advances on lfsx ...

[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 ece33413ebc621f7814bccef71aed79b2c56fe3d..54fdb9e43951fe8ae7e8eac5fd81fddb3bb012d4 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
@@ -13,11 +13,63 @@
 (**************************************************************************)
 
 include "basic_2/relocation/lexs_lexs.ma".
+include "basic_2/static/frees_fqup.ma".
+include "basic_2/static/frees_frees.ma".
 include "basic_2/static/lfxs.ma".
 
 (* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
 
-(* Main properties ******************************************************)
+(* Advanced properties ******************************************************)
+
+lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⦻*[R, T] L2 →
+                      ∀f. L1 ⊢ 𝐅*⦃T⦄ ≡ f → L1 ⦻*[R, cfull, f] L2.
+#R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/
+qed-.
+
+lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+                ∀L1,L2,T. Decidable (L1 ⦻*[R, T] L2).
+#R #HR #L1 #L2 #T
+elim (frees_total L1 T) #f #Hf
+elim (lexs_dec R cfull HR … L1 L2 f)
+/4 width=3 by lfxs_inv_frees, cfull_dec, ex2_intro, or_intror, or_introl/
+qed-.
+
+lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+                          lexs_frees_confluent … R1 cfull →
+                          ∀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
+[ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
+  elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy
+| elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
+  elim (frees_inv_flat … Hg) #y1 #y2 #H #_ #Hy
+]
+lapply(frees_mono … H … Hf) -H #H1
+lapply (sor_eq_repl_back1 … Hy … H1) -y1 #Hy
+lapply (sor_inv_sle_sn … Hy) -y2 #Hfg
+elim (lexs_sle_split … HR1 HR2 … HL12 … Hfg) -HL12 #L #HL1 #HL2
+lapply (sle_lexs_trans … HL1 … Hfg) // #H
+elim (HR … Hf … H) -HR -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 … R1 cfull →
+                          ∀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
+elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
+elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy
+lapply(frees_mono … H … Hf) -H #H2
+lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
+lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
+elim (lexs_sle_split … HR1 HR2 … HL12 … Hfg) -HL12 #L #HL1 #HL2
+lapply (sle_lexs_trans … HL1 … Hfg) // #H
+elim (HR … Hf … H) -HR -Hf -H
+/4 width=7 by sle_lexs_trans, ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
 
 theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T.
                    L1 ⦻*[R, V1] L2 → L1.ⓑ{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
@@ -33,3 +85,26 @@ theorem lfxs_flat: ∀R,I,L1,L2,V,T.
 #R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
 /3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/
 qed.
+
+theorem lfxs_conf: ∀R1,R2.
+                   lexs_frees_confluent R1 cfull →
+                   lexs_frees_confluent R2 cfull →
+                   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
+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
+  lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
+  lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
+  /3 width=5 by ex2_intro/
+| #g #I #K0 #V0 #n #HLK0 #Hgf #V1 #HV01 #V2 #HV02 #K1 #HK01 #K2 #HK02
+  elim (frees_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0
+  lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01
+  lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02
+  elim (HR12 … HV01 … HV02 K1 … K2) /2 width=3 by ex2_intro/
+]
+qed-.