(* *)
(**************************************************************************)
-include "basic_2/relocation/llpx_sn_ldrop.ma".
+include "basic_2/substitution/llpx_sn_ldrop.ma".
include "basic_2/substitution/lleq.ma".
(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
(* Advanced properties ******************************************************)
-lemma lleq_bind_repl_O: â\88\80I,L1,L2,V,T. L1.â\93\91{I}V â\8b\95[T, 0] L2.ⓑ{I}V →
- â\88\80J,W. L1 â\8b\95[W, 0] L2 â\86\92 L1.â\93\91{J}W â\8b\95[T, 0] L2.ⓑ{J}W.
+lemma lleq_bind_repl_O: â\88\80I,L1,L2,V,T. L1.â\93\91{I}V â\89¡[T, 0] L2.ⓑ{I}V →
+ â\88\80J,W. L1 â\89¡[W, 0] L2 â\86\92 L1.â\93\91{J}W â\89¡[T, 0] L2.ⓑ{J}W.
/2 width=7 by llpx_sn_bind_repl_O/ qed-.
-lemma lleq_dec: â\88\80T,L1,L2,d. Decidable (L1 â\8b\95[T, d] L2).
+lemma lleq_dec: â\88\80T,L1,L2,d. Decidable (L1 â\89¡[T, d] L2).
/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-.
lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
- â\88\80L1,L2,T,d. L1 â\8b\95[T, d] L2 →
+ â\88\80L1,L2,T,d. L1 â\89¡[T, d] L2 →
∀L. llpx_sn R d T L2 L → llpx_sn R d T L1 L.
#R #HR #L1 #L2 #T #d #H @(lleq_ind … H) -L1 -L2 -T -d
[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
qed-.
lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
- â\88\80L1,L2,T,d. L1 â\8b\95[T, d] L2 →
+ â\88\80L1,L2,T,d. L1 â\89¡[T, d] L2 →
∀L. llpx_sn R d T L1 L → llpx_sn R d T L2 L.
/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lleq_inv_lref_ge_dx: â\88\80L1,L2,d,i. L1 â\8b\95[#i, d] L2 → d ≤ i →
+lemma lleq_inv_lref_ge_dx: â\88\80L1,L2,d,i. L1 â\89¡[#i, d] L2 → d ≤ i →
∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
- â\88\83â\88\83K1. â\87©[i] L1 â\89¡ K1.â\93\91{I}V & K1 â\8b\95[V, 0] K2.
+ â\88\83â\88\83K1. â\87©[i] L1 â\89¡ K1.â\93\91{I}V & K1 â\89¡[V, 0] K2.
#L1 #L2 #d #i #H #Hdi #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx … H … HLK2) -L2
/2 width=3 by ex2_intro/
qed-.
-lemma lleq_inv_lref_ge_sn: â\88\80L1,L2,d,i. L1 â\8b\95[#i, d] L2 → d ≤ i →
+lemma lleq_inv_lref_ge_sn: â\88\80L1,L2,d,i. L1 â\89¡[#i, d] L2 → d ≤ i →
∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
- â\88\83â\88\83K2. â\87©[i] L2 â\89¡ K2.â\93\91{I}V & K1 â\8b\95[V, 0] K2.
+ â\88\83â\88\83K2. â\87©[i] L2 â\89¡ K2.â\93\91{I}V & K1 â\89¡[V, 0] K2.
#L1 #L2 #d #i #H #Hdi #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn … H … HLK1) -L1
/2 width=3 by ex2_intro/
qed-.
-lemma lleq_inv_lref_ge_bi: â\88\80L1,L2,d,i. L1 â\8b\95[#i, d] L2 → d ≤ i →
+lemma lleq_inv_lref_ge_bi: â\88\80L1,L2,d,i. L1 â\89¡[#i, d] L2 → d ≤ i →
∀I1,I2,K1,K2,V1,V2.
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- â\88§â\88§ I1 = I2 & K1 â\8b\95[V1, 0] K2 & V1 = V2.
+ â\88§â\88§ I1 = I2 & K1 â\89¡[V1, 0] K2 & V1 = V2.
/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
-lemma lleq_inv_lref_ge: â\88\80L1,L2,d,i. L1 â\8b\95[#i, d] L2 → d ≤ i →
+lemma lleq_inv_lref_ge: â\88\80L1,L2,d,i. L1 â\89¡[#i, d] L2 → d ≤ i →
∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
- K1 â\8b\95[V, 0] K2.
