(**************************************************************************)
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/frees_drops.ma".
include "basic_2/static/lfxs.ma".
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
(* Advanced properties ******************************************************)
-lemma lfxs_inv_frees: â\88\80R,L1,L2,T. L1 ⦻*[R, T] L2 →
- â\88\80f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f â\86\92 L1 ⦻*[R, cfull, f] L2.
+lemma lfxs_inv_frees: â\88\80R,L1,L2,T. L1 ⪤*[R, T] L2 →
+ â\88\80f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f â\86\92 L1 ⪤*[cext2 R, cfull, f] L2.
#R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/
qed-.
(* Basic_2A1: uses: llpx_sn_dec *)
lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- â\88\80L1,L2,T. Decidable (L1 ⦻*[R, T] L2).
+ â\88\80L1,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/
+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.
+ lexs_frees_confluent … (cext2 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.
#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
/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.
+ lexs_frees_confluent … (cext2 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.
#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
/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 … R1 cfull →
- â\88\80I,L1,L2,V1,T. L1.â\93\91{I}V1 ⦻*[R1, T] L2 → ∀p.
- â\88\83â\88\83L,V. L1 ⦻*[R1, â\93\91{p,I}V1.T] L & L.â\93\91{I}V ⦻*[R2, T] L2 & R1 L1 V1 V.
+ lexs_frees_confluent … (cext2 R1) cfull →
+ â\88\80I,L1,L2,V1,T. L1.â\93\91{I}V1 ⪤*[R1, T] L2 → ∀p.
+ â\88\83â\88\83L,V. L1 ⪤*[R1, â\93\91{p,I}V1.T] L & L.â\93\91{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 (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 … HR1 HR2 … HL12 … Hfg) -HL12 #Y #H #HL2
+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 #L #V #HL1 #HV0 #H destruct
+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 *)
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 →
+ 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 (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-.
(* 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)) →
- â\88\80p,I,L1,L2,V,T. (L1 ⦻*[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⦻*[R, V] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V ⦻*[R, T] L2.ⓑ{I}V → ⊥).
+ â\88\80p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤*[R, V] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{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)) →
- â\88\80I,L1,L2,V,T. (L1 ⦻*[R, ⓕ{I}V.T] L2 → ⊥) →
- (L1 ⦻*[R, V] L2 â\86\92 â\8a¥) â\88¨ (L1 ⦻*[R, T] L2 → ⊥).
+ â\88\80I,L1,L2,V,T. (L1 ⪤*[R, ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ⪤*[R, V] L2 â\86\92 â\8a¥) â\88¨ (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-.
projects / helm.git / blobdiff