(* *)
(**************************************************************************)
+include "ground_2/lib/star.ma".
include "basic_2/notation/relations/relationstarstar_4.ma".
include "basic_2/static/lfxs.ma".
(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-definition tc_lfxs (R): term → relation lenv ≝ LTC … (lfxs R).
+definition tc_lfxs (R): term → relation lenv ≝ CTC … (lfxs R).
interpretation "iterated extension on referred entries (local environment)"
'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
(* Basic properties *********************************************************)
-lemma tc_lfxs_step_dx: â\88\80R,L1,L,T. L1 ⦻**[R, T] L →
- â\88\80L2. L ⦻*[R, T] L2 â\86\92 L1 ⦻**[R, T] L2.
+lemma tc_lfxs_step_dx: â\88\80R,L1,L,T. L1 ⪤**[R, T] L →
+ â\88\80L2. L ⪤*[R, T] L2 â\86\92 L1 ⪤**[R, T] L2.
#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
qed-.
-lemma tc_lfxs_step_sn: â\88\80R,L1,L,T. L1 ⦻*[R, T] L →
- â\88\80L2. L ⦻**[R, T] L2 â\86\92 L1 ⦻**[R, T] L2.
+lemma tc_lfxs_step_sn: â\88\80R,L1,L,T. L1 ⪤*[R, T] L →
+ â\88\80L2. L ⪤**[R, T] L2 â\86\92 L1 ⪤**[R, T] L2.
#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
qed-.
-lemma tc_lfxs_atom: â\88\80R,I. â\8b\86 ⦻**[R, ⓪{I}] ⋆.
+lemma tc_lfxs_atom: â\88\80R,I. â\8b\86 ⪤**[R, ⓪{I}] ⋆.
/2 width=1 by inj/ qed.
lemma tc_lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⦻**[R, â\8b\86s] L2 â\86\92 L1.â\93\91{I}V1 ⦻**[R, ⋆s] L2.ⓑ{I}V2.
+ L1 ⪤**[R, â\8b\86s] L2 â\86\92 L1.â\93\91{I}V1 ⪤**[R, ⋆s] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
/3 width=4 by lfxs_sort, tc_lfxs_step_dx, inj/
qed.
-lemma tc_lfxs_zero: ∀R. (∀L. reflexive … (R L)) →
- â\88\80I,L1,L2,V. L1 ⦻**[R, V] L2 →
- L1.â\93\91{I}V ⦻**[R, #0] L2.ⓑ{I}V.
+lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
+ â\88\80I,L1,L2,V. L1 ⪤**[R, V] L2 →
+ L1.â\93\91{I}V ⪤**[R, #0] L2.ⓑ{I}V.
#R #HR #I #L1 #L2 #V #H elim H -L2
-/3 width=5 by lfxs_zero, tc_lfxs_step_dx, inj/
+/3 width=5 by lfxs_pair, tc_lfxs_step_dx, inj/
qed.
+lemma tc_lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
+ L1.ⓤ{I} ⪤**[R, #0] L2.ⓤ{I}.
+/3 width=3 by lfxs_unit, inj/ qed.
+
lemma tc_lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⦻**[R, #i] L2 â\86\92 L1.â\93\91{I}V1 ⦻**[R, #⫯i] L2.ⓑ{I}V2.
+ L1 ⪤**[R, #i] L2 â\86\92 L1.â\93\91{I}V1 ⪤**[R, #â\86\91i] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
/3 width=4 by lfxs_lref, tc_lfxs_step_dx, inj/
qed.
lemma tc_lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⦻**[R, §l] L2 â\86\92 L1.â\93\91{I}V1 ⦻**[R, §l] L2.ⓑ{I}V2.
+ L1 ⪤**[R, §l] L2 â\86\92 L1.â\93\91{I}V1 ⪤**[R, §l] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
/3 width=4 by lfxs_gref, tc_lfxs_step_dx, inj/
qed.
-lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (tc_lfxs R T).
-#R #H1R #H2R #T #L1 #L2 #H elim H -L2
-/4 width=3 by lfxs_sym, tc_lfxs_step_sn, inj/
-qed-.
-
lemma tc_lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- â\88\80L1,L2,T. L1 ⦻**[R1, T] L2 â\86\92 L1 ⦻**[R2, T] L2.
+ â\88\80L1,L2,T. L1 ⪤**[R1, T] L2 â\86\92 L1 ⪤**[R2, T] L2.
#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
/4 width=5 by lfxs_co, tc_lfxs_step_dx, inj/
qed-.
(* Basic inversion lemmas ***************************************************)
(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
-lemma tc_lfxs_inv_atom_sn: â\88\80R,I,Y2. â\8b\86 ⦻**[R, ⓪{I}] Y2 → Y2 = ⋆.
+lemma tc_lfxs_inv_atom_sn: â\88\80R,I,Y2. â\8b\86 ⪤**[R, ⓪{I}] Y2 → Y2 = ⋆.
#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/
qed-.
(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
-lemma tc_lfxs_inv_atom_dx: â\88\80R,I,Y1. Y1 ⦻**[R, ⓪{I}] ⋆ → Y1 = ⋆.
+lemma tc_lfxs_inv_atom_dx: â\88\80R,I,Y1. Y1 ⪤**[R, ⓪{I}] ⋆ → Y1 = ⋆.
#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
/3 width=3 by inj, lfxs_inv_atom_dx/
qed-.
-lemma tc_lfxs_inv_sort: â\88\80R,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.
