(* *)
(**************************************************************************)
-include "basic_2/relocation/ldrop_ldrop.ma".
-include "basic_2/relocation/lleq.ma".
+include "basic_2/multiple/lleq_drop.ma".
(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-(* Advanced inversion lemmas ************************************************)
-
-lemma lleq_inv_lref_dx: ∀L1,L2,i. L1 ⋕[#i] L2 →
- ∀I,K2,V. ⇩[0, i] L2 ≡ K2.ⓑ{I}V →
- ∃∃K1. ⇩[0, i] L1 ≡ K1.ⓑ{I}V & K1 ⋕[V] K2.
-#L1 #L2 #i #H #I #K2 #V #HLK2 elim (lleq_inv_lref … H) -H *
-[ #_ #H elim (lt_refl_false i)
- /3 width=5 by ldrop_fwd_length_lt2, lt_to_le_to_lt/
-| #I0 #K1 #K0 #V0 #HLK1 #HLK0 #HK10 lapply (ldrop_mono … HLK0 … HLK2) -L2
- #H destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lleq_inv_lift: ∀L1,L2,U. L1 ⋕[U] L2 →
- ∀K1,K2,d,e. ⇩[d, e] L1 ≡ K1 → ⇩[d, e] L2 ≡ K2 →
- ∀T. ⇧[d, e] T ≡ U → K1 ⋕[T] K2.
-#L1 #L2 #U #H elim H -L1 -L2 -U
-[ #L1 #L2 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H >(lift_inv_sort2 … H) -X
- lapply (ldrop_fwd_length_eq … HLK1 HLK2 HL12) -L1 -L2 -d -e
- /2 width=1 by lleq_sort/ (**) (* full auto fails *)
-| #I #L1 #L2 #K #K0 #W #i #HLK #HLK0 #HK0 #IHK0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H elim (lift_inv_lref2 … H) -H
- * #Hdei #H destruct [ -HK0 | -IHK0 ]
- [ elim (ldrop_conf_lt … HLK1 … HLK) // -L1 #L1 #V #HKL1 #KL1 #HV0
- elim (ldrop_conf_lt … HLK2 … HLK0) // -Hdei -L2 #L2 #V2 #HKL2 #K0L2 #HV02
- lapply (lift_inj … HV02 … HV0) -HV02 #H destruct /3 width=11 by lleq_lref/
- | lapply (ldrop_conf_ge … HLK1 … HLK ?) // -L1
- lapply (ldrop_conf_ge … HLK2 … HLK0 ?) // -Hdei -L2
- /2 width=7 by lleq_lref/
- ]
-| #L1 #L2 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H elim (lift_inv_lref2 … H) -H
- * #_ #H destruct
- lapply (ldrop_fwd_length_eq … HLK1 HLK2 HL12)
- [ /4 width=3 by lleq_free, ldrop_fwd_length_le4, transitive_le/
- | lapply (ldrop_fwd_length … HLK1) -HLK1 #H >H in HL1; -H
- lapply (ldrop_fwd_length … HLK2) -HLK2 #H >H in HL2; -H
- /3 width=1 by lleq_free, le_plus_to_minus_r/
- ]
-| #L1 #L2 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H >(lift_inv_gref2 … H) -X
- lapply (ldrop_fwd_length_eq … HLK1 HLK2 HL12) -L1 -L2 -d -e
- /2 width=1 by lleq_gref/ (**) (* full auto fails *)
-| #a #I #L1 #L2 #W #U #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H elim (lift_inv_bind2 … H) -H
- #V #T #HVW #HTU #H destruct /4 width=5 by lleq_bind, ldrop_skip/
-| #I #L1 #L2 #W #U #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H elim (lift_inv_flat2 … H) -H
- #V #T #HVW #HTU #H destruct /3 width=5 by lleq_flat/
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_dec: ∀T,L1,L2. Decidable (L1 ⋕[T] L2).
