(* *)
(**************************************************************************)
-include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/relocation/llpx_sn_ldrop.ma".
include "basic_2/substitution/lleq.ma".
(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
(* Advanced properties ******************************************************)
-lemma lleq_skip: ∀L1,L2,d,i. yinj i < d → |L1| = |L2| → L1 ⋕[#i, d] L2.
-#L1 #L2 #d #i #Hid #HL12 @conj // -HL12
-#U @conj #H elim (cpys_inv_lref1 … H) -H // *
-#I #Z #Y #X #H elim (ylt_yle_false … Hid … H)
-qed.
-
-lemma lleq_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[#i, d] L2.
-#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ]
-[ lapply (ldrop_fwd_length … HLK1) -HLK1 #H1
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H2
- >H1 >H2 -H1 -H2 normalize //
-| #U @conj #H elim (cpys_inv_lref1 … H) -H // *
- >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU
- [ letin HLK ≝ HLK1 | letin HLK ≝ HLK2 ]
- lapply (ldrop_mono … H … HLK) -H #H destruct elim (IH W)
- /3 width=7 by cpys_subst_Y2/
+lemma lleq_bind_repl_O: ∀I,L1,L2,V,T. L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
+ ∀J,W. L1 ⋕[W, 0] L2 → L1.ⓑ{J}W ⋕[T, 0] L2.ⓑ{J}W.
+/2 width=7 by llpx_sn_bind_repl_O/ qed-.
+
+lemma lleq_dec: ∀T,L1,L2,d. Decidable (L1 ⋕[T, d] L2).
+/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-.
+
+lemma lleq_llpx_sn_trans: ∀R. lleq_transitive R →
+ ∀L1,L2,T,d. L1 ⋕[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/
+|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
+| #I #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn … H … HLK2) -H -HLK2
+ /3 width=11 by llpx_sn_lref/
+| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind … H) -H
+ /3 width=1 by llpx_sn_bind/
+| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat … H) -H
+ /3 width=1 by llpx_sn_flat/
]
-qed.
+qed-.
+
+lemma lleq_llpx_sn_conf: ∀R. lleq_transitive R →
+ ∀L1,L2,T,d. L1 ⋕[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: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ⋕[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: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
+ ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ⋕[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: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I1,I2,K1,K2,V1,V2.
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & K1 ⋕[V1, 0] K2 & V1 = V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma lleq_inv_lref_ge: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 → d ≤ i →
+ ∀I,K1,K2,V. ⇩[i] L1 ≡ K1.ⓑ{I}V → ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ K1 ⋕[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: ∀L1,L2,T,d. L1 ⋕[T, d+1] L2 →
+ ∀I,K1,K2,V. ⇩[d] L1 ≡ K1.ⓑ{I}V → ⇩[d] L2 ≡ K2.ⓑ{I}V →
+ K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
+/2 width=9 by llpx_sn_inv_S/ qed-.
+
+lemma lleq_inv_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
+ L1 ⋕[V, 0] L2 ∧ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref_dx: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
+ ∀I,K2,V. ⇩[i] L2 ≡ K2.ⓑ{I}V →
+ i < d ∨
+ ∃∃K1. ⇩[i] L1 ≡ K1.ⓑ{I}V & K1 ⋕[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_free: ∀L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| → L1 ⋕[#i, d] L2.
-#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12
-#U @conj #H elim (cpys_inv_lref1 … H) -H // *
-#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 … H) -H
-#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
-qed.
+lemma lleq_fwd_lref_sn: ∀L1,L2,d,i. L1 ⋕[#i, d] L2 →
+ ∀I,K1,V. ⇩[i] L1 ≡ K1.ⓑ{I}V →
+ i < d ∨
+ ∃∃K2. ⇩[i] L2 ≡ K2.ⓑ{I}V & K1 ⋕[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: ∀a,I,L1,L2,V,T. L1 ⋕[ⓑ{a,I}V.T, 0] L2 →
+ L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V.
