(* *)
(**************************************************************************)
+include "basic_2/syntax/voids_length.ma".
include "basic_2/static/frees_fqup.ma".
include "basic_2/static/fle.ma".
(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
(* Advanced properties ******************************************************)
-(*
+
lemma fle_refl: bi_reflexive … fle.
-#L #T elim (frees_total L T) /2 width=5 by sle_refl, ex3_2_intro/
+#L #T
+elim (voids_refl L) #n #Hn
+elim (frees_total L T) #f #Hf
+/2 width=8 by sle_refl, ex4_4_intro/
qed.
-*)
+
lemma fle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
-#L1 #L2 #V1 #V2 * -L1 #f1 #g1 #L1 #n #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
+#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
elim (frees_total (L2.ⓧ) T2) #g2 #Hg2
elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-/4 width=8 by fle_intro, frees_bind_void, sor_inv_sle_sn, sle_trans/
+@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/
qed.
-(*
-lemma fle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ →
- ∀p,I,V2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
-#L1 #L2 #T1 #T2 * -L1 #f2 #g2 #L1 #n #Hf2 #Hg2 #HL12 #Hfg2 #p #I #V2
+
+lemma fle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| →
+ ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+#L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2
+elim (voids_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H #_ destruct
+<tls_xn in Hfg2; #Hfg2
elim (frees_total L2 V2) #g1 #Hg1
elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-@(fle_intro … g … Hf2) /2 width=5 by frees_bind_void/
-@(sle_trans … Hfg1) @(sor_inv_sle_sn … Hg)
-
-
-
-/4 width=8 by fle_intro, frees_bind_void, sor_inv_sle_dx, sle_trans/
+@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_bind_void, sor_inv_sle_dx, sor_tls, sle_trans/
qed.
-*)
+
lemma fle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
∀I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #V1 #V2 * -L1 #f1 #g1 #L1 #n #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
+#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
elim (frees_total L2 T2) #g2 #Hg2
elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
-/4 width=8 by fle_intro, frees_flat, sor_inv_sle_sn, sle_trans/
+@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_flat, sor_inv_sle_sn, sor_tls, sle_trans/
qed.
lemma fle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
∀I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #T1 #T2 * -L1 #f2 #g2 #L1 #n #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
+#L1 #L2 #T1 #T2 * #n1 #m1 #f2 #g2 #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
elim (frees_total L2 V2) #g1 #Hg1
elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
-/4 width=8 by fle_intro, frees_flat, sor_inv_sle_dx, sle_trans/
+@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_flat, sor_inv_sle_dx, sor_tls, sle_trans/
qed.