(* 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.
+ 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/
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, #⫯i] 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.
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 →
+lemma tc_lfxs_inv_sort: â\88\80R,Y1,Y2,s. Y1 ⪤**[R, ⋆s] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 &
+ â\88\83â\88\83I,L1,L2,V1,V2. L1 ⪤**[R, ⋆s] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R #Y1 #Y2 #s #H elim H -Y2
[ #Y2 #H elim (lfxs_inv_sort … H) -H *
]
qed-.
-lemma tc_lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⦻**[R, §l] Y2 →
+lemma tc_lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⪤**[R, §l] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻**[R, §l] L2 &
+ â\88\83â\88\83I,L1,L2,V1,V2. L1 ⪤**[R, §l] L2 &
Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
#R #Y1 #Y2 #l #H elim H -Y2
[ #Y2 #H elim (lfxs_inv_gref … H) -H *
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: â\88\80R,I,Y2,L1,V1,s. L1.â\93\91{I}V1 ⦻**[R, ⋆s] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻**[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+lemma tc_lfxs_inv_sort_pair_sn: â\88\80R,I,Y2,L1,V1,s. L1.â\93\91{I}V1 ⪤**[R, ⋆s] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤**[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #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/
]
qed-.
-lemma tc_lfxs_inv_sort_pair_dx: â\88\80R,I,Y1,L2,V2,s. Y1 ⦻**[R, ⋆s] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+lemma tc_lfxs_inv_sort_pair_dx: â\88\80R,I,Y1,L2,V2,s. Y1 ⪤**[R, ⋆s] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤**[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #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/
]
qed-.
-lemma tc_lfxs_inv_gref_pair_sn: â\88\80R,I,Y2,L1,V1,l. L1.â\93\91{I}V1 ⦻**[R, §l] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻**[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+lemma tc_lfxs_inv_gref_pair_sn: â\88\80R,I,Y2,L1,V1,l. L1.â\93\91{I}V1 ⪤**[R, §l] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤**[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #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/
]
qed-.
-lemma tc_lfxs_inv_gref_pair_dx: â\88\80R,I,Y1,L2,V2,l. Y1 ⦻**[R, §l] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻**[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+lemma tc_lfxs_inv_gref_pair_dx: â\88\80R,I,Y1,L2,V2,l. Y1 ⪤**[R, §l] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤**[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #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/
(* 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-.
definition tc_dedropable_sn: predicate (relation3 lenv term term) ≝
λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 →
- â\88\80K2,T. K1 ⦻**[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
- â\88\83â\88\83L2. L1 ⦻**[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
+ â\88\80K2,T. K1 ⪤**[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
+ â\88\83â\88\83L2. L1 ⪤**[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
definition tc_dropable_sn: predicate (relation3 lenv term term) ≝
λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ →
- â\88\80L2,U. L1 ⦻**[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
- â\88\83â\88\83K2. K1 ⦻**[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
+ â\88\80L2,U. L1 ⪤**[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
+ â\88\83â\88\83K2. K1 ⪤**[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
definition tc_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. â\88\80L1,L2,U. L1 ⦻**[R, U] L2 →
+ λR. â\88\80L1,L2,U. L1 ⪤**[R, U] L2 →
∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U →
- â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⦻**[R, T] K2.
+ â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⪤**[R, T] K2.
(* Properties with generic slicing for local environments *******************)
(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
- â\88\80L,V1,V2. LTC â\80¦ R L V1 V2 â\86\92 â\88\80I,T. L.â\93\91{I}V1 ⦻**[R, T] L.ⓑ{I}V2.
+ â\88\80L,V1,V2. LTC â\80¦ R L V1 V2 â\86\92 â\88\80I,T. L.â\93\91{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #H elim H -V2
/3 width=3 by tc_lfxs_step_dx, lfxs_pair, inj/
qed.
lemma tc_lfxs_ind_sn: ∀R. (∀L. reflexive … (R L)) →
∀L1,T. ∀R0:predicate …. R0 L1 →
- (â\88\80L,L2. L1 ⦻**[R, T] L â\86\92 L ⦻*[R, T] L2 → R0 L → R0 L2) →
- â\88\80L2. L1 ⦻**[R, T] L2 → R0 L2.
+ (â\88\80L,L2. L1 ⪤**[R, T] L â\86\92 L ⪤*[R, T] L2 → R0 L → R0 L2) →
+ â\88\80L2. L1 ⪤**[R, T] L2 → R0 L2.
#R #HR #L1 #T #R0 #HL1 #IHL1 #L2 #HL12
@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
qed-.
lemma tc_lfxs_ind_dx: ∀R. (∀L. reflexive … (R L)) →
∀L2,T. ∀R0:predicate …. R0 L2 →
- (â\88\80L1,L. L1 ⦻*[R, T] L â\86\92 L ⦻**[R, T] L2 → R0 L → R0 L1) →
- â\88\80L1. L1 ⦻**[R, T] L2 → R0 L1.
+ (â\88\80L1,L. L1 ⪤*[R, T] L â\86\92 L ⪤**[R, T] L2 → R0 L → R0 L1) →
+ â\88\80L1. L1 ⪤**[R, T] L2 → R0 L1.
