include "basic_2/notation/relations/lazyeqalt_4.ma".
include "basic_2/substitution/lleq_lleq.ma".
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Note: alternative definition of lleq *)
inductive lleqa: relation4 ynat term lenv lenv ≝
| lleqa_sort: ∀L1,L2,d,k. |L1| = |L2| → lleqa d (⋆k) L1 L2
| lleqa_skip: ∀L1,L2,d,i. |L1| = |L2| → yinj i < d → lleqa d (#i) L1 L2
| lleqa_lref: ∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[0, i] L1 ≡ K1.ⓑ{I1}V → ⇩[0, i] L2 ≡ K2.ⓑ{I2}V →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
lleqa (yinj 0) V K1 K2 → lleqa d (#i) L1 L2
| lleqa_free: ∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → lleqa d (#i) L1 L2
| lleqa_gref: ∀L1,L2,d,p. |L1| = |L2| → lleqa d (§p) L1 L2
∀L1,L2,d,i. |L1| = |L2| → yinj i < d → R d (#i) L1 L2
) → (
∀I1,I2,L1,L2,K1,K2,V,d,i. d ≤ yinj i →
- ⇩[O, i] L1 ≡ K1.ⓑ{I1}V → ⇩[O, i] L2 ≡ K2.ⓑ{I2}V →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V → ⇩[i] L2 ≡ K2.ⓑ{I2}V →
K1 ⋕[V, yinj O] K2 → R (yinj O) V K1 K2 → R d (#i) L1 L2
) → (
∀L1,L2,d,i. |L1| = |L2| → |L1| ≤ i → |L2| ≤ i → R d (#i) L1 L2
#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa … H) -H
/3 width=9 by lleqa_inv_lleq/
qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact lleq_inv_S_aux: ∀L1,L2,T,d0. L1 ⋕[T, d0] L2 → ∀d. d0 = d + 1 →
- ∀K1,K2,I,V. ⇩[0, d] L1 ≡ K1.ⓑ{I}V → ⇩[0, d] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
-#L1 #L2 #T #d0 #H @(lleq_ind_alt … H) -L1 -L2 -T -d0
-/2 width=1 by lleq_gref, lleq_free, lleq_sort/
-[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V #HLK1 #HLK2 #HV destruct
- elim (yle_split_eq i d) /2 width=1 by lleq_skip, ylt_fwd_succ2/ -HL12 -Hid
- #H destruct /2 width=8 by lleq_lref/
-| #I1 #I2 #L1 #L2 #K11 #K22 #V #d0 #i #Hd0i #HLK11 #HLK22 #HV #_ #d #H #K1 #K2 #J #W #_ #_ #_ destruct
- /3 width=8 by lleq_lref, yle_pred_sn/
-| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /4 width=7 by lleq_bind, ldrop_ldrop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /3 width=7 by lleq_flat/
-]
-qed-.
-
-lemma lleq_inv_S: ∀T,L1,L2,d. L1 ⋕[T, d+1] L2 →
- ∀K1,K2,I,V. ⇩[0, d] L1 ≡ K1.ⓑ{I}V → ⇩[0, d] L2 ≡ K2.ⓑ{I}V →
- K1 ⋕[V, 0] K2 → L1 ⋕[T, d] L2.
-/2 width=7 by lleq_inv_S_aux/ 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.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind … H) -H
-/3 width=7 by ldrop_pair, conj, lleq_inv_S/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_ge: ∀L1,L2,T,d1. L1 ⋕[T, d1] L2 → ∀d2. d1 ≤ d2 → L1 ⋕[T, d2] L2.
-#L1 #L2 #T #d1 #H @(lleq_ind_alt … H) -L1 -L2 -T -d1
-/4 width=1 by lleq_sort, lleq_free, lleq_gref, lleq_bind, lleq_flat, yle_succ/
-[ /3 width=3 by lleq_skip, ylt_yle_trans/
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d1 #i #Hi #HLK1 #HLK2 #HV #IHV #d2 #Hd12 elim (ylt_split i d2)
- [ lapply (lleq_fwd_length … HV) #HK12 #Hid2
- lapply (ldrop_fwd_length … HLK1) lapply (ldrop_fwd_length … HLK2)
- normalize in ⊢ (%→%→?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/
- | /3 width=8 by lleq_lref, yle_trans/
- ]
-]
-qed-.
-
-lemma lleq_bind_O: ∀a,I,L1,L2,V,T. L1 ⋕[V, 0] L2 → L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
- L1 ⋕[ⓑ{a,I}V.T, 0] L2.
-/3 width=3 by lleq_ge, lleq_bind/ qed.
-
-lemma lleq_bind_repl_SO: ∀I,L1,L2,V,T. L1.ⓑ{I}V ⋕[T, 0] L2.ⓑ{I}V →
- ∀J,W. L1.ⓑ{J}W ⋕[T, 1] L2.ⓑ{J}W.
-#I #L1 #L2 #V #T #HT #J #W lapply (lleq_ge … HT 1 ?) // -HT
-#HT @(lleq_lsuby_repl … HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *)
-qed-.
-
-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.
-/3 width=7 by lleq_bind_repl_SO, lleq_inv_S/ qed-.
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