const k
| Univ i => Univ i
| 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 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.
- 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'_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.
Definition prov_bind {n} p0 A0 B0 (a : Tm n) :=
match a with
| TBind p A B => p = p0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
| _ => False
end.
Definition prov_univ {n} i0 (a : Tm n) :=
match a with
| Univ i => i = i0
| _ => False
end.
Inductive prov {n} : Tm n -> Tm n -> Prop :=
| P_Bind p A A0 B B0 :
rtc Par.R A A0 ->
rtc Par.R B B0 ->
prov (TBind p A B) (TBind p A0 B0)
| P_Abs h a :
(forall b, prov h (subst_Tm (scons b VarTm) a)) ->
prov h (Abs a)
| P_App h a b :
prov h a ->
prov h (App a b)
| P_Pair h a b :
prov h a ->
prov h b ->
prov h (Pair a b)
| P_Proj h p a :
prov h a ->
prov h (Proj p a)
| P_Const k :
prov (Const k) (Const k)
| P_Var i :
prov (VarTm i) (VarTm i)
| P_Univ i :
prov (Univ i) (Univ i)
| P_Bot :
prov Bot Bot.
Lemma ERed_EPar n (a b : Tm n) : ERed.R a b -> EPar.R a b.
Proof.
induction 1; hauto lq:on ctrs:EPar.R use:EPar.refl.
Qed.
Lemma EPar_ERed n (a b : Tm n) : EPar.R a b -> rtc ERed.R a b.
Proof.
move => h. elim : n a b /h.
- eauto using rtc_r, ERed.AppEta.
- eauto using rtc_r, ERed.PairEta.
- auto using rtc_refl.
- eauto using EReds.AbsCong.
- eauto using EReds.AppCong.
- eauto using EReds.PairCong.
- eauto using EReds.ProjCong.
- eauto using EReds.BindCong.
- auto using rtc_refl.
- auto using rtc_refl.
- auto using rtc_refl.
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 RPar_Par n (a b : Tm n) : RPar.R a b -> Par.R a b.
Proof.
move => h. elim : n a b /h; hauto lq:on ctrs:Par.R.
Qed.
Lemma rtc_idem n (R : Tm n -> Tm n -> Prop) (a b : Tm n) : rtc (rtc R) a b -> rtc R a b.
Proof.
induction 1; hauto l:on use:@relations.rtc_transitive, @rtc_r.
Qed.
Lemma EPars_EReds {n} (a b : Tm n) : rtc EPar.R a b <-> rtc ERed.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel, EPar_ERed, rtc_idem, ERed_EPar.
Qed.
Lemma prov_rpar n (u : Tm n) a b : prov u a -> RPar.R a b -> prov u b.
Proof.
move => h.
move : b.
elim : u a / h.
- qauto l:on ctrs:prov inv:RPar.R use:@rtc_r, RPar_Par.
- hauto lq:on ctrs:prov inv:RPar.R use:RPar.substing.
- move => h a b ha iha b0.
elim /RPar.inv => //= _.
+ move => a0 a1 b1 b2 h0 h1 [*]. subst.
have {}iha : prov h (Abs a1) by hauto lq:on ctrs:RPar.R.
hauto lq:on inv:prov use:RPar.substing.
+ move => a0 a1 b1 b2 c0 c1.
move => h0 h1 h2 [*]. subst.
have {}iha : prov h (Pair a1 b2) by hauto lq:on ctrs:RPar.R.
hauto lq:on inv:prov ctrs:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on ctrs:prov inv:RPar.R.
- move => h p a ha iha b.
elim /RPar.inv => //= _.
+ move => p0 a0 a1 h0 [*]. subst.
have {iha} : prov h (Abs a1) by hauto lq:on ctrs:RPar.R.
hauto lq:on ctrs:prov inv:prov use:RPar.substing.
+ move => p0 a0 a1 b0 b1 h0 h1 [*]. subst.
have {iha} : prov h (Pair a1 b1) by hauto lq:on ctrs:RPar.R.
qauto l:on inv:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on ctrs:prov inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
- hauto l:on ctrs:RPar.R inv:RPar.R.
Qed.
Lemma prov_lam n (u : Tm n) a : prov u a <-> prov u (Abs (App (ren_Tm shift a) (VarTm var_zero))).
