(* *)
(**************************************************************************)
+include "basic_2/relocation/lexs_length.ma".
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".
+include "basic_2/static/frees_drops.ma".
+include "basic_2/static/fsle_fsle.ma".
+include "basic_2/static/lfxs_fsle.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-.
+
+lemma frees_lexs_conf: ∀R. lfxs_fsle_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
+@(fsle_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_fsle_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).
+#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-.
+
lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lexs_frees_confluent … R1 cfull →
- â\88\80L1,L2,V. L1 ⦻*[R1, V] L2 → ∀I,T.
- â\88\83â\88\83L. L1 ⦻*[R1, â\91¡{I}V.T] L & L ⦻*[R2, V] L2.
+ lfxs_fsle_compatible R1 →
+ â\88\80L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T.
+ â\88\83â\88\83L. L1 ⪤*[R1, â\91¡{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
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
+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 … R1 cfull →
- â\88\80L1,L2,T. L1 ⦻*[R1, T] L2 → ∀I,V.
- â\88\83â\88\83L. L1 ⦻*[R1, â\93\95{I}V.T] L & L ⦻*[R2, T] L2.
+ lfxs_fsle_compatible R1 →
+ â\88\80L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V.
+ â\88\83â\88\83L. L1 ⪤*[R1, â\93\95{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
+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)) →
+ lfxs_fsle_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
+elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg
+elim (frees_inv_bind … 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_pair_sn … H) -H #V #HV #H1 #H2 destruct
+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)) →
+ lfxs_fsle_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
+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 (frees_lexs_conf … Hf … H0) -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 *)
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_conf: ∀R1,R2.
- lexs_frees_confluent R1 cfull →
- lexs_frees_confluent R2 cfull →
+ lfxs_fsle_compatible R1 →
+ lfxs_fsle_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
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/
-| #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/
+| #g * #I0 [2: #V0 ] #K0 #n #HLK0 #Hgf #Z1 #H1 #Z2 #H2 #K1 #HK01 #K2 #HK02
+ [ elim (ext2_inv_pair_sn … H1) -H1 #V1 #HV01 #H destruct
+ elim (ext2_inv_pair_sn … H2) -H2 #V2 #HV02 #H destruct
+ elim (frees_inv_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) /3 width=3 by ext2_pair, ex2_intro/
+ | lapply (ext2_inv_unit_sn … H1) -H1 #H destruct
+ lapply (ext2_inv_unit_sn … H2) -H2 #H destruct
+ /3 width=3 by ext2_unit, 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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