(**************************************************************************)
include "basic_2/relocation/lexs_lexs.ma".
-include "basic_2/static/frees_drops.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 properties ******************************************************)
+(* Advanced inversion lemmas ************************************************)
lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 →
- â\88\80f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f → L1 ⪤*[cext2 R, cfull, f] L2.
+ â\88\80f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89\98 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).
/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 … (cext2 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 (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
-/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 →
- ∀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 (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
-/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 →
- ∀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 (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 *)
/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 (cext2 R1) cfull →
- lexs_frees_confluent (cext2 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 * #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 *)
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