Proof.
split.
move => h. constructor. move => b. asimpl. by constructor.
inversion 1; subst.
specialize H2 with (b := Const TPi).
move : H2. asimpl. inversion 1; subst. done.
Qed.
Lemma prov_pair n (u : Tm n) a : prov u a <-> prov u (Pair (Proj PL a) (Proj PR a)).
Proof. hauto lq:on inv:prov ctrs:prov. Qed.
Lemma prov_ered n (u : Tm n) a b : prov u a -> ERed.R a b -> prov u b.
Proof.
move => h.
move : b.
elim : u a / h.
- move => p A A0 B B0 hA hB b.
elim /ERed.inv => // _.
+ move => a0 *. subst.
rewrite -prov_lam.
by constructor.
+ move => a0 *. subst.
rewrite -prov_pair.
by constructor.
+ qauto l:on ctrs:prov use:@rtc_r, ERed_EPar, EPar_Par.
+ qauto l:on ctrs:prov use:@rtc_r, ERed_EPar, EPar_Par.
- move => h a ha iha b.
elim /ERed.inv => // _.
+ move => a0 *. subst.
rewrite -prov_lam.
by constructor.
+ move => a0 *. subst.
rewrite -prov_pair.
by constructor.
+ hauto lq:on ctrs:prov use:ERed.substing.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- move => h a b ha iha hb ihb b0.
elim /ERed.inv => //_.
+ move => a0 *. subst.
rewrite -prov_lam.
by constructor.
+ move => a0 *. subst.
rewrite -prov_pair.
by constructor.
+ hauto lq:on ctrs:prov.
+ hauto lq:on ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
- hauto lq:on inv:ERed.R, prov ctrs:prov.
Qed.
Lemma prov_ereds n (u : Tm n) a b : prov u a -> rtc ERed.R a b -> prov u b.
Proof.
induction 2; sfirstorder use:prov_ered.
Qed.
Fixpoint extract {n} (a : Tm n) : Tm n :=
match a with
| TBind p A B => TBind p A B
| Abs a => subst_Tm (scons Bot VarTm) (extract a)
| App a b => extract a
| Pair a b => extract a
| Proj p a => extract a
| Const k => Const k
| VarTm i => VarTm i
| Univ i => Univ i
| Bot => Bot
end.
Lemma ren_extract n m (a : Tm n) (ξ : fin n -> fin m) :
extract (ren_Tm ξ a) = ren_Tm ξ (extract a).
Proof.
move : m ξ. elim : n/a.
- sfirstorder.
- move => n a ih m ξ /=.
rewrite ih.
by asimpl.
- hauto q:on.
- hauto q:on.
- hauto q:on.
- hauto q:on.
- sfirstorder.
- sfirstorder.
- sfirstorder.
Qed.
Lemma ren_morphing n m (a : Tm n) (ρ : fin n -> Tm m) :
(forall i, ρ i = extract (ρ i)) ->
extract (subst_Tm ρ a) = subst_Tm ρ (extract a).
Proof.
move : m ρ.
elim : n /a => n //=.
move => a ha m ρ hi.
rewrite ha.
- destruct i as [i|] => //.
rewrite ren_extract.
rewrite -hi.
by asimpl.
- by asimpl.
Qed.
Lemma ren_subst_bot n (a : Tm (S n)) :
extract (subst_Tm (scons Bot VarTm) a) = subst_Tm (scons Bot VarTm) (extract a).
Proof.
apply ren_morphing. destruct i as [i|] => //=.
Qed.
Definition prov_extract_spec {n} u (a : Tm n) :=
match u with
| TBind p A B => exists A0 B0, extract a = TBind p A0 B0 /\ rtc Par.R A A0 /\ rtc Par.R B B0
| Univ i => extract a = Univ i
| VarTm i => extract a = VarTm i
| (Const i) => extract a = (Const i)
| Bot => extract a = Bot
| _ => True
end.
Lemma prov_extract n u (a : Tm n) :
prov u a -> prov_extract_spec u a.
Proof.
move => h.
elim : u a /h.
- sfirstorder.
- move => h a ha ih.
case : h ha ih => //=.
+ move => i ha ih.
move /(_ Bot) in ih.
rewrite -ih.
by rewrite ren_subst_bot.
+ move => p A B h ih.
move /(_ Bot) : ih => [A0][B0][h0][h1]h2.
rewrite ren_subst_bot in h0.
rewrite h0.
eauto.
