work in progress with voids and lveq (was: the most recent voids)

[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 95b89cd809e2cbca53835b2fe0ae071c9a93768b..89949c4b00ba20b4b43988ab3859a2b3af60005e 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfxs_lfxs.ma
@@ -89,6 +89,26 @@ elim (HR … Hf … H0) -HR -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 →
+                               ∀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
+elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
+elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy
+lapply(frees_mono … H … Hf) -H #H2
+lapply (tl_eq_repl … H2) -H2 #H2
+lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
+lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
+lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg
+elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2
+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
+/4 width=7 by sle_lexs_trans, ex2_intro/ (* note: 2 ex2_intro *)
+qed-.
+
 (* Main properties **********************************************************)
 
 (* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
@@ -145,6 +165,59 @@ elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
 ]
 qed-.
 
+theorem lfxs_trans_gen: ∀R1,R2,R3. 
+                        c_reflexive … R1 → c_reflexive … R2 →
+                        lfxs_confluent R1 R2 → lfxs_transitive R1 R2 R3 →
+                        ∀L1,T,L. L1 ⪤*[R1, T] L →
+                        ∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2.
+#R1 #R2 #R3 #H1R #H2R #H3R #H4R #L1 #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L1 T) -L1 -T
+#G0 #L0 #T0 #IH #G #L1 * *
+[ #s #HG #HL #HT #L #H1 #L2 #H2 destruct
+  elim (lfxs_inv_sort … H1) -H1 *
+  [ #H1 #H0 destruct
+    >(lfxs_inv_atom_sn … H2) -L2 //
+  | #I1 #I #K1 #K #HK1 #H1 #H0 destruct
+    elim (lfxs_inv_sort_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct
+    /4 width=3 by lfxs_sort, fqu_fqup/
+  ]
+| * [ | #i ] #HG #HL #HT #L #H1 #L2 #H2 destruct
+  [ elim (lfxs_inv_zero … H1) -H1 *
+    [ #H1 #H0 destruct
+      >(lfxs_inv_atom_sn … H2) -L2 //
+    | #I #K1 #K #V1 #V #HK1 #H1 #H0 #H destruct
+      elim (lfxs_inv_zero_pair_sn … H2) -H2 #K2 #V2 #HK2 #HV2 #H destruct
+      /4 width=7 by lfxs_pair, fqu_fqup, fqu_lref_O/
+    | #f1 #I #K1 #K #Hf1 #HK1 #H1 #H0 destruct
+      elim (lfxs_inv_zero_unit_sn … H2) -H2 #f2 #K2 #Hf2 #HK2 #H destruct
+      /5 width=8 by lfxs_unit, lexs_trans_id_cfull, lexs_eq_repl_back, isid_inv_eq_repl/
+    ]
+  | elim (lfxs_inv_lref … H1) -H1 *
+    [ #H1 #H0 destruct
+      >(lfxs_inv_atom_sn … H2) -L2 //
+    | #I1 #I #K1 #K #HK1 #H1 #H0 destruct
+      elim (lfxs_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct
+     /4 width=3 by lfxs_lref, fqu_fqup/
+    ]
+  ]
+| #l #HG #HL #HT #L #H1 #L2 #H2 destruct
+  elim (lfxs_inv_gref … H1) -H1 *
+  [ #H1 #H0 destruct
+    >(lfxs_inv_atom_sn … H2) -L2 //
+  | #I1 #I #K1 #K #HK1 #H1 #H0 destruct
+    elim (lfxs_inv_gref_bind_sn … H2) -H2 #I2 #K2 #HK2 #H destruct
+    /4 width=3 by lfxs_gref, fqu_fqup/
+  ]
+| #p #I #V1 #T1 #HG #HL #HT #L #H1 #L2 #H2 destruct
+  elim (lfxs_inv_bind … V1 V1 … H1) -H1 // #H1V #H1T
+  elim (lfxs_inv_bind … V1 V1 … H2) -H2 // #H2V #H2T
+  /3 width=4 by lfxs_bind/
+| #I #V1 #T1 #HG #HL #HT #L #H1 #L2 #H2 destruct
+  elim (lfxs_inv_flat … H1) -H1 #H1V #H1T
+  elim (lfxs_inv_flat … H2) -H2 #H2V #H2T
+  /3 width=3 by lfxs_flat/
+]
+qed-.
+
 (* Negated inversion lemmas *************************************************)
 
 (* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)