(**************************************************************************)
include "basic_2/relocation/lexs_lexs.ma".
+include "basic_2/static/frees_fqup.ma".
include "basic_2/static/lfxs.ma".
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+(* 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-.
+
+(* Advanced properties ******************************************************)
+
+(* 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).
+#R #HR #L1 #L2 #T
+elim (frees_total L1 T) #f #Hf
+elim (lexs_dec (cext2 R) cfull … L1 L2 f)
+/4 width=3 by lfxs_inv_frees, cfull_dec, ext2_dec, ex2_intro, or_intror, or_introl/
+qed-.
+
(* Main properties **********************************************************)
+(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T.
- L1 ⦻*[R, V1] L2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, T] L2.ⓑ{I}V2 →
- L1 ⦻*[R, ⓑ{p,I}V1.T] L2.
+ L1 ⪤*[R, V1] L2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, T] L2.ⓑ{I}V2 →
+ L1 ⪤*[R, ⓑ{p,I}V1.T] L2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
-elim (lexs_fwd_pair … Hf2) -Hf2 #Hf2 #_ elim (sor_isfin_ex f1 (⫱f2))
+lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
qed.
+(* Basic_2A1: llpx_sn_flat *)
theorem lfxs_flat: ∀R,I,L1,L2,V,T.
- L1 ⦻*[R, V] L2 â\86\92 L1 ⦻*[R, T] L2 →
- L1 ⦻*[R, ⓕ{I}V.T] L2.
+ L1 ⪤*[R, V] L2 â\86\92 L1 ⪤*[R, T] L2 →
+ L1 ⪤*[R, ⓕ{I}V.T] L2.
#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_bind_void: ∀R,p,I,L1,L2,V,T.
+ L1 ⪤*[R, V] L2 → L1.ⓧ ⪤*[R, T] L2.ⓧ →
+ L1 ⪤*[R, ⓑ{p,I}V.T] L2.
+#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
+lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind_void, lexs_join, isfin_tl, ex2_intro/
+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 *)
+lemma lfnxs_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
+/4 width=2 by lfxs_bind, or_intror, or_introl/
+qed-.
+
+(* Basic_2A1: uses: nllpx_sn_inv_flat *)
+lemma lfnxs_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀I,L1,L2,V,T. (L1 ⪤*[R, ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1 ⪤*[R, T] L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
+/4 width=1 by lfxs_flat, or_intror, or_introl/
+qed-.
+
+lemma lfnxs_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤*[R, T] L2.ⓧ → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
+/4 width=2 by lfxs_bind_void, or_intror, or_introl/
+qed-.
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