#R #HR #L2 #R0 #HL2 #IHL2 #L1 #HL12
@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
qed-.
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: uses: TC_lpx_sn_fwd_length *)
-lemma tc_lfxs_fwd_length: â\88\80R,L1,L2,T. L1 ⦻**[R, T] L2 → |L1| = |L2|.
+lemma tc_lfxs_fwd_length: â\88\80R,L1,L2,T. L1 ⪤**[R, T] L2 → |L1| = |L2|.
#R #L1 #L2 #T #H elim H -L2
[ #L2 #HL12 >(lfxs_fwd_length … HL12) -HL12 //
| #L #L2 #_ #HL2 #IHL1
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⦻ * [ break term 46 R , break term 46 T ] break term 46 L2 )"
+notation "hvbox( L1 ⪤ * [ break term 46 R , break term 46 T ] break term 46 L2 )"
non associative with precedence 45
for @{ 'RelationStar $R $T $L1 $L2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⦻ * [ break term 46 R1 , break term 46 R2 , break term 46 f ] break term 46 L2 )"
+notation "hvbox( L1 ⪤ * [ break term 46 R1 , break term 46 R2 , break term 46 f ] break term 46 L2 )"
non associative with precedence 45
for @{ 'RelationStar $R1 $R2 $f $L1 $L2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⦻ * * [ break term 46 R , break term 46 T ] break term 46 L2 )"
+notation "hvbox( L1 ⪤ * * [ break term 46 R , break term 46 T ] break term 46 L2 )"
non associative with precedence 45
for @{ 'RelationStarStar $R $T $L1 $L2 }.
qed-.
fact lexs_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ →
- â\88\80f2,L1. L1 ⦻*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≡ f2 →
- â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⦻*[RN, RP, f1] K2.
+ â\88\80f2,L1. L1 ⪤*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≡ f2 →
+ â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⪤*[RN, RP, f1] K2.
#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (lexs_inv_atom2 … H) -H
#H destruct /4 width=3 by lexs_atom, drops_atom, ex2_intro/
(* Basic_2A1: includes: lpx_sn_drop_conf *) (**)
lemma lexs_drops_conf_next: ∀RN,RP.
- â\88\80f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+ â\88\80f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≡ f2 →
- â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
+ â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⪤*[RN, RP, f1] K2 & RN K1 V1 V2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (lexs_inv_next1 … HX) -HX
qed-.
lemma lexs_drops_conf_push: ∀RN,RP.
- â\88\80f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+ â\88\80f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I,K1,V1. ⬇*[b,f] L1 ≡ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≡ f2 →
- â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
+ â\88\83â\88\83K2,V2. â¬\87*[b,f] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⪤*[RN, RP, f1] K2 & RP K1 V1 V2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
#X #HX #HLK2 elim (lexs_inv_push1 … HX) -HX
qed-.
(* Basic_2A1: includes: lpx_sn_drop_trans *)
-lemma lexs_drops_trans_next: â\88\80RN,RP,f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+lemma lexs_drops_trans_next: â\88\80RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
∀f1. f ~⊚ ⫯f1 ≡ f2 →
- â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RN K1 V1 V2.
+ â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⪤*[RN, RP, f1] K2 & RN K1 V1 V2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX
#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma lexs_drops_trans_push: â\88\80RN,RP,f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
+lemma lexs_drops_trans_push: â\88\80RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
∀b,f,I,K2,V2. ⬇*[b,f] L2 ≡ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
∀f1. f ~⊚ ↑f1 ≡ f2 →
- â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⦻*[RN, RP, f1] K2 & RP K1 V1 V2.
+ â\88\83â\88\83K1,V1. â¬\87*[b,f] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⪤*[RN, RP, f1] K2 & RP K1 V1 V2.
#RN #RP #f2 #L1 #L2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf #f1 #Hf2
elim (lexs_co_dropable_dx … HL12 … HLK1 … Hf … Hf2) -L2 -f2 -Hf
#X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX
lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn RN → d_liftable2_sn RP →
- â\88\80f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 →
+ â\88\80f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 →
∀f2. f ~⊚ f1 ≡ ⫯f2 →
- â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⦻*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
+ â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2
elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (lexs_inv_next1 … HX) -HX
lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
d_liftable2_sn RN → d_liftable2_sn RP →
- â\88\80f1,K1,K2. K1 ⦻*[RN, RP, f1] K2 →
+ â\88\80f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
∀b,f,I,L1,V1. ⬇*[b,f] L1.ⓑ{I}V1 ≡ K1 →
∀f2. f ~⊚ f1 ≡ ↑f2 →
- â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⦻*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
+ â\88\83â\88\83L2,V2. â¬\87*[b,f] L2.â\93\91{I}V2 â\89¡ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 V1 V2 & L1.ⓑ{I}V1≡[f]L2.ⓑ{I}V2.
