include "ground_2/relocation/rtmap_id.ma".
include "basic_2/notation/relations/relationstar_4.ma".
+include "basic_2/syntax/lenv_ext2.ma".
include "basic_2/relocation/lexs.ma".
include "basic_2/static/frees.ma".
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition lfxs (R) (T): relation lenv ≝
- λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⪤*[R, cfull, f] L2.
+ λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≡ f & L1 ⪤*[cext2 R, cfull, f] L2.
interpretation "generic extension on referred entries (local environment)"
'RelationStar R T L1 L2 = (lfxs R T L1 L2).
-definition R_frees_confluent: predicate (relation3 lenv term term) ≝
+definition R_frees_confluent: predicate (relation3 …) ≝
λRN.
∀f1,L,T1. L ⊢ 𝐅*⦃T1⦄ ≡ f1 → ∀T2. RN L T1 T2 →
∃∃f2. L ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
-definition lexs_frees_confluent: relation (relation3 lenv term term) ≝
+definition lexs_frees_confluent: relation (relation3 …) ≝
λRN,RP.
∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
∀L2. L1 ⪤*[RN, RP, f1] L2 →
(* Basic properties *********************************************************)
lemma lfxs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
-/3 width=3 by lexs_atom, frees_atom, ex2_intro/
+#R * /3 width=3 by frees_sort, frees_atom, frees_gref, lexs_atom, ex2_intro/
qed.
(* Basic_2A1: uses: llpx_sn_sort *)
-lemma lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #s * /3 width=3 by lexs_push, frees_sort, ex2_intro/
+lemma lfxs_sort: ∀R,I1,I2,L1,L2,s.
+ L1 ⪤*[R, ⋆s] L2 → L1.ⓘ{I1} ⪤*[R, ⋆s] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
+lapply (frees_inv_sort … Hf) -Hf
+/4 width=3 by frees_sort, lexs_push, isid_push, ex2_intro/
qed.
-lemma lfxs_zero: ∀R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
+lemma lfxs_pair: ∀R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
+#R #I1 #I2 #L1 #L2 #V1 *
+/4 width=3 by ext2_pair, frees_pair, lexs_next, ex2_intro/
qed.
-lemma lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #⫯i] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
+lemma lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
+ L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
+/4 width=3 by frees_unit, lexs_next, ext2_unit, ex2_intro/ qed.
+
+lemma lfxs_lref: ∀R,I1,I2,L1,L2,i.
+ L1 ⪤*[R, #i] L2 → L1.ⓘ{I1} ⪤*[R, #⫯i] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
qed.
(* Basic_2A1: uses: llpx_sn_gref *)
-lemma lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #l * /3 width=3 by lexs_push, frees_gref, ex2_intro/
+lemma lfxs_gref: ∀R,I1,I2,L1,L2,l.
+ L1 ⪤*[R, §l] L2 → L1.ⓘ{I1} ⪤*[R, §l] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
+lapply (frees_inv_gref … Hf) -Hf
+/4 width=3 by frees_gref, lexs_push, isid_push, ex2_intro/
qed.
-lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
- L1.â\93\91{I}V ⪤*[R, T] L2.â\93\91{I}V1 →
- ∀V2. R L1 V V2 →
- L1.â\93\91{I}V ⪤*[R, T] L2.â\93\91{I}V2.
-#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
+lemma lfxs_bind_repl_dx: ∀R,I,I1,L1,L2,T.
+ L1.â\93\98{I} ⪤*[R, T] L2.â\93\98{I1} →
+ ∀I2. cext2 R L1 I I2 →
+ L1.â\93\98{I} ⪤*[R, T] L2.â\93\98{I2}.
+#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
/3 width=5 by lexs_pair_repl, ex2_intro/
qed-.
-lemma lfxs_sym: ∀R. lexs_frees_confluent R cfull →
+lemma lfxs_sym: ∀R. lexs_frees_confluent (cext2 R) cfull →
(∀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 (H1R … Hf1 … HL12) -Hf1
-/4 width=5 by sle_lexs_trans, lexs_sym, ex2_intro/
+/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_co *)
lemma lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
-#R1 #R2 #HR #L1 #L2 #T * /4 width=7 by lexs_co, ex2_intro/
+#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by lexs_co, cext2_co, ex2_intro/
qed-.
lemma lfxs_isid: ∀R1,R2,L1,L2,T1,T2.
qed-.
lemma lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, ⋆s] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #Y2 #s * #f #H1 #H2
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
| lapply (frees_inv_sort … H1) -H1 #Hf
elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_sort_gen, ex3_5_intro, ex2_intro, or_intror/
+ elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
+ /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/
]
qed-.
lemma lfxs_inv_zero: ∀R,Y1,Y2. Y1 ⪤*[R, #0] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
-[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
-| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
- /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cext2 R, cfull, f] L2 &
+ Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+#R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
+[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
+| elim (frees_inv_unit … H1) -H1 #g #HX #H destruct
+ elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL12 #H #H2 destruct
+ >(ext2_inv_unit_sn … H) -H /3 width=8 by or3_intro2, ex4_4_intro/
+| elim (frees_inv_pair … H1) -H1 #g #Hg #H destruct
+ elim (lexs_inv_next1 … H2) -H2 #Z2 #L2 #HL12 #H
+ elim (ext2_inv_pair_sn … H) -H
+ /4 width=9 by or3_intro1, ex4_5_intro, ex2_intro/
]
qed-.
lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⪤*[R, #⫯i] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, #i] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
-[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
-| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
- /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R, #i] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
+[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct
+ elim (lexs_inv_push1 … H2) -H2
+ /4 width=7 by ex3_4_intro, ex2_intro, or_intror/
]
qed-.
lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, §l] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #Y2 #l * #f #H1 #H2
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
| lapply (frees_inv_gref … H1) -H1 #Hf
elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
+ elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
+ /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/
]
qed-.
lemma lfxs_inv_bind: ∀R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
L1 ⪤*[R, V1] L2 ∧ L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
-/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+/6 width=6 by sle_lexs_trans, lexs_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_flat *)
(* Advanced inversion lemmas ************************************************)
-lemma lfxs_inv_sort_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⪤*[R, ⋆s] Y2 →
- ∃∃L2,V2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
+lemma lfxs_inv_sort_bind_sn: ∀R,I1,K1,L2,s. K1.ⓘ{I1} ⪤*[R, ⋆s] L2 →
+ ∃∃I2,K2. K1 ⪤*[R, ⋆s] K2 & L2 = K2.ⓘ{I2}.
+#R #I1 #K1 #L2 #s #H elim (lfxs_inv_sort … H) -H *
[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⪤*[R, ⋆s] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
+lemma lfxs_inv_sort_bind_dx: ∀R,I2,K2,L1,s. L1 ⪤*[R, ⋆s] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤*[R, ⋆s] K2 & L1 = K1.ⓘ{I1}.
+#R #I2 #K2 #L1 #s #H elim (lfxs_inv_sort … H) -H *
[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⪤*[R, #0] Y2 →
- ∃∃L2,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
- Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
+lemma lfxs_inv_zero_pair_sn: ∀R,I,L2,K1,V1. K1.ⓑ{I}V1 ⪤*[R, #0] L2 →
+ ∃∃K2,V2. K1 ⪤*[R, V1] K2 & R K1 V1 V2 &
+ L2 = K2.ⓑ{I}V2.
+#R #I #L2 #K1 #V1 #H elim (lfxs_inv_zero … H) -H *
[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
+| #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct
/2 width=5 by ex3_2_intro/
+| #f #Z #Y1 #Y2 #_ #_ #H destruct
]
qed-.
-lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⪤*[R, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
+lemma lfxs_inv_zero_pair_dx: ∀R,I,L1,K2,V2. L1 ⪤*[R, #0] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤*[R, V1] K2 & R K1 V1 V2 &
+ L1 = K1.ⓑ{I}V1.
+#R #I #L1 #K2 #V2 #H elim (lfxs_inv_zero … H) -H *
[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
+| #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct
/2 width=5 by ex3_2_intro/
+| #f #Z #Y1 #Y2 #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lfxs_inv_zero_unit_sn: ∀R,I,K1,L2. K1.ⓤ{I} ⪤*[R, #0] L2 →
+ ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤*[cext2 R, cfull, f] K2 &
+ L2 = K2.ⓤ{I}.
+#R #I #K1 #L2 #H elim (lfxs_inv_zero … H) -H *
+[ #H destruct
+| #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct
+| #f #Z #Y1 #K2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_unit_dx: ∀R,I,L1,K2. L1 ⪤*[R, #0] K2.ⓤ{I} →
+ ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤*[cext2 R, cfull, f] K2 &
+ L1 = K1.ⓤ{I}.
+#R #I #L1 #K2 #H elim (lfxs_inv_zero … H) -H *
+[ #_ #H destruct
+| #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct
+| #f #Z #K1 #Y2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⪤*[R, #⫯i] Y2 →
- ∃∃L2,V2. L1 ⪤*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
+lemma lfxs_inv_lref_bind_sn: ∀R,I1,K1,L2,i. K1.ⓘ{I1} ⪤*[R, #⫯i] L2 →
+ ∃∃I2,K2. K1 ⪤*[R, #i] K2 & L2 = K2.ⓘ{I2}.
+#R #I1 #K1 #L2 #i #H elim (lfxs_inv_lref … H) -H *
[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⪤*[R, #⫯i] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⪤*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
+lemma lfxs_inv_lref_bind_dx: ∀R,I2,K2,L1,i. L1 ⪤*[R, #⫯i] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤*[R, #i] K2 & L1 = K1.ⓘ{I1}.
+#R #I2 #K2 #L1 #i #H elim (lfxs_inv_lref … H) -H *
[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⪤*[R, §l] Y2 →
- ∃∃L2,V2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
+lemma lfxs_inv_gref_bind_sn: ∀R,I1,K1,L2,l. K1.ⓘ{I1} ⪤*[R, §l] L2 →
+ ∃∃I2,K2. K1 ⪤*[R, §l] K2 & L2 = K2.ⓘ{I2}.
+#R #I1 #K1 #L2 #l #H elim (lfxs_inv_gref … H) -H *
[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⪤*[R, §l] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
+lemma lfxs_inv_gref_bind_dx: ∀R,I2,K2,L1,l. L1 ⪤*[R, §l] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤*[R, §l] K2 & L1 = K1.ⓘ{I1}.
+#R #I2 #K2 #L1 #l #H elim (lfxs_inv_gref … H) -H *
[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
qed-.
-lemma lfxs_fwd_dx: ∀R,I,L1,K2,T,V2. L1 ⪤*[R, T] K2.ⓑ{I}V2 →
- ∃∃K1,V1. L1 = K1.ⓑ{I}V1.
-#R #I #L1 #K2 #T #V2 * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
-[ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #K1 #V1 #_ #_ #H destruct
+lemma lfxs_fwd_dx: ∀R,I2,L1,K2,T. L1 ⪤*[R, T] K2.ⓘ{I2} →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1}.
+#R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
+[ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
/2 width=3 by ex1_2_intro/
qed-.
projects / helm.git / blobdiff