+ move => p _ /(_ Bot).
by rewrite ren_subst_bot.
+ move => i h /(_ Bot).
by rewrite ren_subst_bot => ->.
+ move /(_ Bot).
move => h /(_ Bot).
by rewrite ren_subst_bot.
- hauto lq:on.
- hauto lq:on.
- hauto lq:on.
- case => //=.
- sfirstorder.
- sfirstorder.
- sfirstorder.
Qed.
Definition union {A : Type} (R0 R1 : A -> A -> Prop) a b :=
R0 a b \/ R1 a b.
Module ERPar.
Definition R {n} (a b : Tm n) := union RPar.R EPar.R a b.
Lemma RPar {n} (a b : Tm n) : RPar.R a b -> R a b.
Proof. sfirstorder. Qed.
Lemma EPar {n} (a b : Tm n) : EPar.R a b -> R a b.
Proof. sfirstorder. Qed.
Lemma refl {n} ( a : Tm n) : ERPar.R a a.
Proof.
sfirstorder use:RPar.refl, EPar.refl.
Qed.
Lemma ProjCong n p (a0 a1 : Tm n) :
R a0 a1 ->
rtc R (Proj p a0) (Proj p a1).
Proof.
move => [].
- move => h.
apply rtc_once.
left.
by apply RPar.ProjCong.
- move => h.
apply rtc_once.
right.
by apply EPar.ProjCong.
Qed.
Lemma AbsCong n (a0 a1 : Tm (S n)) :
R a0 a1 ->
rtc R (Abs a0) (Abs a1).
Proof.
move => [].
- move => h.
apply rtc_once.
left.
by apply RPar.AbsCong.
- move => h.
apply rtc_once.
right.
by apply EPar.AbsCong.
Qed.
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
R a0 a1 ->
R b0 b1 ->
rtc R (App a0 b0) (App a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.AppCong, @rtc_once.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.AppCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.AppCong, EPar.refl.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.AppCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.AppCong, EPar.refl.
- sfirstorder use:EPar.AppCong, @rtc_once.
Qed.
Lemma BindCong n p (a0 a1 : Tm n) b0 b1:
R a0 a1 ->
R b0 b1 ->
rtc R (TBind p a0 b0) (TBind p a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.BindCong, @rtc_once.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.BindCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.BindCong, EPar.refl.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.BindCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.BindCong, EPar.refl.
- sfirstorder use:EPar.BindCong, @rtc_once.
Qed.
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
R a0 a1 ->
R b0 b1 ->
rtc R (Pair a0 b0) (Pair a1 b1).
Proof.
move => [] + [].
- sfirstorder use:RPar.PairCong, @rtc_once.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.PairCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.PairCong, EPar.refl.
- move => h0 h1.
apply : rtc_l.
left. apply RPar.PairCong; eauto; apply RPar.refl.
apply rtc_once.
hauto l:on use:EPar.PairCong, EPar.refl.
- sfirstorder use:EPar.PairCong, @rtc_once.
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.
sfirstorder use:EPar.renaming, RPar.renaming.
Qed.
End ERPar.
Hint Resolve ERPar.AppCong ERPar.refl ERPar.AbsCong ERPar.PairCong ERPar.ProjCong ERPar.BindCong : erpar.
Module ERPars.
#[local]Ltac solve_s_rec :=
move => *; eapply relations.rtc_transitive; eauto;
hauto lq:on db:erpar.
#[local]Ltac solve_s :=
repeat (induction 1; last by solve_s_rec); apply rtc_refl.
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
rtc ERPar.R (App a0 b0) (App a1 b1).
Proof. solve_s. Qed.
Lemma AbsCong n (a0 a1 : Tm (S n)) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R (Abs a0) (Abs a1).
Proof. solve_s. Qed.
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
rtc ERPar.R (Pair a0 b0) (Pair a1 b1).
Proof. solve_s. Qed.
Lemma ProjCong n p (a0 a1 : Tm n) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R (Proj p a0) (Proj p a1).
Proof. solve_s. Qed.
Lemma BindCong n p (a0 a1 : Tm n) b0 b1:
rtc ERPar.R a0 a1 ->
rtc ERPar.R b0 b1 ->
rtc ERPar.R (TBind p a0 b0) (TBind p a1 b1).
Proof. solve_s. Qed.
