Section Sound_context.
Variable the_rule : Basic_rule.
Let R := Rule the_rule.
Let Rlifts : R_lift R := R_lifts the_rule.
Hint Resolve Rlifts: pts.
Hypothesis
incl_R_le_type :
forall (e : env) (x y : term), ctxt R e x y -> le_type e y x.
simple induction 1; intros; auto with pts.
apply le_type_trans with y0; auto with pts.
Qed.
intros.
apply subtype_in_env with (1 := H).
constructor.
constructor.
red in |- *.
auto with pts.
apply red_incl_le_type.
trivial with pts.
apply wf_var with s; trivial with pts.
Qed.
red in |- *.
simple induction 2; auto with pts; intros.
Inversion_typ H3.
apply typ_conv_wf with (Prod M U); auto with pts.
apply type_conv with (Prod M' U) s; auto with pts.
Inversion_typ H5.
apply type_abs with s3; auto with pts.
apply type_prod with s1 s2; auto with pts.
apply red_in_env with M s1; auto with pts.
apply red_in_env with M s1; auto with pts.
apply incl_R_le_type.
auto with pts.
apply type_correctness with (Abs M N); auto with pts.
Inversion_typ H3.
apply typ_conv_wf with (Prod N U); auto with pts.
apply type_abs with s; auto with pts.
apply type_correctness with (Abs N M); auto with pts.
Inversion_typ H3.
apply typ_conv_wf with (subst M2 Ur); auto with pts.
apply type_app with V; auto with pts.
apply type_correctness with (App M1 M2); auto with pts.
Inversion_typ H3.
apply typ_conv_wf with (subst M2 Ur); auto with pts.
cut (wf_type the_PTS e0 (Prod V Ur)); intros.
inversion_clear H7.
Inversion_typ H8.
apply type_conv with (subst N2 Ur) s2; auto with pts.
apply type_app with V; auto with pts.
apply red_incl_le_type.
unfold subst in |- *.
apply red_subst_l with e0; auto with pts.
replace (Srt s2) with (subst M2 (Srt s2)); auto with pts.
apply substitution with V; auto with pts.
apply type_correctness with M1; auto with pts.
apply type_correctness with (App M1 M2); auto with pts.
Inversion_typ H3.
apply typ_conv_wf with (Srt s3); auto with pts.
apply type_prod with s1 s2; auto with pts.
apply red_in_env with M1 s1; auto with pts.
apply type_correctness with (Prod M1 M2); auto with pts.
Inversion_typ H3.
apply typ_conv_wf with (Srt s3); auto with pts.
apply type_prod with s1 s2; auto with pts.
apply type_correctness with (Prod M1 M2); auto with pts.
Qed.
End Sound_context.
unfold rule_sound in |- *.
intros.
apply H0 with x; auto with pts.
Qed.

Theorem union_sound :
forall P Q : red_rule,
rule_sound P -> rule_sound Q -> rule_sound (reunion P Q).