+lemma tc_lfxs_inv_sort: â\88\80R,Y1,Y2,s. Y1 ⪤**[R, ⋆s] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤**[R, ⋆s] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R #Y1 #Y2 #s #H elim H -Y2
[ #Y2 #H elim (lfxs_inv_sort … H) -H *
- /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
+ /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
| #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H *
- [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
- | #I #L #L2 #V #V2 #HL2 #H #H2 * *
+ [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
+ | #I #I2 #L #L2 #HL2 #H #H2 * *
[ #H1 #H0 destruct
- | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
- /4 width=8 by ex3_5_intro, tc_lfxs_step_dx, or_intror/
+ | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
+ /4 width=7 by ex3_4_intro, tc_lfxs_step_dx, or_intror/
]
]
]
qed-.
-lemma tc_lfxs_inv_gref: â\88\80R,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.
+lemma tc_lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⪤**[R, §l] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤**[R, §l] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
#R #Y1 #Y2 #l #H elim H -Y2
[ #Y2 #H elim (lfxs_inv_gref … H) -H *
- /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
+ /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
| #Y #Y2 #_ #H elim (lfxs_inv_gref … H) -H *
- [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
- | #I #L #L2 #V #V2 #HL2 #H #H2 * *
+ [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
+ | #I #I2 #L #L2 #HL2 #H #H2 * *
[ #H1 #H0 destruct
- | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
- /4 width=8 by ex3_5_intro, tc_lfxs_step_dx, or_intror/
+ | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
+ /4 width=7 by ex3_4_intro, tc_lfxs_step_dx, or_intror/
]
]
]
qed-.
lemma tc_lfxs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
- â\88\80p,I,L1,L2,V,T. L1 ⦻**[R, ⓑ{p,I}V.T] L2 →
- L1 ⦻**[R, V] L2 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\88§ L1.â\93\91{I}V ⪤**[R, T] L2.ⓑ{I}V.
#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
[ #L2 #H elim (lfxs_inv_bind … V ? H) -H /3 width=1 by inj, conj/
| #L #L2 #_ #H * elim (lfxs_inv_bind … V ? H) -H /3 width=3 by tc_lfxs_step_dx, conj/
]
qed-.
-lemma tc_lfxs_inv_flat: â\88\80R,I,L1,L2,V,T. L1 ⦻**[R, ⓕ{I}V.T] L2 →
- L1 ⦻**[R, V] L2 â\88§ L1 ⦻**[R, T] L2.
+lemma tc_lfxs_inv_flat: â\88\80R,I,L1,L2,V,T. L1 ⪤**[R, ⓕ{I}V.T] L2 →
+ L1 ⪤**[R, V] L2 â\88§ L1 ⪤**[R, T] L2.
#R #I #L1 #L2 #V #T #H elim H -L2
[ #L2 #H elim (lfxs_inv_flat … H) -H /3 width=1 by inj, conj/
| #L #L2 #_ #H * elim (lfxs_inv_flat … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
(* Advanced inversion lemmas ************************************************)
-lemma tc_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 (tc_lfxs_inv_sort … H) -H *
+lemma tc_lfxs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤**[R, ⋆s] Y2 →
+ ∃∃I2,L2. L1 ⪤**[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+#R #I1 #Y2 #L1 #s #H elim (tc_lfxs_inv_sort … H) -H *
[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma tc_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 (tc_lfxs_inv_sort … H) -H *
+lemma tc_lfxs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤**[R, ⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤**[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+#R #I2 #Y1 #L2 #s #H elim (tc_lfxs_inv_sort … H) -H *
[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma tc_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 (tc_lfxs_inv_gref … H) -H *
+lemma tc_lfxs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤**[R, §l] Y2 →
+ ∃∃I2,L2. L1 ⪤**[R, §l] L2 & Y2 = L2.ⓘ{I2}.
+#R #I1 #Y2 #L1 #l #H elim (tc_lfxs_inv_gref … H) -H *
[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
-lemma tc_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 (tc_lfxs_inv_gref … H) -H *
+lemma tc_lfxs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤**[R, §l] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤**[R, §l] L2 & Y1 = L1.ⓘ{I1}.
+#R #I2 #Y1 #L2 #l #H elim (tc_lfxs_inv_gref … H) -H *
[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
]
qed-.
(* Basic forward lemmas *****************************************************)
-lemma tc_lfxs_fwd_pair_sn: â\88\80R,I,L1,L2,V,T. L1 ⦻**[R, â\91¡{I}V.T] L2 â\86\92 L1 ⦻**[R, V] L2.
+lemma tc_lfxs_fwd_pair_sn: â\88\80R,I,L1,L2,V,T. L1 ⪤**[R, â\91¡{I}V.T] L2 â\86\92 L1 ⪤**[R, V] L2.
#R #I #L1 #L2 #V #T #H elim H -L2
/3 width=5 by lfxs_fwd_pair_sn, tc_lfxs_step_dx, inj/
qed-.
lemma tc_lfxs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
- â\88\80p,I,L1,L2,V,T. L1 ⦻**[R, ⓑ{p,I}V.T] L2 →
- 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.â\93\91{I}V ⪤**[R, T] L2.ⓑ{I}V.
#R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind … H) -H //
qed-.
-lemma tc_lfxs_fwd_flat_dx: â\88\80R,I,L1,L2,V,T. L1 ⦻**[R, â\93\95{I}V.T] L2 â\86\92 L1 ⦻**[R, T] L2.
+lemma tc_lfxs_fwd_flat_dx: â\88\80R,I,L1,L2,V,T. L1 ⪤**[R, â\93\95{I}V.T] L2 â\86\92 L1 ⪤**[R, T] L2.
#R #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_flat … H) -H //
qed-.
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