Section Kripke.
Context {Sigma : Signature}.
Section Model.
Variable domain : Type.
Class kmodel :=
{
nodes : Type ;
reachable : nodes -> nodes -> Prop ;
reach_refl u : reachable u u ;
reach_tran u v w : reachable u v -> reachable v w -> reachable u w ;
k_interp : interp domain ;
k_P : nodes -> forall P : Preds, Vector.t domain (pred_ar P) -> Prop ;
k_Bot : nodes -> Prop ;
mon_P : forall u v, reachable u v -> forall P (t : Vector.t domain (pred_ar P)), k_P u t -> k_P v t;
mon_Bot : forall u v, reachable u v -> k_Bot u -> k_Bot v
}.
Variable M : kmodel.
Fixpoint ksat u (rho : fin -> domain) (phi : form) : Prop :=
match phi with
| Pred P v => k_P u (Vector.map (@eval _ _ k_interp rho) v)
| Fal => k_Bot u
| Impl phi psi => forall v, reachable u v -> ksat v rho phi -> ksat v rho psi
| All phi => forall j : domain, ksat u (j .: rho) phi
end.
Global Instance kmodel_reach_refl : Reflexive reachable := reach_refl.
Global Instance kmodel_reach_trans : Transitive reachable := reach_tran.
Lemma ksat_mon (u : nodes) (rho : fin -> domain) (phi : form) :
forall v (H : reachable u v), ksat u rho phi -> ksat v rho phi.
Proof.
revert rho. induction phi; intros rho v H; cbn; intuition; eauto using mon_P, mon_Bot, reach_tran.
Qed.
Lemma ksat_iff u rho phi :
ksat u rho phi <-> forall v (H : reachable u v), ksat v rho phi.
Proof.
split.
- intros H1 v H2. eapply ksat_mon; eauto.
- intros H. apply H. eapply reach_refl.
Qed.
End Model.
Notation "rho '⊨(' u ')' phi" := (ksat u rho phi) (at level 20).
Notation "rho '⊨(' u , M ')' phi" := (@ksat _ M u rho phi) (at level 20).
Arguments ksat {_ _} _ _ _, _ _ _ _ _.
Hint Resolve reach_refl.
Section Substs.
Variable D : Type.
Context {M : kmodel D}.
Lemma ksat_ext u rho xi phi :
(forall x, rho x = xi x) -> rho ⊨(u,M) phi <-> xi ⊨(u,M) phi.
Proof.
induction phi in rho, xi, u |-*; intros Hext; comp.
- tauto.
- now rewrite (vec_ext (fun x => eval_ext _ x Hext)).
- split; intros H v Hv Hv'; now apply (IHphi2 v rho xi Hext), (H _ Hv), (IHphi1 v rho xi Hext).
- split; intros H d; apply (IHphi _ (d .: rho) (d .: xi)). all: ((intros []; cbn; congruence) + auto).
Qed.
Lemma ksat_comp u rho xi phi :
rho ⊨(u,M) (subst_form xi phi) <-> (xi >> eval rho (I := @k_interp _ M)) ⊨(u,M) phi.
Proof.
induction phi in rho, xi, u |-*; comp.
- tauto.
- erewrite vec_comp. 2: reflexivity. erewrite vec_ext. 2: apply eval_comp. reflexivity.
- setoid_rewrite IHphi1. now setoid_rewrite IHphi2.
- setoid_rewrite IHphi. split; intros H d; eapply ksat_ext. 2, 4: apply (H d).
all: intros []; comp; rewrite? eval_comp; now comp.
Qed.
End Substs.
Definition kconstraint := forall D, kmodel D -> Prop.
Definition kcon_subs (C C' : kconstraint) := (forall D (M : kmodel D), C' D M -> C D M).
Definition kstandard D (M : kmodel D) := forall v, k_Bot v -> False.
Definition kexploding D (M : kmodel D) := forall v rho phi, rho ⊨(v) (⊥ --> phi).
Definition kexploding' D (M : kmodel D) := forall v rho P t, rho ⊨(v) (⊥ --> Pred P t).
Definition kbottomless D (M : kmodel D) := True.
Lemma kstandard_explodes :
kcon_subs kexploding kstandard.
Proof.
intros D M HC u rho P t v [] % HC.
Qed.
Definition kvalid_L (C : kconstraint) A phi :=
forall D (M : @kmodel D), C _ M -> forall u rho, (forall psi, psi el A -> ksat u rho psi) -> ksat u rho phi.
Definition kvalid_T (C : kconstraint) T phi :=
forall D (M : @kmodel D), C _ M -> forall u rho, (forall psi, psi ∈ T -> ksat u rho psi) -> ksat u rho phi.
Lemma kvalid_L_subs C C' A phi :
kcon_subs C' C -> kvalid_L C' A phi -> kvalid_L C A phi.
Proof.
intros Hsub HC' D M HC % Hsub u rho HA. now apply HC'.
Qed.
End Kripke.
Notation "rho '⊨(' u ')' phi" := (ksat u rho phi) (at level 20).
Notation "rho '⊨(' u , M ')' phi" := (@ksat _ _ M u rho phi) (at level 20).
Notation "A ⊫KE phi" := (kvalid_L kexploding A phi) (at level 20).
Notation "A ⊫KS phi" := (kvalid_L kstandard A phi) (at level 20).
Notation "A ⊫KBL phi" := (kvalid_L kbottomless A phi) (at level 20).
Arguments ksat {_ _} _ _ _ _, {_ } _ _ _ _ _.
Section KSoundness.
Context {Sigma : Signature} {b : bottom}.
Ltac clean_ksoundness :=
match goal with
| [ H : ?x = ?x -> _ |- _ ] => specialize (H eq_refl)
| [ H : (?A -> ?B), H2 : (?A -> ?B) -> _ |- _] => specialize (H2 H)
end.
Lemma ksemantic_explosion D (M : kmodel D) :
kexploding' M -> kexploding M.
Proof.
intros H v rho phi.
induction phi in rho, v |-*; firstorder.
Qed.
Lemma ksoundness A C (phi : form) :
(b = expl -> kcon_subs kexploding C) -> @prv _ intu _ A phi -> kvalid_L C A phi.
Proof.
intros Hexp Hprv D M HM. remember intu as s. induction Hprv; subst; cbn; intros u rho HA.
all: repeat (clean_ksoundness + discriminate). all: (eauto || comp; eauto).
- intros v Hr Hpi. capply IHHprv. intros ? []; subst; eauto using ksat_mon.
- eapply IHHprv1. 3: eapply IHHprv2. all: eauto. reflexivity.
- intros d. apply IHHprv. apply switch_map. intros psi Hpsi. rewrite ksat_comp. apply HA, Hpsi.
- rewrite ksat_comp. eapply ksat_ext. 2: eapply (IHHprv u rho HA (eval rho t)).
asimpl. now intros [].
- apply Hexp in HM. apply (HM u); eauto. apply M.
Qed.
Lemma strong_ksoundness T C phi :
(b = expl -> kcon_subs kexploding C) -> @tprv _ intu _ T phi -> kvalid_T C T phi.
Proof.
intros Hexp (A & HA1 & HA2) D M HM u rho HT. apply (ksoundness Hexp HA2 HM).
firstorder.
Qed.
End KSoundness.
Lemma ksoundness' {Sigma : Signature} A phi :
A ⊢IE phi -> A ⊫KS phi.
Proof.
apply ksoundness. intros _. firstorder.
Qed.