#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I #L1 #V1 #HLK1 #f2 #Hf2
elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
#X #HX #HLK2 #H1L12 elim (lexs_inv_push1 … HX) -HX
qed-.
lemma drops_atom2_lexs_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≡ ⋆ → 𝐔⦃f1⦄ →
- â\88\80f,L2. L1 ⦻*[RN, RP, f] L2 →
+ â\88\80f,L2. L1 ⪤*[RN, RP, f] L2 →
∀f2. f1 ~⊚ f2 ≡f → ⬇*[b, f1] L2 ≡ ⋆.
#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
elim (lexs_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
relation3 rtmap lenv term ≝
λR1,R2,RN1,RP1,RN2,RP2,f,L0,T0.
∀T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- â\88\80L1. L0 ⦻*[RN1, RP1, f] L1 â\86\92 â\88\80L2. L0 ⦻*[RN2, RP2, f] L2 →
+ â\88\80L1. L0 ⪤*[RN1, RP1, f] L1 â\86\92 â\88\80L2. L0 ⪤*[RN2, RP2, f] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
definition lexs_transitive: relation5 (relation3 lenv term term)
(relation3 lenv term term) … ≝
λR1,R2,R3,RN,RP.
- â\88\80f,L1,T1,T. R1 L1 T1 T â\86\92 â\88\80L2. L1 ⦻*[RN, RP, f] L2 →
+ â\88\80f,L1,T1,T. R1 L1 T1 T â\86\92 â\88\80L2. L1 ⪤*[RN, RP, f] L2 →
∀T2. R2 L2 T T2 → R3 L1 T1 T2.
(* Basic inversion lemmas ***************************************************)
-fact lexs_inv_atom1_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+fact lexs_inv_atom1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
#RN #RP #f #X #Y * -f -X -Y //
#f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: â\88\80RN,RP,f,Y. â\8b\86 ⦻*[RN, RP, f] Y → Y = ⋆.
+lemma lexs_inv_atom1: â\88\80RN,RP,f,Y. â\8b\86 ⪤*[RN, RP, f] Y → Y = ⋆.
/2 width=6 by lexs_inv_atom1_aux/ qed-.
-fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g →
- â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+fact lexs_inv_next1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ⫯g →
+ â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J #K1 #W1 #H destruct
| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K1 #W1 #H1 #H2 <(injective_next … H2) -g destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⦻*[RN, RP, ⫯g] Y →
- â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_next1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⪤*[RN, RP, ⫯g] Y →
+ â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
/2 width=7 by lexs_inv_next1_aux/ qed-.
-fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g →
- â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+fact lexs_inv_push1_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K1,W1. X = K1.ⓑ{J}W1 → f = ↑g →
+ â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J #K1 #W1 #H destruct
| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K1 #W1 #_ #H elim (discr_next_push … H)
]
qed-.
-lemma lexs_inv_push1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⦻*[RN, RP, ↑g] Y →
- â\88\83â\88\83K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_push1: â\88\80RN,RP,g,J,K1,Y,W1. K1.â\93\91{J}W1 ⪤*[RN, RP, ↑g] Y →
+ â\88\83â\88\83K2,W2. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
/2 width=7 by lexs_inv_push1_aux/ qed-.
-fact lexs_inv_atom2_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+fact lexs_inv_atom2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
#RN #RP #f #X #Y * -f -X -Y //
#f #I #L1 #L2 #V1 #V2 #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: â\88\80RN,RP,f,X. X ⦻*[RN, RP, f] ⋆ → X = ⋆.
+lemma lexs_inv_atom2: â\88\80RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆.
/2 width=6 by lexs_inv_atom2_aux/ qed-.
-fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g →
- â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+fact lexs_inv_next2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ⫯g →
+ â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
#RN #RP #f #X #Y * -f -X -Y
[ #f #g #J #K2 #W2 #H destruct
| #f #I #L1 #L2 #V1 #V2 #HL #HV #g #J #K2 #W2 #H1 #H2 <(injective_next … H2) -g destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: â\88\80RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
- â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_next2: â\88\80RN,RP,g,J,X,K2,W2. X ⪤*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
+ â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
/2 width=7 by lexs_inv_next2_aux/ qed-.
-fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⦻*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g →
- â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+fact lexs_inv_push2_aux: â\88\80RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J,K2,W2. Y = K2.ⓑ{J}W2 → f = ↑g →
+ â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
#RN #RP #f #X #Y * -f -X -Y
[ #f #J #K2 #W2 #g #H destruct
| #f #I #L1 #L2 #V1 #V2 #_ #_ #g #J #K2 #W2 #_ #H elim (discr_next_push … H)
]
qed-.
-lemma lexs_inv_push2: â\88\80RN,RP,g,J,X,K2,W2. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 →
- â\88\83â\88\83K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_push2: â\88\80RN,RP,g,J,X,K2,W2. X ⪤*[RN, RP, ↑g] K2.ⓑ{J}W2 →
+ â\88\83â\88\83K1,W1. K1 ⪤*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
/2 width=7 by lexs_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2,V1,V2.
- L1.â\93\91{I1}V1 ⦻*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) →
- â\88§â\88§ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
+ L1.â\93\91{I1}V1 ⪤*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) →
+ â\88§â\88§ L1 ⪤*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
#RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_next1 … H) -H
#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
qed-.
lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2,V1,V2.