+ K1 â\89¡[V, 0] K2.
#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
elim (lleq_inv_lref_ge_bi … HL12 … HLK1 HLK2) //
qed-.
-lemma lleq_inv_S: â\88\80L1,L2,T,d. L1 â\8b\95[T, d+1] L2 →
+lemma lleq_inv_S: â\88\80L1,L2,T,d. L1 â\89¡[T, d+1] L2 →
∀I,K1,K2,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
- K1 â\8b\95[V, 0] K2 â\86\92 L1 â\8b\95[T, d] L2.
+ K1 â\89¡[V, 0] K2 â\86\92 L1 â\89¡[T, d] L2.
/2 width=9 by llpx_sn_inv_S/ qed-.
-lemma lleq_inv_bind_O: â\88\80a,I,L1,L2,V,T. L1 â\8b\95[ⓑ{a,I}V.T, 0] L2 →
- L1 â\8b\95[V, 0] L2 â\88§ L1.â\93\91{I}V â\8b\95[T, 0] L2.ⓑ{I}V.
+lemma lleq_inv_bind_O: â\88\80a,I,L1,L2,V,T. L1 â\89¡[ⓑ{a,I}V.T, 0] L2 →
+ L1 â\89¡[V, 0] L2 â\88§ L1.â\93\91{I}V â\89¡[T, 0] L2.ⓑ{I}V.
/2 width=2 by llpx_sn_inv_bind_O/ qed-.
(* Advanced forward lemmas **************************************************)
-lemma lleq_fwd_lref_dx: â\88\80L1,L2,d,i. L1 â\8b\95[#i, d] L2 →
+lemma lleq_fwd_lref_dx: â\88\80L1,L2,d,i. L1 â\89¡[#i, d] L2 →
∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
i < d ∨
- â\88\83â\88\83K1. â\87©[i] L1 â\89¡ K1.â\93\91{I}V & K1 â\8b\95[V, 0] K2 & d ≤ i.
+ â\88\83â\88\83K1. â\87©[i] L1 â\89¡ K1.â\93\91{I}V & K1 â\89¡[V, 0] K2 & d ≤ i.
#L1 #L2 #d #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
qed-.
-lemma lleq_fwd_lref_sn: â\88\80L1,L2,d,i. L1 â\8b\95[#i, d] L2 →
+lemma lleq_fwd_lref_sn: â\88\80L1,L2,d,i. L1 â\89¡[#i, d] L2 →
∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
i < d ∨
- â\88\83â\88\83K2. â\87©[i] L2 â\89¡ K2.â\93\91{I}V & K1 â\8b\95[V, 0] K2 & d ≤ i.
+ â\88\83â\88\83K2. â\87©[i] L2 â\89¡ K2.â\93\91{I}V & K1 â\89¡[V, 0] K2 & d ≤ i.
#L1 #L2 #d #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
qed-.
-lemma lleq_fwd_bind_O_dx: â\88\80a,I,L1,L2,V,T. L1 â\8b\95[ⓑ{a,I}V.T, 0] L2 →
- L1.â\93\91{I}V â\8b\95[T, 0] L2.ⓑ{I}V.
+lemma lleq_fwd_bind_O_dx: â\88\80a,I,L1,L2,V,T. L1 â\89¡[ⓑ{a,I}V.T, 0] L2 →
+ L1.â\93\91{I}V â\89¡[T, 0] L2.ⓑ{I}V.
/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
(* Properties on relocation *************************************************)
-lemma lleq_lift_le: â\88\80K1,K2,T,dt. K1 â\8b\95[T, dt] K2 →
+lemma lleq_lift_le: â\88\80K1,K2,T,dt. K1 â\89¡[T, dt] K2 →
∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- â\88\80U. â\87§[d, e] T â\89¡ U â\86\92 dt â\89¤ d â\86\92 L1 â\8b\95[U, dt] L2.
+ â\88\80U. â\87§[d, e] T â\89¡ U â\86\92 dt â\89¤ d â\86\92 L1 â\89¡[U, dt] L2.
/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-.