Lemma renaming n (a0 a1 : Tm n) m (ξ : fin n -> fin m) :
rtc ERPar.R a0 a1 ->
rtc ERPar.R (ren_Tm ξ a0) (ren_Tm ξ a1).
Proof.
induction 1.
- apply rtc_refl.
- eauto using ERPar.renaming, rtc_l.
Qed.
End ERPars.
Lemma ERPar_Par n (a b : Tm n) : ERPar.R a b -> Par.R a b.
Proof.
sfirstorder use:EPar_Par, RPar_Par.
Qed.
Lemma Par_ERPar n (a b : Tm n) : Par.R a b -> rtc ERPar.R a b.
Proof.
move => h. elim : n a b /h.
- move => n a0 a1 b0 b1 ha iha hb ihb.
suff ? : rtc ERPar.R (App (Abs a0) b0) (App (Abs a1) b1).
apply : relations.rtc_transitive; eauto.
apply rtc_once. apply ERPar.RPar.
by apply RPar.AppAbs; eauto using RPar.refl.
eauto using ERPars.AppCong,ERPars.AbsCong.
- move => n a0 a1 b0 b1 c0 c1 ha iha hb ihb hc ihc.
apply : rtc_l. apply ERPar.RPar.
apply RPar.AppPair; eauto using RPar.refl.
sfirstorder use:ERPars.AppCong, ERPars.PairCong.
- move => n p a0 a1 ha iha.
apply : rtc_l. apply ERPar.RPar. apply RPar.ProjAbs; eauto using RPar.refl.
sfirstorder use:ERPars.AbsCong, ERPars.ProjCong.
- move => n p a0 a1 b0 b1 ha iha hb ihb.
apply : rtc_l. apply ERPar.RPar. apply RPar.ProjPair; eauto using RPar.refl.
hauto lq:on.
- move => n a0 a1 ha iha.
apply : rtc_l. apply ERPar.EPar. apply EPar.AppEta; eauto using EPar.refl.
hauto lq:on ctrs:rtc
use:ERPars.AppCong, ERPars.AbsCong, ERPars.renaming.
- move => n a0 a1 ha iha.
apply : rtc_l. apply ERPar.EPar. apply EPar.PairEta; eauto using EPar.refl.
sfirstorder use:ERPars.PairCong, ERPars.ProjCong.
- sfirstorder.
- sfirstorder use:ERPars.AbsCong.
- sfirstorder use:ERPars.AppCong.
- sfirstorder use:ERPars.PairCong.
- sfirstorder use:ERPars.ProjCong.
- sfirstorder use:ERPars.BindCong.
- sfirstorder.
- sfirstorder.
- sfirstorder.
Qed.
Lemma Pars_ERPar n (a b : Tm n) : rtc Par.R a b -> rtc ERPar.R a b.
Proof.
induction 1; hauto l:on use:Par_ERPar, @relations.rtc_transitive.
Qed.
Lemma Par_ERPar_iff n (a b : Tm n) : rtc Par.R a b <-> rtc ERPar.R a b.
Proof.
split.
sfirstorder use:Pars_ERPar, @relations.rtc_subrel.
sfirstorder use:ERPar_Par, @relations.rtc_subrel.
Qed.
Lemma RPar_ERPar n (a b : Tm n) : rtc RPar.R a b -> rtc ERPar.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel.
Qed.
Lemma EPar_ERPar n (a b : Tm n) : rtc EPar.R a b -> rtc ERPar.R a b.
Proof.
sfirstorder use:@relations.rtc_subrel.
Qed.
Module Type HindleyRosen.
Parameter A : nat -> Type.
Parameter R0 R1 : forall n, A n -> A n -> Prop.
Axiom diamond_R0 : forall n, relations.diamond (R0 n).
Axiom diamond_R1 : forall n, relations.diamond (R1 n).
Axiom commutativity : forall n,
forall a b c, R0 n a b -> R1 n a c -> exists d, R1 n b d /\ R0 n c d.
End HindleyRosen.
Module HindleyRosenFacts (M : HindleyRosen).
Import M.
Lemma R0_comm :
forall n a b c, R0 n a b -> rtc (union (R0 n) (R1 n)) a c ->
exists d, rtc (union (R0 n) (R1 n)) b d /\ R0 n c d.
Proof.
move => n a + c + h.
elim : a c /h.
- sfirstorder.
- move => a0 a1 a2 ha ha0 ih b h.
case : ha.