- L1.â\93\91{I1}V1 ⦻*[RN, RP, ↑f] (L2.ⓑ{I2}V2) →
- â\88§â\88§ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
+ L1.â\93\91{I1}V1 ⪤*[RN, RP, ↑f] (L2.ⓑ{I2}V2) →
+ â\88§â\88§ L1 ⪤*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
#RN #RP #f #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lexs_inv_push1 … H) -H
#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
qed-.
-lemma lexs_inv_tl: â\88\80RN,RP,f,I,L1,L2,V1,V2. L1 ⦻*[RN, RP, ⫱f] L2 →
+lemma lexs_inv_tl: â\88\80RN,RP,f,I,L1,L2,V1,V2. L1 ⪤*[RN, RP, ⫱f] L2 →
RN L1 V1 V2 → RP L1 V1 V2 →
- L1.â\93\91{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2.
+ L1.â\93\91{I}V1 ⪤*[RN, RP, f] L2.ⓑ{I}V2.
#RN #RP #f #I #L2 #L2 #V1 #V2 elim (pn_split f) *
/2 width=1 by lexs_next, lexs_push/
qed-.
(* Basic forward lemmas *****************************************************)
lemma lexs_fwd_pair: ∀RN,RP,f,I1,I2,L1,L2,V1,V2.
- L1.â\93\91{I1}V1 ⦻*[RN, RP, f] L2.ⓑ{I2}V2 →
- L1 ⦻*[RN, RP, ⫱f] L2 ∧ I1 = I2.
+ L1.â\93\91{I1}V1 ⪤*[RN, RP, f] L2.ⓑ{I2}V2 →
+ L1 ⪤*[RN, RP, ⫱f] L2 ∧ I1 = I2.
#RN #RP #f #I1 #I2 #L2 #L2 #V1 #V2 #Hf
elim (pn_split f) * #g #H destruct
[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf
(* Basic properties *********************************************************)
-lemma lexs_eq_repl_back: â\88\80RN,RP,L1,L2. eq_repl_back â\80¦ (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_back: â\88\80RN,RP,L1,L2. eq_repl_back â\80¦ (λf. L1 ⪤*[RN, RP, f] L2).
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I #L1 #L2 #V1 #v2 #_ #HV #IH #f2 #H
[ elim (eq_inv_nx … H) -H /3 width=3 by lexs_next/
]
qed-.
-lemma lexs_eq_repl_fwd: â\88\80RN,RP,L1,L2. eq_repl_fwd â\80¦ (λf. L1 ⦻*[RN, RP, f] L2).
+lemma lexs_eq_repl_fwd: â\88\80RN,RP,L1,L2. eq_repl_fwd â\80¦ (λf. L1 ⪤*[RN, RP, f] L2).
#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
qed-.
qed-.
lemma lexs_pair_repl: ∀RN,RP,f,I,L1,L2,V1,V2.
- L1.â\93\91{I}V1 ⦻*[RN, RP, f] L2.ⓑ{I}V2 →
+ L1.â\93\91{I}V1 ⪤*[RN, RP, f] L2.ⓑ{I}V2 →
∀W1,W2. RN L1 W1 W2 → RP L1 W1 W2 →
- L1.â\93\91{I}W1 ⦻*[RN, RP, f] L2.ⓑ{I}W2.
+ L1.â\93\91{I}W1 ⪤*[RN, RP, f] L2.ⓑ{I}W2.
#RN #RP #f #I #L1 #L2 #V1 #V2 #HL12 #W1 #W2 #HN #HP
elim (lexs_fwd_pair … HL12) -HL12 /2 width=1 by lexs_inv_tl/
qed-.
lemma lexs_co: ∀RN1,RP1,RN2,RP2.
(∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) →
(∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
- â\88\80f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 â\86\92 L1 ⦻*[RN2, RP2, f] L2.
+ â\88\80f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 â\86\92 L1 ⪤*[RN2, RP2, f] L2.
#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
/3 width=1 by lexs_atom, lexs_next, lexs_push/
qed-.
lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2.
(∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
- â\88\80f,L1,L2. L1 ⦻*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
- L1 ⦻*[RN2, RP2, f] L2.
+ â\88\80f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
+ L1 ⪤*[RN2, RP2, f] L2.
#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
#f #I #K1 #K2 #V1 #V2 #_ #HV12 #IH #H
[ elim (isid_inv_next … H) -H //
qed-.
lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
- â\88\80f2,L1,L2. L1 ⦻*[RN, RP, f2] L2 →
- â\88\80f1. f1 â\8a\86 f2 â\86\92 L1 ⦻*[RN, RP, f1] L2.
+ â\88\80f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
+ â\88\80f1. f1 â\8a\86 f2 â\86\92 L1 ⪤*[RN, RP, f1] L2.
#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
#f2 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH
[ * * [2: #n1 ] ] #f1 #H
qed-.
lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) →
- â\88\80f1,L1,L2. L1 ⦻*[RN, RP, f1] L2 →
- â\88\80f2. f1 â\8a\86 f2 â\86\92 L1 ⦻*[RN, RP, f2] L2.
