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Add a new instance of noforbid

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Yiyun Liu 2025-06-19 20:08:24 -04:00
parent b52a3bf3f5
commit 883e851c9e
1 changed files with 76 additions and 27 deletions
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103
cosn.v → theories/cosn.v
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@ -28,11 +28,11 @@ Fixpoint nostuck (a : PTm) :=
CoInductive safe a : Prop :=
safe_intro {safe_nostuck : nostuck a ; safe_red : forall b,RRed.R a b -> safe b}.
safe_intro {safe_nostuck : nostuck a ; safe_red : forall b,RERed.R a b -> safe b}.
Arguments safe_intro {a}.
Lemma safe_coind P : (forall a, P a -> nostuck a /\ (forall b, RRed.R a b -> P b)) -> forall a, P a -> safe a.
Lemma safe_coind P : (forall a, P a -> nostuck a /\ (forall b, RERed.R a b -> P b)) -> forall a, P a -> safe a.
move => h.
cofix ih.
move => a ha. apply h in ha.
@ -80,7 +80,7 @@ Proof.
by eauto using nostuck_antisubstitution.
move => b hr. exists ρ.
inversion ha as [ha0 ha1].
hauto lq:on use:RRed.substing.
hauto lq:on use:RERed.substing.
Qed.
Lemma safe_app_inv0 : forall a b, safe (PApp a b) -> safe a.
@ -182,35 +182,84 @@ Proof.
Qed.
Lemma safe_rred a b :
RRed.R a b -> safe a -> safe b.
RERed.R a b -> safe a -> safe b.
Proof.
sauto lq:on.
Qed.
Lemma safe_abs a : safe a -> safe (PAbs a).
Lemma safe_rered a b :
RERed.R a b -> safe a -> safe b.
Proof.
move E :(PAbs a) => u.
move : u a E.
suff : (forall u, (exists a, PAbs a = u /\ safe a) -> safe u) by firstorder.
apply safe_coind. sauto lq:on.
qauto l:on inv:safe ctrs:safe.
Qed.
Inductive ERed' (a : PTm) : PTm -> Prop :=
| E_Refl :
ERed' a a
| E_Once b :
ERed.R a b ->
ERed' a b.
Lemma safe_ered a b :
ERed.R a b -> safe a -> safe b.
Lemma safe_rereds a b :
rtc RERed.R a b -> safe a -> safe b.
Proof.
move => h.
elim : a b /h.
- move => a. move /safe_abs_inv /safe_app_inv0. substify. apply safe_antisubstitution.
- sfirstorder use:safe_pair_inv0, safe_proj_inv.
- sfirstorder use:safe_abs, safe_abs_inv.
- move => a0 a1 b ha iha happ.
have ha0 : safe a0 by sfirstorder use:safe_app_inv0.
have hb : safe b by sfirstorder use:safe_app_inv1.
induction 1; eauto using safe_rered.
Qed.
Definition tm_omega :=
PApp (PAbs (PApp (VarPTm 0) (VarPTm 0)))
(PAbs (PApp (VarPTm 0) (VarPTm 0))).
Lemma safe_omega : safe tm_omega.
Proof.
move E : tm_omega => u.
move : u E.
apply safe_coind.
move => a ?. subst.
split => //=.
move => b. inversion 1 => //=; subst; sauto q:on.
Qed.
Module Safe_NoForbid <: NoForbid.
Definition P := @safe.
Lemma P_EPar : forall (a b : PTm), EPar.R a b -> P a -> P b.
Proof.
move => a b /EReds.FromEPar /REReds.FromEReds.
apply safe_rereds.
Qed.
Lemma P_RRed : forall (a b : PTm), RRed.R a b -> P a -> P b.
Proof. move => a b /RERed.FromBeta. apply safe_rered. Qed.
Lemma PApp_imp : forall a b, ishf a -> ~~ isabs a -> ~ P (PApp a b).
apply safe_app_imp. Qed.
Lemma PProj_imp : forall p a, ishf a -> ~~ ispair a -> ~ P (PProj p a).
apply safe_proj_imp. Qed.
Lemma PInd_imp : forall Q (a : PTm) b c,
ishf a ->
~~ iszero a ->
~~ issuc a -> ~ P (PInd Q a b c).
apply safe_ind_imp. Qed.
Lemma P_AppInv : forall (a b : PTm), P (PApp a b) -> P a /\ P b.
firstorder using safe_app_inv0, safe_app_inv1. Qed.
Lemma P_PairInv : forall (a b : PTm), P (PPair a b) -> P a /\ P b.
firstorder using safe_pair_inv0, safe_pair_inv1. Qed.
Lemma P_ProjInv : forall p (a : PTm), P (PProj p a) -> P a.
apply safe_proj_inv. Qed.
Lemma P_BindInv : forall p (A : PTm) B, P (PBind p A B) -> P A /\ P B.
firstorder using safe_bind_inv0, safe_bind_inv1. Qed.
Lemma P_SucInv : forall (a : PTm), P (PSuc a) -> P a.
apply safe_suc_inv. Qed.
Lemma P_AbsInv : forall (a : PTm), P (PAbs a) -> P a.
apply safe_abs_inv. Qed.
Lemma P_renaming : forall (ξ : nat -> nat) a , P (ren_PTm ξ a) -> P a.
substify. hauto lq:on use:safe_antisubstitution. Qed.
Lemma P_IndInv : forall Q (a : PTm) b c, P (PInd Q a b c) -> P Q /\ P a /\ P b /\ P c.
qauto l:on use: safe_ind_inv0, safe_ind_inv1,
safe_ind_inv2, safe_ind_inv3.
Qed.
End Safe_NoForbid.