integrating the framework with fle ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / lfxs_lfxs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
index 89949c4b00ba20b4b43988ab3859a2b3af60005e..f827a1f20a9f566ab4e2630b0fb5508e11662c30 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
@@ -12,19 +12,51 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/relocation/lexs_length.ma".
 include "basic_2/relocation/lexs_lexs.ma".
 include "basic_2/static/frees_drops.ma".
+include "basic_2/static/fle_fle.ma".
 include "basic_2/static/lfxs.ma".
 
 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
 
-(* Advanced properties ******************************************************)
+(* Advanced inversion lemmas ************************************************)
 
 lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 →
                       ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⪤*[cext2 R, cfull, f] L2.
 #R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/
 qed-.
 
+lemma frees_lexs_conf: ∀R. lfxs_fle_compatible R →
+                       ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
+                       ∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 →
+                       ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
+#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
+lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
+@(fle_frees_trans_eq … H2L … Hf1) /3 width=4 by lexs_fwd_length, sym_eq/
+qed-.
+
+(* Properties with free variables inclusion for restricted closures *********)
+
+(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
+lemma fle_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
+                      ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
+#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
+elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
+/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lfxs_sym: ∀R. lfxs_fle_compatible R →
+                (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+                ∀T. symmetric … (lfxs R T).
+#R #H1R #H2R #T #L1 #L2
+* #f1 #Hf1 #HL12
+elim (frees_lexs_conf … Hf1 … HL12) -Hf1 //
+/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
+qed-.
+
 (* Basic_2A1: uses: llpx_sn_dec *)
 lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
                 ∀L1,L2,T. Decidable (L1 ⪤*[R, T] L2).
@@ -35,7 +67,7 @@ elim (lexs_dec (cext2 R) cfull … L1 L2 f)
 qed-.
 
 lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                          lexs_frees_confluent … (cext2 R1) cfull →
+                          lfxs_fle_compatible R1 →
                           ∀L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T.
                           ∃∃L. L1 ⪤*[R1, ②{I}V.T] L & L ⪤*[R2, V] L2.
 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
@@ -49,12 +81,12 @@ lapply (sor_eq_repl_back1 … Hy … H1) -y1 #Hy
 lapply (sor_inv_sle_sn … Hy) -y2 #Hfg
 elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
 lapply (sle_lexs_trans … HL1 … Hfg) // #H
-elim (HR … Hf … H) -HR -Hf -H
+elim (frees_lexs_conf … Hf … H) -Hf -H
 /4 width=7 by sle_lexs_trans, ex2_intro/
 qed-.
 
 lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                          lexs_frees_confluent … (cext2 R1) cfull →
+                          lfxs_fle_compatible R1 →
                           ∀L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V.
                           ∃∃L. L1 ⪤*[R1, ⓕ{I}V.T] L & L ⪤*[R2, T] L2.
 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
@@ -65,12 +97,12 @@ lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
 lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
 elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
 lapply (sle_lexs_trans … HL1 … Hfg) // #H
-elim (HR … Hf … H) -HR -Hf -H
+elim (frees_lexs_conf … Hf … H) -Hf -H
 /4 width=7 by sle_lexs_trans, ex2_intro/
 qed-.
 
 lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                          lexs_frees_confluent … (cext2 R1) cfull →
+                          lfxs_fle_compatible R1 →
                           ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤*[R1, T] L2 → ∀p.
                           ∃∃L,V. L1 ⪤*[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤*[R2, T] L2 & R1 L1 V1 V.
 #R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
@@ -85,12 +117,12 @@ elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by
 lapply (sle_lexs_trans … H … Hfg) // #H0
 elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H
 elim (ext2_inv_pair_sn … H) -H #V #HV #H1 #H2 destruct
-elim (HR … Hf … H0) -HR -Hf -H0
+elim (frees_lexs_conf … Hf … H0) -Hf -H0
 /4 width=7 by sle_lexs_trans, ex3_2_intro, ex2_intro/
 qed-.
 
 lemma lfxs_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
-                               lexs_frees_confluent … (cext2 R1) cfull →
+                               lfxs_fle_compatible R1 →
                                ∀L1,L2,T. L1.ⓧ ⪤*[R1, T] L2 → ∀p,I,V.
                                ∃∃L. L1 ⪤*[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤*[R2, T] L2.
 #R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
@@ -105,7 +137,7 @@ elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by
 lapply (sle_lexs_trans … H … Hfg) // #H0
 elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H
 elim (ext2_inv_unit_sn … H) -H #H destruct
-elim (HR … Hf … H0) -HR -Hf -H0
+elim (frees_lexs_conf … Hf … H0) -Hf -H0
 /4 width=7 by sle_lexs_trans, ex2_intro/ (* note: 2 ex2_intro *)
 qed-.
 
@@ -137,8 +169,8 @@ lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
 qed.
 
 theorem lfxs_conf: ∀R1,R2.
-                   lexs_frees_confluent (cext2 R1) cfull →
-                   lexs_frees_confluent (cext2 R2) cfull →
+                   lfxs_fle_compatible R1 →
+                   lfxs_fle_compatible R2 →
                    R_confluent2_lfxs R1 R2 R1 R2 →
                    ∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
 #R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
@@ -146,8 +178,8 @@ lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
 lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01
 elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
 [ #L #HL1 #HL2
-  elim (HR1 … Hf … HL01) -HL01 #f1 #Hf1 #H1
-  elim (HR2 … Hf … HL02) -HL02 #f2 #Hf2 #H2
+  elim (frees_lexs_conf … Hf … HL01) // -HR1 -HL01 #f1 #Hf1 #H1
+  elim (frees_lexs_conf … Hf … HL02) // -HR2 -HL02 #f2 #Hf2 #H2
   lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
   lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
   /3 width=5 by ex2_intro/