new definition of lleq allows to complete the proof of lemma 1000

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / substitution / lleq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma
index 7e48dbca0c27f98d18a3fadea23cf14d6d430a26..577343c6ecbac90fa244c071791078808f1d50ff 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma
@@ -27,6 +27,13 @@ interpretation
 
 (* Basic properties *********************************************************)
 
+lemma lleq_refl: ∀d,T. reflexive … (lleq d T).
+/3 width=1 by conj/ qed.
+
+lemma lleq_sym: ∀d,T. symmetric … (lleq d T).
+#d #T #L1 #L2 * /3 width=1 by iff_sym, conj/
+qed-.
+
 lemma lleq_sort: ∀L1,L2,d,k. |L1| = |L2| → L1 ⋕[⋆k, d] L2.
 #L1 #L2 #d #k #HL12 @conj // -HL12
 #U @conj #H >(cpys_inv_sort1 … H) -H //
@@ -60,12 +67,49 @@ elim (IHV W) -IHV elim (IHT U) -IHT
 /3 width=1 by cpys_flat/
 qed.
 
+lemma lleq_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 → ∀T,d,e. ⇧[d, e] T ≡ U →
+               d ≤ dt → dt ≤ d + e → L1 ⋕[U, d] L2.
+#L1 #L2 #U #dt * #HL12 #IH #T #d #e #HTU #Hddt #Hdtde @conj // -HL12
+#U0 elim (IH U0) -IH #H12 #H21 @conj
+#HU0 elim (cpys_up … HU0 … HTU) // -HU0 /4 width=5 by cpys_weak/
+qed-.
+
+lemma lsuby_lleq_trans: ∀L2,L,T,d. L2 ⋕[T, d] L →
+                        ∀L1. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → L1 ⋕[T, d] L.
+#L2 #L #T #d * #HL2 #IH #L1 #HL12 #H @conj // -HL2
+#U elim (IH U) -IH #Hdx #Hsn @conj #HTU
+[ @Hdx -Hdx -Hsn @(lsuby_cpys_trans … HTU) -HTU
+  /2 width=1 by lsuby_sym/ (**) (* full auto does not work *)
+| -H -Hdx /3 width=3 by lsuby_cpys_trans/
+]
+qed-.
+
+lemma lleq_lsuby_trans: ∀L,L1,T,d. L ⋕[T, d] L1 →
+                        ∀L2. L1 ⊑×[d, ∞] L2 → |L1| = |L2| → L ⋕[T, d] L2.
+/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-.
+
+lemma lleq_lsuby_repl: ∀L1,L2,T,d. L1 ⋕[T, d] L2 →
+                       ∀K1. K1 ⊑×[d, ∞] L1 → |K1| = |L1| →
+                       ∀K2. L2 ⊑×[d, ∞] K2 → |L2| = |K2| →
+                       K1 ⋕[T, d] K2.
+/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-.
+
 (* Basic forward lemmas *****************************************************)
 
 lemma lleq_fwd_length: ∀L1,L2,T,d. L1 ⋕[T, d] L2 → |L1| = |L2|.
 #L1 #L2 #T #d * //
 qed-.
 
+lemma lleq_fwd_ldrop_sn: ∀L1,L2,T,d. L1 ⋕[d, T] L2 → ∀K1,i. ⇩[0, i] L1 ≡ K1 →
+                         ∃K2. ⇩[0, i] L2 ≡ K2.
+#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length … H) -H
+#HL12 lapply (ldrop_fwd_length_le2 … HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
+qed-.
+
+lemma lleq_fwd_ldrop_dx: ∀L1,L2,T,d. L1 ⋕[d, T] L2 → ∀K2,i. ⇩[0, i] L2 ≡ K2 →
+                         ∃K1. ⇩[0, i] L1 ≡ K1.
+/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-.
+
 lemma lleq_fwd_bind_sn: ∀a,I,L1,L2,V,T,d.
                         L1 ⋕[ⓑ{a,I}V.T, d] L2 → L1 ⋕[V, d] L2.
 #a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12