-#T #L1 @(f2_ind … rfw … L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_sort/
-| #i #H1 #L2 elim (eq_nat_dec (|L1|) (|L2|))
- [ #HL12 elim (lt_or_ge i (|L1|))
- #HiL1 elim (lt_or_ge i (|L2|)) /3 width=1 by or_introl, lleq_free/
- #HiL2 elim (ldrop_O1_lt … HiL2)
- #I2 #K2 #V2 #HLK2 elim (ldrop_O1_lt … HiL1)
- #I1 #K1 #V1 #HLK1 elim (eq_bind2_dec I2 I1)
- [ #H2 elim (eq_term_dec V2 V1)
- [ #H3 elim (IH K1 V1 … K2) destruct
- /3 width=7 by lleq_lref, ldrop_fwd_rfw, or_introl/
- ]
- ]
- -IH #H3 @or_intror
- #H elim (lleq_inv_lref … H) -H *
- [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/ ]
- #I0 #X1 #X2 #V0 #HLX1 #HLX2 #HX12
- lapply (ldrop_mono … HLX1 … HLK1) -HLX1 -HLK1
- lapply (ldrop_mono … HLX2 … HLK2) -HLX2 -HLK2
- #H #H0 destruct /2 width=1 by/
- ]
-| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_gref/
-| #a #I #V #T #Hn #L2 destruct
- elim (IH L1 V … L2) /2 width=1 by/
- elim (IH (L1.ⓑ{I}V) T … (L2.ⓑ{I}V)) -IH /3 width=1 by or_introl, lleq_bind/
- #H1 #H2 @or_intror
- #H elim (lleq_inv_bind … H) -H /2 width=1 by/
-| #I #V #T #Hn #L2 destruct
- elim (IH L1 V … L2) /2 width=1 by/
- elim (IH L1 T … L2) -IH /3 width=1 by or_introl, lleq_flat/
- #H1 #H2 @or_intror
- #H elim (lleq_inv_flat … H) -H /2 width=1 by/
-]
--n /4 width=2 by lleq_fwd_length, or_intror/
-qed-.
-
(* Main properties **********************************************************)
-theorem lleq_trans: ∀T. Transitive … (lleq T).
-#T #L1 #L #H elim H -T -L1 -L
-/4 width=4 by lleq_fwd_length, lleq_gref, lleq_sort, trans_eq/
-[ #I #L1 #L #K1 #K #V #i #HLK1 #HLK #_ #IHK1 #L2 #H elim (lleq_inv_lref … H) -H *
- [ -HLK1 -IHK1 #HLi #_ elim (lt_refl_false i)
- /3 width=5 by ldrop_fwd_length_lt2, lt_to_le_to_lt/
- | #I0 #K0 #K2 #V0 #HLK0 #HLK2 #HK12 lapply (ldrop_mono … HLK0 … HLK) -L
- #H destruct /3 width=7 by lleq_lref/
- ]
-| #L1 #L #i #HL1i #HLi #HL #L2 #H lapply (lleq_fwd_length … H)
- #HL2 elim (lleq_inv_lref … H) -H * /2 width=1 by lleq_free/
-| #a #I #L1 #L #V #T #_ #_ #IHV #IHT #L2 #H elim (lleq_inv_bind … H) -H
- /3 width=1 by lleq_bind/
-| #I #L1 #L #V #T #_ #_ #IHV #IHT #L2 #H elim (lleq_inv_flat … H) -H
- /3 width=1 by lleq_flat/
-]
-qed-.
+theorem lleq_trans: ∀l,T. Transitive … (lleq l T).
+/2 width=3 by lleq_llpx_sn_trans/ qed-.
-theorem lleq_canc_sn: ∀L,L1,L2,T. L ⋕[T] L1→ L ⋕[T] L2 → L1 ⋕[T] L2.
+theorem lleq_canc_sn: ∀L,L1,L2,T,l. L ≡[l, T] L1→ L ≡[l, T] L2 → L1 ≡[l, T] L2.
/3 width=3 by lleq_trans, lleq_sym/ qed-.
-theorem lleq_canc_dx: ∀L1,L2,L,T. L1 ⋕[T] L → L2 ⋕[T] L → L1 ⋕[T] L2.
+theorem lleq_canc_dx: ∀L1,L2,L,T,l. L1 ≡[l, T] L → L2 ≡[l, T] L → L1 ≡[l, T] L2.
/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+(* Advanced properies on negated lazy equivalence *****************************)
+
+(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *)
+lemma lleq_nlleq_trans: ∀l,T,L1,L. L1 ≡[T, l] L →
+ ∀L2. (L ≡[T, l] L2 → ⊥) → (L1 ≡[T, l] L2 → ⊥).
+/3 width=3 by lleq_canc_sn/ qed-.
+
+lemma nlleq_lleq_div: ∀l,T,L2,L. L2 ≡[T, l] L →
+ ∀L1. (L1 ≡[T, l] L → ⊥) → (L1 ≡[T, l] L2 → ⊥).
+/3 width=3 by lleq_trans/ qed-.
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