+ move : diamond_R0 h; repeat move/[apply].
hauto lq:on ctrs:rtc.
+ move : commutativity h; repeat move/[apply].
hauto lq:on ctrs:rtc.
Qed.
Lemma R1_comm :
forall n a b c, R1 n a b -> rtc (union (R0 n) (R1 n)) a c ->
exists d, rtc (union (R0 n) (R1 n)) b d /\ R1 n c d.
Proof.
move => n a + c + h.
elim : a c /h.
- sfirstorder.
- move => a0 a1 a2 ha ha0 ih b h.
case : ha.
+ move : commutativity h; repeat move/[apply].
hauto lq:on ctrs:rtc.
+ move : diamond_R1 h; repeat move/[apply].
hauto lq:on ctrs:rtc.
Qed.
Lemma U_comm :
forall n a b c, (union (R0 n) (R1 n)) a b -> rtc (union (R0 n) (R1 n)) a c ->
exists d, rtc (union (R0 n) (R1 n)) b d /\ (union (R0 n) (R1 n)) c d.
Proof.
hauto lq:on use:R0_comm, R1_comm.
Qed.
Lemma U_comms :
forall n a b c, rtc (union (R0 n) (R1 n)) a b -> rtc (union (R0 n) (R1 n)) a c ->
exists d, rtc (union (R0 n) (R1 n)) b d /\ rtc (union (R0 n) (R1 n)) c d.
Proof.
move => n a b + h.
elim : a b /h.
- sfirstorder.
- hecrush ctrs:rtc use:U_comm.
Qed.
End HindleyRosenFacts.
Module HindleyRosenER <: HindleyRosen.
Definition A := Tm.
Definition R0 n := rtc (@RPar.R n).
Definition R1 n := rtc (@EPar.R n).
Lemma diamond_R0 : forall n, relations.diamond (R0 n).
sfirstorder use:RPar_confluent.
Qed.
Lemma diamond_R1 : forall n, relations.diamond (R1 n).
sfirstorder use:EPar_confluent.
Qed.
Lemma commutativity : forall n,
forall a b c, R0 n a b -> R1 n a c -> exists d, R1 n b d /\ R0 n c d.
Proof.
hauto l:on use:commutativity.
Qed.
End HindleyRosenER.
Module ERFacts := HindleyRosenFacts HindleyRosenER.
Lemma rtc_union n (a b : Tm n) :
rtc (union RPar.R EPar.R) a b <->
rtc (union (rtc RPar.R) (rtc EPar.R)) a b.
Proof.
split; first by induction 1; hauto lq:on ctrs:rtc.
move => h.
elim :a b /h.
- sfirstorder.
- move => a0 a1 a2.
case.
+ move => h0 h1 ih.
apply : relations.rtc_transitive; eauto.
move : h0.
apply relations.rtc_subrel.
sfirstorder.
+ move => h0 h1 ih.
apply : relations.rtc_transitive; eauto.
move : h0.
apply relations.rtc_subrel.
sfirstorder.
Qed.
Lemma prov_erpar n (u : Tm n) a b : prov u a -> ERPar.R a b -> prov u b.
Proof.
move => h [].
- sfirstorder use:prov_rpar.
- move /EPar_ERed.
sfirstorder use:prov_ereds.
Qed.
Lemma prov_pars n (u : Tm n) a b : prov u a -> rtc Par.R a b -> prov u b.
Proof.
move => h /Pars_ERPar.
move => h0.
move : h.
elim : a b /h0.
- done.
- hauto lq:on use:prov_erpar.
Qed.
Lemma Par_confluent n (a b c : Tm n) :
rtc Par.R a b ->
rtc Par.R a c ->
exists d, rtc Par.R b d /\ rtc Par.R c d.
Proof.
move : n a b c.
suff : forall (n : nat) (a b c : Tm n),
rtc ERPar.R a b ->
rtc ERPar.R a c -> exists d : Tm n, rtc ERPar.R b d /\ rtc ERPar.R c d.
move => h n a b c h0 h1.
apply Par_ERPar_iff in h0, h1.
move : h h0 h1; repeat move/[apply].
hauto lq:on use:Par_ERPar_iff.
have h := ERFacts.U_comms.
move => n a b c.
rewrite /HindleyRosenER.R0 /HindleyRosenER.R1 in h.
specialize h with (n := n).
rewrite /HindleyRosenER.A in h.
rewrite /ERPar.R.
have eq : (fun a0 b0 : Tm n => union RPar.R EPar.R a0 b0) = union RPar.R EPar.R by reflexivity.
rewrite !{}eq.
move /rtc_union => + /rtc_union.
move : h; repeat move/[apply].
hauto lq:on use:rtc_union.