-lemma lleq_lift_ge: â\88\80K1,K2,T,dt. K1 â\8b\95[T, dt] K2 →
+lemma lleq_lift_ge: â\88\80K1,K2,T,dt. K1 â\89¡[T, dt] K2 →
∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- â\88\80U. â\87§[d, e] T â\89¡ U â\86\92 d â\89¤ dt â\86\92 L1 â\8b\95[U, dt+e] L2.
+ â\88\80U. â\87§[d, e] T â\89¡ U â\86\92 d â\89¤ dt â\86\92 L1 â\89¡[U, dt+e] L2.
/2 width=9 by llpx_sn_lift_ge/ qed-.
(* Inversion lemmas on relocation *******************************************)
-lemma lleq_inv_lift_le: â\88\80L1,L2,U,dt. L1 â\8b\95[U, dt] L2 →
+lemma lleq_inv_lift_le: â\88\80L1,L2,U,dt. L1 â\89¡[U, dt] L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 dt â\89¤ d â\86\92 K1 â\8b\95[T, dt] K2.
+ â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 dt â\89¤ d â\86\92 K1 â\89¡[T, dt] K2.
/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-.
-lemma lleq_inv_lift_be: â\88\80L1,L2,U,dt. L1 â\8b\95[U, dt] L2 →
+lemma lleq_inv_lift_be: â\88\80L1,L2,U,dt. L1 â\89¡[U, dt] L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 d â\89¤ dt â\86\92 dt â\89¤ yinj d + e â\86\92 K1 â\8b\95[T, d] K2.
+ â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 d â\89¤ dt â\86\92 dt â\89¤ yinj d + e â\86\92 K1 â\89¡[T, d] K2.
/2 width=11 by llpx_sn_inv_lift_be/ qed-.
-lemma lleq_inv_lift_ge: â\88\80L1,L2,U,dt. L1 â\8b\95[U, dt] L2 →
+lemma lleq_inv_lift_ge: â\88\80L1,L2,U,dt. L1 â\89¡[U, dt] L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 yinj d + e â\89¤ dt â\86\92 K1 â\8b\95[T, dt-e] K2.
+ â\88\80T. â\87§[d, e] T â\89¡ U â\86\92 yinj d + e â\89¤ dt â\86\92 K1 â\89¡[T, dt-e] K2.
/2 width=9 by llpx_sn_inv_lift_ge/ qed-.
(* Inversion lemmas on negated lazy quivalence for local environments *******)
-lemma nlleq_inv_bind: â\88\80a,I,L1,L2,V,T,d. (L1 â\8b\95[ⓑ{a,I}V.T, d] L2 → ⊥) →
- (L1 â\8b\95[V, d] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V â\8b\95[T, ⫯d] L2.ⓑ{I}V → ⊥).
+lemma nlleq_inv_bind: â\88\80a,I,L1,L2,V,T,d. (L1 â\89¡[ⓑ{a,I}V.T, d] L2 → ⊥) →
+ (L1 â\89¡[V, d] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V â\89¡[T, ⫯d] L2.ⓑ{I}V → ⊥).
/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
-lemma nlleq_inv_flat: â\88\80I,L1,L2,V,T,d. (L1 â\8b\95[ⓕ{I}V.T, d] L2 → ⊥) →
- (L1 â\8b\95[V, d] L2 â\86\92 â\8a¥) â\88¨ (L1 â\8b\95[T, d] L2 → ⊥).
+lemma nlleq_inv_flat: â\88\80I,L1,L2,V,T,d. (L1 â\89¡[ⓕ{I}V.T, d] L2 → ⊥) →
+ (L1 â\89¡[V, d] L2 â\86\92 â\8a¥) â\88¨ (L1 â\89¡[T, d] L2 → ⊥).
/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
-lemma nlleq_inv_bind_O: â\88\80a,I,L1,L2,V,T. (L1 â\8b\95[ⓑ{a,I}V.T, 0] L2 → ⊥) →
- (L1 â\8b\95[V, 0] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V â\8b\95[T, 0] L2.ⓑ{I}V → ⊥).
+lemma nlleq_inv_bind_O: â\88\80a,I,L1,L2,V,T. (L1 â\89¡[ⓑ{a,I}V.T, 0] L2 → ⊥) →
+ (L1 â\89¡[V, 0] L2 â\86\92 â\8a¥) â\88¨ (L1.â\93\91{I}V â\89¡[T, 0] L2.ⓑ{I}V → ⊥).
/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.
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