+ â\88\80f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
+ â\88\80f2. f1 â\8a\86 f2 â\86\92 L1 ⪤*[RN, RP, f2] L2.
#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH
[2: * * [2: #n2 ] ] #f2 #H
qed-.
lemma lexs_sle_split: ∀R1,R2,RP. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- â\88\80f,L1,L2. L1 ⦻*[R1, RP, f] L2 → ∀g. f ⊆ g →
- â\88\83â\88\83L. L1 ⦻*[R1, RP, g] L & L ⦻*[R2, cfull, f] L2.
+ â\88\80f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ⊆ g →
+ â\88\83â\88\83L. L1 ⪤*[R1, RP, g] L & L ⪤*[R2, cfull, f] L2.
#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
[ /2 width=3 by lexs_atom, ex2_intro/ ]
#f #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #y #H
lemma lexs_dec: ∀RN,RP.
(∀L,T1,T2. Decidable (RN L T1 T2)) →
(∀L,T1,T2. Decidable (RP L T1 T2)) →
- â\88\80L1,L2,f. Decidable (L1 ⦻*[RN, RP, f] L2).
+ â\88\80L1,L2,f. Decidable (L1 ⪤*[RN, RP, f] L2).
#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #V1 #IH * ]
[ /2 width=1 by lexs_atom, or_introl/
| #L2 #I2 #V2 #f @or_intror #H
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: includes: lpx_sn_fwd_length *)
-lemma lexs_fwd_length: â\88\80RN,RP,f,L1,L2. L1 ⦻*[RN, RP, f] L2 → |L1| = |L2|.
+lemma lexs_fwd_length: â\88\80RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → |L1| = |L2|.
#RM #RP #f #L1 #L2 #H elim H -f -L1 -L2 //
#f #I #L1 #L2 #V1 #V2 >length_pair >length_pair //
qed-.
theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP) (f):
lexs_transitive RN1 RN2 RN RN1 RP1 →
lexs_transitive RP1 RP2 RP RN1 RP1 →
- â\88\80L1,L0. L1 ⦻*[RN1, RP1, f] L0 →
- â\88\80L2. L0 ⦻*[RN2, RP2, f] L2 →
- L1 ⦻*[RN, RP, f] L2.
+ â\88\80L1,L0. L1 ⪤*[RN1, RP1, f] L0 →
+ â\88\80L2. L0 ⪤*[RN2, RP2, f] L2 →
+ L1 ⪤*[RN, RP, f] L2.
#RN1 #RP1 #RN2 #RP2 #RN #RP #f #HN #HP #L1 #L0 #H elim H -f -L1 -L0
[ #f #L2 #H >(lexs_inv_atom1 … H) -L2 //
| #f #I #K1 #K #V1 #V #HK1 #HV1 #IH #L2 #H elim (lexs_inv_next1 … H) -H
/3 width=3 by/ qed-.
lemma lexs_meet: ∀RN,RP,L1,L2.
- â\88\80f1. L1 ⦻*[RN, RP, f1] L2 →
- â\88\80f2. L1 ⦻*[RN, RP, f2] L2 →
- â\88\80f. f1 â\8b\92 f2 â\89¡ f â\86\92 L1 ⦻*[RN, RP, f] L2.
+ â\88\80f1. L1 ⪤*[RN, RP, f1] L2 →
+ â\88\80f2. L1 ⪤*[RN, RP, f2] L2 →
+ â\88\80f. f1 â\8b\92 f2 â\89¡ f â\86\92 L1 ⪤*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
qed-.
lemma lexs_join: ∀RN,RP,L1,L2.
- â\88\80f1. L1 ⦻*[RN, RP, f1] L2 →
- â\88\80f2. L1 ⦻*[RN, RP, f2] L2 →
- â\88\80f. f1 â\8b\93 f2 â\89¡ f â\86\92 L1 ⦻*[RN, RP, f] L2.
+ â\88\80f1. L1 ⪤*[RN, RP, f1] L2 →
+ â\88\80f2. L1 ⪤*[RN, RP, f2] L2 →
+ â\88\80f. f1 â\8b\93 f2 â\89¡ f â\86\92 L1 ⪤*[RN, RP, f] L2.
#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
#f1 #I #L1 #L2 #V1 #V2 #_ #HV12 #IH #f2 #H #f #Hf
elim (pn_split f2) * #g2 #H2 destruct
(* Basic_2A1: uses: lpx_cpx_frees_trans *)
lemma cpx_frees_conf_lfpx: ∀h,G,L1,T1,f1. L1 ⊢ 𝐅*⦃T1⦄ ≡ f1 →
- â\88\80L2. L1 ⦻*[cpx h G, cfull, f1] L2 →
+ â\88\80L2. L1 ⪤*[cpx h G, cfull, f1] L2 →
∀T2. ⦃G, L1⦄ ⊢ T1 ⬈[h] T2 →
∃∃f2. L2 ⊢ 𝐅*⦃T2⦄ ≡ f2 & f2 ⊆ f1.