Qed.
Lemma pars_univ_inv n i (c : Tm n) :
rtc Par.R (Univ i) c ->
extract c = Univ i.
Proof.
have : prov (Univ i) (Univ i : Tm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
Lemma pars_const_inv n i (c : Tm n) :
rtc Par.R (Const i) c ->
extract c = Const i.
Proof.
have : prov (Const i) (Const i : Tm n) by sfirstorder.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
Lemma pars_pi_inv n p (A : Tm n) B C :
rtc Par.R (TBind p A B) C ->
exists A0 B0, extract C = TBind p A0 B0 /\
rtc Par.R A A0 /\ rtc Par.R B B0.
Proof.
have : prov (TBind p A B) (TBind p A B) by hauto lq:on ctrs:prov, rtc.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
Lemma pars_var_inv n (i : fin n) C :
rtc Par.R (VarTm i) C ->
extract C = VarTm i.
Proof.
have : prov (VarTm i) (VarTm i) by hauto lq:on ctrs:prov, rtc.
move : prov_pars. repeat move/[apply].
apply prov_extract.
Qed.
Lemma pars_univ_inj n i j (C : Tm n) :
rtc Par.R (Univ i) C ->
rtc Par.R (Univ j) C ->
i = j.
Proof.
sauto l:on use:pars_univ_inv.
Qed.
Lemma pars_const_inj n i j (C : Tm n) :
rtc Par.R (Const i) C ->
rtc Par.R (Const j) C ->
i = j.
Proof.
sauto l:on use:pars_const_inv.
Qed.
Lemma pars_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 C :
rtc Par.R (TBind p0 A0 B0) C ->
rtc Par.R (TBind p1 A1 B1) C ->
exists A2 B2, p1 = p0 /\ rtc Par.R A0 A2 /\ rtc Par.R A1 A2 /\
rtc Par.R B0 B2 /\ rtc Par.R B1 B2.
Proof.
move /pars_pi_inv => [A2 [B2 [? [h0 h1]]]].
move /pars_pi_inv => [A3 [B3 [? [h2 h3]]]].
exists A2, B2. hauto l:on.
Qed.
Definition join {n} (a b : Tm n) :=
exists c, rtc Par.R a c /\ rtc Par.R b c.
Lemma join_transitive n (a b c : Tm n) :
join a b -> join b c -> join a c.
Proof.
rewrite /join.
move => [ab [h0 h1]] [bc [h2 h3]].
move : Par_confluent h1 h2; repeat move/[apply].
move => [abc [h4 h5]].
eauto using relations.rtc_transitive.
Qed.
Lemma join_symmetric n (a b : Tm n) :
join a b -> join b a.
Proof. sfirstorder unfold:join. Qed.
Lemma join_refl n (a : Tm n) : join a a.
Proof. hauto lq:on ctrs:rtc unfold:join. Qed.
Lemma join_univ_inj n i j :
join (Univ i : Tm n) (Univ j) -> i = j.
Proof.
sfirstorder use:pars_univ_inj.
Qed.
Lemma join_const_inj n i j :
join (Const i : Tm n) (Const j) -> i = j.
Proof.
sfirstorder use:pars_const_inj.
Qed.
Lemma join_pi_inj n p0 p1 (A0 A1 : Tm n) B0 B1 :
join (TBind p0 A0 B0) (TBind p1 A1 B1) ->
p0 = p1 /\ join A0 A1 /\ join B0 B1.
Proof.
move => [c []].
move : pars_pi_inj; repeat move/[apply].
sfirstorder unfold:join.
Qed.
Lemma join_univ_pi_contra n p (A : Tm n) B i :
join (TBind p A B) (Univ i) -> False.
Proof.
rewrite /join.
move => [c [h0 h1]].
move /pars_univ_inv : h1.
move /pars_pi_inv : h0.
hauto l:on.
Qed.
Lemma join_substing n m (a b : Tm n) (ρ : fin n -> Tm m) :
join a b ->
join (subst_Tm ρ a) (subst_Tm ρ b).