+/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
(* Properties on relocation *************************************************)
lemma lleq_lift_le: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 →
∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀U. ⇧[d, e] T ≡ U → dt ≤ d → L1 ⋕[U, dt] L2.
-#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd
-lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
-#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ]
-elim (cpys_inv_lift1_be … HU0 … HLKA … HTU) // -HU0 >yminus_Y_inj #T0 #HT0 #HTU0
-elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=12 by cpys_lift_be/
-qed-.
+/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-.
lemma lleq_lift_ge: ∀K1,K2,T,dt. K1 ⋕[T, dt] K2 →
∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀U. ⇧[d, e] T ≡ U → d ≤ dt → L1 ⋕[U, dt+e] L2.
-#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt
-lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
-#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 ]
-elim (cpys_inv_lift1_ge … HU0 … HLKA … HTU) /2 width=1 by monotonic_yle_plus_dx/ -HU0 >yplus_minus_inj #T0 #HT0 #HTU0
-elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=10 by cpys_lift_ge/
-qed-.
+/2 width=9 by llpx_sn_lift_ge/ qed-.
(* Inversion lemmas on relocation *******************************************)
lemma lleq_inv_lift_le: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀T. ⇧[d, e] T ≡ U → dt ≤ d → K1 ⋕[T, dt] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd
-lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
-lapply (cpys_lift_be … HT0 … HLKA … HTU … HTU0) // -HT0
->yplus_Y1 #HU0 elim (cpys_inv_lift1_be … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdtd
-#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
-qed-.
+/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-.
+
+lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
+ ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
+ ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ⋕[T, d] K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
lemma lleq_inv_lift_ge: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀T. ⇧[d, e] T ≡ U → yinj d + e ≤ dt → K1 ⋕[T, dt-e] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt
-lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-elim (yle_inv_plus_inj2 … Hdedt) #Hddt #Hedt
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
-lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0 -Hddt
->ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge … (H0 HU0) … HLKB … HTU) // -L1 -L2 -U -Hdedt -Hedt
-#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
-qed-.
+/2 width=9 by llpx_sn_inv_lift_ge/ qed-.
-lemma lleq_inv_lift_be: ∀L1,L2,U,dt. L1 ⋕[U, dt] L2 →
- ∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → d ≤ dt → dt ≤ yinj d + e → K1 ⋕[T, d] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde
-lapply (ldrop_fwd_length_minus2 … HLK1) lapply (ldrop_fwd_length_minus2 … HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA ≝ HLK1 letin HLKB ≝ HLK2 letin H0 ≝ H12 | letin HLKA ≝ HLK2 letin HLKB ≝ HLK1 letin H0 ≝ H21 ]
-lapply (cpys_lift_ge … HT0 … HLKA … HTU … HTU0) // -HT0
-#HU0 lapply (cpys_weak … HU0 dt (∞) ? ?) // -HU0
-#HU0 lapply (H0 HU0)
-#HU0 lapply (cpys_weak … HU0 d (∞) ? ?) // -HU0
-#HU0 elim (cpys_inv_lift1_ge_up … HU0 … HLKB … HTU) // -L1 -L2 -U -Hddt -Hdtde
-#X #HT0 #HX lapply (lift_inj … HX … HTU0) -U0 //
-qed-.
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind: ∀a,I,L1,L2,V,T,d. (L1 ⋕[ⓑ{a,I}V.T, d] L2 → ⊥) →
+ (L1 ⋕[V, d] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, ⫯d] L2.ⓑ{I}V → ⊥).
+/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
+
+lemma nlleq_inv_flat: ∀I,L1,L2,V,T,d. (L1 ⋕[ⓕ{I}V.T, d] L2 → ⊥) →
+ (L1 ⋕[V, d] L2 → ⊥) ∨ (L1 ⋕[T, d] L2 → ⊥).
+/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
+
+lemma nlleq_inv_bind_O: ∀a,I,L1,L2,V,T. (L1 ⋕[ⓑ{a,I}V.T, 0] L2 → ⊥) →
+ (L1 ⋕[V, 0] L2 → ⊥) ∨ (L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V → ⊥).
+/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.
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