#h #G #L1 #T1 @(fqup_wf_ind_eq … G L1 T1) -G -L1 -T1
(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
definition lfxs (R) (T): relation lenv ≝
- λL1,L2. â\88\83â\88\83f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f & L1 ⦻*[R, cfull, f] L2.
+ λL1,L2. â\88\83â\88\83f. L1 â\8a¢ ð\9d\90\85*â¦\83Tâ¦\84 â\89¡ f & L1 ⪤*[R, cfull, f] L2.
interpretation "generic extension on referred entries (local environment)"
'RelationStar R T L1 L2 = (lfxs R T L1 L2).
definition lexs_frees_confluent: relation (relation3 lenv term term) ≝
λRN,RP.
∀f1,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≡ f1 →
- â\88\80L2. L1 ⦻*[RN, RP, f1] L2 →
+ â\88\80L2. L1 ⪤*[RN, RP, f1] L2 →
∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≡ f2 & f2 ⊆ f1.
definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
(relation3 lenv term term) … ≝
λR1,R2,RP1,RP2.
∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- â\88\80L1. L0 ⦻*[RP1, T0] L1 â\86\92 â\88\80L2. L0 ⦻*[RP2, T0] L2 →
+ â\88\80L1. L0 ⪤*[RP1, T0] L1 â\86\92 â\88\80L2. L0 ⪤*[RP2, T0] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
(* Basic properties *********************************************************)
-lemma lfxs_atom: â\88\80R,I. â\8b\86 ⦻*[R, ⓪{I}] ⋆.
+lemma lfxs_atom: â\88\80R,I. â\8b\86 ⪤*[R, ⓪{I}] ⋆.
/3 width=3 by lexs_atom, frees_atom, ex2_intro/
qed.
(* Basic_2A1: uses: llpx_sn_sort *)
lemma 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 * /3 width=3 by lexs_push, frees_sort, ex2_intro/
qed.
-lemma lfxs_zero: â\88\80R,I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 →
- R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, #0] L2.ⓑ{I}V2.
+lemma lfxs_zero: â\88\80R,I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
+ R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 * /3 width=3 by lexs_next, frees_zero, ex2_intro/
qed.
lemma 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, #⫯i] L2.ⓑ{I}V2.
#R #I #L1 #L2 #V1 #V2 #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 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 * /3 width=3 by lexs_push, frees_gref, ex2_intro/
qed.
lemma lfxs_pair_repl_dx: ∀R,I,L1,L2,T,V,V1.
- L1.â\93\91{I}V ⦻*[R, T] L2.ⓑ{I}V1 →
+ L1.â\93\91{I}V ⪤*[R, T] L2.ⓑ{I}V1 →
∀V2. R L1 V V2 →
- L1.â\93\91{I}V ⦻*[R, T] L2.ⓑ{I}V2.
+ L1.â\93\91{I}V ⪤*[R, T] L2.ⓑ{I}V2.
#R #I #L1 #L2 #T #V #V1 * #f #Hf #HL12 #V2 #HR
/3 width=5 by lexs_pair_repl, 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) →
- 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 * /4 width=7 by lexs_co, ex2_intro/
qed-.
lemma lfxs_isid: ∀R1,R2,L1,L2,T1,T2.
(∀f. L1 ⊢ 𝐅*⦃T1⦄ ≡ f → 𝐈⦃f⦄) →
(∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≡ f) →
- L1 ⦻*[R1, T1] L2 â\86\92 L1 ⦻*[R2, T2] L2.
+ L1 ⪤*[R1, T1] L2 â\86\92 L1 ⪤*[R2, T2] L2.
#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
/4 width=7 by lexs_co_isid, ex2_intro/
qed-.
(* Basic inversion lemmas ***************************************************)
-lemma lfxs_inv_atom_sn: â\88\80R,Y2,T. â\8b\86 ⦻*[R, T] Y2 → Y2 = ⋆.
+lemma lfxs_inv_atom_sn: â\88\80R,Y2,T. â\8b\86 ⪤*[R, T] Y2 → Y2 = ⋆.
#R #Y2 #T * /2 width=4 by lexs_inv_atom1/
qed-.
-lemma lfxs_inv_atom_dx: â\88\80R,Y1,T. Y1 ⦻*[R, T] ⋆ → Y1 = ⋆.
+lemma lfxs_inv_atom_dx: â\88\80R,Y1,T. Y1 ⪤*[R, T] ⋆ → Y1 = ⋆.
#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
qed-.
-lemma lfxs_inv_sort: â\88\80R,Y1,Y2,s. Y1 ⦻*[R, ⋆s] Y2 →
+lemma lfxs_inv_sort: â\88\80R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
+ â\88\83â\88\83I,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
[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
]
qed-.
-lemma lfxs_inv_zero: â\88\80R,Y1,Y2. Y1 ⦻*[R, #0] Y2 →
+lemma lfxs_inv_zero: â\88\80R,Y1,Y2. Y1 ⪤*[R, #0] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ â\88\83â\88\83I,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/
]
qed-.