Proof. hauto lq:on unfold:join use:Pars.substing. Qed.
Fixpoint ne {n} (a : Tm n) :=
match a with
| VarTm i => true
| TBind _ A B => false
| App a b => ne a && nf b
| Abs a => false
| Univ _ => false
| Proj _ a => ne a
| Pair _ _ => false
| Const _ => false
| Bot => true
end
with nf {n} (a : Tm n) :=
match a with
| VarTm i => true
| TBind _ A B => nf A && nf B
| App a b => ne a && nf b
| Abs a => nf a
| Univ _ => true
| Proj _ a => ne a
| Pair a b => nf a && nf b
| Const _ => true
| Bot => true
end.
Lemma ne_nf n a : @ne n a -> nf a.
Proof. elim : a => //=. Qed.
Definition wn {n} (a : Tm n) := exists b, rtc RPar'.R a b /\ nf b.
Definition wne {n} (a : Tm n) := exists b, rtc RPar'.R a b /\ ne b.
(* Weakly neutral implies weakly normal *)
Lemma wne_wn n a : @wne n a -> wn a.
Proof. sfirstorder use:ne_nf. Qed.
(* Normal implies weakly normal *)
Lemma nf_wn n v : @nf n v -> wn v.
Proof. sfirstorder ctrs:rtc. Qed.
Lemma nf_refl n (a b : Tm n) (h : RPar'.R a b) : (nf a -> b = a) /\ (ne a -> b = a).
Proof.
elim : a b /h => //=; solve [hauto b:on].
Qed.
Lemma ne_nf_ren n m (a : Tm n) (ξ : fin n -> fin m) :
(ne a <-> ne (ren_Tm ξ a)) /\ (nf a <-> nf (ren_Tm ξ a)).
Proof.
move : m ξ. elim : n / a => //=; solve [hauto b:on].
Qed.
Lemma wne_app n (a b : Tm n) :
wne a -> wn b -> wne (App a b).
Proof.
move => [a0 [? ?]] [b0 [? ?]].
exists (App a0 b0). hauto b:on drew:off use:RPars'.AppCong.
Qed.
Lemma wn_abs n a (h : wn a) : @wn n (Abs a).
Proof.
move : h => [v [? ?]].
exists (Abs v).
eauto using RPars'.AbsCong.
Qed.
Lemma wn_bind n p A B : wn A -> wn B -> wn (@TBind n p A B).
Proof.
move => [A0 [? ?]] [B0 [? ?]].
exists (TBind p A0 B0).
hauto lqb:on use:RPars'.BindCong.
Qed.
Lemma wn_pair n (a b : Tm n) : wn a -> wn b -> wn (Pair a b).
Proof.
move => [a0 [? ?]] [b0 [? ?]].
exists (Pair a0 b0).
hauto lqb:on use:RPars'.PairCong.
Qed.
Lemma wne_proj n p (a : Tm n) : wne a -> wne (Proj p a).
Proof.
move => [a0 [? ?]].
exists (Proj p a0). hauto lqb:on use:RPars'.ProjCong.
Qed.
Create HintDb nfne.
#[export]Hint Resolve nf_wn ne_nf wne_wn nf_refl : nfne.
Lemma ne_nf_antiren n m (a : Tm n) (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
(ne (subst_Tm ρ a) -> ne a) /\ (nf (subst_Tm ρ a) -> nf a).
Proof.
move : m ρ. elim : n / a => //;
hauto b:on drew:off use:RPar.var_or_const_up.
Qed.
Lemma wn_antirenaming n m a (ρ : fin n -> Tm m) :
(forall i, var_or_const (ρ i)) ->
wn (subst_Tm ρ a) -> wn a.
Proof.
rewrite /wn => hρ.
move => [v [rv nfv]].
move /RPars'.antirenaming : rv.
move /(_ hρ) => [b [hb ?]]. subst.
exists b. split => //=.
move : nfv.
by eapply ne_nf_antiren.
Qed.
Lemma ext_wn n (a : Tm n) :
wn (App a Bot) ->
wn a.
Proof.
move E : (App a (Bot)) => a0 [v [hr hv]].
move : a E.
move : hv.
elim : a0 v / hr.
- hauto q:on inv:Tm ctrs:rtc b:on db: nfne.