-lemma lfxs_inv_lref: â\88\80R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
+lemma lfxs_inv_lref: â\88\80R,Y1,Y2,i. Y1 ⪤*[R, #⫯i] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
+ â\88\83â\88\83I,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/
]
qed-.
-lemma lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
+lemma lfxs_inv_gref: â\88\80R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
(Y1 = ⋆ ∧ Y2 = ⋆) ∨
- â\88\83â\88\83I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
+ â\88\83â\88\83I,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
[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
-lemma lfxs_inv_bind: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- L1 ⦻*[R, V1] L2 â\88§ L1.â\93\91{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+lemma lfxs_inv_bind: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ L1 ⪤*[R, V1] L2 â\88§ L1.â\93\91{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/
qed-.
(* Basic_2A1: uses: llpx_sn_inv_flat *)
-lemma 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 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 * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lfxs_inv_sort_pair_sn: â\88\80R,I,Y2,L1,V1,s. L1.â\93\91{I}V1 ⦻*[R, ⋆s] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+lemma lfxs_inv_sort_pair_sn: â\88\80R,I,Y2,L1,V1,s. L1.â\93\91{I}V1 ⪤*[R, ⋆s] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #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/
]
qed-.
-lemma lfxs_inv_sort_pair_dx: â\88\80R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+lemma lfxs_inv_sort_pair_dx: â\88\80R,I,Y1,L2,V2,s. Y1 ⪤*[R, ⋆s] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #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/
]
qed-.
-lemma lfxs_inv_zero_pair_sn: â\88\80R,I,Y2,L1,V1. L1.â\93\91{I}V1 ⦻*[R, #0] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+lemma lfxs_inv_zero_pair_sn: â\88\80R,I,Y2,L1,V1. L1.â\93\91{I}V1 ⪤*[R, #0] Y2 →
+ â\88\83â\88\83L2,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 *
[ #H destruct
]
qed-.
-lemma lfxs_inv_zero_pair_dx: â\88\80R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+lemma lfxs_inv_zero_pair_dx: â\88\80R,I,Y1,L2,V2. Y1 ⪤*[R, #0] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,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 *
[ #_ #H destruct
]
qed-.
-lemma lfxs_inv_lref_pair_sn: â\88\80R,I,Y2,L1,V1,i. L1.â\93\91{I}V1 ⦻*[R, #⫯i] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
+lemma lfxs_inv_lref_pair_sn: â\88\80R,I,Y2,L1,V1,i. L1.â\93\91{I}V1 ⪤*[R, #⫯i] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #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/
]
qed-.
-lemma lfxs_inv_lref_pair_dx: â\88\80R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
+lemma lfxs_inv_lref_pair_dx: â\88\80R,I,Y1,L2,V2,i. Y1 ⪤*[R, #⫯i] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #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/
]
qed-.
-lemma lfxs_inv_gref_pair_sn: â\88\80R,I,Y2,L1,V1,l. L1.â\93\91{I}V1 ⦻*[R, §l] Y2 →
- â\88\83â\88\83L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+lemma lfxs_inv_gref_pair_sn: â\88\80R,I,Y2,L1,V1,l. L1.â\93\91{I}V1 ⪤*[R, §l] Y2 →
+ â\88\83â\88\83L2,V2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
#R #I #Y2 #L1 #V1 #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/
]
qed-.
-lemma lfxs_inv_gref_pair_dx: â\88\80R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
- â\88\83â\88\83L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+lemma lfxs_inv_gref_pair_dx: â\88\80R,I,Y1,L2,V2,l. Y1 ⪤*[R, §l] L2.ⓑ{I}V2 →
+ â\88\83â\88\83L1,V1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
#R #I #Y1 #L2 #V2 #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/
(* Basic forward lemmas *****************************************************)
(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
-lemma 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 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 * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
-lemma lfxs_fwd_bind_dx: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⦻*[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⦻*[R, T] L2.ⓑ{I}V2.
+lemma lfxs_fwd_bind_dx: â\88\80R,p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 â\86\92 L1.â\93\91{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
qed-.
(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
-lemma 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 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 (lfxs_inv_flat … H) -H //
qed-.
-lemma lfxs_fwd_dx: â\88\80R,I,L1,K2,T,V2. L1 ⦻*[R, T] K2.ⓑ{I}V2 →
+lemma lfxs_fwd_dx: â\88\80R,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
definition dedropable_sn: predicate (relation3 lenv term term) ≝
λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 →
- â\88\80K2,T. K1 ⦻*[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
- â\88\83â\88\83L2. L1 ⦻*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
+ â\88\80K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≡ U →
+ â\88\83â\88\83L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≡ K2 & L1 ≡[f] L2.
definition dropable_sn: predicate (relation3 lenv term term) ≝
λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≡ K1 → 𝐔⦃f⦄ →
- â\88\80L2,U. L1 ⦻*[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
- â\88\83â\88\83K2. K1 ⦻*[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
+ â\88\80L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≡ U →
+ â\88\83â\88\83K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≡ K2.
definition dropable_dx: predicate (relation3 lenv term term) ≝
- λR. â\88\80L1,L2,U. L1 ⦻*[R, U] L2 →
+ λR. â\88\80L1,L2,U. L1 ⪤*[R, U] L2 →
∀b,f,K2. ⬇*[b, f] L2 ≡ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≡ U →
- â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⦻*[R, T] K2.
+ â\88\83â\88\83K1. â¬\87*[b, f] L1 â\89¡ K1 & K1 ⪤*[R, T] K2.
(* Properties with generic slicing for local environments *******************)
qed-.
(* Basic_2A1: was: llpx_sn_inv_lift_O *)
-lemma lfxs_inv_lifts_bi: â\88\80R,L1,L2,U. L1 ⦻*[R, U] L2 →
+lemma lfxs_inv_lifts_bi: â\88\80R,L1,L2,U. L1 ⪤*[R, U] L2 →
∀K1,K2,i. ⬇*[i] L1 ≡ K1 → ⬇*[i] L2 ≡ K2 →
- â\88\80T. â¬\86*[i] T â\89¡ U â\86\92 K1 ⦻*[R, T] K2.
+ â\88\80T. â¬\86*[i] T â\89¡ U â\86\92 K1 ⪤*[R, T] K2.
#R #L1 #L2 #U #HL12 #K1 #K2 #i #HLK1 #HLK2 #T #HTU
elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
lapply (drops_mono … HY … HLK2) -L2 -i #H destruct //
qed-.
-lemma lfxs_inv_lref_sn: â\88\80R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
- â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⦻*[R, V1] K2 & R K1 V1 V2.
+lemma lfxs_inv_lref_sn: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≡ K1.ⓑ{I}V1 →
+ â\88\83â\88\83K2,V2. â¬\87*[i] L2 â\89¡ K2.â\93\91{I}V2 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY
#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma lfxs_inv_lref_dx: â\88\80R,L1,L2,i. L1 ⦻*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
- â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⦻*[R, V1] K2 & R K1 V1 V2.
+lemma lfxs_inv_lref_dx: â\88\80R,L1,L2,i. L1 ⪤*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≡ K2.ⓑ{I}V2 →
+ â\88\83â\88\83K1,V1. â¬\87*[i] L1 â\89¡ K1.â\93\91{I}V1 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY
#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: llpx_sn_refl *)
-lemma lfxs_refl: â\88\80R. (â\88\80L. reflexive â\80¦ (R L)) â\86\92 â\88\80L,T. L ⦻*[R, T] L.
+lemma lfxs_refl: â\88\80R. (â\88\80L. reflexive â\80¦ (R L)) â\86\92 â\88\80L,T. L ⪤*[R, T] L.
#R #HR #L #T elim (frees_total L T) /3 width=3 by lexs_refl, ex2_intro/
qed.
lemma lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
- â\88\80L,V1,V2. R L V1 V2 â\86\92 â\88\80I,T. L.â\93\91{I}V1 ⦻*[R, T] L.ⓑ{I}V2.
+ â\88\80L,V1,V2. R L V1 V2 â\86\92 â\88\80I,T. L.â\93\91{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
#R #HR #L #V1 #V2 #HV12 #I #T
elim (frees_total (L.ⓑ{I}V1) T) #f #Hf
elim (pn_split f) * #g #H destruct
(* Forward lemmas with length for local environments ************************)
(* Basic_2A1: uses: llpx_sn_fwd_length *)
-lemma lfxs_fwd_length: â\88\80R,L1,L2,T. L1 ⦻*[R, T] L2 → |L1| = |L2|.
+lemma lfxs_fwd_length: â\88\80R,L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|.
#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/
qed-.
(* 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 ⪤*[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)
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.
+ 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
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.
+ 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
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.
+ 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
(* 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))
/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
(* 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.
(* 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-.
["V"; "𝕍"; "𝐕"; "Ⓥ"; ] ;
["w"; "ω"; "𝕨"; "𝐰"; "𝛚"; "ⓦ"; ] ;
["W"; "Ω"; "𝕎"; "𝐖"; "𝛀"; "Ⓦ"; ] ;
- ["x"; "ξ"; "χ"; "ϰ"; "𝕩"; "𝐱"; "𝛏"; "𝛘"; "𝛞"; "ⓧ"; ] ;
- ["X"; "Ξ"; "𝕏";"𝐗"; "𝚵"; "Ⓧ"; "⦻"; ] ;
+ ["x"; "ξ"; "χ"; "ϰ"; "𝕩"; "𝐱"; "𝛏"; "𝛘"; "𝛞"; "ⓧ"; "⨴"; "⨵"; ] ;
+ ["X"; "Ξ"; "𝕏";"𝐗"; "𝚵"; "Ⓧ"; "⦻"; "⪤" ] ;
["y"; "υ"; "𝕪"; "𝐲"; "ⓨ"; ] ;
["Y"; "ϒ"; "𝕐"; "𝐘"; "𝚼"; "Ⓨ"; ] ;
["z"; "ζ"; "𝕫"; "𝐳"; "𝛇"; "ⓩ"; ] ;
projects / helm.git / commitdiff