- move => a0 a1 a2 hr0 hr1 ih hnfa2.
move /(_ hnfa2) in ih.
move => a.
case : a0 hr0=>// => b0 b1.
elim /RPar'.inv=>// _.
+ move => a0 a3 b2 b3 ? ? [? ?] ? [? ?]. subst.
have ? : b3 = (Bot) by hauto lq:on inv:RPar'.R. subst.
suff : wn (Abs a3) by hauto lq:on ctrs:RPar'.R, rtc unfold:wn.
have : wn (subst_Tm (scons (Bot) VarTm) a3) by sfirstorder.
move => h. apply wn_abs.
move : h. apply wn_antirenaming.
hauto lq:on rew:off inv:option.
+ hauto q:on inv:RPar'.R ctrs:rtc b:on.
Qed.
Module Join.
Lemma ProjCong p n (a0 a1 : Tm n) :
join a0 a1 ->
join (Proj p a0) (Proj p a1).
Proof. hauto lq:on use:Pars.ProjCong unfold:join. Qed.
Lemma PairCong n (a0 a1 b0 b1 : Tm n) :
join a0 a1 ->
join b0 b1 ->
join (Pair a0 b0) (Pair a1 b1).
Proof. hauto lq:on use:Pars.PairCong unfold:join. Qed.
Lemma AppCong n (a0 a1 b0 b1 : Tm n) :
join a0 a1 ->
join b0 b1 ->
join (App a0 b0) (App a1 b1).
Proof. hauto lq:on use:Pars.AppCong. Qed.
Lemma AbsCong n (a b : Tm (S n)) :
join a b ->
join (Abs a) (Abs b).
Proof. hauto lq:on use:Pars.AbsCong. Qed.
Lemma renaming n m (a b : Tm n) (ξ : fin n -> fin m) :
join a b -> join (ren_Tm ξ a) (ren_Tm ξ b).
Proof.
induction 1; hauto lq:on use:Pars.renaming.
Qed.
Lemma weakening n (a b : Tm n) :
join a b -> join (ren_Tm shift a) (ren_Tm shift b).
Proof.
apply renaming.
Qed.
Lemma FromPar n (a b : Tm n) :
Par.R a b ->
join a b.
Proof.
hauto lq:on ctrs:rtc use:rtc_once.
Qed.
End Join.
Lemma abs_eq n a (b : Tm n) :
join (Abs a) b <-> join a (App (ren_Tm shift b) (VarTm var_zero)).
Proof.
split.
- move => /Join.weakening h.
have {h} : join (App (ren_Tm shift (Abs a)) (VarTm var_zero)) (App (ren_Tm shift b) (VarTm var_zero))
by hauto l:on use:Join.AppCong, join_refl.
simpl.
move => ?. apply : join_transitive; eauto.
apply join_symmetric. apply Join.FromPar.
apply : Par.AppAbs'; eauto using Par.refl. by asimpl.
- move /Join.AbsCong.
move /join_transitive. apply.
apply join_symmetric. apply Join.FromPar. apply Par.AppEta. apply Par.refl.
Qed.
Lemma pair_eq n (a0 a1 b : Tm n) :
join (Pair a0 a1) b <-> join a0 (Proj PL b) /\ join a1 (Proj PR b).
Proof.
split.
- move => h.
have /Join.ProjCong {}h := h.
have h0 : forall p, join (if p is PL then a0 else a1) (Proj p (Pair a0 a1))
by hauto lq:on use:join_symmetric, Join.FromPar, Par.ProjPair', Par.refl.
hauto lq:on rew:off use:join_transitive, join_symmetric.
- move => [h0 h1].
move : h0 h1.
move : Join.PairCong; repeat move/[apply].
move /join_transitive. apply. apply join_symmetric.
apply Join.FromPar. hauto lq:on ctrs:Par.R use:Par.refl.
Qed.
Lemma join_pair_inj n (a0 a1 b0 b1 : Tm n) :
join (Pair a0 a1) (Pair b0 b1) <-> join a0 b0 /\ join a1 b1.
Proof.
split; last by hauto lq:on use:Join.PairCong.
move /pair_eq => [h0 h1].
have : join (Proj PL (Pair b0 b1)) b0 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
have : join (Proj PR (Pair b0 b1)) b1 by hauto lq:on use:Join.FromPar, Par.refl, Par.ProjPair'.
eauto using join_transitive.
Qed.